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Interpretation of fluorescence microscopy experiments on cell surface receptor dynamics with stochastic… Herrera Reyes, Alejandra Donaji 2019

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Interpretation of fluorescence microscopy experiments on cell surfacereceptor dynamics with stochastic and deterministic mathematicalmodelsbyAlejandra Donaji Herrera ReyesLic. en Matematicas, Universidad de Guanajuato, 2010M.Sc. Mathematics, The University of British Columbia, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mathematics)The University of British Columbia(Vancouver)December 2019c© Alejandra Donaji Herrera Reyes, 2019The following individuals certify that they have read, and recommend to the Faculty of Gradu-ate and Postdoctoral Studies for acceptance, the dissertation entitled: Interpretation of fluorescencemicroscopy experiments on cell surface receptor dynamics with stochastic and deterministic math-ematical models submitted by Alejandra Donaji Herrera Reyes in partial fulfillment of the require-ments for the degree of Doctor of Philosophy in Mathematics.Examining Committee:Daniel Coombs, MathematicsSupervisorEric Cytrynbaum, MathematicsSupervisory Committee MemberPriscilla E. Greenwood, MathematicsSupervisory Committee MemberGuy Tanentzapf, Cellular and Physiological SciencesSupervisory Committee MemberiiAbstractFluorescence microscopy has provided cellular biologists with quantifiable data, that can be pairedwith mathematical models to discover the mechanics of the imaged processes. We developed math-ematical models to analyze data from two fluorescence techniques: direct Stochastic Optical Re-construction Microscopy (dSTORM) and fluorescence recovery after photobleaching (FRAP).dSTORM is a super-resolution technique that uses photo-switchable fluorophores to achievenanometer resolution images, allowing us to visualize the organization of proteins at nano-scales.However, dSTORM images can suffer from recording a single photo-switchable fluorophore multi-ple times, possibly creating artificial features. This is specially relevant in the analysis of membraneB-cell receptors clustering, where spatial clustering might relate to immune activation. I developeda protocol to estimate the number of unique fluorophores present in the experiment by couplingtheir temporal (with a Markov-chain model) and spatial (with a Gaussian mixture model) dynamicswithin a maximum likelihood framework. Previous studies have used the temporal information, butthey have not coupled it with the spatial information (both localization and localization estimationerror). I tested my protocol on simulated data, well-characterized DNA origami data and B-cellreceptor data with positive results. My model is general enough to apply to other biological sys-tems besides B-cell data and will enhance a microscopy technique that is widely used in biologicalapplications.FRAP can be used to quantify the mobility of membrane proteins. We used it on live Drosophilaorganisms to study the outside-in pathway in cell adhesion to the extracellular matrix (ECM). Wedeveloped an ODE model to describe the recycling of the membrane protein, integrin, in charge ofthe adhesions. We found that both integrin and ECM ligands stabilize outside-in signalling and thatrelevant chemical treatments do not balance mutant integrin activation but stabilize the adhesions incontrol organisms. We also analyzed inside-out activation with a similar ODE model and by labelingthe cytosolic protein talin. We found that talin is sensitive to increases and decreases in appliedforce. Disruptions of the intracellular force negatively affected adhesion stability. Increasing theforce resulted in a faster assembly of new adhesions, whereas decreased forces increased the talinturnover.iiiLay SummaryRecent microscopy techniques determine the precise locations and mobility of cell surface molecules.I used this precise information and novel mathematical models to build our understanding of twodifferent biological systems. In the first case, I provided a method for consolidating multiple lo-cation estimates arising from individual fluorescent molecules, even when the reported locationsoverlap. I applied my method to images of receptors on the surface of immune B cells, showing thatthe molecules aggregate in smaller clusters than previously reported. In the second study, I workedwith fruit fly microscopy data to understand the stability of the bonding between cells and the ex-tracellular matrix. I developed and fitted mathematical models to data obtained from hundreds ofexperiments with different genetic mutations and/or chemical treatments. Overall, my results showhow mathematical modelling coupled with new forms of microscopic imaging gives a deeper levelof understanding of important biological systems.ivPrefaceChapter 5 is adapted from: P. Lo´pez-Ceballos, A. D. Herrera-Reyes, D. Coombs, and G. Tanentzapf,2016: In-vivo regulation of integrin turnover by outside-in activation. Journal of Cell Science,129(15):29122924, 2016. All the figures showing results in this chapter were adapted from thepaper.Chapter 6 is adapted from: G. K. Ha´konardo´ttir, P. Lo´pez-Ceballos, A. D. Herrera-Reyes, R.Das, D. Coombs, and G. Tanentzapf, 2015: In vivo quantitative analysis of talin turnover in responseto force. Molecular Biology of the Cell, 26(22):41494162, 2015. All the figures showing results inthis chapter were adapted from the paper.In Chapter 4, I used DNA origami data from Daniel Nino, University of Toronto previouslypublished in: D. Nino, D. Djayakarsana, and J. N. Milstein. Nanoscopic stoichiometry and single-molecule counting. Small Methods, 2019, p. 1900082. Also in this chapter, I used B-cell data fromthe following publication: J. Scurll, L. Abraham, D. W. Zheng, R. Tafteh, K. Chou, M. R. Gold, andD. Coombs. Stormgraph: An automated graph-based algorithm for quantitative clustering analysisof single-molecule localization microscopy data. bioRxiv, 2019. doi:10.1101/51562.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Fluorescence microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Fluorescence recovery after photobleaching (FRAP) . . . . . . . . . . . . . . . . . 21.2.1 FRAP as a tool in cell biology . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Further discussion of FRAP analysis . . . . . . . . . . . . . . . . . . . . . 41.3 Direct stochastic optical reconstruction microscopy (dSTORM) . . . . . . . . . . . 61.3.1 The diffraction limit of light and single-molecule localization . . . . . . . 61.3.2 direct Stochastic Optical Reconstruction Microscopy (dSTORM) separatesnearby proteins with a temporal approach . . . . . . . . . . . . . . . . . . 81.3.3 dSTORM as a qualitative and quantitative tool . . . . . . . . . . . . . . . 81.3.4 Some limitations of dSTORM . . . . . . . . . . . . . . . . . . . . . . . . 91.3.5 Photoactivated localization microscopy (PALM) . . . . . . . . . . . . . . 91.4 Dissertation organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10vi2 Identifying unique observations in super-resolution microscopy: temporal model . . 112.1 The problem of relating the number of localizations with the number of labelledproteins from dSTORM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Biological motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 B-cell receptor clustering and activation . . . . . . . . . . . . . . . . . . . 132.2.2 Cardiac myocyte receptors . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Previous modelling approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Continous time model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Mean-system behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.2 The distribution of the number of blinks . . . . . . . . . . . . . . . . . . . 202.4.3 The expected time to bleach . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.4 Aggregating in bright and dark classes . . . . . . . . . . . . . . . . . . . . 242.4.5 Likelihood function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.6 Likelihood function using NUobs distribution . . . . . . . . . . . . . . . . . 262.4.7 Problems with the continuous time model . . . . . . . . . . . . . . . . . . 272.5 Discrete-time model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.1 The distribution of the number of blinks . . . . . . . . . . . . . . . . . . . 292.5.2 The distribution of the number of frames to first activation . . . . . . . . . 312.5.3 The distribution of the number of activations . . . . . . . . . . . . . . . . 312.5.4 Likelihood function using D and Bn classes . . . . . . . . . . . . . . . . . 322.5.5 Likelihood when N=1 and estimates for pr, pb, and pd . . . . . . . . . . . 322.6 Bounding the possible number of activations per frame . . . . . . . . . . . . . . . 332.6.1 Transition probability matrix and likelihood function . . . . . . . . . . . . 342.6.2 Case Amax = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Parameter estimation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.7.1 Short review of mixed-optimization algorithms . . . . . . . . . . . . . . . 352.7.2 Maximizing the likelihood for one dataset . . . . . . . . . . . . . . . . . . 372.7.3 Simultaneous fit of data sets with the same number of fluorophores . . . . 382.7.4 Simultaneous fit of data sets with a different number of fluorophores . . . . 382.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.8.1 Simulating data for the discrete-time model . . . . . . . . . . . . . . . . . 402.8.2 Estimates of the kinetic rates depend on the sample size of the data for thediscrete-time model when the number of fluorophores is known . . . . . . 422.8.3 Estimating the number of fluorophores when the transition probabilities areknown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.8.4 Estimating the number of fluorophores assuming no information about theparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47vii2.8.5 Estimating the number of fluorophores while conditioning on the number ofsimultaneous activations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 Divide and conquer to identify unique observations in super resolution microscopy:spatiotemporal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.1 Divide to conquer: pre-processing the spatial data . . . . . . . . . . . . . . . . . . 613.1.1 Stormgraph software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1.2 The last active fluorophore: a constraint on the reactivation probability pr . 623.1.3 Pre-processing algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 Gaussian Mixture models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2.1 Maximizing the likelihood function of a GMM . . . . . . . . . . . . . . . 653.2.2 Proportion parameters pin and prior probabilites . . . . . . . . . . . . . . . 663.2.3 Proportion parameters pin and probabilistic clustering . . . . . . . . . . . . 663.2.4 Expectation-Maximization algorithm (EM) and Gaussian Mixture models . 673.2.5 Estimating the parametes for a general Gaussian Mixture model (GMM) . . 683.2.6 Identifiability problem among mixture order permutations . . . . . . . . . 693.3 dSTORM spatial data as a Gaussian Mixture model . . . . . . . . . . . . . . . . . 703.3.1 Estimating the mean parameters of a Gaussian Mixture model (GMM) withdata (X ,ΣX) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.3.2 dSTORM spatial model identification limitations . . . . . . . . . . . . . . 733.4 dSTORM spatiotemporal model . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.4.1 Likelihood function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.4.2 Constraining the domain of N based on the temporal model . . . . . . . . 763.4.3 An algorithm to correct the spatial model for over-fitting . . . . . . . . . . 783.4.4 Parameter estimation algorithm for the spatiotemporal dSTORM model . . 803.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.5.1 Simulating spatial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.5.2 Pre-pocessing the samples for fitting . . . . . . . . . . . . . . . . . . . . . 833.5.3 Estimating the fluorophores localization when N is known . . . . . . . . . 873.5.4 Fitting simulated data to the full spatiotemporal model . . . . . . . . . . . 913.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944 Identifying unique observations in super resolution experimental data with a spa-tiotemporal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.1 Experimental control: DNA origami . . . . . . . . . . . . . . . . . . . . . . . . . 974.1.1 DNA origami data description . . . . . . . . . . . . . . . . . . . . . . . . 98viii4.2 DNA origami analysis using the spatiotemporal model . . . . . . . . . . . . . . . 994.3 Experimental data test: B-cell receptors (BCR) . . . . . . . . . . . . . . . . . . . 1034.3.1 Data set description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4 Analysis of B-cell A data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.5 BCR data analysis using the spatiotemporal model . . . . . . . . . . . . . . . . . 1094.5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 Analysis of outside-in activation of integrins in cell-extracellular matrix adhesionsusing live Drosophila FRAP data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.1 Cell adhesions background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.1.1 Integrin activation and cell-extracellular matrix (ECM) adhesion . . . . . . 1175.2 Understanding outside activation as an stabilizer of cell-ECM adhesions . . . . . . 1195.2.1 Biological experimental procedures . . . . . . . . . . . . . . . . . . . . . 1195.2.2 FRAP methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.3.1 Assumptions and description . . . . . . . . . . . . . . . . . . . . . . . . . 1235.3.2 Solution, analysis, and theoretical interpretation . . . . . . . . . . . . . . . 1255.4 Data fitting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.5.1 Outside-in integrin activation through chemical induction stabilizes cell-ECM adhesion by decreasing integrin turnover . . . . . . . . . . . . . . . 1285.5.2 Activating integrin mutants regulate turnover with a similar mechanism asectopic integrin activation . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.5.3 Integrin mutations that affect outside-in activation fail to regulate integrinturnover upon ectopic chemical integrin activation . . . . . . . . . . . . . 1305.5.4 ECM reduction increases integrin turnover by increasing integrin endocyto-sis rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.5.5 Integrin turnover is developmentally regulated by integrin activation . . . . 1365.5.6 Rap1 regulates integrin turnover downstream of outside-in activation . . . 1385.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 Analysing talin’s role in inside-out integrin activation, under mechanical forces, us-ing in-vivo Drosophila fluorescence recovery after photobleaching (FRAP) data . . . 1426.1 Inside-out integrin activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.1.1 The integrin adhesion complex (IAC) and its mechanosensors . . . . . . . 1436.1.2 Talin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.2 Understanding talin turnover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.2.1 Biological experimental procedures . . . . . . . . . . . . . . . . . . . . . 146ix6.2.2 FRAP methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.3.1 General model construction . . . . . . . . . . . . . . . . . . . . . . . . . 1486.3.2 Model simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1496.3.3 Parameter symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526.3.4 Analysis and theoretical interepretation . . . . . . . . . . . . . . . . . . . 1526.4 Data fitting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.5.1 Analyzing the turnover of talin under increased force . . . . . . . . . . . . 1576.5.2 Resolving the mechanism by which the turnover of talin is regulated byincreased force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1576.5.3 Analyzing the turnover of talin under decreased force . . . . . . . . . . . . 1606.5.4 Uncovering the mechanism by which the turnover of talin is regulated bydecreased force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.5.5 Reducing focal adhesion kinase (FAK) activity partially mimics the effectof increased force on talin turnover . . . . . . . . . . . . . . . . . . . . . 1646.5.6 Changes in rate constants underlie the developmental regulation of Talinturnover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1656.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1687.1 Identifying unique observations in direct Stochastic Optical Reconstruction Mi-croscopy (dSTORM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697.2 Analysis of fluorescence recovery after photobleaching (FRAP) data to understandcell adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172A Supporting information for Chapter 2: Results of fitting the continuous-time modelto identify the number of unique fluorophores . . . . . . . . . . . . . . . . . . . . . 189A.1 Simulating data for the continuous-time model (Equation 2.1) . . . . . . . . . . . 189A.2 Estimates of the kinetic rates depend on the sample size of the data for the continuous-time model when the number of fluorophores is known . . . . . . . . . . . . . . . 191A.3 Estimating the number of fluorophores when the kinetic rates are known dependson the kinetic parameters for the continuous-time model . . . . . . . . . . . . . . 193A.4 Estimating the number of fluorophores from continuous-time model without assum-ing any information on the parameters. . . . . . . . . . . . . . . . . . . . . . . . . 196xList of TablesTable 2.1 Parameters of the discrete-time and continous-time Markov models for dSTORManalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Table 2.2 Parameter values used to simulate datasets for discrete-time model . . . . . . . 40Table 2.3 Comparing theoretical and simulated statistics of the Nblinks of one fluorophoreunder the discrete-time model . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Table 2.4 Number of samples with ∆n =NLL(n)−NLL(Nˆi) equal to zero and smaller than3, for n = 1, ..,6 all the parameter sets . . . . . . . . . . . . . . . . . . . . . . . 46Table 2.5 Ranges, modes, and means of the number of localizations and the estimatednumber of fluorophores, for samples with N = 20 and all parameter sets . . . . 48Table 2.6 Number of simulations correctly estimating one fluorophore for the differentsample sizes and parameter sets, when estimating the transitions rate as well . . 48Table 2.7 Differences between the negative log likelihood (NLL) function at the MLE forthe full model, the model with the transition information, and the model whenknowing the number of flurophores all parameters sets and N = 1 . . . . . . . . 52Table 2.8 MLE and true valuesof transition probabilites for 1,000 samples of parametersets S2, S3, S5, and S6, when also estimating the number of fluorophores (N = 20) 53Table 3.1 Stormgraph parameters used to cluster the simulated data for all parameter setsand values of N (N = 1,20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Table 3.2 Summary of the pre-processing analysis when N = 1 for all parameter sets . . . 85Table 3.3 Summary of the pre-processing analysis when N = 20 for all parameter sets anddifferent grid sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Table 3.4 Mean and 99% intervals of the Euclidean distances between the true localizationsand the corresponding closest estimated position for pre-processed samples fromN = 20, all parameter sets and all grid sizes . . . . . . . . . . . . . . . . . . . . 89Table 3.5 Estimated temporal parameter values when fitting the spatiotemporal model forone fluorophore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91xiTable 4.1 B-cell A summary of estimates using the spatiotemporal model and differentnoise levels (α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Table 4.2 Summary of the estimates using the spatiotemporal model for five B-cell datasets 111Table 5.1 Integrin mutants used to affect outside-in activation . . . . . . . . . . . . . . . 120Table 5.2 Outside-in integrin activation through chemical induction stabilizes cell-ECMadhesion by decreasing integrin turnover . . . . . . . . . . . . . . . . . . . . . 130Table 5.3 Integrins point mutations that induce activation affect turnover similarly to treat-ment with divalent cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Table 5.4 Mutations that block or strongly induce outside-in integrin activation are insen-sitive to treatment with divalent cations . . . . . . . . . . . . . . . . . . . . . . 132Table 5.5 Reducing the availability of ECM ligands increases integrin turnover . . . . . . 136Table 5.6 Developmental series analysis of activation increasing and deficient integrin mu-tants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Table 5.7 Rap1 regulates integrin turnover downstream of outside-in activation . . . . . . 138Table 6.1 Talin mutants used to affect inside-out activation . . . . . . . . . . . . . . . . . 146Table 6.2 Increased mechanical force at myotendinous junction (MTJ)s stabilizes cell-ECM adhesion by regulating talin turnover . . . . . . . . . . . . . . . . . . . . 157Table 6.3 FRAP analysis of talin mutants uncovers mechanisms that regulate adhesionturnover in response to force . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Table 6.4 Decreased mechanical force at MTJs modifies Talin turnover . . . . . . . . . . 160Table 6.5 The Talin IBS-2 domain is essential to coordinate turnover in response to reducedmechanical force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162Table 6.6 Reduced FAK activity affects integrin turnover in a manner similar to increasedforce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Table A.1 Parameter values used to simulate a single fluorophore . . . . . . . . . . . . . . 190Table A.2 Comparing theoretical and simulated statistics of the Nblinks and Tbleach for onefluorophore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Table A.3 Range intervals of the number of unique observations and the estimated numberof fluorophores, and the number of simulations with less than 20 localizations,for N = 16 and S2, S3 and S4 parameter sets . . . . . . . . . . . . . . . . . . . 195Table A.4 Number of simulations correctly estimating one fluorophore for the differentsample sizes and parameter sets . . . . . . . . . . . . . . . . . . . . . . . . . . 196Table A.5 Differences between the value of the negative log likelihood (NLL) function atthe MLE for the full model and the model with information about the rates or thenumber of flurophores all parameters sets and N = 1 . . . . . . . . . . . . . . . 200xiiList of FiguresFigure 1.1 Schematic of FRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 Schematic of single-molecule localization . . . . . . . . . . . . . . . . . . . . 7Figure 1.3 Schematic of dSTORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 2.1 Schematic of states of fluorophores with transition rates . . . . . . . . . . . . 17Figure 2.2 Schematic of overlapping blinks of two fluorophores . . . . . . . . . . . . . . 23Figure 2.3 Example of a time series divided in bright and dark classes . . . . . . . . . . . 25Figure 2.4 Example of a time series divided in D, B1 and B2 classes . . . . . . . . . . . . 26Figure 2.5 Distribution of the number of blinks of the simulated data with N = 1 for pa-rameter sets S1, S3 and S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 2.6 MLE for the parameter sets S1, S2 and S3 for different sample sizes with N = 1fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 2.7 MLE for the parameter sets S4, S5 and S6 for different sample sizes with N = 1fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 2.8 Estimates of the number of fluorophores with known transition probabilities forall the parameter sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 2.9 Estimates of the number of fluorophores with known transition probabilities andN = 20, for all parameter sets . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 2.10 MLE for the parameter sets S1, S2 and S3 for different sample sizes whenestimating also N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 2.11 MLE for the parameter sets S4, S5 and S6 for different sample sizes whenestimating also N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 2.12 Estimates of the number of fluorophores for the complete sample with 10,000simulations for each parameter set S1-S6 without fixing the transition parameters 51Figure 2.13 Number of fluorophores estimated for 1,000 simulations for parameter sets S2,S3, S5, and S6 without fixing the transition parameters (N = 20) . . . . . . . . 53Figure 2.14 Comparing the distribution of the estimated number of fluorophores when N =20 for parameter sets S2, S3, S5, and S6 . . . . . . . . . . . . . . . . . . . . . 54xiiiFigure 2.15 Difference in estimates between conditioning the model and not conditioningon the maximum number of simultaneous activations, for N = 1 . . . . . . . . 55Figure 2.16 Comparing Nfit distribution estimated by conditioning and not conditioning overAmax for N = 1 and S1, S2 and S3 parameter sets . . . . . . . . . . . . . . . . 56Figure 2.17 Comparing Nfit distributions from conditioning and not conditioning over Amaxfor N = 20 and S2, S3, S5, and S6 parameter sets . . . . . . . . . . . . . . . . 57Figure 2.18 Comparing Nfit distribution estimated by conditioning and not conditioning overAmax for N = 20 when simulating conditioned model with S5 parameter set . . 58Figure 3.1 Example of spatial separation of dSTORM data . . . . . . . . . . . . . . . . . 61Figure 3.2 Schematic of a time series that ends with one active fluorophore during one frame 63Figure 3.3 Schematic of Gaussian symmetry and example of one flurophore with multiplelocalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 3.4 Schematic of multiple data points fitted to a GMM with three Gaussians . . . . 71Figure 3.5 Analysis of the number of unique estimated mixtures for the dSTORM spatialmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 3.6 Schematic of the likelihood of two independent models . . . . . . . . . . . . . 76Figure 3.7 Schematic of the likelihood of two independent models when approximatingone within its confidence interval . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 3.8 Spatial variances simulated from a Gamma distribution . . . . . . . . . . . . 82Figure 3.9 Simulated localizations of one sample with a single fluorophore for the differentparameter sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 3.10 Simulated localizations of one sample with twenty fluorophore with temporalparameters S3 and different spatial separation . . . . . . . . . . . . . . . . . . 84Figure 3.11 Example of clustering pre-processing of one sample simulated with twenty flu-orophores and temporal parameters S3 for different spatial separation . . . . . 86Figure 3.12 Estimated localization of one sample with a single fluorophore using the spatialdSTORM model for the different parameter sets . . . . . . . . . . . . . . . . . 88Figure 3.13 Distribution of the Euclidean distance between estimated and true positions forone fluorophore for all parameter sets . . . . . . . . . . . . . . . . . . . . . . 89Figure 3.14 Example of data and estimated positions of twenty fluorophores using the spa-tial dSTORM model for all grid sizes and parameter sets S1, S3, and S5 . . . . 90Figure 3.15 Estimated number of fluorophores for 10,000 spatiotemporal simulations of afluorophore for each parameter set S1-S6 . . . . . . . . . . . . . . . . . . . . 92Figure 3.16 Comparing the estimated number of fluorophores for 10,000 simulations usingthe spatiotemporal model and the temporal model for S3, S4, and S5 parametersets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92xivFigure 3.17 Estimated number of fluorophores of simulated data at different spatial separa-tion using the spatiotemporal model and parameter set S3 (N = 20) . . . . . . 93Figure 4.1 DNA origami data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Figure 4.2 DNA origami data pre-processed using the mean spatial error (8nm) . . . . . . 99Figure 4.3 Estimated number of fluorophores for DNA origami sub-sampled data using ourspatiotemporal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 4.4 Estimated postion of the fluorophores for the DNA origami sample . . . . . . . 102Figure 4.5 Example of the estimated postion of the fluorophores for four DNA origamis . 103Figure 4.6 Density plot of region of interest of five B-cell dSTORM data . . . . . . . . . 105Figure 4.7 dSTORM data summary from the resting B-cell A data . . . . . . . . . . . . . 107Figure 4.8 Stormgraph analysis for B-cell A with different values of α . . . . . . . . . . 108Figure 4.9 Estimated number of fluorophores for B-cell A sub-sampled data using our spa-tiotemporal model and α = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . 110Figure 4.10 B-cell A data and the estimated possition of the fluorophores from our spa-tiotemporal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Figure 4.11 Examples of the fits B-cell A data with the estimated possition of the fluo-rophores from our spatiotemporal model . . . . . . . . . . . . . . . . . . . . . 112Figure 4.12 Comparing the distribution of the number of localizations per estimated numberof fluorophores for all B-cell data sets . . . . . . . . . . . . . . . . . . . . . . 113Figure 4.13 Distribution of the number of localizations in all B-cell sub-sampled data usingour spatiotemporal model and α = 0.05 . . . . . . . . . . . . . . . . . . . . . 114Figure 4.14 Distribution of the estimated number of fluorophores for all B-cells using ourspatiotemporal model and α = 0.05 . . . . . . . . . . . . . . . . . . . . . . . 115Figure 5.1 Integrin activation schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Figure 5.2 Example of FRAP curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Figure 5.3 Schematic of the mathematical model for integrin dynamics . . . . . . . . . . 123Figure 5.4 Example of data fitting for integrin turnover . . . . . . . . . . . . . . . . . . . 127Figure 5.5 Outside-in integrin activation through chemical induction stabilizes cell-ECMadhesion by decreasing integrin turnover . . . . . . . . . . . . . . . . . . . . . 129Figure 5.6 Integrins point mutations that induce activation affect turnover similarly to treat-ment with divalent cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Figure 5.7 Mutations that induce outside-in integrin activation are insensitive to treatmentwith divalent cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Figure 5.8 Mutations that block outside-in integrin activation are insensitive to treatmentwith divalent cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Figure 5.9 Reducing the availability of ECM ligands increases integrin turnover . . . . . . 135xvFigure 5.10 Developmental series analysis of wild-type (WT) and activating integrin mutants 137Figure 5.11 Developmental series analysis of activation deficient integrin mutants . . . . . 139Figure 5.12 Rap1 regulates integrin turnover downstream of outside-in activation . . . . . . 140Figure 6.1 Schematic of talin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Figure 6.2 Schematic off all iterations between talin and integrin on the membrane or in-tracellular, and bound or unbound . . . . . . . . . . . . . . . . . . . . . . . . 150Figure 6.3 Dynamics of fluorescent talin as described by Equation 6.6 . . . . . . . . . . . 150Figure 6.4 Double fitting protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Figure 6.5 Increased mechanical force at MTJs stabilizes cell-ECM adhesion by regulatingtalin turnover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Figure 6.6 FRAP analysis of talin mutants uncovers mechanisms that regulate adhesionturnover in response to force . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Figure 6.7 Decreased mechanical force at MTJs modifies Talin turnover . . . . . . . . . . 161Figure 6.8 The Talin IBS-2 domain is essential to coordinate turnover in response to re-duced mechanical force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Figure 6.9 Reduced FAK activity affects integrin turnover in a manner similar to increasedforce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Figure 6.10 Talin turnover is developmentally regulated through distinct mechanisms . . . 165Figure A.1 Distributions of the number of blinks from the simulated data with N = 1 forparameter sets S1, S3 and S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 190Figure A.2 MLE for the parameter sets S1, S2 and S3 for different sample sizes when N = 1is fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Figure A.3 MLE for the parameter sets S4 and S5 for different sample sizes when N = 1 isfixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Figure A.4 Estimates of the number of fluorophores with known kinetic parameters for allthe parameter sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194Figure A.5 Difference between the number of localizations and Nfit for S4 and S5 parametersets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Figure A.6 Estimates of the number of fluorophores with known kinetic parameters for allthe parameter sets and N = 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 196Figure A.7 MLE of kinetic parameters for the parameter sets S1, S2 and S3 for differentsample sizes with N not fixed . . . . . . . . . . . . . . . . . . . . . . . . . . 197Figure A.8 MLE of kinetic parameters for the parameter sets S4 and S5 for different samplesizes with N not fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Figure A.9 Estimates of the number of fluorophores for the complete sample with 10,000simulations for each parameter set S1-S5 without fixing the kinetic parameters 199xviFigure A.10 MLE of kinetic parameters for the parameter sets S3 and S5 for different samplesizes with N = 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Figure A.11 Estimates of the number of fluorophores for the complete sample with 10,000simulations for N = 16 and parameter sets S3 and S5 without fixing the kineticparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202xviiGlossaryMicroscopy terms:dSTORM direct Stochastic Optical Reconstruction MicroscopyFRAP fluorescence recovery after photobleachingGFP green fluorescent proteinSTORM Stochastic Optical Reconstruction MicroscopyPALM Photoactivated Localization MicroscopyPSF point spread functionBiological terms:ABS actin binding siteBCR B-cell receptorsBrkd breakdanceDMSO dimethyl sulfoxideDYN dynasoreECM extracellular matrixFAK focal adhesion kinaseFERM F for 4.1 protein, E for ezrin, R for radixin and M for moesinIAC integrin adhesion complexIBS-1 integrin binding site 1IBS-2 integrin binding site 2xviiiMTJ myotendinous junctionpara paralyticTHATCH talin/HIP1R/Sla2p Actin-Tethering C-terminal HomologyWT wild-typeMathematical and statistical terms:AIC Akaike information criterionEM Expectation-Maximization algorithmGMM Gaussian Mixture modeliid independent and identically distributedMLE maximum likelihood estimatorNLL negative log likelihoodODE ordinary differential equationsSEM standard error of the meanSSR sum of square residualsxixAcknowledgmentsI want to acknowledge my gratitude towards all the faculty, staff and fellow graduate students thathelped, inspired and encouraged me to succeed in this job. I also want to acknowledge my fundingagencies: The Pacific Institute for Mathematical Science (PIMS, Canadian), the National Councilof Science and Technology (Consejo Nacional de Ciencia y Tecnologia, CONACyT, Mexican),The University of British Columbia (UBC, Canadian), and the Natural Sciences and EngineeringResearch Council (NSERC, Canadian, throughout Daniel Coombs grads).In particular, I wish to highlight all the support, help and advice given by my supervisor Dr.Daniel Coombs. His creativity and passion showed me a greater world beyond mathematics. Also,I thank all the past and current members of the Coombs group whose discussions strength this dis-sertation. In no relevant order, they are: Bernhard Konrad, Rebeca Cardim Falcao, Catherine Byrne,Sarafa Iyaniwura, Joshua Scurll, Libin Abraham, Raibatak Das, Jessica Conway, Mike Irvine, andClaire Guerrier.I would like to thank the Tanentzapf lab, without them I would not discover the pleasure ofworking directly with experimentalists. I learned from our collaboration how mathematics learnfrom biology as much as biology from mathematics. In particular, I want to thank Guy Tanentzapf,Pablo Lopez-Ceballos and Katrin Hakonardottir.I owe special thanks to my family in Mexico, Vancouver, and around the globe. They werealways there to support me emotionally, academically and sometimes financially. I would not havegotten here without you! THANK YOU!xxChapter 1IntroductionIf and to what degree fluorescence microscopy will widen the possibilities ofmicroscopic imaging only the future will show.— Oskar Heimstadt, 1911Fluorescence microscopy has proven itself to be extremely useful for cell biology research [135].It has allowed biologists to examine fixed or live cells and tissues, replacing qualitative representa-tive observations by high-throughput quantitative results [59, 221]. The development and applica-tion of these microscopic techniques have been a beautiful example of collaborative work amongphysicists, chemists, mathematicians and biologists [126, 135].The application of these microscopic techniques, along with mathematical models and statisticalanalysis of the data, has been essential to understand cell adhesions and B-cell signalling, two of thebiological process that I consider in this dissertation. To quantitatively understand these biologicalprocesses, I have used different deterministic and stochastic mathematical models to analyze dataobtained from two microscopy techniques: fluorescence recovery after photobleaching (FRAP) anddirect Stochastic Optical Reconstruction Microscopy (dSTORM).This introduction chapter will begin with a brief introduction to fluorescence microscopy andrecently developed super-resolution techniques, including in particular FRAP and dSTORM andtheir application to biological systems. The organization of this dissertation is described at the endof this chapter.Throughout, I will illustrate the biological context using analogies and simplified examples.1.1 Fluorescence microscopyIn the sixteenth century, the microscope was invented by Robert Hooke and Antonie van Leeuwen-hoek [59, 98, 125]. Leeuwenhoek saw life in a drop of water, an early example of how microscopycan change biology [59]. Nowadays, we can see the proteins on a cell membrane!1The microscope, and in particular the fluorescence microscope, has since become essential inbiology, pathology, and medicine [59, 221]. Oskar Heimstadt developed the first fluorescent mi-croscope in 1911 [59, 90]. He was pessimistic about the future use of this technology given itsmultiple caveats. Since then, many advances have taken fluorescence microscopy to study biologi-cal molecules, live cells and tissues and even molecular pathways [59, 60, 221]. Fluorescence mi-croscopy nowadays is compatible with fixed and live samples, is minimally invasive, and can allowexperiments to be performed under near-physiological conditions [60]. To get here, fluorescence mi-croscopy needed the many advances in microscopy, protein labelling and optical techniques, suchas improvements in the lenses, microscopes and cameras; the discovery and use of fluorescent an-tibodies (with fluorescein isothiocyanate dye [41, 42]) and genetically encoded fluorescent proteins(green fluorescent protein (GFP) [37, 202]); the development of techniques to remove out-of-focuslight from the image (confocal microscope [56, 135]), to select the place of excitation (two-photonmicroscopy [50, 221]), and to use evanescent waves to observe thin layers of a sample (total internalfluorescence microscopy (TIRF) [10]).Methods have also been developed to use fluorescence microscopy to investigate dynamic phe-nomena across time and space. By taking advantage of the photokinetics of the fluorophores andvarying labelling concentrations, single particles can be tracked and kinetic reactions can be mea-sured. By labelling at high concentrations, and then bleaching parts of the sample, you can measurethe diffusional or recycling properties of membrane proteins. This is the main idea behind fluores-cence recovery after photobleaching (FRAP). FRAP is the main microscopy technique underlyingChapter 5 and Chapter 6 of this dissertation, and I will spend the first part of this chapter describingit.Light microscopy has an inherent resolution limit. Super-resolution fluorescence microscopytechniques allow biologists to observe cellular structures smaller than the limit of light diffraction(< 200nm) [126]. These methods can be used in living cells and even for 3D imaging. Thereare different methods to achieve super-resolution. I will focus on single-fluorophore-based super-resolution where the location of individual fluorophores is estimated via fitting a Gaussian distri-bution to the observed distribution of photons. When the molecules are not well separated, super-resolution can be achieved by activating only a few fluorophores at the same time. In Section 1.3,I will give an overview of super-resolution and focus on direct Stochastic Optical ReconstructionMicroscopy (dSTORM).1.2 Fluorescence recovery after photobleaching (FRAP)Fluorescence recovery after photobleaching, or FRAP, is a fluorescence microscopy technique usedto quantify molecular mobility inside or on the surface of a cell [11, 46, 46, 55, 116, 156, 157, 164,211]. The molecules of interest are labelled at high density with a fluorescent tag and imaged underthe microscope. FRAP can be performed with a confocal microscope [46, 116] or via total internal2Figure 1.1: Schematic of FRAP (a) The muscle-tendon regions are labelled in a Drosophila embryo.The yellow rectangle is showed in detail in (a’). The sample is bleached in the circular area. Thefluorescence intensity is recorded in the measured region (orange rectangle) over time (a’-a”).Scale bar, 5µm.reflection fluorescence microscopy for thin samples [12, 100, 192, 200, 221]. During the imaging,a region of the cell containing the labelled molecules is selected and photo-bleached (Figure 1.1)[23, 46, 116]. The fluorescence intensity in the bleached region is then recorded for enough time tocapture the change of intensity in the bleached area [23, 46, 116, 211].To understand FRAP better, imagine you are filming a field covered by fireflies. The fireflies flyaround at random and your camera captures the light they emit. Then, someone traps every fireflyin a small region and removes all those fireflies from the filming region. You keep your camera onwhile other fireflies fly into the empty area. You can then estimate firefly mobility by measuring therecovery of light in the capture region. In terms of FRAP, the fireflies are the tagged molecules, thecapture region is the bleached region, and the movie corresponds to the imaging record. The onlydifference is that you “turn off” the tag from the molecules by photobleaching, instead of removing3them from the sample.Thus, the recovery of fluorescence in the bleached area gives us a lot of information about themobility properties of the molecules [46, 100, 116, 164, 211]. There is no recovery on immobilesamples, but otherwise, molecules moving into the bleached region will produce an increase influorescence. The fluorescence does not necessarily recover to pre-bleaching levels [46, 116, 164,211]. The difference between pre-bleaching and post-bleaching fluorescent intensity is interpretedas representing an immobile fraction of molecules [46, 116, 211].In the recovery region, the intensity could come from different recovery pools depending onthe dynamics of the molecules. Samples can be analyzed with FRAP to quantify if the protein isfreely diffusing, changing diffusion rates, trapped in spatial regions, bound to immobile proteins, ora combination of these [24, 46, 54, 100, 116, 164, 186, 211].The average fluorescence intensity in the photo-bleached region in each image is measuredover time and is called the recovery curve [46, 116, 211]. Commonly, the recovery curves arethen analyzed with a simple model to estimate the fraction of mobile proteins and the meantimeof recovery [116, 121, 211]. Quantitative conclusions about the dynamics and differences betweendifferent samples are quantified by the change in the parameter fits.In Chapter 5 and 6, we approach the fitting from a modelling perspective rather than as purelya curve-fitting exercise. We used biological assumptions about the labelled protein and constructeda mathematical model to describe their dynamics. Our model, therefore, has meaningful biologicalparameters allowing a direct mechanistic interpretation of recovery.1.2.1 FRAP as a tool in cell biologyFRAP was developed in the 1970s [11, 55, 156], but it was not until the discovery and increased useof GFP [37, 202, 203] that FRAP became a popular tool in cell biology [46, 116, 157, 164, 211, 221].Before GFP, fluorescent proteins were micro-injected into the cells [221]. This procedure disruptedliving cells, compared to GFP which can be genetically encoded [37, 202, 203, 221]. In general,GFP doesn’t affect the function of the tagged proteins, is photostable and bleaches with a high-intensity laser. Because of these characteristics, GFP and other analogous fluorescent probes areessential allies to FRAP in probing live cell dynamics [46, 116].Using GFP, FRAP has become an essential tool for the in-vivo study of cell biology. Experimen-talists may now watch and quantify diffusion, assembly in the membrane and within the cytoplasm,recycling and degradation, among other transport dynamics of tagged proteins [46, 116].1.2.2 Further discussion of FRAP analysisFRAP experiments and analysis have experimental and theoretical limitations [46, 116, 211]. Inthis section, I will discuss some of those limitations.The loss of spatial information caused by averaging recovery intensities over the bleached region4is a limitation on interpretation. Consider a membrane protein that binds to an immobile moleculethat is not homogeneously distributed on the membrane. In this case, the spatial information isessential, and averaging over space can cause a misinterpretation of the data. The protein will diffusedifferently when bound versus when free in the membrane but the average intensity will not capturesuch changes [116, 186]. Multiple experiments to study different aspects of the dynamics of theproteins, along with a customized mathematical model, are recommended to avoid misinterpretation[54, 116, 159, 186, 211]. However, taking a spatial average will improve the signal-to-noise ratioand thus has benefits in the absence of spatial structure.On the experimental side, photoswitching of the fluorophores can be a problem [46, 116]. Pho-toswitching here refers to the ability of a fluorophore to recover from the bleached state. A centralassumption of FRAP is that proteins become invisible after bleaching, but photoswitching violatesthis assumption. Photoswitching would represent a means of recovery, independent of the biolog-ical system. GFP and similar probes show photoswitching at low levels (less than 15% in normalexperimental conditions) [46, 116]. So, we cannot rule out this possibility but we may assume thatthe photoswitching affects multiple samples similarly and thus derive conclusions about differencesbetween various experimental conditions. However, exact parameter estimates should be interpretedcarefully if photoswitching is believed to have occurred.The energy to activate the fluorescence comes from laser illumination [54, 116]. Therefore,fluorophore activation and photobleaching increase with the intensity of the laser. This is a charac-teristic exploited to bleach the region of interest by using a brief period of high-intensity laser light.Rapid bleaching reduces the problem of the immigration of proteins from outside the bleach regionduring bleaching [116]. But high-intensity light cause heating and thus changes in biological func-tion [46, 116, 164]. Experimental protocols should consider a careful balance between bleachingintensity and sample damage.Giving that exposure to laser light causes photobleaching, photobleaching also occurs whilerecording the recovery of fluorescence. This photobleaching is slower than the initial burst usedto bleach the region because the intensity of the laser is lower. Most studies do not account forthe slow background photobleaching [46, 116]. There have been some theoretical works showinghow to correct for bleaching using an exponential decay [35, 185, 217]. In our work, we added aparameter to account for the photobleaching occurring by the experimental laser observation.To simplify the description, in Chapters 5 