Mathematical modeling in cellular immunology: T cell activation and parameter estimation by Omer Dushek A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Mathematics) The University Of British Columbia December, 2008 c© Omer Dushek 2008 ii Abstract A critical step in mounting an immune response is antigen recognition by T cells. This step proceeds by pro- ductive interactions between T cell receptors (TCR) on the surface of T cells and foreign antigen, in the form of peptide-major-histocompatibility-complexes (pMHC), on the surface of antigen-presenting-cells (APC). Anti- gen recognition is exceedingly difficult to understand because the vast majority of pMHC on APCs are derived from self-proteins. Nevertheless, T cells have been shown to be exquisitely sensitive, responding to as few as 10 antigenic pMHC in an ocean of tens of thousands of self pMHC. In addition, T cells are extremely specific and respond only to a small subset of pMHC by virtue of their specific TCR. To explain the sensitivity of T cells to pMHC it has been proposed that a single pMHC may serially bind mul- tiple TCRs. Integrating present knowledge on the spatial-temporal dynamics of TCR/pMHC in the T cell-APC contact interface, we have constructed mathematical models to investigate the degree of TCR serial engagements by pMHC. In addition to reactions within clusters, the models capture the formation and mobility of TCR clusters. We find that a single pMHC serially binds a substantial number of TCRs in a TCR cluster only if the TCR/pMHC bond is stabilized by coreceptors and/or pMHC dimerization. In a separate study we propose that serial en- gagements can explain T cell specificity. Using Monte Carlo simulations, we show that the stochastic nature of TCR/pMHC interactions means that multiple binding events are needed for accurate detection of foreign pMHC. Critical to our studies are estimates of TCR/pMHC reaction rates and mobilities. In the second half of the thesis, we show that Fluorescence Recovery After Photobleaching (FRAP) experiments can reveal effective diffu- sion coefficients. We then show, using asymptotic analysis and model fitting, that FRAP experiments can be used to estimate reaction rates between cell surface proteins, like TCR/pMHC. Lastly, we use FRAP experiments to investigate how the actin cytoskeleton modulates TCR mobility and report effective reaction rates between TCR and the cytoskeleton. iii Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Co-Authorship Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Models of T cell activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 The serial engagement hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 The kinetic proofreading model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Combining kinetic proofreading and serial engagement . . . . . . . . . . . . . . . . . . . 6 1.1.4 The equilibrium model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 The immune synapse is the site of T cell activation . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 T cell receptor clusters in T cell activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Modeling relies on parameter estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.1 FRAP experiments reveal protein diffusion coefficients . . . . . . . . . . . . . . . . . . . 11 1.4.2 FRAP can determine binding constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Brief chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Contents iv 2 pMHC dynamics in TCR microclusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Four state escape time formulation from a mobile TCR cluster . . . . . . . . . . . . . . . 22 2.2.2 Mean escape time and reaction rates determine the number of hits . . . . . . . . . . . . . 24 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 TCR/pMHC binding events in a mobile TCR cluster . . . . . . . . . . . . . . . . . . . . 25 2.3.2 Coreceptors augment TCR/pMHC interactions . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.3 T cell stimulation is correlated to pMHC transport to cSMAC . . . . . . . . . . . . . . . 32 2.3.4 Potential effects of pMHC dimers on the APC . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.5 TCR/pMHC binding events with multiple TCR clusters . . . . . . . . . . . . . . . . . . . 35 2.3.6 Effects of multiple stationary clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3.7 Effects of multiple mobile clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 T cell receptor clustering is essential for accurate ligand discrimination . . . . . . . . . . . . . . . 51 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.1 Simulation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.1 T cells cannot make a direct and reliable estimate of KD . . . . . . . . . . . . . . . . . . 55 3.3.2 Multiple binding events are required to determine the TCR/pMHC reaction rates . . . . . 56 3.3.3 TCR enrichment impacts pMHC detection . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.4 TCR enrichment with coreceptors improves pMHC detection for weakly binding pMHC . 58 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4 Improving parameter estimation for cell surface FRAP data . . . . . . . . . . . . . . . . . . . . . 64 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.1 Experimental protocols for cell surface FRAP . . . . . . . . . . . . . . . . . . . . . . . . 65 Contents v 4.2.2 An improved protocol for cell surface FRAP . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3.1 Error estimates for common FRAP protocols . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3.2 Utility of the 2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.3 Practical recommendations for cell surface FRAP experiments . . . . . . . . . . . . . . . 76 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5 Analysis of membrane-localized binding kinetics with FRAP . . . . . . . . . . . . . . . . . . . . . 80 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2.1 Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2.2 Reduced models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.3 Evaluation of theoretical FRAP recovery curves . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.4 Structure of parameter space and model reductions . . . . . . . . . . . . . . . . . . . . . 87 5.3 Proposed experimental protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Simulated experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.1 Simulated FRAP titrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.4.2 Fitting simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5 Experimental outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6 Effects of intracellular calcium and actin cytoskeleton on TCR mobility . . . . . . . . . . . . . . . 103 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.3.1 TCR mobility is similar in naive and activated human T cells . . . . . . . . . . . . . . . . 105 6.3.2 [ Ca2+ ] i increase affects TCR mobility by an actin cytoskeleton dependent mechanism . . 106 6.3.3 [ Ca2+ ] i increase induces polymerization of the actin cytoskeleton . . . . . . . . . . . . . 107 6.3.4 Modeling reveals effective TCR binding parameters to the actin cytoskeleton . . . . . . . 107 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Contents vi 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.1 Serial engagement in T cell activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.1.1 Serial engagements in TCR clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.1.2 Localized serial engagements as a mechanism of pMHC discrimination . . . . . . . . . . 117 7.2 Parameter estimation using FRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.2.1 Estimating diffusion coefficients using FRAP . . . . . . . . . . . . . . . . . . . . . . . . 117 7.2.2 Estimating binding rates between cell membrane proteins using FRAP . . . . . . . . . . . 118 7.2.3 Estimating effective binding rates between cell membrane proteins and the actin cytoskele- ton using FRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 A Matched asymptotic solution for multiple stationary TCR clusters (Chapter 2) . . . . . . . . . . . 125 B Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) . . . . . . . . . . . . . . . . 130 B.1 Model of pMHC and TCR cluster dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 B.1.1 Simulation Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 B.2 Determining ψ and χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 B.2.1 PDE governing pMHC dynamics in a TCR cluster . . . . . . . . . . . . . . . . . . . . . 133 B.2.2 Laplace transform solution of the PDE system . . . . . . . . . . . . . . . . . . . . . . . . 135 B.2.3 Rapid evaluation of ψ and χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 B.2.4 Definitions of Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 B.3 Connection between P (t), p(t, θ), and MFPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 C Analysis of errors due to geometric approximations (Chapter 4) . . . . . . . . . . . . . . . . . . . 142 D Asymptotic reductions of the full model (Chapter 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 144 D.1 Full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 D.2 Dimensionless form of full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 D.3 Pure diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 D.4 Weighted diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 D.5 Independent diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 D.6 Reaction dominant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Contents vii E Details of model fitting to simulated data (Chapter 5) . . . . . . . . . . . . . . . . . . . . . . . . . 151 F Materials and methods (Chapter 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 F.1 Cell isolation and culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 F.2 Fluorescence recovery after photobleaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 F.3 Fitting procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 F.4 Quantification of F-actin by phalloidin staining and flow cytometry . . . . . . . . . . . . . . . . . 155 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 viii List of Tables 2.1 Parameter Definitions and Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Estimates of TCR hits and pMHC transport by a mobile TCR cluster . . . . . . . . . . . . . . . . 29 2.3 CD8 augments low affinity TCR/pMHC interactions . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 TCR hits and pMHC transport by multiple mobile TCR clusters . . . . . . . . . . . . . . . . . . 40 4.1 Fitting the 1D and 2D models to the sphere simulations (Figure 4.4, solid lines) . . . . . . . . . . 74 5.1 Parameters used for numerically simulating FRAP recovery curves shown in Figure 5.3. . . . . . 92 5.2 Parameter values used in numerical simulations, and the estimates from fitting the simulated FRAP titration data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.1 Diffusion coefficient and mobile fraction under various pharmacological treatments . . . . . . . . 105 6.2 Model fitting to FRAP recovery curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 B.1 The state of pMHC determines the possible reactions . . . . . . . . . . . . . . . . . . . . . . . . 133 D.1 Summary of Asymptotic Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 E.1 Model Fitting to Simulated Experiment A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 E.2 Model Fitting to Simulated Experiment B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 E.3 Model Fitting to Simulated Experiment C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 ix List of Figures 1.1 Interactions of TCR and pMHC at the T cell-APC contact junction . . . . . . . . . . . . . . . . . 2 1.2 Early signaling events in T cell activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Three models of T cell activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Early studies of the immune synapse reveal protein segregation . . . . . . . . . . . . . . . . . . . 8 1.5 Recent studies of the immune synapse reveal the existence of many TCR clusters . . . . . . . . . 9 1.6 Schematic representation of Fluorescence Recovery After Photobleaching (FRAP) experiments . . 11 2.1 The most general reaction scheme we consider . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Reactions schemes we consider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Residence time and total TCR hits in a TCR cluster as a function of the initial position . . . . . . 28 2.4 Maximum dissociation constant (KCD) of coreceptors required to achieve transport of agonist pMHC to cSMAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5 Number of pMHC engagements to clustered TCR is almost independent of cluster size . . . . . . 38 2.6 Probability of cSMAC arrival by pMHC depends on kon and koff. . . . . . . . . . . . . . . . . . . 42 3.1 Models of TCR-pMHC reactions with and without coreceptor-MHC interactions . . . . . . . . . 53 3.2 Comparisons of stochastic simulation and PDE computations . . . . . . . . . . . . . . . . . . . . 54 3.3 TCR concentration impacts the accuracy in the estimation of kon and engagement localization . . 57 3.4 Total time required to achieve 10 binding events . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5 Clustering of coreceptors with TCR improves ligand detection . . . . . . . . . . . . . . . . . . . 58 3.6 Confidence in the estimation of koff is independent of kon and TCR concentration . . . . . . . . . 59 4.1 Experimental geometry for protocol 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Experimental geometry for protocol 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Accuracy of the 1D model and the closed form circularly symmetric solution of Soumpasis . . . . 72 4.4 FRAP simulations based on the geometry of protocol 2 . . . . . . . . . . . . . . . . . . . . . . . 73 4.5 Statistical signifiance of the 2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1 The geometry of a typical confocal FRAP experiment . . . . . . . . . . . . . . . . . . . . . . . . 83 List of Figures x 5.2 Quantifying full model limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.3 Numerically simulated FRAP titration experiments demonstrate the effect of increasing ligand density on FRAP recovery dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.1 TCR mobility does not depend on the activation state of T cells . . . . . . . . . . . . . . . . . . . 106 6.2 TCR mobility is modulated by intracellular calcium via the actin cytoskeleton . . . . . . . . . . . 107 6.3 [ Ca2+ ] i increase induces actin polymerization in PBL . . . . . . . . . . . . . . . . . . . . . . . 108 6.4 Fitting the diffusion and binding model to FRAP recovery curves . . . . . . . . . . . . . . . . . . 109 7.1 Possible relationship between 3D and 2D reaction rates . . . . . . . . . . . . . . . . . . . . . . . 119 B.1 Illustration of model geometry and the interaction between diffusing pMHC and advecting TCR clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 B.2 Flow chart illustrating the main loop in the simulation . . . . . . . . . . . . . . . . . . . . . . . . 133 C.1 Potential source of error in using the 2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 xi List of Abbreviations APC Antigen presenting cell AIC Akaike’s information criterion BCR B cell receptor CAC Cortical actin cytoskeleton CD4/CD8 T cell coreceptors CLSM Confocal laser scanning microscope cSMAC Central supramolecular activation cluster FCS Fluorescence correlation spectroscopy FRAP Fluorescence recovery after photobleaching FRET fluorescence resonance energy transfer ICAM1 Inter-cellular adhesion molecule 1 IL-2 Interleukin 2 IS Immunological synapse ITAMs Immunoreceptor tyrosine-based activation motifs LCK Lymphocyte-specific protein tyrosine kinase LFA1 Leukocyte function-associated antigen 1 MHC Major-histocompatibility-complex PDE Partial differential equation pMHC Peptide-major-histocompatibility-complex pSMAC Peripheral supramolecular activation cluster SPR Surface Plasmon Resonance TCR T cell receptor TIRF Total internal reflection fluorescence (microscopy) xii Glossary Antigen A substance that can cause an immune response. In the context of the present work, an antigen is usually displayed as a pMHC on an APC. Antigen presenting cell (APC) Cell displaying antigen, in the form of pMHC, to immune cells. CD4/CD8 T cell coreceptors that bind the MHC molecule directly and may help stabilize the TCR-pMHC bond. Coreceptors are also important for signaling. Supramolecular activation cluster (SMAC) A region on the membrane enriched with signaling molecules and receptors. In the immune synapse, a central-SMAC (cSMAC) and a peripheral-SMAC (pSMAC) have been characterized in detail. Fluorescence recovery after photobleaching (FRAP) An experimental technique used to quantify the mobil- ities of proteins (e.g. by determining a diffusion coefficient). Inter-cellular adhesion molecule 1 (ICAM1) See LFA1. Immunological synapse A term used to describe the interface between a T cell and an antigen presenting cell (APC). Interleukin 2 (IL-2) A molecule that is secreted by activated T cells to mobilize an immune response. Leukocyte function-associated antigen 1 (LFA1) An adhesion molecule found on T cells and other immune cells that binds to ICAM1 on APCs. The LFA1/ICAM1 interaction facilitates cell-cell adhesion and the formation of the immunological synapse. Lymphocyte-specific protein tyrosine kinase (Lck) Lck iniates a signaling cascade by phosphorylating the signaling modules of TCR that are bound to pMHC. Peptide-major-histocompatibility-complex (pMHC) Complexes derived from self and foreign proteins dis- played to T cells on the surface of antigen presenting cells (APCs, see above). Glossary xiii Surface plasmon resonance (SPR) A technique used to measure binding parameters between proteins. In typical experiments, one protein is confined to a two-dimensional chip while a second protein is in solution. T cell receptor (TCR) The primary receptor used by T cells to scan the surfaces of antigen presenting cells (APCs, see above) for foreign material in the form of pMHC (see above). Total internal reflection fluorescence (TIRF) microscopy A technique used to obtain high resolution images of thin surfaces, like the plasma membrane. It is often used to study the T cell membrane during the formation of the immune synapse. xiv Acknowledgements The work presented here is the culmination of my PhD studies at the University of British Columbia (UBC). Many people and institutions, in various capacities, have contributed to my work. I wish to thank the Natural Sciences and Engineering Research Council, Mathematics of Information Technology and Complex Systems, UBC, the International Graduate Training Centre in Mathematical Biology, and Green College for providing financial assis- tance which allowed me to dedicate the majority of my time to research. I thank the Department of Mathematics and the Institute of Applied Mathematics for providing excellent facilities and I thank my engaging colleagues for all our discussions. I would also like to thank the administrative staff in the Mathematics Department for their assistance and extreme efficiency. The Mathematical Biology group has been of tremendous support and provided invaluable feedback throughout my time at UBC. I hope to find myself working in such an atmosphere in the future. I would also like to thank several people who went out of their way to assist me with my research. I thank Michael Ward and Eric Cytrynbaum for sitting on my proposal committee, assisting with mathematical analysis, and for providing feedback on my modeling endeavors. I thank Alan Lindsay, Yoichiro Miro, Anmar Khadra, and Sasha Jilkine for mathematical advice. I thank James Bailey for helpful discussions about theoretical immunology. I thank Talia Cohen for helping me illustrate figures. I thank David Depoil, Sebastien Soubies, Sabina Mueller, and Salvatore Valitutti for introducing me to the experimental side of immunology. I thank Michael Gold and Leah Keshet for being on my proposal committee and my thesis committee. In addition, I thank Leah Keshet for her encouragement ever since I got to UBC. I would like to thank Raibatak (Dodo) Das for exhaustively sharing his knowledge of biochemistry and immunology during our many discussions. Daniel Coombs, my PhD advisor, has been a tremendous force over the past few years and I am indebted to him for taking the time to mentor me. Finally, I wish to thank my parental units, Yoram and Ilana Dushek, and my sibling unit, Hana Dushek, for their continual support in the form of love and rugelach. xv Dedication To my parental units, Yoram and Ilana Dushek. xvi Co-Authorship Statement My PhD. supervisor, Daniel Coombs, has been involved in all projects presented in this thesis. He has pro- vided feedback, assisted in solving analytical and computational difficulties, analyzed results, and has edited all manuscripts. All computations and simulations were performed by me using Matlab or C. I produced all plots using Matlab. Schematics were designed by me and illustrated by me. Chapter 2 Daniel Coombs and I identified and designed the research on serial engagement in the context of TCR clusters. I constructed, parameterized, and analyzed the mathematical model and subsequently wrote a draft manuscript. Daniel Coombs edited the manuscript. Chapter 3 I proposed that serial engagement can explain T cell specificity by allowing for multiple samples of the reaction parameters of a given pMHC. Together with Daniel Coombs, I designed a set of numerical exper- iments to support the hypothesis. I constructed and analyzed the necessary simulations and subsequently wrote a draft manuscript. Daniel Coombs edited the manuscript. Chapter 4 Daniel Coombs and I felt that FRAP analysis of protein mobility on cell membranes could be improved and together identified an improved protocol. I performed the necessary calculations and simulations and subsequently wrote a draft manuscript. Daniel Coombs rigorously edited the manuscript. Chapter 5 I proposed that FRAP can be used to determine binding reactions between membrane proteins. Through discussions with Raibatak Das and Daniel Coombs we designed the research. I constructed the math- ematical model, performed model analysis, and fit simulated data. Raibatak Das and I wrote a draft manuscript which was edited by Daniel Coombs. Chapter 6 Salvatore Valitutti, David Depoil, and I designed the research to probe the interplay between intracellular calcium, the actin cytoskeleton, and TCR mobility. David Depoil and I designed the experimental protocol. All experimental data was obtained in Toulouse (France) by Sabina Mueller, Sebastien Soubies, and David Depoil. I performed all data fitting. All authors on this work analyzed the data. Salvatore Valitutti, Sebastien Soubies, and Daniel Coombs wrote the draft manuscript which I edited. 1Chapter 1 Introduction Thymus derived lymphocytes, or T cells, play a central role in cellular immunity (1). Continually circulating be- tween peripheral lymphoid tissues (e.g. lymph nodes), naive T cells scan the surfaces of dendritic cells for specific antigens. Antigens are proteins that are recognized by the adaptive immune system. Upon encountering a specific antigen, the T cell becomes activated and begins a program of proliferation and differentiation that culminates in the generation of many mature effector T cells that can recognize the specific antigen. Effector T cells can acti- vate specific B cells to make antibody and may leave the lymphoid tissue to perform targeted killings of infected cells and activate infected macrophages. These effector functions also prevent viruses from efficiently replicating within cells and spreading between them. Therefore, T cells are critical for the elimination of viral infections but, in addition, they play an important role in clearing many other pathogens. Like the initial activation of naive T cells, activation of effector functions is only carried out upon recognition of specific antigen on the surface of the relevant cell. Consequently, how a T cell recognizes a specific antigen is a central question in contemporary immunology (2; 3). T cells use their antigen receptor, the T cell receptor (TCR), to scan the surfaces of dendritic cells, macrophages, and B cells, collectively known as antigen presenting cells (APC), for foreign antigen, see Figure 1.1. A given T cell has thousands of identical TCR on its cell surface and in general, each T cell in the body has a different TCR that is able to detect a different set of antigens. In other words, all TCR on a single T cell have the same specificity for antigen. Foreign antigen is displayed on the surface of APCs in the form of peptide fragments, derived from foreign proteins, attached to major-histocompatibility-complex (MHC). If the foreign protein is produced within the cell, fragments of it will be displayed on class I MHC along with peptides derived from all other proteins produced within the cell. If the foreign protein is endocytosed, fragments of it will be displayed on class II MHC along with all other peptides endocytosed by the cell. In certain APCs, displayed peptides on a particular class of MHC can arise by both routes, a phenomena known as cross-presentation (4; 5). In addition to the APCs mentioned above that express both MHC class I and II, all cells in the body express MHC class I. In each case a different subset of T cells detect the peptide-MHC (pMHC) complex: CD8 T cells detect peptides bound to MHC class I while CD4 T cells detect peptides bound to MHC class II. The labels CD8 and CD4 refer to cell surface proteins that, respectively, bind to MHC class I and MHC class II directly. In both cases, it is expected that the Chapter 1. Introduction 2 vast majority of displayed pMHC will not be antigenic but will represent pMHC derived from self-proteins (i.e. proteins used by cells in the body for normal functions). Indeed, experiments have shown that T cells can detect as few as 1-10 antigenic pMHC among tens of thousands of self-pMHC (6; 7; 8; 9; 10), in part by direct binding of TCR to antigenic pMHC. T CELL APC TCR Agonist pMHC Endogenous pMHC Figure 1.1: Interactions of TCR and pMHC at the T cell-APC contact junction. Binding events between TCR and pMHC take place at the contact junction between a T cell and APC. The two proteins are mobile on their respective cell membranes. Each T cell has ∼40 000 identical copies of TCR while each APC contains roughly 10 000 different pMHC, although relatively few pMHC are recognized by any specific T cell in an individual. The early signaling events following the engagement of TCR by pMHC have been extensively studied (11). A simplified signaling diagram is shown in Figure 1.2, which focuses on four signaling molecules: Lck, ZAP70, LAT, and SLP76. The engagement of TCR by pMHC results in the phosphorylation (phosphate addition) of specific tyrosine residues within specialized signaling motifs (ITAMs) on the cytoplasmic domains of the TCR- associated signaling modules by leukocyte-specific protein tyrosine kinase (Lck). In Figure 1.2 the signaling mod- ules are shown as CD3 and ζ-chain on which specific ITAMs reside. Once phosphorylated, ζ-chain-associated protein kinase 70 (ZAP70) can bind to the ITAMs. The next step in the signaling cascade is the phosphorylation of ZAP70 by Lck. Phosphorylated ZAP70 then activates linker of activated T cells (LAT) and SH2-domain- containing leukocyte protein of 76 kDa (SLP76). LAT and SLP76 form a large signaling platform which recruits many additional signaling molecules and subsequently activates multiple downstream responses, such as actin polymerization, integrin activation, TCR internalization, calcium release, and ultimately full effector function. Chapter 1. Introduction 3 The process of signaling is augmented by the action of the CD4 and CD8 coreceptors which bind the MHC molecule directly. Lck is constitutively associated with the cytoplasmic tails of coreceptors and therefore core- ceptors help localize Lck to the TCR-pMHC complex (12; 13; 14; 15; 16; 17). However, the spatial and temporal orchestration of signaling in T cells is complex and does not proceed in a simple linear cascade. Rather, various feedback loops have been described (11). Although the interactions between many signaling molecules have been characterized, the quantitative rates have generally not been measured. Lck CD3 TCR CD4 ZAP70 67PLS T A L P P P FURTHER DOWNSTREAM SIGNALING APC T CELL CD8/ pMHC ζ - Chain P P Figure 1.2: Early signaling events in T cell activation. Upon engagement of TCR by pMHC, lymphocyte-specific protein tyrosine kinase (Lck) phosphorylates specific immunoreceptor tyrosine-based activation motifs (ITAMs) located on the CD3 and ζ chains of the TCR. Lck is constitutively associated with the cytoplasmic domain of the T cell coreceptors CD4 and CD8. The next step in the signaling cascade involves the binding of ZAP70 to phosphorylated ITAMs. Lck then phosphorylates ZAP70 which proceeds to phosphorylate SLP76 and LAT, two critical signaling molecules that mediate many downstream responses. The biophysical parameters governing mobility and interactions for TCR and pMHC have been extensively studied. Reactions between TCR and pMHC have been shown to obey the law of mass-action (18) and have been measured to be weak, with solution dissociation constants, KD, in the range of 0.1-100 µM and half-lives in the Chapter 1. Introduction 4 range of 1-50s (19; 20; 21; 22). For comparison, typical specific interactions between membrane proteins have KD = 0.1 µM (23). TCR and pMHC have been shown to undergo effective diffusion on their respective cell membranes with D ≈ 0.05 µm2/s (24; 25; 26; 27), which is comparable to the diffusion coefficients of other cell surface proteins (28). The concept that T cell activation in response to antigenic pMHC can follow the productive interaction with a single APC has existed since the 1980s (29) and detailed studies of the T cell-APC contact region first appeared in 1998 (30) and 1999 (21). These latter studies revealed a large and homogenous enrichment of TCR and pMHC in the center of the contact region. Given the weak reactions between TCR and pMHC, this central accumulation was thought to mediate T cell activation by increasing the probability of TCR/pMHC engagement. As we shall discuss below, subsequent studies utilized higher resolution microscopy to paint a much more dynamic picture of the contact region (31; 32; 33; 34; 35; 36). The term “immunological synapse” (or immune synapse) is now in common use to describe the T cell-APC contact region because it has many structural and functional similar- ities to the neurological synapse (37). Structurally, both synapses are composed of closely opposing membranes with many receptor-ligand interactions. Functionally, both synapses utilize their close membrane opposition for directed secretion (37; 38). In summary, CD4 and CD8 T cells are thought to form immunological synapses with APCs in order to facil- itate the interaction between TCR and pMHC. Antigenic pMHC is extremely diluted by thousands of irrelevant pMHC on a given APC yet the T cell is able to recognize and become activated by few antigenic pMHC. These observations form one of the central questions in modern immunology: how can a few pMHC having low affinity (i.e. weak reaction kinetics) to the TCR drive the activation of a T cell? In what follows, I will discuss previous experimental and theoretical work aimed at answering this question. I will then discuss recent experimental work characterizing the spatial and temporal dynamics of TCR and pMHC in the immune synapse. My own work on the subject, presented in Chapters 2 and 3, focuses in part on how these recent experimental observations impact the previous theoretical work aimed at understanding T cell activation. 