and Chapter 6, I will use the terms “natural photo-bleaching” or “natural bleaching” to refer to the photobleaching of fluorophores caused by the obser-vation process, and the terms “photobleaching” or “bleaching” to refer to the intentional bleachingstep at the start of the FRAP protocol.51.3 Direct stochastic optical reconstruction microscopy (dSTORM)Direct stochastic optical reconstruction microscopy, or dSTORM, is a super-resolution fluorescencemicroscopy technique used to achieve≈ 20nm lateral resolution images of individual fluorescently-labelled proteins [88, 126, 169]. dSTORM uses single-molecule localization to surpass the lightdiffraction limit of ≈ 150nm [88, 169]. The single-molecule resolution is produced by separatingthe proteins in a temporal sense using the fluorescence intermittency (“blinking”) property of thedies [88, 169]. The image is reconstructed by putting together the single localizations.dSTORM was developed in 2008 by Heilemann and collaborators [88]. It is a simplification ofthe technique developed by Rust and colleagues in 2006 called Stochastic Optical ReconstructionMicroscopy (STORM) [169]. In the 2006 protocol, STORM was performed using two fluorescentmarkers to label the specimen [169]. One of the fluorophore switches between luminous and darkstates with aid of the second fluorophore [15, 169]. Heilemann and collaborators simplified themethod by activating one fluorescent tag directly, hence the name “direct” STORM, or dSTORM.This super-resolution technique has been extended to multiple labelling (using multiple colourtags [16]), and to achieve three-dimensional data [101], among other extensions [93, 102, 126, 215].In this thesis, we focus only on a two-dimensional acquisition of a specific protein labelled with asingle colour.1.3.1 The diffraction limit of light and single-molecule localizationThe resolution of a microscope is given by its ability to distinguish nearby objects [21, 91, 92, 94,102]. While improving lens technology, Abbe discovered in 1873 that the diffraction of light alsoaffects the resolution [1, 92, 94]. When a point is imaged, it produces a blurred spot [21, 26, 94, 102].The blurred image of a point source of light is represented by the point spread function (PSF). If thedistance between multiple objects is smaller than the width of their PSF, their blurred images willoverlap and it will be impossible to distinguish them[102].The size of the PSF depends, in part, on the wavelength of light [1, 21, 26, 94, 95]. LordRayleigh derived an approximation to calculate the width of the PSF [91, 162]. This relationshipstates that the resolution is approximately half the wavelength of the light divided by the numericalaperture of the lens. Visible light studies are thus restricted, even with the best light microscope, toa resolution of ≈ 150nm [91, 92, 95].The desire to resolve positional information below the diffraction limit has encouraged the de-velopment of other imaging techniques [91, 91, 92, 102, 115, 126]. Such methods include mod-ifying the imaging wave as in electron microscopy, or the optics in near-field optical microscopy[91, 94, 126]. In general, this increased the resolution but resulted in the loss of the ability to image3D and live specimens, which are not limitations to fluorescent microscopy [102, 126]. Therefore,a higher resolution technique based on fluorescent microscopy was sought.6Figure 1.2: Schematic of single-molecule localization. One labelled molecule is imaged showing itscorresponding point spread function (green) and the corresponding enhance localization precisionfrom the number of photons collected during imaging (blue).During the last two decades, interdisciplinary teams developed different methods to take thelight microscope to nanometre scales [93, 95, 102, 126]. We will focus on the single-moleculelocalization method, which is the base method for dSTORM.Single-molecule localization develops from the observation that resolution is not the same aslocalization precision [94, 201, 215]. The resolution restricts the details we can see within thePSF, but we could determine the centre of such PSF and increase the localization precision ofthe molecule (Figure 1.2) [201]. We know that the PSF follows an Airy distribution that is wellapproximated with a Gaussian distribution [20, 201]. Therefore, if we image a single molecule, themean of the Gaussian fit to the PSF is the localization of such molecule [20, 93, 102, 126, 201, 215].There will still exist an error in the estimated localization. Assuming that the background noiseis small, the localization precision depends on the number of photons collected during imaging[20, 201]. The PSF is the probability distribution of observing a photon given the exact localizationof the molecule. Then each photon collected is a sample from the distribution [91, 201]. The local-ization error is approximately the microscope resolution divided by the square root of the numberof captured photons [93, 102, 126, 201, 215].To best estimate the centre of the PSF, the fluorophores or molecules need to be far apart andtheir PSF should not overlap. That is the reason single-molecule localization was used first insparse or low-density labelling systems, including single-particle tracking [201]. In the early 2000s,this method was extended to create several super-resolution techniques even in non-sparse environ-ments. These techniques include Photoactivated Localization Microscopy (PALM), STORM, anddSTORM [93, 102, 126, 215]. Such super-resolution techniques use the blinking properties of thefluorophores to separate them temporally.7Figure 1.3: Schematic of dSTORM. At each frame, only a few fluorophores are activated (green)and their localization is estimated to single-molecule resolution (blue). Then, all the frames areoverlapped to generate a super-resolution image.1.3.2 dSTORM separates nearby proteins with a temporal approachdSTORM separates proteins that are closer than the resolution limit by observing only a few at atime [88]. Heilmann et al. labelled the sample with fluorophores that transition back and forth, orphotoswitch, between fluorescent and dark states, like cyanine dyes Cy5 and Alexa 647 [88, 209].This photoswitchable characteristic allowed them to modulate the number of active fluorophores ateach time using activation lasers.The imaging sequence starts with all fluorophores turned off. An imaging cycle lasts for a timedetermined by the camera and lasers. During that time, the laser activates a small and randomnumber of fluorophores [88, 215]. By activating a small sample of fluorophores across the wholeimaging region, the probability of activating two overlapping fluorophores is close to zero. Thefluorescence is recorded and one imaging cycle ends [88]. The imaging cycle is then repeated manytimes to generate many temporally-separated snapshots. Figure 1.3 illustrates the concept.I refer to each imaging cycle as a frame. At each frame, computer software is used to resolvethe fluorophore localizations using single-molecule methods. Next, the collection of nano-resolvedframes are merged to reconstruct a final super-resolution image [88, 215]. Heilemann et al. usedthis technique to image microtubules and actin filaments in mammalian cells with 21-nanometreresolution [88].1.3.3 dSTORM as a qualitative and quantitative toolHistorically, scientists have relied on images to expand the knowledge of many biological systems.With the development of super-resolution techniques, we have achieved a view of the nano-worldthat was previously only accessible through electron microscopy. In particular, dSTORM only8depends on fluorescence tags and is a noninvasive method. These characteristics have allowed theuse of dSTORM to qualitatively improve our understanding of many biomolecular structures. It hasbeen used to image microtubules, actin filaments, and cellular cytoskeleton, among other proteins[87, 88]. DNA was also imaged to a 40nm resolution thanks to similar techniques [61, 87].Beyond providing high-resolution images, dSTORM images give information on the organiza-tion of the proteins imaged [87, 89, 153, 208]. The high definition provides insight into denselypacked proteins that were previously impossible to distinguish [88, 208]. Scientists had used suchinsights to analyze the distribution and density of biomolecules, like relating T-cell activation withmembrane receptors spatial distribution [209].1.3.4 Some limitations of dSTORMdSTORM, along with other single-molecule super-resolution methods, has to balance the labellingdensity of the proteins. A high final resolution of the images requires high labelling of proteinsto ensure that they are all observed [87]. Also, the chosen labels should have long dark periodsto reduce the probability of activating nearby proteins in the same frame [87]. Moreover, if thelabelling density is too high, it may cause energy transfer between fluorophores or spurious sponta-neous activations [102, 215]. Those unexpected activations could compromise the single-moleculelocalization assumption. Therefore, fluorophores with stable and long dark states are desired [102].One more limitation comes from taking many individual images to gain super-resolution. Sam-ples undergo a temporal drift coming from physical movements of the microscope stage at thenanoscale. An excellent approach to reduce this problem is to perform an automated drift correc-tion with the use of a software-controlled stage.In contrast to imagining fixed cells, dSTORM temporal resolution could cause biases whenimaging live-cell samples [87, 215]. The cell or the labelled molecule might move faster than theframe size, thus impeding our ability to reconstruct an accurate image. New methods are beingdeveloped mixing dSTORM with other labelling or imaging techniques to overcome this limitation[215].In Chapters 2 and 3, we focus on another problem inherent in dSTORM: determining the num-ber of fluorescent labels that are present, from the recorded localizations. The blinking properties ofthe fluorophores might cause a single fluorescent tag to appear in multiple frames with slightly dif-ferent spatial localization. We develop a spatiotemporal model to recognize the unique fluorescenttags in a given dSTORM data set.1.3.5 Photoactivated localization microscopy (PALM)Photoactivated Localization Microscopy (PALM) is another super-resolution technique. PALM anddSTORM share the same idea: separate the observations in time and use single-molecule localiza-tion at each time step. The main difference is in the labelling proteins. PALM uses photo-activable9fluorescent labels (PA-FP). PA-FP can be activated and transition to and from dark and light statesusing lasers [126]. The initial activation of the fluorophores is not needed in dSTORM. PALM hasbeen used in live samples, for 3D imaging, and with multiple colours. Rollins et al. developed astochastic model of the dynamics of PA-FPs for PALM, to quantify the molecules imaged [167].The model of Rollins et al. provided a framework for analyzing the temporal aspects of fluorophoreblinking. We exploit this framework in Chapters 2 and 3.1.4 Dissertation organizationWe could describe this dissertation in two parts divided by their main topics: dSTORM and FRAPdata. The first part comprising Chapters 2-4 focuses on dSTORM analysis with applications toexperimental data. The second part regards the analysis and modelling of FRAP data to understandcell adhesions, in Chapters 5 and 6.Our goal in the analysis of dSTORM data is to estimate the unique number of fluorophorespresent in a sample. To do this, we developed a continuous-time and a discrete-time Markov modelin Chapter 2. In that chapter, we focus on the theoretical and numerical side of the model. Wevalidate our approach using simulated data. We improved our model by adding spatiotemporalinformation about the data. This is described in Chapter 3. Again, we focused on the methods,algorithms and tests using simulated data. Finally, Chapter 4 uses our model from Chapter 3 toestimate the number of distinct fluorophores on real experimental data coming from experimentsusing labelled DNA origami and B-cell receptors.In Chapter 5, we study the relationship between membrane integrin proteins and the extracel-lular matrix. In this chapter, we apply a linear ordinary differential equation model to describe thedynamics of the system in individual Drosophila. We then apply a similar model in Chapter 6 tostudy the interaction of the cytoplasmic protein talin with integrins and the consequences of talin-integrin interactions on cell adhesion. To extract useful mechanistic insights from the data, wemixed mathematical analysis and biological experiments to identify the parameters.Chapter 7 contains a brief conclusion and outlook on possible future directions in these areas ofthe dissertation.10Chapter 2Identifying unique observations insuper-resolution microscopy: temporalmodelBe that self which one truly is— Soren KierkegaardIn this chapter, we analyze super-resolution microscopy data with the goal of identifying uniquefluorescent objects in the images. We focus on the super-resolution method called dSTORM (seeSection 1.3). dSTORM achieves≈ 20nm resolution images by separating the proteins in a temporalsense using photoswitchable fluorophores [126, 169]. Photoswitchable fluorophore blinking prop-erties are essential for fluorophore temporal separation but they can appear (blink) multiple timesduring an experiment. These blinks will usually be localized at different spatial locations, leadingto the problem that the number of unique fluorophores is not known. With our model, we seek toimprove the quantitative analysis of super-resolution data by effectively correcting for this problem.We developed Markov type models to describe the dynamics of each fluorescent protein. As-suming independence between the fluorophores in the sample, we can extend our single fluorophoremodel to N fluorophores. Then, we use maximum likelihood techniques to estimate the number offluorophores in a sample. We present tests of our method using simulated data. Results obtainedfrom real data will be presented in Chapter 4.This chapter is organized as follows: Section 2.1 describes the difference between the numberof localizations and the number of labelled proteins in dSTORM data; Section 2.2 presents a biolog-ical motivation to obtain unique observations from potentially over-counted super-resolution data;Section 2.3 summarizes previous modelling strategies for similar super-resolution microscopy tech-niques; in Section 2.4 and Section 2.5, we build continuous and discrete-time models and presentanalytical and simulated results. Section 2.6 resolves an important technical issue by conditioning11the discrete-time model on a maximum number of simultaneous observations. Section 2.7 describesthe optimization problem and its solution. The chapter concludes with results based on simulateddata and a general discussion.2.1 The problem of relating the number of localizations with thenumber of labelled proteins from dSTORM dataIt is important to recognize that the number of localizations in a dSTORM image is not equivalentto the number of labelled proteins present. There are many factors that can differentiate the countof proteins and observations. Some of those factors come from the labelling methods, while othersfrom the single-molecule localization algorithms.First, there is a direct difference between observations and biomolecules. Since dSTORMdoes not use genetically encoded fluorophores, what we observe is the fluorescent tag and notthe biomolecule [89]. Moreover, it is common to rely on antibodies to adhere the label to thebiomolecule of interest [102]. The antibodies may have more than one fluorophore attached, so oneprotein could have more than one fluorescent tag. Therefore, we can only relate the observationsfrom dSTORM data to fluorophore counts and not to proteins.Focusing on relating the fluorophores and the observations, we have two main problems. Wecould underestimate the number of fluorophores due to the presence of multiple simultaneous local-izations with overlapping point spread function (PSF), or incomplete activation of fluorophores, orwe could overestimate the number due to multiple blinking of individual fluorophores [52, 215].The under-observation problem arises from the accidental activation of nearby fluorophores orfrom the incomplete activation of the fluorophores [52, 215]. It has been observed that only afraction of the fluorophores activate [52]. This lack of activation could be caused by misfoldingor incomplete maturation [52]. On the other hand, if two nearby fluorophores activate at the sametime, their corresponding PSFs will overlap. In this situation, single-molecule localization cannotbe achieved and the observation may be discarded or recorded as a single localization [52].Overestimation arrives from the blinking characteristics of the fluorophores [3, 52, 124]. Sincethe fluorophores are changing between dark and bright states, we may count the same fluorophoremultiple times. The localizations of those multiple observations will be nearby, forming small arti-ficial clusters. In principle, if the meantime in the dark state is a lot smaller than the duration of theexperiment, then the time series of localization observations will also cluster temporally [52]. Onecorrection to over-counting is thus to perform clustering of the data in time. This clustering mightlead to incorrectly removing fluorophores if the temporal cluster size is not accurate. In particular,we cannot differentiate between bleached fluorophores or long dark states. Thus, applying a clus-tering threshold is likely to affect the identification and probably discard more localizations thannecessary.In 2012, Lee et al. introduced a correction for the blinking of fluorophores in Photoactivated12Localization Microscopy (PALM) [124]. PALM is a super-resolution microscopy technique thatalso uses single-molecule localization to achieve nanoscale resolution. Lee et al. modelled thedynamics of the fluorophores using Markov properties and independence. Others have improvedand implemented Lee’s model (see Section 2.3 for more detail). We used similar assumptions tocorrect for the temporal aspects of blinking of fluorophores in dSTORM.2.2 Biological motivationIn this section, we will introduce two biological systems where over-counting could lead to a mis-interpretation of dSTORM data. The need to correct these datasets motivated this project. In bothcases, identifying the correct number of fluorescent objects will enhance the meaning and quantita-tive interpretation of the experimental images.2.2.1 B-cell receptor clustering and activationB-cells are an important part of our adaptive immune system [145]. B-cells capture and processantigens, leading to antibody production and T-cell activation. B-cells have B-cell receptors (BCR)in the cell membrane. These receptors bind certain antigens and trigger the activation signallingpathway of B-cells. There are different isotypes of BCR, such as IgD and IgM with subtly differentfunctions. Therefore, the study of BCR is one of the main points to understand the function of theimmune system.Previous super-resolution imaging studies have shown the existence of BCR nano-clusters [136].Mattila et al. observed clusters of 60-80 nm radius with around 30-120 IgD and and 20-50 IgM(BCR types) [136]. They also showed a correlation between the intensity of BCR signalling andthe size of the clusters [51, 136], with no signal in the absence of BCR clusters. The BCR nano-clusters change to micro-clusters when a B-cell encounters an antigen, leading to B-cell activation.By correcting over-counting on our own B-cell data, we seek to better understand the relation of thecluster size with the signalling level.2.2.2 Cardiac myocyte receptorsCardiac myocytes are the muscle cells of the heart. They have ryanodine receptors (RyR receptors)that control their calcium channels [8, 18]. The RyR receptors are arranged in tetramers and Asghariet al. hypothesized that the spatial organization of the tetramers is related to the level of health ofthe cell [8]. Therefore, experimentalists desire to have an exact localization of the channels, butthe aggregation of the RyR receptors (tetramers and not monomers) will affect the super-resolutionimaging. Given the technical nature of dSTORM, the reconstructed image might have spatial biasescoming from multiple observations of the same receptor, or by missing receptors in a tetramer.I did not attack this project, but it provides additional motivation for resolving super-resolution13localizations and could be a subject for further study.2.3 Previous modelling approachesIn Section 2.1, we mentioned some solutions found in the literature for the over-counting problemin super-resolution. Here, we will describe these methods further and relate them to the presentwork.As mentioned before, repeated blinking of the fluorophores could lead to multiple observationsof the same fluorophore [3, 4, 52]. The first method to differentiate fluorophores constrained the datain time for PALM data [4]. Assuming that the time in a dark state is shorter than the experimentduration, the data should aggregate in time with a high probability. Thus, selecting a threshold intime will separate localizations coming from different fluorophores [4]. The resulting count willvary when the threshold changes. This method is not well-suited to the fluorescent proteins used indSTORM since those fluorescent tags have prolonged dark-state dwell times [103, 207].Other methods corrected over-counting with Kalman filters [123], spatial pair correlation func-tion [180, 210], or Fourier ring correlation analysis [14, 147]. They were developed for single-molecule localization super-resolution techniques, like PALM or STORM. Those methods usedboth temporal dynamics and spatial localization information [52].Lee et al. presented a kinetic model describing the individual dynamics of the fluorophoresin PALM [124]. They assumed that each fluorophore can transition between 4 states: non-active,active, dark or photo-bleached. Assuming independence between the fluorophores, they supposedthat each fluorophore follows a continuous-time Markov chain. They simplified the model to 4random variables to estimate from the data: the number of times a molecule blinks Nblink, thetime it stays on in each blink Ton or between blinks Toff, and the time to photobleaching of themolecule Tbleach. They found that Nblink follows a negative binomial distribution, Ton an exponentialdistribution, and Toff a double exponential. By estimating the latter three random variables, theyfitted for the kinetic rates and use the time to bleach to validate the model.With their kinetic model, they related the empirical threshold in time model, mentioned above[4], to the probability of observing unique molecules. They proposed an algorithm that iteratesbetween calculating the number of molecules with a fixed time threshold and estimating the bestthreshold for the fixed number of molecules, to estimate the unique localizations. Lee et al. alsocorrected for under-counting by assuming that most of the molecules activate at the start of theexperiment. An experimental method to use the activation laser to decrease the poor time separationof localizations at the beginning of the imaging acquisition was also proposed.Lee et al. performed an in-vitro experiment where they controlled the distance between fluo-rescent molecules. This distance was large enough to detect each molecule individually. Underthose settings, they estimated individual rates for two photo-activatable fluorescent proteins, used inPALM. To estimate their random variables Nblink, Ton and Toff, Lee et al. assumed that each dataset14has the same number of fluorophores. This assumption constrains the uses of the model.Fricke et al. used Lee’s model to differentiate between monomers, dimers, oligomers, or amix of those [65]. Lee’s model has also been incorporated in software for quantitative analysis ofsuper-resolution data [134]. The software uses clustering in time of the sample to speed up thecomputation of the number of fluorescent particles and was applied by Kruger et al. in a study ofT-cell receptor density on live T-cells [120]. As mentioned above, this might not be very suitablefor dSTORM data [103, 207].In 2014, Rollins et al. proposed a likelihood-based method to fit the model of Lee et al. byaggregating the states of the Markov chain into classes [167]. The model has 4 states (inactive,active, dark and photo-bleached) for each fluorophore, giving (N +3)(N +2)(N +1)/3 states of achain with N fluorophores. They aggregated the states in two classes: bright and dark. Bright statesare those with at least one fluorophore active; every other state belongs to the dark class. Theyreduced the dimension of the bright class further by assuming a maximum number of simultaneousactive fluorophores, Amax. This assumption is valid for single-molecule super-resolution since weexpect to see sparse-in-time localizations. They also generalized the likelihood function to correctfor missing data. Rollins fitted the data from [124] and their results generally agree with those ofLee et al.Rollins’s algorithm becomes slower as the number of localizations increases. However, it givesthe number of unique fluorophores without assuming the same number of fluorophores in all thesamples. Thus, this model can be used for small heterogeneous samples.The approaches of Lee and Rollins assume continuous time. However, the data is fundamentallydiscrete in time. In 2012, Gunzenha¨user et al. presented a simple discrete-time model correcting forthe total number of observed molecules discrete in time [79]. They assumed that the probability ofactivating a fluorophore is constant in time and calculated the cumulative number of localizations[52, 79].In 2016, Hummer et al. presented an almost-model-free functional for the distribution of thenumber of fluorophores [103]. They neglected the exact kinetic dynamics to estimate the real num-ber of molecules in a computational-friendly and generalizable way. Their assumptions only rely ona unique active state and apply to PALM or STORM indiscriminately. The model is a generalizationof [124]. By ignoring the dynamics of the fluorophores, the estimation depends on multiple sampleswith the same number of fluorophores, as in [124].More recently, Nino et al. created a model to estimate the number of molecules in a single-molecule super-resolution sample [148]. Their model was like those of Lee et al. and Hummer etal. [103, 124]. They sought to estimate the number of real molecules, even when a molecule hadmultiple labels attached, by adding a correction to the estimated number of fluorescent proteins.This showed that to estimate over-counting due to blinking, redundant information increases theaccuracy of the estimation [148]. They also founded that the coefficient of variation of the number15of molecules scales inversely with the square root of the mean number of labels on a target.In this thesis, we develop a kinetic model similar to that of Lee et al. [124] but specially adaptedto dSTORM data in continuous time (Section 2.4). We apply a similar class aggregation to that ofRollins et al. [167] to estimate the parameters using a likelihood function. Given the inherent dis-cretization in time from the data, we then propose a discrete-time version of the model (Section 2.5).Finally, we constrain our parameter estimation to the largest number of observed simultaneous acti-vations.Neither Lee et al. nor Rollins et al. attempted to include the spatial localization information intheir fits. We believe that this is an important source of extra information that should be exploited toimprove the quality of estimation of parameters. Thus, we couple our temporal model with a spatialmodel to account for this extra information. This is a novel feature incorporated in our approach.This part of the model is described in Chapter 3.2.4 Continous time modelLet’s assume that a fluorescent protein has unique dark and active states, as well as a bleachedstate from which it cannot be reactivated. We also assume that each fluorophore changes betweenthe states independently of other fluorophores. This might not be true if the distance between thefluorophores is sufficiently small that there is energy dissipation between fluorophores [80, 215].However, we ignore such effects here. Let the transition rates from one state to the next dependonly on the present states. In other words, the fluorophores do not have memory beyond theirpresent state and follow a Markov process. To simulate dSTORM data, we start the model with allthe potentially observable fluorophores in the dark state.We also assume that the experiment continues for long enough to allow all the fluorophoresto bleach. This assumption is strong. Performing the parameter inference under this assumptioncould inflate the bleaching rate and the number of unique fluorophores. A possible correction mightinclude modelling the probability distribution of ending points or including partial bleaching offluorophores. We leave this for future work.Putting these assumptions together, we model the fluorophores as a continuous time Markovprocess. Let N be the number of fluorophores, and kr, kd and kb the rates of transition for acti-vation, deactivation or bleaching per fluorophore, per second. Our process can be then describedby a three-dimensional non-negative random process {Nd(t),Nb(t),Na(t)}, where each coordinaterepresents the number of fluorophores in the dark, bleached or active states respectively, and N =Na(t)+Nb(t)+Nd(t) for all t. The possible transitions between Na, Nb and Nd are summarized in16krkdkbFigure 2.1: Schematic of states of fluorophores with transition rates. A fluorophore can transitionfrom active (yellow circle) to dark (black circle) with rate kd , or to bleach (empty circle) with ratekb. When in the dark state, it can become active at rate kr. Any fluorophore in the bleached statewill remain bleached. We assume that all fluorophores are in the dark state at the beginning ofthe experimentEquation 2.1 and Figure 2.1.{Nd ,Nb,Na}→ {Nd ,Nb+1,Na−1} at rate kb,{Nd ,Nb,Na}→ {Nd +1,Nb,Na−1} at rate kd , (2.1){Nd ,Nb,Na}→ {Nd−1,Nb,Na+1} at rate kr.Let pa,d,b(t) describe the transition probability of the process given an initial state Na(0) = a0,Nd(0) = d0, Nb(0) = b0 to state Na(t) = a, Nd(t) = d, Nb(t) = b at time t. Then the forwardKolmogorov equation is given as follows:p′a,d,b(t) = kb((a+1)pa+1,d,b−1(t)−apa,d,b(t))+ kd((a+1)pa+1,d−1,b(t)−apa,d,b(t))+ kr((d+1)pa−1,d+1,b(t)−d pa,d,b(t)), (2.2)with initial condition pa,d,b(0) = δa0,aδd0,dδb0,b. In matrix form, the system can be described bydPdt= P(t)Q,where Q is the transition rate matrix and P(t) is the probability vector with entries pa,d,b(t) for allvalues of a,d,b.This representation allows a simple expression of the probability distribution:P(t) = P(0)exp(Qt). (2.3)The dimension of Q is equal to the number of states of the system (N+2)(N+1)/2. To defineQ, we must first order the states. Let’s call si any state of the process. We order first all the stateswith zero active flurophores in descending order of the number of fluorophores in the dark state,17followed by those with one fluorophore active, then two fluorophores, and so on. Then:si ={N+1− i, i−1,0} 1≤ i≤ N+1,{2N+1− i, i−N−2,1} N+2≤ i≤ 2N+1,... ...{0,0,N} i = (N+2)(N+1)2 .(2.4)Equivalently, the state si = {N−b−a,b,a} corresponds to i = a(2N+3−a)2 +b+1.Let qi j be the {i, j} entry of Q. Then qi j is the transition rate from state si into state s j. Observethat the rows of Q sum to zero. If si = {N−b−a,b,a}, then qi j is given byqi j =(N−b−a)kr j = i+N+1−a,akd j = i− (N+2−a),akb j = i− (N+1−a),−(N−b−a)kr−a(kd + kb) j = i,0 otherwise.(2.5)To illustrate the notation, here are the states and the matrix Q for N = 3, corresponding to threefluorophores and 10 states.s1 = {3,0,0},s2 = {2,1,0}, s5 = {2,0,1},s3 = {1,2,0}, s6 = {1,1,1}, s8 = {1,0,2},s4 = {0,3,0}, s7 = {0,2,1}, s9 = {0,1,2}, s10 = {0,0,3}.(2.6)s1 s2 s3 s4 s5 s6 s7 s8 s9 s10Q =−3kr 0 0 0 3kr 0 0 0 0 00 −2kr 0 0 0 2kr 0 0 0 00 0 −kr 0 0 0 kr 0 0 00 0 0 0 0 0 0 0 0 0kd kb 0 0 −Σ 0 0 2kr 0 00 kd kb 0 0 −Σ 0 0 kr 00 0 kd kb 0 0 −Σ 0 0 00 0 0 0 2kd 2kb 0 −Σ 0 kr0 0 0 0 0 2kd 2kb 0 −Σ 00 0 0 0 0 0 0 3kd 3kb −Σ, (2.7)18where Σ is just the sum of the rest of the entries on each row, to ensure that each row sums to zero.2.4.1 Mean-system behaviourUsing the linearity of Equation 2.2 and the definition of expectation, we can easily show that theexpectation of the system satisfies the following system of ordinary differential equations (ODE).E(Nd)dt=kdE(Na)− krE(Nd),E(Nb)dt=kbE(Na), (2.8)E(Na)dt=krE(Nd)− (kd + kb)E(Na).Let k = kd +kb+kr. In order to have blinks (activation and bleaching), kr and kb should be positive,while it sufies kd ≥ 0. Thus, the corresponding eigenvalues are real and non-positive (Equation 2.9),with eigenvectors defined in Equation 2.10.λ0 = 0, λ± =−k±√k2−4krkb2, (2.9)w0 = (0,1,0) , w± =(−1− kbλ±,kbλ±,1). (2.10)Then, the general solution of the mean system is given by(E(Nd(t)),E(Nb(t)),E(Na(t))) = a1w0+a2w+eλ+t +a3w−eλ−t , (2.11)where ai depends on the initial conditions of the system. In our case, we assume that all the N fluo-rophores start in the dark state, i.e (E(Nd(0)),E(Nb(0),E(Na(0))) = (N,0,0). Let ∆=√k2−4krkb.Thusa1 = N, a2 =Nλ−λ+kb(λ+−λ−) =Nkr√k2−4krkb=Nkr∆, a3 =−a2,and E(Nd(t))E(Nb(t))E(Na(t))=0N0+ 1λ+−λ−−N(kr +λ−)Nλ−Nkreλ+t −−N(kr +λ+)Nλ+Nkreλ−t .As the experiment goes by, all the fluorophores move to the bleaching state. All the averages aregreater or equal to zero (E(Na(t)), E(Nd(t)), E(Nb(t)) ≥ 0) for all times. Thus, E(Nb(t)) has apositive derivative and increases to N as time goes to infinity. On the other hand, E(Na(t)) andE(Nd(t)) decay to zero as time goes by, but they start with a positivive and negative derivative19respectebly. The critical time where E(Na(t∗)) stops increasing and its maximum value are givenbyt∗ =− 1∆log(λ+λ−), E(Na(t∗)) =Nkrλ−(λ+λ−)− λ−∆.The expected number of simultaneous activations grows and then decreases. On the other hand, theexpected number of fluorophores in the dark state will always decrease.2.4.2 The distribution of the number of blinksFollowing a similar approach to that of Lee et al. [124], we now calculate the distribution of the num-ber of blinks for one fluorophore. Recall that Lee’s model was constructed for the super-resolutiontechnique PALM, which requires an initial activation step that does not happen in dSTORM. Thus,Lee’s model had four compartments and it differs from our model as dSTORM differs from PALM.Altought, we still can use the methods used by Lee et al. to find our blinking distribution. We definea blink to be when a fluorophore transits from active to dark or bleaching. For our system, the min-imum number of blinks per fluorophore is one since we always start at the dark state. We used theLaplace transform for the calculation and generalized for the case of N independent and identicallydistributed (iid) fluorophores.For any transition rate x = r,d,b, the probability of one transition is given bypkx(t) = kxe−kxt .Thusqkx(t) = 1−∫ t0pkx(t)dt = e−kxtis the probability that the transition does not occur up to time t [62].The probability that a fluorescent molecule is photobleached at time t after one blink, P(1, t), isequal to the probability of it activating by time τ , then bleaching without first passing through darkfor all possible τ smaller than t. In other words,P(1, t) =∫ t0pkr(τ)pkb(t− τ)qkd (t− τ)dτ =∫ t0pkr(τ)w2(t− τ)dτ = (pkr ∗w2)(t),where w2(t) = pkb(t)qkd (t). Let’s also define w1(t) = pkd (t)qkr(t).Now, the probability of observing two blinks before bleaching at time t, P(2, t), is equal to theprobability of activating by time t1, then dark and no bleach by time t2, then active again by t3, and20finally bleach without transition to dark for all possible positive t1, t2, t3 and t1+ t2+ t3 < t. Thus,P(2, t) =∫ t0pkr(t1)∫ t−t10w1(t2)∫ t−t1−t20pkr(t3)w2(t− t1− t2− t3)dt3dt2dt1=∫ t0pkr(t1)∫ t−t10w1(t2)(pkr ∗w2)(t− t1− t2)dt2dt1=∫ t0pkr(t1)(w1 ∗ pkr ∗w2)(t− t1)dt1 = (pkr ∗w1 ∗ pkr ∗w2)(t),where the last integral can be interpreted as the probability of having one blink finishing in the darkstate and one blink finishing in the bleach state by time t.Similarly, the probability of blinking three times before bleaching is equal to the probability ofhaving two blinks going dark and one bleached by time t.P(3, t) = ((pkr ∗w1 ∗ pkr ∗w1)∗ pkr ∗w2)(t).We can now see that the probability of having n≥ 1 blinks before bleaching by time t is:P(n, t) =((pkr ∗w1 ∗ ...∗ pkr ∗w1)︸ ︷︷ ︸(n−1 times)∗pkr ∗w2)(t), (2.12)where the first convolutions happen n−1 times.Let Nblinks be the random variable of the number of blinks. Then the probability of observingn≥ 1 blinks at any time is the integral of Equation 2.12 over all t. In other words:P(Nblinks = n) =∫ ∞0P(n, t)dt =∫ ∞0((pkr ∗w1 ∗ ...∗ pkr ∗w1)∗ pkr ∗w2)(t)dt.We use the Laplace transform to solve for P(Nblinks = n). Let P˜(n,s) =L (P(n, t)), then:P˜(n,s) =L (P(n, t)) =∫ ∞0P(n, t)e−stdt = (p˜kr(s)w˜1(s))n−1 p˜kr(s)w˜2(s),were p˜kr(s), w˜1(s), and w˜2(s) are the Laplace transforms of pkr(t), w1(t), and w2(t) respectively.These are given byw˜1(s) =kds+ kd + kb, p˜kr(s) =krs+ kr, w˜2(s) =kbs+ kd + kb.Therefore,P˜(n,s) =(krs+ kr)n( kds+ kd + kb)n−1 kbs+ kd + kb, (2.13)21andP(Nblinks = n) =P˜(n,0) =(kdkd + kb)n−1 kbkd + kb= (1−η)ηn−1. (2.14)We see that Nblinks has a geometric distribution with domain n≥ 1 and parameter(1−η) = kbkd + kb.The expected number of blinks is given byE(Nblinks) =11−η = 1+kdkb.We expect to observe at least one blink, otherwise, we will have no data from this fluorophore. Thisis an inevitable limitation imposed by the experimental approach: we cannot obtain informationfrom fluorophores that never become active. Any extra blinks depend on the relationship betweenthe transitions to the dark state vs the bleached state.The variance of the number of blinks is given byVar(Nblinks) =η(1−η)2 =kd(kd + kb)k2b=kdkb+(kdkb)2.Finally, the total number of blinks for N fluorophores follows a negative binomial distributionwith parameters N and η , domain n≥ N.P(N∑i=1Niblink = n)=(n−1n−N)ηn−N(1−η)N , n = N,N+1, ... (2.15)This distribution has mean and varianceE(N∑i=1Niblink)= N+Nkdkb, Var(N∑i=1Niblink)= N(kdkb+(kdkb)2).Lee et al. used the equivalent result from their model to estimate the number of blinks [124].But, if one blink from a fluorophore overlaps with the blink with another fluorophore, it will countas only one observable total blink (Figure 2.2). Therefore, the total number of observed blinks ofthe sample will be smaller than the sum of the individual blinks. Using the sum as an approximationto the number of blinks works if the probability of overlapping blinks is low, for example, wheneither kr is low or kd and kb are high and each blink is short lived.Nblinks represent the number of times we observe a single fluorophore. Therefore, the sum of theblinks of N independent fluorophores will be equal to the number of unique observations. Then, the22Figure 2.2: Schematic of overlapping blinks of two fluorophores. The observed blink correspond tothe sum of the blinks of the fluorophores. Thus, the real value of number of blinks is 2 while theaggregating obsevations result into one blink.real number of unique observations, NUobs follows a negative binomial distribution with parametersN and η , as described in Equation 2.15. This is true because the probability of two transitionshappening at the same time is negligible in this model. For example, if the observation changesfrom Na = x to Na = x+ 1, we say the difference came from one activation happening and notthe simultaneous activation and deactivation of several other fluorophores. In the real data, thetransitions are recorded only at discrete times. Therefore we will not have full information and wecannot be certain if multiple transitions occurred or not, between recorded observations.2.4.3 The expected time to bleachUsing the same tools as in the last section, we can calculate the expected time to bleach for onefluorophore, Tbleach. This expectation can be calculated as the sum of the expectation of bleachingafter n blinks for all n≥ 1. In other words:E(Tbleach) =∞∑n=1∫ ∞0tP(n, t)dt =−∞∑n=1∂ P˜(n,s)∂ s∣∣∣∣s=0. (2.16)The partial derivative of P˜(n,s) with respect to s is:∂ P˜(n,s)∂ s=− knr kn−1d kbn(1(s+ kb+ kd)(s+ kr))n+1(2s+ kb+ kd + kr) .23Thus,∂ P˜(n,s)∂ s∣∣∣∣s=0=−n kbkb+bd(kb+ kd + kr)kr(kb+ kd)(kdkd + kb)n−1.Therefore, the mean time to bleaching is given byE(Tbleach) =(kb+ kd + kr)krkb=1kr+1kb+kdkrkb. (2.17)Notice that the mean time to one activation is 1/kr and to one bleach is 1/kb. Thus the mean timeto bleaching is the time it takes to activate, plus the time it takes to bleach from activation, plus thetime it remains dark.When comparing the real data to the model, we assume that all the fluorophores are bleachedby the end of the experiment. This means that we require kr and kb to be reasonably high, and kdreasonably small if the experiment duration is short and vice versa.2.4.4 Aggregating in bright and dark classesGiven the model and the data, we want to find the parameters N, kr, kd , kb that maximize thelikelihood of the data. To achieve this, we would need to exponentiate the matrix Q of dimensionO(N2), where N is an unknown parameter. In order to simplify the calculations, we aggregate thestates of our Markov process in two classes, following Rollins et al. [167]. We aggregate the statesinto a bright class (when at least one fluorophore is observed), and a dark class (when nothing isobserved).For those classes, we can divide Q into submatrices depending on the aggregated states (Equa-tion 2.18).Q =(QDD QDBQBD QBB). (2.18)The submatrices Qxy encode rates of transition from class x to class y. B represents the aggregatedbright classes and D the dark classes. To clarify the notation, I will use capital B and D to refer tobright and dark classes, with lowercase d and b reserved for dark and bleach states. In particular,all states where there is at least one fluorophores active, Na > 0, belong to class B, while states withNa = 0 belong to class D, regardless of the number of fluorophores in dark or bleached states.By aggregating states in our model, we can define dark and bright times in our time series, andthe transitions between those times. This allows us to replace the transition probabilities of the fullprocess with transitions between the two classes.Let GDB(t) be the probability density of starting in class D and dwelling there for time t, and24brigthdarktimebrightdark t1t2t3t4tfinalFigure 2.3: Example of a time series divided in bright and dark classes. We have no observations fort1 seconds after starting the experiment. Then we observe the bright state (at least one fluorescentobject) for t2 seconds, then the dark state for t3 seconds and the bright state for t4 seconds. There-fore we are in the dark class for t1 and t3 seconds with two transitions to the bright class lasting t2and t4 seconds. Notice that it is not relevant how many fluorophores were active during t2 and t4.then transiting to B. ThereforeGDB(t) = exp(QDDt)QDB.We define GBD(t) equivalently.Let’s focus for a moment on a small time series where we observe no bright fluorophores for timet1, observe some for time t2, nothing for time t3, and again observe some for time t4 (Figure 2.3).Assume that we know that we start with all fluorophores in the dark state s1 = {N,0,0} and allending in the bleached state sN+1 = {0,N,0} (Equation 2.4). Then, the probability of our timeseries, Equation 2.3, translate using the aggregated classes into:P({t1, t2, t3, t4}) = PintGDB(t1)GBD(t2)GDB(t3)GBD(t4)Pend,where Pint and Pend refer to the initial and ending conditions: start at s1 and end at sN+1.We have thus reduced the problem from exponentiating a matrix of dimension O(N2), to ex-ponentiating a diagonal matrix, QDD, and another matrix with dimension O(N2). But we have notincorporated all the information we have yet. Our time series actually contains information abouthow many fluorophores we are observing at a given time. Then we can aggregate in even moreclasses, similar to Rollins et al. [167].Let D be, as before, the dark class with all states where Na = 0, and let Bn be the bright-nclass with all state where Na = n. We define the rate transition matrix Qxy, and the class transitionprobability density Gxy(t) similar as before but for x,y any of D, Bn for n = 1...N.Let’s take the small time series from Figure 2.3 again, but now including how many fluorophoreswe have at different time points (Figure 2.4). In this case, we have two observations during t2 and25brigthdarktimedark t1t2t3 tfinalt4b2 b1brigthdarktimeFigure 2.4: Example of a time series divided in D, B1 and B2 classes. This example has the samebright and dark times as the example from Figure 2.3, but in here we also know the number ofobservations during t2 and t4. We observed 2 fluorophores during time t2 and one fluorophoreduring time t4 and we incorporate this information. The times in dark class remain the same, t1and t3 seconds. But with the sub-bright classification, we define that during t2 the chain is in classB2, with exactly 2 observations. Similarly, the process belongs to class B1 during t4 seconds.one observation during t4. Then the probability density of the data isP({t1, t2, t3, t4}) = PintGDB2(t1)GB2D(t2)GDB1(t3)GB1D(t4)Pend. (2.19)With these new classes we reduced the matrix sizes to O(N). The dimension of class D remainN+1, but now the dimension of class Bn is given by N+1−n.We can bound the number of classes by the maximum value of Na observed in the experimentaltime series. Therefore, by extending the concept of aggregated classes, we have reduced the problemto calculating a product of matrices of order N.2.4.5 Likelihood functionTo estimate the parameter values, we used the maximum likelihood estimator. Under this method,the best parameters are those that maximize the likelihood of observing the data given the model.The likelihood function corresponds to the density function, which for our model is given in Equa-tion 2.20:L (N,kd ,kb,kr|{ti}) = P({ti}|N,kd ,kb,kr) = PintGDX1(t1)∏iGXi−1Xi(ti)Pend, (2.20)where Xi correspond to the class of the i transition.2.4.6 Likelihood function using NUobs distributionWe can use the distribution NUobs to obtain a maximum likelihood estimator (MLE) for N andη (Equation 2.15). These estimators are not sufficient since we cannot distingush kb and kd or26get an estimate for kr, but allow us to estimate N and thus constrain the likelihood function inEquation 2.20. The likelihood function for a dataset with n blinks and η fixed isLUobs (N|n,η) =(n−1n−N)ηn−N(1−η)N = Γ(n)ηn−N(1−η)NΓ(n−N+1)Γ(N) .The negative log likelihood is thereforeNLLUobs (N|n,η) = log(Γ(n−N+1))+ log(Γ(N))− log(Γ(n))− (n−N) log(η)−N log(1−η)=n−N∑i=1log(i)+N−1∑i=1log(i)−n−1∑i=1log(i)+N log(η1−η)−n log(η).2.4.7 Problems with the continuous time modelThe dSTORM data consists of localizations observed at each frame. The length of the framesdepends on the microscope. As an example, data in our group is acquired at 50 frames per second.Therefore, we do not have precise visibility times for each blink. A fluorophore could be active forshorter or longer times than one frame.The continuous-time model implicitly assumes that the time resolution is so high that no tran-sitions happen between frames. There is a relation between the frame length and the laser intensityand the cameras, and there is a limitation on the laser intensity to preserve the biological samplewithout damage. Thus, higher time-resolutions (shorter time frames) are constrained by microscopyresources and biological preservation. In conclusion, we might not be able to increase the temporalresolution experimentally, and such data might miss observations and bias the parameter estimates.Rollins and colleagues added a correction to their Markov process to account for missing transitions[167]. However, this correction requires the exponential of large matrices because it accounts forany transitions outside the observed class. Therefore, the correction increases the computationaldifficulty of the problem.Instead of implementing either of these two solutions, we created a discrete-time version of themodel. The discrete-time model better reflects the quantized nature of the problem. We did not findany reference in the literature that model this problem using a discrete-time Markov chain. Thusthis is a novel approach in the analysis of dSTORM data.2.5 Discrete-time modelWe use similar assumptions as in Section 2.4 to develop a discrete-time Markov chain model forfluorophore states. In this case, each time step will correspond to a frame from the data. Eachfluorophore can be active (a), dark (d) and bleached (b) and we assume that they are independent andidentically distributed. Let N be the number of fluorophores, and {Nd(i),Nb(i),Na(i)} the number of27dark, bleached and active fluorophores in frame i, satisfying Nd(i)+Nb(i)+Na(i) = N for all i. Asbefore, we have (N+1)(N+2)/2 possible states of the chain and we order them as in Equation 2.4.When we have only one fluorophore, we have only three possible states: dark or s1 = {1,0,0},bleached or s2 = {0,1,0}, and active or s3 = {0,0,1}.We assume that there is no transition between the dark and bleached states, and that the bleachedstate is absorbent. We suppose that a fluorophore in the dark state can transition to the activestate with probability pr or stay dark with probability 1− pr. Similarly, an active fluorophore cantransition to dark with probability pd , to the bleached state with probability pb, or remain active forthe next frame with probability 1− pd− pb. Let P be the transition matrix of the process. For N = 1,P is given as follows:s1 s2 s3dark bleach activeP =1− pr 0 pr0 1 0pd pb 1− (pd + pb) s1 darks2 bleachs3 active.(2.21)P is a stochastic matrix with rows adding to 1 and dimension (N+1)(N+2)/2 by (N+1)(N+2)/2. The (i, j) entry of P represents the transition probability from state i to state j in one frame.Equation 2.22 gives the transition probability from si = {Nd ,Nb,Na} to s j = {Nd − k,Nb + b,Na +k− b} for b and k described next. b is the number of active fluorophores that bleach. Thus, b ispositive and no more than the number of active fluorophores, 0 ≤ b ≤ Na. k is the total changebetween fluorophores going dark or getting activated; k is negative when changing to a state withmore deactivations than activations, and positive in the opposite case. The values of k are boundedby the number of fluorophores either active or dark, b−Na ≤ k ≤ Nd . For any other case psi,s j = 0.We can thus calculate an explicit expression of the probability transtition from si = {Nd ,Nb,Na} tos j = {Nd− k,Nb+b,Na+ k−b}:psi,s j = pi, j =min(k+Na−b,Nd)∑a=max(k,0)(Nda)par (1− pr)Nd−aNa!pbb pa−kd (1− pb− pd)Na−b−a+kb!(a− k)!(Na−b−a+ k)! . (2.22)Notice that in Equation 2.22 we can separate the transitions as coming from dark or activestates. In that case the probability is a product of binomial and multinomial densities. The bino-mial describes the activation of fluorophores from the dark state, with parameters Nd and pr. Themultinomial part corresponds to any transition from the active state, with parameters Na, pb pd , and1− pd− pb.The state sN+1 = {0,N,0} is the only absorbing state of the chain. Thus, the first eigenvalue hasdimension one, and the chain will converge to the absorbing state as time goes to infinity.28By taking powers of P we can calculate the transitions between states in t frames, with a corre-sponding initial and final condition (Equation 2.23).p({Nd(t),Nb(t),Na(t)}|{N,0,0}) = P0PtPend . (2.23)We can easily relate this to the continous time model. The discrete-time transition matrix P isequivalent to exp(Q∆t) where Q is the transition rate matrix and ∆t is the time step correspondingto one frame.2.5.1 The distribution of the number of blinksLet’s again define a “blink” as the event of a fluorophore transiting from active to dark, or bleach-ing, as in Section 2.4.2. Again, we must have at least one blink in order to be able to record thefluorophore. Let Nblink be the random variable describing the number of blinks observed beforebleaching. Then,P(Nblinks = n) =∞∑i=2nPi(n),where Pi(n) is the probability that one fluorophore blinks n times and bleaches exactly at frame i.Notice that i≥ 2n, since it takes at least two frames to have one blink.For n = 1, Pi(1) is the probability of having only one blink and bleaching at frame i. Thus,P2(1) = (activate in frame one, and bleach in frame two) = pr pb.P3(1) =(activate, stay active, bleach; or stay dark, activate, bleach)=pr(1− pb− pd)pb+(1− pr)pr pb = pr pb1∑k=0(1− pr)k(1− pb− pd)1−k.In general, Pi(1) will have one activation and one bleach transition and the remaining transitionswill be split between sojourns in the dark and active states. Thus,Pi(1) = pr pbi−2∑k=0(1− pr)k(1− pb− pd)i−2−k =⇒ P(Nblinks = 1) = pbpb+ pd .For n= 2, the probability of having 2 blinks is equal to the probability of having one blink goingto dark in the first k frames and then having one more blink that bleaches eventually, for all possiblek. Let Pˆi(n) be the probability that a fluorophore ends in the dark state at frame i after n blinks.29Since the only thing that changes from Pi(n) is the last frame, we have that Pˆi(n) = Pi(n)pdpb. Thus,P(Nblinks = 2) =∞∑k=2Pˆk(1)P(Nblinks = 1) = P(Nblinks = 1)pdpb∞∑k=2Pk(1)=pdpb(P(Nblinks = 1))2 =(pdpb+ pd)(pbpb+ pd).We can repeat the same argument to obtain the distribution of Nblinks for all n ∈ N:P(Nblinks = n) =(pdpb+ pd)n−1( pbpb+ pd)= ηn−1(1−η). (2.24)Nblinks has a geometric distribution with domain n≥ 1 and parameter 1−η = pbpb+pd (Equation 2.24).As in Section 2.4.2, we can find the expectation and the variance of the number of blink of onefluorophore.E(Nblinks) = 1+pdpb, Var(Nblinks) =pdpb+(pdpb)2.We always expect to observe at least one blink. And as with the continuous case, any extra blinksdepend on the ratio of possible transitions out the active state.Also as in the continous model, NUobs for N independent and identically distributed fluorophoresfollows a negative binomial distribution with parameters N and η , domain n≥ N, density in Equa-tion 2.15, and first moments:E(NUobs) = N+N pdpb, Var(NUobs) = N(pdpb+(pdpb)2). (2.25)Similar to Section 2.4.2, if one blink from a fluorophore overlaps with a blink with anotherfluorophore, it will count as only one observable total blink (Figure 2.2). Therefore, the total numberof observed blinks of the sample will be smaller than the sum of the individual blinks.We could try to extend again the number of blinks to be the number of unique observationsNUobs. However, for the discrete model, we cannot say that the sum of the individual blinks isequal to NUobs as we did with the continuous model. Under discrete time, the probability of twotransitions happening at the same time step is not negligible. It is possible that one fluorophoretransits to active in the same frame that another goes out of the active state, leaving us to observea longer blink instead of two blinks. Thus, if we have the same number of localizations acrosstwo frames, Na(t) = Na(t + 1) = n, we cannot distinguish if nothing changed or if there was thesame number of deactivations than of activations at frame t +1. The only way to distinguish is byknowing the complete time series of each fluorophore, which cannot be found from the aggregatetime series. To resolve this issue, we will use spatial information in the next chapter.302.5.2 The distribution of the number of frames to first activationOne part of the time series where we know exactly in which state is each fluorophore are thoseframes to the first activation. Before the first activation, all our fluorophores are in the dark stateand at the frame of activation the observed ones are in the active state, and the rest remain in thedark state. After that first activation, we have no control over the unobserved fluorophores going todark or bleached states. In here, we calculate the distribution of the number of frames to the firstactivation.For one fluorophore, the probability of activation in frame i follows a geometric distributionwith parameter pr and domain i = 1,2,3... . It will activate in frame one with probability pr, inframe two with (1− pr)pr, in frame three with (1− pr)2 pr, and so for. Thus, the meantime toactivation is 1/pr.For N independent fluorophores, we want to know the probability to the first time at least onefluorophore is active. Notice that adding the independent times of each one to activation is not whatwe are looking for, but the minimum of those independent times.The probability of all N fluorophores staying in the dark state is (1− pr)N . Thus, the probabilityof at least one active fluorophore is 1− (1− pr)N . Thus, the probability of the number of frames tothe first activation will also follow a geometric distribution with domain i = 1,2,3... and parameterηA = 1− (1− pr)N . We call TA the random variable modelling the first time to activation. Thus,xP(TA = i|N) = (1− pr)N(i−1)(1− (1− pr)N), (2.26)E(TA|N) = 11− (1− pr)N , Var(TA|N) =(1− pr)N(1− (1− pr)N)2 .2.5.3 The distribution of the number of activationsLet’s first assume, we are observing only one fluorophore. We define tbright to be the total number offrames the fluorophore is active. Thus, the domain of tbright is i = 1,2,3, ... Notice that the time inthe dark state does not affect tbright because of the Markov property. Thus, tbright follows a geometricdistribution with parameter pb and domain i = 1,2,3... If i is an integer greater or equal to zero,thenP(tbright = i) = (1− pb)i−1 pb, (2.27)E(tbright) =1pb, Var(tbright) =1− pbp2b.This probability tell us that the fluorophore is active and does not bleach for i−1 frames and thenit transits to the bleached state from the last activate frame. Thus, by independence among thefluorophores, the total number of activations for N flurophores would follow a negative binomial31distribution with parameters N and (1− pb) and domain i ∈ {N,N + 1, ...}. Notice that the totalnumber of acvations is the same as the number of localizations from the data Nloc. Thus,P(Nloc = i) =(i−1i−N)(1− pb)i−N pNb , i = N,N+1, ... (2.28)E(Nloc) =Npb, Var(Nloc) =N(1− pb)p2b.2.5.4 Likelihood function using D and Bn classesWe extend the definition of aggregated classes from Section 2.4.4 to the discrete model. Recall Dcorresponds to states where Na = 0, and Bn to those with Na = n. Changing from class x to classy in one frame has probability Pxy. Pxy is a submatrix of the transition matrix P containing theprobabilities of transitions from any state si ∈ x to any state s j ∈ y (Equation 2.22). To simplifynotation, let Px = Pxx be the probability of staying within any class x.Let’s redefine ti to be in units of numbers of frames. In the Figure 2.4 example, we have noactive fluorophores for t1, record two fluorophores for t2, nothing in the next t3, and one fluorophorefor t4. Thus, similar to Equation 2.19, the probability of our time series for the discrete modelaggregated in classes is:P({t1, t2, t3, t4}) = PintP(t1−1)D PDB2P(t2−1)B2 PB2DP(t3−1)D PDB1P(t4−1)B1 PB1DPend.We can generalize this equation to the probability of any time series {ti}:P({ti}) = PintP(t1−1)D(∏iPXi−1XiP(ti−1)Xi)Pend.By definition, the likelihood of a set of parameters given the data given is proportional this proba-bility. Thus, For a data series {ti}, the most likely parameters minimize the following expressionL (N, pd , pb, pr|{ti}) = P({ti}|N, pd , pb, pr) = PintP(t1−1)D(∏iPXi−1XiP(ti−1)Xi)Pend (2.29)2.5.5 Likelihood when N=1 and estimates for pr, pb, and pdFor N = 1 and initial condition {1,0,0}, Equation 2.29 for a time series T = {t1, ..., tn} with ntransitions becomes:L (1, pd , pb, pr|T ) =(1− pr)t1−1 pr(1− pd− pb)t2−1 pd(1− pr)t3−1 · · · pr(1− pd− pb)tn−1 pb.32Since we have only one fluorophore, then odd time points t2k−1 are times in the dark state andeven time points t2k are the times the fluorophore is active. The total time the fluorophore spendin the dark, tdark, is the sum of the odd time points times, and similarly tbright is the sum of theeven time points. Thus, if we factor the products in L (1, pd , pb, pr|T ) and take the negative logtransformation, we obtain:NLL(1, pd , pb, pr|T ) =(tbright− n2)log(1− pd− pb)+ n−22 log(pd)+(tdark− n2)log(1− pr)+ n2 log(pr)+ log(pb),where tdark = ∑i oddti, tbright = ∑i eventi.The parameters that most likely generate the data are those that maximize the likelihood func-tion, or minimize the negative log likelihood. We take derivatives and set to zero to find the mini-mum values of pˆr, pˆb, and pˆd . Therefore the MLEs of the model when N = 1 are:pˆr =n2tdark, pˆb =1tbright, pˆd =n−22tbright.Having n total transitions implies that half of them are from dark to bright and half of them arebright to dark. Thus, we have n/2 blinks. Thus, the estimated transition probabilities depend onlyon the number of blinks and the total time in each state.2.6 Bounding the possible number of activations per framedSTORM experiments are designed so that, as far as possible, fluorescent blinks are well-separatedin time and space. This allows individual blinks to be accurately localized and nanometer resolutionto be achieved. Therefore, in dSTORM data, it is not common to have many observations in thesame frame.We can exploit this experimental, spatial constraint by conditioning the possible number ofactive fluorophores per frame, thus reducing the complexity of the model. This conditioning doesnot affect the parameters of the model (N, pr, pb, and pd) and their domian, but the possible statesin the Markov chain. In the full model, Na(i) could be anything from 0 to N, defining different statesof the Markov chain. But the conditioned process will have a smaller domain of possible values ofNa(i).Let Amax be the maximum number of observations in any frame,Amax = maxi(Na(i)).Thus, we condition the model by making the transition probability to and from any state where33Na(i) > Amax is zero. Ordering the states of the chain as in Equation 2.4, the states satisfyingNa ≤ Amax are s j, where1≤ j ≤ N(Amax+1)− Amax(Amax−1)2 +1.We can transition from any state si = {Nd ,Nb,Na}, where Na ≤ Amax, to s j = {Nd − k,Nb +b,Na + k− b} where 0 ≤ b ≤ Na, as in Equation 2.22, but the range of k changes. Now the valuesof k are also bounded by Amax, thus b−Na ≤ k ≤min(Amax−Na+b,Nd). It is also useful to recallthat for a state si = {Nd ,Nb,Na}, the index i is given by i = Nb + 1+(Na(2N + 3−Na))/2. WhenAmax is small, the number of states in the conditioned chain is thus order N. This is a much smallerchain, compared to N2 states of the full model.2.6.1 Transition probability matrix and likelihood functionLet P¯ be the transition probability matrix of the conditioned chain with entries p¯i j. The transitionprobabilities in the conditioned chain are equal to those of the full model times a normalizationfactor, for states Na ≤ Amax, and zero for any other state. In other words,p¯i j =pi j∑ j pi j=pi jwi1≤ j ≤ (Amax+1)N− Amax(Amax−1)2 +1.0 otherwise.(2.30)with pi j as in Equation 2.22. Notice that the chain has the same stationary state sN+1 = {0,N,0},where all fluorophroes bleached. The likelihood function will be the same as in the discrete generalmodel but substituting the matrix P and its corresponding submatrices with the conditional matrixP¯. Thus, the likelihood with classes D, B1, . . . ,BAmax is given by Equation 2.29.2.6.2 Case Amax = 1If the data shows only one observation per frame, then Amax = 1. In this case we only have 2N+1states, and classes D and B1. Let si ∈D, then si = {N− i+1, i−1,0} with 1≤ i≤N. If i=N+1, siis the absorbing state and p¯i = 1. Otherwise, si stays to itself or changes to sN+1+i = {N− i, i−1,1}.In this case,p¯i j =(1−pr)N+1−iwi= 1−pr1+(N−i)pr , j = i,(N+1−i)pr(1−pr)N−iwi= (N+1−i)pr1+(N−i)pr , j = i+N+1,0, otherwisewhere wi = (1− pr)N+1−i + (N + 1− i)pr(1− pr)N−i is the normalization constant. If sl ∈ B1,sl = sN+1+i = {N− i, i− 1,1} with 1 ≤ i ≤ N. In this case, sN+1+i can transition to itself, to si =34{N − i+ 1, i− 1,0}, and to si+1 = {N − i, i,0}. If i < N, sN+1+i can also transition to sN+2+i ={N− i−1, i,1}.p¯l j =(1−pb−pd)(1−pr)N−i+pd(N−i)pr(1−pr)N−i−1wij = l = N+1+ i,pd(1−pr)N−iwi= pd(1−pr)1+pr((N−i)(pd+pb)−1) , j = i,pb(1−pr)N−iwi= pb(1−pr)1+pr((N−i)(pd+pb)−1) , j = i+1,pb(N−i)pr(1−pr)N−i−1wi= pb(N−i)pr1+pr((N−i)(pd+pb)−1) , j = N+2+ i and i < N,0, otherwisewhere wi = (1− pr)N−i−1(1− pr + pr(N − i)(pd + pb1i<N)) is the normalization constant. Thisdefine the normalized matrix for the conditional case with Amax = 1.It is simple but not especially illuminating to derive formulae for the situation where Amax = 2,thus we will leave them out of this manuscript.2.7 Parameter estimation algorithmIn both the continuous and discrete-time models, we have four parameters: the number of fluo-rophores N, and three kinetic rates (continuous model) or three transition probabilities (discretemodel). In this section, we will define Θ to represent the state transition parameters. Thus, Θ =(pr, pb, pd) in the discrete-time model and Θ= (kr,kb,kd) in the continuous one. To find the MLE,we cannot use standard maximization algorithms because one of our parameters, N, is a discreteparameter (Table 2.1). We are confronting a mixed-optimization problem. We consider the possiblenumber of fluorophores in a sample to be at least Amax (since simultaneous observations are unique)and no more than Nloc, the total observations in the sample (since we assume all the N fluorophoresmust activate at least once during the experiment). The definition and domain of each parameter issummarized in Table 2.1.2.7.1 Short review of mixed-optimization algorithmsMixed-optimization problems deal with computational problems that are not present in continuousor discrete optimization. Algorithms developed to solve either continuous or discrete optimizationsdo not work well when the domain is mixed. Therefore, many algorithms have been developed oraltered to solve non-linear mixed-optimization problems [7, 17]. Some of those algorithms requirelittle user input, for example, the branch-and-bound algorithm [71, 122], genetic algorithm [75, 81],simulated annealing [114, 140], and other Bayesian algorithms. Other methods constrain the dis-crete domains with prior knowledge, approximate the problem with continuous and linear versions,or include a penalization to the continuous problem or the Lagrangian to force solutions onto the35Parameter Description DomainN number of fluorophores in thesample{Amax,Amax+1, ...,Nloc}Continuous modelkr transition rate from dark to active (0,∞)kd transition rate from active to dark (0,∞)kb transition rate from active tobleach(0,∞)Discrete modelpr probability of activating from darkstate(0,1)pd probability of going to dark statefrom active(0,1) and pd + pb < 1pb probability of bleaching of an ac-tive fluorophore(0,1) and pd + pb < 1Table 2.1: Parameters of the discrete-time and continous-time Markov models for dSTORM analysis.Name, meaning and numerical domain included. Nloc is the total number of observations in thedataset. Amax is the maximum number of obserevations in any frame.discrete domain [7, 72].The branch-and-bound method divides the problem into branches. It solves the problem in eachof those branches and then cuts out those branches with unfeasible solutions [7, 17, 165, 171]. Themethod uses a continuous approximation to solve the problem in each branch [165, 171]. Noticethat our problem cannot be described as a continuous problem since the discrete parameter N definesthe dimension of the transition matrix.Simulated annealing and genetic methods are implicit enumeration stochastic procedures [7].They do not require a continuous approximation and are therefore more feasible for our problem.Simulated annealing is a Metropolis type of optimization, and relies on many iterations of the prob-lem [113, 114, 140]. Genetic algorithms mimic the mutation and evolution of genes and need tocompute many mutation iterations to ensure convergence [75, 81, 129, 191]. These algorithmsconverge to a global optimum but they are slow.Thinking about our problem as a stochastic process, we could introduce latent variables [19, 70].These latent variables would represent the exact fluorophores active at each observation and theprocess would become a finite mixture model. Each mixture would correspond to one fluorophore.For a fixed number of mixtures, we could then use the Expectation-Maximization algorithm [19,34, 70, 112]. We will explain the Expectation-Maximization algorithm in the next chapter. Whenthe number of mixtures is unknown, we could use non-parametric Bayesian methods [70, 182].The translation to a non-parametric process is not straightforward and we will not develop it in thepresent work. This would be an interesting area to explore in future work.36We use the knowledge that our parameter range is “small”, and we iterated between the contin-uous and discrete problems. This is easy in the case of one dataset or even when fitting multiplesamples with the same number of fluorophores (see Section 2.7.2 and Section 2.7.3). For the generalproblem, we use parallel computing over the discrete parameters to find the optimum (Section 2.7.4).Parallelizing the problem changes the multiplicative dimension of the discrete space to an additiveone. This is still an extensive search algorithm but, if the range of possible values of N is small, thisalgorithm is very likely to be faster than a genetic or simulated annealing approach.2.7.2 Maximizing the likelihood for one datasetWe separate the problem into two: a discrete optimization and a continuous one. They are not inde-pendent problems since the dimension of the transition probability matrix depends on the numberof fluorophores. But if we have one solution, we can solve for the other. If we know N, we can usecontinuous optimization algorithms to estimate the rest of the parameters Θ. And when we knowΘ, finding the most likely N consists only of evaluating the likelihood function on the domain ofN (Table 2.1). The likelihood functions are given by Equation 2.20 for the continuous model andEquation 2.29 for the discrete one. Recall that the possible values of N vary from Amax, the maxi-mum number of observations in any frame, up to Nloc, the total observations in the sample. Thus,we can iterate between solving the continuous part at discrete N values, and then take the maximumover N (Algorithm 2.1).Algorithm 2.1 Maximum likelihood estimators (MLE) of the dSTORM modelsRequire: Time series T = {ti}, Nloc = total number of localizations.Ensure: During t1 and tend the fluorophore is not active.1: Define Θ= (pd , pb, pr) or Θ= (kd ,kb,kr) . Discrete or continous time2: for N ∈ {Amax, ...,Nloc} do3: LN = maxΘ(L (N,Θ|T )) . Equation 2.29 or 2.204: pMLE(:,N) = argmaxΘ(L (N,Θ|T ))5: Nˆ = argmaxN(LN) , Θˆ= pMLE(:, Nˆ).We used MATLAB to construct the models and likelihood functions (Equation 2.29 or 2.20 forcontinuous or discrete versions). When Θ is known, we calculate the likelihood function for all thepossible values of N (extensive search) to find the most likely value. We use the MATLAB functionfmincon to minimize the negative logarithm of Equation 2.20 or Equation 2.29 w.r.t. Θ when N isknown. Thus, we can solve the problem by finding Θˆ for a fixed N, repeat for all N, then take themaximum value of the likelihood function over N (Algorithm 2.1).372.7.3 Simultaneous fit of data sets with the same number of fluorophoresTo improve the parameter fits, we can estimate the parameters of multiple independent and identi-cally distributed samples. In that case, the likelihood for all the samples is the product of the likeli-hoods of each sample. We transform those products into sums by using the logarithm transformationof the likelihood. Taking the negative of the sum, we reduce the problem from the maximizationof a product to the minimization of a sum. It is numerically easier to calculate than a product oflikelihoods. Let T1,T2, ...,TM be the M iid time series, then the negative log likelihood (NLL) of thesample is given byNLL(Θ,N|T1, ...,TM) =−M∑i=1log(L (N,Θ|Ti)) .We still can handle this problem using extensive search over N and minimizing w.r.t. the otherparameters for a fixed N (Algorithm 2.2).Algorithm 2.2 MLE of the dSTORM models for M iid samplesRequire: M samples, T1, ...TM, Nloc vector with localizations per sampleEnsure: During t1 and tend the fluorophore is not active for all samples.1: for N ∈ {max(Amax), ...,min(Nloc)} do2: for i ∈ {1, ..,M} do3: LN(i) = maxΘ(L (N,Θ|Ti)) . Equation 2.20 or 2.294: pMLE(:,N) = argmaxΘ(NLL(Θ,N|T1, ...,TM))5: Nˆ = argmaxN(LN) , Θˆ= pMLE(:, Nˆ).2.7.4 Simultaneous fit of data sets with a different number of fluorophoresA more realistic assumption would be to allow each sample to have different numbers of fluo-rophores while maintaining the iid assumption over all fluorophores. In other words, Θ is the samefor all samples but the i’th sample has Ni fluorophores. In that case, the NLL isNLL(Θ,N1, ...,NM|T1, ...,TM) =−M∑i=1log(L (Ni,Θ|Ti)) . (2.31)Here, we confront a more complex domain. We need to optimize for M discrete and 3 continuousparameters. If we were to solve the continuous problem for all the possible values in the discretedomain as in Algorithm 2.2, we will have to compute as many calculations as the combinations ofthe possible number of fluorophores for each sample size. Therefore, an extensive search over thediscrete domain is not feasible anymore. We use parallel computing to overcome this problem.38To parallelize the problem by iterating between solving the continuous part at discrete Ni valuesit is not straightforward since all the samples share Θ. Instead, we change the order of the iteration.If we know Θ, we can minimize the NLL over Ni for each sample i = 1, ...,M in isolation and sumall to find the total NLL. Thus, parallelization of the total NLL is easier when iterating betweenminimizing the discrete problem and updating the continuous one. As described in Algorithm 2.3,we fix Θ, find the best estimated of Ni(Θ) for that set of Θ independently for each sample, and thenupdate Θ.Algorithm 2.3 MLE of the discrete model for M with different number of fluorophoresRequire: M samples, T1, ...,TM time series of each sample, Nloc vector with localizations per sampleEnsure: During t1 and tend the fluorophore is not active for all samples.1: function MIN-NLL(M,T1, ...TM,Nloc)2: for i ∈ {1, ..,M} do . Compute in parallel over M3: L (i) = maxAmax(i)≤ j≤Nloc(i)(L ( j,Θ|Ti)) . Equation 2.20 or 2.294: Ni(Θ) = argmaxj(L ( j,Θ|Ti))5: Minimize NLL(Θ) = sum(L (i))return Θˆ=min argument, and Nˆi = Ni(Θˆ)We could use the likelihood from the distribution of NUobs with fixed Θ to further reduce thediscrete domain. We will still need to estimate the NLL from Equation 2.20 or 2.29 since NUobscannot differentiate the Θ parameters. We did not implement this idea for this thesis work, but itwill be implemented in the future.2.8 ResultsTo test our models and inference algorithms, we used simulated data. We simulated each of themodels and fit the parameter values. For each model, we started by simplifying the optimization.Regardless of the domain in time (continuous or discrete models), we fixed the number of flu-orophores and estimated the transition parameters and vice versa. After testing each part of theoptimization, we estimated all the parameters with the parallel algorithm for each dataset. Eachmodel was used to generate simulated data and the corresponding likelihood function was used tofit the data. We repeated this for different parameter sets to simulate different fluorophore dynamics.The dSTORM data is discrete in time and the time resolution will affect the continuous-timemodel fitting. Here, we show the results of the discrete-time model (described in Equation 2.23)with and without conditioning on the maximum number of possible activation allowed. The fits ofsimulated data from the continuous-time model are presented in Appendix A. We wrote all the codenecessary to simulate the data and estimate the parameters in MATLAB. The simulations for thecontinuous-time model were coded in C to speed up the Gillespie algorithm.39Parameter S1 S2 S3 S4 S5 S6N 1pr .1pd 0.8 0.8 0.4 0.5 0.1 0.1pb 0.05 0.1 0.1 0.5 0.5 0.8Table 2.2: Parameter values used to simulate datasets for discrete-time model. We used six differentsets of parameters to define different blinking properties of the fluorophores. All the parametersets have one fluorophore, N = 1, and a fixed probability of activation pr = .1. The parameter setsare called S1, S2, S3, S4, S5, and S6. They are ordered by increasing bleaching probability.S1 S2 S3 S4 S5 S6Nloc S 167,447 91,576 48,797 20,191 12,003 11,207E(Nblinks)T 17 9 5 2 1.2 1.12S 16.74 9.16 4.88 2.02 1.2 1.12SD of NblinksT 16.49 8.49 4.47 1.41 0.49 0.37S 16.03 8.59 4.35 1.44 0.49 0.37Nblinks 99 percT 76 40 21 7 3 3S 74 41 20.5 7 3 3η (nbinfit) T 0.94 0.89 0.80 0.5 0.17 0.11S 0.94 0.89 0.80 0.51 0.17 0.11N (nbinfit)T 1 1 1 1 1 1S 1.01 1 0.98 0.98 0.96 0.95Table 2.3: Comparing theoretical and simulated statistics of the Nblinks of one fluorophore under thediscrete-time model. Each parameter set (S1-S6) was simulated 10,000 times. Nloc refers to thetotal number of localizations obtain in the simulations. The theoretical values, in T rows, werecalculated using Equation 2.24, and the corresponding percentile value of a geometric distribution.The simulated values, in S rows, sumarized the mean, standard deviation, and percentile functionsto the observed number of blinks. The last two rows, η and N, were calculated using the functionnbinfit from MATLAB, which calculates the MLE for a negative binomial distribution.2.8.1 Simulating data for the discrete-time modelThe simulated data was created by iterating the Markov chain with transition matrix P (given asin Equation 2.22) until all fluorophores bleached. We simulated multiple data sets with one fluo-rophore, N = 1, with different transition probabilities. For simplification, we kept the probability ofactivation pr = .1 for all the parameter sets. Varying the deactivation and bleaching probabilities tochange the blinking properties, we created six different parameter sets for testing (Table 2.2). Sim-ulated fluorophores with parameters from S1 have the most blinking cycles and smallest bleachingprobability, while those from S6 have the least blinks and largest bleaching probability (Figure 2.5).We created 10,000 replicates of the model for each parameter set S1-S6.To test the accuracy of our simulations, we estimated the parameter η , expectation, varianceand 99 percentile of the distribution of the number of blinks, Nblinks, and compared them to the40(a) S10 50 100 150Number of activations00.010.020.030.040.050.06probabilityDataNB at MLE(b) S30 20 40 60Number of activations00.050.10.150.20.25probabilityDataNB at MLE(c) S50 5 10Number of activations00.20.40.60.81probabilityDataNB at MLEFigure 2.5: Distribution of the number of blinks of the simulated data with N = 1 for parameter sets(a) S1, (b) S3, and (c) S5. The histograms (blue) show the normalized frequency of the numberof observations in each of the 10,000 simulations. The red line corresponds to the probabilitydistribution of a negative binomial with parameters given by the MLE for N and η (Table 2.3).41theoretical values (Table 2.3). We know that Nblinks ∼NB(N,η) when all the samples have the samenumber of fluorophores N, (see Equation 2.24). Here, one blink corresponds to the transition of afluorophore from dark to active and back to dark. Thus, Nblinks = NUobs when N = 1. For theoreticalvalues, we calculated the mean, variance, and 99 percentile values using the real parameter valuesand the geometrical distribution (case N = 1 of the negative binomial distribution). We estimatedN and η using a custom-made code and the function nbinfit in MATLAB. Figure 2.5 illustrates theagreement of theory and simulations with the normalized histogram of the simulated data, and theprobability density function of the negative binomial distribution for datasets S1, S3, and S5.2.8.2 Estimates of the kinetic rates depend on the sample size of the data for thediscrete-time model when the number of fluorophores is knownTo test the continuous part of our optimization algorithm, we estimated the transition probabilitiespr, pd and pb while fixing the number of fluorophores to N = 1. The N range is set to be Amax(i) =Nloc(i) = 1 in Algorithm 2.3 for all the samples.Giving that the samples from a parameter set share the transition probabilities, we can increasethe accuracy of the estimates by fitting multiple samples at a time. Furthermore, we expect that theMLE will converge to the real values as we increase the number of samples. We test this by groupingour 10,000 simulated samples into groups of size M = 10,100,1000,10000, and finding the MLEof the transition parameters for each group. We had 100 groups with M = 10 and 100 samples,10 groups with M = 1000 samples, and one group with M = 10,000 (all simulated samples in onegroup).Using that we have multiple groups with M = 10,100 and 1000, we summarized the correspond-ing estimates distributions using box plots (Figure 2.6, 2.7). Parameter estimates with parametersset S1, S2, and S3 are in Figure 2.6, while Figure 2.7 has the results from parameter sets S4, S5, andS6. The box plots corresponding to the estimates of pr, pd , and pb are in the first, second and thirdcolumns in both figures. All box plots had 100 estimates calculated by fitting 10 or 100 samplessimultaneously, 10 estimates by fitting 1000 samples, and one estimate of fitting all the samplestogether. The boxes show the mean with a red line, 25 and 75 percentiles as a blue box, and the 95%confidence interval as error bars. The red dots represent outliers.For all the parameter sets, the MLE converges to the real value (blue horizontal line) as weincrease the sample size (M). We observe that our accuracy is high even when M = 1000. There-fore, higher sample size is encouraged if the computational power is available. Notice that theparallelization part of Algorithm 2.3 is not used in this part of the fitting since we fixed N = 1.42(a) S1 pr pd pb10 100 1000 10000Sample size0.090.10.110.12Parameter value10 100 1000 10000Sample size0.70.750.80.85Parameter value10 100 1000 10000Sample size0.040.060.080.10.12Parameter value(b) S2 pr pd pb10 100 1000 10000Sample size0.080.090.10.110.120.130.14Parameter value10 100 1000 10000Sample size0.70.750.80.85Parameter value10 100 1000 10000Sample size0.050.10.150.20.25Parameter value(c) S3 pr pd pb10 100 1000 10000Sample size0.070.080.090.10.110.120.130.14Parameter value10 100 1000 10000Sample size0.250.30.350.40.450.5Parameter value10 100 1000 10000Sample size0.050.10.150.2Parameter valueFigure 2.6: MLE for the parameter sets (a) S1, (b) S2 and (c) S3 for different sample sizes with N = 1fixed. Box plots of the MLE and real values (blue line) of pr, pd and pb are in the first, second andthird column respectively. Sample size M = 10 and M = 100 were repeated 100 times, M = 100010 times and M = 1000 only one time.43(a) S4 pr pd pb10 100 1000 10000Sample size0.050.10.150.20.25Parameter value10 100 1000 10000Sample size0.10.20.30.40.50.60.7Parameter value10 100 1000 10000Sample size0.30.40.50.60.70.80.9Parameter value(b) S5 pr pd pb10 100 1000 10000Sample size0.050.10.150.2Parameter value10 100 1000 10000Sample size00.050.10.150.2Parameter value10 100 1000 10000Sample size0.30.40.50.60.70.80.91Parameter value(c) S6 pr pd pb10 100 1000 10000Sample size0.050.10.150.2Parameter value10 100 1000 10000Sample size00.050.10.150.20.25Parameter value10 100 1000 10000Sample size0.60.70.80.91Parameter valueFigure 2.7: MLE for the parameter sets (a) S4, (b) S5, and (c) S6 for different sample sizes with N = 1fixed. Box plots of the MLE and real values (blue line) of pr, pd and pb are in the first, second andthird column respectively. Sample size M = 10 and M = 100 were repeated 100 times, M = 100010 times and M = 1000 only one time.44S1 S2 S3 S4 S5 S600.10.20.30.40.50.60.70.80.91probability12345NfitFigure 2.8: Estimates of the number of fluorophores with known transition probabilities for all theparameter sets. Bars are grouped by the different parameter sets S1-S6. All the samples wereestimated to have from 1 (dark blue bars) to 5 (yellow bars) distinct fluorophores. All the modeswere at Nfit = 1.2.8.3 Estimating the number of fluorophores when the transition probabilities areknownWe now look at estimating the number of fluorophores when the transition probabilities are known.We assume each sample has its own number of fluorophores Ni. Thus, Algorithm 2.3 will computethe values of the likelihood function for the possible values of Ni and select Nfit(i) that minimize thelikelihood function. Each calculation can be done in parallel, since each sample is independent ofeach other, and share only the known transition rates.We cannot differentiate if apparent activations in continuous frames came from a long blink ofone fluorophore or if they are formed by mergers of multiple independent blinks. Thus, we wereconservative and allowed the algorithm to vary the parameter Ni from one to the sum of the numberof active frames, Nloc(i). Notice that Nloc(i) varies a lot depending on the transition probabilitiesfrom large values in S1 to less than 8 in S6. Recall that Nloc ≤ Nblink, but still, there are more blinksin S1 and less in the samples form S6. The mean number of blinks for each parameter set is inTable 2.3.Figure 2.8 summarizes the values of the most likely number of fluorophores for each sample,grouped by parameter sets S1-S6. Each colour bar corresponds to the estimated number of fluo-rophores, with Nfit = 1 in dark blue all the way to Nfit = 5 in yellow. The high of the bar represents45n=1 n=2 n=3 n=4 n=5 n=6S1∆n = 0 9,129 868 3 0 0 0∆n < 3 9,992 8,064 2,072 129 3 0S2∆n = 0 8,550 1,425 25 0 0 0∆n < 3 9,983 8,218 3,178 477 30 0S3∆n = 0 8,406 1,553 41 0 0 0∆n < 3 9,999 7,709 3,734 1,053 128 3S4∆n = 0 7,205 2,565 224 6 0 0∆n < 3 9,980 4,920 2,097 638 118 8S5∆n = 0 8,535 1,277 174 12 2 0∆n < 3 9,897 4,465 1,458 412 94 26S6∆n = 0 8,667 1,179 143 10 1 0∆n < 3 9,901 1,952 357 74 10 2Table 2.4: Number of samples with ∆n = NLL(n)−NLL(Nˆi) equal to zero and smaller than 3, forn = 1, ..,6 all the parameter sets. All the values are out of a total of 10,000 samples.the normalized frequency of samples with the same estimated Nfit.Only for the parameter set S1, the samples estimated to have one fluorophore were more than90%. The rest of the parameter sets yielded estimates of one fluorophore on 70-90% of the samples.However, most of the estimates lie between 1 and 2 fluorophores for all the samples and parametersets (Table 2.4).To determine whether the likelihood function was concave enough to differentiate between dif-ferent values of N, we calculated the differences ∆n = NLL(n)−NLL(Nˆi), where Nˆi is the MLE forsample i and n = 1, ...,6 (Table 2.4). This method is equivalent to applying the likelihood-ratio test,and ∆n = 0 when n is the MLE. We recorded the number of samples per parameter sets with Nfit = n( ∆n = 0) for n = 1, ...,6. We also registered how many samples have ∆n < 3. Negative likelihooddistances smaller than 3, ∆n < 3, are usually interpreted to be within the 95% confidence interval,and thus we cannot discard that the value of N is n.Using ∆n < 3 as our cut off, we found that 99% of the samples have N = 1 within the confidenceinterval for the number of fluorophores for all the parameter sets. All the samples seem to havebetween 1-3 fluorophores with 95% confidence, and the number of fluorophores could be 4 only in10% of the samples in S3, around 5% in S2 and S4, and less than 1.5% in S1.The story is different for parameter sets S4-S6. We cannot distinguish the observations from thefluorophores as precisely in these samples. The frequencies of Nfit estimates for S4 could mislead usto think S4 is harder to estimate, given ≈ 70% accurate results (Figure 2.8). In fact, parameter setsS5 and S6 have more samples showing unique localizations, implying a higher chance to estimateone fluorophore. The algorithm is choosing the number of observations to be the most likely numberof fluorophores, and S5-S6 have a greater probability of observing only one fluorophore.To test if our fitting procedure works over a larger number of fluorophores, we created 1,00046S1 S2 S3 S4 S5 S600.050.10.150.20.25probability 14161820222426NfitFigure 2.9: Estimates of the number of fluorophores with known transition probabilities and N = 20,for all parameter sets. Bars are grouped by the different parameter sets (S1-S6).There are 1,000estimates for each parameter set. The samples fits ranged from 13 to 32 distinct fluorophores forall parameter sets with modes at Nfit = 20 for S2, S3 and S6, at Nfit = 19 for S4 and Nfit = 21for S1 and S5. Dark blue bars correspond to Nfit ≤ 14, and yellow to Nfit ≥ 28. The rest of thecolours vary from Nfit = 15 (blue) to 27 (orange).simulations with N = 20 fluorophores for all parameter sets. We estimated the number of distinctfluorophores, again assuming known transition probabilities (Figure 2.9). Our estimated numberof fluorophores ranged from 13 to 32 distinct fluorophores with a mean of 20 (Figure 2.9 and Ta-ble 2.5).The huge difference between the number of observations and the estimated number of fluo-rophores points to a good convergence. As expected from the results with one fluorophore, the S6sample’s estimates were close to the number of localizations. With these results, we show that ouralgorithm can estimate the number of distinct fluorophores when we know the transition probabili-ties and the number of observations is not too large.2.8.4 Estimating the number of fluorophores assuming no information about theparametersAfter checking the optimization over N and over pd , pb, and pr independently, we estimated thenumber of fluorophores for each parameter set assuming no information about the transition proba-bilities. We used Algorithm 2.3 to estimate all the parameters (N, pd , pb, and pr). Here, we have tofit multiple samples simultaneously, since all the fluorophores follow the same transition probabili-47Parameter Nfit NlocSet Range Mode (mean) Range Mode (mean)S1 [15,25] 21 (20) [204,778] 442 (404)S2 [14,28] 20 (20) [93,358] 167 (203)S3 [13,29] 20 (20) [82,387] 192 (200)S4 [14,28] 19 (20) [25,66] 41 (40)S5 [13,32] 21 (20) [25,65] 39 (40)S6 [16,28] 20 (20) [20,35] 24 (25)Table 2.5: Ranges, modes, and means of the number of localizations and the estimated number offluorophores, for samples with N = 20 and all parameter sets. The total number of simulations is1,000 for parameter all parameter sets.Parameter set M = 10 M = 100 M = 1000 M = 10000 Nloc = 1S1 869 8797 8843 8821 481S2 603 7463 7527 7559 1014S3 586 6461 6638 6619 1078S4 517 4981 4981 4981 4981S5 820 8326 8327 8330 5001S6 844 8123 8026 8020 8020Table 2.6: Number of simulations correctly estimating one fluorophore for the different sample sizesand parameter sets, when estimating the transitions rate as well. The total number of samples withexactly one localization is given in the last column as a reference for the fits. Column two to fivehave the number of samples with Nfit = 1 for different sample sizes M. The second column is outof 1,000 samples (M = 10), the rest of them are out of 10,000.ties. As in Section 2.8.2, we tested how the estimates changed when varying the sample size fromM = 10 to M = 10,000. We had 100 sub-samples of size M = 10 and M = 100, 10 sub-samples ofM = 1000, and one of M = 10,000. The estimates show that both the sample size and the transitionrates (parameter sets) played a role in the fit accuracy.We summarize the parameter estimates in Figure 2.10 and Figure 2.11 for pr, pd , and pb, andin Table 2.6 for N. The box plots in Figure 2.10 and Figure 2.11 show the distribution over 100(for M = 10 and 100), 10 (for M = 1,000) or 1 (for M = 10,000) parameter estimation as describedbefore in Section 2.8.2. Again, the estimated distributions of pr, pd and pb are in the first, secondand third columns in both figures. Table 2.6 summarize the number of samples where the estimatednumber of fluorophores is one (Nfit = 1), and those with only one observation (Nloc = 1). Thecolumn corresponding to samples fitted to groups of 10 (M = 10) is out of 1,000 samples. All theother columns are out of 10,000.Samples simulated from parameter sets S1, S2 and S3 showed a fairly good convergence of theparameter estimates as we increased from 10 to 10,000 samples (Figure 2.10 and Table 2.6). Fig-ure 2.10 shows that even when M = 10,000, we underestimate the recovery and dark probabilities48(a) S1 pr pd pb10 100 1000 10000Sample size0.070.080.090.10.110.12Parameter value10 100 1000 10000Sample size0.70.750.80.85Parameter value10 100 1000 10000Sample size0.040.060.080.10.120.140.16Parameter value(b) S2 pr pd pb10 100 1000 10000Sample size0.020.040.060.080.10.12Parameter value10 100 1000 10000Sample size00.20.40.60.8Parameter value10 100 1000 10000Sample size0.20.40.60.81Parameter value(c) S3 pr pd pb10 100 1000 10000Sample size0.020.040.060.080.10.120.14Parameter value10 100 1000 10000Sample size00.10.20.30.40.5Parameter value10 100 1000 10000Sample size0.10.20.30.40.50.6Parameter valueFigure 2.10: MLE for the parameter sets (a) S1, (b) S2 and (c) S3 for different sample sizes whenestimating also N. Box plots of the MLE and real values (blue line) of pr, pd and pb are in thefirst, second and third column respectively. Sample size M = 10 and M = 100 were repeated100 times, M = 1000 10 times and M = 10,000 only one time.49(a) S4 pr pd pb10 100 1000 10000Sample size0.050.10.150.2Parameter value10 100 1000 10000Sample size00.10.20.30.40.5Parameter value10 100 1000 10000Sample size0.50.60.70.80.91Parameter value(b) S5 pr pd pb10 100 1000 10000Sample size0.050.10.150.2Parameter value10 100 1000 10000Sample size00.020.040.060.080.1Parameter value10 100 1000 10000Sample size0.40.50.60.70.80.91Parameter value(c) S6 pr pd pb10 100 1000 10000Sample size0.050.10.150.2Parameter value10 100 1000 10000Sample size00.020.040.060.080.1Parameter value10 100 1000 10000Sample size0.650.70.750.80.850.90.951Parameter valueFigure 2.11: MLE for the parameter sets (a) S4, (b) S5 and (c) S6 for different sample sizes whenestimating also N. Box plots of the MLE and real values (blue line) of pr, pd and pb are in thefirst, second and third column respectively. Sample size M = 10 and M = 100 were repeated100 times, M = 1000 10 times and M = 10,000 only one time.50S1 S2 S3 S4 S5 S600.10.20.30.40.50.60.70.80.9probability123456>6NfitFigure 2.12: Estimates of the number of fluorophores for the complete sample with 10,000 simulationsfor each parameter set S1-S6 without fixing the transition parameters. Bars are grouped bythe different parameter sets (S1-S6). All the samples were estimated to have 1 to 14 distinctfluorophores. Colour scale corresponds to the number of fluorophores estimated Nfit from 1(darkblue) to 6 (dark yellow), or > 6 (bright yellow).pr and pd , and overestimate the bleaching probability, pb. A system with a smaller dark rate andlarger bleach rate will have more unique observations. Also, a system with a smaller reactivationrate will have fewer blinks but these will usually last longer. Thus, the model is balancing pd andpb and underestimating pr because is hard to identify continuous blinks (one fluorophore on andone off, and then change to the opposite for the next frame) from a larger blink (one fluorophore onfor two consecutive frames). We will address this problem in the next chapter where we integratethe spatial information into the process. Meanwhile, the parameter estimates are in line with theestimated number of fluorophores, which increased compared to the values achieved with knownprobabilities. We obtained around 88% accuracy in the estimated number of fluorophores for S1,75% for S2, and 66% for S3, but with a small error. All the samples are estimated to have less than5 fluorophores when M = 10,000 (Figure 2.12).On the contrary, samples from parameter sets S4, S5, and S6 were not sufficient to differentiatethe number of fluorophores from the number of blinks (Figure 2.11, and Table 2.6). This is clearfrom the value estimated for the transition probability to the dark state, pd . Our algorithm estimatedthe dark probability to be near zero, implying that the fluorophores will only activate and bleach(Figure 2.11). Since pd + pb = 0.6 in S5 (Table 2.2), the fluorophore will remain active for more51S1 S2 S3 S4 S5 S6Full fit 0 0 0 0 0 0Fit only N -115.4 -351.7 -608.1 -6,745.2 -1,943.1 -1,852.2Fit only rates -715.4 -1,355.7 -1,448.8 -8,718.9 -4,332.9 -4,012.3Table 2.7: Differences between the NLL function at the MLE for the full model, the model with thetransition information, and the model when knowing the number of flurophores all parameters setsand N = 1. We take the difference between the value of the NLL when fitting all the parameters(full fit) minus the NLL fitting only N (known transition probabilites), and the NLL fitting only thetransition probabilities (N known). The differences are calculated only for when all the samplesare fitted (M = 10,000). In all the parameter sets, the minimum occurs when fitting the full modelsince it has the most degrees of freedom.than one frame, and those will have longer blinks and a smaller number of samples with Nloc = 1.That is the reason for the difference in value between the last two columns of S5 in Table 2.6.We checked if the misfit for S4, S5, and S6 could result from an optimization error. We com-pared the values of the likelihood function with those from previous sections, and we found theseresults to be correct (Table 2.7). In S4-S6, we expect to see a fluorophore no more than two times,resulting in a lack of information required to differentiate the fluorophores. Samples from S4, S5,and S6 have less than half of the localizations seen in samples from S3, and almost 15 times lessthan those from S1 (Table 2.3). To improve the results, we would require many more samples fromparameter sets S4, S5, and S6 to sample the tail of the distribution and capture the