1.1 Models of T cell activation 1.1.1 The serial engagement hypothesis A partial solution to how T cells become activated by few antigenic pMHC was first proposed by Valitutti et al. (39) in 1995. In experiments where a T cell is in contact with an APC bearing as few as 100 pMHC, Valitutti Chapter 1. Introduction 5 observed the internalization of thousands of TCRs from the T cell surface. Assuming each internalized TCR has contacted a pMHC, Valitutti estimated that a single pMHC serially engages roughly 200 TCRs in the immune synapse. Mathematically, the rate of serial engagement by a single pMHC or the “hitting rate”, is simply the reciprocal of the total time for a single TCR-pMHC binding cycle, hitting rate = 1 1/ (konT ) + 1/koff = koffkonT koff + konT (1.1) where T is the concentration of free TCR (in units of µm−2), kon is the two-dimensional bimolecular reaction constant (in units of µm2/s), and koff is the reaction off-rate (in units of s−1). The hitting rate is plotted in Figure 1.3a as a function of koff. In order to determine the total number of TCR hits from the hitting rate, an estimate of the time that a pMHC is able to interact with TCR is required. Wofsy et al. (40) used a TCR/pMHC reaction- diffusion framework to estimate the mean escape time, defined as the mean time a single pMHC is expected to remain in the immune synapse (and interact with TCRs). During the time of their research (1999-2000), available data indicated that the TCR and pMHC were roughly homogenously distributed in the immune synapse (21; 30). Using this assumption they computed the mean escape time and multiplied it by the hitting rate to find that a typical pMHC can engage≈ 5−35 TCRs in the immune synapse before diffusing out. Although smaller than the estimates of Valitutti, they found that substantial serial engagement of TCR is plausible. Taken together, these ex- perimental and theoretical results suggest that a single pMHC can initiate multiple signaling cascades by serially binding multiple TCRs. The summation of TCR signaling from few serially binding pMHC could therefore be sufficient to activate the T cell. A further prediction from the serial engagement hypothesis is that the stimulatory potency of pMHC improves with increasing kon and koff. −3 −2 −1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 log10(koff) (s −1) H itt in g Ra te (s − 1 ) Serial Engagement Hypothesis −3 −2 −1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 log10(koff) (s −1) Pr ob . o f P ro du ct iv e En ga ge m e n t The Kinetic Proofreading Model −3 −2 −1 0 1 2 3 0 0.01 0.02 0.03 0.04 log10(koff) (s −1) Pr od uc tiv e H itt in g Ra te (s − 1 ) Serial Engagement and Kinetic ProofreadingA) B) C) Figure 1.3: Three models of T cell activation. The serial engagement model (A) predicts that the hitting rate, the number of TCR engaged by a single pMHC per unit time, is an important determinant of T cell activation. In contrast, the kinetic proofreading model (B) predicts that the quality of each hit is important (i.e. the duration of binding). Combining the two models (C) reveals a peak in T cell stimulation as a function of koff. In contrast, the equilibrium model predicts that T cell stimulation will be a strictly decreasing function with koff (not shown). Parameters: kon = 0.01µm2/s, T = 100µm−2, kp = 0.5s−1, N = 5. Chapter 1. Introduction 6 1.1.2 The kinetic proofreading model In the same year that Valitutti proposed the serial engagement hypothesis, McKeithan (41) proposed a kinetic proofreading mechanism of pMHC discrimination based on the work of Hopfield (42). Kinetic proofreading, as applied to the T cell, postulates that upon pMHC binding to the TCR, a series of biochemical modifications begin to accumulate at the TCR (e.g. phosphorylation of ITAMs, binding of ZAP70, etc.). However, if the pMHC dissociates before a critical modification has occurred, the TCR-pMHC engagement does not result in a downstream signal and the TCR reverts to a resting unmodified state. Assuming there are N modification steps and that the rate of each modification is equal to kp, an ordinary differential equation (ODE) formulation can be used to show that the probability of a productive engagement, fkp, is (43), fkp = ( kp kp + koff )N . (1.2) In Figure 1.3b we plot fkp as a function of koff. We see from the Figure and from equation 1.2 that when koff kp, the probability of a productive engagement approaches unity. The kinetic proofreading mechanism, in contrast to the serial engagement hypothesis, predicts that the stimulatory potency of pMHC will improve with decreasing koff. Weak evidence for this model can be found in studies showing that pMHC having smaller KD (= koff/kon) are more stimulatory than pMHC with larger KD (18; 20; 21; 22). 1.1.3 Combining kinetic proofreading and serial engagement The kinetic proofreading and serial engagement models were combined in 2002 by Coombs et al. (43). Their reaction-diffusion partial differential equation (PDE) model of the immune synapse captured serial engagement, kinetic proofreading, and TCR internalization. One simple result from their study is that the productive hitting rate is, productive hitting rate = koffkonT koff + konT ( kp kp + koff )N (1.3) which is simply the hitting rate, equation 1.1, multiplied by the probability of a productive engagement, Equation 1.2. The balance between the hitting rate and productive engagement reveals that the stimulatory potency of a pMHC is maximized when koff ≈ kp/(N − 1), see Figure 1.3c. This observation was also made in a Monte Carlo study of TCR on a lattice that incorporated both serial engagement and kinetic proofreading (44). The existence of a peak in T cell stimulation as a function of koff has been observed experimentally (43; 45). Chapter 1. Introduction 7 1.1.4 The equilibrium model The combined serial engagement and kinetic proofreading model has been challenged over the years (20; 22; 46; 47; 48). In particular, the equilibrium model of T cell activation argues that T cells discriminate pMHC based on KD, which at equilibrium determines the number of TCR-pMHC bonds. Experimental evidence for the model comes from studies showing a significant negative correlation betweenKD and T cell stimulation (20; 22; 46; 47). For example, Holler and Kranz (20) allowed specific T cells to interact with APCs bearing pMHC known to stim- ulate that particular T cell. After several hours of incubation they measured the release of interleukin 2 (IL-2), a cytokine secreted by T cells upon activation. In separate experiments they used Surface Plasmon Resonance (SPR) to measure the solution binding parameters between the TCR and pMHC and found a good correlation between KD and T cell stimulation. Similar results were obtained by Andersen et al. (46) and Tian et al. (22). The combined kinetic proofreading and serial engagement model and the equilibrium model both assume a standard mass action reaction and diffusion model for TCR and pMHC in the immune synapse. These assumptions were considered reasonable based on the early studies of the immune synapse that showed diffusive enrichment of TCR and pMHC to the contact region (21; 30). However, our understanding of the spatial-temporal dynamics of TCR and pMHC at the immune synapse has dramatically improved over the past ten years and this new under- standing can give clues into the T cell activation process. 1.2 The immune synapse is the site of T cell activation Given that TCR/pMHC interactions take place within the immune synapse there has been considerable interest in characterizing the spatial and temporal dynamics of TCR and pMHC within the immune synapse in the hope that they can be used to uncover important mechanisms of T cell activation. Using fluorescent markers and digital three-dimensional microscopy, Monks et al. (30) were able to image mature T cell-APC junctions on fixed cells. Fixed cells are obtained from live cells by treatment with a fixative agent (e.g. formaldehyde) that preserves the morphological structure at the time of fixing. In these junctions they found two distinct domains that collectively formed a bulls eye, see Figure 1.4. In the central domain, which they called the central supramolecular activation cluster (cSMAC), they observed a large aggregation of TCRs. In the second outer domain, a region known as the peripheral SMAC (pSMAC), they observed a ring of leukocyte function-associated antigen 1 (LFA1) surrounding the cSMAC. LFA1 is a cell surface receptor involved in cell adhesion. Grakoui et al. (21) confirmed these results by showing that fluorescently labeled pMHC and ICAM1, Chapter 1. Introduction 8 the complementary receptors to TCR and LFA1, aggregated in the c and p SMAC, respectively. In summary, these studies showed that cell surface receptors and intracellular proteins localize to either the p or cSMAC at the T cell-APC junction. EN FACE VIEW T CELL APC cSMAC pSMAC TCR Agonist pMHC Endogenous pMHC LFA1/ICAM1 TCR/pMHC Figure 1.4: Early studies of the immune synapse reveal protein segregation. In the center of the contact region (cSMAC) aggregation of TCR and pMHC is observed while in the periphery of the contact region (pSMAC) aggregation of the adhesion molecules LFA1 and ICAM1 is observed. Grakoui et al. (21) made several observations that allowed them to surmise a purpose for the immune synapse. First, they observed that the extracellular length of TCR are small (∼15nm (49)) compared to many other cell surface receptors (LFA1∼42nm, CD45∼80nm (49)). Synapse formation would then allow for close membrane opposition in the cSMAC allowing TCRs to easily interact with pMHC molecules. Second, the low affinity of the TCR-pMHC complex would be enhanced by confinement within such a small space. Further, aggregation of these molecules in the cSMAC increases their concentrations leading to further increases in signaling. Thus, synapse formation seemed essential for “sustained TCR engagement and signaling” (21). I briefly mention that soon after these experimental studies appeared, mathematical modeling of synapse for- mation showed that purely passive processes, including diffusion, reactions, and membrane tension and bending rigidity, could lead to observed molecular patterns during synapse formation through a spontaneous self-assembly process (50; 51; 52; 53; 54). The basic idea behind these models is that minimization of membrane bending and tension will faciliate protein sorting/aggregation according to their extracellular bond-length. Two years following the publication of the studies by Monks et al. (30) and Grakoui et al. (21), two studies showed that the signaling important for T cell activation is confined to the pSMAC and that minimal signaling Chapter 1. Introduction 9 occurs in the cSMAC, despite the fact that it contains a large concentration of TCR and pMHC. Krummel et al. (55) showed that the coreceptor CD4 colocalizes with TCR in the early stages of immune synapse formation but is largely absent from the cSMAC, suggesting that Lck cannot efficiently localize with TCR in the cSMAC. Lee et al. (56) stained for global tyrosine phosphorylation, phosphorylated Lck, and phosphorylated ZAP70 during immune synapse formation. They found that all phosphorylation takes place in the pSMAC and concluded that the cSMAC is not the site of T cell activation. With these observations in hand, studies began using higher resolution microscopy to resolve the dynamics of TCR and pMHC at the site of T cell activation: the pSMAC. 1.3 T cell receptor clusters in T cell activation The concept that TCR clustering can facilitate T cell activation (57) existed for many years before T cell receptor clusters were first observed (36). Over the past three years, experiments utilizing high resolution total internal reflection fluorescence (TIRF) microscopy of the T cell membrane resting on a supported planar bilayer have shown that TCR aggregate into sub-micron clusters that form in the pSMAC and are actively transported to the cSMAC (31; 32; 33; 34; 35), see Figure 1.5. EN FACE VIEW T CELL cSMAC pSMAC t = 0 - 60s t > 60s TCR Agonist pMHC Signaling TCR Cluster Non-Signaling TCR Cluster PLANAR BILAYER Figure 1.5: Recent studies of the immune synapse reveal the existence of many TCR clusters. Interactions between TCR and pMHC are confined to the contact interface between the T cell and APC. Studies of the contact interface use a surrogate APC, in the form of a planar bilayer, because the contact interface remains flat over long timescales. Within 60 s of contact the interface begins to accumulate with newly formed TCR clusters. After∼ 60s the clusters begin to aggregate in the center of the interface (forming the cSMAC) by traveling along radially oriented actin filaments. Generation of new signaling TCR clusters in the pSMAC and their subsequent translocation to the cSMAC continues for hours. The emerging picture of TCR cluster dynamics is that they form very rapidly (∼ 5s) after T cell stimulation with antigenic pMHC. After a brief initial period (≈ 60s), they begin to translocate along radially aligned actin filaments towards the synapse center where they coalesce to form the cSMAC, see Figure 1.5. Generation of Chapter 1. Introduction 10 new TCR clusters in the pSMAC and their subsequent translocation to the cSMAC continues for hours. Effective TCR cluster velocities have been measured to be ≈ 0.05µm/s (31; 34). Each TCR cluster contains ≈ 50 − 150 TCRs (31; 32) and covers an area ≈ 0.35 − 0.50µm2 (32). Therefore the concentration of TCR in a cluster is = 100/0.35 ≈ 285µm−2 which is almost three times greater than typical concentration of TCR on resting T cells. The spatial and temporal dynamics of the early signaling molecules shown in Figure 1.2 have also been in- vestigated. Rapid localization of Lck, ZAP70, SLP76, and LAT to newly formed pSMAC TCR clusters has been observed (31; 32). As with earlier studies, none of these signaling molecules were found in the cSMAC which is presently believed to be the site of TCR internalization (33). In summary, TCRs form clusters in the periphery of the immune synapse (pSMAC) in response to agonist pMHC. Signaling relevant to T cell activation takes place within the peripheral TCR clusters during their journey to the synapse center (cSMAC), where signaling is extin- guished and TCRs are internalized. Chapter 2 examines the degree of serial engagement in the context of TCR clustering. I find that TCR/pMHC binding alone does not allow for substantial serial engagement of TCR and that the pMHC molecules are usually not transported to the center of the contact region by a single TCR cluster. I show that the presence of TCR core- ceptors such as CD4 and CD8, or pMHC dimerization on the APC, can substantially increase serial engagement and directed transport of pMHC by a single TCR cluster. Finally, I analyze effects of multiple TCR clusters. Chapter 3 takes an alternative view of serial engagement. By considering the stochastic nature of TCR/pMHC interactions, I suggest that serial engagement is used by the T cell to sample the reaction parameters of pMHC. Integrating multiple binding events allows the T cell to accurately probe the reaction parameters of a pMHC and hence discriminate between pMHC. I further show that TCR clustering is essential to this process, in part, by allowing for rapid and localized binding events. 1.4 Modeling relies on parameter estimates Mathematical modeling of biological processes is heavily dependent on accurate estimates of various parameters. Typical parameters used in biological modeling are diffusion coefficients and reaction rates. Specific to T cell biology, models of T cell activation and immune synapse formation typically rely on estimates of diffusion coef- ficients for TCR and pMHC, reaction rates between TCR and pMHC, and effective reaction rates between TCRs and the actin cytoskeleton. Chapter 1. Introduction 11 1.4.1 FRAP experiments reveal protein diffusion coefficients A common tool used to investigate protein mobility on the cell membrane is Fluorescence Recovery After Pho- tobleaching (FRAP), see Figure 1.6. In this experimental technique, a fluorescent marker is attached to a protein of interest. When viewed under a confocal microscope, a homogenous distribution of fluorescence is observed at the cell membrane (t < 0). High laser intensity in a confined region bleaches (denatures) the fluorescent marker, reducing the fluorescence to background levels (t = 0). Over time, the fluorescence intensity in the bleached region recovers due to motion of unbleached proteins (t > 0). Using image analysis software one can integrate the total fluorescence from the bleached region over time to obtain the iconographic FRAP recovery curve. To obtain an estimate of the diffusion coefficient, a mathematical solution of the diffusion equation that captures the experimental protocol (e.g. the shape and dimensions of the bleach region) is fit to the FRAP recovery curve. 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 1.2 Time (s) N or m al iz ed Fl uo re sc en ce In te ns ity D=0.005 µm2/s D=0.05 µm2/s D=0.5 µm2/s t < 0 t = 0 t > 0 Large t Figure 1.6: Schematic representation of Fluorescence Recovery After Photobleaching (FRAP) experiments. Be- fore the experiment (t < 0), a uniform distribution of fluorescence is observed indicating that the fluorescently labeled protein is homogenously distributed on the cell membrane. At t = 0 high laser intensity is applied to a small localized region of the cell membrane, termed the bleached region (white rectangle), bringing the fluo- rescence intensity to background levels. Over time, t > 0, the fluorescence intensity recovers and returns to its original level after a long time (large t). The FRAP recovery curve is obtained by integrating the fluorescence intensity in the bleached region over time. The speed of recovery depends, in part, on the diffusion coefficient. Chapter 4 shows that common mathematical models and experimental protocols used for estimating diffusion coefficients for cell surface proteins give rise to appreciable errors. I propose an improved protocol that utilizes a small bleach region on the cell membrane and provide the relevant solution of the two-dimensional diffusion equation required for fitting. This FRAP protocol is used in the theoretical study presented in Chapter 5 and in experiments probing TCR mobility in Chapter 6. 1.4.2 FRAP can determine binding constants In recent years the use of FRAP has been extended to study the binding properties of proteins in vivo (58; 59; 60; 61). For example, Sprague et al. (58) used FRAP to study the binding kinetics of diffusing glucocorticoid receptor (in the nucleus) binding to an immobile substrate. The basic tenet of these experiments is that FRAP recovery will Chapter 1. Introduction 12 be slower in the presence of binding reactions. Having an independent estimate of the free diffusion coefficient, one can determine whether the FRAP recovery curve for the protein of interest is consistent with free diffusion or if binding reactions need to be invoked to explain slower recovery compared to free diffusion. Inspired, in part, by the work of Sprague et al. (58), in Chapter 4 we show how FRAP can be used to determine the reaction parameters governing the TCR/pMHC bond. Briefly, we propose a set of FRAP experiments in the immune synapse formed in response to various concentration of pMHC. We show how fitting the resulting set of FRAP experiments to a mathematical model can uncover the reaction off-rate and the two-dimensional bimolec- ular reaction on-rate. The importance of this protocol is underlined by the fact that the majority of experiments characterize TCR/pMHC reaction rates in solution and not in the context of the plasma membrane (2; 62). Over the past decade there has been mounting work suggesting that cell surface proteins can be regulated by the actin cytoskeleton (63; 64; 65; 66; 67). In the T cell system, several groups have shown that TCR can undergo directed motion by the cytoskeleton (32; 33; 34; 68). The study of Kaizuka et al. (34) provided indirect evidence that TCRs undergo dynamic reactions with the cytoskeleton by comparing the directed velocity of TCR towards the cSMAC and the rate of actin treadmilling. They showed that the actin treadmilling rate is twice as fast as the TCR velocity, a result implying that TCRs are not constantly bound to actin filaments but cycle between bound and unbound states. Mathematical models attempting to explain immune synapse formation have assumed that TCR can undergo mass action reactions with the cytoskeleton (52; 54). However, effective reaction rates between the TCR and the cytoskeleton have not been reported. In Chapter 6 we establish a role for intracellular calcium in regulating the interaction between surface TCR and the cytoskeleton. Using the FRAP protocol developed in Chapter 4, in addition to other experimental tools, we show that TCR mobility decreases upon T cell activation by the cortical actin cytoskeleton. Using the binding and fitting procedures developed in Chapters 4 and 5, we are able to extract the first estimates of the effective reaction rates between TCR and the cytoskeleton. 1.5 Brief chapter summary My thesis is composed of two major themes: serial engagement in T cell activation (Chapters 2 and 3) and pa- rameter estimation using FRAP (Chapters 4, 5, and 6). A table of abbreviations and a glossary of terms used frequently throughout my thesis can be found in the prefactory pages (page xi). Chapter 1. Introduction 13 Serial engagement in T cell activation Chapter 2 Several models of T cell activation rely on serial engagement as a mechanism of signal ampli- fication. Given recent experimental observations showing that signaling is confined to mobile TCR clusters, I investigate the degree of serial engagement within TCR clusters. Chapter 3 In this chapter I propose that serial engagements may explain the specificity of T cells. I propose that multiple TCR engagements allow the T cell to sample the biochemical parameters of a given pMHC and hence discriminate between pMHC. I show that TCR clustering is required, in part, to achieve rapid and localized binding events. Parameter estimation using FRAP Chapter 4 In this chapter I investigate the use of FRAP to determine diffusion coefficients for cell surface proteins. I find that established methods give rise to appreciable errors. I propose an alternative protocol and provide the equation required for FRAP fitting which minimizes errors for cell surface FRAP experiments. Chapter 5 Motivated by the lack of experimental methods to determine reaction parameters between mem- brane proteins, I investigated how FRAP experiments can be used to uncover these parameters. I propose an experimental protocol and provide the mathematical analysis required to fit experimental data. Using simulated data I confirm the utility of the method. Chapter 6 In collaboration with experimental laboratory of Salvatore Valitutti (INSERM U563, Toulouse, France), we use a set of experiments to show that TCR mobility is modulated by the actin cytoskeleton. Using the FRAP protocol developed in Chapter 4, I fit experimental FRAP recovery data for TCR when T cells are treated with various pharmacological agents. 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Golan. 2006. Cytoskeletal regulation couples LFA-1 conformational changes to receptor lateral mobility and clustering. Immunity. 25:297–308. [68] Moss, W., D. Irvine, M. Davis, and M. Krummel. 2002. Quantifying signaling-induced reorientation of T cell receptors during immunological synapse formation. Proc Natl Acad Sci U S A. 99:15024–9. 20 Chapter 2 Analysis of serial engagement and peptide-MHC transport in T cell receptor microclusters1 2.1 Introduction T cells play a central role in adaptive immunity by regulating immune responses and performing targeted killing of infected cells. In order for T cells to carry out these functions they must be stimulated by antigen-presenting cells (APC) bearing cognate antigen. Stimulation is mediated by interactions between T cell receptors (TCR) and specific antigen presented in the form of peptide-major-histocompatibility complexes (pMHC), in a tight adhesion region between the T cell and APC (69; 70). In this region, known as the immunological synapse (IS), fewer than 10 agonist pMHC molecules have been shown to transduce sufficient intracellular signaling to cause measurable stimulation of T cells (71; 72; 73). Further, the TCR-pMHC bond is weak, with solution KD usually in the µM range (74). A partial explanation of the sensitivity of T cells to such weak stimuli was proposed by Valitutti et al. (75). TCR downregulation was measured over the course of several hours of T cell interaction with APC carrying a known number of antigenic pMHC. The ratio of downregulated TCR to the number of pMHC was found to be as high as 200. Assuming that every internalized TCR has previously bound pMHC, this suggests that pMHC sequentially bind hundreds of TCR in the IS. This is known as the “serial engagement” hypothesis. Similar re- sults were found by Itoh et al. (76) but the notion of serial engagement was weakened by findings that TCR that have never bound antigenic pMHC can be internalized in a pMHC-dependent manner (77). Further inves- tigation revealed that when few pMHC are present on the APC, TCR are downregulated so rapidly that a strict serial engagement model could not fit the data (78; 79). Rather, a model allowing for downregulation of nearby, 1A version of this chapter has been published. Dushek O, Coombs D (2008) Analysis of serial engagement and peptide-MHC transport in T cell receptor microclusters. Biophys J. 94:3447-3460 Chapter 2. pMHC dynamics in TCR microclusters 21 unstimulated TCR, in a TCR-density-dependent manner, provided a more plausible explanation (79). The impor- tance of serial engagement of TCR for T cell activation has been further challenged by data of Holler and Kranz (80) which did not show a decline in T cell activation by high affinity pMHC. We re-analyze parts of this data here. We see that the rate of engagements (the “hitting rate”) between TCR and pMHC is of great importance. Using a simple mathematical model, Wofsy et al. (81) calculated the hitting rate for a single pMHC moving diffusively on the APC surface. Using measured parameters for the TCR-pMHC bond, it was found, for instance, that a single agonist pMHC can engage 5-35 TCR during one sojourn in the IS. The highest rates of serial engagement were achieved by those pMHC that bind TCR very transiently. On the other hand, fast-dissociating pMHC would lead to only weak signaling of individual TCR. These considerations led to a model of overall pMHC signaling efficacy based on the lifetime of the TCR-pMHC bond, where the optimal pMHC has a lifetime that is high enough for reliable signaling of individual TCR, but low enough for serial engagement to proceed efficiently (75; 82; 83). These predictions were confirmed experimentally and theoretically in a series of experimental and theoretical pa- pers (84; 85; 86). In the calculations of Wofsy et al. (81) it was assumed that TCR are distributed homogeneously within the immune synapse. It has since been shown that TCR aggregate into half-micron sized clusters, often referred to as TCR microclusters, in T cell synapses formed with both APC (87) and suspended planar bilayers (87; 88; 89) bearing agonist pMHC. These TCR clusters form in the periphery of the immune synapse, a region known as the peripheral supramolecular activation cluster (pSMAC), and migrate towards the center of the immune synapse, (the central-SMAC (cSMAC)) (87). Various molecules in the signaling cascade that begins with a pMHC inter- acting with a TCR have been shown to localize in pSMAC clusters and not in the cSMAC (87; 88). TCR clusters are observed to form when bilayer pMHC densities are as low as 0.2µm−2 (89). Weak TCR-pMHC interactions can also be stabilized by more complex binding reactions. For instance, the cell-surface coreceptors CD4 and CD8 (90) are thought to bind the nonpolymorphic regions of the pMHC molecule (91), strengthening the TCR-pMHC interaction. The cytoplasmic tails of CD8 and CD4 are also associ- ated with Lck, a molecule known to interact with the cytoplasmic tail of engaged TCR, perhaps further stabilizing the complex (92). In a similar vein, pMHC are thought to form multimeric complexes that may allow for stabi- lization of bonds formed by the individual pMHC (further discussion below). In this paper we revisit the calculations of Wofsy et al. (81) in the context of moving TCR clusters and stabi- lization of the TCR-pMHC bond by coreceptors and pMHC dimerization. We show that serial engagement, in the Chapter 2. pMHC dynamics in TCR microclusters 22 context of mobile clusters, is negligible unless the TCR-pMHC bond is stabilized by coreceptors and/or pMHC dimerization, in which case substantial serial engagement of TCR by a single pMHC molecule will occur. We also show that stabilization is necessary for pMHC transport to the cSMAC within a single cluster and that pMHC transport to the cSMAC is well correlated with T cell stimulation. Finally, we investigate effects of multiple TCR clusters. In the case of immobile TCR clusters we are able to obtain an analytical solution while in the case of mobile TCR clusters we perform a full Monte Carlo simulation of an immune synapse containing multiple mobile TCR clusters. 2.2 Mathematical model We derive a model for the escape time of a pMHC molecule diffusing within the immunological synapse and potentially binding and unbinding from TCR. We do not resolve individual TCR in this model, nor do we con- sider TCR signaling following pMHC engagement of TCR. Principally, we are concerned with understanding how physical parameters of the TCR-pMHC interaction are related to pMHC motion within the immune synapse and serial engagement of TCR by pMHC. TCR signaling and more complex effects within this model could be addressed using an agent-based approach (93). 