full dynamics.Figure 2.12 and Table 2.7 summarize the estimated number of fluorophores when fitting 10,000samples simultaneously, compared to the likelihood values from previous sections. We lost accuracyby not knowing the transition probabilities, especially for parameter sets S3 and S4 (Figure 2.12 vsFigure 2.8). Thus, if we have any extra information about the transitions, we could improve ourestimates. This is further illustrated when comparing the values of the NLL when estimating allthe parameters (full fit), only N, or only the transitions (Table 2.7). The full model has the smallestvalue of NLL, followed by the case when the transition rates are known, and the largest happenswhen the probabilities are estimated with N known. Table 2.7 contains the differences between theNLL at the MLE for the different cases.As in Section 2.8.3, we next simulated data with twenty fluorophores, N = 20, and estimatedthe parameters. Given our computational limitations, we did the fit of all the parameters only for1,000 samples from parameter sets S2, S3, S5 and S6. The transition probabilities behave similarlyas when N = 1. Fits for S2 and S3 converged but fits for S5 and S6 estimated pd = 0 (Table 2.8).Figure 2.13 shows the number of fluorophores estimated for the 1,000 samples grouped by param-eter set. We noticed that fits for S2 and S3 estimated a lower number of fluorophores compared toS5 and S6 (blue colours vs yellowish colours in Figure 2.13). The difference in the mean of thedistributions is more clear in Figure 2.14, where we group the estimates by the value of Nfit. Themodes are 20 for S2, 21 for S3, and 25 for S5, 24 for S6.52S2 S3 S5 S6pr 0.1 0.1 0.1 0.1pˆr 0.097 0.096 0.082 0.087pd 0.8 0.4 0.1 0.1pˆd 0.8 0.4 0.00 0.00pb 0.1 0.1 0.5 0.8pˆb 0.10 0.10 .63 1.00Table 2.8: MLE and true values of transition probabilites for 1,000 samples of parameter sets S2, S3,S5, and S6, when also estimating the number of fluorophores (N = 20). The estimates (pˆr, pˆd , pˆb)were calcualted by fitting 1,000 samples for each parameter set. True values as in Table 2.2.S2 S3 S5 S600.050.10.150.2probability 14172023262932 35NfitFigure 2.13: Number of fluorophores estimated for 1,000 simulations for parameter sets S2, S3, S5,and S6 without fixing the transition parameters (N = 20). Bars are grouped by the differentparameter sets (S2, S3, S5, and S6). All the samples were estimated to have between 13 to 40distinct fluorophores. Colour scale corresponds to the number of fluorophores estimated Nfit.With these results, we show that our method can estimate the number of distinct fluorophores ifthe fluorophores blink several times. The estimation is harder when the fluorophores typically blinkonly once or very few times.2.8.5 Estimating the number of fluorophores while conditioning on the number ofsimultaneous activations.Usually, dSTORM data have only a few localizations in the same frame. This is because dSTORMneeds a minimal distance between the point spread functions of observations to distinguish them(as explained in Section 1.3). We implemented this constraint by conditioning our model to have amaximum number of active fluorophores at each frame, Amax.5314 16 18 20 22 24 26 28 30 32 34 36 38050100150200frequencyS2S3S5S6Figure 2.14: Comparing the distribution of the estimated number of fluorophores when N = 20 forparameter sets S2, S3, S5, and S6. Bars are grouped by Nfit from 13 to 38, with the last barwith Nfit = 40. Colours refer to the parameter sets (S2 dark blue, S3 blue, S5 green, S6 yellow).Modes at 20 for S1, 21 for S2, 24 for S6 and 25 for S5. Results over 1,000 samples for eachparameter set.To test if we will get correct estimates on the number of fluorophores even if the real dynamicsof the fluorophores were coming from the non-conditional model, we estimated the number offluorophores using data simulated with the model without normalization, by applying our modelconditioned to have no more than Amax number of active fluorophores at each frame. We willestimate all the parameters, N, pr, pd and pb. The estimates for the transition probabilities willnot necessarily correspond to the true dynamics of the fluorophore since we are not fitting thesame model. This is not an important problem in the larger sense since we are most interested inestimating N, not the dynamics of a fluorophore.We fitted the normalized model to the simulated data of one fluorophore (N = 1) for all param-eter sets S1-S6, and of twenty fluorophores (N = 20) for parameter sets S2, S3, S5 and S6. ForN = 1, we divided the data into groups of M = 10,100,1000,10000 as in Section 2.8.4. For N = 20we fitted 1,000 simulations simultaneously.Amax corresponds to the maximum number of simultaneous observations over all frames and allsamples,Amax = maximaxtNa(t)i.When we simulated only one fluorophore, we obviously never observed more than one localizationper frame, and thus Amax = 1 when N = 1. For N = 20, we found different values of Amax for each540 1 2 3  d 7difference00.10.20.30.40.50.60.70.80.91probabilityS1S2S3S4S5S6parameter setFigure 2.15: Difference in estimates between conditioning the model and not conditioning on themaximum number of simultaneous activations, for N = 1. Bars are grouped by values of thedifference. Colours refer to the parameter sets S1 (dark blue), S2 (blue), S3 (light blue), S4(green), S5 (orange) and S6 (yellow). A difference of zero means the estimated was the samefor both models for that sample. The maximum distance was 7. Data are shown for the fit whenM = 10,000.parameter set:Amax =7 for S69 for S2 and S510 for S3(2.32)We compared the number of fluorophores estimated from the conditioned model to the val-ues found using the model without conditioning (as got in Section 2.8.4). To do so, we took thedifference between the estimated values with or without conditioning.Figure 2.15 shows the difference between the estimated numbers of fluorophores from all thesamples (M = 10,000) and parameter sets when N = 1. This difference is positive since all theestimates from the conditional model were greater than the ones from the unconstrained model.Thus, conditioning for the maximum number of activations will generally cause an overestimationof the number of fluorophores.Samples from parameter sets S4 and S6 generated the same estimates over 10,000 samples(green and yellow bars in Figure 2.15). There were only 32 samples with one more estimated55(a)1 2 3 4Nfit0200040006000800010000frequencyS1Not conditionedConditioned(b)1 2 3 4 5Nfit02000400060008000frequencyS2Not conditionedConditioned(c)1 2 3 4  5Nfit01000200030004000500060007000frequencyS3Not conditionedConditionedFigure 2.16: Comparing Nfit distribution estimated by conditioning and not conditioning over Amaxfor N = 1 and (a) S1, (b) S2 and (c) S3 parameter sets. Bars grouped by estimated number offluorophores Nfit. Blue bars are the results with the model without conditioning (not norm) andyellow ones correspond to results with conditioning the simultaneous activations (normalized).The mode of Nfit for S1 and S2 is 1 with both models, for S3 is 1 without normalizing and 2when normalizing. The total number of samples is 10,000 for all parameter sets.fluorophore from S5 (orange bar in Figure 2.15). This agrees with the results from previous sec-tions since S4-S6 samples did not have enough information to distinguish between the observations(Figure 2.12).Conditioning produced a noticeable difference when we fit data with one fluorophore createdfrom parameter sets S1, S2, and S3 (Figure 2.12 and Figure 2.16). In these parameter sets, theoverestimate was obvious. In Figure 2.16, blue bars represent the distribution of the estimates withthe unconstraint model and yellow bars the distribution of the constraint model. The shift to the rightin the estimated number of fluorophores is clear in those figures. All sets (S1-S3) had fewer sampleswith Nfit = 1 and increased the samples with ≥ 2 estimated fluorophores in the constrained model,compared with the unconstrained one (Figure 2.16). Moreover, the distribution for S3 changed itsshape. The mode of the estimated number of fluorophores was 2 when conditioning to have no morethan Amax simultaneous activations, and the samples had up to 7 fluorophores.The results are different when using the datasets with twenty fluorophores (N = 20). In herethe difference in estimates between the conditioned model and the full model was almost none(Figure 2.17a). There was no difference between the conditioned and the full model for S6, onedifference for S3 and two for S5. The biggest difference was in S2, where 18 out of the 1,000simulations had one more fluorophore with the conditional than with the full model. Figure 2.17bshows the distribution of both models for 1,000 simulations and parameter set S2.The datasets of N = 20 had Amax ≤ 7, while the N = 1 datasets had Amax = 1. Thus, the valueof pˆi j ≈ pi j for the states si and s j with Na ≤ Amax since Amax is larger (Equation 2.22 and Equa-560 1difference00.20.40.60.81probabilityS2S3S5S6parameter set(a) Nfit differences between models14 17 20 23 26 29Nfit050100150200frequencyS2Not conditionedConditioned(b) Nfit distributions for S2Figure 2.17: Comparing Nfit distributions from conditioning and not conditioning over Amax for N =20 and S2, S3, S5, and S6 parameter sets. (a) Differences in estimates between the models witheach bar being the values of the difference and colour-coded by parameter sets (S2 (dark blue),S3 (light blue), S5 (green) and S6 (yellow)). A difference of zero means the estimated wasthe same for both models for that sample. (b) Comparing the distribution of Nfit from the full(blue bars) and the conditioned (yellow bars) models with S2 parameter set. The total numberof samples is 10,000 for all parameter sets.tion 2.30). Therefore, the models have almost the same transition probabilities and the estimatescoincide regardless of the conditioning. In other words, the probability of observing more than Amaxsimultaneous simulations becomes small as we increase Amax.In our data sets, the transition probability from a state with Na ≤ Amax to a state with Na > Amaxis smaller than 0.054 for all S2, S3, S5 and S6 (Equation 2.32).We also tested the opposite: when we assumed the data comes from the conditioned model andcompared the estimated when fitting with the model without conditioning. We expect that by usingthe unconstrained model we will underestimate the number of fluorophores of data coming from theconstrained model. The underestimation will be more remarkable if Amax is small.We simulated 500 repetitions with N = 20 while constraining to observe no more than Amax= 2fluorophores active in the same frame. We did this only for the parameter set S5. We compared ourresults when fitting the data with the conditioned model versus using a model with no conditioning(Figure 2.18). We found that the unconstrained model (blue bars in Figure 2.18) was estimatedto have half of the true number of fluorophores in the simulated samples (N = 20). On the otherhand, the constrained model (yellow bars in Figure 2.18) behaves similarly to the case of fitting theunconstrained data to the unconstrained model (green bars in Figure 2.14).57S50 10 20 30 40Nfit050100150200frequencyNot conditionedConditionedFigure 2.18: Comparing Nfit distribution estimated by conditioning and not conditioning over Amax forN = 20 when simulating conditioned model with S5 parameter set. Blue bars show estimatesfor the model without conditioning and yellow ones correspond to results with conditioning ofsimultaneous activations (conditioned). The total number of samples is 500.2.9 DiscussionOur algorithm correctly estimates the number of fluorophores in a sample when we know the physi-cal properties of the fluorophores, both in continuous-time and discrete-time. In their paper, Rollinset al. fit for five fluorophores since the computation is slow for larger sets [167]. By using paral-lelization, we speeded up the estimation, being able to fit simulations coming multiple fluorophores.We presented results for 20 fluorophores with up to 350 data points.With unknown transition parameters, the continuous-time model works better at estimating thenumber of fluorophores but it requires the exact transition times. In reality, we only have approxima-tion times binned in discrete periods. Rollins et al. already showed that the number of localizationsestimates will change with ∆t (the discretization over the continuous-time) [167]. We showed thatour algorithm worked as well for the discrete version of the model.Moreover, we found that conditioning on the maximum number of observed simultaneous ac-tivations (Amax) often gives better estimates than estimating without conditioning if the transitionparameters are unknown. This result occurs since either (a) the data was constrained to have amaximum number of activations, or (b) we have good information on the probability of simulta-neous activations (e.g. large datasets). In the first case, the inherent model of the data is alreadyconditioned and using the incorrect model will cause an underestimate of the number of unique58observations. In the second case, we can define Amax to be the greater observed simultaneous ac-tivation on our data, in other words, to be the greatest number of simultaneous activations with apositive probability. In this case, having more than Amax simultaneous activation will be unlikelyand thus the conditioning model will be a good approximation to the unconditioned one. Havinga good fit with the conditioned model when the sample comes from unconditioned data causes tooverestimate the number of fluorophores if the value of Amax is not accurate.From real data, we know that dSTORM data presents a tiny number of simultaneous activations(order 1-10) versus the number of total observations (order 104). This highlights the utility of aconditioned model. We will not know if dSTORM data is or not conditioned and our samples couldbe heterogeneous over the number of fluorophores. In our results, we got the value of Amax usingour large datasets. The homogeneity on the number of fluorophores of our samples also affectedAmax (samples with either one or twenty fluorophores). Thus, we may require a larger number ofsamples to have a good value of Amax.We based all our results on the assumption that the algorithms for the single-molecule localiza-tions have no errors. This can bring another source of biases [52], and that bias could be avoidedby coupling our temporal models during the single-molecule localization estimation. We will notaddress this but we will incorporate the localization estimates in the next section.2.10 SummaryWe have presented a set of algorithms for estimating the number of fluorophores presented in a smalldSTORM dataset. Our discrete algorithm estimated the number of fluorophores of samples with upto ≈ 350 localizations.We characterized estimation errors and showed that they depend on the tran-sition probabilities of the fluorophores, being smaller for samples with low bleaching probability.With low bleaching, fluorophores blink many times before bleaching and samples have localizationsproviding more information. Improving the calculation of the likelihood will enhance our algorithmto fit bigger samples. However, another approach will consist of using the spatial information of thelocalizations to separate larger samples into smaller datasets, where we can differentiate localiza-tions that are far from each other. This idea will be implemented in the next chapter.Our approach does not assume fixed values of the kinetic dynamics of the fluorophores, andthus our estimates will account for the inherent variability coming from the imagining settings.Nonetheless, our temporal models provided insight and will enhance the interpretation of amicroscopy technique that is already widely used in biological applications. As with any model, theunderlying assumptions are only approximations to the true dynamics.59Chapter 3Divide and conquer to identify uniqueobservations in super resolutionmicroscopy: spatiotemporal modelEvery good, successful player, especially an attacking player, has a well-developedsense of space and time.— Thomas Mu¨ller, professional footballerdSTORM data contains specific spatial localizations as well as a time series of fluorophore ac-tivation events. In Chapter 2, we focused on the time series information and developed Markovmodels. In this chapter, we will incorporate the spatial information to develop an integrated ap-proach.We will pre-process the data by separating those localizations unlikely to come from the samefluorophore. This clustering process will account for the observed error of each localization. Smallersamples will be faster to analyze, and will obey the same temporal dynamics. To control for over-fitting, we also apply a constraint on the temporal parameter pr, following similar results to thosepresented in Section 2.5.2.The multiple localizations of one fluorophore are assumed to be drawn from a spatial Gaussianprocess centred at the true positions of the fluorophore. Using the independence of fluorophores, wecan use a Gaussian Mixture model (GMM) to estimate the real position of multiple fluorophores.We then couple our GMM to the Markov model and find the best estimates of the parameters ofthe joint model to the pre-clustered data. The resulting spatiotemporal model will enhance ourestimation of the number of different fluorophores in a data set.We start this chapter by introducing the pre-processing method and the pr constraint in Sec-tion 3.1. We then explain the GMM and the Expectation-Maximization algorithm (EM) in Sec-tion 3.2. This algorithm is needed to calculate the likelihood of the GMM. We then describe60100 nm60 nm100 nm60 nmFigure 3.1: Example of spatial separation of dSTORM data. This figure is a schematic of spatialdSTORM data in a 100 by 60 nm box. Top panel shows only the localizations while bottom panelincludes the spatial errors. Yellow dots represent the localizations and the ellipsoidal dotted linesrepresent two standard deviations from the observation. There are 25 points in the left cluster and39 in the right cluster.dSTORM localization data and develop the corresponding GMM in Section 3.3. In Section 3.4, wecouple the spatial and temporal models. Finally, the results and discussion based on simulated dataare in Section 3.5 and Section 3.6. Real experimental data will be analyzed in Chapter 4.3.1 Divide to conquer: pre-processing the spatial dataFollowing the super-resolution algorithm, it is unlikely that a fluorophore produces two points sep-arated by over 3 times their spatial uncertainty. Remember that fluorophores are immobile and thatthe uncertainty follows an Airy distribution, which can be well approximated by a 2-D Gaussiandistribution. Figure 3.1 illustrates such a case. Notice that in that case, we can separate the sam-ple into two disjoint pieces. We incorporate such information into our algorithm by running oursamples through a clustering algorithm. Such an algorithm should always account for the spatialuncertainty of the data. Otherwise, we might separate localizations coming from one fluorophorethat was poorly resolved during imaging, leading to false division of localizations coming from that61fluorophore.By separating a sample into smaller sub-samples, we change the problem of finding N for onesample to finding several N’s across sub-samples. Thus, we decrease the size of the domain of ourinteger parameter, which will usually speed up our computations.There are many clustering algorithms in the literature and such algorithms are not the focus ofthis thesis work. Thus, we used a recently developed user-friendly Matlab clustering analysis soft-ware created in the Coombs group by Joshua Scurll [179]. We will briefly introduce the algorithmin the next section without further discussion.3.1.1 Stormgraph softwareScurll et al. developed Stormgraph in Matlab to spatially analyze single-molecule super-resolutiondata [179]. The software is available upon direct request to the authors. The input data includes theobservation coordinates in 2 or 3 dimensions, and importantly their respective spatial errors. In ourcase, we have two-dimensional data with their corresponding spatial errors.Stormgraph generates Monte Carlo simulations to decided if two points belong to the samecluster accounting for their spatial uncertainty. To include this analysis, we set the parameteruse localization errors to true. The software simulates one-dimensional normal distributions withmean the observation coordinate and standard deviation the corresponding spatial error. It assumesthat the x and y coordinates are independent. The parameter p controls how conservatively the clus-ters are created. In our case, we want to be conservative since we prefer larger clusters and wishespecially to avoid observations from the same fluorophore separated in different clusters. Thus, werecommend p≤ 0.05 for our analysis. This value of p corresponds to retaining points with at least5% of “spatial overlapping” in the same cluster.Stormgraph also includes a correction for those localizations that come from the same fluo-rophore. Those localizations are highly valuable for our analysis and, thus, we set the parametermultiple blinking to false. This will indicate to the software to keep all clusters, including those thatcould come from only one molecule.The software has a minimal number of points below which it rejects clusters. Again, we want toconserve most of the data, even when only a few localizations are recorded. The minimum allowedvalue in the software is 3, but we modified the code to allow MinCluSize = 2. We will use either 2or 3 depending on the data sets.3.1.2 The last active fluorophore: a constraint on the reactivation probability prPre-processing our samples into smaller sub-samples could lead to over-fitting of the temporal pa-rameters, especially because the new sub-samples have a smaller number of localizations. We usedthe distribution of the time to the last activation of a fluorophore to control the estimates. To calcu-late the time to the last activation, we work similarly to Section 2.5.2, where we found that the time62Figure 3.2: Schematic of a time series that ends with one active fluorophore during one frame. Thex-axis represents time and the y-axis represents the number of activations. Notice that only thelast two transition times are illustrated, hence the three dots at the left of the blue time series.During 4 frames the time series is in the Dark class tend−1 = 4 and the experiment ends with oneframe in the Bright-1 class (tend = 1 and NA(end) = 1). Classes defined as in the previous chapter.to the first activation follows a geometric distribution with parameter 1− (1− pr)N (Equation 2.26).The key assumption for our constraint is that all the fluorophores bleach by the end of theexperiment. Thus, we know the are zero non-bleached fluorophores after the last time we detect anyactivations. In a specific case where the last localization lingers for exactly one frame (tend = 1) andcontains only one active localization (NA(end) = 1), we had only one fluorophore in the dark state,while the rest had bleached already. We illustrate this in Figure 3.2 where we graph the last twotransition times of a time series, tend and tend−1. It takes tend−1 frames for the last fluorophore to beactivated. And similar to Section 2.5.2, tend−1 will follow a geometric distribution with parameterpr and domain i ∈ {1,2,3, ...}.Notice that this distribution does not depend on the value of N and thus is not affected by oursub-sampling. Thus, we could estimate the parameter pr by fitting those samples with only oneframe active at the end (tend = 1 and NA(end) = 1). Since such an estimate does not account forthe full-time series information, we compute the 95% confidence interval for pr from this data andincorporate that interval as a constraint on the value of pr when optimizing the temporal parameterswith Algorithm 2.3. If the number of samples that satisfy the last time to activation conditionstend = 1 and NA(end) = 1 is small, we will not use this constraint.3.1.3 Pre-processing algorithmAlgorithm 3.1 summarizes the protocol for pre-processing multiple dSTORM data. We cluster thespatial information (localizations and errors) using Stormgraph (Algorithm 3.1, line 4) for eachsample in parallel. With the labels gotten from the spatial clustering, we format the time series ofeach sample, Tm, into the time series of the cluster, Tm,k (Algorithm 3.1, line 6).Once we have the data sub-samples, Tm,k, we take those samples that have only one fluorophore63Algorithm 3.1 Pre-processing M samples by clustering and pr constraintRequire: M samples, {Xm,ΣXm ,Tm}Mm=1 localization, errors, and time series of each sample, andMtolpr (min number of samples to create pr constraint).Ensure: Data formatted as required by Algorithm 2.3, and Stormgraph.1: function PRE-PROCESSING(M,{Xm,ΣXm ,Tm}Mm=1)2: Define Stormgraph parameters Θsg . Section 3.1.13: for i ∈ {1, ..,M} do . Compute in parallel over M4: function STORMGRAPH(Xm,ΣXm ,Θsg)5: GET: Km number of clusters, labelm vector with cluster label per localization in Xm.6: Format Xm,ΣXm , and Tm into cluster data {Xm,k,ΣXm,k ,Tm,k}Kmk=1.7: for j ∈ {1, ...,Km} do8: if Last transition in Tm, j satisfies tend = 1 and NA(end) = 1 then9: Append tend−1 into Tpr .10: if length(Tpr )> Mtolpr then11: [lpr ,Upr ] = 95% confidence interval of Geo(pr) fit of Tpr . . Section 3.1.312: else13: [lpr ,Upr ] = [0,1]return Mc = ∑Km, {Xm,k,ΣXm,k Tm,k}Km,Mk,m , [lpr ,Upr ], {labelm}Mm=1active at the last frame, Tpr (Section 3.1.3, Algorithm 3.1, lines 7-9). If we have more than Mtolprsub-samples that satisfy tend = 1 and NA(end) = 1, we compute the 95% confidence interval of prwith the MATLAB function nbinfit.Last, we use our new data set {Tm,k}Km,Mk,m and the pr constraint to estimate the temporal param-eters Θˆ and the number of fluorophores in each cluster Nˆm,k. For the estimation, we will coupleAlgorithm 2.3 with a Gaussian Mixture model and obtain the most likely number and position offluorophores given the data. In the next sections, we will describe the theory and the protocol thatwill be used for our parameter estimation.3.2 Gaussian Mixture modelsA Gaussian Mixture model (GMM) is defined as a linear combination of N Gaussian densities(Equation 3.1). Let pin be the proportion of mixture n relative to the rest. The proportions pin shouldbe positive and add up to one so that the resulting function is a probability density. In general,mixture models could have other densities but we will focus solely on the case of φ ∼N (µ,Σ).Formally,p(x) =N∑n=1pinφn(x), φn ∼N (µn,Σn),N∑n=1pin = 1 0≤ pin ≤ 1. (3.1)64This linear combination of densities has enough freedom in its parameters that it can approximatealmost any continuous density by adding together a sufficient number of mixtures [19]. We also findGMM useful from their modeling interpretation. Each mixture represents a Gaussian distributedgroup, and the group’s aggregated observations will be represented by a GMM [188]. Thus, asample coming from a GMM is formed by a number of subpopulations, each of which can beestimated from a independent, but not necessarily identical, Gaussian distribution [70]. A samplewith a multi-mode distribution can thus be well modeled with a GMM.3.2.1 Maximizing the likelihood function of a GMMFor given data, we might want to find the parameters that maximize the likelihood of a GMM.The MLE of a single Gaussian distribution is easy to calculate, but this is not the case for a linearcombination of Gaussians.To calculate the MLE of a normally distributed sample, we minimize the negative logarithmof the likelihood. This transformation simplifies the exponential term in the normal density andconverts the product of independent observations into a sum. If our data is X = {x1,x2, ...,xK} andwe know it is normally distributed, the parameters that maximize the likelihood of observing X arecalculated as follows:L (µ,σ |X) =K∏k=11√2piσ2e−(xk−µ)22σ2 =⇒ NLL(µ,σ |X) = 12K∑k=1(log(2pi)+2log(σ)+(xk−µ)2σ2)=⇒ µˆ = 1KK∑k=1xk, σˆ2 =1KK∑k=1(xk− µˆ)2.In fact, this is exactly the case of a GMM with only one mixture (N = 1). In the case of one mixture,the proportion parameter pi = 1.Now, take the data X to consist of points coming from N > 1 normally distributed subpopu-lations. If we knew the distribution of subpopulations, then our GMM could be simplified into Nindependent Gaussians with parameters depending only on the points from each subpopulation. LetXn be the points coming from subpopulation n, such that X = ∪Nn=1Xn, and define Θ to be the set ofall parameters, Θ= {µn,σ2n ;n = 1, ...,N}. Now, the MLEs are as follows:L (Θ|X) =K∏k=1N∑n=11xk∈Xnφn(xk) =⇒ NLL(Θ|X) =−K∑k=1N∑n=11xk∈Xn log(φn(xk)) (3.2)=⇒ µˆn = 1|Xn||Xn|∑k=1xk, σˆ2n =1|Xn||Xn|∑k=1(xk− µˆn)2, n = 1, ...N. (3.3)However, GMMs are used for data when the subpopulations are not known and part of theproblem is finding the most likely subpopulation each points belong to. To maximize the likelihood,65we could try to apply the negative logarithm transformation. But in this case, the solution is not aseasy. We obtain a sum inside the logarithm, complicating the optimization:L (Θ|X ,Π) =K∏k=1N∑n=1pinφn(xk) =⇒ NLL(Θ|X ,Π) =−K∑k=1log(N∑n=1pinφn(xk)). (3.4)There are several methods that work around this problem. Here, we will focus on the Expectation-Maximization algorithm (EM) (Section 3.2.4). The intuition needed to simplify the optimizationcomes by relating the proportion parameters pin to the probability that a given point belongs tomixture n. I will describe two different ways to interpret that relationship, and use this to explainthe EM algorithm.3.2.2 Proportion parameters pin and prior probabilitesLet’s think back to the subpopulation separation of the GMM. The proportion pin represents theprobability that a point comes from the subpopulation Xn, thus pin = P(x ∈ Xn). It is usual to assumethat all subpopulations are independent. Thus, if we want to simulate a new point from the dataset, we can select a subpopulation at random (n with probability pin) and then draw x from thecorresponding normal distributionN (µn,Σn). We can therefore rewrite Equation 3.1 asp(x) =N∑n=1pinφn(x) =N∑n=1P(Xn)p(x|Xn). (3.5)Here, we can interpret pin = P(Xn) as the prior probability of population n and p(x) = p(x|Xn) as theprobability of x conditioned on Xn [19]. Next, using Bayes theorem we can find the probability of asubpopulation given the data as:p(Xn|x) = p(x|Xn)P(Xn)p(x) =p(x|Xn)P(Xn)∑Nn=1 P(Xn)p(x|Xn). (3.6)This equation represents the posterior probabilites, also known as responsibilities [19]. We willrelate this further to EM, but first, let’s relate the subpopulation probabilities to labeling the data.3.2.3 Proportion parameters pin and probabilistic clusteringAs in the previous section, we extend ideas from defined subpopulations to their mixture. If weknew the subpopulation to which a point belongs, we could define a label for that point. The labelwill be a binary vector Z of dimension N (the number of mixtures), where an entry zn = 1 if thepoint belongs to subpopulation n, and zn = 0 otherwise [19, 70, 183]. Since the label is in generalunobserved, we define it to be a random variable. The density of this variable will depend on the66proportion parameters, as follows:p(zn = 1) = pin = P(Xn), p(zn = 0) =N∑i6=npii = 1−pin. (3.7)In other words, Z follows a multinomial distribution, Z ∼Multi(1;Π), and this defines the subpop-ulation x belongs to. Thus, the joint probability density of the data and the labels is given byp(x,Z) =N∏n=1(pinφn(x))zn .The log likelihood function of the joint probability isNLL(Θ,Π|x,Z) =−N∑n=1zn (log(pin)+ log(φn(x))) . (3.8)Notice that by using Bayes theorem we can find the probability of x,p(x) =N∑n=1p(x|zn = 1)p(zn = 1) =N∑n=1pinφn(x),which is exactly the same as Equation 3.1. We have thus defined the GMM in terms of the latentvariable.Moreover, the expectation of the labels given the data is given by:E(Z|x,Θ,Π) = 0 ·P(z1 = 0|x,Θ,Π)+1 ·P(z1 = 1|x,Θ,Π)...0 ·P(zN = 0|x,Θ,Π)+1 ·P(zN = 1|x,Θ,Π)=P(z1 = 1|x,Θ,Π)...P(zN = 1|x,Θ,Π) ,where, using Bayes theorem,P(zn = 1|x,Θ,Π) = pinφn(x)p(x|Θ,Π) =pinφn(x)∑Ni=1piiφi(x)= γ(zn). (3.9)This is exactly the same as the responsibilities (or posteriors) given by Equation 3.6. We use thenotation γ(zn) as in Bishop’s book [19].3.2.4 Expectation-Maximization algorithm (EM) and Gaussian Mixture modelsAs shown in Equation 3.4, the minimization of the NLL of a GMM does not have a simple solution.The Expectation-Maximization algorithm (EM) is a popular approach to this problem [19, 49, 70,138, 183, 188]. The EM algorithm is a general iterative algorithm used for mixture models since the1970s [49, 183]. This algorithm iterates between two steps: the Expectation step (E-step) and the67Maximization step (M-step) (Algorithm 3.2). The two steps come from the observation that if weknew the labels, we could estimate the mixture parameters, and if we knew the parameters of eachmixture, we could estimate the labels. In other words, the algorithm is trying to convert the problemdescribed by Equation 3.4 into that of Equation 3.2. To achieve this, EM uses the latent variable Zdefined in the previous section.Algorithm 3.2 EM algorithmRequire: Data X ; initial guess for the mixtures parameters, Θ0,Π0; tolerance error.1: function EM(X ,Θ0,Π0, tolerr)2: while error > tolerr do3: Zˆ = E-step(X ,Θ0,Π0) . Equation 3.94: (Θˆ,Πˆ) = M-step(X , Zˆ) . Equation 3.105: NLL =L (Θˆ,Πˆ|X , Zˆ)6: Update error, and Θ0 = Θˆ, Π0 = Πˆreturn NLL, Θˆ, PˆiIn the E-step, we calculate the expectation of the label values given the data and an initial guessof the parameters. We want to know the label that a point will be expected to have given the cor-responding density of the initial parameters. This expectation corresponds to γ(zn) ( Equation 3.9)and is easily generalized for multiple data points γ(zkn) = P(zn = 1|xk,Θ,Π).The M-step consists of finding a local maximum of the likelihood given the labels estimatedwith the E-step. This step does not calculate the exact likelihood function of the data, but theexpectation of the log likelihood of the full data (x,Z), with respect to the labels Z. In other words,using Equation 3.8 we get:EZ(log(x,Z)) =N∑n=1E(zn)(log(pin)+ log(φn(x))) . (3.10)E(zn) is known from the E-step ( Equation 3.9).EM converges to a point that locally maximizes the likelihood function of Gaussian Mixturemodel [19, 45, 49, 70, 86, 146]. We will not describe the details of the convergence of the algorithm,but focus on its implementation.3.2.5 Estimating the parametes for a general GMMGiven the data X = {x1, ...xK}, we want to estimate the parameters of a GMM with N mixtures usingthe EM algorithm. The calculation of the γkn from Equation 3.9, as well as finding Θˆ and Πˆ thatmaximize Equation 3.10, are well documented in the literature and summarized in Algorithm 3.3[19, 70, 183, 188].In Algorithm 3.3, we must assume we know the number of mixtures N. If N is unknown, we68could estimate it by repeating EM for all possible values of the number of mixtures N. That is theway we do it in this dissertation. We will explain how the iteration over N is done and how wepenalize for overfitting as we add mixtures. There are alternatives to this approach. A possiblefruitful alternative could be to explore efficient Bayesian methods that allow us to estimate thenumber of mixtures along with the other parameters. Such methods include variational inferenceand non-parametric methods [19, 70]. Bayesian methods will automatically balance the overfitting[19]. We leave the implementation of such methods to future work.Algorithm 3.3 EM algorithm for a GMMRequire: Data X ; initial guess for the parameters, Θ0 = {µ 0,n,Σ0,n}Nn=1,Π0 = {pi0,1,pi0,2, ...,pi0,N};tolerance error.1: function EM(X ,Θ0,Π0, tolerr)2: while error > tolerr do3: function E-STEP(X ,Θ0,Π0)4:γˆkn =pi0,nφn(xk)∑Ni=1pi0,iφi(xk), φi ∼ N(µ 0,i,Σ0,i) (3.11)return Zˆ = [γˆkn] . Matrix with all γˆkn5: function M-STEP(X , Zˆ)6: for n = 1, ...,N doKn =K∑k=1γˆkn, pˆin =KnK, (3.12)µˆ n =1KnK∑k=1γˆknxk, Σˆn =1KnK∑k=1γˆkn(xk− µˆ n)(xk− µˆ n)T (3.13)return Θˆ= {µˆ 1, Σˆ1, ...., µˆN , ΣˆN},Πˆ= {pˆi1, ..., pˆiN}7: NLL =L (Θˆ,Πˆ|X , Zˆ)8: Update error, and Θ0 = Θˆ, Π0 = Πˆreturn NLL, Θˆ, Πˆ3.2.6 Identifiability problem among mixture order permutationsNotice that interchanging the order of the labels does not change the probability distribution, andthus the likelihood functions are also unchanged [70, 183]. This is a problem of parameter identifi-ability: any permutation of the labels will give you the same likelihood value. For example, we canlabel the mixtures from left to right, or from right to left in a one-dimensional GMM and the modelwill remain the same.This trivial identification problem is not important. One can correct for it by defining an orderingof labels to remove the ambiguity. However, that ordering is not needed in order to estimate the rest69of the parameters.3.3 dSTORM spatial data as a Gaussian Mixture modelAs explained in Section 1.3, dSTORM data consists of localizations, either in two or three dimen-sions. Focusing on the two-dimensional case, we will model the dSTORM spatial information as aGMM assuming the following:1. We may observe one fluorophore multiple times, and we assume that all those observationsfollow a normal distribution centred at the true position of the fluorophore.2. Fluorophores behave independently.3. Each element of the GMM will represent the true position of a different fluorophore.In adition to positional information, each localization has an error depending on the number ofphotons captured in the observation frame. In principle, dSTORM experiments provide a couple(x,∆x), where x is the localization and ∆x captures the error in that localization. Given that the errorof the localizations is approximately Gaussian, we can use the symmetry of the Gaussian distributionto relate ∆x to the covariance of the mixture model (Figure 3.3). By Gaussian symmetry, drawing µfrom a normal distribution N(x,Σx) has the same density as drawing x from N(µ ,Σx) (Figure 3.3a).Thus, we model N fluorophores as a GMM with mean µ (true position) and covariance matrixgiven by the observation errors. In this case, each data point xk has its own covariance matrix Σk(Figure 3.4). The covariance information is obtained from the data during the localization-detectionstep of image processing. The covariances of the mixtures vary between localizations. Equation 3.1becomesp(xk,Σk) =N∑n=1pinφn(xk,Σk) where φn(xk,Σk) = φnk(xk), (3.14)φnk ∼ N(µ n,Σk), Σk =(σ21,k 00 σ22,k).Usually, the error of the dSTORM localization is calculated only for each coordinate direction.Thus, we assume Σk is diagonal to match the data. Our method could be extended to a full covariancematrix, if such information was available.3.3.1 Estimating the mean parameters of a GMM with data (X ,ΣX)We will use EM to find the most likely position of the fluorophores, assuming that they follow theGMM given in Equation 3.14. The expectation step remains as in Equation 3.9, while the M-step,70(a) Gaussian symmetry(b) Example of one fluorophore with multiple activationsFigure 3.3: Schematic of Gaussian symmetry and example of one flurophore with multiple localiza-tions. (a) Illustration of the symmetry of the Gaussian centred at x (green dot) with standarddeviation ∆x (green dotted elipse), with a Gaussian centred at µ (white star) and same standarddeviation ∆x. (b) Example of one fluorophore (white star) with two observed localizations (greendots). Each localization has a corresponding error (dotted ellipses represent the standard de-viation of position). In the left panel, we show the dSTORM data and on the right panel thecorresponding Gaussian fit µ .Figure 3.4: Schematic of multiple data points fitted to a GMM with three Gaussians. Left panel showsthe data and right panel the corresponding GMM fit. The three mixture means are shown asyellow stars (right panel). The localizations and standard deviations are shown as dots and dottedellipses with matching colours.71Algorithm 3.4 EM algorithm for dSTORM spatial modelRequire: Data X ,ΣX ; initial guess for the parameters, µ 0 = {µ 0,n}Nn=1,Π0 = {pi0,1,pi0,2, ...,pi0,N};tolerance error.1: function EM(X ,ΣX ,Π,µ 0, tolerr)2: while error > tolerr do3: function E-STEP(X ,ΣX ,µ 0)4:γˆkn =pi0,nφnk(xk)∑Ni=1pi0,iφik(xk), φik ∼ N(µ 0,i,Σk) (3.15)return Zˆ = [γˆkn] . Matrix with all γˆkn5: function M-STEP(X , Zˆ)6: xk = (x1,k,x2,k).7: for n = 1, ...,N and i = 1,2 doKn =K∑k=1γˆkn, pˆin =KnK, (3.16)µˆi,n =(K∑k=1γknσ2i,kxi,k)/(K∑k=1γknσ2i,k)(3.17)8: µˆ n = (µˆ1,n, µˆ2,n).return µˆ = {µˆ 1, ...., µˆN},Πˆ= {pˆi1, ..., pˆiN}9: NLL =L (µˆ ,Πˆ|X ,ΣX , Zˆ)10: Update error, and µ 0 = µˆ , Π0 = Πˆreturn NLL, µˆ , ΠˆEquation 3.10, changes. The algorithm is summarized in Algorithm 3.4. In this case, the M-step isgiven byEZ(log(X ,Z)) =K∑k=1N∑n=1γkn (log(pin)+ log(φn(xk,Σk)))=K∑k=1N∑n=1γkn(log(pin)− 12 log(det(2piΣk))−12(xk−µ n)TΣ−1k (xk−µ n)).We need to minimize this function to estimate the means. To illustrate the calculation, we willassume xk ∈ R for all k.∂EZ(log(X ,Z))∂µn=K∑k=1−2(xk−µn)γkn2σ2k= 0 =⇒ µˆn =(K∑k=1xkγknσ2k)/(K∑k=1γknσ2k)(3.18)If the covariance matrix is diagonal, Equation 3.18 works the same for the n-dimensional case. Inparticular, it works for our two dimensional data and so we will use Equation 3.18 as our M-step.72Notice that the estimation of the proportion parameters Π does not change when adding theerror (ΣX ) data. In that case, the estimate Πˆ will be as in Equation 3.12.3.3.2 dSTORM spatial model identification limitationsWhen we have the same number of data points and mixtures, K, we would expect that the best fitwould simply be that the MLE coincides exactly with our data, µn = xn. In fact, this is not alwaysthe case since the MLE also depends on ΣX . Thus, the MLE can be “biased” by the data. Forexample, it is possible to data from have multiple mixtures with overlapping errors that produce oneunique position estimate.It is complicated to calculate the exact MLE, so we show this counter-intuitive point with anumerical example (Figure 3.5). In this example, we have three localizations {x1,x2,x3} with theircorresponding Σi covariance matrices. The three data points lie on a circle of radius R. We startwith x1 = (1,0), x2 = (−1,0) and x3 = (0,1), with standard deviations given by ∆x1 = (0.23,0.5),∆x2 = (0.39,0.4), and ∆x3 = (0.45,0.61). We let Σi be fixed but we decrease the distance betweenthe data points xi by changing the radius of the circle. Specifically, we allow R to vary between 0.1to 1 in steps of 0.1 units. We estimate the mean of the mixtures when N = 1,2,3 for all the valuesof R and check how many unique µ estimates we get. Results are summarized in Figure 3.5.In Figure 3.5a, the data points (yellow stars) lie on a circle of radius R = .9 with dotted elipsesrepresenting the standard deviation of possition for that localization. We see they are well separated.The same subfigure shows the MLE positional estimates for three mixtures (red dots) and observethat they align fairly well with the data points. We also plotted the MLE for 1 and 2 mixtures,(N = 1,2), for illustrative purposes.As described above, we repeated the estimation for different values of the radius R. Figure 3.5bshows the value of the log-likelihood at the MLE for the different models (with one, two or threemixtures). We observe that for larger values of R, when the data points are far apart, the model withthree mixtures has the larger log-likelihod values than one or two mixtures. But as R decreases,the log-likelihod values collapse. Moreover, Figure 3.5c shows the number of different estimatesobtained when fitting the two and three mixture models. The trend is observed here again: for datapoints far apart (larger R), each model has the same number of mixtures and unique estimators; butas the data points get closer to each other (smaller R) the estimators collapse to only one. In otherwords, the multiple mixtures have the same estimated mean.The spatial model alone cannot reliably differentiate between the data points when the fluo-rophores are close together. To improve the fitting, we will create a protocol to include the temporalinformation.73(a)-1 0 1x-1-0.500.511.52y(b)(c)Figure 3.5: Analysis of the number of unique estimated mixtures for the dSTORM spatial model. Weshow the data in yellow, and results from one mixture in green, two mixtures in light blue andthree in red. (a) Example of a data set with its corresponding mean estimates for three differentmixtures. The three data points (yellow stars) are at equal distance from the origin, on a circleof radius R = 0.9. x1 = (.9,0),x2 = (−.9,0),x3 = (0, .9). The yellow dashed lines represent thestandard deviation of the localization. (b) Log-likelihoods at the MLE of the GMM with one(green squares), two (light blue pentagrams), or three (red diamonts) mixtures, and at the truepositions (yellow stars) when we vary the distance between the three data points. The distancebetween the points is controlled by changing the radius R from 0.1 to 1 in steps of 0.1 units. (c)Number of unique mean estimates when changing the distance between the three data points forone, two and three mixtures.743.4 dSTORM spatiotemporal modeldSTORM data consists of localizations and times of observation of single molecules (Figure 1.3).We have developed models to represent each part, spatial and temporal, to identify unique fluo-rophores (Figure 2.4 and Figure 3.4). Each model has its deficiencies where the identification isimpossible but by combining the data, we should be able to improve our ability to reconstruct thetrue numbers and positions of fluorophores. In this section, we describe how to couple our temporaldiscrete model from Chapter 2 and the GMM from Section 3.3 to better estimate the number ofunique fluorophores present in dSTORM data.The key assumption in our coupling is that the fluorophores being imaged in dSTORM are notmoving. A fluorophore localization is recorded when visible (Active), but it does not move wheninactive. The error in each spatial observation, Σn, results from the light diffraction and photonemission, not from changes in the sample. The activation of the fluorophore is assumed to beuniformly at random on the whole sample. Thus, where you see a flurophore does not change bywhen you see it, and vice versa. We assume that the fluorophore localization and its activation timeare independent. Thus, if we knew the number of fluorophores, N, we could estimate the temporalkinetics with the temporal model independently of the most likely positions with our GMM.We use this assumption to fit the temporal and spatial models iteratively and find the best esti-mates of the parameters of the combined model. We also include the data pre-processing methoddefined in Section 3.1. Our protocol will be explained in the next sections with the complete al-gorithm summarized in Section 3.4.4. With this protocol, we will find the most likely number ofunique fluorophores in the sample when accounting for the spatial and temporal information.3.4.1 Likelihood functionSince we assume the temporal and spatial data to be independent, the likelihood of the spatiotem-poral model is given by the product of the temporal and spatial models, as illustrated in Figure 3.6.Thus, the likelihood and NLL functions of the full model are given byL (N,µ ,Π, pd , pb, pr|X ,ΣX ,T ) = PT (T |N, pd , pb, pr)× pS(X ,ΣX |N,µ ,Π),NLL(N,µ ,Π, pd , pb, pr|X ,ΣX ,T ) =− log(PT (T |N, pd , pb, pr))− log(pS(X ,ΣX |N,µ ,Π)) (3.19)= NLLT (N, pd , pb, pr|T )+NLLS(N,µ ,Π|X ,ΣX).Here, pS describes the spatial model (Equation 3.14) and PT the discrete temporal part (Equa-tion 2.29). X and ΣX represent the spatial localizations, and T the activation times. The num-ber of spatial localizations and activations are the same since the only way to localize the fluo-rophores is when they are active. Let K be the total number of localizations, thus X = {x1, ...,xK},ΣX = {Σ1, ...,ΣK}, and T = {t1, ..., tK}.75likelihoodModel1Model2Figure 3.6: Schematic of the likelihood of two independent models. Assuming that one model followsthe red distribution and one follows the blue distribution, the most likely value of the parameterΘ is given by the green dot.Furthermore, we can calculate the likelihood and NLL functions for M multiple samples bymultiplying or adding the respective function at each sample, similar to Equation 2.31. We assumeall the samples follow the same temporal dynamics, but we let the localizations vary from sample tosample. Thus, we will get estimates for µ i,Πi for each sample, but single estimates for pd , pb, andpr over all samples.N is the only parameter that both models share for each sample. Moreover, the estimates of thetemporal model take into account the dynamics of all the samples while the spatial model estimationis done at each sample individually. We also know that the spatial separation was partially analyzedwhen reconstructing the dSTORM image, whereas the temporal resolution has not. Thus, we trustour estimates of the temporal model to be more informative and we used them to restrict the domainof N for the spatial model. This is illustrated in Figure 3.7, where the value of the likelihood ofΘ is replaced by zero outside the confidence interval for the temporal model. The blue dotted linerepresents the original likelihood function and it is replaced with the likelihood approximation (solidblue line).3.4.2 Constraining the domain of N based on the temporal modelWhen we know the transition probabilities (pr, pb, pd), the temporal likelihood depends only onN. In that case, we can use the likelihood-ratio test to define a confidence interval over N. Thelikelihood-ratio test is a hypothesis test used to compare the MLE, θˆ , with other parameter values.The test compares the ratio between the likelihood at the testing value, θ0, versus at the MLE, and76likelihoodModel1Model2Model1 approxFigure 3.7: Schematic of the likelihood of two independent models when approximating one withinits confidence interval. The distribution of Model 1 is the dotted blue line, and its approximationis the blue line. The distribution of Model 2 is the red line, as in Figure 3.6. The most likely valueof the parameter Θ will lie inside the confidence interval of Model 1.rejects the null hypothesis if the ratio is smaller than p.L (θ0|x)supθ 6=θ0L (θ |x)=L (θ0|x)L (θˆ |x) ≤ pWe can transform the likelihood-ratio into a difference of the negative log-likelihood. Then, thelikelihood-ratio test reduces to calculate the difference between NLL(θ0) and NLL(θˆ). We reject θ0if the difference is greater than c =− log(p).∆(θ0) = NLL(θ0)−NLL(θˆ)≥− log(p) = cThus, we cannot reject θ0 if ∆(θ0)< c. We define our confidence interval by the union of all those θ0values that are not rejected. Usually for one parameter, c= 3 is related to a 95% confidence and thatwill be our default value. For our temporal model, we will not reject any number of fluorophores nsuch that ∆(n)< c when compared to the estimate for N.In reality, we are also estimating the transition probabilities (pˆr, pˆb, pˆd) by using all the infor-mation of the multiple samples whereas the value of Ni depends only on each sub-sample (Algo-rithm 2.3). As a first approximation to the likelihood, we assume that those probabilities are fixedand profile the likelihood over Ni to find a confidence interval for the number of fluorophores for77each sub-sample i. We then define temporal likelihood approximation to beNLLTa (n|T, pˆd , pˆb, pˆr) =NLLT (n, pˆd , pˆb, pˆr|T ) n ∈ confidence interval,∞ otherwise. (3.20)This approximation does not capture the variability over the transition probabilities, which affectsthe likelihood of N. In future work, we could include such variability. Algorithm 3.5 summarizesthe construction of the confidence interval.Algorithm 3.5 Confidence interval of N for the temporal model with fixed (pˆr, pˆb, pˆd)Require: M samples, {Tm}Mm=1 time series, NMLE, Θˆ= (pˆr, pˆb, pˆd), c treshold (default c = 3).Ensure: Previously fitted data using Algorithm 2.3.1: function TEMPORAL Nˆ DOMAIN(M,{Tm}Mm=1,NMLE,Θˆ,c)2: for i ∈ {1, ..,M} do . Compute in parallel over M3: for j ∈ {Amax(i), ...,Nloc(i)} do4: ∆ j = NLLT(j,Θˆ|Ti)−NLLT (NMLE(i),Θˆ|Ti) . NLLT as in Equation 3.195: a = minj(∆ j < c) , b = maxj(∆ j < c)6: CIN(i) = [a,b]return CIN3.4.3 An algorithm to correct the spatial model for over-fittingThere is a problem with finding the number of fluorophores by varying the number of fluorophoresin Algorithm 3.4. Since the number of mixtures represents the number of fluorophores, we have avariable number of mixtures and we could be over-fitting the data. The extra degrees of freedomfrom more mixtures will allow the likelihood to increase, but this is not always viable and couldlead to over-fitting. We did not have to worry about this with the temporal model since the numberof parameters, and thus degrees of freedom, do not change. Moreover, if we compare the value ofthe likelihood at the optimal (µ i,Πi) among all possible values of mixtures Ni, we would comparedifferent models since the number of mixtures changes in the GMM.Selecting the correct number of mixtures appropriatly for a dataset is a general problem withmixture models [19, 70]. Efforts have been made to solve this problem through Bayesian methods,including variational inference [19]. We decided to apply a straightforward method and use theAkaike information criterion (AIC) and the AIC weights [2, 31].The Akaike information criterion (AIC) is commonly used for model selection. It is one ofthe model selection criterion based on information theory. It tries to balance the fitting and themodel complexity by penalizing the likelihood function with a factor linked to the number of pa-rameters (Equation 3.21). Among a pool of models, the best model will have the smallest value of78Equation 3.21, where θˆ stands for the MLE of each model.AIC = 2p−2log(L(θˆ)). (3.21)We will include the AIC of the spatial model. For this, we will find the parameters µˆn and Πˆnthat minimize NLLS(n|X) for each possible number of fluorophores n and then compare the AICamong them. The spatial model has n+(n−1) = 2n−1 parameters: n means and n−1 proportions.Thus, the AICS(n) is given byAICS(n) = 2NLLS(n, µˆn ,Πˆn |X ,ΣX)+2(2n−1).To estimate how much better the model with minimum AIC is compared to other possible mod-els, we use AIC differences and AIC weights. AIC differences are defined as the difference in AICvalue of the test model minus the minimum possible value corresponding to the best model, AIC∗S,∆AIC(n) = AICS(n)−AIC∗S.Usually, any model with an AIC difference of more than 10 is considered to have essentially noempirical support compared to the one with minimum AIC [31].To relatively scale all the possible models, we used AIC weights. The AIC weights representthe probability of the model being the true model if we assume that the true model is among them[31]. They are given bywn =exp(−∆AIC(n)2)∑i exp(−∆AIC(i)2) .Moreover, we can include prior information by multiplying the weights by the probability that modeln is the true model, τn [31]. This leads to the generalized AIC weights, given bywn =τn exp(−∆AIC(n)2)∑i τi exp(−∆AIC(i)2) . (3.22)This is not a true Bayesian method since we are not iterating between prior and posterior distributionbut it allows us to incorporate information known about each model. In the case of our dSTORMmodel, we can include the probability of the temporal model into the selection of the spatial model.There are a few disadvantages of using AIC in our algorithm. Theoretically, AIC is not justifiedfor mixture models (model redundacy when mixtures collapses, MLE at parameter boundaries, etc.)[31, 70]. Moreover, we lose the uncertainty on the estimated number of mixtures (fluorophores) for79other inferences, like those over the temporal parameters [70]. In other words, we will be able toget inferences on the temporal parameters for a given number of fluorophores but such inferencescould be different when we change N.Any information criterion that compares among models will have this inference problem wherethe uncertainty over N is lost. Therefore, we chose to use AIC given its simplicity, rather than otheroptions such as the Bayesian information criterion or the Watanabe-Akaike information criterion,that might be more theoretically justified for the problem.3.4.4 Parameter estimation algorithm for the spatiotemporal dSTORM modelTo estimate our parameters, we use a divide-and-conquer strategy. There are six main steps on ourapproach:1. Pre-process using Stormgraph and the constraint over pr as in Section 3.1.2. Fit the pre-processed samples with Algorithm 2.3 to obtain (pˆd , pˆb, pˆr).3. Compute confidence interval CIi for each Ni using Algorithm 3.5.4. Calculate the NLLS for all possible values of ni ∈CIi in parellel using Algorithm 3.4.5. Calculate the AIC weights wn as in Equation 3.22 by defining τn the probability of n fluo-rophores given the data clusterd as the GMM and the temporal parameters (pˆd , pˆb, pˆr).6. Select the best N values by comparing the wn.This protocol assumes that the spatial separation was partially analyzed when reconstructing thedSTORM image, whereas the temporal resolution has not. Thus, we believe that the temporal modelwill provide better information about the true value of N. The spatial fit will be used to distinguishamong those values of N, refining the output of the temporal model (steps 1-4).Notice that steps 4 and 5 can be calculated independently for each sample since the localiza-tion estimations of one sample are not related to the rest of the samples and (pˆd , pˆb, pˆr) are fixed.Moreover, we can use the labels, ln, of the clusters obtained with the GMM in step 4 to separate thesample n into clusters of one flurophore and calculate τn as folowsτn =n∏i=1P(X ∈ li|N = 1, pˆd , pˆb, pˆr).Equivalently, we can transform τn using the negative of the logarithm and relate it to the NLLT− log(τn) =n∑i=1NLLT (1|X ∈ li, pˆd , pˆb, pˆr) =n∑i=1NLLT (1|X ∈ li,Θˆ).80Thus, Equation 3.22 becomeswn =exp(−∆AIC(n)2 −∑ni=1 NLLT (1|X ∈ li,Θˆ))∑i exp(−∆AIC(i)2 −∑ij=1 NLLT (1|X ∈ l j,Θˆ)) . (3.23)This last expresion might look really complicated, but we can easily calculate each part and it isnumerically stable. This is writen as pseudocode in Algorithm 3.6.Algorithm 3.6 Estimating the number of unique fluorophores with the spatiotemporal dSTORMRequire: M samples, {Xm,ΣXm ,Tm}Mm=1 localization, errors, and time series of each sample, Nlocvector with localizations per sample, tolerance error for EM, c treshold for temporal CI.Ensure: Data formated as required by Algorithm 2.3, and Algorithm 3.4.1: function DSTORMNEST(M,{Xm,ΣXm ,Tm}Mm=1,Nloc, tolerr,c)2: function MIN-NLL(M,T1, ...TM,Nloc) . Algorithm 2.33: MLE: Θˆ= (pˆd , pˆb, pˆr), and Nˆm,m ∈ 1, ...,M4: for m ∈ {1, ..,M} do . Compute in parallel over M5: function TEMP Nˆ DOMAIN(M,{Tm, Nˆm}Mm=1,Θˆ,c) . Algorithm 3.56: New [Nmin(m),Nmax(m)] domain.7: for j ∈ {Nmin(m), ...,Nmax(m)} do8: (NLLS(m, j), µˆ (m, j),Πˆ(m, j), l(m, j)) = EM(Xm,ΣXm , tolerr) . Algorithm 3.49: AIC( j) = 2∗ (NLLS(m, j))+2(2 j−1)10: Re-format data following labels l j.11: Compute wm, j . Equation 3.2312: Nm(Θˆ) = argmaxj(wm, j)return Θˆ, Nˆm = Nm(Θˆ), µˆm = µˆ (m,Nm(Θˆ)),Πˆm = Πˆ(m,Nm(Θˆ)), lm = l(m,Nm(Θˆ)).3.5 ResultsIn this section, we test the spatiotemporal model fitting using simulated data. We extend the discrete-time simulated data from Chapter 2 by adding spatial localization information. For all our simulateddatasets, we estimate the spatial localizations while fixing the number of fluorophores in order totest our spatial procedure (Algorithm 3.4). Next, we fit the complete data, localizations and timeseries, from several datasets using our spatiotemporal model. Finally, we compare the output of thediscrete-time algorithm from Chapter 2 with the results in this section. All the algorithms used inthis chapter were implemented using MATLAB.81Figure 3.8: Spatial variances simulated from a Gamma distribution with parameters a = 2.1 and b =11.8nm2. The left panel shows the probability density function of a Gamma(2.1,11.8) in red andsimulations drawn from it in blue bars. The right panel shows the square root of the simulatedvariances.3.5.1 Simulating spatial dataThe number of localizations to simulate depends on the temporal data. We can only observe fluo-rophores when they are active. In Chapter 2, we simulated the temporal dynamics for six differentsets of temporal parameters with different numbers of activations per fluorophore (Table 2.2 and2.3). We recorded the activation observations with labels differentiating each active fluorophore.We now simulate a spatial localization at each activation for each fluorophore across all parametersets (S1-S6).All the fluorophores in a sample are independent from each other and we simulate them sep-arately. We assume each fluorophore has a fixed location µ , but that the spatial error may varybetween activations. The squared spatial errors are assumed independent and identically distributedfollowing a Gamma distribution with parameters a = 2.1 and b = 11.8nm2 (Figure 3.8).Gamma(x|a,b) = 1baΓ(a)xa−1e−x/b. (3.24)We selected the parameters of the Gamma distribution to mimic the observed errors from realdSTORM data. A localization is thus drawn from a Gaussian distribution N(µ ,Σ), where Σ isdiagonal with entries from the Gamma distribution.As in Chapter 2, we first simulated 10,000 samples with one fluorophore (N = 1) for eachparameter set (S1-S6). To review the differences in dynamics between the parameter sets, refer toSection 2.8.1 and Figure 2.5. Recall that Nloc(i) is the number of activations in sample i, thus forsamples with one fluorophore the number of activations is the same as the number of spatial points,8242 nm42 nmS142 nm42 nmS242 nm42 nmS342 nm42 nmS442 nm42 nmS542 nm42 nmS6Figure 3.9: Simulated localizations of one sample with a single fluorophore for the different parametersets. The white star represents the true position of the fluorophore and the yellow dots are thelocalizations recorded at each activation of the fluorophore. On average, parameter set S1 willhave more activations than S2, and so on up to S6. All the examples have the same scale, 42by 42 nanometers. All the parameter sets follow the same Gamma distribution of spatial errors.Spatial errors are not shown.K = Nloc(i). Figure 3.9 shows one sample from each parameter set.We also simulated the spatial localizations of twenty fluorophores, N = 20, for all parametersets. We maintain the same Gamma distribution for the spatial error but we changed the spacebetween the twenty fluorophores. For each sample, we select the fluorophore true positions equallyseparated in a rectangular grid of 4 by 5 fluorophores inside rectangular domains of side 200nm,300nm, 500nm and 1000nm (see Figure 3.10). The domain sizes were selected so that at 200nmmost fluorophores overlap, while at 300nm and 500nm the fluorophores begin to separate, and at1000nm the fluorophores are very well separated spatially. We simulated the spatial data at eachactivation from a normal distribution as explained before. Figure 3.10 shows one sample fromparameter set S3, simulated at different grid sizes.3.5.2 Pre-pocessing the samples for fittingBefore estimating N, we pre-processed the data as described in Section 3.1. We used the spatiallocalizations and uncertainties to cluster and sub-sample the data, and the last time of activation83167 nm200 nm200 nm box250 nm300 nm300 nm box417 nm500 nm500 nm box834 nm1000 nm1000 nm boxFigure 3.10: Simulated localizations of one sample with twenty fluorophore with temporal parame-ters S3 and different spatial separation. The yellow dots are the localizations recorded at eachactivation of the fluorophores. Spatial errors are drawn from the same Gamma distribution andare not shown.to constrain the estimates of pr. By clustering the data, we accounted for the spatial informationpresent in the data but we also increased the degrees of freedom of the fitting by adding an inde-pendent value of N per new sample (each cluster). By constraining pr, we controlled the degrees offreedom over N gained with the sub-sampling.We used Stormgraph to cluster the data sets. For all simulated sets, we used the Stormgraph pa-rameters as described in Table 3.1. Stormgraph should be used allowing for small clusters, withoutrejecting multiple blinking and including the spatial errors. Hence, we selected the values in the firstfour rows of the table as described in Section 3.1.1. For these simulations, we also set the parameterα to 1, since we know that the simulations do not include false observations of background noise.The value of α might change for real data sets where noise is relevant. The rest of the Stormgraphparameters were set to default values.As explained in Section 3.1.3, we used the last time to activation to constrain pr. We separatedall the samples in which the last activation lasted only one frame and had one observation. Thetime to last activation of these samples will follow a geometric distribution with parameter pr (Sec-tion 3.1.3). Thus, we used these data and the MATLAB function mle for the geometric distributionto calculated the constraint interval for pr (Algorithm 3.1). We will use the pr estimate found hereas our initial condition for the temporal fitting and the corresponding 95% confidence interval as theinterval where pr should be optimized.Table 3.2 summarizes the pre-processing results of the simulated data with one fluorophore,N = 1. We pre-processed 10,000 samples and none of them were separated into sub-samples (therewas only one cluster per sample). This shows that Stormgraph could keep all the localizations ofeach fluorophore together. We also found that every constraint interval for pr contained its truevalue (Table 2.2: pr = 0.1; Table 3.2’s fourth column). This was true even for the parameter set S3,where only about half of the samples were used in the estimation (fifth column of Table 3.2).When N = 20, the clustering depended on the spatial distribution of the fluorophores (Table 3.3).84Parameter Value CommentsMinCluSize 2 To allow as few as two blinks per fluorophore.multiple blinking false To keep data with only one fluorophore.use localization errors true To include the spatial errors in the Stormgraph analysis.p 0.05 Conservative choise to allow “spatial overlapping”.α 1 By construction, we test noise free data in this section.Table 3.1: Stormgraph parameters used to cluster the simulated data for all parameter sets and valuesof N (N = 1,20). Stormgraph parameters not listed here were set at default values.N = 1Stormgraph prclusters MLE CI samplesS1 10,000 0.1024 (0.096, 0.109) 8,414S2 10,000 0.0986 (0.093, 0.104) 9,011S3 10,000 0.1025 (0.094, 0.111) 5,080S4 10,000 0.0988 (0.093, 0.104) 10,000S5 10,000 0.0994 (0.092, 0.107) 6,005S6 10,000 0.0991 (0.093, 0.105) 8,980Table 3.2: Summary of the pre-processing analysis when N = 1 for all parameter sets. We clusteredand calculate the pr constrain of 10,000 samples for each parameter set. Stormgraph found onecluster per sample always, which matches the data (N = 1, second column). We calculated theMLE of pr from the last time to activation (third column) and the corresponding 95% confidenceinterval (fourth column) to constraint our temporal fitting as explained in Algorithm 3.1. Thevalues were obtained using the function mle in MATLAB for the geometric distribution and thosesamples that satisfy the assumptions in Section 3.1.3 (fifth column).We pre-processed 1,000 simulations with 20 fluorophores for grid sizes 200nm, 300nm, 500nm and1,000nm and all temporal parameter sets S1-S6. Fluorophores from S4-S6 activate twice on averagemeaning that their possible clusters are too small, and thus, we pre-processed the 1,000nm grid onlyfor S1-S3. Figure 3.11 shows the clustering result from Stormgraph for the same sample but atdifferent grid sizes. In that figure, we plotted each cluster with a different colour. Cluster numberzero (corresponding to white dots in Figure 3.11) represents the points which Stormgraph foundisolated, and thus each point represents a sub-sample. These isolated points happen more often forparameter sets S4-S6 since those fluorophores blink fewer times.In Figure 3.11, there is only one cluster when the spatial separation is small (200nm grid), andthere are twenty when the spatial separation is greater (1,000nm grid). This trend was obviousby looking at the number of Stormgraph clusters (column 3 in Table 3.3). For S1-S3, there arearound 1,000 clusters (same as samples) in 200nm grids, while there are 20,000 clusters (same asfluorophores) in 1,000nm grid samples. In a 200nm grid the fluorophores are usually not separatedspatially while in the 1,000nm they are. In other words, pre-processing 1,000nm grids resulted insingle fluorophore samples and thus we will focus on 200, 300 and 500 nanometer grids.With N = 20, Stormgraph clustering was not perfect and split a small number of fluorophores85167 nm200 nm200 nm box250 nm300 nm300 nm box417 nm500 nm500 nm box834 nm1000 nm1000 nm box0123456789101112131415161718Figure 3.11: Example of clustering pre-processing of the sample in Figure 3.10. Each colour repre-sents a different cluster. Cluster number zero represents those points that Stormgraph identifyas isolated. There is a single cluster for 200nm grid, 8 clusters for 300nm grid, 15 clusters and2 isolated points for 500nm grid (total of 16 sub-samples), and 18 clusters and 2 isolated pointsfor 1,000nm grid (total of 20 sub-samples). Spatial errors are not shown.86N = 20 GridStormgraph prclusters split MLE CI samplesS1200 nm 1021 1 0.1023 (0.0959, 0.1087) 872300 nm 5316 0 0.1003 (0.0975, 0.1030) 4564500 nm 18949 1 0.1008 (0.0993, 0.1023) 160741000 nm 20000 0 0.1005 (0.0990, 0.1019) 16954S2200 nm 1130 2 0.0982 (0.0924, 0.1040) 992300 nm 7516 2 0.1006 (0.0983, 0.1029) 6783500 nm 19550 0 0.1003 (0.0989, 0.1017) 175361000 nm 20000 0 0.1005 (0.0991, 0.1019) 17940S3200 nm 1109 2 0.0982 (0.0924, 0.1040) 556300 nm 7673 1 0.1006 (0.0983, 0.1029) 4089500 nm 19613 0 0.1003 (0.0989, 0.1017) 97441000 nm 20000 0 0.1005 (0.0991, 0.1019) 9911S4200 nm 2943 4 0.0999 (0.0964, 0.1034) 2796300 nm 9761 1 0.1003 (0.0984, 0.1022) 9334500 nm 12064 0 0.1000 (0.0983, 0.1017) 12008S5200 nm 2797 4 0.1097 (0.1048, 0.1146) 1688300 nm 9589 0 0.1060 (0.1034, 0.1085) 5778500 nm 11786 0 0.1003 (0.0981, 0.1025) 6928S6200 nm 2522 0 0.1024 (0.0982, 0.1066) 2064300 nm 3649 0 0.1053 (0.1018, 0.1088) 3092500 nm 4190 0 0.1018 (0.0986, 0.1049) 3557Table 3.3: Summary of the pre-processing analysis when N = 20 for all parameter sets and differentgrid sizes. We clustered and calculate the pr constrain of 1,000 samples for each parameter setand for 200, 300 and 500 nanometer grids. The 1,000 nm grid was pre-processed only for S1-S3. Parameter sets S4-S6 have few localizations and thus are not clustered in the 1,000nm grid.Stormgraph split a few fluorophores when clustering, with most splits happening for parametersets S4 and S5 on the 200 nm grid. The MLE for pr and its confidence interval (columns 5 to 7)was calculated as in Table 3.2.into multiple clusters (column 4 in Table 3.3). Most of these errors happen on 200nm samples andwere caused by the spatial proximity between clusters. There were 4 or fewer fluorophores splitamong 20,000 and we consider this an acceptable error in the analysis.Regarding our constraint over pr, most grid samples contained the true value of pr = 0.1 withinthe constraint interval for pr (columns 5-6 in Table 3.3). This was false for the S5 set with a 200nmrectangular grid and 300nm grids, and for S6 in the 200nm grid. This was not surprising since S5and S6 sets provide less information than the rest of the parameter sets (fewer blinks).3.5.3 Estimating the fluorophores localization when N is knownOur approach fits the spatial localization and uncertainties to a GMM to estimate the true positionsof the fluorophores. In this section, we test the GMM fit when we know the number of fluorophores8742 nm42 nmS142 nm42 nmS242 nm42 nmS342 nm42 nmS442 nm42 nmS542 nm42 nmS6Figure 3.12: Estimated localization of one sample with a single fluorophore using the spatialdSTORM model for the different parameter sets. These samples are the same as those in Fig-ure 3.9. Again, the white star represents the true position of the fluorophore and the yellowdots are the localizations recorded at each activation of the fluorophore. The estimated positionobtained from Algorithm 3.4 with one mixture is represented with a blue open circle. Spatialerrors are not shown.(N = 1 and the pre-processed samples of N = 20).We fitted 10,000 samples from each parameter set using Algorithm 3.4, assuming each samplehas one fluorophore. Figure 3.12 shows the data from Figure 3.9 and the estimate localization ofthe fluorophore (N = 1). We compared the estimated position, µˆ , and the true position, µ , using theEuclidean distance between them (Figure 3.13). On average, all the localizations were from 1.68nm(S1) to 5.5nm (S6) away from the real position. The estimated localizations were less than 10nmaway from the true localization (excluding outliers) for parameters sets S1-S3, and less than 17nmfor S4-S6. These results show the influence on the number of localizations per fluorophore on theposition estimation. The number of activations is smaller in parameters sets S4-S6 (≤ 2 activationsper fluorophore on average), compared with samples from S1-S3 (≥ 5 on average). Thus, theestimate variability increased in samples with less information (S4-S6).Notice that Algorithm 3.4 depends on the initial condition µ 0. To ensure convergence, we repeatthe fit 50 times by selecting µ 0 at random from the spatial observation. We also use the k-meansalgorithm to inform the initial condition, µ 0. Then, we compared the quality of the estimation for88S1 S2 S3 S4 S5 S6010203040spatial error (nm)Figure 3.13: Distribution of the Euclidean distance between estimated and true positions for a singlefluorophore for all parameter sets. Box plots were taken over 10,000 simulations.200nm 300nm 500nmS1 3.18 (0.15, 16.63) 2.66 (0.14, 15.87) 2.52 (0.15, 11.97)S2 3.80 (0.22, 17.37) 3.28 (0.20, 17.18) 3.14 (0.20, 13.50)S3 3.83 ( 0.21, 17.52) 3.29 (0.20, 17.16) 3.16 (0.18, 14.00)S4 5.31 (0.39, 17.23) 5.09 (0.32,19.28) 5.03 (0.33,17.82)S5 5.30 (0.32, 17.23) 5.06 (0.35, 18.54) 5.00 (0.31, 18.14)S5 5.72 (0.37, 17.07) 5.72 (0.41, 18.91) 5.74 (0.39, 19.10)Table 3.4: Mean and 99% intervals of the Euclidean distances between the true localizations and thecorresponding closest estimated position for pre-processed samples from N = 20, all parametersets and all grid sizes. The .5 and 99.5 percentiles are in parentheses. Calculations are done overthe number of clusters given in the third column of Table 3.3.all of the 51 different initial conditions and selected the best one.We also fitted the spatial localizations of those samples with twenty fluorophores, N = 20, pre-processed in the previous section. As mentioned before, we did not analyze the 1,000 nm grid sinceit is equivalent to analyze samples with N = 1 (done above). Figure 3.14 illustrates the results forone sample on different grid sizes and from parameter sets S1, S3 and S5. Table 3.4 summarizesthe Euclidean distance between the real positions and their closest estimated position. We enlist themean, .5 and 99.5 percentiles of the distances to describe the differences for the different parametersets and spatial separations (Table 3.4). The error distances decreased as the spatial separation waslarger for parameter sets S1-S3, but not by much. For S4-S6, the distribution did not change bychanging the spatial separation. This agrees with the fact that we have less information in thosesamples. Notice that the error distributions for all the parameter samples in Table 3.4 are similar tothose with a single fluorophore (Figure 3.13). In other words, our algorithm is good at estimating89(a) S1(b) S3(c) S5Figure 3.14: Example of data and estimated positions of twenty fluorophores using the spatialdSTORM model for all grid sizes and parameter sets (a) S1, (b) S3, and (c) S5. The yellow starrepresents the real position of the fluorophore and the red open circle is the estimated positionobtained from Algorithm 3.4. The localizations recorded at each activation are semitransparentdots colour coded according to their Stormgraph clustering. Note that each grid size has itsown scale. All the parameter sets follow the same Gamma distribution of spatial errors. Spatialerrors are not shown.90Parameter S1 S2 S3 S4 S5 S6pr 0.1 0.1 0.1 0.1 0.1 0.1pˆr 0.098 0.093 0.095 0.093 0.092 0.093pd 0.8 0.8 0.4 0.5 0.1 0.1pˆd 0.798 0.789 0.376 0.151 0.0001 0pb 0.05 0.1 0.1 0.5 0.5 0.8pˆb 0.057 0.123 0.132 0.849 0.604 1.0Table 3.5: Estimated temporal parameter values when fitting the spatiotemporal model for one fluo-rophore. The rows show pr, pd and pb true values (as in Table 2.2) and estimates (identified witha .ˆ).the positions of the fluorophores regardless of the number of them present in the sample. Theseresults confirm the convergence of our algorithm for multiple fluorophores in the same sample.3.5.4 Fitting simulated data to the full spatiotemporal modelWe tested the ability of our spatiotemporal model to estimate the number of fluorophores usingAlgorithm 3.6. We fitted 10,000 samples with one fluorophore simulated and pre-processed asin the previous section. The number of fluorophores was allowed to vary from Amax to the totalnumber of activation Nloc for each sample while fitting. Figure 3.15 shows the estimated numberof fluorophores and Table 3.5 shows the estimated temporal parameter values for all parametersets. The distribution of the estimated number of fluorophores for S1-S4 improved compared to theestimations with only the temporal model (Figure 3.15 versus Figure 2.12). Parameter sets S1-S4had at most 4 estimated fluorophores, while their estimates from the temporal model were largerthan 4. In particular, the estimates for the parameter set S3 and S4 improved substantially by addingthe spatial information (Figure 3.16). On the other hand, parameter sets S5 and S6 showed no muchdifference in the estimated number of fluorophores by adding the spatial information (Figure 3.16).This resulted from the smaller temporal information (fewer blinks) in these sets.We also fitted samples with twenty fluorophores for the parameter set S3 simulated and pre-processed in previous sections for grid sizes 200, 300 and 500 nm. There were 1,000 samples foreach grid size, split into clusters as summarized in Table 3.3. To compare the temporal model onlyand the spatiotemporal model at different grid sizes, we added the estimated number of fluorophoresof the clusters coming from each of the 1,000 original samples. Figure 3.17 shows the distributionsof the estimated number of fluorophores out of twenty true fluorophores. Again, we found anincrease in accuracy by adding spatial information to the temporal model. Our estimates rangedfrom 13 to 30 when using only the temporal model, while the range of the spatiotemporal modeldepended on the grid size. The smallest range ([20,26]) was for the 500nm grid size, equivalentto the grid size with the most information. The 200nm and 300nm grids had range of [15,36] and[18,30] respectively. As expected, the estimates for the 200nm grid were less accurate since this91S1 S2 S3 S4 S5 S600.10.20.30.40.50.60.70.80.91probability1234NfitFigure 3.15: Estimated number of fluorophores for 10,000 spatiotemporal simulations of a fluo-rophore for each parameter set S1-S6. Bars are grouped by the different parameter sets (S1-S6).All the samples were estimated to have 1 to 7 distinct fluorophores. colour scale corresponds tothe number of fluorophores estimated Nfit from 1 (dark blue) to at least 4 (yellow).S31 2 3 4Nfit0200040006000800010000frequencyT+STS41 2 3 4Nfit0200040006000800010000frequencyT+STS51 2 3 4  5Nfit0200040006000800010000frequencyT+STFigure 3.16: Comparing the estimated number of fluorophores for 10,000 simulations using the spa-tiotemporal model (blue bars) and the temporal model (yellow bars) for S3, S4, and S5 parame-ter sets. The spatiotemporal model showed a substantial improvement in estimating the numberof fluorophores for the parameter set S3 and S4. Results for parameter set S5 were mostly thesame for both models.92T 200nm 300nm 500nm00.10.20.3probability 1517192123252729 31Nfit16 18 20 22 24 26 28 30Nfit00.050.10.150.20.250.3probability500300200NONGrid size (nm)Figure 3.17: Estimated number of fluorophores of simulated data at different spatial separation usingthe spatiotemporal model and parameter set S3 (N = 20). In the upper panel, the bars aregrouped by grid size. The Nfit = 20 bar has a dark red box. T represents the temporal modelestimates without spatial information. colour scale corresponds to the number of fluorophoresestimated (Nfit). In the lower panel, the bars are grouped by NFit values colour-coded by gridsize: NON (only temporal, red bars), 200nm (yellow bars), 300nm (light blue bars), and 500nm(blue bars). The estimates were taken over 1,000 simulations of the spatiotemporal model.added almost no spatial information to the system.There were some cases where the AIC weights could not be computed because their numeratorvalues were numerically zero. In those cases, we selected N as the value that maximized−∆AIC(n)2−NLLT (n)(see Equation 3.23). The selection is equivalent and avoids numerical problems.933.6 DiscussionWe found our spatiotemporal model improves the estimation of the number of fluorophores for allour samples compared to the temporal model alone. The spatial information was essential whenfluorophores activated a few times (parameter sets S4-S6). While we observed that the temporalmodel alone could not differentiate the number of activations from the real number of fluorophoresin S4 and S5, the spatial information could cluster activations together and improved the estimates.In S6, most fluorophores only activated once, providing so little spatial and temporal informationthat the estimates did not change much.The spatial error played an important part in our fit since it balanced the distances betweenspatial points. If two observations had overlapping errors, the chances of them coming from thesame fluorophore increased. If spatial points were widely separated, the pre-processesing protocoldistinguished them. So, adding the spatial uncertainties refined the temporal and spatial information,thus improved the parameter estimates.We balanced the best information, either temporal or spatial, from the data to improve the es-timates, compared to using only the temporal or spatial model. The models provide disjoint in-formation to the process but they share a common parameter: N. We restricted the domain of thespatial model by using the temporal information and thus estimated N that satisfied both models.This approach could be improved by using multi-model inference instead, such as model averagingto predict the value of N [31]. Those techniques might not be straightforward to implement fornon linear models but they would reduce the bias caused by using the temporal information into thespatial one.Our divide-and-conquer strategy was essential to couple the temporal and spatial models. Itallowed us to weight the spatial estimation considering the temporal dynamics. Using AIC gen-eralized weights compensated for the spatial model over-fitting while enhancing the coupling. Weselected this criterion given its simplicity, but it has not been theoretically justified for these typeof models [70]. We could use other information criterion with a better theoretical foundation, butwe will still lose the uncertainty over N [70]. A more general solution to address over-fitting of thespatial model would be to use Bayesian methods, such as variational methods or Chinese Restaurantprocess models [70].Moreover, our iterative method does not account for the estimated uncertainty of the temporaltransition probabilities. That can be included by extending the profile of the temporal likelihood tothose parameters or by bootstrapping the data. Such methods are likely to computationally intensive.Another solution could be to translate the complete spatiotemporal model to variational method ora non-parametric Bayesian model [70]. That approach would define a new protocol with the sameunderlying modeling of the fluorophores.We assume our fluorophore dynamics to be independent and Markovian. There could be someenergy transfer among nearby fluorophores or the bleaching probability could increase as the fluo-94rophore activates. In that case, our assumption of independence between the spatial and temporaldata would not be accurate. A way to model this would be with a self-exciting spatiotemporal pro-cess [163]. The rates in these types of processes depend on previous observations of the process(non-Markovian). Self-exciting can account for the activation of fluorophores given transfer of en-ergy and the bleaching probability increasing with multiple fluorophore activations. These generalprocesses have not been implemented in this thesis project.95Chapter 4Identifying unique observations in superresolution experimental data with aspatiotemporal modelFeeling unique is no indication of uniqueness— Douglas Coupland.In this Chapter, we test our spatiotemporal STORM model to two experimental data sets arisingfrom DNA origami and B-cell receptors (BCR). The DNA origami data was obtained by directcommunication with Daniel Nino at the University of Toronto and was used in a recent paper onmolecular identification in dSTORM data [149]. The DNA origami experiments are equivalent toa biological control since the experimentalists can control the number of fluorescent molecules thatare imaged and their positions. Thus, succeeding with this data will imply that our model fits wellunder experimental conditions.On the other hand, the BCR data provides a realistic application of our algorithm for true bi-ological data. In this case, we do not know the true number of fluorophores to compare with ourestimation. BCR are receptors in the membrane of B-cells. They recognize antigens and triggerB-cell activation. BCR have been shown to form into spatial clusters following B-cell signaling. Bycorrecting for over-counting of the B-cell data, we expect to better understand the relationship ofthe cluster size with the signaling background. This data is published in [179].We will start by describing the DNA origami experiments in Section 4.1, followed by the resultsobtained by fitting the DNA origami with our spatiotemporal model, in Section 4.2. Then, we willdescribe the BCR data sets in Section 4.3 and present the corresponding fitting results in Section 4.4and Section 4.5.964.1 Experimental control: DNA origamiThe first super-resolution microscopy methods were tested by imaging microtubules, cellular fila-ments, and other biological systems [101, 126, 169]. Those experiments are not easily reproducible[176]. To have a more standard way to test super-resolution microscopy techniques, Steinhauer etal. proposed to use DNA origami to calibrate super-resolution techniques [189].DNA origami was first developed in the 1980s but it was with the developments made by Rothe-mund that it gained popularity as a controllable nanostructure [168]. Rothemund named the methodorigami since he produced 2D nanostructures by “folding” a long single DNA strand and “sta-pling” it with multiple short oligonucleotide strands. The specificity of the base pairing on DNAallows the correct and automatic assembling of the structure as long as all the pieces are interacting[168, 170]. In his 2006 Nature paper, Rothemund illustrated the DNA origami method by creatingdiverse nanostructures, from simple square grids to complex settings like smiles [168]. He showedsome designed patterns (like a world map) on top of DNA origami and suggested using the DNAorigami as a “nanobreadboard” with specific attaching places for particles or materials [168].This last suggestion of Rothemund is what led others to use DNA origami as molecular rulersfor super-resolution microscopy [88, 128, 189]. In the paper introducing dSTORM, Heilemann etal. used double-stranded DNA labeled with Cy5 dyes to test the reversibility of the dyes [88]. TheDNA strands were immobilized and imaged. This was a novel use of DNA nanostructures to testsuper-resolution methods [88, 189].In 2009, Steinhauer et al. showed the use of DNA origami as molecular rulers for several super-resolution techniques [189]. DNA origami structures have, by construction, a fixed size and preciseaddressability, and can be immobilized and fluorescently labeled [161, 168, 189]. Steinhauer etal. constructed DNA origami grids with fluorescent tags at specific locations of the DNA strands,thus controlling the distance between the fluorescent probes [189, 222]. Steinhauer et al. showedthat fluorescently labeled DNA origami have stable imagining characteristics: the intensity andemission of the fluorescence grow linearly with the number of fluorescent dyes [189]. Thus, themethod of Steinhauer et al. provided a mechanism to test the characteristics of super-resolutiontechniques, such as dSTORM, and improvements of this method are nowadays a standard way totest the resolution of new super-resolution fluorescent microscopy protocols [88, 128, 176, 177, 189,209].When designing the DNA origami rulers, we have control over the localization and the numberof fluorescent binding sites [149, 177]. Thus, DNA origami rulers provide an experimental controlfor counting algorithms like ours. Recently, Nino et al. tested their model using DNA origami gridswith a single, or up to four, fluorescent binding sites [149]. We have described their theoreticalmethods previously (Section 2.3, [148]). They estimated the number of molecules and proved thatdifferent experiment settings and localization methods will cause different fluorophore kinetic ratesand molecular counting results [149]. By direct communication with Daniel Nino, we have access973380 nm4560 nmDNA origami020406080100120140160180Figure 4.1: DNA origami data. There are 86 DNA origami on the lattice. Nino et al. pre-processedand organized the DNA origami onto the lattice. Each DNA origami has a trapezoidal shape withone binding side at each corner. Thus, there are between 1 and 4 fluorophores in each latticepoint.to some of their DNA origami data and will use it as an experimental control for our model.4.1.1 DNA origami data descriptionNino et al. constructed DNA origami with a trapezoidal shape measuring: 69nm, 40nm, 60 nm, and40nm along each side [149]. These origamis simulate one molecule with multiple fluorescent labelsby having one binding site for Alexa-647 at each corner (see Supplementary Figure 2 in [149]).Thus, each trapezoid will have at most four dyes, at least 40nm apart from each other. The distancebetween labels allows the measurement of the labeling efficiency directly from the super-resolutionimages After labeling, the trapezoids were distributed at random on a coverslip for imaging. Ninoet al. imaged the sample using a home-built microscope (based on an Olympus, APON 60XOTIRF)and reconstructed using rapidSTORM software [149]. They then processed the data to identifyand differentiate the trapezoidal DNA origamis on the coverslip and arranged them on a lattice.They discarded imaged trapezoids that were not easily identified (due to poor imaging, small spatialseparation, etc [149]), (Figure 4.1).983380 nm4560 nm(a) Stormgraph clusters (b) Number of localizations per clusterFigure 4.2: DNA origami data pre-processed using the mean spatial error (8nm). (a) Stormgraphclustering using ∆x=∆y= 8 for all localizations in the sample. Each colour represents a differentcluster, being a total of 88 sub-samples. Two DNA origami were split into sub-samples (fromtop to bottom: (row 3, column 5), and (row 10, column 7)). (b) Histogram of the number oflocalizations in each cluster from (a).Nino provided us with the data of 86 trapezoidal DNA origami (Figure 4.1). The origamis areprocessed so that they appear on a lattice, but we analyzed them without that information so as totest our pre-processing algorithm. Each DNA origami has one to four dye molecules, so there arebetween 86 and 344 fluorophores in the data.The DNA origami data does not have the spatial uncertanties at each individual point as neededby our algorithm. Nino et al. estimated the spatial uncertanty of the sample using nearest neigh-bor analysis. The mean spatial error they found is 8nm. To implement this information into ourapproach, we used the mean spatial error (8nm) as the uncertanty for all localizations and ran ouranalysis. The estimated number of fluorophores is described in the next section.4.2 DNA origami analysis using the spatiotemporal modelWe fixed the spatial error for all localizations to be 8nm, and pre-processed the data. We set theStormgraph parameters as in Table 3.1 and obtained 88 clusters (Figure 4.2a). Stormgraph keptall the localizations coming from the same DNA origami together, except for two cases. Visualinspection reveled, in the two sub-sampled DNA origami objects, the fluorophores were sufficientlyfar appart to be spatially identified. The number of observations per clusters varied from 4 to 348(Figure 4.2b).99To constrain pr, we used only those samples whose time series of activations ended within thefirst half of the observation period. Remember that our constraint assumes that the samples havebeen bleached, hence this cut-off ensure us to use fluorophores that had truly bleached by the endof the experiment. The resulting 99.9% confidence interval for pr was [0.00004,0.00172]. Thisestimate was calculated with only 12 clusters out of the 88 in the data. Only those 12 clustershave their last localization for exactly one frame within the first half of the experiment total time,assumptions needed to constraint pr (Section 3.1.3). To balance the possible bias arising fromsuch a small sample, we increased the size of the confidence interval when profiling our temporallikelihood over N. The larger the profile-likelihood interval the more uncertanty we add to ourresults from the temporal information. The width of the interval is controlled by parameter c inAlgorithm 3.6. Thus, we will use c = 10 in this chapter. In summary, pr will be constrainted to bewithin [0.00004,0.00172] for all our samples and our profile-likelihood interval will be larger.Figure 4.3 summarizes the results of our analysis. First, Figure 4.3a shows the distribution of thenumber of fluorophores per sample. This distribution looks uniform for 1 to 4 unique fluorophoresand decreases for 5 to 9 fluorophores. 