2.2.1 Four state escape time formulation from a mobile TCR cluster We want to calculate the length of time that a pMHC is expected to remain within a TCR cluster before escaping. We model a particle (i.e. a pMHC molecule) advecting and diffusing in a 2D membrane. The particle transitions between four states (i=A, B, C and D) by a Markov process with first order reactions between the states, as shown in Figure 2.1. We define the escape time from some domain, for a particle in state i at position (x, y), to be ti(x, y). This is calculated for a diffusing particle (without reactions) by solving the Poisson equation on the region of interest using Dirichlet (zero) boundary conditions (see Appendix A of (94) for a heuristic derivation based on a two-dimensional random walk). Following the procedure in (81; 94), we derive a system of equations governing Chapter 2. pMHC dynamics in TCR microclusters 23 A B D C λ -1 λ +1 λ +3 λ -3 λ +2 λ -2 λ -4 λ +4 Figure 2.1: The most general reaction scheme we consider. Transitions between the four states occur with first order reactions, with transition rates given by λ±i. Detailed balance (microscopic reversibility) requires that λ1λ2λ3λ4 = λ−1λ−2λ−3λ−4. Specific reaction schemes are illustrated in Figure 2.2. the ti(x, y): DA∇2tA − ~VA · ∇tA + λ+1 (tB − tA) + λ−4 (tD − tA) = −1 DB∇2tB − ~VB · ∇tB + λ+2 (tC − tB) + λ−1 (tA − tB) = −1 DC∇2tC − ~VC · ∇tC + λ+3 (tD − tC) + λ−2 (tB − tC) = −1 DD∇2tD − ~VD · ∇tD + λ+4 (tA − tD) + λ−3 (tC − tD) = −1. (2.1) Di are the diffusion coefficients of the particle in each state, ~Vi are the advection velocity vectors, λi are the transition rates, and we set ti = 0 on the domain boundary. Although we will apply the general reaction scheme illustrated in Figure 2.1 to several scenarios, we will always identify state i = A with a free pMHC on the APC or supported planar bilayer and states i = B,C,D with pMHC bound to molecule(s) on the T cell (e.g. TCR, CD4, CD8), see Figure 2.2. Since the pMHC molecule undergoes directed motion towards the cSMAC only when bound to molecules within a cluster, we can make the equations simpler by working in the cluster reference frame (i.e. ~VA = −~V and ~VB = ~VC = ~VD = 0) and picking the cluster velocity vector along a coordinate axis ~V = (V, 0). We further assume that, when bound, the pMHC has a negligible rate of diffusion (i.e. DB = DC = DD = 0), to find DP∇2tA + V ∂tA/∂x+ λ+1 (tB − tA) + λ−4 (tD − tA) = −1 λ+2 (tC − tB) + λ−1 (tA − tB) = −1 λ+3 (tD − tC) + λ−2 (tB − tC) = −1 λ+4 (tA − tD) + λ−3 (tC − tD) = −1 (2.2) Chapter 2. pMHC dynamics in TCR microclusters 24 where we set DA = DP , the free pMHC diffusion coefficient. We use the last three equations to solve for tB and tD in terms of tA and so obtain a single equation for tA: DP∇2tA + V ∂tA ∂x = −β. (2.3) β = 1+Λ1 +Λ1Λ2 +Λ1Λ2Λ3, where Λi = λi/λ−i are the transition affinities. We eliminated Λ4 from β using the principle of detailed balance (microscopic reversibility) which requires that Λ1Λ2Λ3Λ4 = 1. For simplicity we will assume that the cluster is a square of side b. We formulate the solution to Equation 2.3 as tA(x, y) = βΦ(x, y), with Φ(x, y) = ∞∑ n=1 qn(x)sin (npiy b ) . (2.4) After substituting in, we obtain qn(x) = 2b2 ((−1)n − 1) DP (npi)3 [ exp (−x (V + wn) 2DP ) − 1 +exp ( V (b− x) 2DP )( 1− exp (−b(V + wn) 2DP )) sinh ( xwn 2DP ) /sinh ( bwn 2DP )] (2.5) where wn = √ V 2 + 4D2P (npi/b) 2 . We note that the escape time is only dependent on transition affinities, through the parameter β, and not on the individual forward and backward reaction rates. We obtain the mean escape time by averaging over all possible starting positions of the pMHC, < t >= β < Φ(x, y;V,DP , b) >= β b2 ∫ b 0 ∫ b 0 Φ(x, y;V,DP , b)dxdy. (2.6) This averaging gives us a simple measure of the time a pMHC will spend in a single cluster, allowing us to easily understand the effects of changing the parameters of the TCR-pMHC interaction. Finally, the mean distance a pMHC molecule is transported by the cluster is then < L >= V < t >. 2.2.2 Mean escape time and reaction rates determine the number of hits We now derive expressions for the number of transitions between each state in Figure 2.1 during a sojourn in a TCR cluster. We begin by calculating the probability, Pi, of finding the particle in state i by considering the Chapter 2. pMHC dynamics in TCR microclusters 25 transition matrix associated with Figure 2.1 augmented by the conservation equation PA + PB + PC + PD = 1, −λ1 − λ−4 λ−1 0 λ4 λ1 −λ2 − λ−1 λ−2 0 0 λ2 −λ3 − λ−2 λ−3 λ−4 0 λ3 −λ4 − λ−3 1 1 1 1 PA PB PC PD = 0 0 0 0 1 The solution of this system is (PA, PB, PC, PD) = (1/β)(1,Λ1,Λ1Λ2,Λ1Λ2Λ3). We can now use the Pi to approximate the mean number of transitions between each state. As an example, consider the mean number of transitions between state i = A and state i = B. The total amount of time the particle spends in state A is < t > PA. The particle transitions out of state A into state B or D with a total rate λ1 + λ−4 and therefore the number of transitions is < t > PA(λ1 + λ−4). We can calculate how many of these transitions were strictly to state B by noting that the probability of transitioning, given that a transition occurred, from A to B is λ1/(λ1+λ−4). Therefore, the number of transitions from A to B is [< t > PA(λ1 + λ−4)] [λ1/(λ1 + λ−4)] = λ1 < t > PA. A similar calculation shows that hits(B → A)=hits(A → B). In this way we can estimate the mean number of transitions (hits) along any arrow in the reaction network, while the particle (pMHC) stays in the cluster: hits(A B) = λ1 < t > PA, (2.7a) hits(B C) = λ2 < t > PB, (2.7b) hits(C D) = λ3 < t > PC, (2.7c) hits(D A) = λ4 < t > PD. (2.7d) 2.3 Results 2.3.1 TCR/pMHC binding events in a mobile TCR cluster We begin our study of the TCR/pMHC dynamics by focusing on a single TCR cluster that forms in the pSMAC and travels towards the cSMAC. We first calculate an upper bound on the number of TCR a pMHC can hit (bind to) during one visit to that cluster. Yokosuka et al. (87) experimentally measured the mean velocity of 81 TCR clusters as a function of their initial formation location. They found a mean velocity of V = 0.0249µm/s for pS- MAC clusters traveling towards the cSMAC and found that these clusters migrated a maximum of Lmax = 4.5µm towards the cSMAC (Figure S1 in their work). Therefore the maximum journey time is tmax ≈ 181 s. If during Chapter 2. pMHC dynamics in TCR microclusters 26 this time a pMHC molecule can continually engage TCR (i.e. the pMHC molecule cannot escape the cluster) and the mean time for a free pMHC to bind a TCR is negligible (i.e. the reaction on-rate is very large) then the number of TCR engagements, or ‘hits’, can be approximated by dividing tmax by the mean TCR-pMHC bond lifetime, 1/koff. Typical off-rates for agonist pMHC are 0.03-0.3 s−1 (69) and therefore the maximum number of possible hits if the pMHC remains in the cluster for 181 s is in the range of∼5-54. Parameters are summarized in Table 2.1. This rough calculation suggests that, in principle, a single pMHC molecule can engage a substantial number of TCR in a cluster before arriving at the cSMAC. However, the number of engagements can be significantly reduced by considering the effects of pMHC diffusion, TCR cluster composition and mobility, and the finite re- action on-rate. In the next section we will use our escape time formulation to quantify the decrease in the number of hits when these effects are considered. Table 2.1: Parameter Definitions and Estimations Parameter Description Reported parameter ranges Amc TCR microcluster area 0.35-0.5 µm2 (88) b TCR microcluster dimension Amc=b2, b = 0.59− 0.71µm or Amc=pib2, b = 0.33− 0.40µm R Synapse radius 5-6 µm (69) NT TCR number in a microcluster 40-150 (87; 88) Tmc TCR concentration in microclusters 80-430 µm−2 T0 TCR concentration outside microclusters 50 µm−2 (87) Cmc Coreceptor concentration in microclusters Not known KD TCR/pMHC dissociation constant 0.005-1500µM (see Table 2.2) KCD Coreceptor/pMHC dissociation constant ∼ 200µM (95; 96) V TCR microcluster velocity 0.0249 µm/s (87) DP pMHC diffusion coefficient 0.03 µm2/s (81; 97) Two state escape time formulation from a mobile TCR cluster To investigate the behavior of a single pMHC within a mobile TCR cluster, we consider the pMHC to be in one of two states; unbound (i=A) or bound (i=B) to a TCR; see Figure 2.2 (basic model). The reaction scheme is A B, with a forward rate λ1, backward rate λ-1, and transition affinity Λ1 = λ1/λ−1. We define the number of “hits” for this simple model to be the number of times (on average) the pMHC binds a TCR during its time in the cluster. We compute the mean escape time, the pMHC transport distance, and the number of hits by reducing the four state model described earlier to a two state model. This reduction is achieved by setting Λ2 = Λ3 = Λ4 = 0 in Equations 2.6 and 2.7a. The number of hits in this two state model is then hits(A B) =< t > λ1 1 + Λ1 (2.8) Chapter 2. pMHC dynamics in TCR microclusters 27 A B C D B) Coreceptor Heterodimerization Model A B A) Basic Model TCR pMHC Coreceptor C) Dimer Model DCBA Agonist-Agonist pMHC / Endogenous-Agonist pMHC Figure 2.2: Reactions schemes we consider. (a) two state reaction scheme between single pMHC and TCR. Definition of states: A - pMHC is unbound from TCR, B - pMHC is bound to TCR. (b) inclusion of coreceptors requires a four state reaction scheme. Definition of states: A - pMHC is unbound from TCR and coreceptor, B - pMHC is bound only to TCR, C - pMHC is bound to both TCR and coreceptor, and D - pMHC is bound only to coreceptor. (c) inclusion of pMHC dimers also requires a four state reaction scheme. Definition of states: A - Both pMHC are unbound from TCR, B - Agonist is bound to TCR, C - Both pMHC are bound to TCR, and D - Endogenous pMHC (or the other Agonist pMHC, depending on the dimer) is bound to TCR. We plot the mean escape time, tA, as a function of the initial position of the pMHC in Figure 2.3 for different values of V . We find that microcluster velocities < 0.1µm/s do not substantially affect the escape time. Estimating the pMHC transport distance and total TCR engagements Before we can determine the number of TCR engagements by a pMHC molecule in a cluster we need to estimate parameters. Several recent studies have characterized TCR cluster composition and mobility (87; 88; 89). By Chapter 2. pMHC dynamics in TCR microclusters 28 0 0.1 0.2 0.3 0.4 0.50 5 10 15 20 25 0 0.3 0.6 0.8 1.1 1.4 H its x (µm) t A (s) V=0 µm/s V=0.05 µm/s V=0.1 µm/s V=0.5 µm/s V=1 µm/s Figure 2.3: Residence time and total TCR hits in a TCR cluster as a function of the initial position. We plot tA(x, y) as a function of x for y = b/2 for several values of the microcluster velocity V . We use reaction parameters for the MCC88-103 peptide (Table 2): b = 0.59µm, Tmc = 286µm−2, DP = 0.03µm2/s, kon = 0.0057µm2/s and koff = 0.057s−1 comparing background and cluster anti-TCR-fab fluorescence intensity, Campi et al. (88) determined that 140 TCR are contained in a single cluster. Consistent with these findings, Yokosuka et al. (87) reported 40-150 CD3ζs per cluster. Cluster area was observed to be 0.35-0.5 µm2 (88). The concentration range of TCR in a cluster is then Tmc = 80 − 430µm−2. We also take the diffusion coefficient of pMHC to be DP = 0.03µm2/s (81). These parameters are summarized in Table 2.1. We summarize experimentally determined reaction rates between various TCR and pMHC in Table 2.2. The transition rates in our model are related to these using the relations, λ1 = konTmc , λ−1 = koff (2.9) where kon is the 2D on-rate which can be related to the experimentally measured 3D on-rate using a confinement length (see Table 2.2) (98; 99). The two-dimensional forward rate constant kon for the peptide MCC88-103 was calculated from its koff value and the two-dimensional dissociation constant determined in (69). For the other peptides, kon was calculated by assuming that the proportionality constant was the same as for MCC99-103. In relating the transition rates to the reaction rates in this way (equation 2.9) we have assumed that there is no com- petition among pMHC for TCR. This assumption is reasonable because pMHC concentration is low (< 10µm−2) compared to the cluster TCR concentration (> 100µm−2). Chapter 2. pMHC dynamics in TCR microclusters 29 Table 2.2: Estimates of TCR hits and pMHC transport by a mobile TCR cluster < t >* < L >* KD kon kon koff (s) (µm) Hits* TCR pMHC (µM) (M−1s−1 ) (µm2s−1) (s−1) CR− CR+ CR− CR+ CR− CR+ Data from Grakoui et al. (69) 2B4 MCC88-103 60.2 900 0.0057 0.057 12 121 0.30 3.0 0.67 6.7 2B4 T102S 238 1500 0.0095 0.36 3.5 35 0.087 0.87 1.1 11 2B4 T102G 1520 3400 0.022 5.0 0.91 9.2 0.023 0.23 2.5 25 3.L2 Hb64-76 12.1 5557 0.035 0.064 65 180 1.6 4.5 4.1 12 3.L2 N72T 9.90 15374 0.097 0.136 84 181 2.1 4.5 11 25 3.L2 N72I 14.9 16600 0.11 0.248 50 181 1.2 4.5 12 45 Data from Krogsgaard et al. (100) 2B4 102S 90 2240 0.014 0.20 8.7 87 0.22 2.2 1.7 17 2B4 PCC 32 1080 0.0068 0.035 23 181 0.58 4.5 0.80 6.2 2B4 MCC95-103 8.7 2200 0.014 0.019 86 181 2.1 4.5 1.6 3.4 2B4 K2 8.7 6670 0.042 0.058 86 181 2.1 4.5 4.9 10 2B4 K3 33 2120 0.013 0.071 23 181 0.56 4.5 1.6 12.6 2B4 K5 2.9 4900 0.031 0.014 181 181 4.5 4.5 2.5 2.5 Data from Garcia et al. (101) 172.10 MBP1-11 5.9 37200 0.24 0.219 126 181 3.1 4.5 28 39 1934.4 MBP1-11 31 5130 0.032 0.160 24 181 0.60 4.5 3.8 28 Data from Willcox et al. (102) JM22z HLFA-A2 17 69000 0.44 1.2 43 181 1.1 4.5 51 215 * We calculate the mean time the pMHC is expected to remain within the TCR cluster (< t >), the total distance the cluster transports the pMHC (< L >), and the number of TCR serially bound by pMHC (hits) as described in the main text in the absence (CR−) and prescence (CR+) of coreceptors. The 3D molar values are converted to 2D values using a confinement length of 0.262 µm (81). Parameters: b = 0.59µm, Tmc = 100TCR/(0.59µm)2 = 286µm−2, Cmc = Tmc, KCD = 200µM, V = 0.0249µm/s, and DP = 0.03µm2/s. We can now compute the mean escape time < t > (Equation 2.6, β = 1 + Λ1), pMHC transport distance < L >, and total hits (using Equation 2.8). We summarize these three results for specific TCR and pMHC in the the CR− columns of Table 2.2. We can draw two main conclusions: (1) pMHC interactions with mobile TCR in a cluster are insufficient to consistently transport pMHC to the cSMAC in a single journey (< L > < 4.5µm); (2) most pMHC molecules engage less than 5 TCR in the cluster. However, exceptions to these conclusions exist. For example, clusters comprised of 172.10 TCR transport MBP1-11 to the cSMAC and MBP1-11 can engage 40 TCR. There are also examples of substantial TCR engagement even though the pMHC is not transported to the cSMAC (e.g. 172.10/MBP1-11, JM22z/HLFA-A2). In constructing Table 2.2 we have imposed the total journey time, tmax = 181s, as an upper limit to < t >, which subsequently imposes an upper bound on < L > and hits. We impose this upper bound because once in the cSMAC, TCR begin to be internalized (89), an effect not accounted for in our model. We also focus on the cluster journey because experiments have shown that pMHC dependent signaling through the TCR occurs during the journey to the cSMAC and not in the cSMAC itself (87; 88; 89; 103). Chapter 2. pMHC dynamics in TCR microclusters 30 We can also determine the required TCR/pMHC affinity to achieve pMHC transport to the cSMAC by a single mobile cluster from Equation 2.6. Using the parameters given above, we calculate < Φ >= 0.41s (equation 2.4). Setting< t >= tmax, we find that on average the pMHC remains in the microcluster for tmax provided β > 442.2. For a TCR concentration of Tmc = 286µm−2 this corresponds to KD < 0.65µm−2. The maximum 3D dis- sociation constant that permits transport to the cSMAC, based on the TCR-pMHC interaction alone is therefore K∗D = 4.1µM. We note that estimates of TCR numbers in clusters and estimates of cluster size, discussed above, rely on opti- cal fluorescence microscopy (87; 88). Since clusters cannot be resolved by optical microscopy, the latter estimate is probably an upper bound. We find that variations in cluster size do not have a significant effect on < t > and hits, provided we fix the number of TCR per cluster, because a decrease (increase) in cluster size is proportionally balanced by a larger (smaller) reaction on-rate. 2.3.2 Coreceptors augment TCR/pMHC interactions There is a growing body of evidence suggesting that T cell activation by pMHC molecules is dependent upon coreceptors CD4 (69; 73; 104; 105) or CD8 (90; 95; 106; 107). In cases where the pMHC concentration is low (69; 73; 108) or the pMHC exhibits small affinity to TCR (107), T cells lacking coreceptors have been observed to be less likely to form an immune synapse (69; 73), flux calcium (73; 108), proliferate (69), or secrete IL-2 (107). How coreceptors facilitate the activation of T cells remains largely unknown, in part because the experimentally determined affinity between CD4 or CD8 and MHC is very weak (KD ∼ 200µM) (95; 96) (reviewed in (109)). CD4 and CD8 are known to associate with TCR via the signaling molecules Lck and ZAP-70 (110; 111), possibly providing additional stabilization to the TCR-pMHC-coreceptor complex. In this section we will show that coreceptors, although having weak binding to MHC, sufficiently augment the TCR/pMHC interactions such that substantial TCR engagement and pMHC transport to the cSMAC are achieved. Four state escape time formulation from a mobile TCR cluster To determine the degree to which coreceptors augment the TCR/pMHC interaction we will need to consider the full four state escape time problem. In this reaction scheme, the pMHC molecule can exist in four states: unbound from both TCR and coreceptor (i = A), bound to TCR (i = B), bound to TCR and coreceptor (i = C), and bound to a coreceptor (i = D), see Figure 2.2 (coreceptor model). There are two fundamental reactions in this Chapter 2. pMHC dynamics in TCR microclusters 31 scheme: TCR/pMHC (transition rates: λ1,λ−1) and coreceptor/pMHC (transition rates: λ2,λ−2). We assume the reactions occur independently and therefore identify the transitions between states C and D with those of A and B (i.e. λ−3 = λ1, λ3 = λ−1) and transitions between states D and A with those of B and C (i.e. λ−4 = λ2, λ4 = λ−2). As before, the mean escape time is given by equation 2.6, with β = (1 + Λ1)(1 + Λ2) in this case. We also keep track of the total number of times the pMHC binds TCR or “hits” (in this case determined by summing equations 2.7a and 2.7c. The Pi in equations 2.7 can be simplified to the form PA PB PC PD = 1 β 1 Λ1 Λ1Λ2 Λ2 . (2.10) Coreceptors increase pMHC transport distance and total hits The only additional parameter we introduce by including coreceptors is their transition affinity for the MHC molecule, Λ2. This parameter is related to the equilibrium dissociation constant as follows, Λ2 = Cmc/K C D (2.11) where Cmc is the coreceptor concentration in the cluster and K C D is the 2D dissociation constant between pMHC and coreceptors. The 3D dissociation constant has been reported to be ∼ 200µM between class I MHC and CD8αα (95) and ∼ 199µM between class II MHC and CD4 (96). As before, we convert 3D values to 2D values using a confinement length (see Table 2.2). The results when coreceptors are incorporated into the TCR/pMHC model are summarized in the CR+ columns of Table 2.2. We find that in most cases, the addition of coreceptors maximizes the mean escape time, which results in the transport of pMHC to the cSMAC. Furthermore, pMHC molecules that exhibit few TCR engagements in the absence of coreceptors are able to engage a substantial number of TCR in the presence of coreceptors. These increases are observed despite the large dissociation constant for the coreceptor-MHC interac- tion. Chapter 2. pMHC dynamics in TCR microclusters 32 2.3.3 T cell stimulation is correlated to pMHC transport to cSMAC The importance of coreceptors to T cell stimulation was highlighted in a study by Holler and Kranz (107). Briefly, they measured IL-2 release following interactions between APC bearing a particular pMHC and CD8− or CD8+ T cells. In separate experiments, they determined the binding constants for some of the TCR-pMHC pairs in their experiments. They found that coreceptors were necessary for T cell stimulation only when the dissociation constant for the TCR/pMHC bond roughly exceeded 5µM. This is comparable to our previous estimate for the minimal affinity required to retain pMHC within a TCR cluster for travel to the cSMAC (described above) of 4.1µM. In Table 2.3 we summarize the reaction rates for TCR/pMHC combinations used in their study and indicate the particular experiments where coreceptors were required for T cell stimulation. In this table we also compute < t >, < L >, and total hits in the absence (CR−) and presence (CR+) of coreceptors. We find a striking corre- lation between the experimental determination of CD8 dependence and our theoretical determination of whether the pMHC can be expected to travel to the cSMAC. We also find that, if T cell stimulation is independent of CD8, the predicted number of hits is also independent of the presence of coreceptors, but the opposite is true when CD8 is required for T cell stimulation. Table 2.3: CD8 augments low affinity TCR/pMHC interactions < t > < L > KD kon kon koff (s) (µm) Hits TCR pMHC (µM) (M−1s−1) (µm2s−1) (s−1) CR− CR+ CR− CR+ CR− CR+ T cell stimulation is CD8 independent 2C QL9/L 3.9 6300 0.040 0.025 181 181 4.5 4.5 4.5 4.5 m6α QL9/L 0.0055 155000 0.98 0.0008 181 181 4.5 4.5 0.15 0.15 m6α Y5/L 0.0051 115000 0.73 0.0006 181 181 4.5 4.5 0.11 0.11 m6α M5/L 0.034 147000 0.93 0.005 181 181 4.5 4.5 0.90 0.90 m6α H5/L 0.0778 NA - NA 181 181 4.5 4.5 - - m6α Q5/L 0.167 NA - NA 181 181 4.5 4.5 - - m67α SIYR/K 0.0159 277000 1.75 0.44 181 181 4.5 4.5 79 79 T cell stimulation is CD8 dependent 2C dEV8/K 84.1 2200 0.014 0.185 9.2 92 0.24 2.3 1.6 16 m33α dEV8/K 38 NA - NA 20 181 0.49 4.5 - - m67α dEV8/K 6.57 86000 0.54 0.567 113 181 2.8 4.5 64 102 2C SIYR/K 31.9 2300 0.015 0.075 23 181 0.58 4.5 1.7 13 Data taken from Holler and Kranz (107). Calculations and parameters are described in Table 2.2. We can write a simple formula to estimate the maximum pMHC-TCR dissociation constant, K∗D, that gives pMHC transport to the cSMAC. Rearranging β = (1 + Λ1)(1 + Λ2) and substituting the physical parameters for Chapter 2. pMHC dynamics in TCR microclusters 33 Λ1 and Λ2 we obtain, K ∗ D = Tmc ( K C D + Cmc ) βK C D − ( K C D + Cmc ) (2.12) We obtain K∗D = 6.65µm−2 (using parameters β = 442.2,K C D = 31.6µm −2 , and Cmc = Tmc = 286µm−2). The 3D value is then K∗D = 42.1µM, an order of magnitude larger than in the case without coreceptors. Examining Tables 2.2 and 2.3 we can see that many TCR/pMHC have a lower KD than this, permitting pMHC transport and maximal hits when coreceptors are present. In our analysis of the effects of coreceptors on pMHC transport and TCR hits we have assumed equal concen- trations of coreceptors and TCR. Lower concentration of coreceptors would decrease K∗D. In Figure 2.4 we plot KD versusK C D using Equation 2.12 forCmc = [Tmc, Tmc/2, Tmc/5, Tmc/10]. We find that at lower coreceptor con- centration, a dissociation constant of ∼ 200µM is too large to achieve pMHC transport to cSMAC. We conclude that if coreceptors facilitate pMHC transport to the cSMAC, they must be present in clusters at concentrations comparable to that of TCR. 20 40 60 80 100 200 400 600 800 1000 1200 K D (µM) K DC (µ M ) 1:1 1:2 1:5 1:10 Figure 2.4: Maximum dissociation constant (KCD) of coreceptors required to achieve transport of agonist pMHC to cSMAC. The dissociation constant between the agonist pMHC and TCR is given on the x-axis (KD). Results are shown for different ratios of coreceptor:TCR in the microcluster. 2.3.4 Potential effects of pMHC dimers on the APC Biochemical assays have provided evidence that MHC class II molecules form dimers (112; 113) and it is rea- sonable to suppose that they may form dimers in experiments using supported bilayers or APC. pMHC dimers have also been shown to be the minimal unit required for T cell activation in an assay where soluble multimeric pMHC complexes were used to stimulate T cells (114), although it is not clear to what extent this result informs Chapter 2. pMHC dynamics in TCR microclusters 34 the physiological situation where binding occurs at a cell-cell interface. Furthermore the coreceptor and the TCR that it associates with may bind different pMHC complexes (105; 115). This forms a “pseudodimer” model of TCR triggering, proposed in part to explain the observation that a single agonist pMHC complex can lead to TCR triggering. It was suggested that, when a TCR with an associated coreceptor binds to an agonist-pMHC, the core- ceptor binds a distinct self/null-pMHC complex. A TCR pseudodimer is formed when a second TCR binds to this self-pMHC complex. We will show that pMHC dimers can boost the effective affinity of their constituent pMHC for the cluster and thus allow enhanced pMHC transport to the cSMAC. Therefore, higher-order complexes such as pseudo-dimers will also allow enhanced transport. To study the effects of pMHC dimers we use the full four state escape time formulation, see Figure 2.2 (dimer model). This is exactly analogous to the coreceptor theory, with Λ2 in this case related to the second pMHC molecule in the pMHC dimer, Λ2 = Tmc/K (2) D (2.13) where K(2)D is the dissociation constant for the TCR-second pMHC bond. In the case of homodimers (identical presented peptide and MHC molecule), we ignore any cooperative effects and set K(2)D = KD. In the case of heterodimers,K(2)D will be different from KD. As in the previous cases, we find a simple relationship to determine the maximum dissociation constant, K∗D, required for pMHC transport to the cSMAC. Using the definition of β we find K ∗ D = Tmc ( K (2) D + Tmc ) βK (2) D − ( K (2) D + Tmc ) . (2.14) This equation is equivalent to Equation 2.12 when Cmc = Tmc and is shown in Figure 2.4 (1:1 case). In the case of pMHC homodimers, we find that for KD < 80µM the dimer will be transported to the cSMAC. Consequently, transport to the cSMAC is expected for almost all homodimers of pMHC listed in Tables 2.2 and 2.3. The observation that endogenous peptides accumulate in the IS (108) and that heterodimers of agonist/endogenous pMHC stimulate T cells (115) suggest that endogenous peptides may play a role in pMHC transport to the cS- MAC and in serial triggering. Endogenous peptides generally have an undetectable affinity for TCR and therefore have a dissociation constant > 200µM. Equation 2.14 (shown in Figure 2.4, 1:1 case)) indicates that endoge- Chapter 2. pMHC dynamics in TCR microclusters 35 nous peptides are able to transport the heterodimer only when the dissociation constant of the agonist peptide is stronger than ∼ 40µM. The physical reason for such transport is that the endogenous pMHC is able to anchors the heterodimer to clustered TCR and therefore increase the transport distance. In cases when the agonist pMHC dissociation constant is > 40µM, we predict that the addition of endogenous peptides alone will not sufficiently augment the interaction to result in pMHC transport to the cSMAC and substantial serial engagement of TCR. 2.3.5 TCR/pMHC binding events with multiple TCR clusters In the previous sections we have shown that agonist pMHC can be expected to escape from TCR clusters during the journey to the cSMAC unless the reaction is augmented by coreceptors or pMHC dimers. Substantial TCR engagements and transport to the cSMAC could be possible in the absence of these factors if a single pMHC molecule visits multiple TCR clusters whose collective motion sieve the pMHC molecules into the cSMAC. 2.3.6 Effects of multiple stationary clusters We leave a complete examination of pMHC dynamics in a field of moving clusters for the next section, focusing here on the question of how many engagements (hits) a pMHC would experience during a single sojourn in an IS that holds a number of immobile clusters enriched in TCR as well as a background concentration of non-clustered TCR. Varma et al. (89) generated such synapses by treating T cells with latrunculin A, a drug that prevents actin polymerization, shortly after synapse formation. The result, shown in Figures 6F-6J of their work, is an immobile and stable field of TCR clusters. In Figures 3C and 3D of their work they also illustrate the rapid (∼ 60s) reduction in calcium signaling upon administering latrunculin A to T cells forming an IS. In order to form a clear picture of the potential effects of multiple TCR clusters, in what follows we will not include coreceptors. We reduce system 2.1 by setting λ±i = 0, for all i 6= 1, and removing the advective field by setting ~Vi = 0, for all i. We take the synapse to be a disc of radius R containing N disc-shaped clusters centred at the points r1, r2, ...rN and having a radius of b. Each cluster contains NT TCR. These simplifications allow us to write a single equation governing the escape time tsyn, DP∇2tsyn = −1− λ1 λ−1 I(r)− λ ∗ 1 λ−1 (1− I(r)) (2.15) I(r; r1, r2, ...rN ) = 1 | r− rj |≤ b 0 | r− rj |> b (2.16) Chapter 2. pMHC dynamics in TCR microclusters 36 where the superscript, syn, indicates the escape time is from the entire synapse. The boundary condition is tsyn = 0 when |r| = R. To account for the difference between background and clustered TCR concentrations we have split the reaction term into two parts using the indicator function I , which is zero everywhere except within clusters where it is equal to one. Therefore the term with λ1/λ−1, captures reactions within clusters having a forward transition rate λ1. The term with λ∗1/λ−1 captures reactions outside of clusters with a forward rate λ∗1. The tran- sition rates are related to the physical parameters by equation 2.9 and the relation λ∗1 = konT0 where T0 is the TCR concentration outside clusters, which we take to be 50µm−2 (87). The TCR concentration within clusters is NT /(pib 2). As discussed earlier, there is uncertainty in the amount of area covered by TCR in clusters. Consequently, in the analysis that follows we fix the number of TCR per cluster,NT, and vary the cluster size, b. We can decompose the solution to Equation 2.15 into three parts, tsyn = t1 + t2 + t3, satisfying DP∇2t1 = −1, (2.17a) DP∇2t2 = − λ1 λ−1 I(r), (2.17b) DP∇2t3 = − λ ∗ 1 λ−1 (1− I(r)). (2.17c) All the ti = 0 on the synapse boundary (i.e. ti(r = R) = 0). t1(r) = (R2 − r2)/(4D) is the escape time from a synapse without any TCR, leading to the average time of escape < t1 >= R2/8DP (averaged over all possible starting positions). t2 is the time spent bound to clustered TCR, and t3 is the time spent bound to non- clustered TCR. By linearity, t3 = (λ∗1/λ−1) t1−(λ∗1/λ1) t2. Substituting in for the physical parameters we obtain t3 = (T0kon/koff)t1+(T0pib 2/NT )t2. This shows that t3 ≈ (T0kon/koff)t1 as b→ 0, in agreement with the theory for a uniform distribution of TCR presented in (81). We can obtain the mean number of transitions in this case by dividing the total time spent bound to TCR, t2 + t3, by the mean time per binding event, 1/λ−1 (i.e. hits = λ−1 < t2 + t3 >). Weak dependence of escape time on cluster size If b/R 1 then we can use matched asymptotics to obtain t2 as a power series in the small parameter = b/R, see Appendix A for detailed calculations. Averaging over all starting positions of the pMHC, we find, to first Chapter 2. pMHC dynamics in TCR microclusters 37 order in , < t2 > = konNT 4pikoffDP N∑ k=1 ( 1− ∣∣∣rk R ∣∣∣2) (2.18) < tsyn > = ( 1 + T0kon koff ) R2 8DP + ( 1− T0pib 2 NT ) konNT 4pikoffDP N∑ k=1 ( 1− ∣∣∣rk R ∣∣∣2) (2.19) We see that, to first order in , < tsyn > depends only on physical parameters and the locations of the clusters relative to the synapse boundary, and that centrally located clusters have the biggest impact on the mean time a pMHC spends in the synapse. We can also consider the case where we assume that the pMHC starts within a cluster. In this case, we average over all possible starting positions within clusters to obtain the mean time spent bound to clustered TCR: < tin2 > = NT k̄on pikoffDP 1 8 − log(b/R) 2 + 1 N N∑ j=1 Aj,0 (2.20) where the Aj,0 are constants that depend only on the cluster positions (see Appendix A). This formulation reveals the weak (logarithmic) dependence of escape time on TCR cluster size, b. pMHC engagement of clustered TCR is nearly independent of cluster size We begin by randomly placing TCR clusters within a synapse of radius 5.5 µm. In Figure 2.5a we show the locations, rj, of N=5 (black), N=25 (black + dark grey), and N=50 (all discs) TCR clusters. In Figure 2.5b we plot the total hits to clustered TCR (solid line) and to non-clustered TCR (dashed line) as a function of b for the three values of N. We obtain < t2 >, and hence the total hits, by using a central difference scheme for the Laplacian in equation 2.17 when b > 0.07µm. When b < 0.1µm we use the asymptotic solution, Equation 2.18, which agrees with the numerical solution on the overlapping range of b. We take reaction rates for a typical TCR-pMHC interaction: kon = 0.05µm2/s, koff = 0.05s−1. We find that the total hits to clustered TCR increases with N but is independent of the cluster size. Indeed, the asymptotic solution, Equation 2.18, is independent of b. As b → 0, for all values of N , the total hits to TCR external to clusters asymptotes to the value it would have if there were no clusters, as it should. For larger values of b, the total hits to non-clustered TCR decreases as N increases. The small changes in the number of hits at large b ' 1µm occur because, in the simulation, parts of the microclusters end up outside the idealized synapse region. Chapter 2. pMHC dynamics in TCR microclusters 38 (a) (b) 0 2 4 6 8 10 120 2 4 6 8 10 12 x (µm) y (µm ) −2 −1.5 −1 −0.5 00 100 200 300 400 500 600 log10(b) (µm) < hi ts > N=5 N=5 N=25 N=25 N=50 N=50 hits to MC TCR hits to non−MC TCR (c) (d) 0 0.05 0.1 0.15 0.2 0.25200 250 300 350 400 450 500 550 600 total microcluster area / synapse area < hi ts > hits to MC TCR hits to non−MC TCR Figure 2.5: Number of pMHC engagements to clustered TCR is almost independent of cluster size. We compute the number of binding events (hits) in two scenarios. In (a) we use an idealized disc synapse of radius 5.5 µm containing N randomly distributed TCR microclusters each of radius b and containing NT = 100 TCR. The positions of N=5 (black), N=25 (black + dark grey), and N=50 (all discs) TCR clusters is shown. In (b) we use the TCR microcluster distribution from a) to compute the total hits to clustered (solid line) and non-clustered (dashed line) TCR for the three values of N as a function of the microcluster size, b. In (c) we use experimental data (Figure 6H in (89)) to obtain a physiological microcluster distribution, see main text for details. Also shown in (c) are three steps along the multiple erosion operations performed on the microcluster stencil. Using these microcluster distributions we compute the total hits (d) as a function of the total microcluster area. Parameters: k̄on = 0.05µm 2/s, koff = 0.05s −1 , DP = 0.03µm 2/s. As discussed earlier, Varma et al. (89) created synapses containing immobile TCR clusters. In order to obtain a physiological TCR microcluster distribution, we performed simple image analysis on figure 6H of their work. We acquired a high resolution version of figure 6H and performed thresholding to convert the grayscale TIRF image into a binary (black/white) image. Morphological open and close operations were performed to remove isolated pixels. The resulting TCR microcluster stencil, shown in Figure 2.5c, was used as the indicator function for a numerical solution of Equation 2.17. The synapse boundary, where ti = 0, was obtained by alternate thresh- olding and multiple morphological open operations on the TIRF image (see black outline in Figure 2.5c). To examine the effect of cluster size we use a morphological operation that removes pixels that are not sur- rounded. Applying this morphological operation multiple times on the TCR cluster stencil progressively decreases Chapter 2. pMHC dynamics in TCR microclusters 39 the total TCR cluster area. We show the TCR cluster stencil at three instances in Figure 2.5c. We keep the to- tal number of TCR in clusters fixed at 5000, equivalent to 100 TCR distributed across 50 clusters (N=50 in the idealized synapse). We take kon, koff, and DP as in the idealized synapse calculation. In Figure 2.5d we plot the total hits as a function of the microcluster to synapse area ratio (synapse area is fixed at 85µm2). The total hits to clustered TCR remains almost unchanged. The main trend observed in Figure 2.5, that the number of hits to clustered TCR is only weakly dependent on cluster size, is a result of keeping the number of TCR per cluster constant. Decreases in cluster area are balanced by increases in the reaction on-rate within clusters leaving the number of hits unchanged. The number of hits that a single pMHC makes with clustered and non-clustered TCR during a single sojourn in the IS is substantial, even without the aid of bond stabilization by coreceptors or dimerization. 