88% of the sample estimates agree with the construction ofthe DNA origami data with 1 to 4 fluorophores, 8% had five estimated fluorophores, and only 5%had over five estimated fluorophores. The last 5% can be explained either by spatial outliers or bythe accuracy of the temporal parameters (Figure 4.3c). This could probably be improved if we hadthe spatial error at each localization (instead of only the mean uncertainty for the sample), and moreDNA origami samples.The ratio between the number of localizations and the estimated number of fluorophores foreach sample is 40 on average (s.d. 30 Figure 4.3b). Thus, there are 40 localizations per estimatedfluorophore on average, i.e., we are observing each fluorophore in the active state for 40 frames onaverage. This does not mean that we have only one blink for 40 frames, it means that the total timea fluorophore is observed is 40 frames. The number of blinks (transitions between on/off states) isrelated to the temporal parameters.As described in previous chapters, our analysis starts by fitting all the samples with the temporalmodel to reduce the possible values N to use in the spatial model (Figure 4.3c). Using all thesamples, we found the estimated value of the temporal transition probabilitiespˆd = 0.092, pˆb = 0.0555, pˆr = 0.0017. (4.1)We observed that pˆr is basically at the upper bound of the pr constraint (less than 10−10 difference).This might result from the small sample used to calculate the pr constraint as explained above,might cause the outliers of the Nfit distribution, and should be improved with access to a largersample set. Using those estimates for the transition probabilities, we decreased the range over Nper sample by using Likelihood ratio test. For each sample, the range of N plotted in Figure 4.3c100(a) Estimated number of fluorophores (b) Localizations per estimated fluorophores(c) Estimated number of fluorophores (crosses) and range of possible values (red lines) per sample0 10 20 30 40 50 60 70 80sample510152025num fluorophoresRangeTNfitFigure 4.3: Estimated number of fluorophores for DNA origami sub-sampled data using our spa-tiotemporal model. (a) Distribution of the estimated number of fluorophores for the DNA origamidata. The distribution looks uniform from 1 to 4. There are 7 samples with 5 estimated fluo-rophores, 2 samples with 6, and one with 8 and 9 estimated fluorophores. (b) Distribution of thenumber of localizations per estimated number of fluorophores. This ratio informs the number ofblinks coming from the same fluorophore (mean 40). (c) Constraint domain of N from the tem-poral model (red lines) and the final estimated number of fluorophores (blue crosses) per sample.The range of the number of fluorophores is the profile-likelihood resulting confidence intervalover the temporal model estimation.1013380 nm4560 nmFigure 4.4: Estimated postion of the fluorophores for the DNA origami sample. The white dotts repre-sent the DNA origami data and the red open circles are the estimated position of the fluorophoresobtain with our spatiotemporal model.contains all possible values of N with their negative log-likelihood value less than c= 10 units fromthe minimum negative log-likelihood of the temporal model.Following Equation 2.24, we know that the expected number of blinks per fluorophore will be1+ pd/pb. Thus, our estimates predict each fluorophore to blink on average 2.7 (s.d. 2.1) timesbefore bleaching. This value agrees with the estimate found by Nino et al. (parameter λ in [149]).In their paper, they report a mean number of blinks per fluorophore of 3.5 (calculated with ≈ 350samples) and showed that it varies from 2.3 to 5 for their bootstrap estimations.Figure 4.4 and Figure 4.5 show the data along with our estimated fluorophore position for the102700 nm700 nmFigure 4.5: Example of the estimated postion of the fluorophores for four DNA origamis. The datacorrespond to samples 72, 33, 11, and 68 from previous figures (ordered from left to right, andup to down). The white dotts represent the DNA origami data and the red open circles are theestimated position of the fluorophores obtain with our spatiotemporal model.whole sample and for four DNA origamis respectably. The data is shown as white filled dots andthe estimates in red open circles. The samples in Figure 4.5 correspond to samples 72, 33, 11, and68 from previous figures in this section (ordered from left to right, and up to down).Sample 11 is one of the samples that have over five estimated fluorophores (Nfit = 8). Thisestimated was carried on from the temporal fit already. The lower bound on the range from thetemporal fit is 6 (Figure 4.3c). Moreover, there are 3 fluorophore estimated positions that overlapwith localizations (left lower cluster in Figure 4.5). We can think of such localizations as outliers,probably caused by a higher spatial uncertainty than the mean. Again, this highlights the importanceof knowing the spatial uncertainties at each localization and of having a larger data set.Overall, the results in this section show that our algorithm can recover the number of fluo-rophores in a sample. Thus, we have tested our spatiotemporal model over simulated data and DNAorigami control data with satisfactory results. Now, we analyze images from B-cell receptors wherethe fluorophore dynamics are a-priori unknown.4.3 Experimental data test: B-cell receptors (BCR)As mentioned in Section 2.2.1, there is a correlation among the spatial aggregation of B-cell re-ceptors (BCR) on the B-cell membrane and B-cell activation, thus relating B-cell receptors (BCR)organization to immune activation [51, 136]. Moreover, Mattila et al. showed that the spatial organi-zation of BCR also correlates with the survival of B-cells [136]. They found small BCR clusteringwas needed to create a small (tonic) signal without which the B-cell enter apoptosis [136]. It iscommon to refer to the small “tonic” clusters as nano-clusters and those clusters necessary to B-cell103activation as micro-clusters[51, 136]. All these correlations were observed using super-resolutionimaging [51, 136].Focusing on the nano-clusters, Mattila et al. observed 30-120 or 20-50 BCR within a radius of60-80nm, (each range corresponding to two different BCR isotypes as explained in Section 2.2.1)[136]. For those quantifications, they used a spatial threshold to correct for multiple observationscoming from the same molecule. The correction for over-counting of fluorophores is crucial inthese small clusters since it could lead to fictitious clustering. To define unique fluorophores, theyexperimentally tested for multiple blinking by changing the labeling conditions [136]. They alsoused threshold techniques and correlation analysis to correct for multiple blinking [4, 136, 180]. Wewill test our algorithm in similar resting B-cell samples.We present our analysis of BCR data published in [179]. Dr. Libin Abraham, who is part of theCoombs lab, collected and proportioned the data. The sample was collected from ex-vivo murinesplenic B-cells [48, 179]. BCR were imaged using dSTORM in a super-resolution microscope atUBC [194, 195]. The single-molecule localization analysis provided us with spatial positions andtheir associated uncertainties. To define the BCR clusters, we used Stormgraph [179]. To avoid biascoming from variations in receptor density at the edge of the imaged cell, we used a rectangularregion of interest inside the imaged cell as described by Scurll et al. [179].4.3.1 Data set descriptionIn [179], Scurll et al. showed the clustering results of 28 ex-vivo B-cells samples in the resting state.All the samples were labeled using Alexa Fluor R© 647 dyes. We selected five samples at random totest our algorithm. We have the 2-D super-resolution coordinates (x1,x2), their corresponding ∆x1and ∆x2 standard deviations of the super-resolution fit, and the frame number of the observation foreach fluorescent observation [179]. There are 21,994, 4,297, 15,347, 9,632 and 42,298 data points(localizations) inside each region of interest of the cells in our sample.We defined the covariance matrix for each spatial observation asΣx =(σ2x1 00 σ2x2)=((∆x1)2 00 (∆x2)2), (4.2)where σ2i is the variance and ∆xi is the observed error in the ith direction. We assume the variance tobe identically distributed in both directions, and thus we treat them as one distribution of variancesσ2 for each cell.Figure 4.6 shows the density plot of a rectangular region of interest inside each cell to analyze.In each density plot, we counted the number of observations inside boxes from a 30nm square gridto simulate a pixel. The colour code defines a density scale different for each cell and varies fromblack to white depending on the number of observations within the pixel. In this figure, we can104(a)B-cell A4052 nm3825 nm020406080(b) B-cell B1462 nm1463 nm051015202530(c) B-cell C2474 nm2137 nm05101520253035(d) B-cell D1576 nm1914 nm0102030405060(e) B-cell E4162 nm4612 nm050100150200Figure 4.6: Density plot of region of interest of five B-cell dSTORM data. The density of the spatialpositions of the observations calculated by aggregating the observations inside 30nm box grids(imitating pixels). colour scale varies with the density, with white/yellow meaning high densityand red/black low density. BCR(a) will be the main example in this section.105α Noise Clusters Nloc/Nfit (pˆd , pˆb, pˆr) E(Nblink)0.05 4,121 335 (Nloc < 400) 7.9 (s.d. 4.1) (0.59, 0.08, 0.0004) 8.8 (s.d. 8.2)0.50 1,715 318 (Nloc < 500) 7.0 (s.d. 3.7) (0.59, 0.07, 0.0004) 8.8 (s.d. 8.3)1.00 101 257 (Nloc < 400) 5.3 (s.d. 3.2) (0.57, 0.09, 0.0004) 7.2 (s.d. 6.7)Table 4.1: B-cell A summary of estimates using the spatiotemporal model and different noise levels(α). For α = 0.005,0.5,1, we find the number of localizations categorized as noise, the number ofclusters analyzed, the estimated number of localizations per fluorophore, the temporal parameterestimates and the estimated number of blinks per fluorophore. We notice that our analysis is robustover the different noise levels.observe the spatial heterogeneity of the BCR on the membrane.We will perform our analysis for each cell independently and then we will analyze the resultingestimates over all the data sets. To exemplify our analysis, we will first show a detailed descriptionof the results for Figure 4.6a. To simplify the notation, we will refer to this cell as B-cell A.4.4 Analysis of B-cell A data setFigure 4.7 summarises the spatial (density and variances of 2-D dSTORM positions) and temporal(same-frame observations) data in B-cell A. This sample has 21,994 localizations. All the observa-tions are inside a rectangular region of 3,825nm by 4,052nm (spatially), and within 40,000 frames(temporally). There are up to eight same-frame observations, with most frames having only onelocalization (Figure 4.7c). Thus, our test data set agrees with the assumption mentioned in previouschapters, the dSTORM data has few observations in each frame.The variance distribution σ2 has a mean value of 25nm2, equivalent to a 5nm spatial error onaverage, (Equation 4.2, Figure 4.7b). We used the gamfit function from MATLAB to estimate theparameters of a gamma distribution to the spatial variances data. The MLE values are a= 2.1 (2.09,2.14) and b= 11.8 (11.6, 11.95) (Equation 3.24) and the corresponding probability density functionis shown in red in Figure 4.7b. We used these parameter values to create spatial synthetic data inSection 3.5.To pre-process the data, we used the same parameters as in Table 3.1 except for parameter α .This parameter controls the amount of data considered outliers (noise) and discards them from theclustering settings. dSTORM data usually have localizations coming from noise in the imagingprocess and we should correct our sample accordingly. In previous analyses, we assumed the datasets were noise-free (in simulated data) or had been previously cleaned (DNA origami data). Forthis data set, we will use different values of α check the robustness of our algorithm. Scurll etal. reported their results using the default α value (α = 0.05, [179]).We tested our algorithm using α = 1,0.5, and 0.05 (Figure 4.8 and Table 4.1). There wereover 4,000 localizations categorized as noise when α = 0.05, contrary to only 101 singletons whenα = 1. By removing the noise points, the Stormgraph clusters were smaller for α = 0.05 since some106(a)(b) (c)Figure 4.7: dSTORM data summary from the resting B-cell A data. (a) The density of the spatialpositions of the observations calculated by aggregating the observations inside 30nm box grids(imitating pixels). colour scale varies with the density, with yellow meaning high density andred low density. (b) Spatial variances calculated by Equation 4.2. The histogram bars representthe data and the red line represents the gamma fit calculated with gamfit MATLAB function(a = 2.1,b = 11.8). (c) Frequency of simultaneous observations. The histogram summaries thenumber of observations at a given frame. Most of the frames have only one observation, and thereis only one frame with eight observations.107(a) α = 0.054052 nm3825 nm(b) α = 0.54052 nm3825 nm(c) α = 14052 nm3825 nm(d)4052 nm3825 nm(e)4052 nm3825 nm(f)4052 nm3825 nm(g) Nloc < 4004052 nm3825 nm(h) Nloc < 5004052 nm3825 nm(i) Nloc < 4004052 nm3825 nmFigure 4.8: Stormgraph analysis for B-cell A with different values of α . (a, d, g) α = 0.05. (b, e,h) α = 0.5. (c, f, i) α = 1. (a), b, (c) Comparing data considered noise (gray) versus not-noise(white) for each value of α . (d, e, f) Data to be analyzed (white) removing large clusters andnoise (gray). We considered as large any cluster with Nloc > 400 for α = 0.05,1, and Nloc > 500for α = 0.5. (g, h, i) Data to be analyzed colour-coded by clusters.108of those points kept localizations clustered. We analyzed all clusters with less than 400 localizations(and one over 400 for the set with α = 0.5). The temporal parameter estimates and the propertiesof the fluorophores were similar for the three distinct values of α (columns 4-6 in Table 4.1). Thus,we will use α = 0.05 for the rest of our B-cell data, as Scurll et al. [179], and we will describe indetail the results for B-cell A and α = 0.05.When α = 0.05, there were 4,121 localizations in the B-cell A data considered as noise byStormgraph. The noise-free set was split into 336 clusters. Most of those clusters had less than400 localization, but one of them with 4,403 localizations. We performed the analysis for the 335clusters with up to 400 localizations. We can extrapolate the information from the smaller clusters tothe large cluster after estimating the parameters following our assumption of identically distributedfluorophores.There were 13,470 localizations in the 335 clusters, and we found 1,534 unique fluophores(Figure 4.9). The mean number of localizations and the mean number of blinks per flurophore areapproximatelly 8 (Figure 4.9c and Table 4.1). Thus, each blink had length 1 frame on average. Themedian number of fluorophore was 3, thus there were many clusters with one fluorophore (Fig-ure 4.9b, 193 clusters with Nfit ≤ 3). Thus, the majority of the clusters have less than 10 estimatedfluorophores (Figure 4.9d). Moreover, given that each BCR could have up to multiple labeling fluo-rophores, this cluster distribution could indicate BCR monomers and dimers instead of nanoclusters.This preliminary result will need to be tested with other experimental replicates, and with a propercluster analysis over the estimated possition of the fluorophores (Figure 4.10). Figure 4.11 showssome examples of the data and the estimated position of the flurophores.Finally, we estimate that the large cluster with 4,403 localizations will have around 550 uniquefluorophores (s.d. 1100).4.5 BCR data analysis using the spatiotemporal modelAfter setting α = 0.05, we analyzed the BCR dSTORM data from five B-cells. The BCR densityinside the region of interest for each B-cell is in Figure 4.6. All the data sets were imaged for 40,000frames. The smallest region of interest of approximately 1,460nm2 has 4,297 localizations (B-cellB), and the largest of 4,162 by 4,612nm has 42,298 (B-cell E). All the data sets had a mean spatialerror of 5nm with similar Gamma fits as the estimated for B-cell A. Similar to B-cell A, most of theframes had only one observation and at most 6 for B-cell E.For all samples, the temporal parameter estimates were similar (Table 4.2). There were around7 to 12 localizations and between 9 to 13 blinks per fluorophore. The distribution of the numberof localizations and the estimated number of fluorophores are in Figure 4.13 and Figure 4.14 re-spectively. In Figure 4.12, we observed that the number of localization per estimated fluorophore isrobust over the five different data sets.109(a) Localizations per sample (b) Estimated number of fluorophores(c) Localizations per estimated fluorophores (d) Estimated number of fluorophores, Nloc ≤ 50Figure 4.9: Estimated number of fluorophores for B-cell A sub-sampled data using our spatiotemporalmodel and α = 0.05. (a) Distribution of the number of localizations per cluster. (b) Distributionof the estimated number of fluorophores for the BCR in B-cell A. (c) Distribution of the numberof localizations per estimated number of fluorophores (mean 7.9, s.d. 4.1). (d) Distribution of theestimated number of fluorophores for the samples with up to 50 localizations (Nloc ≤ 50).1104052 nm3825 nmFigure 4.10: B-cell A data (red dots) with the estimated possition of the fluorophores from our spa-tiotemporal model (yellow dots).B-cell Noise (Nloc) Clusters Nloc/Nfit (s.d) (pˆd , pˆb, pˆr) E(Nblink) (s.d.)A 4,121 (42,298) 335 7.9 ( 4.1) (0.59, 0.08, 0.0004) 8.8 (8.2)B 1,375 (4,297) 50 8.3 (3.8) ( 0.72, 0.06 0.0005) 13.5 (13)C 5,521 (15,347) 143 8.0 (3.4) (0.65, 0.06, 0.0004) 11.0 (10.5)D 2,272 (9,632) 93 11.7 (4.4) (0.52, 0.05, 0.0004) 11.8 (11.3)E 11,420 (42,298) 484 7.3 (2.8) (0.63 0.08, 0.0004) 9.2 (8.7)Table 4.2: Summary of the estimates using the spatiotemporal model for five B-cell data sets. Thenoise data was discarded using Stormgraph parameter α = 0.05. We did our analysis using allStormgraph clusters with up to 400 localizations. Thus, we found the estimated number of local-izations per fluorophore, the temporal parameter estimates and the estimated number of blinks perfluorophore.111383 nm298 nm116 nm81 nm246 nm80 nm91 nm36 nm32 nm19 nm24 nm12 nmFigure 4.11: Examples of the fits B-cell A data (white dots) with the estimated possition of the flu-orophores from our spatiotemporal model (red dots). Sample numbers: 3, 20, 30, 70, 200 and250 (left to right, up to down).1121 2 3 4 5B-cell102030Nloc per NfitFigure 4.12: Comparing the distribution of the number of localizations per estimated number of fluo-rophores for all B-cell data sets. We observe a consistent distribution over the five B-cells.4.5.1 DiscussionOur estimated number of fluorophores is roughly a tenth of the original number of localizations.Thus, the density of BCR nano-clusters might be of the order of 10 instead of the 30-120 and20-50 found previously by Mattila et al. [136]. We cannot argue against or in favour of nano-clusters in tonic signal with such a small sample size. We will need to do a thorough analysisusing our method to correct for over-counting of BCR, including a larger sample size and possiblere-analyzing the data published by Mattila et al. [136]. Still, these preliminary results highlightthe potential importance of correcting over-counting of fluorophores if we wish to achieve accurateconclusions about receptor clustering from super-resolution images.113(a) B-cell B (b) B-cell C(c) B-cell D (d) B-cell EFigure 4.13: Distribution of the number of localizations in all B-cell sub-sampled data using our spa-tiotemporal model and α = 0.05.114(a) B-cell B (b) B-cell C(c) B-cell D (d) B-cell EFigure 4.14: Distribution of the estimated number of fluorophores for all B-cells using our spatiotem-poral model and α = 0.05.115Chapter 5Analysis of outside-in activation ofintegrins in cell-extracellular matrixadhesions using live Drosophila FRAPdataBe sure you put your feet in the right place, then stand firm.— Abraham LincolnA version of the following project has been published in [132].We combined mathematical modeling with mutation analysis to understand mechanistically thedynamics of transmembrane integrins in live Drosophila embryos/larvae. Integrins are the maindriver of cell adhesion to the extracellular matrix (ECM). Cell-ECM adhesions play an importantrole in morphogenesis and tissue maintenance. In particular, we focused on the outside-in activationof integrins and we identified new in-vivo roles for outside-in activation, based on experiments indeveloping embryos.The chapter will contain a short introduction to cell adhesions (Section 5.1) and a descriptionof the biological problem (Section 5.2). I will describe the experimental methods in Section 5.2.1and Section 5.2.2. Section 5.3 will contain the mathematical model description and solution, whileSection 5.4 will summarize the parameter estimation and data fitting methods. Finally, I will presentthe results in Section 5.5 and the discussion in Section 5.6 [132].The main microscopy technique here and in Chapter 6 is fluorescence recovery after photo-bleaching. A description of FRAP in the context of Drosophila embryos/larvae data can be foundin Section 1.2.1165.1 Cell adhesions backgroundCell adhesions allow communication among cells and with the extracellular environment by intraand extracellular signals. Such signals can trigger morphogenetic or behavioral changes in the cells.Controlled cell adhesions allow multicellular organisms to maintain a fixed form where each of theirorgans and tissues is well defined and bounded.The organismal structure is not only determined by cells but also by the ECM. Thus, there aretwo types of adhesions essential in morphogenesis: cell to cell, and cell to ECM. In non-biologicalterms, we can imagine cells as pieces of a jigsaw puzzle which match each other (cell-cell) and theECM to be the glue (or net) that holds all the pieces together (cell-ECM). Without the glue, we stillcan form the puzzle, but we can not move it or join it to other puzzles; if we do, it will crack. Inother words, the puzzle is not stable without the glue. The ECM can be roughly portrayed as a mesharound the cells.Here, we will focus on cell-ECM adhesions. As described above, cell-ECM adhesions areessential for tissue maintenance and stability, cellular compaction and rearrangement. They enhancetissue structure and stability despite constant strain, tension, and other extracellular mechanicalforces.More than separating and holding the cells, the ECM also enhances communication betweenthem. One path for this communication is triggered by signals from the ECM adhesion to cells. Theadhesions signal changes the ECM, which in turn trigger signals to the cell. We will define this inmore detail in the next section.We will study an important character in this adhesion complex, a transmembrane protein calledintegrin [29]. It has been observed in Drosophila development that when integrin adhesion is absentmuscles can form, but shortly after, the tissue detaches [220].5.1.1 Integrin activation and cell-ECM adhesionIntegrins are critical for cell adhesion, cell migration, inflammatory and immune signaling, amongother biological process [150]. Different types of integrins perform each of those actions but allof them preserve a general structure across cells and species [131]. Integrins have two subunits,referred as α and β units, that form extracellular, transmembrane and cytoplasmic domains (Fig-ure 5.1). These domains allow integrins to send bidirectional signals into and out of the cell[150, 184]. The conformational state of these domains determines integrin function by changingthe affinity of binding to its partners [184].Broadly, intracellular integrin domains create a large protein complex that attaches to the cy-toskeleton and forms the integrin adhesion complex (IAC), while the extracellular domains act asan anchor between cells and the ECM [69, 83, 104, 106, 109]. These are the two steps of integrinactivation (Figure 5.1). The activation starts with the “inside-out” step caused by the assembly of117Figure 5.1: Integrin activation schematic. Integrins are shown in blue, ECM in green and IAC inbrown. The intracellular molecules bind to integrin and activate it (inside-out activation). Talinlinks the IAC with the cytoskeleton. Activated complex binds to the ECM proteins and trigger asecond activation of integrins (outside-in activation)the IAC in the cytosolic domain. Then, the “outside-in” step is triggered and the extracellular do-main binds to the ECM and sends an external signal from the ECM to the cell. The affinity andduration of the cell-ECM adhesions vary from low or transient to high or long-lasting depending onthe activation. These different characteristics allow integrin activation to mediate cell migration andtissue maintenance [30, 131, 220].In the cytoplasm, the integrin domain is ∼8nm and both subunits are connected with a saltbridge (Figure 5.1). Integrin binds to cytosolic proteins including talin, filamin, vinculin and α-actin to form the IAC [29, 131]. Most of the binding sites are on the β subunit. The IAC forms andtriggers a conformational change of the external domains of integrins, allowing activated integrins tobind to the ECM ligands. It has been shown that talin binding to integrin causes such conformationalchanges by disrupting the salt bridge between the integrin subunits [5]. Thus, talin is essential tointegrin activation [193] and will be an important protagonist in Chapter 6. In this chapter, we willrefer to the IAC as one entity.The extracellular domain has a length of approximately 11nm up to 19− 23nm depending onits conformational state (Figure 5.1). The usual analogy for this domain is to see it as a body with“head, knees and legs, with the legs by the membrane, knees at the middle, and head at the end.Following the metaphor, the head is the fundamental part where ECM ligands bind to the domain.The knees (which I imagine more like hips) can bend or be straight and are bent when integrinsare inactive. ECM ligands cannot access the head when close to the membrane, thus the kneeshave to stretch out to allow the binding (Figure 5.1). The inside-out activation signaling produces aconformational change that unbends them, and then ECM ligands can bind to the head.After the ECM ligands bind to integrins a second conformational change takes place (Fig-118ure 5.1). The ligand binding extends the knees of the integrins further. This second conforma-tional change increases the affinity of ECM-integrin adhesion by an outside-in activation signal.Extended integrins expose more binding sites enhancing binding to ECM ligands. Inside the cell,the aggregated integrins on the membrane reinforce binding to the cytoskeleton.The second activation signal can be also achieved by treatment with the divalent cations Mn+2and Mg+2 . On one hand, Mn+2 is indispensable for integrin activation [67, 131, 144], while Ca+2 isinhibitory. Moreover Mg+2 increases integrin activation but might only do so at low concentrationsof Ca+2 [53].Integrin activation, therefore, contains bidirectional and complementary signals. Overall, adhe-sion constitutes a global and robust response of cells underlying morphogenesis and tissue stability.5.2 Understanding outside activation as an stabilizer of cell-ECMadhesionsTo become activated, integrins must be at the membrane. That sentence sounds like a tautology butthen why do integrins undergo internalization and therefore turnover? At first sight, this might seemlike a bad idea but it adds dynamic behavior to integrin-mediated adhesion: fewer integrins in themembrane might imply weaker adhesions [25, 27, 155]. So integrin turnover supports different be-haviors like cell migration vs tissue stability. However, the relationship between integrin activation,turnover, and cell-ECM adhesions is still not fully understood.Here, we aim to better understand how the outside-in signal is involved in the stability of cell-ECM adhesion and in regulating integrin turnover. We combined mathematical modeling and bio-logical experiments to analyze integrin turnover using activation mutants and chemically enhancedor inhibited activation environments. Pablo Lopez, from the Tanentzapf’s lab at UBC, performedthe biological experiments on live Drosophila embryos and larvae. I create the mathematical modeland fit its corresponding parameters.We used FRAP assays of Drosophila embryos and larvae as our biological model and ODEas our mathematical model. The turnover dynamics are described with a linear ODE system ac-counting for the disassembly and recycling of integrin from the cytosol to the membrane. The restof this section, Section 5.3 and Section 5.4 present further details of the methods for biologicalexperiments, mathematical modeling, and parameter estimation.5.2.1 Biological experimental proceduresTo analyze the regulation of the IAC, we used FRAP experiments in live Drosophila embryos andlarvae. Drosophila is one of the most popular biological models; its genetical analysis is simplerthan other organisms. In particular, integrins in Drosophila are an excellent biological model giventhat they are homologous to those in vertebrates, allowing for generalization [29, 76, 131]. Areas of119Mutation Domain Molecular functionL211I I-like Activating. Promotes extracellulardomain extension [108, 133]G792N Transmembrane Activating. Increases avidity(clustering) [127]N828A Membrane-proximal NPxY Deactivating. Perturbs binding ofsignaling molecules (talin) [33, 57,193, 196, 197, 214]N840A Membrane-distal NPxY Deactivating. Perturbs binding ofsignaling molecules (kindlin) [198]S196F ECM binding domain Deactivating. Binds some ECMligands but it cannot undergo con-formational changes [13, 39, 107,108]Table 5.1: Summary of the integrin mutants used to affect outside-in activation via FRAP analysis.Mutation domain and description is included.muscle-tendon attachment (or myotendinous junctions (MTJs)) localize integrins and their adhesioncomplexes in thin lines between muscles [159, 220]. Their strong localization and visibility with aconfocal microscope make them perfect for FRAP experiments (Figure 1.1).We altered integrin turnover at MTJs using a genetic and a chemical approach. The first pro-cedure uses several integrin activation mutants to either reduce or enhance integrin activation (Ta-ble 5.1). The second method modulates integrin activation using Mg+2 and Mn+2 , and Ca+2 . Bothmethods use experimental protocols and mutants previously used or developed in the Tanentzapflab [57, 83, 158].We used three types of genetic mutants: integrin activation enhancers, integrin activation in-hibitors, and ECM ligands reducers. The mutations selected to induce conformational changes con-sistent with integrin activation were L211I and G792N. L211I promotes activation by stabilizing theconformation of the extracellular integrin domain [133]. G792N induces activation by increasingintegrin clustering at the membrane [158]. We used N828A and N840A mutants in the cytoplasmicdomain and the S196F mutant in the extracellular domain to block outside-in activation. N828Arelates to talin binding, while N840A relates to kindlin binding (kindlin and talin are cytoplasmicproteins necessary for integrin activation) [33, 57, 85, 143, 158, 193, 214]. The S196F mutationprevents the conformational changes essential to transmit the outside-in signal [13, 38, 39, 97, 158].The ECM ligands selected to test extracellular influence in the adhesions were laminin and Colla-gen IV. We used a laminin mutation on the gene lanB1 and a collagen mutation on the gene viking[28, 29]. Such mutations reduce ligand availability to integin.We used Mn+2 and Mg+2 to ectopically activate integrins. As mention above, Mn+2 and Mg+2mimic the activation signal. The experimentalist delivered the chemicals in developmental stage 16120live embryos using a published permeabilization protocol [83, 178]. The protocol allows pharma-ceutical therapy in live embryos without jeopardizing the membrane. To control for osmolarity, weconcurrently delivered NaCl.We investigated changes in outside-in activation over development by comparing stage 16 and17 embryos and third instar larva. Stage 16 occurs near the end of embryogenesis and completionof major morphogenetic processes. In comparison, stage 17 is the last stage of embryogenesis whenthe embryo transitions to larva. The third larval instar occurs 72 hours after hatching and is theculmination and final larval stage before metamorphosis.5.2.2 FRAP methodsDrosophila embryos and larvae possess a transparent epidermis that allows non-invasive visualiza-tion of proteins near the epidermis with confocal microscopy [29]. Labeled integrins in the MTJ arethen simple to visualize experimentally [29, 76, 131].FRAP assays were conducted with an inverted confocal microscope (Olympus Fluoview, -FV1000). FRAP curves were established for each experiment as the average of fluorescent intensitymeasured across the region of interest, for 825 frames. The frame rate was 2.25fps. Bleaching wasdone using the Tornado scanning tool (Olympus) with a 473nm laser at 5% power for 2s, at 100µs/pixel. The bleached section varied from individual to individual, being on average a quarter ofthe MTJ of interest. We measured the fluorescence in several non-bleached regions to control formovement of the organism. If movement was observed the sample was discarded.The animals were washed with water and mounted in phosphate-buffered saline solution. Beforethat, embryo samples were dechorionated with 50% bleach for 4 min. FRAP was performed aftera waiting period (1-1.5h) in a diverse population of MTJ [159]. As shown by [159], selecting adiverse population of MTJ average the variations among the different muscle types [159].The MTJ fluorescence was recorded in the region of interest for 30 sec before photobleachingto establish a reference level and to estimate the bleaching rate caused by the imaging process(Figure 5.2). In Section 5.3, we will refer to this pre-bleaching reference as the “reference region”.We then monitored the recovery of fluorescence in the bleached region for almost 5 minutes and wecalled this part of the FRAP curves “recovery region (Figure 5.2).Each experimental setting had samples from 4 to 11 different embryos and 6-8 different larvae(median 6). One to three different FRAP curves were taken per organism (median 3) to a total of14-30 experiments for embryos (median 15.5) and 14-16 for larvae (median 15). We consider everyFRAP experiment in the sample to be an independent and identical realization, even for replicatestaken from the same organism.1210 60 120 180 240 300 360time (s)100150200250300FluorescenceRaw data0 15 30time (s)0.70.80.91.01.11.21.3Normalized fluorescenceReference0 60 120 180 240 300time (s)0.00.10.20.30.40.50.6Normalized fluorescenceRecoveryFigure 5.2: Example of FRAP curves of WT Drosophila integrins, color coded by experiment.The top panel shows the average intensity at the imagined region versus time. Lower panels showthe same average intensity after normalization so that the total intensity is equal to one at thebeginning of the experiment and zero at the time of induced photobleaching. Let the referenceregion be any fluorescence recorded before photobleaching, and the recovery region anythingafter photobleaching.122Figure 5.3: Schematic of the mathematical model for integrin dynamics. Integrins transit from/to thecell membrane to/from internal vesicles at rates kexo and kendo. The fluorescence is lost at rate δdue to background bleaching.5.3 Mathematical modelPrevious experiments have shown that diffusive mobility of integrins in the MTJs is negligible overthe time scales of our FRAP experiments [220]. This observation allows us to focus only on theexternalization and internalization of integrins due to intracellular transport processes [27, 155].5.3.1 Assumptions and descriptionWe use a compartmental model of linear ordinary differential equations (ODE) to describe the in-tegrin turnover (Figure 5.3). We assume that labeled integrins are on the membrane or on vesiclesin the cytoplasm. Let M be the concentration of labeled integrins on the membrane, and V the con-centration on vesicles inside the cell. We assume that the FRAP data corresponds only to labeledintegrins in the membrane M, but V is not observable.The populations can either (a) undergo exocytosis at rate kexo, or (b) form vesicles via endo-cytosis at a rate kendo (Figure 5.3). When integrins are on the membrane, the label tags lose theirfluorescence due to imaging at rate δ (Figure 5.3). There is no natural photobleaching affectingthe vesicles since image acquisition only affects the membrane components. We assume that theconcentration of fluorescent integrin starts at equilibrium at the beginning of the experiment, but itgoes to zero at the membrane at the moment of the photobleaching (see Section 5.2.2 for regiondefinitions).123Then the ODE system is:dMdt=−(kendo+δ )M+ kexoV,dVdt= kendoM− kexoV,with initial conditionsM(0) =0 recovery regionMeq reference region , V (0) =Veq.We assume that prior to the start of the experiment the system is at chemical equilibrium. Thenthe pre-experiment equilibrium condition is given by the steady state equation:dMdt=−kendoM+ kexoV = 0 =⇒ MeqVeq =kexokendo.This equilibrium condition allows us to set the initial conditions of the system in terms of theparameters.Recall that FRAP experiments consist of independent repetitions under the same experimentalconditions. However, different replicates do not show the same intensity levels. In order to comparerecovery of fluorescence between replicates, we normalize the FRAP curves by the intensity atthe beginning of the experiment, Meq. Therefore, we non-dimensionalize the system by lettingm = MMeq and v =VMeq, where m and v represent the normalized labeled concentration of integrinscorresponding to M and V (Equation 5.1).dmdt=−(kendo+δ )m+ kexov, (5.1)dvdt= kendom− kexov,with initial conditionsm(0) =0 recovery region1 reference region , v(0) = VeqMeq = kendokexo .1245.3.2 Solution, analysis, and theoretical interpretationGiven that, the data only gives us m, we simplify the system to achive a solution. We transform thesystem to a second order ODE on m, with solution in Equation 5.2.m(t) =c(eλ+t − eλ−t) recovery regionc+eλ+t + c−eλ−t reference region, (5.2)whereλ± =12(−k±√k2−4δkexo), k = kendo+ kexo+δ ,c =kendoλ+−λ− , c− =λ++δλ+−λ− , c+ = 1− c−.Because k2 > k2− 4δkexo = k2endo + 2kendo(kexo + δ ) + (kexo− δ )2 > 0, we conclude that theeigenvalues are always real and negative for our parameter regime (positive rates). Therefore, thesystem has an asymptotically stable steady state at zero. Then m, the amount of fluorescently labeledintegrins in the membrane, approaches zero as time goes to infinity. This is in accordance with theloss of fluorescence during the experiment.In the reference region, the solution decays exponentially. This can be proven analytically since|k−2δ | <√k2−4δkexo, showing that c−,c+ > 0, and therefore the derivative is always negative.The decay in this region is given by fluorescence lost due to bleaching.On the other hand, the recovery region starts with a positive derivative. This initial increaseis led by the integrins endocytosis, but it eventually decreases as the fluorescence is lost due tophotobleaching. The maximum concentration ismmax =kendoλ+(λ−λ+) λ+√∆,obtained attmax = log((λ−λ+) 1√∆).The second derivative zero is att = 2tmax =2√k2−4δkexolog(λ−λ+).We assume that the total integrin concentration (labeled and unlabeled) does not change duringthe experiment timescale. Therefore, in the absence of natural bleaching (δ = 0), we have con-servation of m+ v over time. The corresponding conservation law leads to an expression of the125concentration of integrin in vesicles (v) with respect to membrane integrin concentrations (m).m(t)+ v(t) = m(0)+ v(0) = 0+kendokexo=⇒ v(t) = kendokexo−m(t).To estimate the fraction of integrins moving from one compartment to the other, we simplyextrapolate the dynamics to unlabeled proteins. That extrapolation is equivalent to assume thatthe labeling does not change the protein dynamics. In mathematical terms, the dynamics followEquation 5.1 with δ = 0. We can also calculate the mobile fraction, corresponding to the steadystate solution:mˆ(t) =kendokendo+ kexo.5.4 Data fitting methodsTo find parameter estimates, we minimized the sum of square residuals (SSR) between the modelsolution of m from (Equation 5.2) and the observed FRAP data, employing a nonlinear least-squarealgorithm (Figure 5.4). We minimized the total SSR by separating the residuals into two regions:before and after the photobleaching. We assumed that the residuals from the region the fluorescenceis recovering, recovery region, were more informative than those before the bleaching (referenceregion). Thus, we defined the total square residuals to be a linear combination of the SSR fromeach region, where those from the recovery region weighted more than those from before the pho-tobleaching. The results use a 1/10 ratio between the residuals before and after photobleaching. Iwrote the fitting procedure code in Python.To find confidence intervals for the parameter values, we created synthetic sample sets by boot-strapping. We assumed independent and identically distributed residuals (Figure 5.4b). Bootstrapdata was generated by sampling the residuals with replacement and adding them to the best estimatesolution. We analyzed the resulting bootstrap data as a new sample and obtained new parameter es-timates. Iterating the sampling and fitting process 1000 times yields the bootstrap distribution foreach parameter. These distributions are approximations to the true distributions of the parametersand we use them to generate confidence intervals for each estimated parameter value.Once we have fit the parameters to each data set, we compare the estimates between differentmutations and chemical environments by examining the bootstrap distributions for the parameters.To compare two distributions, it suffices to take the difference between them and analyze if zero liesin the interval center at the mean and containing a given percentage, 100∗α , of the distribution. Incase that zero is within the interval, there is not sufficient evidence to support the hypothesis that theparameters are different with probability 1−α . In our case, we use the largest confidence intervalnot containing zero to define the significance level of the difference.126(a)0.70.80.91.01.11.21.3normalized fluorescenceReference0.00.10.20.30.40.50.6normalized fluorescenceRecovery0 15 30time (s)0.70.80.91.01.11.21.31.4normalized fluorescence0 60 120 180 240 300time (s)0.000.050.100.150.200.250.300.350.40normalized fluorescenceboots_WT/BPS-integrinYFP_mysXG43_25_Controle16EXTRA(b)0.015 0.020 0.025kexo0100200300Count0.0055 0.0060 0.0065kendo0100200300Count0 0.0005 0.001δ0100200300400CountFigure 5.4: Example of data fitting for integrin turnover. Here, we show the data corresponding toWT integrin tagged FRAP curves as in Figure 5.2. (a) Black line represents the best model fit,blue boxplots represent the FRAP data (diamonds are data means and error bars correspond to 25and 75 percentiles). The distances from the black line to the diamonds are the residuals, whichare assumed independent and identically distributed and used for bootstrapping. (b)Bootstrapsdistributions for the model parameters127In the results section the mobile fraction was also roughly calculated by fitting an exponentialmodel typically used for FRAP data and ignoring the natural photobleaching [116, 121, 187, 200,211]. In that case, the recovery is assumed to follow f (t)= fmax(1−e− tτ ) and this formula was fittedto each replicate individually. The mobile fraction for the complete sample mˆ was then defined asthe average of the estimates for fmax. Using Equation 5.1 with δ = 0, we find a relationship betweenfmax, τ and kendo, kexo.fmax =kendokendo+ kexo, τ =1kendo+ kexo.5.5 ResultsResearchers have used the mobile fraction as a primary quantitative tool to describe changes inFRAP data, but with our mathematical model, we can determine the mechanism driving the changes.I described the mathematical model and the experimental methods in Section 5.2.1 and Section 5.3.I fitted the different data (control, mutations and chemical induction) to the mathematical model toinvestigate changes in the integrin turnover after outside-in integrin activation.Changes between experimental conditions could result from an increase in the attachment ratekendo or a reduction in the adherence of new proteins to the membrane kexo, or a mix of both, andthis can be analyzed by looking at the rate constant estimates. Moreover, the ratio between therates will determine if the change in the turnover is affected equally by the change in the rates,or if it is primarily driven by one effect. The ratio will also allow us to differentiate cases wherethe mobile fraction remains constant but the rates balance their changes. By calculating the ratioof kendo/kexo we are able to see shifts that favor increased turnover, build up, or breakdown of theadhesion complex.5.5.1 Outside-in integrin activation through chemical induction stabilizes cell-ECMadhesion by decreasing integrin turnoverAs explained in Section 5.2.1, we exposed Drosophila embryos to media containing magnesiumand manganese and used FRAP to see if integrin turnover was altered. Given the osmotic changeswe exposed the embryos to, we first controlled for the effects of osmotic conditions on integrinturnover, using simple NaCl. Our analysis found no substantial differences between organismsunder NaCl protocol vs control experiments. There were no statistical differences in the mobilefraction and integrin endocytosis rate (Figure 5.5). The only statistical change we detected wasa higher exocytosis rate compared to untreated experiments (Figure 5.5a), but without affectingmuch the kendo/kexo ratio (Table 5.2b). With these controls in place, we repeated the protocol andintroduce MgCl2 and MnCl2 to Drosophila embryos.Adding MnCl2 and MgCl2 resulted in a significant change of integrin turnover (Figure 5.5b,c, Table 5.2). We discovered that mobility was reduced compared to untreated control in both128(a) Na(b) Mn(c) MgFigure 5.5: Outside-in integrin activation through chemical induction stabilizes cell-ECM adhesionby decreasing integrin turnover. FRAP curves, mobile fraction values and rate constant valuesfor integrin WT treated only with buffer (light green), and buffer and (a)NaCL (osmolarity con-trol), (b) MnCl2, or (c)MgCl2 (dark green). Each point in each curve is the mean of n separateFRAP experiments. Error bars represent SEM for FRAP curves and 95% confidence intervals forparameter values. “NS” not statistically significant, “∗” 95% significance, “∗∗” 99% significance,“∗∗∗” 99.5% significance.129(a) Relative changeBuffer vs: kendo kexoOsm. control (Na) NS 10%Activation (Mn) -16% 26%Activation (Mg) -28% NS(b) Relative ratiokendo/kexoBuffer control 0.32Osm. control (Na) 0.3Activation (Mn) 0.21Activation (Mg) 0.22Table 5.2: Outside-in integrin activation through chemical induction stabilizes cell-ECM adhesion bydecreasing integrin turnover. (a) Relative change in rate constants between buffer-only and buffercontaining NaCL (osmolarity control), MnCl2, or MgCl2. (b) Relative rate ratios. “NS” not statis-tically significant.cases. The rate kendo significantly decreased with both treatments, while exocytosis increased onlywith MnCl2. It was noticeable that the parameter estimates showed similar distributions betweentreatments. The ratio of the constants showed a decrease compare to control and treatment withsodium, consistent with the decrease of kendo.These results suggest that the activation of integrin via outside-in signaling stabilizes integrinsin the membrane by reducing the rate of their removal via endocytosis.5.5.2 Activating integrin mutants regulate turnover with a similar mechanism asectopic integrin activationFollowing our results using Mg+2 and Mn+2 , we questioned if mutating integrin activation domainswill archive integrin stabilization at the membrane (Section 5.2.1). We used L211I and G792Nmutations, expected to promote integrin activation (see Section 5.2.1).The activation mutants L211I and G792N showed significant changes in turnover (Figure 5.6,Table 5.3), reminiscent of those observed under Mn+2 treatment. The mobile fraction and kendowere lower, and kexo higher compare to wild-type. Moreover, the ratio between endocytosis andexocytosis rates decreased approximately 70% and 30% for L211I and G792N compared to WT(Table 5.3b). As with the drug treatments, these changes would favor the accumulation and stabi-lization of integrins at the membrane lowering their turnover.Taken together, these data link outside-in activation to integrins reduced endocytosis and in-creased exocytosis. Such changes stabilize membrane integrins and reduce the mobile fraction.5.5.3 Integrin mutations that affect outside-in activation fail to regulate integrinturnover upon ectopic chemical integrin activationThe obvious next step was to compare how chemical treatments will affect turnover in integrinactivation mutants. We collected FRAP data for different outside-in integrin mutants under theMn+2 chemical protocol.We analyzed the activator promoters L211I and G792N, and the activator obstructers N828A,130(a) L211I(b) G279NFigure 5.6: Integrins point mutations that induce activation affect turnover similarly to treatment withdivalent cations. FRAP curves, mobile fraction values and rate constant values for WT in blackand for activation mutants in maroon( (a) L211I, (b) G792N). Each point in each curve is the meanof n separate FRAP experiments. Error bars represent SEM for FRAP curves and 95% confidenceintervals for parameter values. “NS” not statistically significant, “∗” 95% significance, “∗∗” 99%significance, “∗∗∗” 99.5% significance.(a) Relative changekendo kexoWT vs L211I -45% 88%WT vs G792N -31% 20%(b) Relative ratiokendo/kexoWT 0.66L211I 0.19G792N 0.38Table 5.3: Integrins point mutations that induce activation affect turnover similarly to treatment withdivalent cations. (a) Relative change in rate constants between WT and induce activation mutants(b) Relative rate ratios. “NS” not statistically significant.131(a) Relative changekendo kexoWT -16% 26%L211I -8% NSG792N -16% -28%N840A NS NSN828A NS NSS196F NS 20%(b) Relative ratiokendo/kexoBuffer MnCl2WT 0.32 0.21L211I 0.26 0.23G792N 0.33 0.38N840A 0.33 0.31N828A 0.39 0.39S196F 0.38 0.34Table 5.4: Mutations that block or strongly induce outside-in integrin activation are insensitive to treat-ment with divalent cations. (a) Relative change in rate constants between buffer only control andbuffer containing MnCl2. (b) Relative rate ratios within each experimental condition. “NS” notstatistically significant.N840A, and S196F (see Section 5.2.1). For the promoter mutations, the mobile fraction did notchange when treated with Mn+2 , compared to untreated mutants (Figure 5.7). For the activationmutant L211I, the only significant change in the kinetic rates was a decrease in the endocytosisrate, smaller than that on the WT background (Figure 5.7a). On the other hand, the G792N mu-tant showed a decrease in both reaction rates, but with an increase in their ratio (Figure 5.7b andTable 5.4). The simultaneous decrease of the exocytosis and endocytosis rates might explain theunchanged mobile fraction. Overall, the Mn+2 treatment did not dramatically change the ratio ofkendo/kexo in either activated integrin mutant (Table 5.4b). The data of the block activation mutationalso showed no change in the mobile fraction (Figure 5.8). None of the rate constants change underMn+2 treatment for N828A or N840A. Only kexo increased for the S196F mutant but did not notablychange the ratio kendo/kexo (Table 5.4b).All these data together point out that exposure to divalent cations, such as Mn+2 , has little effecton blocker and enhancer activation mutants. Those mutations are already active, or cannot transduceoutside-in activation.5.5.4 ECM reduction increases integrin turnover by increasing integrin endocytosisrateHaving shown that increased outside-in activation lowers turnover, we wondered if the opposite wastrue: does lowering outside-in activation increase turnover? To test this hypothesis, we reduced ac-tivation by controlling the availability of the ECM proteins laminin and collagen (see Section 5.2.1).As hypothesized, the mobile fraction in our FRAP experiments increased upon reduction ofECM ligands (Figure 5.9 Table 5.5). In particular, the rate constant kendo increased compared tocontrol, thus the ratio between endocytosis and exocytosis increased (Table 5.5b). Overall, this datashows that ECM ligands induce outside-in activation and cause the stabilization of integrins at themembrane.132(a) L211I(b) G792NFigure 5.7: Mutations that induce outside-in integrin activation are insensitive to treatment with diva-lent cations. FRAP curves, mobile fraction values and rate constant values for activation mutantsunder treatment with buffer only (light red) or with buffer containing MnCl2 (dark red) ( (a)L211I, (b) G792N). Each point in each curve is the mean of n separate FRAP experiments. Er-ror bars represent SEM for FRAP curves and 95% confidence intervals for parameter values.