2.3.7 Effects of multiple mobile clusters The analysis above focused on a stationary (steady-state) distribution of signaling TCR clusters generated by treat- ing T cells with a cytoskeleton poison. However, in healthy synapses new TCR clusters are continually formed in the periphery of the synapse and signal as they migrate towards the cSMAC, where signaling is abolished (87; 88; 89). The time-dependent positions of TCR clusters precludes the use of an escape time formulation to study effects of multiple mobile TCR clusters. Instead, we use a Monte Carlo method to simulate the formation of TCR clusters in the pSMAC, at a rate of kmc, and their mobility towards the cSMAC. The simulation begins with the pMHC unbound in a random location within a TCR cluster located 4.5 µm away from the cSMAC. As before, the pMHC undergoes free diffusion unless it binds to molecules within a TCR cluster in which case it undergoes directed motion. In contrast to the single TCR cluster calculations, once the pMHC escapes the cluster it continues to diffuse and may subsequently bind to another TCR cluster. The simulation is terminated once the pMHC has escaped the synapse or has entered the cSMAC. The simulation efficiently and accurately captures the multiple time and space scales involved in the process. For clarity, the full description of the simulation is given in Appendix B. The only additional parameter introduced by incorporating multiple TCR clusters is their formation rate, kmc. Assuming there is roughly a constant number of TCR clusters in the pSMAC (∼ 50) we expect the rate of cluster formation to equal the rate of cluster arrival in the cSMAC. For an individual cluster, the latter quantity is simply V/L = 0.0249µm/s/4.5µm= 0.0055s−1. Therefore, kmc ≈ (50)(0.0055) = 0.28s−1. Parameters related to the geometry of the synapse (e.g. cSMAC radius, pSMAC radius, etc), which have been experimentally measured Chapter 2. pMHC dynamics in TCR microclusters 40 (69; 87; 89), are discussed in Appendix B. Few TCR clusters are required for pMHC transport to cSMAC and substantial TCR engagements In Table 2.4 we summarize the results of the Monte Carlo simulations for all pMHC found in Table 2.2. In contrast to the escape time calculations that focused on a single mobile TCR cluster, we find that pMHC have a large probability, > 0.7, of entering the cSMAC when we consider multiple TCR clusters. For comparison, the probability of cSMAC arrival in the absence of TCR clusters is ∼ 0.2, which can be computed by solving the diffusion equation on an annulus with zero Dirichlet boundary conditions (not shown). We further find that an individual pMHC may escape and subsequently enter > 5 TCR clusters. However, we find that re-entry into the same TCR cluster occurs frequently. In fact, many pMHC listed in Table 2.4 have a large probability of cSMAC arrival, > 0.9, but only utilize ∼ 2 unique TCR clusters to achieve this large probability. Table 2.4: TCR hits and pMHC transport by multiple mobile TCR clusters kon koff Prob. of Cluster Unique Total Hits per TCR pMHC (µm2s−1) (s−1) Transporta Visitsb Visitsb Hits Clusterc Data from Grakoui et al. (69) 2B4 MCC88-103 0.0057 0.057 0.77 5.4 2.5 14.0 2.6 2B4 T102S 0.0095 0.36 0.59 17.5 4.6 49.4 2.8 2B4 T102G 0.022 5.0 0.23 33.5 5.1 109.5 3.3 3.L2 Hb64-76 0.035 0.064 0.93 4.1 1.8 20.4 5.0 3.L2 N72T 0.097 0.136 0.94 4.8 1.7 42.7 8.9 3.L2 N72I 0.11 0.248 0.93 7.9 2.1 65.9 8.3 Data from Krogsgaard et al. (100) 2B4 102S 0.014 0.20 0.74 11.8 3.4 39.2 3.3 2B4 PCC 0.0068 0.035 0.84 3.7 2.0 10.1 2.7 2B4 MCC95-103 0.014 0.019 0.93 2.2 1.4 8.2 3.7 2B4 K2 0.042 0.058 0.94 3.6 1.6 20.0 5.6 2B4 K3 0.013 0.071 0.85 5.6 2.3 18.3 3.3 2B4 K5 0.031 0.014 0.95 1.8 1.3 9.6 5.3 Data from Garcia et al. (101) 172.10 MBP1-11 0.24 0.219 0.96 5.2 1.8 81.1 15.6 1934.4 MBP1-11 0.032 0.160 0.88 8.1 2.4 37.2 4.6 Data from Willcox et al. (102) JM22z HLFA-A2 0.44 1.2 0.91 14.6 2.4 275.9 18.9 a The probability of transport refers to the fraction of the simulations that terminated with the pMHC transported to the cSMAC. b These values correspond to the number of clusters a pMHC molecule visits before the simulation is terminated by pMHC arrival in the cSMAC or pMHC exiting the synapse. Unique visits refers to the number of different clusters the pMHC visits. c Obtained by dividing the Total Hits by Cluster Visits. Reported values are averages over 1000 simulations. Calculations and parameters are described in Table 2.2 and in the main text. See Appendix B for simulation details We find that by visiting multiple TCR clusters, a single pMHC can engage a substantial number of TCRs, see Total Hits in Table 2.4. However, we find that the average number of TCR hits per cluster visit is small. A Chapter 2. pMHC dynamics in TCR microclusters 41 greater number of TCR hits per cluster visit can be achieved by the use of coreceptors, see Table 2.2. Therefore a trade-off exists between the number of localized TCR engagements and the total number of TCR engagements. pMHC discrimination by cSMAC arrival A recent study found that B cell receptor (BCR) clustering and mobility can discriminate antigens in B cells. Fleire et al. (116) observed the time-course of B cell synapse formation and found that antigens with certain reaction rates to the BCR efficiently accumulated in the B cell cSMAC. Using Monte Carlo simulations that capture the essential features of the experiments, they showed that antigen collection to the B cell cSMAC by BCR clusters is dependent on the BCR-antigen affinity and proposed that this mechanism allows the B cell to discriminate antigen. Motivated by this observation, we investigated the possibility that T cells can discriminate pMHC using mobile TCR clusters. We used our Monte Carlo simulation, as described in the previous section, with the exception that the initial position of pMHC was not confined to a TCR cluster. In Figure 2.6a we plot the probability that the pMHC enters the cSMAC. For a given parameter set, we per- formed the simulation 500 times and record the fraction of the simulations that terminated by pMHC arrival in the cSMAC. As expected, we find that the probability of cSMAC arrival depends on the reaction rates. For high affinity interactions we find that the probability saturates to ∼ 0.9, reflecting the probability of finding a TCR cluster. Indeed, in these simulations we find that a single TCR cluster visit is sufficient for cSMAC transport (data not shown). For low affinity interactions we see that the probability of cSMAC arrival approaches ∼ 0.2, which reflects the random, diffusive, probability of entering the cSMAC. In these simulations we find that pMHC rarely bind to TCR clusters (data not shown). Including the effects of coreceptors, see Figure 2.6b, we find that the probability of cSMAC arrival is nearly independent of the pMHC reaction rates because the pMHC remains in a single TCR cluster (in agreement with the escape time calculations above). Therefore, pMHC discrimination by the probability of cSMAC arrival is only possible in the absence of coreceptors. We note that the results presented for the Monte Carlo model involving coreceptors apply equally well to the model of agonist pMHC dimerization with an endogenous pMHC (Figure 2.2, dimer model) if we make the assumptions that 1) the agonist pMHC diffuses as a dimer and 2) that the reaction rates between the endogenous pMHC/TCR are similar to the rates between coreceptors and MHC. Chapter 2. pMHC dynamics in TCR microclusters 42 (a) (b) −4 −3 −2 −1 0 1 0 0.2 0.4 0.6 0.8 1 log10(kon) (µm 2/s) Pr ob . o f c SM AC A rri va l −4 −3 −2 −1 0 1 0 0.2 0.4 0.6 0.8 1 log10(kon) (µm 2/s) Pr ob . o f c SM AC A rri va l k off=0.01 s −1 k off=0.1 s −1 k off=1 s −1 k off=10 s −1 Figure 2.6: Probability of cSMAC arrival by pMHC depends on kon and koff. Results are shown in the absence (a) and presence (b) of coreceptors. Legend for panel (a) is shown in panel (b). The probability of cSMAC arrival in the absence of TCR clusters is computed to be 0.19 (dotted black line) using an analytical solution of the diffusion equation (not shown). Error bars indicated standard error in the mean. Simulation performed as described in the main text. 2.4 Discussion Experimental observations show pMHC-dependent signaling in TCR clusters and the accumulation of pMHC in the cSMAC. Using a series of mathematical models we have analyzed serial engagement of TCR and transport of pMHC by mobile TCR clusters. We have shown that the TCR-pMHC interaction alone does not support substan- tial serial engagement of TCR or pMHC transport to the cSMAC by a single cluster but that if the TCR-pMHC bond can be stabilized (for instance, by coreceptor molecules such as CD4/8 or by dimerized pMHC, or by com- bined effects thereof), transport to the cSMAC and serial engagement within a single cluster can be expected. We have calculated minimum affinities of the TCR-pMHC bond that allow pMHC transport to the cSMAC to proceed efficiently in each scenario. We found evidence that coreceptors CD4/8 must be present in concentrations comparable to that of clustered TCR for pMHC transport in a coreceptor-dependent manner. Using experimental data (107) we were able to correlate predicted pMHC transport to the cSMAC with T cell stimulation as measured by IL-2 production. We also analyzed the role of multiple clusters in trapping pMHC in the synapse and boosting serial engagement. Our conclusions are based on a number of modeling assumptions and suggest future directions of experimental and theoretical enquiry. We discuss these in turn. Parameter estimation Our results underline the importance of measuring the kinetic parameters for TCR- pMHC bonds. The parameters which have the largest uncertainty in our model are probably the two-dimensional dissociation constant, KD, and two-dimensional on-rate, kon. We obtained 2D values by converting their respec- tive experimentally measured 3D values using a constant factor (confinement length) determined for a specific TCR/pMHC interaction. In a review, Davis et al. (74) discuss the importance of directly determining 2D rates, as they can be substantially different from their respective 3D values. Recently, we have proposed a method to Chapter 2. pMHC dynamics in TCR microclusters 43 directly determine 2D affinities and on-rates using live cells based on fluorescence recovery after photobleaching (FRAP) (O. Dushek, R. Das and D. Coombs, see Chapter 5). In this paper we used the macroscopic or long range diffusion coefficient for pMHC which can be substan- tially smaller than the microscopic diffusion coefficient (117). Increases in DP would decrease our predictions of < t >, pMHC transport distance and hits. We also remark that we have assumed that pMHC binding to TCR creates TCR-pMHC complexes that cannot diffuse. Diffusion of these complexes within the cluster would also decrease < t >. In the absence of other uncertainties concerning the binding parameters, these considerations would mean that the values of< t > in Tables 2.2, 2.3, and 2.4 would be upper bounds. FRAP experiments could indicate the extent to which diffusion constants change following TCR-pMHC binding. Role of clusters in TCR signaling Our calculations of numbers of TCR-pMHC engagement (“hits”) in single clusters are smaller than those calculated for a homogeneous TCR distribution (Wofsy et al. (81)) and a clus- tered TCR distribution (this work) across the whole synapse. However, despite the relatively small numbers of engagements, the spatial and temporal proximity of TCR binding events probably has a major impact on sig- nal transduction, perhaps due to spatiotemporal localization of signaling proteins in the cluster. In this work we looked only at physical stabilization of the TCR-pMHC bond by coreceptors and null pMHC. The presence of the coreceptor (and associated Lck) at the TCR during pMHC binding has been shown, experimentally and the- oretically, to boost TCR signaling (105). Furthermore, it has been shown that stronger agonists are able to take better advantage of the pool of null or endogenous pMHC present on the APC (115). Further experimental and theoretical analysis will refine our understanding of the costs and benefits of TCR clustering on antigen detection and effector function. Multiple cluster effects We have calculated the degree of TCR engagement by a diffusing pMHC in a field of immobile TCR clusters and found that substantial engagement of clustered and non-clustered TCR is expected. However, experiments show no change in Zap-70 recruitment to non-clustered TCR once the TCR-pMHC in- teraction is blocked (89) but do show large accumulation of Zap-70, Lck, SLP-76, and LAT to clustered TCR (87; 88; 89), suggesting that only clustered TCR signal. Using a full Monte Carlo simulation of mobile TCR clusters we find that pMHC achieve the maximum number of TCR engagements in the absence of coreceptors or pMHC dimerization by visiting multiple TCR clusters. However, the average number of TCR engagements per cluster is smaller without coreceptors or pMHC dimerization (compare TCR hits in Table 2.2 to hits per cluster in Table 2.4). We therefore report a trade-off between the total number of TCR hits and the number of localized Chapter 2. pMHC dynamics in TCR microclusters 44 TCR hits. Using the full Monte Carlo simulation we have also shown that pMHC transport to the cSMAC is possible without the aid of coreceptors or dimerization. In this case, we find that the probability that a given pMHC will arrive in the cSMAC (after many simulations) depends on the TCR/pMHC reaction rates and may serve to discrim- inate pMHC. However, discrimination will only be possible when pMHC are presented in large concentrations, as is the case for presented antigen to B cells. When discrimination is required based on 1-10 presented pMHC, as is the case for T cells, the probability of cSMAC arrivial, which is an ensemble quantity, will not be well correlated to the reaction parameters. Although a plausible mechanism for antigen discrimination in B cells, we suggest that such large scale (across the whole immune synapse) discrimination is unlikely to be taking place in T cells. Future particle tracking studies of labeled pMHC molecules (on a cell or supported bilayer) during T cell contact will inform on the duration that they spend within clusters and will resolve whether pMHC move between clusters. Such studies will reveal whether maximizing the total number of TCR hits or the number of TCR hits per cluster visit is important. Our work has highlighted the role of coreceptors in pMHC trapping in a single TCR cluster and suggest that blocking the action of coreceptors (e.g. by antibody blocking or mutations in the coreceptor binding domain to MHC) will lead to dramatic changes in the spatial-temporal dynamics of pMHC in the IS. Importance of pMHC-TCR bond affinity Fundamentally, T cell detection of pMHC rests on the TCR-pMHC interaction and is parameterized by just a few measured constants - kon, koff and/or KD = koff/kon. The im- portance of koff has been emphasized in some studies (for instance, (84; 85; 86; 118)) where an intermediate “optimal” lifetime is proposed to balance serial engagement with effective TCR signaling, suggesting that theKD of the interaction is less important. On the other hand, (107) presents striking evidence for the importance of KD. Here, we have found maximum values forKD that allow effective pMHC transport, independent of the kinetic rate constants, and a potentially important role for factors strengthening the TCR-pMHC bond. The data presented in tables 2 and 3 show no correlation between koff and hits in the absence of coreceptors, but good correlation (for data from (115) and (107)) in the presence of coreceptors. More work is definitely needed to elucidate the importance of koff vs KD in T cell signaling. Importance of serial engagement of TCR In this paper, we have focused on calculating levels of serial en- gagement of TCR by pMHC. The importance of serial engagement to T cell activation, however, remains unclear. Chapter 2. pMHC dynamics in TCR microclusters 45 It has been shown that very few agonist pMHC are able to stimulate signaling downstream from the TCR within 5-10 s of contact (73; 119). Furthermore, cytotoxic T cell killing can occur extremely rapidly, before the forma- tion of the full immune synapse (120). On the other hand, sustained TCR signaling during a prolonged interaction (many hours) has been shown to be important for cytokine production and cellular proliferation (121). We suggest that there are different levels of TCR signaling, leading to various cellular responses, and that serial engagement of TCR by pMHC may be important for sustained signaling on the timescale of minutes to hours. Furthermore, the density of agonist pMHC for a particular T cell is probably lower in vivo than in the usual experimental situations and the TCR-pMHC bond may be weaker. Understanding signal amplification by various mechanisms (coreceptor action, pMHC dimers and pseudodimers, serial engagement of TCR by pMHC, and serial encounters of T cells with APC) is a major theme in T cell acti- vation research. 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T cell killing does not require the formation of a stable mature immunological synapse. Nat. Immunol. 5:524–530. [121] Huppa, J. B., and M. M. Davis. 2003. T-cell-antigen recognition and the immunological synapse. Nat. Rev. Immunol. 3:973–983. 51 Chapter 3 T cell receptor clustering is essential for accurate ligand discrimination1 3.1 Introduction T cells are responsive to small numbers of particular pMHC complexes presented on APC. However, the bonds formed between TCR and agonist pMHC are weak and transient (122; 123; 124), with solution dissociation con- stants, KD, in the range of 0.1-100 µM and half-lives in the range of 1-100 s. Experiments using a variety of TCR, pMHC, altered peptide ligands (122; 123; 124; 125; 126) and point mutations in the CDR3 loops of the TCR (127) demonstrate clear correlations between T cell stimulation, as measured by cytokine production and cytotoxicity, and the solution reaction parameters governing the TCR-pMHC interaction. For example, Holler and Kranz (123) measured IL-2 production from T cells incubated with APC pulsed with a particular peptide. In separate experiments they measured the solution reaction parameters between the TCR and pMHC using surface plasmon resonance (SPR), finding a good correlation between IL-2 production and KD. Other work has shown that modest variations (factors of ∼ 1.5) in the measured binding parameters lead to measurable changes in the cellular response (122; 126). TCR aggregation into sub-micron scale clusters has been shown to occur in T cells upon stimulation by pMHC presented on a suspended planar bilayer (128; 129; 130) and on an APC (129). Many signaling molecules impor- tant to T cell activation have been shown to localize to these TCR clusters (128; 129; 130) and there is evidence that the coreceptors CD4 and CD8 are present with Lck, an important signaling molecule associated with the cytoplasmic tail of these coreceptors, in TCR clusters (128; 131). The function of TCR clustering is discussed controversially. Normally, experiments are undertaken to correlate the binding parameters of a given TCR-pMHC interaction with the cellular response. Here, we take an alternative approach by asking how well a T cell can measure the 1A version of this chapter has been submitted for publication. Dushek O, Coombs D (2008) T cell receptor clustering is essential for accurate ligand discrimination. Chapter 3. T cell receptor clustering is essential for accurate ligand discrimination 52 binding parameters of the TCR-pMHC interaction. With this idea in mind, we imagine that the T cell seeks sim- ply to measure the binding parameters, and the decision to act (making the pMHC a de facto agonist) follows this measurement. As we shall see, since TCR-pMHC bonds form and break rapidly and stochastically, pMHC must be repeatedly sampled (by binding TCR) before the reaction parameters can be confidently estimated. We will show that TCR clustering is required, in part, to ensure that the multiple binding events are localized, allowing for rapid and accurate detection of presented pMHC. 3.2 Methods 3.2.1 Simulation method The modeling approach we have used in this study is the next reaction Monte Carlo simulation, originally pro- posed by Gillespie (132). The primary purpose of our simulations is to record the duration of binding events and the time interval between binding events as a function of several parameters. We chose the next reaction method because, unlike other Monte Carlo simulations that rely on iteration numbers or fixed time steps, this method advances time in a natural way that is in good agreement with an underlying continuum model (132). A Monte Carlo approach is used, instead of a PDE, because we wish to investigate how stochastic binding events (duration between binding events and duration of individual binding events) may be used by the T cell to determine kinetic parameters. Our spatial reaction-diffusion simulation is based on the work of Isaacson and Peskin (133). We simulate a single pMHC diffusing on the APC membrane and binding to TCR on the T cell membrane, see Figure 3.1 (basic model). We use a lattice simulation which allows for reactions when the pMHC is in the same lattice site as a TCR. The binding rate in this case is kmicro = kon/h2, where kon is the experimentally measured (e.g. by SPR) bimolecular reaction constant and h is the lattice constant. The unbinding rate is koff and diffusion is captured by a first order reaction, of rate D/h2, to nearest neighbours lattice sites. We take h to be roughly the size of a TCR, h = 10 nm, in our simulations making it possible to capture both reaction-limited and diffusion-limited regimes. Depending on the state of the pMHC only certain reactions are possible and the quantity a is the combined rate. In the basic model, Figure 3.1, three states are possible: pMHC is unbound and not above a TCR (a = 4D/h2), pMHC is unbound above TCR (a = 4D/h2 + kon/h2), or pMHC is bound (a = koff). Based on the overall rate, a, we compute the time for the next reaction, τnext = 1 a ln ( 1 r1 ) Chapter 3. T cell receptor clustering is essential for accurate ligand discrimination 53 where r1 is a uniformly distributed random number between 0 and 1. Knowing τnext, we use a second random number, r2, to determine which reaction actually took place. Finally, we simulate the reaction, advance time by τnext, recompute a, and repeat. A B C D TCR pMHC Coreceptor Coreceptor Heterodimerization Model A B Basic Model Figure 3.1: Models of TCR-pMHC reactions with and without coreceptor-MHC interactions. In each simulation we record the location in the membrane of each binding event, the time between binding events (τon), and the duration of binding at each event (τoff). The pMHC is initially bound and the simulation is stopped after 10 binding events. We compute the percent error in estimating kon from τon as Eon = |1 − (1/〈τon〉)/(kon/h2)|×100 and the percent error in estimating koff from τoff asEoff = |1− (1/〈τoff〉)/(koff)|×100. Angled brackets indicate the mean after 10 binding events. We collect statistics for each set of parameters by repeating the simulation 500 times. Confidence estimates are defined as the fraction of simulations that result in Eon (or Eoff) less than 50%. We tested the accuracy of our stochastic simulations by comparing them to a partial differential equation (PDE) model. We performed simulations, as described above, but terminated the simulation when the pMHC reached a distance R. This minor change converted our stochastic simulation to a stochastic first passage time simulation. We performed 1000 simulations and in Figure 3.2 we show the binned (and normalized) first passage time (grey line). The PDE describing this first passage time process is, ∂fA ∂t = −konTfA + kofffB +D∇2fA, ∂fB ∂t = konTfA − kofffB (3.1) with fA = 0 on the boundary (at r = R), fA(t = 0) = δ(r)/r, and fB(t = 0) = 0. To obtain the probability of a first passage we numerically solve the above PDE and compute the total flux through the boundary, p, as a function of time. We plot p in Figure 3.2 (dotted black line) and find that it is in good agreement with the Chapter 3. T cell receptor clustering is essential for accurate ligand discrimination 54 stochastic simulation in both the reaction- and diffusion-limited regimes. Good agreement is also observed over a wide range of parameters. (a) (b) 0 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 time (s) pr ob ab ilit y of fi rs t p as sa ge , p (t) Stochastic simulation PDE computation 0 0.5 1 1.5 2 x 105 0 1 2 3 4 x 10−5 time (s) pr ob ab ilit y of fi rs t p as sa ge , p (t) Stochastic simulation PDE computation Figure 3.2: Comparisons of stochastic simulation and PDE computations. In a) we simulated a reaction-limited process (kon = 0.005µm2/s) and in b) we simulate a diffusion limited process (kon = 5µm2/s). Parameters: R = 1µm, DP = 0.05µm 2/s, koff = 1s−1. We investigated an alternate stochastic simulation that allowed pMHC to bind TCR when it was in one of eight nearest neighbour lattice sites, in addition to being centered on the TCR. In contrast to the hard-shell simulations, the overall binding rate in these soft-shell simulations is smeared over nine lattice sites. The rate of binding in the neighbouring lattice sites decreased according to a Gaussian with a full-width-half-maximum of h, the grid spacing. We normalize the binding rates such that the sum of the rates in all nine sites is equal to kon. We obtained good agreement with the PDE and repeating all simulations using this model we find no substantial variation in the results. 3.2.2 Simulation parameters The simulations we have used rely on several parameters, many of which have been experimentally determined. The solution reaction parameters for many TCR/pMHC have been measured using SPR (122; 123; 124; 126). Solution on-rates have been reported in the range of 500-300,000 M−1s−1 and off-rates 0.0001-5 s−1. SPR mea- surements have revealed similar equilibrium binding constants between CD4 or CD8 and MHC (134; 135). Wyer et al. (134) report the solution on-rate to be > 100, 000 M−1s−1 and the off-rate to be > 18 s−1. In our simula- tions we have used the solution off rates directly. We note that the actual value of koff for TCR-pMHC only alters the total time in Figure 3.4 and does not affect results in other figures. We have simulated reactions between membrane proteins and hence the bimolecular reaction parameter, kon, is a 2D quantity. However, reaction measurements between TCR-pMHC and CD4/CD8-MHC using SPR provide Chapter 3. T cell receptor clustering is essential for accurate ligand discrimination 55 solution or 3D on-rates. To obtain estimates of 2D quantities we multiplied the 3D on-rate (in units of M−1s−1) by a factor of 1015/NA (where NA is Avagadro’s number) to obtain units of µm3/s. We next divide this 3D quan- tity by a confinement length to obtain the 2D on-rate in units of µm2/s. We use a confinement length of 0.262 nm (136). This conversion indicates a range of ∼ 0.001− 2µm2/s for TCR-pMHC on-rates and ∼ 0.1µm2/s for CD4/CD8-MHC on-rates. The accuracy of this method is unknown and we therefore explore a larger range in the TCR-pMHC on-rate (see x-axis in figures). We have taken the diffusion coefficient of pMHC to be 0.05µm2/s (137). Simulations with D = 0.1µm2/s or D = 0.01µm2/s give similar results. The diffusion of TCR also warrants some discussion. Our goal was to create a simple stochastic simulation to characterize τon, the time between binding events, as a function ofD, T , kon, and koff. We have assumed that upon binding to a TCR, the diffusion coefficient of the TCR-pMHC complex is zero. We have also assumed a diffusion coefficient of zero for TCR. These assumptions, although quantitatively altering our results, do not alter our conclusions. When initially unbound, a pMHC molecule diffuses around looking for another TCR. The probability of encountering another TCR depends on the TCR concentration and simulating TCR mobility will not alter the time it takes to find another TCR, on average. We neglect TCR diffusion for com- putational efficiency. On a grid of 5 by 5 µm at a TCR concentration of 5000 µm−2 it would be very expensive to simulate diffusion of 125 000 TCR. 3.3 Results 3.3.1 T cells cannot make a direct and reliable estimate of KD Given the observed correlation betweenKD and T cell stimulation (123; 126), we first consider whether the T cell can determine KD directly. KD is an equilibrium quantity determining the concentration of pMHC-TCR bonds, according to KD = PT/B where B is the concentration of bound TCR, T of free TCR, and P of free pMHC in the contact region. There are substantial obstacles to making an accurate measurement of KD within a reason- able timeframe. First, agonist pMHC may be present at low numbers (138), and may enter and leave the contact area during the sample time, so there will be large fluctuations in the number of TCR that are bound. Second, non-specific (null) pMHC are likely present in large numbers, weakly binding with TCR and interfering with agonist binding (as well as possibly generating weak signals that must be ignored). In summary, TCR occupancy alone cannot be used to estimate KD and so pMHC quality must be assessed by some means that uses the kinetic parameters of the reactions (kon and koff), possibly allowing subsequent estimation of KD. Chapter 3. T cell receptor clustering is essential for accurate ligand discrimination 56 3.3.2 Multiple binding events are required to determine the TCR/pMHC reaction rates We now imagine an APC bearing a single pMHC that binds and unbinds from TCR on the T cell surface. How many times must a pMHC bind to TCR in order that the true unbinding rate koff can be estimated with a given accuracy? Since unbinding is an exponential stochastic process, the maximum likelihood estimate of koff is the re- ciprocal of the mean binding time. Even with perfect measurement of pMHC-TCR bond durations, the number of samples needed to have 85% confidence that the estimate of koff is within 50% of the actual value can be calculated as 10 (139). In order to ensure that these multiple binding events arise from the same pMHC and for effective integration of signals by the intracellular signalling machinery, the binding events presumably must be local- ized, for instance within TCR microclusters (128; 129; 130). pMHC are expected to bind multiple TCR within a single cluster only if the TCR-pMHC bond is stabilized, for instance by the TCR coreceptors CD4 and CD8 (136). Continuing with this line of reasoning, we ask how well the binding rate of the pMHC can be estimated from the time between binding events. This rate depends on the two-dimensional chemical forward rate constant (kon), the diffusion coefficient of the pMHC (D) and the TCR concentration (T ). The forward rate constant kon is measured in units of µm2/s, and is related to the rate of formation of bonds between a single pMHC-TCR pair, kmicro, by kmicro = kon/h2 where h ∼ 10 nm is the reaction radius. Reported values of kon for agonist pMHC range from 0.001 − 2µm2/s (see methods). We performed spatial Monte Carlo simulations of a pMHC diffus- ing (D = 0.05µm2/s (137)) through the contact area and interacting with a homogeneous distribution of TCR (132; 133) (Figure 3.1: basic model) and took the estimate of kmicro to be the reciprocal of the mean time between binding events. Simulations begin with the pMHC in the bound state and are stopped after 10 binding events. In Figure 3.3a we report the statistical confidence that the estimate of kon is within 50% of the actual value, for varying TCR concentrations. We observe that, at physiological TCR concentrations (T ∼ 100µm−2 for resting T cells and ∼ 500µm−2 within TCR clusters (128; 129; 136)), the confidence in estimating kon is low unless kon ≥ 1µm2/s.The estimates of kon for slowly-binding pMHC at physiological TCR concentrations are poor be- cause they spend a lot of time diffusing between binding events. In fact, if kon is smaller thanD then the system is reaction-rate-limited and direct estimates of kon might be made using information about the TCR concentration. However, in such cases, pMHC binding events are infrequent and widely distributed, making it hard to imagine how the TCR signaling machinery might integrate them successfully or how it is possible to distinguish the ago- nist signals from those due to null pMHC. Chapter 3. T cell receptor clustering is essential for accurate ligand discrimination 57 (a) (b) −4 −3 −2 −1 0 1 0 20 40 60 80 100 log(k on ) (µm2/s) % C on fid en ce in E st im a te o f k o n [TCR]=100 µm−2 [TCR]=500 µm−2 [TCR]=1000 µm−2 [TCR]=5000 µm−2 −4 −3 −2 −1 0 1 0 2 4 6 8 log(k on ) (µm2/s) M ea n D is ta nc e (µm ) [TCR]=100 µm−2 [TCR]=500 µm−2 [TCR]=1000 µm−2 [TCR]=5000 µm−2 Figure 3.3: TCR concentration impacts the accuracy in the estimation of kon and engagement localization. (a) statistical confidence that the estimate of kon from 10 binding events is within 50% of the true value. (b) mean distance between 10 TCR engagements. Parameters are koff = 0.1 s−1, D = 0.05µm2/s (simulation methods and model parameters are discussed in the supplementary information) 3.3.3 TCR enrichment impacts pMHC detection The mean distance and time between binding events scale as (D/konT )1/2 and 1/konT respectively, so increases in TCR concentration and kon improve spatial and temporal localization of signals (Figure 3.3b and 3.4a). How- ever, even at clustered TCR concentrations of 500 µm−2, the mean distance and time between events exceed the cluster size (∼0.5 µm) and cluster lifetime (100s) if kon < 10−2µm2/s (129). Summarizing, we have shown that membrane organization, in the form of TCR enrichment, impacts the accuracy, localization, and speed of pMHC detection and discrimination. However, TCR clustering as observed in vitro does not allow accurate discrimina- tion of kon during a 10-sample period or confinement of 10 binding events within a single cluster. (a) (b) −4 −3 −2 −1 0 1 200 400 600 800 1000 1200 1400 log(k on ) (µm2/s) Si m ul at io n Ti m e (s) [TCR]=100 µm−2 [TCR]=500 µm−2 [TCR]=1000 µm−2 [TCR]=5000 µm−2 −4 −3 −2 −1 0 1 90 100 110 120 130 140 150 log(k on ) (µm2/s) Si m ul at io n Ti m e (s) [TCR]=100 µm−2 [TCR]=500 µm−2 [TCR]=1000 µm−2 [TCR]=5000 µm−2 Figure 3.4: Total time required to achieve 10 binding events. The simulation is performed as described in the main text in the absence (a) and prescence (b) of coreceptors. As expected, for large kon the total simulation time is dependent only on koff and plateaus to 10*(1/koff)≈100 s. In the absence of coreceptors and TCR enrichment we see that the detection time of pMHC with small kon is substantial. Parameters: D = 0.05µm2/s, koff = 0.1 s−1. Chapter 3. T cell receptor clustering is essential for accurate ligand discrimination 58 3.3.4 TCR enrichment with coreceptors improves pMHC detection for weakly binding pMHC We now investigate whether bond stabilization by the TCR coreceptors CD4 and CD8 can improve pMHC detec- tion and discrimination. To this end, we extend our previous simulation to allow for coreceptor binding to pMHC (Figure 3.1). We assume a 1:1 TCR:coreceptor stoichiometry and use experimentally determined reaction param- eters (134; 135) (see methods). Repeating the simulations we obtain Figures 3.5a-b and Figure 3.4b. We find a dramatic improvement in the accuracy of kon (Figure 3.5a) when the TCR concentration is ∼ 500µm−2 or larger. Under these conditions, we also find that pMHC binding events are well localized (< 0.5µm) and rapid (10 events within 100 s). Coreceptors are not required to achieve accuracy, localization, and rapid detection of pMHC having large kon. We note that other mechanisms of bond stabilization (for instance, dimerization of agonist/self-pMHC (140)) could create more stable complexes and achieve similar results. However, our results provide a simple explanation for the observations of Holler and Kranz (123), who found that TCR coreceptors must be present for pMHC detection when the TCR-pMHC bond has large KD. (a) (b) −4 −3 −2 −1 0 1 0 20 40 60 80 100 log(k on ) (µm2/s) % C on fid en ce in E st im at e of k o n [TCR]=100 µm−2 [TCR]=500 µm−2 [TCR]=1000 µm−2 [TCR]=5000 µm−2 −4 −3 −2 −1 0 1 0 0.5 1 1.5 2 log(k on ) (µm2/s) M ea n D is ta nc e (µm ) [TCR]=100 µm−2 [TCR]=500 µm−2 [TCR]=1000 µm−2 [TCR]=5000 µm−2 Figure 3.5: Clustering of coreceptors with TCR improves ligand detection. Identical to Figure 3.3 expect that the coreceptor model is used, see Figure 3.1. Reaction rate between coreceptors and MHC was taken to be 0.1µm2/s (on-rate) and 50 s−1 (off-rate) TCR clustering and the presence of coreceptors effectively reduce the mobility of pMHC enabling the T cell to accurately, locally, and rapidly, probe the kinetic parameters of a given pMHC using TCR. Consistent with these observations we find that a very small pMHC diffusion coefficient (D < 0.001µm2/s in contrast to mea- surements of∼ 0.05µm2/s (137)) would allow for pMHC detection in the absence of coreceptors and substantial TCR enrichment (results not shown). Similarily, increasing the on-rate for CD4/CD8-MHC by a factor of 102 allowed coreceptors alone, without the aid of TCR enrichment, to immobilize pMHC (results not shown). Lastly, the confidence in estimating koff in our simulations is independent of kon, the TCR concentration, and the presence of coreceptors, see Figure 3.6. Chapter 3. T cell receptor clustering is essential for accurate ligand discrimination 59 (a) (b) −4 −3 −2 −1 0 1 20 40 60 80 100 log(k on ) (µm2/s) % C on fid en ce in E st im at e of k o ff [TCR]=100 µm−2 [TCR]=500 µm−2 [TCR]=1000 µm−2 [TCR]=5000 µm−2 −4 −3 −2 −1 0 1 20 40 60 80 100 log(k on ) (µm2/s) % C on fid en ce in E st im at e of k o ff [TCR]=100 µm−2 [TCR]=500 µm−2 [TCR]=1000 µm−2 [TCR]=5000 µm−2 Figure 3.6: Confidence in the estimation of koff is independent of kon and TCR concentration. Statistical confi- dence that the estimate of koff, from the recipercol of the mean waiting time after 10 engagements, is within 50% of the true value in the absence (a) and presence (b) of coreceptors. The confidence is ≈ 85% as expected (see main text). Parameters: D = 0.05µm2/s, koff = 0.1s−1. In applying our results to TCR clusters we have ignored the directed motion of TCR clusters to the center of the contact region (129). We have previously shown that the diffusion coefficient is the primary determinant of pMHC residence time within a cluster and that the cluster velocity only determines the transport distance (136). Consequently, the time between binding events and hence our results will not be altered by TCR cluster mobility. 3.4 Discussion The present work suggests a model where pMHC binding to TCR (141) can trigger the formation of a TCR cluster. Using a simple model where TCR diffuse and are trapped in a localized region (142), thus forming a microcluster, we estimate that a TCR cluster can form in ∼ 1s, which is smaller than typical TCR-pMHC bond half-lives. After the cluster is formed, the reaction rates of the pMHC are rapidly and efficiently measured via a series of spatially localized binding events. Self-pMHC that may subsequently enter the small cluster can be shown to transiently bind few TCR (during the lifetime of the cluster). Discrimination and signaling within a cluster may then be explained by existing nanoscale models of TCR signaling (143; 144; 145). Having outlined the necessity of TCR clustering to detect the TCR-pMHC biochemical parameters, it remains to be determined how this information is translated, via TCR signaling or otherwise, into differential release of cytokines (122; 123; 124; 126). The serial engagement hypothesis (146) proposes that multiple TCR binding by a single pMHC is important for signal amplification. Here, we have found that serial engagement is essential for cells to determine biochemi- cal rate constants rapidly and robustly. In contrast to analyses based on serial engagement (147; 148), which have focused largely on the role of the off-rate of the TCR-pMHC bond, however, we have found a role for a large kon in allowing resampling of the pMHC within a single TCR cluster. These effects will be important in future Chapter 3. 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Motion (thermal or otherwise) of unbleached proteins from outside the bleached region causes fluorescence in the bleached region to recover over time. The exact form of the recovery is governed by the mobility properties of the labeled protein itself, modulated by the details of the experimental protocol (i.e. size and shape of the bleached region, amount of bleaching, etc.). Excellent reviews of the technique are given in (154; 155; 156; 157). In particular, the book chapter by Rabut and Ellenberg (157) provides a complete general description of the technique, including numerous experimental practicalities. As described in the review of Goodwin and Kenworthy (156), a common method to obtain the diffusion coef- ficient (D) of the labeled protein and the fraction of these proteins that are mobile (Mf) is to fit an equation to the fluorescence recovery in the bleached region. The equation is derived from a model that captures the experimental protocol and the underlying physical processes. Commonly used models are described by Ellenberg et al. (158) and Axelrod et al. (159). We will show here that for the space and time-scales of FRAP experiments for cell sur- face proteins, certain key assumptions of the common models are violated resulting in significant errors in D and Mf. We will then present a refined experimental protocol and fitting procedure for use in obtaining the diffusion coefficient and mobile fraction of cell surface molecules. 1A version of this chapter has been published. Dushek O, Coombs D (2008) Improving parameter estimation for cell surface FRAP data. J. Biochem. Biophys. Methods. 70:1224-1231 Chapter 4. Improving parameter estimation for cell surface FRAP data 65 4.2 Methods 4.2.1 Experimental protocols for cell surface FRAP In what follows, we assume, as is usually the case for a CLSM, that the bleaching beam and monitoring beam are parallel to each other and to the z-axis. There are two common protocols for performing confocal FRAP for cell surface molecules using a CLSM. Protocol 1, the simpler of the two, is more suitable if the cell is able to spread considerably on a substrate. In this case, the focal plane of the microscope (depicted as the xy plane in Figure 4.1) is adjusted to obtain fluorescence from membrane proteins that are directly above the substrate. A region of the cell membrane, generally a rectan- gle measuring Lxb by Lyb, is selected for bleaching and in a subset of this region, which we call the monitoring region, the fluorescence is integrated to obtain the iconographic FRAP recovery curve. The dimensions of the monitoring region are Lxm by Lym. In this case, it is quite easy to visualize both the bleached and monitoring regions. Studies where this protocol is employed are (156; 160; 161; 162) and (163). This protocol cannot always be used because many cells do not spread sufficiently and further, light scattering can cause problems when the microscope is focused close to the substrate. Figure 4.1: Experimental geometry for protocol 1 (not to scale). Shown here is a schematic of a cell that has spread on a substrate. The focal plane of the CSLM is adjusted to obtain fluorescence from membrane anchored proteins proximal to the substrate. The bleached region is a rectangle measuring Lxb by Lyb and the monitoring region is a subset of this region measuring Lxm by Lym. The geometry is more complicated in protocol 2 because the focal plane of the microscope is set to the equa- torial plane of the cell. We assume the cell is spherical and centered at the origin of a Cartesian coordinate system. A ring of fluorescence, represented as a circle in the xy plane, is observed from the equatorial plane of the cell (Figure 4.2a). The thickness of this ring is dependent on the confocal pinhole, which controls the depth, Wz, Chapter 4. Improving parameter estimation for cell surface FRAP data 66 about the focal plane where fluorescence is collected as shown in Figure 4.2b. The box (Wx by Wy) in Figure 4.2a represents a typical region selected for bleaching. Unlike protocol 1, here one cannot directly observe the bleached region on the cell surface. However, since we do know that the bleaching beam penetrates the entire depth of the cell (157), we can safely assume that the bleached region is a projection of the box of Figure 4.2a onto the cell surface. Therefore the bleached region is a stripe, shown in Figure 4.2c, measuring Lxb by Lyb. The monitoring region in this case is the intersection of the focal plane and the bleached stripe. Its dimensions are Lxm by Lym as indicated in Figure 4.2c. The fluorescence in the monitoring region is integrated to obtain the FRAP recovery curve. Examples where this protocol is employed are (151; 152) and (153). Note that, in describing protocol 2, we label experimentally adjustable parameters with a W and the quantities describing the actual shape of the bleached and monitoring regions by L. The most common equation used to fit FRAP recovery curves generated by protocol 1 or 2 is the empirical equation presented by Ellenberg et al. (158). We restate their 1D approximation as F (t) Fp = ( 1− Fo Fp ) Mf ( 1− ( 4piDt (Lyb)2 + 1 )− 1 2 ) + Fo Fp (4.1) where F (t) is the time-dependent fluorescence intensity during the recovery phase, Fp the fluorescence prior to bleaching, Fo the fluorescence immediately after bleaching, D the diffusion coefficient, and Lyb the width of the stripe. The mobile fraction, Mf, is defined as Mf = F∞ − Fo Fp − Fo (4.2) where F∞ is the asymptotic fluorescence (reached after a long recovery period). The exact analog of the empirical formula given in Equation 4.1 can be derived by modeling the cell surface as an infinite (flat) plane of diffusing labeled proteins. The initial fluorescence distribution models the experimental geometry, i.e. a stripe of width Lyb and infinite length (Lxb =∞) with zero intensity and a constant intensity everywhere else. This initial condition reduces the problem from 2D to 1D. The solution obtained from the diffusion equation is then integrated over the width of the stripe and is normalized. This solution agrees well with the empirical Equation 4.1 (158). Henceforth we will refer to Equation 4.1 as the 1D model. To achieve an effective 1D diffusion on a finite cell, an effectively infinite stripe must be bleached on the cell surface. This can be achieved by bleaching a stripe that wraps around the cell. In protocol 1 this is accomplished by extending the bleached rectangle, shown in Figure 4.1, through the entire length of the cell and in protocol 2 it is accomplished by pickingWx = 2R. In both protocols, a symmetric 1D recovery is ensured. However, as we discuss in the results, several sources of error Chapter 4. Improving parameter estimation for cell surface FRAP data 67 (a) (b) (c) Figure 4.2: Experimental geometry for protocol 2 (not to scale). The focal plane of the CSLM is adjusted to capture an equatorial section of the cell, depicted as the xy plane in (a). The thickness of the focal plane, labeled Wz in (b), is set by the pinhole. In the focal plane a rectangular box (Wx by Wy) is selected for bleaching. Since the bleaching beam penetrates the entire depth of the cell, the bleached rectangle translates into a bleached stripe (Lyb by Lxb) on the surface of the cell, as shown in c). The fluorescence recovery is monitored from a subset of this bleached region: the intersection of the focal plane and the bleached region. The monitoring box measures Lxm by Lym. We note that the bleaching and monitoring lasers are both parallel to the z-axis. (a) and (b) depict planar slices of the cell. Experimentally adjustable quantities are labeled W and the quantities L describe the actual dimensions of the bleached and monitoring regions on the curved cell surface. Chapter 4. Improving parameter estimation for cell surface FRAP data 68 exist when a large stripe is bleached. 4.2.2 An improved protocol for cell surface FRAP As we will show in the results section, an improvement to the common experimental protocols for cell surface FRAP can be made by bleaching a sufficiently small region and using a 2D model of diffusion. Below we derive the necessary equations and describe the protocol. We begin by deriving the 2D equation to use when a finite rectangular region is bleached, as shown in Figures 4.1 and 4.2. We begin by solving the 2D infinite plane diffusion equation, ∂f(x, y, t) ∂t = D ( ∂2f(x, y, t) ∂x2 + ∂2f(x, y, t) ∂y2 ) (4.3) We assume that the initial intensity is unity everywhere except in the bleached rectangle, of dimensions Lxb by Lyb, where the intensity is zero. Equation 4.3 can be solved by using Fourier transforms. The solution, after applying the initial conditions and performing an inverse Fourier transform is f(x, y, t) = 1− 1 4 [ erf ( Lxb/2− x√ 4Dt ) + erf ( Lxb/2 + x√ 4Dt )] ×[ erf ( Lyb/2− y√ 4Dt ) + erf ( Lyb/2 + y√ 4Dt )] (4.4) erf is the usual error function. This equation gives the spatial time course of the recovery. To obtain the theoretical FRAP recovery curve this equation must be integrated and normalized, a procedure analogous to averaging in this case. The solution is integrated over the monitoring region, a rectangle measuring Lxm by Lym centered at the origin, and then normalized by the total intensity in the monitoring region prior to the bleaching event. The FRAP recovery curve, when bleaching is complete and the mobile fraction is 1, is G(t,D) = 1− 18piLxmLym {√ piLxm [ erf ( Lxb−Lxm 4 √ Dt ) + erf ( Lxb+Lxm 4 √ Dt )] + √ piLxb [ erf ( Lxb+Lxm 4 √ Dt ) − erf ( Lxb−Lxm 4 √ Dt )] +4 √ Dt [ exp ( − (Lxm+Lxb)216Dt ) − exp ( − (−Lxm+Lxb)216Dt )]} × {√ piLym [ erf ( Lyb−Lym 4 √ Dt ) + erf ( Lyb+Lym√ 16Dt )] + √ piLyb [ erf ( Lyb+Lym 4 √ Dt ) − erf ( Lyb−Lym 4 √ Dt )] +4 √ Dt [ exp ( − (Lym+Lyb)216Dt ) − exp ( − (−Lym+Lyb)216Dt )]} (4.5) Chapter 4. Improving parameter estimation for cell surface FRAP data 69 Allowing for the possibility of incomplete bleaching and an immobile fraction, we have F (t) Fp = ( 1− Fo Fp ) Mf G(t,D) + Fo Fp (4.6) This equation, with G(t,D) as defined in Equation 4.5 and all other quantities as defined earlier, can easily be fit directly to FRAP data when rectangular bleach and monitoring regions are used. The use of Equation 4.6 to analyze data obtained using the geometry of protocol 1 is straightforward. A rect- angle (Lxb by Lyb) is chosen for bleaching, as shown in Figure 4.1. In every subsequent frame, the intensity in the rectangular monitoring region (Lxm by Lym) is integrated to obtain the experimental FRAP recovery curve. This integration can be accomplished with standard imaging software. Equation 4.6 can then be fit directly to experimental FRAP recovery curves to obtain the diffusion coefficient and mobility fraction. The use of Equation 4.6 with the geometry of protocol 2 is more complicated. Here the bleached and moni- toring regions are not directly observable and therefore the quantities Lxb, Lyb, Lxm, and Lym in Equation 4.5 are unknown. Protocol 2 is generally used when cells are unable to spread and maintain a roughly spherical shape, for example unstimulated T cells (152). By approximating the cell as a sphere (at least near the bleached and monitoring regions) the unknown quantities can be related to the known quantities by simple geometry, Lxb = 2R arccos(1−Wx/R) Lyb = 2R arcsin(Wy/2R) Lxm = 2R arcsin(Wz/2R) Lym = 2R arcsin(Wy/2R) (4.7) where Wx, Wy, and Wz are chosen (as shown in Figure 4.2) and R is the cell radius. Knowing the quantities describing the dimensions of the bleach and monitoring regions, Equation 4.6 can be fit directly to the integrated fluorescence from the monitoring region for FRAP experiments utilizing protocol 2. There are two potential sources of error in this method which we explore in detail in Appendix C. We ultimately find that they are negli- gible for relevant experimental parameter regimes. 4.3 Results 4.3.1 Error estimates for common FRAP protocols For most models used in fitting FRAP recovery curves, there are three requirements that must be satisfied: 1. Total time of bleaching event compared to characteristic diffusion time must be small. Chapter 4. Improving parameter estimation for cell surface FRAP data 70 2. Size of bleached region compared to total surface area should be small. 3. Effects due to the finite area of the cell surface are not important. When bleaching a large region, as in the case of a large stripe, it is difficult to satisfy all these requirements. The first requirement that most models assume is that the bleaching event is instantaneous. It has been recommended that the bleaching time be at least 10 times smaller than the characteristic diffusion time (157; 164). The characteristic diffusion time when bleaching a large stripe is ∼ 150 s (calculated as LxbLyb/4D = (30µm)(2µm)/4(0.1µm2/s)). Typical times required to bleach a large stripe extending through the entire cell are tens of seconds, depending on the particular FRAP protocol and apparatus being used. In that case, we would expect appreciable recovery before the bleaching event is complete, leading to errors in D. The correction for the finite bleach time has been investigated in detail in (165). The second requirement addresses the fact that most models assume that, in the absence of an immobile fraction, recovery should be 100%. However, bleaching a large region can remove upwards of 20% of the total fluorescence of the cell (LxbLyb/4piR2 = (30µm)(2µm)/4pi(5µm)2) preventing recovery to pre-bleach levels. If not corrected for, an error of 20% is expected in Mf when fitting Equation 4.1 to the recovery curve. Of course, Mf and D are fit simultaneously to the data and so the error in the estimate of Mf may be reduced, at the expense of introducing additional errors in the estimate of D. Lastly, the third requirement (previously considered by Carrero et al. (155) for cytosolic FRAP and Wey et al. (166) for disk membranes ) addresses the fact that most models assume an infinite plane of fluorescence. This assumption is valid provided the characteristic time for molecules to diffuse from the unbleached poles (x = 0, y = ±R, z = 0 in Figure 4.2a for protocol 2 and at maximum y-values in Figure 4.1 for protocol 1) into the bleached stripe is longer than the time-scale of the FRAP experiment. That is, the derivation of the equa- tion assumes that boundary effects are minor during the time-scale of the experiment. The distance, S, from the unbleached poles to the bleached stripe is roughly 7 − 15µm for cells of radius 5 − 10 µm. The characteristic diffusion time is then S2/4D ≈ 150 − 1000 s for D = 0.05 − 0.1 µm2/s (typical of cell surface molecules). This range is well within the time-scale of typical FRAP experiments and so the infinite plane of fluorescence assumption is violated for both protocols. In fact, we estimate a factor of 3 error in D under certain conditions (see Table 4.1 and discussion in a subsequent section). The three requirements are not satisfied, for either protocol, because a large stripe that extends through the entire length of the cell is bleached (Figure 4.2c with Lxb ≈ 2piR). One obvious way to satisfy these requirements Chapter 4. Improving parameter estimation for cell surface FRAP data 71 is to bleach a sufficiently small region by decreasingLxb. In this case, the bleaching event will be shorter, the ratio of bleach area to total cell surface area will be smaller, and the infinite plane assumption will be valid for a longer period of time since the effective boundaries will be further away. We provide quantitative recommendations in a subsequent section. Many authors bleach a smaller region using both protocols and analyze their results with either the 1D model, Equation 4.1 (as in (152; 160; 161; 162; 163)), or the circularly symmetric model outlined in Axelrod et al. (159) (used by Fabra et al. (153)). When Lxb is less than the cell circumference the fluorescence recovery into the monitoring region is 2D. Continuing to use the 1D model in this case leads to appreciable errors in D. In Figure 4.3 we explore the percent difference in D (|(1−Dfit/D)× 100%|) by fitting the 1D model to the solution of the 2D infinite-plane diffusion equation, for various aspect ratios (Lxb/Lyb) of the bleached stripe. All fitting procedures in this paper were carried out using the Matlab function nlinfit. We label quantities arising from the fiting procedure with subscript fit (i.e. Dfit and Mf-fit). As an example, consider an aspect ratio of ≈ 5 (Lxb ≈10 µm and Lyb ≈2 µm) typical of a small bleach region. The 1D model gives rise to a percent difference of 130% (or Dfit/D = 2.3) in either protocol 1 or 2. For aspect ratios in the range of 5-12 the 1D model gives rise to percent differences in D of 50− 130%. In making these calculations, we have ignored potential errors introduced by the finite extent of the cell surface. These error contributions are discussed in the next section but ultimately do not affect the main conclusion drawn from Figure 4.3, which is that the 1D model is an inaccurate model for these experimental protocols. In the case of protocol 2, it is problematic to use the closed form variant of Axelrod’s formula presented by Soumpasis (167) because it is impossible, in this geometry, to bleach a circle on the surface of the cell and to monitor the fluorescence solely from this circle. The closest shape that can be achieved, in theory, is a square, with Lxb = Lyb = Lxm = Lym, which gives an error inD of 40% when the square side length is 1 µm. In practice, the signal to noise ratio is too low when such a small region is bleached. The dotted line in Figure 4.3 describes the error in D when the Soumpasis solution is used, as a function of the aspect ratio. Since it is more natural for most CSLM to bleach a rectangular region, the use of a circularly symmetric model for protocol 1 has not been widely used. We see that substantial errors are introduced in estimates of D by the use of Equation 4.1. Chapter 4. Improving parameter estimation for cell surface FRAP data 72 Figure 4.3: Accuracy of the 1D model and the closed form circularly symmetric solution of Soumpasis (167). We solve the 2D diffusion equation on an