“NS” not statistically significant, “∗” 95% significance, “∗∗” 99% significance, “∗ ∗ ∗” 99.5%significance.133(a) N840(b) N828(c) S196Figure 5.8: Mutations that block outside-in integrin activation are insensitive to treatment with diva-lent cations. FRAP curves, mobile fraction values and rate constant values for activation deficientmutants under treatment with buffer only (light blue) or with buffer containing MnCl2 (dark blue)( (a) N840, (b) N828, (c) S196). Each point in each curve is the mean of n separate FRAP exper-iments. Error bars represent SEM for FRAP curves and 95% confidence intervals for parametervalues. “NS” not statistically significant, “∗” 95% significance, “∗∗” 99% significance, “∗ ∗ ∗”99.5% significance.134(a) Laminin(b) CollagenFigure 5.9: Reducing the availability of ECM ligands increases integrin turnover. FRAP curves, mo-bile fraction values and rate constant values for WT in light orange and (a) laminin or (b) collagenmutants in dark orange. Each point in each curve is the mean of n separate FRAP experiments.Error bars represent SEM for FRAP curves and 95% confidence intervals for parameter values.“NS” not statistically significant, “∗” 95% significance, “∗∗” 99% significance, “∗ ∗ ∗” 99.5%significance.135(a) Relative changekendo kexoWT vs laminin 48% NSWT vs collagen 61% NS(b) Relative ratiokendo/kexoWT 0.18Laminin 0.3Collagen 0.27Table 5.5: Reducing the availability of ECM ligands increases integrin turnover. (a) Relative changein rate constants between WT and laminin or collagen mutants mutants (b) Relative rate ratioswithin each experimental condition. “NS” not statistically significant.(a) Relative changekendo kexoWT (e16-eL3) -28% 117%L211I (e16-eL3) -9% NSG792N (e16-eL3) -18% 59%N840A (e16-eL3) -9% 21%N828A (e16-eL3) -34% NSS196F (e16-eL3) 49% 28%(b) Relative ratiokendo/kexoe16 e17 L3WT 0.66 0.28 0.22L211I 0.19 0.19 0.27G792N 0.38 0.24 0.19N840A 0.38 0.31 0.25N828A 0.35 0.39 0.26S196F 0.22 0.26 0.26Table 5.6: Developmental series analysis of activation increasing and deficient integrin mutants. (a)Relative change in rate constants between embryonic stage 16 (e16) and third larval instar (L3) forall genotypes (b) Relative rate ratios within each experimental condition at progressive develop-mental stages: embryonic stages 16, 17 and third larval instar. “NS” not statistically significant.5.5.5 Integrin turnover is developmentally regulated by integrin activationOrganisms develop and maintain a stable architecture in part by modulating the dynamics of cell-ECM adhesions. To gain insight into these dynamics, we analyzed the outside-in activation ofintegrins in stage 16 and 17 embryos, and third instar larva (see Section 5.2.1). The experimentprocedures were the same as described above.We discovered a decrease in the integrin mobile fraction during development, with a greater dropduring the embryo to larva transition (Figure 5.10a, Table 5.6) [159, 220]. Our mathematical modelrevealed a reduction of kendo between stage 16 and 17, and a rise of kexo with each developmentalstage (Figure 5.10a). These rate variations decreased the kendo/kexo ratio by over 57% betweenstages 16 and 17 and suggest that integrins stabilize over development (Table 5.6a).We also tested how development affects dynamic activation mutants. G792N showed the sametrends as WT (Figure 5.10c), although the effects were weaker. L211I’s mobile fraction was lowerthan WT for the embryo stages (Figure 5.6, Figure 5.10c). Also, this mutant had similar mobilefractions between stages 16 and 17, different from the WT and G792N results. L211I showed a dif-ferent development trend in kexo compared to WT, decreasing between stage 16 and 17 and increas-ing when progressing to the larval stage (Figure 5.10a,b). These results implied L211I stabilizesintegrins at the MTJs.136(a) WT(b) L211I(c) G792NFigure 5.10: Developmental series analysis of WT and activating integrin mutants. FRAP curves, mo-bile fraction values and rate constant values for (a) WT and (b, c) activation integrin mutants atprogressive developmental stages: embryonic stages 16 (e16, green) and 17 (e17, purple) andthird larval instar (L3, blue). Each point in each curve is the mean of n separate FRAP experi-ments. Error bars represent SEM for FRAP curves and 95% confidence intervals for parametervalues. “NS” not statistically significant, “∗” 95% significance, “∗∗” 99% significance, “∗∗∗”99.5% significance.137(a) Relative changekendo kexoControl vs Rap1-DN NS NSControl vs Rap1-DN Mn NS NSRap1-DN vs Rap1-DN Mn NS NS(b) Relative ratiokendo/kexoControl 0.21Rap1-DN 0.19Rap1-DN Mn 0.21Table 5.7: Rap1 regulates integrin turnover downstream of outside-in activation. (a) Relative changein rate constants between WT, mutant, and mutant treated with MnCl2. (b) Relative rate ratios.“NS” not statistically significant, “∗” 95% significance, “∗∗” 99% significance, “∗ ∗ ∗” 99.5%significance.On the other hand, the integrin mutants that disrupt outside-in activation didn’t stabilize integrinduring development (Figure 5.11, Table 5.6). The mobile fraction of N828A and S196F did notchange, while N840A showed a smaller drop compared to WT (Figure 5.10a, Figure 5.11). Theexocytosis rate did not change between embryo stages, and it increased from embryo to larva tran-sition only in the N840A mutant. Estimated endocytosis rates showed different changes in eachmutant. They decreased over development for N840A (Figure 5.11a) and increased for S196F (Fig-ure 5.11c). For the N828A mutant, the rate was constant during embryo stages and decreased onthe larval stage (Figure 5.11b).The variability of integrin turnover during development in activating and activation-defectivemutants suggests that modulation of activation is important in regulating turnover during animaldevelopment.5.5.6 Rap1 regulates integrin turnover downstream of outside-in activationTo analyze the intracellular delivery of outside-in signaling, we used a dominant negative mutantof Rap1 (Rap1-DN). The small GTPase Rap1 is known to regulate cell-ECM adhesion and to actdownstream of outside-in activation [64, 204]. It promotes the IAC activation reinforcing the adhe-sions. As with other mutants, we fluorescently labeled integrins on the Rap1 mutant and performedFRAP experiments with and without Mn+2 treatment.We discovered that the Rap1-DN mutant did not exhibit altered integrin turnover compared toWT (Figure 5.12, Table 5.7). We observed no difference between the Rap1-DN mutant treatedwith Mn+2 and the untreated. Thus, Mn+2 did not stabilize integrins in the absence of Rap1. Theseresults suggested that Rap1 signaling is important for controlling turnover downstream of outside-inactivation.5.6 DiscussionIn this work, I performed a quantitative analysis of imaging data for Drosophila’s integrins to an-alyze cell-ECM adhesions in the context of a live, developing animal. We altered the outside-in138(a) N840(b) N828(c) S196Figure 5.11: Developmental series analysis of activation deficient integrin mutants. FRAP curves,mobile fraction values and rate constant values for activation deficient mutants ( (a) N840, (b)N828, (c) S196) at progressive developmental stages: embryonic stages 16 (e16, green) and17 (e17, purple) and third larval instar (L3, blue). Each point in each curve is the mean of nseparate FRAP experiments. Error bars represent SEM for FRAP curves and 95% confidenceintervals for parameter values. “NS” not statistically significant, “∗” 95% significance, “∗∗”99% significance, “∗∗∗” 99.5% significance.139Figure 5.12: Rap1 regulates integrin turnover downstream of outside-in activation. FRAP curves, mo-bile fraction values and rate constant values for WT in black, Rap1 negative mutant in light pink,and Rap1 negative mutant treated with MnCl2 in dark pink. Each point in each curve is the meanof n separate FRAP experiments. Error bars represent SEM for FRAP curves and 95% confi-dence intervals for parameter values. “NS” not statistically significant, “∗” 95% significance,“∗∗” 99% significance, “∗∗∗” 99.5% significance.integrin activation by treating embryos with divalent cations, by introducing point mutations in in-tegrin, or by reducing the levels of ECM ligands. We identified roles for outside-in activation inregulating turnover and suggested the relevance of this signal in stabilizing tissue architecture to-wards the end of embryogenesis. We were able to increase our knowledge of integrin-mediatedadhesion by integrating quantitative and descriptive mathematical models with biological experi-ments. Our findings added an important in-vivo context to the existing theory of a role for integrinactivation in regulating cell-ECM adhesion.Our mathematical model allowed quantification of the transition rates of integrins, and thusrepresent a substantial improvement over the usual exponential fit used to analyze FRAP data. Wecaptured changes in dynamics under different modifications of integrin activation. By correcting forimaging bleaching, we found imaging independent, generalizable and more realistic parameters.Several groups have previously investigated links between activation and integrin trafficking[6, 172, 205, 206]. Particularly, Arjonen and colleagues used conformation-specific antibodies anda quenching-based assay to visualize and compare active and inactive β1 integrin traffic through theendocytic machinery [6]. They found that active integrins have a more efficient internalization anda higher endocytosis rate than the inactive integrins. They also presented a sorting scheme whereinactive integrins belonged in the faster recycling compartment [6]. These findings are very muchin line with our observation that activation decreases the rate of integrin endocytosis and its overallturnover.Links between outside-in activation and mechanical force are well-documented, but our results140deepen these links [66, 118, 213]. An earlier collaboration between the Tanentzapf and Coombslabs showed that increasing mechanical force regulates integrin turnover by lowering the rate ofendocytosis [159]. Moreover, integrin mutants defective in ligand binding or outside-in signalingcannot modulate turnover in response to force [159]. It is therefore likely that activation of outside-in signaling also regulates integrin response to force and the resulting adhesion stabilization. Herewe report that outside-in activation lowers the mobile fraction and the endocytosis rate, kendo, similarto previous results [159]. In particular, Mg+2 treatment lowers mobile fraction solely through kendoreduction. However, unlike mechanical force, outside-in activation modulates turnover also throughchanges in kexo indicating that these two regulatory mechanisms diverge at certain points.As described in [131], we propose the following model. During the late stages of fly develop-ment, MTJs, once formed, encounter increasingly powerful muscle contractions [43]. This increasesthe mechanical force on ligand-bound integrins at the plasma membrane and changes their confor-mation. The conformational change then triggers an increase in outside-in signaling, which activatesRap1. Rap1 increases integrin trafficking to the membrane and this leads to a rise in integrin ex-ocytosis [36, 139]. At the same time, higher forces lower the endocytosis rate of active integrins,increasing integrins at the membrane [6]. Taken together, the endocytosis and exocytosis rates bal-ance integrins at the membrane and thus decrease the mobile fraction. This mechanism providesa framework for translating environmental cues (such as ECM ligands availability or mechanicalforces) into changes in the stability of integrin-based cell adhesion to ECM.Using mathematical models along with biological experiments, we showed that outside-in ac-tivation also regulates the turnover of integrins. This was a novel discovery of the in-vivo role foroutside-in activation and allowed us to develop a better understanding of tissue consolidation andlong-term preservation. This project shows how simple mathematics can contribute to new discov-eries in complex biological systems.141Chapter 6Analysing talin’s role in inside-outintegrin activation, under mechanicalforces, using in-vivo Drosophila FRAPdataWhat we achieve inwardly will change outer reality.— PlutarchI adapted the published paper [83] to write this chapter. It shares the common goal with [132]of understanding cell-ECM adhesions.We studied the integrin adhesion complex (IAC) formation and the first step in integrin ac-tivation driven by the intracellular protein talin. Earlier collaboration between the Coombs andTanentzapf labs investigated inside-out signaling from the perspective of integrins [159]. Here,we used a similar approach but focused on talin. We analyzed talin turnover under different forceschemes to test the sensitivity of talin and the IAC to force changes. We coupled the force modifica-tion mutants with integrin and talin point mutations to find pathways to stable adhesions. Our resultssupport the hypothesis that mechanical forces stabilize tissue architecture by reinforcing cell-ECMadhesion. We also studied how variations in talin turnover affect Drosophila development and foundthat mechanical force is not its only regulatory component.We merged experimental and analytical analysis in this study. Using FRAP in live Drosophilalarvae and embryos, Katrin Hakonardottir and Pablo Lopez performed the experiments. I createdthe mathematical model and the fitting protocol. I computed the parameter values and distributions.This chapter contains the following sections. In Section 6.1, I resume describing the cell-ECMadhesions from Section 5.1, but this time focusing on the integrin adhesion complex and inside-out activation. This is followed in Section 6.2 with a description of how talin influences IAC, and142experimental details. In Section 6.3 and Section 6.4, I describe the mathematical models and datafitting protocols. Section 6.5 and Section 6.6 contain all results and discussion. The reader shouldsee the Introduction for a FRAP description.6.1 Inside-out integrin activationAs mentioned in Section 5.1, integrins are transmembrane proteins mediating cell to ECM adhe-sions. Integrins have to go over a sequence of conformational changes to form stable bonds withthe ECM ligands (Figure 5.1) [9, 69]. Those conformational changes activate integrins via two sig-nals: inside-out, and outside-in. In Chapter 5, I explained the outside-in activation. I will focus oninside-out activation in this section.Returning to the “head, knees and legs” analogy of the extracellular integrin domain (see Sec-tion 5.1.1), integrins need to “stand up” to be accessible for ECM binding (Figure 5.1). In order forintegrins to extend, a conformational change has to occur. We call this part of the integrin activationthe “inside-out” activation because it is triggered by intracellular proteins. These proteins bind tothe intracellular domain of integrins and form the so-called integrin adhesion complex (IAC) (Fig-ure 5.1, [29, 131]). One important characteristic of integrins and other IAC proteins is their abilityto sense mechanical forces. Proteins with such characteristics are known as mechanosensors.6.1.1 The integrin adhesion complex (IAC) and its mechanosensorsThe integrin adhesion complex is a dynamic network of cytosolic proteins that link the cytoplasmictails of integrins to the cytoskeleton. Its composition differs depending on the ECM stiffness, tensileforce and maturation state of the IAC [216, 223]. The IAC components serve as bonds, scaffolds,mechanosensors, and signaling triggers [82, 159]. Some mechanosensors of the complex are talin,p130Cas, α-actin, paxillin and focal adhesion kinase (FAK) [47, 69, 174, 219].Integrins cannot themselves generate force, but they use the complex as an anchor to the cy-toskeleton to resist mechanical force [69, 104, 106, 109]. Together, the IAC and integrins behaveas one entity reacting and responding to foreign forces, either external (ECM pulling) or internal(actomyosin contractility) [69, 69, 109]. Integrins and the other mechanosensors react and translateforce into signaling, communicating in and out of the cell [82]. In this way, force sensing creates aself-induced reaction that reinforces the robustness of the adhesion.In particular, integrin-ECM bonds are catch bonds that create pulling forces between integrinsand ECM ligands [40, 66, 117]. These pulling forces change the architecture of integrins and thecatch bonds strengthen the adhesions [104]. Catch bonds are akin to a toy finger trap where yourfinger gets trapped as you pull it away. The trap or bond becomes stronger as the force appliedincreases. These extracellular bonds induce the engagement of intracellular components which thencreate a complementary internal force ultimately connecting to the cells actin-myosin network.143Figure 6.1: Schematic of talinAmong the proteins of the IAC, talin is a mechanosensor essential to integrin activation [29, 30,193]. It takes part in both outside-in and inside-out signaling. In the absence of talin, the Drosophilaphenotype resembles that of integrin negative mutants. I will describe this essential protein in moredetail in the next section.6.1.2 TalinTalin plays an important role in activating integrins and linking them to the actin cytoskeleton[99, 137]. It is a large and mechanosensitive cytoplasmic protein centrally positioned in the IAC.Its central position allows it to sense external and internal mechanical forces and coordinate IACformation and signaling [47, 69, 104].A caricature of talin would show a head domain connected to an elastic rod domain (Figure 6.1).Talin’s head, or globular N-terminal FERM (F for 4.1 protein, E for ezrin, R for radixin and M formoesin), has four sub-domains (F0, F1, F2, and F3) [73]. The head contains a binding site for theβ -integrin cytoplasmic tails, called integrin binding site 1 (IBS-1). IBS-1 is essential to activateintegrins. There are also binding sides for FAK and PIPK1γ 90. These are important players in thedynamics of focal adhesions. One binding site in talin’s head binds to the GTPase Rab1, a regulatorof the cell-ECM adhesions [22].Linked to the head, the second part of talin comprise a flexible C-terminal rod. 62 α-helicesform the rod domain and are organized into 13 amphipathic helical bundles (R1-R13) [199]. The rodhas binding sides for integrins (integrin binding site 2 (IBS-2)), vinculin, and actin [44]. The actinbinding site (ABS) contains a THATCH (talin/HIP1R/Sla2p Actin-Tethering C-terminal Homology)domain, which enables talin-actin binding by dimerizating talin [73]. THATCH domain is alsorequired to link the cytoskeleton and integrins-ECM bounds [110].The rod of talin has a domain called auto-inhibition domain since it allows talin head and rodbound to each other forming a “loop” that hides the integrin binding sites [74]. The auto-inhibitiondomain in the rod binds to the FERM domain in the head head and talin changes to a “fold” state[74]. When talin is folded, the head and rod enclose the IBS-1. Thus, integrins cannot bind to talin144in the folded state and integrin activation is inhibited [74]. Goult and Goksoy showed a competitionbetween the auto-inhibitory talin domain and integrins for binding sites located in talin’s head [74,77]. When the auto-inhibitory talin domain is not present (auto-inhibition mutant), there is a higherintegrin activation and a higher rate of focal adhesion assembly [74, 119]. Thus in auto-inhibitionmutant case, integrin has no competition and binds to talin more often.Integrin to talin binding triggers “inside-out” integrin activation [32]. First, talin must interactwith plasma membrane lipids to correct its orientation to one more pertinent for binding [5]. Then,talin’s IBS-1 subunit binds to the integrin β subunit and breaks the salt bridge between the α andβ integrin subunits [5, 99, 111, 137]. With the bridge disrupted, the α and β transmembrane inte-grin domains separate and change the extracellular inclination of integrin [105]. In simple words,talin binding separate the integrin subunits in the membrane and extends the extracellular domain.Extended integrins expose additional binding sites for ECM ligands [5]. Conversely, a lack of talindestabilizes adhesions. Talins importance in achieving stable integrin-mediated adhesions cannotbe overstated [193, 196, 218].Inside-out activation is also reliant on intracellular dynamics. Talin links the cytoskeleton tointegrins, anchoring integrins within the cell. Talin binds to actin via talin’s ABS, or by recruitingother actin-binding proteins like vinculin [225]. It is important to note that the direct and indirectmechanisms are probably complementary and simultaneous. Binding through vinculin requiresmechanical stretching of talin to exposes the vinculin binding sides [47, 69], highlighting againthe role of force in forming the IAC. The THATCH domain is the most studied domain for directbinding between talin and cytoskeleton. As mentioned above, THATCH domain links talin withactin and reinforces the integrin-ECM bounds. Thus, it is relevant for in and outside activations[110].In simple terms, talin connects the cytoskeleton to the membrane by attaching integrin to actin,and by recruiting complementary proteins. Subsequently, talin triggers the changes necessary toconnect the membrane and the ECM by increasing integrins affinity to ECM ligands. This summaryhas highlighted talin’s relevance ito integrin activation and adhesion stability. This motivates ourquantitative investigations into the role of talin and forces in controlling cell-ECM adhesion stability.6.2 Understanding talin turnoverIn earlier studies, the Tanentzapf lab showed that talin undergoes turnover in myotendinous junction(MTJ) in live Drosophila [82, 159, 220]. They also showed how forces regulate integrin turnoverin-vivo [159]. Del Rio et al. showed that force is necessary to unfold talin and expose the vinculinbinding sides needed for actin binding [47]. We therefore hypothesized that force could regulatetalin and IAC turnover, either by altering talin or by changing talin’s ability to bind to other proteins[83].As the biological system, we used live Drosophila embryos and larvae, focusing on the MTJ.145Mutation Domain Molecular functionL334R FERM/IBS-1 Integrin activation mutant.Blocks integrin activation butnot talin head binding to integrincytoplasmic tail. [84]R367A FERM/IBS-1 IBS-1 mutant. Blocks talin headbinding to integrin cytoplasmictail; blocks integrin activation. [57,68, 196]K2094D/S2098DIBS-2 IBS-2 mutant. Blocks talin rodbinding to integrin; affects IAClink to integrin [57, 141, 166]K2450D/K2451D/K2452DTHATCH/ABS Actin binding mutant. Reducesactin binding to talin. [63, 73]Table 6.1: Summary of the talin mutants used to affect inside-out signalling via FRAP analysis. Mu-tation domain and description is included.The Tanentzapf lab developed a protocol to manipulate the force at the MTJ while using FRAP withlabeled integrins [159]. Therefore, we adopted similar protocols to analyze in-vivo talin turnoverby tagging talin, see Section 6.2.2. Likewise, we investigated how the increased/decreased forcebackgrounds affect different talin mutations, and the resulting alteration on IAC turnover. Table 6.1has a summary of the mutations. I will describe the biological methods further in Section 6.2.1.We modelled talin fluorescence recovery as resulting from two independent pathways. Therecovery could be due to disassembly and recycling of the integrins or due to talin binding the IAC.The corresponding mathematical model was a three-compartment linear ODE, but the two cytosolicequations are not distinguishable with the FRAP data. To overcome this, we established a protocolto disrupt one of the recovery pathways experimentally (Section 6.2.1). We then used analyticaltools to identify the parameters.6.2.1 Biological experimental proceduresThe experimental protocols were investigated, developed and carried out by Katrin Hakonardottirand Pablo Lopez. As established in Section 5.2.1, the MTJs are excellent areas to analyze integrin-mediated adhesions in live Drosophila. In Section 5.2.1, we tagged integrins at the MTJ and ob-served their recovery using fluorescence microscopy to understand integrin turnover. The MTJ hasserved as a model for the study of talin and other proteins of the IAC [63, 130, 224]. We exploitedthe advantages of MTJs by adopting integrin protocols, developed in the Tanentzapf lab, to studytalin turnover in live Drosophila [57, 58, 63, 159].The resulting talin protocol could not distinguish the cytosolic pathways that drive the recovery146of the fluorescence (see below, Section 6.3.3). To separate the pathways, we used the endocytosisinhibitory drug dynasore (DYN) in live embryos and larvae [178]. Schulman et al. developed aprotocol to deliver DYN solution into live flies [178]. Pablo Lopez used this protocol to deliver150-200 µM DYN solution into live stage 17 embryos and third instar larvae [83]. Section 6.4provides more details on this protocol.To alter the force environment, we used the temperature-shift protocol established by Pines etal. [159]. The protocol uses the temperature-sensitive mutations Brkdj29 (breakdance (Brkd)) andparats2 (paralytic (para)) [78, 142, 173]. Those mutants behave as wild-type (WT) embryos orlarvae at 25◦C but they reveal their phenotype at 37◦C. To turn on the mutation, we heat shockedthe sample at 37◦C for 1.5 hours before FRAP and kept it in an incubator during imaging [83].The Brkd mutation increases the force that acts at the MTJ by inducing muscle hyper-contraction[142, 159]. para caused muscle relaxation with a marked effect in the larval stages [159]. Thus, weused larvae for para background experiments and embryos for Brkd ones.We modified integrin inside-out activation using additional targeted talin mutations already gen-erated and analyzed in the Tanentzapf lab (Table 6.1) [57, 58, 63]. We selected the mutations tomodify the essential functions of talin in integrin activation. The mutant selected to block inside-out activation without disturbing the talin-integrin binding was TalinL334R [58]. I will call it eitherL334R or integrin activation mutant. To affect talin-integrin binding, we used TalinR367A in theIBS-1 [196] and TalinK2094D/S2098D in the IBS-2 [57]. Remember that IBS-1 joins talin and in-tegrins and triggers inside-out activation, while IBS-1 keeps the IAC attached to integrin [57, 196].Finally, we picked the K2450D/V2451D/K2452D mutant to block talin’s capacity to bind to actin[63]. None of the mutations affect muscle integrity. We experimented with the mutants under WTand force backgrounds.FAK is another mechano-activator in the IAC that regulates integrin by sensing forces [154, 181,212]. FAK traduces the force into a signaling cascade that phosphorite paxillin [154], which thenadheres to the IAC and helps steady the adhesions [175]. We perturbed inside-out activation usinga FAK mutant. In particular, we used the Y430F mutant which inhibits the activation of FAK. Wecompared FAK mutant only to the increased force background.After comparing different force backgrounds on each mutant, we investigated the effect of forcewhen the fly is growing. We performed a developmental series experiment in WT talin heterozygousbackground for embryonic stages 15, 16, and 17, first instar larvae, and third instar larvae. Thisanalysis clarified the dynamics of talin as the organism develops.6.2.2 FRAP methodsTo analyze the turnover of the IAC, we used FRAP experiments at MTJ in live Drosophila embryosand larvae. Talin was labeled with a fluorescent tag and visualized with a confocal microscope. Asstated in Section 5.2.2, the experimentalist imaged the MTJ for approximately 30 sec before pho-147tobleaching to establish a fluorescence reference. After photobleaching a region, we monitored therecovery of fluorescence in the bleach region for almost 5 minutes. Each control and mechanosensormutant sample stayed in an incubator at 37◦C in the dark for 1.5 hours prior to imaging to inducechange in phenotype.Samples came from 5 to 14 different embryos (median 8) and 5-10 different larvae (median 8)per mutation. One to three different FRAP curves were taken per organism (median 3 for embryos,and 2 for larvae) to get 15-34 experiments for embryos (median 18) and 9-27 experiments for larvae(median 18) per mutation. We consider every FRAP experiment in the sample to be an independentand identical realization of the process.We assumed that the observed fluorescence came from talin bound to integrin and neglected freetalin protein diffusing near the membrane. We also assumed that talin and integrin concentrationsare independent. If incorrect, these assumptions would lead us to overestimate the level of talin-integrin binding at the membrane.6.3 Mathematical modelAs described above, our FRAP data represents labeled talin near the membrane. It seems unlikelythat talin proteins remain unbound for a long time while close to membrane integrins. So we ad-ditionally assume that the fluorescence comes from talin bound to membrane integrins. Thus, themobility of integrins defines the mobility of observed talin in the membrane. We assumed thatmembrane diffusion of integrins is negligible (as in Section 5.3 [220]). Under these assumptions,we constructed a compartmental ODE model.The dynamics of internalization and externalization of labeled talin are more complex thanthose assumed for integrins in Section 5.3. We develope a general model describing talin dynamics,follow by a simplified version. The simplifications are done by grouping talin pathways that aremathematically equivalent even if they are biologically different. The model has direct parameterinterpretation, and is essentially mechanistic in nature.6.3.1 General model constructionWe describe talin dynamics as interactions between concentrations of free and labeled cytosolicproteins: talin P, the integrin adhesion complex (IAC) at the membrane Cm, integrin-talin boundcomplex at the membrane Bm, or intracellular Bc, and unbound integrins at the membrane Im orin vesicles Ic (Figure 6.2). We assume the recovered fluorescence comes from two independentmechanisms related to integrin localization and constructed a compartmental ODE model.The first mechanism arises from membrane integrins (Im). Free and fluorescent talin (P) bindsto integrin on the membrane (Im) at a rate k∗on to form the bound complex Bm. The integrin-talincomplex Bm then forms a full IAC (Cm) at rate α or disassociates from the IAC at a rate β . We148assume that talin can unbind from integrins at the membrane (from a formed IAC, or from anintegrin-talin complex Bm) at a rate k∗off.The second mechanism focuses on integrins inside the cell (Ic). Here, talin-integrin complex(Bc) does not break or form. The IAC assembles and externalizes at different rates dependingif talin is bound to integrins or not. It will externalize at rate k∗asm1 when talin binds to integrinright before externalization, or at rate k∗asm2 when the talin comes from Bc complex. Similarly,the IAC disassembles due to internalization of integrins with talin disassociated at rate kdis1, or bymaintaining talin bound, at rate kdis2. The membrane talin-integrin complex Bm internalizes at ratekin2 without being disrupted, or at rate kin1 when talin disassociation happens. Bm externalizes eitherfrom Bc at rate kex2 or by externalization of integrin and intermediate binding of talin at rate kex1.Similar to Section 5.3, we assume the experiments did not affect the other IAC componentconcentrations or chemical interactions. Thus, we considere all other chemical interactions to beat steady state during the experiments. In particular, the membrane and cytosol concentrations ofintegrins remain at the pre-bleaching equilibrium during the recovery phase. Thus, Im and Ic remainconstants and equal to Iˆm, Iˆc. We simplify the system by setting kon1 = k∗onIˆm, kasm1 = k∗asm1Iˆc andk∗ex = kex1Iˆc. Other IAC proteins dynamics are implicitly in the parameters β , kasm1, and k∗asm2.Under the steady state assumption of integrin concentrarion, the model in Figure 6.2 can bedescribed with the following system of equations.dPdt=−(kon1+ k∗ex+ kasm1)P+ k∗off(Bm+Cm)+ kin1Bm+ kdis1Cm, (6.1)dBmdt= (kon1+ k∗ex2)P+ kexBc+βCm− (k∗off+ kin1+ kin2+δ +α)Bm, (6.2)dCmdt=−(k∗off+ kdis1+ kdis2+δ +β )Cm+αBm+ k∗asm2Bc+ kasm1P, (6.3)dBcdt= kdis2Cm+ kin2Bm− (k∗asm2+ kex2)Bc. (6.4)6.3.2 Model simplificationThe observed fluorescence at the membrane M is the sum of Bc and Cm. Since the internalizationof Bm and the disassembly and internalization of the IAC depends mostly on the internalizationof integrins, we assume kdis1 ≈ kin1 and kdis2 ≈ kin2. Thus, the dynamics of the fluorescence data,Equation 6.2 and Equation 6.3, simplifies to the following equationdMdt= (kon1+ k∗ex+ kasm1)P+(kex2+ k∗asm2)Bc− (k∗off+ kdis1+ kdis2+δ )M (6.5)To simplify Equation 6.5, let kon = kon1+ k∗ex+ kasm1, kasm = kex2+ k∗asm2, koff = k∗off+ kdis1, andkdis = kdis2. This change of notation also simplifies the equations for P and Bc, leaving the talin149CmδδBmαβk∗onk∗o f fk∗o f fIm+BcP+k∗asm2 kdis2kex1 kin1Ick∗asm1 kex2kin2 kdis1Figure 6.2: Schematic off all iterations between talin and integrin on the membrane or intra-cellular, and bound or unbound. Bold letters refer to: talin P, the IAC at the membraneCm, integrin-talin bound complex at the membrane Bm, or intracellular Bc, and unboundintegrins at the membrane Im or in vesicles Ic. Arrows refer to reactions, dotted arrowsrefer to photobleaching. Refer to text for description of all the variables.Figure 6.3: Dynamics of fluorescent talin as in Equation 6.6. Labeled talin (dark green) can bind orunbind from membrane integrins (blue) or can come to the membrane by forming a complex.Talin loses fluorescence (light green) only at the membrane when bound to integrin.150turnover dynamics as followsdMdt=−(koff+ kdis+δ )M+ konP+ kasmBc,dPdt=−konP+ koffM, (6.6)dBcdt= kdisM− kasmBc,with initial conditionsM(0) =0 recovery region(M)eq reference region , P(0) = Peq, Bc(0) = (Bc)eq.The simplified parameters kon and koff are the raets of talin binding and unbinding to and frommembrane integrins. When integrins become internalized, the IAC breaks down and disassemblesat a rate kdis. Talin binds to integrin before externalization at a rate kasm, possibly forming a newIAC.Let Meq, Peq and (Bc)eq be the pre-bleaching concentration (equilibrium) of each compartment.These represent the system concentration before the experiment and remain constant before FRAP.Thus, we achive a relationship between the parameters and the equilibrium concentrations as fol-lows:dPdt=−konP+ koffM = 0 =⇒ PeqMeq =koffkon,dBcdt= kdisM− kasmBc = 0 =⇒ (Bc)eqMeq =kdiskasm.To compare between experiments, we normalize the system and definem =MMeq, p =PMeq, c =BcMeq(see Section 5.3). The normalized fluorescence recovery is then described by Equation 6.7.dmdt=−(koff+ kdis+δ )m+ kon p+ kasmc,d pdt=−kon p+ koffm, (6.7)dcdt= kdism− kasmc.151with initial conditionsm(0) =0 recovery region1 reference region , p(0) = PeqMeq = koffkon , c(0) = (Bc)eqMeq = kdiskasm .6.3.3 Parameter symmetryThe model from Equation 6.7 is linear and describes one observed variable (m) and two hidden ones(p and c). An essential feature of this system is that the missing information about the cytosolicvariables p and v creates an inherent symmetry. If you exchange kdis ↔ koff and kasm ↔ kon, youwill find the same equation for m but with opposite interpretations for p and c. Therefore, we havetwo independent sets of parameters that are equally valid, if the only available data is the time seriesof observations of m.We worked together with our experimental collaborators from the Tanentzapf’s lab at UBC todevelop an experimental protocol that will allow us to identify the parameters and circumvent thissymmetry. This protocol and the fitting procedure will be explained in Section 6.4.6.3.4 Analysis and theoretical interepretationThe characteristic polynomial of Equation 6.7 is a degree three polynomial with all its coefficientspositive.λ 3+ k2λ 2+ k1λ + k0 = 0 (6.8)where: k2 = koff+ kdis+ kasm+ kon+δ .k1 = kon(kdis+δ )+ kasm(koff+δ )+ konkasm,k0 = kasmkonδ .To find the stability of the system it is enough to find the sign of the eigenvalues. If all theroots of Equation 6.8 are real, then they have to be negative since all the coefficients are positive (byDescartes’ rule of signs).If there are one real and two complex roots, then the real one has to negative (because thecoefficients are positive). Let −λ1 and λ2± iµ be the roots of the polynomial, where λ1,µ > 0 andwe make no assumptions on the sign of λ2. Then, the characteristic polynomial is:0 =(λ +λ1)(λ −λ2− iµ)(λ −λ2+ iµ)=λ 3+(−2λ2+λ1)λ 2+(µ2−2λ2λ1+λ 22 )λ +λ1µ2+λ 22 λ1152Balancing coefficients we get:k2 =−2λ2+λ1, k1 = µ2−2λ2λ1+λ 22 , k0 = λ1µ2+λ 22 λ1.From here, we can find a cubic equation in λ2 with coefficients related to k0, k1, and k2. Removeµ by factorizing µ2+λ 22 from k0 and k1, then find the value of λ1 in terms of λ2 and k2. We knowthat λ2 is real and is a solution of:k0 =k1(k2+2λ2)+2λ2(k2+2λ2)2=k1k2+2k1λ2+2λ2k22 +8k2λ22 +2λ32⇐⇒ 0 =2λ 32 +8k2λ 22 +(2k1+2k22)λ2+ k1k2− k0.Notice that all the coefficients of the final polynomial are positive. Therefore, λ2 < 0 and all the rootsof Equation 6.8 are real or have negative real parts. Zero is an asymptotically stable steady state andthe solution will decay regardless of the initial conditions. This reflects the loss of fluorescence dueto background bleaching over time, and is precisely what we expected.From Equation 6.7, it is easy to show that the derivative of m begins positive in the recoveryregion and negative in the reference region. Then, the model shows the natural loss of fluorescencein the reference region. The model also shows the increase of fluorescence in the recovery regioncoming from the mobile fraction of talin. In both regions, the fluorescence will eventually be lostover time.6.4 Data fitting methodsThe parameter estimation was done by minimizing the SSR between the solution of m and the data,followed by creating the confidence intervals via bootstrap, as in Section 5.4. As described above,given the symmetry between p and c, the resulting bootstrap distribution was bi-modal for all ourdatasets (Figure 6.4a, b).To break the symmetry and differentiate disassembly from disassociation and assembly frombinding, we asked our experimental collaborators from the Tanentzapf’s lab at UBC if there was apossible experiment that will distrup any of the symetrical pathways. Pablo Lopez used an integrinendocytosis inhibitor drug called dynasore (DYN) to create new data sets that will help break theoriginal model symetry. Endocytosis refers to the internalization of integrins from the membraneand is necessary for the disassembly of the IAC. We assume that by using DYN to inhibit endocyto-sis we decrease the disassembly rate. We also assume that this inhibition affected none of the otherinteractions. For different talin mutants, we require different concentrations of DYN to inhibit theendocytosis and disturb the disassembly.To use DYN, we needed to add the solvent dimethyl sulfoxide (DMSO). We found no significant153(a) (b)(c) (d)0.00 0.02 0.04 0.06 0.080500100015002000Double fitSingle fitKasm(s-1)Count(double fit)(e)Figure 6.4: Double fitting protocol to estimate kasm. (a) Recovery curve of raw data for WT-Talin-GFPat 25◦C (blue), fit adjustment of Equation 6.7 (black line) and photobleaching adjustment (orangeline) calculated with the same parameters as black line, but without bleaching. (b) bootstrapdistribution for kasm from fitting the model as in Equation 6.7. The distribution is bimodal, i.e.,the parameters kon and kasm are not identifiable. (c) Data averages (dots) and fit adjustment (lines)for control (blue) and DYN (orange) assuming they share all the parameter but kdis. (d) Bootstrapdistributions for kasm fitting the control data by itself (single fit, blue) or along with DYN data(double fit, orange). (e) Final bootstrap distribution for kasm selected by the overlap with thedouble fit bootstrap distribution. The protocol is analogous for the other parameters.154difference between a sample on DMSO and a wild-type sample. Thus we used the solvent alone asour control sample. Thus, we were able to use the endocytosis inhibition data to break the symmetryand estimate the correct distribution of each rate.To distinguish the parameters, I compared DYN experiments using control experiments from thecontrol sample. Assuming that the only parameter different between them is the disassembly rateallows us to find unimodal parameter estimates. Let kdisC and kdis be the disassembly rates for thecontrol and the drug data respectively. I fitted both datasets, control and dynasore, to Equation 6.9.Notice that this equation is the same as Equation 6.7 with kon, koff, kasm and δ held the same fordatasets, but kdisC, kdis different. Then I repeated the simultaneous fitting for the bootstrap samples.For notation, I will refer to this analysis as “double fitting”, and the analysis of fitting the model toone data set as “single fitting”.DYN datadmdt=−(koff+ kdis+δ )m+ kon p+ kasmc,Control datadmCdt=−(koff+ kdisC +δ )mC + kon pC + kasmcC. (6.9)Once we find the bootstraps distributions, we assume that the parameters real distributions com-puted by fitting one dataset should be similar to those from the double fitting procedure. Thus, weselect the parameter’s range from the single fit that was closer to the double fit distribution.We repeated the protocol to each talin mutant under WT background, creating DYN and controldatasets. I find the correct parameter intervals by matching the single fit with the double fit for eachmutation. Notice we used the DYN dataset only to compute the double fitting.We could not do likewise with the mechanosensitive backgrounds. Exposing the individual tothe drug protocol along with the temperature change required to activate mechanosensitivity com-promised the organism’s health. To overcome this, we assume that the endocytosis range shouldbe similar between the mechanosensor data and the control data parameter estimates (at least theyshould be the same order of magnitude). Thus, we compare parameter distributions of mechanosen-sor mutants for each of the talin mutants with their reciprocal control double fitting distribution. Ichoose the parameter distribution closer to the one of the control double fitting.As in Section 5.4, results here report the mobile fraction from the simple exponential recoveryFRAP model and neglect the background photobleaching.6.5 ResultsTo compare the rate constants when varying the temperature from 25◦C to 37◦C, we use the relativechange given byk(37◦C)x − k(25◦C)xk(25◦C)x, (6.10)155(a) Embryo temperature control(b) BrkdFigure 6.5: Increased mechanical force at MTJs stabilizes cell-ECM adhesion by regulating talinturnover. FRAP curves, mobile fraction values and rate constant values for WT temperaturecontrol (blue, (a)) and Brkd background (red, (b)) at 25◦C (light color) and 37◦C (dark color).Each point in curves is the mean of n separate FRAP experiments. FRAP curves shown includephotobleaching. Error bars represent SEM for FRAP curves and 95% confidence intervals for pa-rameter values. “NS” not statistically significant, “∗” 95% significance, “∗∗” 99% significance,“∗∗∗” 99.5% significance.where kx represents any of the turnover parameters. Remember that the force mutations Brkd andpara present a WT phenotype at room temperature (25◦C) and a force change phenotype at 37◦C.The relative changes in rate constants tell us how increased force Brkd or decreased force (para)affects individual rate constants, giving mechanistic insight into talin turnover.We also calculated the ratios kasm/kdis and koff/kon at 25◦C and 37◦C. These ratios show thedifference that the force makes in the relationship between the assembly and disassembly of thecomplex and the binding and unbinding of talin to integrin.As in Chapter 5, the force mutations Brkd and para are active at different stages of Drosophiladevelopment. Experiments with the increased force mutation Brkd were obtained from stage 17embryos. Force reduction experiments with para were performed using third instar larvae.156(a) Relative changekon koff kdis kasmControl 99% 57% -14% -15%Brkd 117% NS -44% 263%(b) Relative ratiokoff/kon kasm/kdis25◦C 37◦C 25◦C 37◦CControl 0.08 0.07 3.88 3.84Brkd 0.10 0.06 1.28 8.32Table 6.2: Increased mechanical force at MTJs stabilizes cell-ECM adhesion by regulating talinturnover. (a ) Relative change in rate constants in percentages for the temperature control andthe Brkd experiment (Equation 6.10). (b ) Rate constant ratios for the temperature control and theBrkd experiment. “NS” not statistically significant.6.5.1 Analyzing the turnover of talin under increased forceIncreased force mutation Brkd embryos showed a reduction in the mobile fraction from 37◦C com-pared to 25◦C showing that talin responds to mechanical forces (Figure 6.5b). On the other hand,the temperature change did not affect the mobile fraction in the control data, corroborating earlierstudies that 37◦C is tolerable to WT-Talin-GFP control embryos (Figure 6.5a).The mathematical model points out explicit changes in the mechanism by detecting changesin the parameter trends. All rate constants changed with the temperature shift, probably due tometabolic changes (Figure 6.5, Table 6.2). Despide this, the temperature variation caused compara-ble trends in the model parameters for both control and Brkd, except kasm. The assembly rate kasmshowed a decrease from 25◦C to 37◦C in control, but an increase from 25◦C to 37◦C in the Brkd ex-periment (Figure 6.5, Table 6.2). We quantified the parameter trends by comparing the temperaturesusing Equation 6.10 (Table 6.2a).The ratio koff/kon was similar at both temperatures for control and Brkd experiments (Ta-ble 6.2b). The contrast showed up in the kasm/kdis ratio. This ratio remained constant at bothtemperatures in control data (≈ 3.8), but increased in the Brkd experiments (1.28 at 25◦C vs 8.32 at37◦C). The ratio variability was consistent with a higher rate of adhesion complex assembly underincreased force. Overall, these data revealed that an increased assembly rate stabilizes talin.6.5.2 Resolving the mechanism by which the turnover of talin is regulated byincreased forceWe sought to achieve further mechanistic detail by performing force-increase esperiments usingmutated talin (see Section 6.2). All the talin mutants responded to increased force similar to WTBrkd talin. They presented a reduction in the mobile fraction when the temperature increased (Fig-ure 6.6, Table 6.3). The mobile fraction reduction was significant in Brkd WT and all the mutations,157(a)(b)(c)(d)Figure 6.6: FRAP analysis of talin mutants uncovers mechanisms that regulate adhesion turnover inresponse to force. FRAP curves, mobile fraction values and rate constant values for Brkd back-ground at 25◦C (light colors) and 37◦C (dark colors) in (a) integrin activation mutant L334, (b)inside-out activation mutant IBS-1, (c) talin binding site IBS-2, (d) actin binding mutant. Eachpoint in curves is the mean of n separate FRAP experiments. FRAP curves shown include photo-bleaching. Error bars represent SEM for FRAP curves and 95% confidence intervals for parame-ter values. “NS” not statistically significant, “∗” 95% significance, “∗∗” 99% significance, “∗∗∗”99.5% significance.158(a) Relative changekon koff kdis kasmControl 99% 57% -14% -15%Brkd WT 117% NS -44% 263%Activation mutant NS -17% -58% NSIBS-1 mutant 42% -16% -49% NSIBS-2 mutant 116% 51% -58% -28%ABS mutant NS -48% NS NS(b) Relative ratiokoff/kon kasm/kdis25◦C 37◦C 25◦C 37◦CWT 0.10 0.06 1.28 8.32Activation mutant 0.16 0.14 0.74 1.65IBS-1 mutant 0.20 0.12 0.66 1.27IBS-2 mutant 0.06 0.04 2.90 4.96ABS mutant 0.08 0.05 3.03 3.56Table 6.3: FRAP analysis of talin mutants uncovers mechanisms that regulate adhesion turnover inresponse to force. (a) Relative change in rate constants (Equation 6.10). (b) Rate constant ratios,calculated for Brkd wild-type data and Brkd mutants at 25◦C and 37◦C. “NS” not statisticallysignificant.except the activation mutant (Figure 6.5 and Figure 6.6a).We analyzed the estimated parameter values in detail. WT and mutations showed similar trendsover temperature shift for kdis (Table 6.3a). All experiments also showed an increase (or no change)in kon with the temperature modification (Figure 6.6a and Table 6.3a). Surprisingly, kasm showedeither a non-significant difference or a slight increase from 25◦C to 37◦C, similar to the control butopposite to the WT Brkd talin (Table 6.3a). Nonetheless, the activation, IBS-1, and actin bindingmutants displayed a decrease in the koff rate perhaps explaining their ability to modulate turnover inresponse to increased force (Figure 6.6a,a).The ratios between the rates acted differently for the two possible pathways, on-off vs asm-dis. The koff/kon ratio had a similar range for both temperatures for WT Brkd talin and the talinmutants, except the IBS-1 mutant. In the IBS-1 mutant, the ratio was reduced almost by half withthe temperature shift (Table 6.3b). koff always became proportionally smaller than kon at 37◦C. Theratio kasm/kdis differed for WT Brkd talin and the mutants at 25◦C as compared to 37◦C, althoughthe mutants showed a smaller difference. This indicated that the kasm/kdis ratio is dysregulated inthe mutants under increased force, and so the adhesion assembly is not as favored as in the wild-type (WT).Taken together, these results showed that talin stabilization in response to increased force oc-curred by increasing the rate of assembly of new adhesions. This increase in the rate of assemblywas not present on the mutants, highlighting the assembly dependency on the ability of talin to ac-159(a) Relative changekon koff kdis kasmControl -56% -54% -62% -72%para NS -26% NS NS(b) Relative ratiokoff/kon kasm/kdis25◦C 37◦C 25◦C 37◦CControl 0.03 0.03 13.06 9.56para 0.05 0.03 9.51 12.07Table 6.4: Decreased mechanical force at MTJs modifies Talin turnover. (a) Relative change in rateconstants for temperature control and para mutant (Equation 6.10). (b) Rate constant ratios. “NS”not statistically significant.tivate integrin, reinforce the adhesion complex, and link to actin. Other compensatory mechanismscould still stabilize talin mutants under Brkd, like a smaller unbinding rate for the activation, IBS-1and actin binding mutants. In other words, the mutants stabilize talin by decreasing unbinding fromalready assembled complexes. However, those compensatory mechanisms resulted in a less robustresponse to increased force, since the mobile fraction difference is smaller than that of the WT-Brkdtalin experiments.6.5.3 Analyzing the turnover of talin under decreased forceWe investigated the turnover of the IAC under reduced force compared to wild-type using the treat-ment in third instar larvae as described in Section 6.2. As expected, there were no differences inthe mobile fraction for the larvae temperature control (Figure 6.7a). As in earlier integrin observa-tions [159], there was no significant change in the mobile fraction of talin upon induction of para(Figure 6.7b).Despite the overall constant mobile fraction, the estimated parameter values changed in eachdataset (Figure 6.7,Table 6.4). The temperature control data showed a uniform, across-the-boarddecrease with the temperature increment in all four rate constants (Figure 6.7a,c, Table 6.4a). Inthe para background, the only significant change happened in the koff rate, which decreased withthe temperature shift (Figure 6.7b,c, Table 6.4a). This indicated that kon, kasm, and kdis increased inthe para experiment compared to control. When comparing the rate ratios, we observed only minordifferences between the control and para data (Table 6.4b). This observation explains how changesin the individual rates occurred while the mobile fraction remained constant. These experimentssuggest that reducing force produces a more rapid talin turnover. These insights cannot be obtainedfrom simple mobile fraction analysis. We needed our modeling procedures and the parameter fittingto find out the difference between control and para treated individuals.160(a) Larvae temperature control(b) para(c)Figure 6.7: Decreased mechanical force at MTJs modifies Talin turnover. FRAP curves, mobile frac-tion values and rate constant values for WT talin tagged with GFP at 25◦C (light colors) and 37◦C(dark colors) in (a) WT background (larvae temperature control) and in (b) para background. (c)Relative change in rate constants illustrated in a graph (Equation 6.10). Each point in curves is themean of n separate FRAP experiments. FRAP curves shown include photobleaching. Error barsrepresent SEM for FRAP curves and 95% confidence intervals for parameter values. “NS” notstatistically significant, “∗” 95% significance, “∗∗” 99% significance, “∗∗∗” 99.5% significance.161(a) Relative changekon koff kdis kasmTemp control -56% -54% -62% -72%para WT NS -26% NS NSActivation mutant NS NS NS NSIBS-1 mutant NS -22% 15% NSIBS-2 mutant 155% 160% NS 80%ABS mutant NS NS NS NS(b) Relative ratiokoff/kon kasm/kdis25◦C 37◦C 25◦C 37◦Cpara 0.05 0.03 9.51 12.07Activation mutant 0.06 0.04 6.07 7.06IBS-1 mutant 0.06 0.06 4.72 4.08IBS-2 mutant 0.02 0.02 11.37 23.10ABS mutant 0.04 0.04 9.27 9.63Table 6.5: The Talin IBS-2 domain is essential to coordinate turnover in response to reduced me-chanical force. (a) Relative change in rate constants for all the mutations under para tratement(Equation 6.10) (b) Rate constant ratios at 25 and 37◦C within each experimental condition. “NS”not statistically significant.6.5.4 Uncovering the mechanism by which the turnover of talin is regulated bydecreased forceUpon induction of para, the mobile fraction of wild-type talin and mutants was unaltered, except bythe IBS-2 mutant where the mobile fraction was reduced (Figure 6.8, Table 6.5). For the activationmutant and the actin-binding mutant, the decreased force affected none of the rate constants sig-nificantly (Table 6.5a). For the IBS-1 mutant, koff and kdis displayed negative and positive changesrespectively, with no variation in the rest of the parameters. The IBS-2 mutant produced the highestresponse to decreased force. It exhibited 155% relative change in kon, 160% in koff, and 80% inkasm, contrasting to virtually no change in WT (Table 6.5a). The IBS-2 mutation affected adhesioncomplex assembly and stability [57].Moreover, the koff/kon ratio remained rather constant with the temperature shift for all the mu-tations, showing that the decreased force background did not affect the binding rates. However,the kasm/kdis ratio doubled in the IBS-2 mutant upon induction of the para phenotype. Thus theincrease in the overall assembly of adhesion caused adhesion stability in the IBS-2 mutant. Theseresults showed that reduced force increases the rate constants in a harmonized fashion, allowing theturnover to increase while maintaining talin at the membrane in a stable fashion. This depends onthe activity of the IBS-2 talin domain.162(a)(b)(c)(d)Figure 6.8: The talin IBS-2 domain is essential to coordinate turnover in response to reduced mechan-ical force. FRAP curves, mobile fraction values and rate constant values for para backgroundtalin tagged with GFP at 25◦C (light colors) and 37◦C (dark colors) in (a) integrin activationmutant L334, (b) inside-out activation mutant IBS-1, (c) talin binding site IBS-2, (d )actin bind-ing mutant. Each point in curves is the mean of n separate FRAP experiments. FRAP curvesshown include photobleaching. Error bars represent SEM for FRAP curves and 95% confidenceintervals for parameter values. “NS” not statistically significant, “∗” 95% significance, “∗∗” 99%significance, “∗∗∗” 99.5% significance.163Figure 6.9: Reduced FAK activity affects integrin turnover in a manner similar to increased force.FRAP analysis, mobile fractions and rate constant estimations performed to talin control flies,and flies expressing the nonactivatable FAK-Y430F mutant. Each point in FRAP curves is themean of n separate FRAP experiments. FRAP curves shown include photobleaching. Error barsrepresent SEM for FRAP curves and 95% confidence intervals for parameter values. “NS” notstatistically significant, “∗” 95% significance, “∗∗” 99% significance, “∗∗∗” 99.5% significance.(a) Relative changekon koff kdis kasmFAK-Y430F 150% NS -32% 27%(b) Relative ratiokoff/kon kasm/kdisControl Y430F Control Y430FFAK-Y430F 0.061 0.035 5.6 10.6Table 6.6: Reduced FAK activity affects integrin turnover in a manner similar to increased force. (a)Relative change in rate constants between control and FAK mutant (Equation 6.10). (b) Rateconstant ratios from control and FAK-Y430F-expressing flies. “NS” not statistically significant.6.5.5 Reducing FAK activity partially mimics the effect of increased force on talinturnoverAs described in Section 6.2, focal adhesion kinase (FAK) is an essential regulator of integrin re-sponse to force. Thus, we investigated if FAK activity produces comparable effects as the Brkdphenotype. We used the FAK-Y430F mutant that blocks FAK activation.The FAK mutant shows a lower mobile fraction compare to WT (Figure 6.9, Table 6.6). Whenanalyzing the parameter estimates we found an increase in kon and kasm, and a decrease in thedisassemble rate (Table 6.6a). The ratio of assembly vs disassembly raised by from 5.6 to 10.6,while the binding rate ratio (koff/kon) declined by 42% (Table 6.6b). Overall, the data showed thatpreventing FAK activation promoted the assembly and suppressed the breakdown of the junction.The FAK mutant showed the same patterns in parameters relative changes as those observedwith Brkd (Table 6.2 and Table 6.6). The difference in the mobile fraction of the FAK mutant was164Figure 6.10: Talin turnover is developmentally regulated through distinct mechanisms. FRAP anal-ysis of WT-Talin-GFP, mobile fraction and rate constants values at progressive developmentalstages: embryonic stages 15 (e15), 16 (e16), and 17 (e17) and first (L1) and third (L3) larvalinstars. Each point in curves is the mean of n separate FRAP experiments. FRAP curves showninclude photobleaching. Error bars represent SEM for FRAP curves and 95% confidence inter-vals for parameter values. “NS” not statistically significant, “∗” 95% significance, “∗∗” 99%significance, “∗∗∗” 99.5% significance.not as that of the Brkd treatment, probably since the assemble rate increased 10 folds more underincreased force. Our experiments suggest that FAK inactivation affects turnover similarly as theincreased force treatment.6.5.6 Changes in rate constants underlie the developmental regulation of TalinturnoverTo analyze the changes in talin dynamics as the organism develops, we repeated the WT talin ex-periments on embryonic stage 16, embryonic stage 17, first instar larvae and third instar larvae(see Section 6.2). The analysis identified different mechanisms for MTJ stabilization between theembryos and larvae. First, the mobile fraction displayed an obvious decrease between embryonicstages, but an increase between larval instars (Figure 6.10). For embryos, the rate constants declinedbetween stages 15 and 16, but the binding and assembly rates remained constant from stage 16 to17 (Figure 6.10). During larval development, the assembly rate increased again, while the bind-165ing/unbinding constants remained unchanged. The data showed that, IAC assembly increases inlate development stages while talin unbinding and IAC disassembly is faster in early development.We, therefore conclude that stage-specific modulation of the different turnover processes ac-complishes stable cell-ECM adhesions at the MTJ. During early development, when muscles beginto contract, a reduction in all the rate constants stabilizes the adhesions, while in later embryonicstages, the adhesion complex disassembly and talin unbinding rates decline. In contrast, in larvalstages, the binding rate of talin increases, stabilizing cell-ECM adhesions.6.6 DiscussionIn this project, we investigated the response of talin turnover to different forces or developmentalstages, and the corresponding effect in integrin-based adhesions. We used genetic and imaging toolspreviously established for integrin turnover with some substantial technical upgrades [159, 220].Modeling talin turnover proved to be challenging since talin has multiple pathways to and from themembrane. This is a challenge for any intracellular protein that binds to a membrane. I overcamethis challenge by creating a novel two-step “double-fitting” strategy.The “double-fitting” strategy interweaved mathematical and cell biological tools to study themechanisms affecting talin turnover. The method used a well-characterized pharmacological per-turbation along with a costum mathematical treatment of the system. This mix of methods was es-sential to understand and differentiate between the membrane recovery pathways of talin. We usedthe parmacological agent dynasore (DYN) to disrupt one recovery pathway. We then compared tothe drug-free experiment by estimating the mathematical parameters under each conditions, thusidentifying the characteristics of the pathways. This approach is versatile and could be customizedto study other membrane-bound complexes.Applying the double-fit protocol and Brkd treatment, the data indicated an increase of the as-sembly rate under elevated force. We analyzed activation, integrin bingind, and actin binding mech-anisms to investigate the increased assembly rate trigger. Integrins react similarly in response toincreased force [160], and thus talin turnover inherited integrin behavior. Moreover, the assemblyrate did not increase in integrin activation mutants under a higher force. Focusing on talin, mechani-cal forces stretch and expose the binding sites [47, 174]. Thus, more force could translate into morebinding sites, leading to higher recruitment and assembly of the IAC. Our analysis with actin-talinand integrin-talin binding mutants showed no increase in the assembly rate, consistent with this sec-ond mechanism. These results are in line with earlier work where a higher force on MTJs correlatedwith increased availability of integrins at the membrane [159].Talin turnover did not show a straightforward interpretation under reduced force in para exper-iments. We found a constant mobile fraction but higher turnover rates. Again, integrins showedsimilar results [159]. We speculated that higher rates with a constant pool of proteins increase theturnover without compromising IAC stability. Our results using IBS-2 mutant with weakened force,166along with earlier culture experiments, uphold this speculation [57, 96].To our surprise, the data suggested that mechanical force is not the only factor regulating theturnover during Drosophila development. Such a conclusion comes from observing the changesin rate constants over development. Comparing WT and increased force mutants, kasm does notchange in embryonic stages and shows only a small change in larvae. This is contrary to the earlierobservation where the assembly rate is the main respondent to increased force conditions. Theseresults resemble those from cell culture where increased force by itself is not always sufficient togenerate stable focal adhesions [151, 152, 190].Based on our results and earlier studies, we propose the following model for the effects of forceon the adhesion complex turnover in live fly MTJs. Higher mechanical force decreases the removalof integrins from the membrane, resulting in a pool of stable integrins in IAC at the membrane[159]. Higher forces also produce conformational changes in talin, recruiting more componentsto the IAC. Thus, outside-in signaling is robust and the IAC is reinforced. With lower force, theturnover rates rise in a harmonized manner without altering the global pool of integrin and talin thatundergoes turnover. The integrin binding site 2 of talin plays a crucial role in regulating turnover inresponse to decreased force [159]. Our data show an inverse effect of the forced induced turnover ofIAC components: more force inhibits turnover, while less force promotes turnover. Taken together,these mechanisms optimize tissue strength and stability under variations of mechanical stress.167Chapter 7ConclusionNow this is not the end. It is not even the beginning of the end.But it is, perhaps, the end of the beginning— Winston Churchill, 1942.This dissertation focused on analyzing, interpreting and understanding biological systems usingmicroscopy data and mathematical models. In particular, we analyzed direct Stochastic Optical Re-construction Microscopy (dSTORM) and fluorescence recovery after photobleaching (FRAP) datausing stochastic and deterministic models choosen according to the characteristics of the system.Understanding the spatial distribution of B-cell receptors (BCR) on the membrane of resting B-cellsmotivated the dSTORM chapters, whereas FRAP analysis was exploited to study the dynamics ofmembrane proteins in the cell to extracellular matrix (ECM) adhesion of Drosophila embryos andlarvae. Here, we innovated by designing and implementing new methods to connect the fluorescencemicroscopy with the biology, through changes in interpretation (FRAP) or with a new quantitativemethod (dSTORM).We divided the thesis into two parts, reflecting a division between of these microscopy tech-niques, and, implicitly, a particular biological motivation and an original mathematical view. Thus,this dissertation contains two major results, one for each part of the thesis. Each result exhibitsthe applicability of mathematical thinking to reveal otherwise hidden information within the mi-croscopy data and so enhance our biological knowledge. In summary, in this thesis:1. We developed a method to identify unique observations in dSTORM super-resolution mi-croscopy.2. We created mechanistic models to describe the outside-in and inside-out signalling behindcell-ECM adhesion and analyzed FRAP curves to infer behavioural changes in mutant organ-isms.We discuss results separately in the following sections.1687.1 Identifying unique observations in direct Stochastic OpticalReconstruction Microscopy (dSTORM)As described in the introduction, dSTORM, and similar super-resolution techniques, allow us to seenanometer resolution images for the first time with almost no disruption to the biological system.There are several downsides to the methods used to reconstruct those nanometer images. We fo-cused on one of those problems: multiple observations (blinking) arising from the same fluorescenttag. This problem is important since observing a fluorescent tag multiple times might form smallartificial clusters of localizations and could lead to a misinterpretation of the underlying biology.Thus, analyzing the dynamics of one fluorescent tag will enhance the spatial interpretation of thesuper-resolution image, and hence change the biological interpretation.Our research question was: How many unique fluorescent labels are present in a dSTORMimage? We answered this question by creating a spatiotemporal model as described in Chapter 2and Chapter 3. We described the blinking properties of the fluorophores with a discrete stochas-tic Markov chain model to model the transitions between visible/invisible states, and a GaussianMixture model (GMM) to account for a single fluorescent tag appearing in multiple frames withslightly different spatial localization. We simulated our underlying model and then tested our al-gorithm by recovering all the defining parameters of the model (Chapter 2 and Chapter 3). We didall our parameter estimation using negative log-likelihood minimization. When all the fluorophoredynamics are known, such as the transition probabilities, our protocol recovers the number of fluo-rophores almost perfectly (Chapter 2). That case is of course utopian since that information is rarelyavailable.For the interesting case, when parameters of the system are unknown a-priori, we recovered thenumber of fluorophores in each sample by coupling the temporal and spatial information from thedata (Chapter 3). In particular, for those samples with few observations, our estimates improvedhighly when we also included the spatial information (localizations and uncertainties, Chapter 3).The estimated spatial errors played an important part in our fitting since they balanced the distancesbetween localizations. If two localizations had overlapping errors, the chances of them coming fromthe same fluorophore increased. On the other hand, if they have widely separated relative to theirspatial errors, the pre-processing protocol distinguished them reliably.After extensive testing of our method over different simulated data sets, we used our algorithmto estimate the number of fluorophores in an experimental control: DNA origami (Chapter 4). Weexpected our samples to have between one to four fluorophores, and we recovered those values onalmost 90% of our available samples. The use of this experimental control data allows us to validateour method before application to a biological data set where no information is available ahead oftime. We believe our estimates could be improved by adding more data sets and thus we trust ourmodel assumptions and our estimate values.Finally, we applied our protocol to identify unique fluorophores in dSTORM images of BCR.169We analyzed 5 different resting B-cells and found similar estimated distributions of the numberof dSTORM localizations per fluorophore. Each fluorophore is estimated to appear on averagearound 10 different times in the reconstructed super-resolution image. Thus, apparent clusters mightcontain only a tenth of the number of molecules we might imagine. This highlights the importanceof identifying individual molecules before drawing conclusions about the spatial distribution. Inthis thesis work, we conclude nothing about the size of nano-clusters in BCR, we only prove theusefulness of our algorithm. To draw conclusions, we will need to analyze a larger data set. Thisis an obvious future application of our algorithm. We will also like to compare the distribution ofBCR between resting and activated B-cells. We will need to analyze activated B-cell data, correct tounique observations, and then we will apply clustering methods to determine features of the spatialdistribution of BCR that triggers the characterize activated versus resting B-cells.In an ideal world, a variant of the temporal part of our model should be included in the initialanalysis of row dSTORM data to improve the reconstruction results. This is needed since thereare some discarded observations during the reconstruction process and these might bias the local-ization output data. That will be an extension of our algorithm, along with the possible Bayesianimplementations or the self-activated point process (mentioned in the discussion of Chapter 3).7.2 Analysis of fluorescence recovery after photobleaching (FRAP)data to understand cell adhesionIn Chapter 5 and Chapter 6, we developed mechanistic models to analyze and interpret FRAP curvesfrom developing Drosophila organisms. Usually, FRAP curves are seen as a simple exponential-type function whose parameters relate to the fraction of mobile proteins and the mean recoverytime of the fluorescence. In contrast, we constructed our models via biological assumptions aboutbehaviour of the labelled protein.Using Drosophila as the biological model and FRAP as our experimental method, we answeredtwo important research questions using linear ordinary differential equations and parameter esti-mation. In both chapters, we imaged the muscle-tendon junctions of live Drosophila embryos andlarvae. The main experimental difference was that in Chapter 5 we fluorescently labelled the mem-brane protein integrin, while in Chapter 6 we labelled the cytosolic protein talin. In all our models,we also added a parameter measuring the loss of fluorescence (photobleaching) during the experi-ment.1. How does outside-in activation stabilize cell-ECM adhesions? (Chapter 5)Outside-in signalling is the secondary step in cell-ECM adhesion, and it is triggered by theinteractions between the integrin adhesion complex (IAC) and ECM ligands (Figure 5.1). Toinvestigate the importance of this signalling, we analyze and compared FRAP curves obtainedfrom fluorescently labelled integrins under different biological settings (integrin and ECM170ligands mutants) and environmental settings (magnesium, manganese and calcium chemicalmodifications). We described the system as the dynamics of two populations: integrins atthe membrane and integrins inside the cell. By fitting this simple model to the FRAP curves,we found that increasing integrin activation (either with integrin mutants or through chemicalinduction) resulted in more stable cell-ECM adhesions. The reduction of integrin turnovercaused this stabilization. Moreover, we found that exposure to divalent cations (manganeseand magnesium) has little to no effect on integrin mutants. In particular, the ectopic chemi-cal fails to regulate integrins in the presence of integrin disrupting mutants. Finally, we alsodiscovered that reducing the ECM ligands that interact with integrin resulted in less stableadhesions because of an increased integrin internalization rate. Our experiments confirm thatboth integrin and ECM ligands are required for outside-in signalling, and that ectopic chem-ical treatment will not be enough to balance integrin activation of mutants but it stabilizesthe adhesions for control organisms. The mathematical models were essential to discover theintegrin pathway behind the stabilization of the adhesion.2. How does the cytosolic protein talin affect the inside-out integrin signalling in cell-ECMadhesions under different force schemes? (Chapter 6)We continued our work on cell-ECM adhesion analysis by now focusing on the inside-outsignalling of integrins. We studied this signalling from the perspective of talin, which is oneof the essential proteins in inside-out signalling. Our mathematical model described the in-teraction between three talin populations: free talin, talin-integrin bound at the membraneand talin-integrin bound inside the cell. FRAP experiments were not enough to differenti-ate between free talin and cytosolic talin-integrin complexes, thus we improved our analysisby adding a comparing set to our control data where we assumed that the population of cy-tosolic talin-integrin bound decreased. With this double-fitting approach, we found out thattalin turnover is reduced by increased force, which translates into less stable adhesions. Weexplicitly found that talin under increased force had a higher rate of assembly of new adhe-sions compared to regular force settings. We did not observe that increase in assembly ratefor talin mutant organisms but a decrease in the unbinding of assembled complexes. Theresponse of talin mutants to increased force was less robust than the control sample. Withdecreasing force, we found similar mobile fractions; but our double-fitting approach showedthat there were dynamic changes in the samples: they had a more rapid talin turnover. Rapidtalin turnover also translates into less stable adhesions. By analyzing talin mutants, we foundthat reduced force increases the rate constants in a harmonized fashion, allowing the turnoverto increase while maintaining talin at the membrane.171Bibliography[1] E. Abbe. 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We vary the rates such that we will observeone fluorophore blinking many times (kb < kd) or once time (kb > kd). We defined 5 differentparameter sets as described in Table A.1, with S1 the set with the smallest bleaching rate and largestdeactivation rate, up to S5 which has the opposite characteristics. For simplification, we kept theactivation rate kr = 1 per second for all of S1-S5.Using the Gillespie algorithm, we created 10,000 replicates of the model for each parameterset S1-S5. We wrote the Gillespie algorithm in C and the data analysis and parameter estimationin MATLAB. The code ensured that all fluorophore beaching and transition times were recordedexactly.We can calculate the expected time to bleach for each parameter sets. We used Equation 2.17as our theoretical value. The simulated value was calculated by taking the mean over the 10,000simulations of the total time of each simulation.We also estimated the parameters and main statistics for the distribution of the number ofblinks, Nblinks. Assuming all the samples have the same number of fluorophores N, we know thatNblinks ∼NB(N,η) (see Equation 2.14). Since in here we simulated only one fluorophore, we definea blink to be one activation. Thus, Nblinks = NUobs = Nloc. For theoretical values, we calculated the189Parameter set S1 S2 S3 S4 S5N 1kr (s−1) 1kd (s−1) 1.6 1.3 1 0.5 0.1kb (s−1) 0.1 0.2 0.5 1 1.6Table A.1: Parameter values used to simulate a single fluorophore. We used 5 different sets of param-eters to define different blinking properties of the fluorophore. All the parameter sets have a fixedactivation rate (kr = 1 per second). The parameter sets S1-S5 are ordered by increasing bleachingrate and decreasing deactivation rate.(a) S1 (b) S3 (c) S5Figure A.1: Distributions of the number of blinks from the simulated data with N = 1 for parame-ter sets S1, S3 and S5. The histograms (blue) correspond to the normalized frequency of thenumber of observations for the 10,000 simulations. The red line corresponds to the probabilitydistribution of a negative binomial with parameters given by the MLE for N and η .mean, variance, and 99 percentile values using the real parameter values and the negative binomialdistribution. We calculated the mean, variance and 99th percentile of the number of localizationsper sample and reported as our simulated values. To estimate N and η , we used a custom-made codeand the function nbinfit in MATLAB. Figure A.1 shows the normalized histogram of the simulateddata, and the NB probability distribution at the MLE for N and η for datasets S1, S3, and S5. Also,we notice the differences between the parameter sets in the histograms, where fluorophores withparameters from S1 have the most blinking cycles while those from S5 have the fewest.In Table A.2, we compared the theoretical values calculated with Equation 2.14 and Equa-tion 2.17, and the values estimated from the 10,000 simulations for S1-S5. Table A.2 shows that thetheoretical and estimated parameters are consistent.190S1 S2 S3 S4 S5Nloc S 170,480 76,015 29,997 14,983 10,652E(Tbleach)T 27s 12.5s 5s 2.5s 1.68sS 26.99s 12.67s 4.97s 2.49s 1.70sE(Nblinks)T 17 7.5 3 1.5 1.06S 17.05 7.60 3 1.50 1.07SD of NblinksT 16.49 6.98 2.45 0.87 0.26S 16.27 7.01 2.47 0.86 0.26Nblinks 99 percT 76 33 12 5 2S 73.5 32.5 12 5 2η (nbinfit) T 0.94 0.87 0.67 0.33 0.06S 0.94 0.87 0.67 0.32 0.07N (nbinfit)T 1 1 1 1 1S 0.995 1.02 0.98 1.04 0.90Table A.2: Comparing theoretical and simulated statistics of the Nblinks and Tbleach for one fluorophore.Each parameter set S1-S5 was simulated 10000 times. Nloc refers to the total number of localiza-tions obtain in the simulations. The theoretical values, in T rows, were calculated using Equa-tion 2.14, Equation 2.17 and the corresponding percentile value of a geometric distribution. Thesimulated values, in S rows, using the mean, standard deviation, and percentile MATLAB func-tions to the observed number of blinks and time before bleaching. The last two rows, η andN, were calculated using the function nbinfit from MATLAB, which gives you the MLE for anegative binomial distribution.A.2 Estimates of the kinetic rates depend on the sample size of thedata for the continuous-time model when the number offluorophores is knownTo test our optimization for the kinetic parameters, we fix the number of possible fluorophores inthe fitting algorithm to match the true value N = 1. This is equivalent to setting Amax(i) = Nloc(i) =N = 1 in Algorithm 2.3 for all the samples.The kinetic parameters are the same for all the samples from each parameter set. Thus, the MLEshould converge to the real values as we increase the sample size. We test this by sub-sampling our10,000 simulated samples in groups of size M = 10,100,1000, and 10000. This gives 100 sub-samples of size M = 10,100 but only one sample of size M = 10000 and 10 of size M = 1000. Wefit each sub-sample independently to obtain the corresponding MLE of the kinetic parameters.Figures A.2 and A.3 show boxplots of the MLE, varying the sample size. kr, kd and kb are inthe first, second and third column respectively. Parameter estimates with parameters sets S1, S2 andS3 are in Figure A.2, while Figure A.3 has the results from parameter sets S4 and S5. For all theparameter sets, the MLE converges to the real value (blue horizontal line) as we increase the samplesize (M). We observe that the accuracy is high even with M = 100.The fit for 10,000 simulations has high accuracy, but it takes more computational power. For this191(a) MLE for S110 100 1000 10000Sample size0.911.11.2Parameter valuekr10 100 1000 10000Sample size11.21.41.61.82Parameter valuekd10 100 1000 10000Sample size0.050.10.150.2Parameter valuekb(b) MLE for S210 100 1000 10000Sample size0.811.21.4Parameter valuekr10 100 1000 10000Sample size11.21.41.6Parameter valuekd10 100 1000 10000Sample size0.20.40.60.8Parameter valuekb(c) MLE for S310 100 1000 10000Sample size11.52Parameter valuekr10 100 1000 10000Sample size0.60.811.21.41.6Parameter valuekd10 100 1000 10000Sample size0.511.5Parameter valuekbFigure A.2: MLE for the parameter sets (a) S1, (b) S2 and (c) S3 for different sample sizes whenN = 1 is fixed. Box plots of the MLE and true values (blue line) of kr, kd and kb are in the first,second and third column respectively. Sample size M = 10 and M = 100 were repeated 100times, M = 1000 was repeated 10 times and M = 1000 was repeated only one time.192(a) MLE for S410 100 1000 10000Sample size0.60.811.21.41.61.8Parameter valuekr10 100 1000 10000Sample size00.511.5Parameter valuekd10 100 1000 10000Sample size0.511.522.533.5Parameter valuekb(b) MLE for S510 100 1000 10000Sample size0.511.52Parameter valuekr10 100 1000 10000Sample size00.10.20.30.40.50.6Parameter valuekd10 100 1000 10000Sample size11.522.533.54Parameter valuekbFigure A.3: MLE for the parameter sets (a) S4 and (b) S5 for different sample sizes when N = 1 isfixed. Box plots of the MLE and true values (blue line) of kr, kd and kb are in the first, second andthird column respectively. Sample size M = 10 and M = 100 were repeated 100 times, M = 1000was repeated 10 times and M = 1000 was repeated only one time.model, taking 10,000 samples in subgroups of 100, 1000 or 10,000 did not reduce the speed of thecalculation. Therefore, a higher sample size is encouraged if the computational power is available.A.3 Estimating the number of fluorophores when the kinetic ratesare known depends on the kinetic parameters for thecontinuous-time modelWe now assume the kinetic rates to be known and test the ability of the algorithm to correctly predictN = 1. In this case, the likelihood function varies with the number of fluorophores only, and thuseach estimated value Ni dependes only on the infomation from sample i. We compute likelihoods193S1 S2 S3 S4 S500.20.40.60.81probabilityN fit =1 N fit =2 N fit =3 N fit =4Figure A.4: Estimates of the number of fluorophores with known kinetic parameters for all the pa-rameter sets. Bars are grouped by the different parameter sets S1-S5. All the samples wereestimated to have 1 (dark blue bars), 2 (light blue bars), 3 (green bars), or 4 (yellow bars) distinctfluorophore(s).in parallel over the samples ( Algorithm 2.3).We found the value of Ni that minimizes the likelihood function for each one of the 10,000samples of each parameter sets S1-S5 (Figure A.4). The possible domain of Ni ranges from Amax(i),equal to one for all samples, to Nloc(i). Note that Nloc(i) varies a lot depending on the parameterkinetics, from large values in S1 to values smaller than 4 in S5 (Table A.2).Figure A.4 shows the estimates of the number of fluorophores for each sample (color bars) foreach parameter set. For parameter sets S1, S2, and S3, more than 90% of the samples correctlyestimated Nfit = 1. We calculated the equivalent likelihood-ratio test for the NLL and found thatthe differences NLL(Nˆi)−NLL(1) < 3 for all samples from S1-S3 where Nˆi is the MLE value forsample i. The distance of between the negative likelihood smaller than 3 is usually interpreted tobe within the 95% confidence interval, and thus we cannot discard that the true value is N = 1.Therefore, our optimization approach essentially estimates the correct number of fluorophores forS1, S2, and S3 parameter sets.For parameter sets S4 and S5, we cannot distinguish the observations from the fluorophoresas precisely. The estimates for Nfit are 80% accurate for S4, while most of the samples form S5have Nfit = Nloc (Figure A.5). Thus, S4 and S5 are less accurate, with S5 localizations being almost1940 1 2 3 4 5 6 70200040006000800010000frequencyS4S5Figure A.5: Difference between the number of localizations and Nfit for S4 (blue bars) and S5 (yellowbars) parameter sets. Most of the samples obtained using S5 parameters did not have enoughinformation to distinguish the correct number of fluorophores from the number of observations.More than 90% of the estimates of samples from parameter set S4 matched the number of obser-vations minus 0 or 1.Parameter set Range of Nfit Range of NUobs Samples NUobs < 20S2 [9,22] [43,245] 0S3 [10,23] [20,104] 1S5 [14,22] [16,24] 9,954Table A.3: Range intervals of the number of unique observations and the estimated number of fluo-rophores, and the number of simulations with less than 20 localizations, for N = 16 and S2, S3and S4 parameter sets. The total number of simulations for each paramter set is 10,000.indistinguishable for the model (Figure A.5).To test if our fitting procedure works with multiple simulated fluorophores, we created 10,000simulations with N = 16 fluorophores using parameter sets S2, S3 and S5. We excluded S1 param-eters since the number of localizations will be too large (≈ 3× 107), and S4 since we expect it tobehave similarly to S5. We verified that the number of unique observations, NUobs, followed a neg-ative binomial distribution as theoretically prescribed. The total NUobs for the 10,000 simulationswas equal to 1,199,482 for S2, 479,827 for S3, and 166,956 for S5.We estimated the number of distinct fluorophores, again assuming the known kinetic rates (Fig-ure A.6). The estimated number of fluorophores ranged from 9 to 23 distinct fluorophores with amode of 16 (Figure A.6 and Table A.3). The huge difference between the number of observationsand the estimated number of fluorophores point to a good convergence of the data for S2 and S3parameter sets. As expected from the results with one fluorophore, the samples from S5 were moresimilar to the number of unique observations.To summarize, these results show that our algorithm can estimate the number of distinct fluo-195S5 S3 S200.050.10.150.20.250.30.350.40.450.5probability10121416182022NfitFigure A.6: Estimates of the number of fluorophores with known kinetic parameters for all the param-eter sets and N = 16. Bars are grouped by the different parameter sets (S5, S3 and S2). Colorscale corresponds to the number of fluorophores estimated Nfit. All the samples were estimatedto have between 9 (dark blue bars) and 23 (yellow bars) distinct fluorophores. The bar corre-sponding to 16 distinct fluorophores is the tallest bar for all parameter sets (green-aqua bars).Parameter set M = 10 M = 100 M = 1000 M = 10000 Nloc = 1S1 999 9990 9992 9992 619S2 977 9898 9910 9911 1301S3 721 8402 8707 8690 3354S4 702 6655 6652 6652 6652S5 925 9390 9390 9390 9390Table A.4: Number of simulations correctly estimating one fluorophore for the different sample sizesand parameter sets. The total number of samples with exactly one localization is given in the lastcolumn as a reference for the fits. Column two to five have the number of samples with Nfit = 1for different sample sizes M. The second column is out of 1,000 samples (M = 10), the rest ofthem are out of 10,000.rophores when the kinetic dynamics of a fluorophore are known.A.4 Estimating the number of fluorophores from continuous-timemodel without assuming any information on the parameters.We now estimate the number of fluorophores for each parameter set without assuming any infor-mation about the kinetic rates. We use Algorithm 2.3 to simultaneoustly estimate the kinetic rates196(a) MLE for S110 100 1000 10000Sample size0.911.11.2Parameter valuekr10 100 1000 10000Sample size11.21.41.61.82Parameter valuekd10 100 1000 10000Sample size0.050.10.150.20.25Parameter valuekb(b) MLE for S210 100 1000 10000Sample size0.40.60.811.2Parameter valuekr10 100 1000 10000Sample size00.511.5Parameter valuekd10 100 1000 10000Sample size0.511.5Parameter valuekb(c) MLE for S310 100 1000 10000Sample size0.511.52Parameter valuekr10 100 1000 10000Sample size00.511.5Parameter valuekd10 100 1000 10000Sample size0.511.52Parameter valuekbFigure A.7: MLE of kinetic parameters for the parameter sets (a) S1, (b) S2 and (c) S3 for differentsample sizes with N not fixed. Box plots of the MLE and real values (blue line) of kr, kd and kbare in the first, second and third column respectively. Sample size M = 10 and M = 100 wererepeated 100 times, M = 1000 10 times and M = 1000 only one time.197(a) MLE for S410 100 1000 10000Sample size0.511.5Parameter valuekr10 100 1000 10000Sample size00.20.40.60.8Parameter valuekd10 100 1000 10000Sample size1234Parameter valuekb(b) MLE for S510 100 1000 10000Sample size0.60.811.21.41.61.8Parameter valuekr10 100 1000 10000Sample size00.020.040.060.080.1Parameter valuekd10 100 1000 10000Sample size11.522.533.54Parameter valuekbFigure A.8: MLE of kinetic parameters for the parameter sets (a) S4 and (b) S5 for different samplesizes with N not fixed. Box plots of the MLE and real values (blue line) of kr, kd and kb are inthe first, second and third column respectively. Sample size M = 10 and M = 100 were repeated100 times, M = 1000 10 times and M = 1000 only one time.and the number of fluorophores. Contrary to Section A.3, we now have to fit all the samples si-multaneously because we assume all the fluorophores have the same kinetic rates. We tested howthe fit changes when changing the sample size as we did in Section A.2. Again, we divided the10,000 simulations into 100 groups of size M = 10 and M = 100, 10 groups of M = 1000, and oneof M = 10000. Both the sample size and the parameter set played a role in the accuracy of the fit.Samples simulated from parameter sets S1 and S2 converged well as we increased the informa-tion from M = 10 to 10000 (Figure A.7a,b and Table A.4). The kinetic rates estimates are shown inFigure A.7a,b. The estimates converged to the real values as the sample size increases. The qualityof the estimate for the number of fluorophores N is summarized in Table A.4, where each columnhas the number of samples with Nfit = 1 for the different sample sizes. Notice that the first column198S1 S2 S3 S4 S500.20.40.60.81probability123456789NfitFigure A.9: Estimates of the number of fluorophores for the complete sample with 10,000 simulationsfor each parameter set S1-S5 without fixing the kinetic parameters. Bars are grouped by thedifferent parameter sets (S1-S5). All the samples were estimated to have either 1 to 9 distinctfluorophores. Color scale corresponds to the number of fluorophores estimated Nfit from 1(darkblue) to 9 (yellow).is out of 1000 samples instead of 10,000. For 99% of the samples, we obtain the correct estimatenumber of fluorophores in both parameter sets when fitting 1,000 or 10,000 samples simultaneously.Even in the case of fitting 10 or 100 samples, we achieved an accuracy of more than 95% and 98%respectively.On the contrary, samples from parameter sets S4 and S5 did not converge as well (Figure A.8,and Table A.4). The most likely number of fluorophores found for these parameter sets was equalto the total number of localizations and did not change much when increasing the sample size(Table A.4 and Figure A.9). This is also evident from the estimated value of kd , the rate to transitionto the dark state. Regardless of the number of samples fitted simultaneously, the value of kd wasestimated to be zero, or equivalently, all the fluorophores only activate and then immediatelly bleach(Figure A.8). For S1 and S4, the average number of observations of one fluorophore is less than one,and thus the probability of observing one fluorophore many times is really small. The number oflocalizations of these datasets is within {1,2, ...,9} for S4, and {1,2,3,4} for S5. In order to improveour estimates, we will need many more datasets to be able to sample the tail of the distribution andcapture the real kinetics.Parameter set S3 was more interesting (Figure A.7c and Table A.4). Under this parameter set,199S1 S2 S3 S4 S5Full fit 0 0 0 0 0Fit only N -2.93 -0.76 -106 -3359 -643Fit only rates -2.65 -25.51 -439 -4719 -1901Table A.5: Differences between the value of the NLL function at the MLE for the full model andthe model with information about the rates or the number of flurophores all parameters sets andN = 1. We take the difference between the value of the NLL when fitting all the parameters (fullfit) minus the NLL fitting only N (kinetic rates known, second column), or the NLL fitting onlythe kinetic rates (N known, third column).we will observe each fluorophore on average three times (Table A.2). The kinetic rate estimateswere not perfect, but they improved as we added more information (Figure A.7c). We observe anunderestimation of the activation and deactivation rates, kr and kd , while the bleach rate, kb wasoverestimated. This implies that the probability of going dark is estimated to be smaller than thereal value. A system with a smaller deactivation rate will have more unique observations. This isin line with the estimated number of fluorophores, which increased compared to the estimates whenthe kinetic rates are known ( Table A.4, Figure A.4 and Figure A.9).For all the parameter sets, we observed that the number of correctly estimated fluorophore num-bers did not change much when changing the sample size from 1,000 to 10,000 (Table A.4). Thus,we could control the amount of data to fit simultaneously. This is important for our algorithm sincelarger datasets require longer computational time.Figure A.9 and Table A.5 summarize the estimated number of fluorophores and their accuracywhen fitting 10,000 samples simultaneously. We lost accuracy by not knowing the correct kineticrates, specially for parameter sets S3 and S4 (Figure A.9 vs Figure A.4). Thus, if we had any extrainformation about the kinetic rates, we could improve our estimates. This is further illustrated inthe difference of NLL when estimating all the parameters (full fit) and when estimating N knowingthe rates (Table A.5). For the parameter sets where the kinetic rate estimates did not converge (S3,S4, and S5) the difference is much bigger than in the cases where the estimates rates converged (S1and S2). Moreover, the fit for the kinetic rates, when N = 1 is known, showed always a greaterdifference in NLL compare to the full model.As in Section A.3, we fit the model to simulated data with sixteen fluorophores, N = 16. Givencomputational limitations, we only did the fit of N and the kinetic parameters for the parameter setsS3 and S5 for the different sample size. The kinetic rates converged for simulations of the parameterset S3 as we increased the number of samples M (Figure A.10a). As expected, the samples from S5did not converge to the real values, and kd was estimated to be zero (Figure A.10b). Thus, we didnot observe much difference on the estimated number fo fluorophores as we changed the samplesize. The number of fluorophores estimated when fitting the 10,000 sample simultaneously is inFigure A.11.200(a) MLE for S310 100 1000 10000Sample size0.811.21.4Parameter valuekr10 100 1000 10000Sample size0.80.911.1Parameter valuekd10 100 1000 10000Sample size0.40.450.50.550.60.650.7Parameter valuekb(b) MLE for S510 100 1000 10000Sample size0.70.80.911.1Parameter valuekr10 100 1000 10000Sample size00.020.040.060.080.1Parameter valuekd10 100 1000 10000Sample size1.41.61.82Parameter valuekbFigure A.10: MLE of kinetic parameters for the parameter sets (a) S3 and (b) S5 for different samplesizes with N = 16. Box plots of the MLE and real values (blue line) of kr, kd and kb are in thefirst, second and third column respectively. Sample size M = 10 and M = 100 were repeated100 times, M = 1000 10 times and M = 1000 only one time.With these results, we have shown that our method can estimate the number of distinct flu-orophores if the fluorophores blink several times. The estimation is harder when the fluorophoresblink only one or a few times. Furthermore, dSTORM data is discrete in time and the time resolutionwill also affect our fitting (see Chapter 2).201S5 S300.050.10.150.20.250.30.350.4probability1012141618202224NfitFigure A.11: Estimates of the number of fluorophores for the complete sample with 10,000 simula-tions for N = 16 and parameter sets S3 and S5 without fixing the kinetic parameters. Bars aregrouped by the different parameter sets (S3 and S5). All the samples were estimated to havebetween 10 to 24 distinct fluorophores. Color scale corresponds to the number of fluorophoresestimated Nfit.202

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