infinite plane using as initial condition unit intensity everywhere except on a rectangle (Lxb by Lyb) where the intensity is set to zero, simulating the bleached region. We integrate the solution over a subset of this rectangle (Lxm by Lym), i.e. the monitoring region. We then fit the 1D and Soumpasis solutions to the generated recovery curve for various values of Lxb. Each recovery curved is generated using 200 equally spaced time points reaching a recovery of 99%. The graph shows the percent difference in D (|(1−Dfit/D)× 100%|) for both models. As Lxb increases, the percent difference decreases when using the 1D model and increases when using the solution of Soumpasis, as expected. When using a small bleach region with protocol 1 or 2, a typical aspect ratio of ≈ 5 (Lxb ≈ 10 µm and Lyb ≈2 µm for example) is used. The 1D model gives rise to a percent difference of 130% (or Dfit/D = 2.3) and the Soumpasis solution gives rise to a percent difference of 80% (or Dfit/D = 0.2). Here we have ignored the finite extent of the cell surface. We note that this graph is independent of the actual value of D. 4.3.2 Utility of the 2D model In the previous section we stated that bleaching a sufficiently small region satisfies the three requirements impor- tant for FRAP experiments but that continuing to use the 1D model, even though 2D diffusion occurs, resulted in significant errors in the estimate of D. In section 2.2 we described a 2D model of diffusion that can be used with a small bleach region. To confirm the utility of the 2D model we simulated FRAP experiments for cell surface proteins according to protocol 2. We numerically solved the diffusion equation on the surface of a sphere using an explicit finite difference scheme with 13000 grid points. We replicate the initial conditions of a bleached stripe and integrate the solution over the monitoring region, both as shown in Figure 4.2. All simulations were carried out in Matlab. We used D = 0.1 µm2/s, Mf = 1, R =5 µm, Wy =2 µm, Wz =2 µm, and varied Wx. Figure 4.4 shows the results of the simulations along with the predicted FRAP recovery curves from the 2D and 1D models. Figures 4.4 a), b), and c) reveal that the 1D model is inaccurate for all Wx and that the 2D model is the most accurate when Wx is small (Figure 4.4c). We fit the 2D and 1D models directly to the simulated FRAP recovery curves (Figure 4.4, solid lines). We Chapter 4. Improving parameter estimation for cell surface FRAP data 73 (a) (b) (c) Figure 4.4: FRAP simulations based on the geometry of protocol 2. We solve the diffusion equation on the surface of a sphere using an explicit scheme with 13000 grid points. We replicate the initial conditions of a bleached stripe and integrate the solution over the monitoring region, both as shown in Figure 4.2. We use D = 0.1µm2/s, Mf = 1, R = 5µm, Wy = 2µm, and Wz = 2µm. In (a), (b), and (c) the solid line shows the simulated FRAP recovery while the dotted lines show the predictions of the 2D and 1D models with the same simulation parameters except that the mobile fraction is chosen to be Mf − LxbLyb/4piR2 (corrected for loss of fluorescence during bleach). In (a) we set Wx = 10µm, bleaching a stripe that wraps around the cell. In (b), Wx = 4.5µm, and in (c) Wx = 1µm. In all cases we see that the 1D model is a poor predictor of the recovery. The 2D model is accurate when Wx is small because the three requirements important for FRAP for cell surface proteins are satisfied. Chapter 4. Improving parameter estimation for cell surface FRAP data 74 fit for the diffusion coefficient Dfit and the mobile fraction Mf-fit; results shown in Table 4.1. We find percent differences inD of∼ 200% for both models whenWx = 2R = 10µm (Figure 4.4a). DecreasingWx substantially reduces the error in the estimate of D to 1% (Dfit/D = 1.01) but only for the 2D model. We would expect this error reduction because decreasing the size of the bleach region addresses all three requirements in the previous section. Table 4.1: Fitting the 1D and 2D models to the sphere simulations (Figure 4.4, solid lines) 2D Model 1D Model Wx Dfit % err Mf r2 Dfit % err Mf r2 (µm) (µm2/s) (µm2/s) 10.0 (fig. 4.4a) 0.36 260 1.0 0.93 0.27 173 1.0 0.94 4.5 (fig. 4.4b) 0.14 35 1.0 0.99 0.12 20 1.1 0.98 1.0 (fig. 4.4c) 0.10 1 1.0 1.00 0.15 48 1.1 0.96 a. As described in Figure 4.4, the sphere simulations were carried out with D=0.1 µm2/s and Mf = 1. b. For each fit, we correct the fitted value of Mf to reflect the loss of fluorescence during the bleaching process (Mf =Mf-fit + LxbLyb/4piR2). c. Degree-of-freedom adjusted coefficient of determination, r2, is a goodness-of-fit measure. We note that in all the cases in Table 4.1 the fitted diffusion coefficient, Dfit, was overestimated. This effect is counterintuitive as one would expect Dfit to be underestimated because boundary effects (due to the finite cell size) slow diffusion into the monitoring region. However, we are simultaneously fitting for Dfit and Mf-fit and this introduces a problem. In the simulations, Mf = 1. However, because a finite fraction of the cell surface is bleached, the fitting procedure will necessarily estimate Mf-fit as less than Mf. Given this fact, Dfit will be estimated as greater than D to match the initial slope of the FRAP curve. We refer to this effect as the mobile fraction effect. For the 2D model, the mobile fraction effect decreases as the ratio of bleached area to total surface area de- creases. The decrease in this effect is observed in Table 4.1 since Dfit converges to its true value and we can always correct Mf-fit by the fraction of lost fluorescence during bleaching (giving the correct value Mf = 1 for all cases with this model). We use this correction fraction because the bleaching event effectively removes a portion of the labeled protein. In the 1D model, as the mobile fraction effect decreases, errors introduced by 2D diffusion prevent accurate determinations of both Dfit and Mfit. Although the 1D model gives a reasonable estimate for D when Wx = 4.5µm, it overestimates Mf by 10%. We note that in all cases both models provided excellent fits with r2 (degree-of-freedom adjusted coefficient of Chapter 4. Improving parameter estimation for cell surface FRAP data 75 determination) statistic close to one. Therefore, goodness of fit with an inappropriate model should not be relied upon in fitting FRAP data on cell surfaces. We next investigated the effects of instrumental noise. Since some amount of experimental noise is expected in all FRAP experiments, we wanted to ensure that the 1D and 2D models gave results that were statistically different. To this end, we added 15% normally distributed noise to FRAP curves generated using Equation 4.5 for various D values. For each D value we generated 10 noisy FRAP curves, simulating 10 FRAP experiments. We then fit each of the 10 curves for a given D value using the 1D model 4.1 and 2D model 4.6. In Figure 4.5, grey up-triangles show the results for the 1D model and black down-triangles show the results for the 2D models. Dotted lines show the mean value of Dfit. For all D values the 1D and 2D models give statistically different mean value of Dfit (using a paired t-test at the 5% level). We found that when the means were different by a factor of ∼ 2 and a large amount of noise (15%) was present, the mean diffusion coefficients predicted by each model were statistically different with 10 experiments. Performing only 5 experiments, we occasionally found that values were not statistically different. However, most authors make ≥ 10 measurements. We also note that the spread in Dfit/D for a given D value is large for the 1D model and much smaller for the 2D model. Figure 4.5: Statistical signifiance of the 2D model. 15% normally distributed noise was added to FRAP curves generated using Equation 4.5 for various D (x-axis). For each D value, 10 noisy FRAP curves were fit using the 1D model, Equation 4.1, and 2D model, Equation 4.6. Grey up-triangles show the ratio of Dfit to the actual D for the 1D model and black down-triangles show the ratio for the 2D model. Dotted lines show the means. For all D values the 1D and 2D models give statistically different mean value of Dfit (using a paired t-test at the 5% level). We ultimately conclude that it is not advisable to use the 1D model in situations where the three requirements Chapter 4. Improving parameter estimation for cell surface FRAP data 76 discussed in the previous section are violated. The 2D model combined with a small bleach region provides a more accurate protocol. 4.3.3 Practical recommendations for cell surface FRAP experiments In order to satisfy the three requirements important for cell surface FRAP experiments Equation 4.6 with protocols 1 or 2 should be used. We recommend bleaching the smallest region possible while maintaining a good signal to noise ratio. As rough guidelines, we recommend that the bleaching event take less than 3 seconds (characteristic diffusion times for an aspect ratio of≈ 5 are > 45 s) and that the ratio of bleached region area to total cell surface area be less than 5%. These rough guidelines can be made more precise for a particular FRAP experiment by examining the arguments presented above. We also recommend that the bleach region should be in the center of the cell for protocol 1 and that protocol 2 be used for cells most resembling spheres. We hope that the use of the 2D model, Equation 4.6, in conjunction with protocols 1 or 2 will provide more consistent results for the diffusion coefficient and mobile fraction within and between laboratories. 4.4 Discussion We have used theoretical arguments to show that common analytic methods used to obtain the diffusion coeffi- cient and mobile fraction are generally not ideal for cell surface FRAP. When the 1D model is used appropriately by bleaching a large stripe wrapping around the cell, it is difficult to satisfy the three requirements important for FRAP experiments listed earlier. Ignoring possible errors arising from a finite bleach time, the diffusion coeffi- cient in this case will be overestimated by a factor of ∼ 3 due to the mobile fraction effect. Bleaching a smaller region will reduce this effect but new errors (factor of ∼ 1.5) will be introduced if the 1D model is used because 2D diffusion into the monitoring region now takes place. The 1D model with a large stripe can be used for cell surface proteins provided the three requirements important for FRAP are satisfied, as may be the case for very large cells. We have presented a more accurate equation to use in the case of a small bleach region where 2D diffusion occurs and we have provided guidelines for the use of this equation when using protocol 1 and 2 for cell surface proteins. We note that this equation can be used for rectangular geometries with in vitro and in vivo studies of protein mobilities, analogous to the circularly symmetric solution of Soumpasis (167) based on the theory of Ax- elrod et al. (159). Chapter 4. Improving parameter estimation for cell surface FRAP data 77 Our method for fitting FRAP data relies on Equation 4.5, which is derived from an approximate model. In deriving the equation, we replaced the curved, finite, cell surface with an infinite flat plane. An alternative to Equation 4.5 could be found by analytically solving the FRAP problem on a finite, spherical surface, yielding a solution in the form of an infinite series of special functions. This solution would be difficult to work with in fitting data, and in any case would provide only a very slight improvement in the estimate ofD in situations where the three FRAP requirements are met. As we have shown, Equation 4.5 represents a substantial improvement over existing formulae for fitting FRAP data without adding much computational effort in fitting. Improvements in the absolute determination of D, as reported here, are useful in comparing different experi- mental systems and in physical modeling. Ultimately, the use of the 2D equation along with a small bleach region should lead to more consistent reports of diffusion coefficients. 78 References [151] Sloan-Lancaster, J., J. Presley, J. Ellenberg, T. Yamazaki, J. Lippincott-Schwartz, and L. E. Samelson. 1998. ZAP-70 association with T cell receptor ζ (TCRζ): Fluorescence imaging of dynamic changes upon cellular stimulation. J. Cell. Biol. 143:613–624. [152] Favier, B., N. J. Burroughs, L. Wedderburn, and S. Valitutti. 2001. TCR dynamics on the surface of living T cells. Int. Immunol. 13:1525–1532. [153] Fabra, S., V. Lang, J. Harriague, A. Jobart, T. G. Unterman, A. Trautmann, and G. Bismuth. 2005. 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Immune synapse forma- tion can also be induced by substrate immobilized lipid bilayers containing a cognate pMHC and the adhesion molecule ICAM-1 (168). Substrate-immobilized lipid bilayers containing a cognate ligand have also been used to study stimulated activation of mast cells through the IgE-receptor complex (169). In order to properly under- stand these phenomena, in immunology and elsewhere, we require good experimental estimates of kinetic binding parameters for the ligand-receptor interaction. For instance, T cell activation has been shown to depend on TCR- pMHC binding kinetics (reviewed in (170)). Measurement of kinetic parameters for ligand-receptor binding can, for example, be performed using a Bia- core optical sensor (Biacore, AB, Uppsala, Sweden). The receptor of interest is immobilized on the surface of a chip, and ligands flow through a cell above the chip leading to changes in its optical properties (171; 172). It is then possible to estimate the on- and off-rates of the ligand-receptor interaction. However, a limitation of Biacore, and indeed of most conventional techniques for estimating binding kinetics, is that one or both binding partners are freely diffusing in a three dimensional volume, whereas in a cell-cell or a cell-bilayer localized interaction they are restricted to diffuse on two dimensional surfaces. Thus, Biacore measurements give three dimensional (3D) reaction rates, that depend on the the molar concentration in solution, rather than 2 dimensional (2D) rates that depend on the surface densities of the the binding partners. Moreover, the receptor is immobilized on the chip surface and therefore, the potentially important influence of the plasma membrane is neglected. 1A version of this chapter has been published. Dushek O, Das R, Coombs D (2008) Analysis of membrane-localized binding kinetics with FRAP. Eur Biophys J. 37(5):627-638 Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 81 A few alternative approaches have been used, with varying degrees of success, to estimate 2D binding affini- ties and reaction rates. Using quantitative fluorescence microscopy, the 2D affinity of the T cell surface adhesion molecules, CD2 and CD28 to their ligands has been determined (173; 174; 175) as well as the 2D affinity of certain TCR/pMHC interactions (168). Fluorescence resonance energy transfer (FRET) is another fluorescence microscopy technique that can be used to observe the association between two fluorophores (176). Because of its exquisite sensitivity to the distance between two fluorophores, FRET is ideal for measuring nanometer scale changes in spatial separation. FRET measurements have been primarily used for assaying qualitative changes in the association between interacting proteins (177; 178). Another possibility for quantifying binding interactions is to use dual color fluorescence correlation spectroscopy (FCS) which measures correlated fluctuations in the signals from two different fluorophores to infer a binding interaction between them (179; 180). Both FRET and FCS measurements are, at the present time, technically challenging to implement. Moreover, these techniques require distinct fluorescent tags on both the binding partners of an interacting pair. These difficulties may explain why there are very few measurements of kinetic parameters for membrane-bound protein interactions. In this paper we focus on the technique of fluorescence recovery after photobleaching (FRAP), most com- monly used to measure diffusion coefficients of biomolecules (181; 182), reviewed in (183). FRAP has recently been used to measure kinetic parameters of 3D binding interactions between a fluorescently labeled ligand and its unlabeled binding partner (184; 185; 186; 187; 188), comprehensively reviewed in (189). These studies show that FRAP can yield 3D kinetic parameters in a variety of settings, but subject to certain assumptions that will not apply in the case of membrane-restricted binding interactions. For instance, (184; 185; 187) assumed simple geometries and immobile binding sites for the labeled receptor. More recently, Braga et al. extended this analysis to the case when the binding sites are also mobile and applied their analysis to examine the diffusion of mRNA in the the nucleus (188). A potential limitation of all these studies, discussed further in (190), is that they yield compound parameters such as the true equilibrium constant multiplied by the free receptor concentration. Be- cause the receptor is commonly assumed to be in excess in these studies, some care is required in interpreting the compound parameters obtained. Here, we present an experimental protocol that can be utilized to measure true (not compound) kinetic parame- ters for a binding interaction between two mobile proteins diffusing on nearby membranes. We allow for diffusion of both the labeled receptor and the unlabeled ligand, and perform our analysis using the rectangular geometry that is commonly used in cell-surface experiments with a laser confocal scanning microscope. We systematically identify regions of parameter space in which the kinetic parameters can (theoretically) be measured, as well as Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 82 those in which they cannot. Finally, we fit a set of simulated FRAP recovery curves using the outlined protocol, and we provide guidelines for fitting experimental FRAP recovery curves to infer kinetic binding parameters. 5.2 Modeling 5.2.1 Full model We begin our analysis with a reaction-diffusion system that governs the dynamics of two diffusing species on a 2-dimensional surface which bind with a 2-dimensional association rate kon, and unbind at a rate koff. One of the two species, the receptor, is labeled with a fluorophore, and it binds to a complementary ligand that is unlabeled. After photobleaching there is a subpopulation of unlabeled receptors and unlabeled receptor-ligand complexes. We use the following notation to describe the surface densities of all the relevant species: f = [labeled receptor], f ′ = [unlabeled (photobleached) receptor], c = [labeled complex], c′ = [unlabeled (photobleached) complex], s = [unlabeled ligand], (5.1) These quantities evolve according to the PDE system ∂f/∂t = Df∇2f − konfs+ koffc, (5.2a) ∂c/∂t = Dc∇2c+ konfs− koffc, (5.2b) ∂f ′/∂t = Df∇2f ′ − konf ′s+ koffc′, (5.2c) ∂c′/∂t = Dc∇2c′ + konf ′s− koffc′, (5.2d) ∂s/∂t = Ds∇2s− konf ′s+ koffc′ − konfs+ koffc, (5.2e) where Df, Ds, and Dc are diffusion coefficients of the free receptors, free ligands and bound complexes, re- spectively. We assume that the system is at chemical equilibrium prior to the FRAP experiment and that the fluorophores in the bleaching region are instantaneously photobleached at time t = 0. Thus, for t < 0, we have f(t) = Feq, f ′(t) = 0, c(t) = Ceq, c′(t) = 0, and s(t) = Seq where the subscript eq indicates the equilibrium density of the three species. The photobleaching event at t = 0 creates a new subpopulation of unlabeled receptors and unlabeled complexes in a localized region, i.e. f ′(t) > 0 and c′(t) > 0 for t > 0. Moreover, because photo- bleaching only removes the label and does not disturb the chemical equilibrium, we can assume that s(t) = Seq Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 83 for all t. Therefore, we define a pseudo on-rate k∗on = konSeq, and rewrite Equations 5.2a and 5.2b as: ∂f/∂t = Df∇2f − k∗onf + koffc, (5.3) ∂c/∂t = Dc∇2c+ k∗onf − koffc. (5.4) To obtain theoretical FRAP recovery curves we must solve Equations 5.3 and 5.4 with appropriate initial and boundary conditions reflecting the experimental geometry, and integrate the total fluorescence f + c over the recovery region. The resulting quantity, Gfm(t), gives the idealized FRAP recovery curve and is a function of the bleaching geometry, diffusion coefficients, and reaction rates. In a subsequent section we describe how to efficiently evaluate Gfm. Note that because s(t) is invariant in time, the dynamics of the labeled species are decoupled from those of the unlabeled species, resulting in a simpler system of equations to analyze. With these assumptions, Equations 5.3 and 5.4 govern the dynamics of the two labeled species and by solving them we obtain expressions for the ob- served fluorescence recovery (see below). As illustrated in Figure 5.1, we assume a rectangular bleaching region of dimensions Xb and Yb along the x and y directions, respectively, and a monitoring region of dimensions Xm by Ym that is a smaller subset of this bleaching region. X b X m Y b Y m x y monitoring bleached region region Figure 5.1: The geometry of a typical confocal FRAP experiment. 5.2.2 Reduced models The full model defined by Equations 5.3 and 5.4 can be well-approximated by four simpler models in certain situations. These model reductions are briefly described below, and formally derived in Appendix D. In each of Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 84 these reduced models (1-4 below), the spatio-temporal dynamics are governed by simple diffusion equations of the form ∂u/∂t = D∇2u (5.5) where D is a relevant diffusion coefficient for that model. The problem of fitting FRAP data to estimate D can then be solved as usual for the experimental geometry that is used (181; 191; 192; 193). The solution for the geometry shown in Figure 5.1 is described below. 1. Weighted diffusion: One limit arises when k∗on and koff are not well-separated, but the lifetimes of the free and bound states are much smaller than their characteristic diffusive timescales. In this regime, the labeled receptor switches rapidly between its unbound form, with diffusion coefficient Df, and its bound form with diffusion coefficient Dc. The FRAP recovery curve Gwd(t) = U(t,Dwd) (Equation 5.9) then depends only on the effective (or weighted) diffusion coefficient: Dwd = 1 1 +K∗ Df + K∗ 1 +K∗ Dc (5.6) where K∗ = k∗on/koff. In this regime, only Dwd can be determined from the data. 2. Independent diffusion: When the diffusive timescales are much shorter than the lifetimes of the free and bound states, a labeled receptor remains in its (un)bound state on the timescale of the FRAP experiment. In this case, there is no exchange between the pools of free and bound fluorophores, and FRAP recovery is a result of “independent diffusion” of these two pools with diffusion coefficient Df and Dc, respectively. In this case, the FRAP recovery curve Gid(t) is simple a weighted average of two independent FRAP recovery curves, (Gpf(t) for free receptors and Gpc(t) for bound receptors) : 1 1 +K∗ Gpf(t) + K∗ 1 +K∗ Gpc(t). (5.7) 3. Receptor diffusion dominant: When koff k∗on, the bound complex rapidly dissociates, and the labeled receptor is almost always in the free state. The FRAP recovery curve Gpf(t) is then a function of Df alone. This regime is a subset of both the weighted and independent diffusion regimes. 4. Complex diffusion dominant: When k∗on koff, the labeled receptor is almost always in the bound state. The FRAP recovery curve Gpc(t) is then a function of Dc alone. Like the receptor diffusion dominant regime, Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 85 this regime is a subset of both the weighted and independent diffusion regimes. The crucial difference between the full model and the reduced models is that, for the full model, but not any of the reduced models, all four parameters, Df, Dc, k∗on and koff are required to completely describe the FRAP re- covery curves. In contrast, in parameter regimes where the full model is reduced to one of these effective models, FRAP recovery curves are well-described by a smaller subset of parameters. Thus, in the weighted and indepen- dent diffusion regimes, the necessary parameters are Df, Dc and K∗, and in the two diffusion dominant regimes, only Df or Dc are sufficient to fully describe the FRAP recovery curves. Therefore in parameter regimes where one of these reduced models is applicable, one cannot, with any statistical confidence, estimate all four model pa- rameters. Specifically, in the two diffusion dominant regimes no binding parameters can be estimated, and in the weighted and independent diffusion regimes, only the binding affinity K∗ can be estimated. This categorization of the FRAP recovery curves is the basis of our proposed protocol described below. 5.2.3 Evaluation of theoretical FRAP recovery curves Simple FRAP analysis In typical surface photobleaching experiments, the photobleached area is much smaller than the total area of the plasma membrane. Therefore, we can assume that the domain is effectively infinite in extent without introducing appreciable error. For the parameter regimes that reduce to simple diffusion (cases 1-3), the FRAP recovery curve is determined by Equation 5.5 with initial conditions u(t = 0) = 1− [H(x+Xb/2)−H(x−Xb/2)] [H(y + Yb/2)−H(y − Yb/2)] (5.8) where H(x) is the Heaviside step-function. This is easily solved by Fourier transform and the desired FRAP recovery curve is then obtained by integrating u over the monitoring region (−Xm/2 ≤ x ≤ Xm/2, −Ym/2 ≤ y ≤ Ym/2), to obtain the following expression for the theoretical FRAP recovery curve (193): U(t,D) = 1− √ Dt 2piXmYm {√ pi [X+ erf(X+)−X− erf(X−)] + [ exp(−X2+)− exp(−X2−) ]} × {√pi [Y+ erf(Y+)− Y− erf(Y−)] + [exp(−Y 2+)− exp(−Y 2−)]} (5.9) where we define X± = Xb ±Xm 4 √ Dt and Y± = Yb ± Ym 4 √ Dt . (5.10) Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 86 This expression can be fit to data to determine the parameterD. FRAP with binding kinetics We are now concerned with solving Equations 5.3 and 5.4. Taking the Fourier transform (denoted with hats), we obtain the linear system ∂f̂/∂t = Df(−4pi2q2x − 4pi2q2y)f̂ − k∗onf̂ + koffĉ, (5.11a) ∂ĉ/∂t = Dc(−4pi2q2x − 4pi2q2y)ĉ+ k∗onf̂ − koffĉ. (5.11b) We define qf = 4pi2Df(q2x + q2y) + k∗on and qc = 4pi2Dc(q2x + q2y) + koff to obtain ∂f̂ ∂t = −qf f̂ + koffĉ, (5.12a) ∂ĉ ∂t = k∗onf̂ − qcĉ, (5.12b) whose solutions are f̂ = 1 v [ (1/2)f̂(t = 0) (exp(D1t)(qc − qf + v) + exp(D2t)(−qc + qp + v)) + koffĉ(t = 0)(exp(D1t)− exp(D2t))] , (5.13a) ĉ = 1 v [(1/2)ĉ(t = 0) (exp(D1t)(−qc + qf + v) + exp(D2t)(qc − qf + v)) + k∗onf̂(t = 0)(exp(D1t)− exp(D2t)) ] (5.13b) where v = [(qc − qf )2 + 4k∗onkoff]1/2, D1 = (−qc − qf + v)/2, D2 = (−qc − qf − v)/2. f̂(t = 0) and ĉ(t = 0) are Fourier transforms of the initial conditions, f̂(t = 0) = −Feq sinc(piXbqx) sinc(piYbqy) + Feqδ(qx)δ(qy) (5.14a) ĉ(t = 0) = −Ceq sinc(piXbqx) sinc(piYbqy) + Ceqδ(qx)δ(qy). (5.14b) where Feq = koff/(k∗on + koff) and Feq = k∗on/(k∗on + koff). We obtain these relationships by assuming that the sys- tem is in equilibrium prior to the FRAP experiment (k∗onFeq = koffCeq) and the FRAP recovery can be normalized to unity (Feq + Ceq = 1), see (185). It is not possible to obtain an analytical inverse Fourier transform of these equations, and therefore theoretical FRAP recovery curves must be numerically computed. These are obtained by integrating the sum f + c over the monitoring region, and dividing by the area of the monitoring region to obtain the mean fluorescence intensity Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 87 over this area. Thus, for a rectangular monitoring region of dimensions Xm(≤ Xb) by Ym(≤ Yb), the recovery curve is given by Gfm(t) = (∫ Xm/2 −Xm/2 dx ∫ Ym/2 −Ym/2 dy [f(t, x, y) + c(t, x, y)] ) /(XmYm) , (5.15) and, by definition of the inverse Fourier transform, Gfm(t) = 1 4XmYm ∫ Xm −Xm dx ∫ Ym −Ym dy ∫ ∞ −∞ dqx ∫ ∞ −∞ dqy [ f̂(t, qx, qy) + ĉ(t, qx, qy) ] × exp(2piıqxx) exp(2piıqyy) = ∫ ∞ −∞ dqx ∫ ∞ −∞ dqy [ f̂(t, qx, qy) + ĉ(t, qx, qy) ] sinc(piqxXm) sinc(piqyYm) = 4 ∫ ∞ 0 dqx ∫ ∞ 0 dqy [ f̂(t, qx, qy) + ĉ(t, qx, qy) ] sinc(piqxXm) sinc(piqyYm), (5.16) which we compute by numerical integration. We found integrating to 15-20 in qx and qy yields sufficient accu- racy for our purposes. Using the Matlab function quadl, the above integral can be evaluated 100 times (typical number of images in FRAP experiments) in approximately 3 seconds, allowing parameter fits in a reasonable timeframe using a standard fitting procedure. All fits presented in this work were carried out using the Matlab function lsqcurvefit. 5.2.4 Structure of parameter space and model reductions In the model reductions described above, we have made no assumptions about the relative magnitudes of Df and Dc. If Df = Dc then we can see by adding Equations 5.3 and 5.4 that the full system reduces to a single diffusion equation and no binding information can be obtained. More generally, if Df and Dc are close to each other, the region in parameter space where the full model applies is small. On the other hand, if the two diffusion coefficients are well separated then the analysis of Sprague et al (185) applies (see Appendix D). We illustrate how the separation between Df and Dc impacts the regions of parameter space where the model reductions are relevant in Figure 5.2. We evaluate the sum of the squared residuals (SSR) between the full model and either the weighted or the independent diffusion reductions. In Figure 5.2a-c we plot SSR contours as a function of k∗on and koff for biologically relevant values of Df and Dc (194; 195). Following (185), we consider the model reduction a good estimate to the FRAP recovery if SSR< 1. We note that the closer the two diffusion coefficients are to each other, the smaller is the region where only the two-parameter full model provides a better fit to the data (compare Figure. 5.2, a and b). When the diffusion coefficients differ only by a factor of two, the Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 88 region where only the two-parameter full model disappears (Figure. 5.2 c). The structure of parameter space is also illustrated in Figure 5.3d where all four regimes are shown (discussed below). As discussed in the following section, we utilize the existence of these different regions in the k∗on − koff parameter-space to quantify binding kinetics of the receptor-ligand interaction. (a) (b) 0.1 0.1 0. 1 0.1 0.1 0.1 0.1 1 1 1 1 1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1 1 1 1 1 1 log(k off) lo g( k o n * ) Df=0.50 µm 2/s D c =0.05 µm2/s −4 −3 −2 −1 0 1 2 3−4 −3 −2 −1 0 1 2 3 Weighted Diffusion Independent Diffusion 0.1 0.1 0.10.1 0.1 0.1 1 1 1 1 1 0.1 0.1 0.1 0.1 0.1 0.1 1 1 1 1 1 log(k off) lo g( k o n * ) Df=0.50 µm 2/s D c =0.10 µm2/s −4 −3 −2 −1 0 1 2 3−4 −3 −2 −1 0 1 2 3 Weighted Diffusion Independent Diffusion (c) 0.1 0.1 0. 1 0.1 0.1 0.1 0.5 0.5 0.5 0.5 0.1 0.1 0.1 0.1 0.1 0.5 0.5 0.5 0.5 log(k off) lo g( k o n * ) Df=0.50 µm 2/s D c =0.25 µm2/s −4 −3 −2 −1 0 1 2 3−4 −3 −2 −1 0 1 2 3 Weighted Diffusion Independent Diffusion Figure 5.2: Quantifying full model limits. We compute the sum of the squared residuals (SSR) between the full model, Equation 5.16, and cases 1 and 2 for 200 equally spaced time points reaching 99% recovery. We plot SSR contours as a function of k∗on and koff. In (a-c) we see that the region where the full model is valid progressively decreases as Dc approachesDf. Parameters: Xb = Yb = Xm = Ym = 2µm. 5.3 Proposed experimental protocol In the previous section we identified four parameter regimes, in which the combined reaction-diffusion system (Equations 5.3 and 5.4) is effectively reduced to a single diffusion equation (Equation 5.5). As discussed above, in those parameter regimes, it is not possible to estimate the binding parameters k∗on and koff independently. To do so, we must first identify regions of parameter space where the full model fits the data, i.e. outside regimes 1-4. In this section, we propose an experimental protocol that, under suitable conditions, can be used to obtain these binding parameters by fitting a series of FRAP recovery curves. We consider an experimental system consisting Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 89 of a cell bearing fluorescently labeled receptors on its surface, that is in contact with a suspended planar bilayer bearing unlabeled ligands at concentration ST. We assume that the system is at chemical equilibrium before each experiment (achieved by waiting a sufficiently long time after the initial contact between the cell and the bilayer). A. Measuring diffusion coefficients The first step is to establish the diffusion coefficients Df and Dc, if they are not already known. This can be done by performing a straightforward FRAP experiment in the absence of the ligand (i.e. ST = 0), and fitting the recovery curve to Equation 5.5 to establish Df. The FRAP experiment can then be repeated with a saturating density of the ligand (such that all the receptors are engaged) to establish Dc. These two FRAP experiments correspond to the two pure-diffusion limits of the full model (cases 3 and 4), respectively. If Df is found to be close to Dc then successful measurement of the binding kinetic parameters is not expected in part B. B. Ligand titration experiment Having established the diffusion coefficients of the free and bound forms, the key experiment is to ‘titrate’ the ligand density, by performing FRAP experiments with a range of intermediate ligand densities. As the total ligand density, ST , increases, the equilibrium free ligand concentration, Seq, also increases according to the following equation (derived from a conservation equation): Seq = K(ST − FT )− 1 + √ [1 +K(ST + FT )]2 − 4STFT 2K , (5.17) where FT is the total receptor concentration and K = kon/koff is the 2D equilibrium constant. As a result, the pseudo-on rate, k∗on = konSeq, also increases, and the system traverses the k∗on − koff parameter-space parallel to the k∗on axis (Figure 5.3d). As shown in Figure 5.3d, for different k∗on values, a different model will best describe the FRAP recovery curve. In other words, as the total ligand concentration, ST , is varied in a controlled fashion, each resulting FRAP recovery curve is best fitted by a particular model, depending on the location of the system on the k∗on − koff phase plane. In Figure 5.3a-c the recovery dynamics are progressively slower as the ligand density increases because receptor-ligand binding effectively slows down the diffusion of the receptors, Dc < Df. The idea is to perform a series of FRAP experiments traversing the k∗on−koff phase plane (Figure 5.3d) and, for each experiment, establish which model best describes the FRAP recovery curve. For example, for the recovery curves shown in Figure 5.3b, with koff = 0.01s−1, the system traverses from case 3 (receptor diffusion dominant), via the full model regime, to case 4 (complex diffusion dominant). Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 90 C. Data fitting The goal is now to classify each of the recovery curves for intermediate ligand densities into case 1, case 2 or the full model. Each recovery curve is initially fit using all three different models: the two-parameter full model (Equation 5.16), a one-parameter diffusion model (case 1;Equations 5.6,5.9), and a one-parameter independent diffusion model (case 2; Equations 5.7,5.9). For each fit, we determine the best-fit value of the parameter(s) for that model. The relative quality of the three fits are then compared on the basis of the sum-of- squared-residuals (SSR), where a lower SSR indicates a better fit. However, to compensate for the difference in number of parameters, we use Akaike’s Information criterion (AIC) as a metric for comparing the quality of fits (196; 197). AIC, defined as: AIC = n ln(SSR/n) + 2P (5.18) where n is the number of data points, and P is the number of parameters, balances the gain due to a lowered SSR with the penalty for an increase in the number of parameters. The model with the lowest AIC value is considered to be the best descriptor of the experimental data. We use the quality of fits, as quantified by AIC (Equation 5.18), to select the appropriate model(s) for param- eter estimation. For each recovery curve, we identify the fit with the lowest AIC value. The other two fits are quantified by the difference in their AIC value from this fit. The difference, ∆AIC, is a measure of the probability of the other model being an equally likely descriptor for the FRAP recovery curves. The probability of each model is given by p = exp(−∆AIC/2) 1 + exp(−∆AIC/2) . (5.19) In practice, a fit with ∆AIC ≥ 10 indicates that the model under consideration is extremely unlikely compared to the best model, and a fit with ∆AIC ≤ 2 indicates that the model is nearly as likely as the best model (197). We use the relative quality-of-fit for the three models to determine (for each one of the the FRAP experiments) whether or not k∗on, koff and K∗ can be found. Three scenarios emerge: Scenario 1. ∆AIC < 10 for both one-parameter models. The system is in a pure diffusion limit and no estimates of binding parameters can be obtained. Scenario 2. ∆AIC < 10 for exactly one of the one-parameter models. The system is in the regime best described by that model, and the best-fit value of the pseudo-affinityK∗ is obtained. Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 91 Scenario 3. ∆AIC ≥ 10 for both the one-parameter models. The system is in the full model regime and best-fit values of k∗on and koff are obtained. This procedure rejects any fits with ∆AIC ≥ 10 as being too unreliable for parameter estimation. Categoriz- ing the fits into these three scenarios is motivated by the existence of different model regimes in the k∗on − koff parameter space. As shown in Figure 5.3d, the points corresponding to the recovery curves for a titration will transition from one diffusion-dominant regime to another diffusion-dominant regime as the ligand concentration is increased. In the diffusion-dominant regimes, the recovery dynamics are well-described by either of the two one-parameter models (scenario 1). For intermediate concentrations, however, the system is in either the full model regime, or in a one-parameter model regime. When the system is in a weighted-diffusion or independent- diffusion regime, the recovery dynamics are best fitted exclusively by the corresponding one-parameter model, and not the other one (scenario 2). Finally, in scenario 3, the system is in the full model regime, and neither of the two one-parameter reductions are appropriate descriptors of recovery dynamics. D. Parameter estimation In scenario 1, the system is in a regime where no binding information can be obtained. In scenario 2, we determine the pseudo-affinity K∗ using the one-parameter model with ∆AIC < 10. If this is the independent diffusion model, then the parameterK∗ is directly determined from the fit. If this is the weighted diffusion model, then the fitted parameter is Dwd, and K∗ is obtained by rearranging Equation 5.6: K∗ = Df −Dwd Dwd −Df (5.20) Additionally, in scenario 2, it is also possible that ∆AIC < 10 for the full model, and we have another estimate for K∗ from the full model fit, given by the ratio k∗on/koff. We combine the parameter estimates from the two fits by using the probability, p (Equation 5.19), of each model as a weight, and calculate the weighted mean of the best-fit K∗. In scenario 3 fitting the full model provides estimates of k∗on and koff, as well as K∗ = k∗on/koff. Estimates of the pseudo-affinity,K∗ (scenario 2) and the pseudo-association rate, k∗on (scenario 3), can be con- verted to the true 2D affinity,K = K∗/Seq and association rate, kon = k∗on/Seq using the free ligand concentration Seq, given by: Seq = ST − K ∗ 1 +K∗ FT , (5.21) where ST and FT are the known total ligand and total receptor concentrations, respectively, and we use the best-fit estimate of K∗ from the fits. However, the estimate of the true parameters is unreliable when Seq ≈ 0 because of the small value of the denominator in the expressions for kon and K . In practice, we find that when the ratio Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 92 Seq/ST becomes small, that is, at equilibrium only a negligible fraction of the total ligand is unbound, the esti- mates of the true parameters can deviate significantly from their actual value. 5.4 Simulated experiments 5.4.1 Simulated FRAP titrations To test our protocol, we simulated three sets of numerical experiments, and applied the above data-fitting and parameter-estimation procedure to the simulated datasets. These simulations model the interaction between a receptor-ligand pair, with a fixed kon, and three different koff values, representing, for example, a range of bind- ing affinities of pMHC for a particular TCR. Table 5.1 lists the parameter values used to generate the simulated data. We use biologically relevant reaction rates between cell surface proteins (168; 198), typical diffusion coeffi- cients for membrane proteins (194; 195), and commonly used sizes for the bleach and monitoring regions (195). The Einstein-Stokes relation predicts that if two particles with individual diffusion coefficients Df and Ds bind together, the complex will have diffusion coefficient DfDs/(Df +Ds) (assuming their drag coefficients are addi- tive). Supposing Df ' Ds for similar surface proteins, this estimate yields that the complex will be roughly half as diffusive. However, proteins diffusing in a lipid membrane may not behave as ideal Brownian particles (199), and upon binding the resulting complex, that may undergo a conformational change or begin signaling, may have a drag coefficient different from the sum of the individual drag coefficients. We therefore take this estimate as only an upper bound on Dc, and assume Dc = 0.1Df in our simulations. Table 5.1: Parameters used for numerically simulating FRAP recovery curves shown in Figure 5.3. Parameter Value Df 0.5 (µm)2 s−1 Dc 0.05 (µm)2 s−1 FT 100 (µm)−2 Xb 6 µm Yb 3 µm Xm 2 µm Ym 3 µm kon 0.01 (µm)2s−1 10−4 s−1 (Experiment A, Figure 5.3a) koff 10 −2 s−1 (Experiment B, Figure 5.3b) 1 s−1 (Experiment C, Figure 5.3c) Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 93 For each experiment, we chose a set of increasing ST values to simulate the effect of titrating in an increasing density of pMHC into the lipid bilayer. The corresponding recovery curves are shown in Figures 5.3a-c. Individ- ual recovery curves were generated using the full model for the given choice of parameter values. 10% Gaussian noise was added to the solution, to model typical experimental errors in FRAP measurements. Each FRAP recov- ery curve shown in Figures 5.3a-c is the average of ten such individual recovery curves. The location of points on the k∗on − koff parameter-space are indicated by corresponding colored dots in Figure 5.3d. The SSR contours in Figure 5.3d indicate that for different ST values (i.e. different k∗on values through Equation 5.17) a different model will best describe the FRAP recovery. (a) (b) 0 20 40 60 80 1000 0.2 0.4 0.6 0.8 1 Time (s) N or m al iz ed In te ns ity , G (t) k off=0.0001 s −1 k on =0.01µm2s−1 ST=0 µm −2 ST=1µm −2 ST=40µm −2 ST=80µm −2 ST=200µm −2 ST=600µm −2 0 20 40 60 80 1000 0.2 0.4 0.6 0.8 1 Time (s) N or m al iz ed In te ns ity , G (t) k off=0.01 s −1 k on =0.01µm2s−1 ST=0 µm −2 ST=1µm −2 ST=40µm −2 ST=80µm −2 ST=200µm −2 ST=600µm −2 (c) (d) 0 20 40 60 80 1000 0.2 0.4 0.6 0.8 1 Time (s) N or m al iz ed In te ns ity , G (t) k off=1 s −1 k on =0.01µm2s−1 ST=0 µm −2 ST=1µm −2 ST=50µm −2 ST=200µm −2 ST=1000µm −2 ST=5000µm −2 log(k off) lo g( k o n * ) Df=0.5 µm 2/s D c =0.05 µm2/s Diffusion Dominant Weighted Diffusion Independent Diffusion Diffusion Dominant FullModel −6 −4 −2 0 2 4−6 −4 −2 0 2 4 Figure 5.3: Numerically simulated FRAP titration experiments demonstrate the effect of increasing ligand density on FRAP recovery dynamics. FRAP curves (a-c) were numerically simulated as described in the text, with parameter values listed in Table 5.1, and the total ligand density ST indicated in the legend. The corresponding points on the k∗on− koff parameter space are shown in (d), along with the contours indicating the applicable model regimes. Only contours of SSR=1 are shown (grey - pure diffusion (free), dotted grey - pure diffusion (bound), black - weighted diffusion, dotted black - independent diffusion). The contour lines were generated as described in Figure 5.2. 5.4.2 Fitting simulated data We applied the fitting procedure described above to the simulated FRAP titrations. For each experiment, we first fit the recovery curves with ST = 0 to Equation 5.5, and determined the best-fit values of Df = [0.50, 0.50, 0.49] Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 94 (µm)2s−1, for experiments A-C (Figure 5.3, a-c), respectively. Similarly, for each experiment, we also fit the titrations with the highest ST value and obtained the best-fit values of Dc = [0.050, 0.051, 0.059] (µm)2s−1, re- spectively. Having established the diffusion coefficient of the free receptor and bound complex, we next fit the interme- diate FRAP recovery curves in each experiment to all three models, i.e. weighted diffusion (case 1), independent diffusion (case 2), and the full model. We used the best-fit values of Df and Dc, as determined from fitting the two limiting FRAP curves, as known constants in these equations. Thus, for each intermediate FRAP curve in a titration we obtained the best-fit value of the the model parametersDwd, K∗, k∗on and koff. These best-fit parameter values, the SSR from the corresponding fits, and the resulting AIC statistics for the intermediate FRAP recovery curves from all three titrations are shown in Appendix E. Based on the ∆AIC values of the three fits for each curve, we identify the particular scenario that is applicable for parameter estimation. As described in detail above, for a fit in scenario 2 we can determine the pseudo-affinity, K∗, and for a fit in scenario 3 we can also determine the pseudo-on rate k∗on and the off-rate koff. In Table 5.2 we list the identified fitting scenario for each FRAP curve, and parameter estimates for fits in scenarios 2 and 3. Comparing the fitted parameter values with those used for simulating the data, we note that for all fits under scenarios 2 and 3, the fitted parameter values well match their actual values. We also list the ratio Seq/ST in Table 5.2. This is determined using the known ST , and FT values, and the fitted K∗ values. When this ratio Seq/ST > 0.01, we are additionally able to estimate the true 2D on rate kon (Experiment B), and the true 2D affinityK∗ (Experiments, B and C) for the ligand-receptor pair. The ratio Seq/ST measures the fraction of ligands that are in an unbound state at equilibrium. Thus, this fraction is small for high affinity ligands, and also at low ligand densities. Thus, under these conditions, we are unlikely to be able to re- liably determine the true 2D kinetic parameters. In the numerical simulations above, the receptor-ligand affinity is the greatest for experiment A, for which we observe the smallest values of this ratio. For the lower affinities in experiments B and C, Seq/ST is large enough for progressively higher ligand concentrations, allowing reliable estimates of the true 2D parameters. 5.5 Experimental outlook In the previous sections we proposed a confocal FRAP-based protocol to measure true 2D binding parameters be- tween a membrane-associated receptor-ligand pair, and tested its applicability with numerically simulated FRAP Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 95 Table 5.2: Parameter values used in numerical simulations, and the estimates from fitting the simulated FRAP titration data. Simulation parameters koff K ST K ∗ k∗on Fit Scenario s−1 (µm)−2 (µm)−2 s−1 1 0.0101 1.01×10−6 1 0.0001 100 40 0.666 6.66×10−5 2 (Experiment A) 80 3.99 3.99×10−4 2 200 1.00×10−4 1.00 1 1 0.0100 1.00×10−4 1 0.01 1 40 0.649 6.49×10−3 3 (Experiment B) 80 3.29 0.0329 3 200 1.01×10−2 1.01 1 1 5.01×10−3 5.01×10−3 1 1 0.01 50 0.281 0.281 2 (Experiment C) 200 1.41 1.41 2 1000 9.10 9.10 2 Fitted Parameters Expt. ST K∗ k∗on koff Seq/ST K kon (µm)−2 s−1 s−1 (µm)−2 (µm)2 s−1 A 1 40 0.667 1.25× 10−4 80 3.88 0.00631 200 B 1 40 0.692 7.67×10−3 0.0111 -0.0222 80 3.31 0.0309 0.00933 0.0401 1.03 0.00963 200 C 1 40 0.269 0.576 0.00934 80 1.46 0.703 0.0104 200 11.082 0.908 0.0122 Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 96 recovery curves. Here we point out some experimental challenges that need to be overcome for a successful ap- plication of this protocol. A key requirement of our protocol is the prior knowledge of the diffusion coefficients of free and bound forms of the receptor. We suggest conducting FRAP experiments in the absence of any ligand to establish the former, and in the presence of a saturating concentration of the ligand, to establish the latter. A limitation of this approach may be the low surface density of free receptors resulting in a low signal to noise ratio in the FRAP ex- periments. Nonetheless, translational diffusion coefficients of a variety of cell-surface proteins, including a GFP conjugated TCR, have been successfully measured using FRAP (194; 195), and we expect that for an experimen- tal system of interest such measurements will be feasible. Alternatively, overexpressing the receptor of interest could further enhance the signal to noise ratio for receptors with low surface density. When the receptors are engaged, possible complications in FRAP measurement may arise from clustering of receptors into microclusters, observed for TCR-pMHC interactions (200) or a substantial immobile fraction (194). However, these effects are cytoskeleton-dependent, and are abrogated in the presence of pharmacological inhibitors of actin polymerization such as latranculin-A. The more challenging requirement of our protocol is the accurate and reproducible measurement of the sur- face densities of both binding partners. In cells expressing a genetically encoded fluorescently-labeled protein, one often observes a heterogenous population with varying expression levels, that may be difficult to quantify, without a detailed knowledge of the photophysical properties of the fluorophore in an intracellular environment. Fluorescent calibration beads have been successfully used to quantify the surface density of TCR on cells in simi- lar experiments (200). Further, it is possible to maintain a relatively uniform expression level by isolating a single clone and proliferating it, or alternatively by sorting cells on the basis of fluorescence intensity and retaining a population with a relatively homogenous expression level. Likewise, by loading liposomes with varying concen- trations of the specific ligand during bilayer preparation, it is possible to control the ligand surface density on a lipid bilayer (168; 201). Lastly, as with all FRAP experiments care must be taken to minimize errors due to the finite bleach time (202; 203), the shape of the bleaching beam (202), the finite domain (193). Incomplete FRAP recovery, in the absence of binding, could arise from an immobile fraction (203). By defining Mf to be the fraction of proteins that are mobile, we can correct for situation when Mf < 1 by simplying multiplying the FRAP recovery by this fraction (see (193; 203)). Additionally, our method and indeed most common FRAP analyses are not applicable when there is appreciable membrane curvature. For a detailed discussion of the applicability of FRAP to curved Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 97 membranes see (204). 5.6 Conclusions We have outlined an experimental protocol for determination of kinetic parameters for membrane-restricted bind- ing of a ligand and receptor, based on titrating the density of the unlabeled substrate (ligand) and measuring FRAP recovery dynamics of the labeled receptor. As the ligand density increases, the system transitions from a diffusion-dominant regime where the protein is predominantly unbound, to another diffusion-dominant regime where the protein is predominantly bound. The passage between these two diffusion dominant regimes is through an intermediate regime where either the full model, or one of either the weighted-diffusion model, or the indepen- dent diffusion model are applicable. As described above, and illustrated in Figure 5.2, the extent of this intermediate regime in parameter space is determined by the separation between the diffusion coefficients of the free and the bound forms of the fluo- rophore. Thus, when the diffusion coefficients of the bound and free forms of a protein are sufficiently different, this method allows for an estimation of the true two-dimensional on and off rates for the binding or the true two- dimensional affinity. Alternatively, if the diffusion coefficients are not sufficiently different, this technique cannot discern any kinetic information. Due to this constraint, it may be necessary to slow down the diffusion of the ligand (and therefore, presumably of the bound complex), for instance via adhesion to a large, or slowly-diffusing particle. A distinct advantage of our technique is the need to fluorescently label only one of the binding partners. Fur- ther, the feasibility of our approach is at least partially evident from a number of recent papers where FRAP has been used to measure 3D binding constants in a variety of settings (e. g. (184; 185; 187; 188)). The study of Sprague et al. assumed that the unlabeled binding partner was immobile, whereas we analyze a more general model without this constraint. Further, for sufficiently different diffusion coefficients, our method allows us to fit for the 2D rate constant kon rather than the compound rate k∗on = konSeq. We additionally specialize to the commonly used geometry of confocal FRAP on cell surfaces. Our method also represents an alternative to that of (188), where FRAP measurements were used to elucidate the binding-mediated diffusion of mRNA fragments in the nucleus of HeLa cells. In that study, a relatively complex method was used to fit the data, involving repeated numerical solution of the partial differential equations for the diffusing species. Our fitting technique is concep- tually simpler, involving fitting only a relatively simple expression to the data. Chapter 5. Analysis of membrane-localized binding kinetics with FRAP 98 We also point out some experimental challenges for a successful implementation of our protocol. One of the key requirement of our method is an accurate estimation of the surface density of the two binding partners, and we foresee this to be a key experimental challenge toward a successful implementation. 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Simulations of (an)isotropic diffusion on curved biological surfaces. Biophys. J. 90:878–885. 103 Chapter 6 Effects of intracellular calcium and actin cytoskeleton on TCR mobility measured by fluorescence recovery1 6.1 Introduction The activation of T lymphocytes by antigenic ligands displayed on the surface of antigen presenting cells (APC) is the central event in developing an adaptive immune response. During the T cell/APC interaction, T cell recpetors (TCR) bind to peptide-major-histocompatibility-complexes (pMHC) present on the surface of the APC (205). T cells are observed to respond sensitively and specifically to antigenic stimulation (206; 207). During the interaction between T cells and APC, TCR and accessory molecules travel towards the T cell/APC contact site, forming a signaling area at the interface that has been named the immunological synapse (208; 209). TCR movement towards the immune synapse has been thoroughly documented and shown to reflect the intensity of antigenic stimulation (210; 211) but we do not have a complete picture of how surface receptor signaling mod- ulates TCR motion. TCR motion on live cells has been studied using fluorescence recovery after photo-bleaching (FRAP) and fluorescence loss in photo-bleaching (FLIP) techniques (212). In a primary study, Sloan-Lancaster et al. showed that GFP-tagged CD25 ζ-chain (i.e. fluorescently labeled TCR) chimeras expressed on the surface of HeLa cells exhibit very low mobility (212). Following this observation, we employed a TCR β-chain deficient Jurkat T cell line transfected with a GFP/fusion TCR β-chain to measure lateral mobility of fully assembled TCR/CD3 com- plexes using FRAP. We showed that TCR are mobile on the surface of Jurkat T cells (213). Using a different approach, M. Krummel and colleagues also showed that TCR are mobile on the surface of murine T cells and that 1Dushek O∗, Mueller S∗, Soubies S, Depoil D, Caramalho I, Coombs D∗, Valitutti S∗ (2008) Effects of intracellular calcium and actin cytoskeleton on TCR mobility measured by fluorescence recovery. PLoS One. in press. ∗ indicates equal contribution Chapter 6. Effects of intracellular calcium and actin cytoskeleton on TCR mobility 104 they undergo directed transport towards the immune synapse during antigen recognition (214). More recently, it has been shown that TCR dynamically form micrometer-scale clusters that move in a directed fashion towards the center of the immune synapse during antigen recognition (215; 216; 217). TCR recruitment to the center of the immune synapse has been associated with productive signaling (209; 218) and with TCR internalization (217; 219). The directed motion of TCR requires an energy-consuming mechanism and a natural candidate for this mech- anism is an interaction between TCR and the cortical actin cytoskeleton (CAC). T cells need a functional actin cytoskeleton to form productive conjugates with APC (217; 220; 221) and cortical actin has further been impli- cated in the transport of immune synapse components to the T cell/APC contact (222; 223; 224). However, an explicit link between the actin cytoskeleton and TCR dynamics has not been established at the molecular level. In particular it is not clear how activation signals triggered by TCR engagement might modulate TCR mobility via CAC-dependent mechanisms. Here, we report on efforts to characterize TCR mobility in human T lymphocytes using fluorescence recovery techniques. We stimulated T cells using ionomycin, a drug known to increase the concentration of intracellular calcium, [ Ca2+ ] i . We show that [ Ca2+ ] i increase markedly reduces TCR mobility on the T cell surface via an actin cytoskeleton-dependent mechanism. We also show that actin polymerization increases following [ Ca2+ ] i increase, suggesting a direct link between calcium induced CAC polymerization and constraints on TCR mobility on the T cell surface. 6.2 Materials and methods A full discussion of the materials and methods can be found in Appendix F. Briefly, CD4 T cells were isolated from the blood of healthy donors. The TCR of these cells was fluorescently labeled and visualized using a confocal laser scanning microscope. Cells were either untreated or treated with various pharmacological agents before performing FRAP experiments. The protocol used for FRAP experiments is described in (225) (Chapter 4) and Matlab was used for model fitting. The amount of actin in a filamentous form was quantified using flow cytometry by fixing cells and staining them with phalloidin. Chapter 6. Effects of intracellular calcium and actin cytoskeleton on TCR mobility 105 6.3 Results 6.3.1 TCR mobility is similar in naive and activated human T cells We initially focused on studying TCR mobility on unstimulated T cells. We used three CD4+ T cell popula- tions: naive T lymphocytes freshly isolated from cord blood (CBTL); T cells from a line obtained by culturing for two weeks in vitro naive T cells in the presence of magnetic beads coated with anti-CD3 plus anti-CD28 mAb (activated CBTL); freshly isolated T lymphocytes from peripheral blood of adult healthy donors (PBL). We stained cells from each population using anti-CD3 Fab antibodies labeled with Cy5 (211; 226) and performed FRAP experiments. We applied the method described in Dushek at al. (225) to analyze FRAP recovery curves and extracted the mobility parameters of the TCR. Two parameters were calculated: the diffusion coefficient (D, generally expressed in µm2/s) and the mobile fraction (Mf ), indicating the fraction of TCR that are mobile on the timescale of the experiment. The results of this analysis are presented in Figure 6.1 and Table 6.1. TCR had a diffusion coefficient (D, expressed as mean ± SEM)) of 0.048 ± 0.008 µm2/s in CBTL, 0.035 ± 0.006 µm2/s in activated CBTL and 0.061± 0.008 µm2/s in PBL. The values of D were not found to be significantly different (p>0.05). The mobile fraction (Mf ) of TCR was very high in all cell populations, supporting the notion that TCR are constitutively mobile on living T cells (Table 6.1). These results show that the basic parameters of TCR dynamics on the surface of T cells are not significantly different in T cells at different stages of activation, indicating that TCR mobility is not involved in allowing acti- vated T cells to be more responsive to antigenic stimulation. Table 6.1: Diffusion coefficient and mobile fraction under various pharmacological treatments Diffusion Coefficient Mobile Fraction P-Value of Diffusion Condition n (µm2/s) (Mf ) Relative to PBL∗ PBL 61 0.061±0.008 0.96±0.04 1.00 CBTL 20 0.048±0.008 1.00±0.05 0.40 Activated CBTL 20 0.035±0.006 1.09±0.04 0.08 PBL + Cyto D 20 0.068±0.014 0.93±0.05 0.66 PBL + Lat B 10 0.082±0.016 0.95±0.06 0.34 PBL + Iono 37 0.017±0.002 1.00±0.04 0.0001 PBL + Iono + Cyto D 25 0.043±0.005 0.92±0.04 0.17 PBL + Iono + Lat B 10 0.045±0.009 0.99±0.05 0.45 ∗ P-values were obtained by comparing diffusion coefficients from each experiment to the untreated PBL data using a two-sample T-test. With the exception of PBL + Iono, no signifi- cant differences in the diffusion coefficient were observed (P-values > 0.05). No significant differences in the mobile fraction were observed (all P-values > 0.05). Chapter 6. Effects of intracellular calcium and actin cytoskeleton on TCR mobility 106 0 10 20 30 40 50 60 70 80 90 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) N or m a liz e d In te n si ty PBL CBTL Activated CBTL Figure 6.1: TCR mobility does not depend on the activation state of T cells. Mean FRAP recovery curves are shown for peripheral blood T Lymphocytes (PBL, blue curve n=61), Cord Blood T Lymphocytes either freshly isolated (CBTL, green curve n=20) or activated in culture (activated CBTL, red curve n=20) were submitted to FRAP experiments. Data are from 2 (for CBTL) and 4 (for PBL) independent experiments. 6.3.2 [ Ca2+ ] i increase affects TCR mobility by an actin cytoskeleton dependent mechanism We next investigated whether calcium signaling in T cells could affect TCR mobility. To this end, T cells were loaded with the calcium probe FLUO-4 (a dye that increases its light emission when bound to Ca2+) before staining with anti-CD3/Cy5 Fab antibodies (the fluorescent tag that binds TCR). This allowed us to detect sig- nal transduction in parallel with TCR dynamics (211; 227). We stimulated T cells with ionomycin to mimic the [ Ca2+ ] i increase that follows TCR productive engagement (228). Our results show that [Ca2+] i increase results in a substantial decrease in TCR lateral diffusion in PBL, as measured by a reduction of D, see Figure 6.2 and Table 6.1. Pre-treatment of T cells with 10 µM cytochalasin D (a drug that inhibits actin cytoskeleton function) abolished the effect of ionomycin on TCR mobility, Figure 6.2a. The values of D were 0.017 ± 0.002 µm2/s in PBL treated with ionomycin, 0.068 ± 0.014 µm2/s in PBL treated with cytochalasin D only and 0.043 ± 0.004 µm2/s in PBL treated with both cytochalasin D and ionomycin. Similar results were obtained when T cells were treated with 50 nM latrunculin B instead of cytochalasin D, Figure 6.2b. Interestingly, cytochalasin D/latranculin B treatements did not significantly affect basal lateral mobility of TCR indicating that this process is not dependent on actin cytoskeleton function. Ionomycin treatment, although affecting TCR lateral diffusion (D values) did not affect the mobile fraction of TCR that was again close to 1 in the different samples (Table 6.1). Our results indicate that the activation of the calcium signaling pathway reduces TCR mobility in T cell by an actin cytoskeleton-dependent mechanism. They also show that [ Ca2+ ] i increase, although reducing the effective mobility of individual TCR, does not reduce the number of mobile molecules. Chapter 6. Effects of intracellular calcium and actin cytoskeleton on TCR mobility 107 (a) (b) 0 50 100 150 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) N or m a liz e d In te ns ity PBL PBL+CytoD PBL+Iono PBL+CytoD+Iono 0 50 100 150 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (s) N or m a liz e d In te ns ity PBL PBL+Lat B PBL+Iono PBL+Lat B+Iono Figure 6.2: TCR mobility is modulated by intracellular calcium via the actin cytoskeleton. a. Mean FRAP recov- ery curves are shown for Peripheral blood T Lymphocytes (PBL, blue curve n=61), PBL treated with cytochalasin D (PBL + CytoD, green curve n=20), PBL treated with ionomycin (PBL + iono, orange curve n=37), and PBL treated with ionomycin and cytochalasin D (PBL + Iono + CytoD, light blue curve n=25). b. Mean FRAP recov- ery curves are shown for PBL (blue line, n=61) PBL treated with latrunculin B (PBL + Lat B, green curve n=10), PBL treated with ionomycin (PBL + iono, orange curve n= 37), and PBL treated with ionomycin and latrunculin B (PBL + Iono + LatB, light blue curve n=10). 6.3.3 [ Ca2+ ] i increase induces polymerization of the actin cytoskeleton The above results suggested that, following [ Ca2+ ] i increase, the actin cytoskeleton becomes a constraint for TCR mobility. To define a link between [ Ca2+ ] i increase and actin cytoskeleton function, we used FACS anal- ysis to measure F-actin (polymerized actin). PBL were either untreated or treated with 2.5 µg/ml ionomycin or increasing concentrations of latrunculin B (as a control of actin cytoskeleton de-polymerization) for 15 min- utes at 37 ◦C. Cells were fixed, permeablized, and stained with Alexa-488 labelled phalloidin to selectively tag F-actin. As shown in Figure 6.3, treatment with latrunculin B decreased the amount of F-actin in T cells in a dose-dependent fashion, supporting the notion that FACS analysis is suitable for quantification of F-actin (229). Treatment of T cells with ionomycin induced a moderate increase in F-actin. These results indicate that [ Ca2+ ] i increase augments the amount of F-actin in T cells and suggest that the increase in actin polymerization could be responsible for the observed effect of [ Ca2+ ] i increase on TCR mobility. 6.3.4 Modeling reveals effective TCR binding parameters to the actin cytoskeleton We have shown that the observed decrease in TCR mobility following ionomycin treatment is mediated by the cortical actin cytoskeleton. We can extend the pure diffusion model, used to obtain the results in Table 6.1, to Chapter 6. Effects of intracellular calcium and actin cytoskeleton on TCR mobility 108 Figure 6.3: [ Ca2+ ] i increase induces actin polymerization in PBL. PBL were treated for 15 minutes with lantran- culin A (50 nM, 100 nM or 500 nM) or with ionomycin (2.5 µg/ml). F-actin cellular content was measured by FACS analysis. Data are from one representative experiment out of three. include binding interactions between TCR and the actin cytoskeleton as follows, ∂T/∂t = D∇2T − k∗onT + koffC, ∂C/∂t = k∗onT − koffC. (6.1) where T is the concentration of diffusing TCR, C is the concentration of TCR bound to the cytoskeleton, D is the diffusion coefficient of TCR, koff is the unbinding rate, and k∗on is the effective binding rate. The effective binding rate can be written as k∗on = k̄onS, where k̄on is the intrinsic binding rate of TCR directly to the cytoskeleton or binding through a putative mediator that links the TCR to the cytoskeleton. S is the concentration of binding sites available to diffusing TCR which are assumed to be homogenously distributed. This binding model has been previously used in FRAP analysis to extract binding parameters (230). In Table 6.2 we summarize the results of fitting the pure diffusion model and the binding model to the PBL data and the PBL + ionomycin data. When fitting the binding model we only fit for the on- and off-rates and set D=0.0609 µm2/s, the free diffusion coefficient. The fitting procedure we used is described in Dushek et al. (231). We find that both the pure diffusion and binding models accurately fit both data sets, Figure 6.4. Using a statistical test to compare the explanatory power of each model (Akaike’s Information Criterion, AIC) we find that both models explain the PBL data equally well. However, we find that the binding model has significant additional power in explaining the PBL + ionomycin data. This suggests that TCR binding is quite possibly re- sponsible for the slow FRAP recovery after ionomycin treatment as opposed to explanations involving a smaller mobile fraction or a change in the underlying TCR diffusivity. In this case, the binding model reveals rapid bind- ing kinetics between the TCR and the putative cytoskeletal binding partner with k∗on = 0.22 s−1 and koff = 0.05 s−1. Put another way, a TCR might be expected to bind the cytoskeleton on average every 5 s and unbind after 20 s. Chapter 6. Effects of intracellular calcium and actin cytoskeleton on TCR mobility 109 Table 6.2: Model fitting to FRAP recovery curves PBL Data PBL + Ionomycin Data SSR AIC 4AIC Prob. SSR AIC 4AIC Prob. Pure Diffusion Model 0.0449 -680.06 0.799 0.40 0.0281 -722.27 38.71 0.00 Diffusion + Binding Model 0.0445 -680.86 0.000 0.60 0.0183 -760.97 0.00 1.00 SSR - Sum squared residuals, AIC - Akaike Information Criterion, Prob - Probability that the given model is the best descriptor of the FRAP recovery based on4AIC. 0 50 100 150 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (s) N or m a liz e d In te n si ty PBL PBL+Ionomycin Pure Diffusion Model Binding + Diffusion Model Figure 6.4: Fitting the diffusion and binding model to FRAP recovery curves. We show the fit of the pure diffusion model (red) and the binding model (black) to the PBL data and to the ionomycin-treated PBL. Both models provide a good fit to the recovery curves but a statistical test reveals that the binding model is significantly better at explaining the FRAP recovery when the PBL are treated with ionomycin (Table 6.2). 6.4 Discussion In the present work we have shown, in agreement with previous studies (213; 214), that TCR are constitutively mobile on the surface of T cells. This lateral mobility (with a diffusion coefficient in the region of 0.05-0.1 µm2/s) is in principle sufficient to allow individual TCR to patrol the entire T cell surface within about 30 minutes. The diffusivity of TCR is similar in T cells at different stages of activation, indicating that this parameter is not in- volved in allowing activated T cells to be more responsive to antigenic stimulation. Fast lateral mobility and rapid cell surface patrolling by TCR may be instrumental in resting conditions to al- low T cells to rapidly sense antigenic stimuli. However it seems likely that, upon antigenic stimulation, regulated TCR motion would be needed to permit spatially localized signaling. Accordingly we find that [ Ca2+ ] i increase (a central and early signal in T cell activation) induces a significant decrease of TCR mobility. Our results also show that this reduction in TCR mobility requires a functional actin cytoskeleton. This suggests that following[ Ca2+ ] i increase, TCR become associated with actin cytoskeleton or with actin binding proteins. Fitting a model for TCR diffusion and binding to an immobile partner to our data suggests that this may indeed be the case. Our observations, by showing that TCR may be bound or trapped by F-actin, raise the question of how they Chapter 6. Effects of intracellular calcium and actin cytoskeleton on TCR mobility 110 could move towards the immune synapse. It is well established that F-actin is enriched at the immune synapse (232). Moreover, work from M.M. Davis and colleagues showed that cortical F-actin moves towards the T cell/APC contact site in antigen stimulated T cells (223). Our present work points towards an important role of actin cytoskeleton in control of TCR motion although we did not focus in this study on TCR mobility during T cell/APC cognate interaction. We also showed that increase [ Ca2+ ] i by a pharmacological agent results in increased levels of F-actin in T cells. The concentration of ionomycin at which a clear shift in F-actin cellular content was observed was 2.5 µg/ml. In the FRAP experiments, 0.5 µg/ml ionomycin (a concentration previously used to activate T cells (220)) was sufficient to induce a reduction of TCR mobility. The reason for this discrepancy is presently elusive. It is possible that the effect on TCR mobility precedes significant actin polymerization. Alternatively, differences in sensitivity between the two experimental methods could be the reason for the observed discrepancy. This result suggests that upon initial conjugation of a T cell with a cognate APC the productive and localized engagement of a small number of TCR may lead rapidly, via [ Ca2+ ] i increase and actin polymerization, to global control of TCR motion. Mathematical models of immune synapse formation commonly include TCR transport towards the center of the immune synapse, driven by the actin cytoskeleton (218; 233; 234; 235; 236). Important parameters in these models are effective reaction rates between TCR and the actin cytoskeleton, which to our knowledge have not been previously reported. For example, in the study of Burroughs et al. (234), it was assumed that the effective on- and off-rates were identical and equal to 0.1 s−1. It would be interesting to investigate the effects of a larger on-rate and smaller off-rate, as reported in the present work, on the results of this and other models. Several lines of evidence established a link between CAC polymerization and TCR signaling (237). Treatment of T cells with drugs affecting actin cytoskeleton such as cytochalasin D, aborts antigen induced [ Ca2+ ] i increase indicating that actin polymerization is required to sustain signaling in T cells (220). Moreover, deficiency or mu- tation in several molecules regulating the actin cytoskeleton such as Vav-1, Cdc42 and Wiskott-Aldrich syndrome protein (WASP) affect IS formation (238). Some of these proteins such Nck and WASP are recruited to the TCR signaling area together with polymerized actin (239). Interestingly, while actin cytoskeleton polymerization is usually considered to be required for the activation of calcium pathway (220; 237), here a reverse mechanism is proposed: [ Ca2+ ] i increase was shown to enhance actin cytoskeleton polymerization and in turn modulate TCR mobility. Chapter 6. Effects of intracellular calcium and actin cytoskeleton on TCR mobility 111 We speculate that the binding or trapping of TCR by the actin cytoskeleton network could also be instrumen- tal in favoring the assembly of the TCR associated signaling cascade. The notion that TCR move very rapidly is difficult to reconcile with the necessity of assembling complexes of adaptor molecules and signaling components at the immune synapse. The observation that TCR mobility is affected by global [ Ca2+ ] i increase is compatible with the formation of localized signaling scaffolds favoring complex molecular interactions. This hypothesis is in agreement with reported data by M. Dustin and colleagues showing that a functional actin cytoskeleton is required to allow TCR micro-clusters to signal and that high concentration of polymerized actin is confined to TCR clusters (217). It is tempting to speculate that global [Ca2+] i increase induced by the engagement of a few TCR at the T cell/APC contact site (via polimerization of CAC) might lead to the rapid formation of a much greater number of TCR clusters in the nascent immune synapse, If so, this would equip T cells with pre-formed signaling units ideal for sensitive detection of a limited number of antigenic ligands. Our work outlines a previously unexpected link between the calcium pathway, the actin cytoskeleton and TCR mobility. Although we have not examined the detailed mechanisms of this triangular interaction our results high- light the ability of TCR population to control their own mobility via calcium signaling. 112 References [205] Davis, M., M. Krogsgaard, M. Huse, J. Huppa, B. Lillemeier, and Q. Li. 2007. T cells as a self-referential, sensory organ. Annu Rev Immunol. 25:681–95. [206] Harding, C., and E. Unanue. 1990. 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Biophys J. 87:3665–3678. [236] Figge, M. T., and M. Meyer-Hermann. 2006. Geometrically repatterned immunological synapses uncover formation mechanisms. PLoS Comput Biol. 2:e171. [237] Billadeau, D., J. Nolz, and T. Gomez. 2007. Regulation of T-cell activation by the cytoskeleton. Nat Rev Immunol. 7:131–43. [238] Tooley, A., J. Jacobelli, M. Moldovan, A. Douglas, and M. Krummel. 2005. T cell synapse assembly: proteins, motors and the underlying cell biology. Semin Immunol. 17:65–75. [239] Barda-Saad, M., A. Braiman, R. Titerence, S. Bunnell, V. Barr, and L. Samelson. 2005. Dynamic molecular interactions linking the T cell antigen receptor to the actin cytoskeleton. Nat Immunol. 6:80–9. 115 Chapter 7 Discussion In my thesis I investigated two major themes: serial engagement in T cell activation (Chapters 2 and 3) and parameter estimation using FRAP (Chapters 4, 5, and 6). These will be discussed in turn. 7.1 Serial engagement in T cell activation The serial engagement hypothesis has received much attention over the years (240; 241; 242; 243; 244; 245; 246). The fact that serial engagements are inevitable in systems where mobile proteins undergo reactions, as in the case for TCR/pMHC, will ensure that it remains a central component in models of T cell activation. Serial engage- ments are probably necessary but not sufficient for T cell activation, as the serial engagement hypothesis, in its standard form, does not provide a mechanism for TCR triggering or accurate pMHC discrimination (246). The serial engagement hypothesis predicts that the total number of TCRs engaged by a single pMHC will determine the stimulatory potency of that pMHC. Experimentally, it is difficult to quantify the number of binding events between pMHC and TCR because fluorescence microscopy, the most commonly used tool to probe the immune synapse (IS), provides information only on the spatial dynamics of proteins. Therefore, mathematical modeling is necessary to estimate the number of serial engagements. 7.1.1 Serial engagements in TCR clusters In Chapter 2, I used Monte Carlo simulations and escape time formulations (or mean first passage time cal- culations) to explore serial engagement in the context of TCR clusters. Using a simple two-state escape time formulation I showed that 1) pMHC are expected to escape from a mobile TCR cluster before the cluster reaches the cSMAC and 2) pMHC engage few TCRs before exiting the TCR cluster. I was able to capture the effects of CD4 and CD8 coreceptors or pMHC dimerization by extending the model to include 4 states. Including corecep- tors or pMHC dimerization allowed the pMHC to remain in the TCR cluster until it arrives in the cSMAC and in addition, allowed for substantially more TCR engagements. Chapter 7. Discussion 116 The role of coreceptors in T cell activation has remained controversial (246; 247). Here, I have shown that coreceptors are predicted to increase the trapping time of pMHC in TCR clusters. These predictions can be exper- imentally tested. Using single particle tracking (SPT) of pMHC molecules in the immune synapse will reveal if pMHC move between clusters or remain in a single cluster. If the latter is true, my work suggests that coreceptors and/or pMHC dimerization play an important role in such confinement. Antibody blocking of coreceptors when presented pMHC are monomeric should allow the pMHC to easily escape the TCR cluster. In addition to trapping pMHC, I have shown that coreceptors allow pMHC to make multiple localized engagements. The association of Lck with coreceptors (246; 248) suggest that coreceptor facilitated TCR engagements are important in signal amplification. By examining a single TCR cluster I have shown that, in the absence of coreceptors and dimerization, the pMHC will escape from a TCR cluster well before the cluster arrives in the cSMAC. I therefore examined the effects of multiple TCR clusters. First I examined an IS that contains multiple immobile TCR clusters. I was able to obtain an asymptotic solution in the limit that the TCR clusters are small relative to the IS and used a numerical solution when the TCR clusters are large. I found that substantial serial engagements are possible but that engagements are no longer spatially localized because the pMHC visits multiple TCR clusters before leaving the IS. I then examined a more relevant model that accounted for the mobility of TCR clusters. By using a fully stochastic Monte Carlo simulation I showed that pMHC have a high probability of entering the cSMAC and of achieving substantial TCR engagements by visiting multiple TCR clusters. Mathematically, there are several possible extensions of the TCR cluster models I have investigated. I have assumed that interactions between TCR and pMHC in TCR clusters obey the law of mass-action. However, molecular crowding within the clusters and/or steric hindrances by molecules surrounding the TCR cluster may trap pMHC within a TCR cluster without the aid of pMHC dimerization or coreceptors. These effects can be incorporated into a nano-scale Monte Carlo simulation of dynamics within a single TCR cluster, similar to the model presented in Chapter 3. Future work is also possible on the models of multiple TCR clusters that I have presented. First, I was only able to obtain asymptotic solutions in the case of immobile TCR clusters. It may be possible to obtain asymptotic solutions for a PDE model of multiple mobile TCR clusters by using matched asymptotic analysis in the reference frame of the TCR clusters. As the number of TCR clusters increases the asymptotic solution will break down and full PDE computations will be required. In the limit of many TCR clusters, a mean field approach may be possible. This analysis will be mathematically interesting and studying how the number of TCR clusters alters the Chapter 7. Discussion 117 spatial-temporal dynamics of diffusing pMHC is immunologically relevant. 7.1.2 Localized serial engagements as a mechanism of pMHC discrimination In Chapter 3 I investigated an alternative role for serial engagement of TCRs by a single pMHC. Given that down- stream responses of T cells are highly sensitive to the dissociation constant, KD, governing the TCR/pMHC bond (249; 250), I investigated how the T cell can determine KD. I found that a direct estimate of KD, for example by the fraction of TCR bound to pMHC at equilibrium, is unlikely and that estimates of the kinetic parameters governing the TCR/pMHC bond are required to allow the T cell to estimate KD. I then showed that the stochastic nature of the TCR/pMHC interaction requires multiple samples of the pMHC reaction rates. Multiple samples are achieved by serial engagements of TCR by a single pMHC. I showed that large TCR concentrations, similar to the concentration of TCR in TCR clusters, and coreceptors (for weakly binding pMHC) are required for localized engagements. I suggest that localized samples are important to ensure that binding events arise from the same pMHC and for effective integration of signals that arise from each binding event. By viewing serial engagements as a sampling process, I proposed that membrane organization, in the form of TCR clusters, is required for accurate pMHC discrimination. This view of serial engagement poses several chal- lenges regarding the molecular mechanisms that store and integrate signals from multiple binding events. Analysis of the experimentally determined time course of activity for early signaling molecules like Lck, ZAP-70, LAT, and SLP-76 in response to various pMHC having different reaction parameters may elucidate the mechanism of pMHC discrimination. The feasibility of this analysis is at least partially evident by recent work that characterized the phosphorylation time course of LAT (251). A long term, and a more ambitious goal is to understand how the complex signaling network in T cells (252) faithfully translates information about the reaction rates between TCR and pMHC into downstream responses, such as cytokine release. 7.2 Parameter estimation using FRAP 7.2.1 Estimating diffusion coefficients using FRAP Fluorescence Recovery After Photobleaching (FRAP) is a widely used technique to study the mobility of proteins in the cytoplasm and the cell membrane (253; 254; 255). In Chapter 4, I investigated the application of FRAP to cell surface proteins. I found that two commonly used protocols to estimate the diffusion coefficient of cell surface proteins give rise to appreciable errors. The first utilitizes a large bleach region but does not account for Chapter 7. Discussion 118 the finite extent of the cell membrane. The second uses a small bleach region but the equation used to fit the FRAP recovery curve does not capture the experimental geometry. I proposed a protocol that is simple to implement and provided an analytical equation to fit to the FRAP curve. This protocol provides improved parameter estimation for diffusion coefficients and was successfully implemented in the study of TCR mobility in Chapter 6. There have been several other attempts to improve the absolute accuracy in estimating the diffusion coefficient from FRAP experiments. One study improved parameter estimation by capturing the diffusion of highly mobile proteins during the bleaching event (256). Other studies focused on the inhomogeneous environment of diffus- ing proteins and implemented a numerical solution of the diffusion equation to improve parameter estimation (257; 258). Although these studies provide improved parameter estimates, few FRAP studies have implemented these improvements, probably because implementation is nontrivial. The protocol I proposed to improve parame- ter estimates is simple, primarily because I provide an analytical formula that can be fit directly to the experiment data. Reports of diffusion coefficients for membrane proteins span roughly two orders of magnitude,D = 0.005− 0.5µm2/s (259). Even for the same protein, estimates in the diffusion coefficient range over an order of magni- tude. The reason for these large differences is probably due to different experimental apparatus and, in addition, because of appreciable errors in the fitting procedure. Minimizing the errors in the fitting procedure, by the pro- tocol I proposed or otherwise, will allow for more consistent reports of diffusion coefficients between laboratories. 7.2.2 Estimating binding rates between cell membrane proteins using FRAP Communication between nearby cells in the immune system, and elsewhere, relies on interactions between pro- teins on opposing membranes. The strength of these interactions often determines the subsequent activity of each cell. In the case of T cells interacting with APCs bearing antigenic pMHC, the reaction rates between TCR and pMHC have been shown to govern T cell activation and the amount of cytokine released by the T cell (249; 250; 260; 261). Hence, measuring reaction rates between membrane proteins is critical to the understanding of cellular activity following cell-cell (and hence protein-protein) interactions. In Chapter 5 I outlined an experimental protocol for determining reaction rates between receptors and their ligands on opposing membranes, based on titrating the density of the ligand and performing a series of FRAP experiments in the junction between a cell and a planar bilayer. Using asymptotic analysis and model fitting, I showed that as the ligand density increases the system transitions between a diffusion-dominant regimes where Chapter 7. Discussion 119 the receptor is primarily unbound to another diffusion-dominant regime where the receptor is primarily bound to the ligand. In both of these diffusion-dominant regimes no binding information can be obtained. However, in the regime that exists between the transition from these diffusion-dominant regimes it is possible to obtain the 2D bond affinity (independent diffusion or weighted diffusion regimes) or the 2D reaction on-rate and off-rate (full model regime). I confirmed these theoretical predictions using simulated data. A key advantage of this FRAP-based method to determine reaction rates, compared to other existing meth- ods, is that the reaction rates are determined in vivo. The most widely used method for determining reaction rates between proteins is Surface Plasmon Resonance (SPR) (262; 263; 264). However, this method requires that the proteins of interest be removed from their native environment (i.e. the plasma membrane) and placed in solution. Therefore, SPR provides solution or 3D reaction rates whereas the actual rates are 2D because the proteins are confined to the plasma membrane, which is a 2D surface. The relationship between experimentally measured 3D reaction rates and the physiologically relevant 2D rates is presently unknown (265). Most studies assume a linear relationship but non-linear relationships may arise, see figure 7.1. For example, taking into ac- count conformational changes, Qi et al. (266) have shown that the relationship between 2D and 3D off rates is not a simple linear function. By using the FRAP-based method to directly determine 2D reaction rates for pro- teins whose 3D rates have already been measured, the effective relationship between these rates can be uncovered. 3D (Solution) Reaction Rate 2D (M em bra ne ) R ea cti on R ate Linear Quadratic Square Root Figure 7.1: Possible relationship between 3D and 2D reaction rates. The simplest relationship is that 3D and 2D rates are related by a constant factor (linear model). However, non-linear effects may be introduced into the conversion (quadratic and square root model) by considering effects of the plasma membrane, molecular crowding, and/or cooperative effects. The actual relationship between 3D and 2D reaction rates is presently unknown. An arbitrary scale has been used. I briefly mention that a FRAP-based method for determining 2D reaction parameters has recently been pub- lished (267; 268). In this work, the authors perform FRAP experiments in the contact junction between a cell and planar bilayer and report, for the first time, direct measurements of the 2D reaction on- and off-rates between membrane proteins. Interestingly, they found that the reaction off-rate was 100 times smaller than the reaction Chapter 7. Discussion 120 off-rate measured by SPR. The analysis and concepts of using FRAP to determine binding kinetics developed in Chapter 5 can be ap- plied to many scenarios. For example, the asymptotic analysis was performed without imposing any geometry and therefore can be directly applied to studies that use circular bleaching profiles instead of the rectangular pro- files often used for cell membrane FRAP. Although applied to the interaction between a receptor and ligand on opposing membranes, the procedure can readily be applied to interacting proteins on the same membrane or to interacting proteins in the cytosol. The procedure can also be used to determine effective reaction parameters between membrane proteins and lipid rafts (269). 7.2.3 Estimating effective binding rates between cell membrane proteins and the actin cytoskeleton using FRAP It is now well established that integrins regulate the actin cytoskeleton and that the cytoskeleton, in turn, regu- lates the mobility of integrins (270; 271). However, quantitative estimates for the effective binding reactions have not been measured. In Chapter 6 I presented work, performed in collaboration with the experimental laboratory of Salvatore Valitutti, aimed at characterizing the relationship between TCR mobility, the cytoskeleton, and in- tracellular calcium. Using the protocol developed in Chapter 4 we performed a series of FRAP experiments to determine the mobility of TCR when T cells were treated with various pharmacological agents. In particular, we showed that treatment with ionomycin decreases TCR mobility in an actin cytoskeleton dependent manner. We fit a diffusion-binding model, using the tools developed in Chapter 5, and extract the first estimates of the effective binding parameters governing the TCR and cytoskeleton interaction. We find an effective on-rate of 0.22 s−1 and an effective off-rate of 0.05 s−1. These estimates for the effective on- and off-rates are relevant to studies of immune synapse formation. The study of Qi et al. (272) argued that TCR enrichment in the cSMAC can come about by passive processes alone while the study of Burroughs et al. (273) argued that the actin cytoskeleton is required to transport TCR to the cSMAC. In the latter study, it was assumed that TCR undergo reversible first order reactions with the actin cy- toskeleton and that when bound to the cytoskeleton, TCR are actively transported to the cSMAC. The reaction rates used by the authors (273) are similar to the experimentally determined values in our study. The FRAP experiments we have performed provided ensemble estimates of the reaction rates between TCR and the actin cytoskeleton. In the future, it will be important to confirm the interaction between TCR and the Chapter 7. Discussion 121 cytoskeleton using single particle tracking (SPT) (259). Trajectories of individual TCR using SPT will reveal if they are undergoing free diffusion or in addition, binding to the actin cytoskeleton. Our results suggests that whichever nanoscale model governs the mobility of TCR, on the micron scale TCR are effectively diffusing and binding to the actin cytoskeleton. In the future, it will be important to confirm these results in a more physiological setting by activating T cells with an APC bearing antigenic pMHC. These experiments pose a challenge for FRAP analysis. Unlike treatment with soluble pharmacological agents, stimulation with an APC will likely polarize the actin cytoskeleton. The FRAP recovery curve will no longer represent binding of TCR to an immobile actin cytoskeleton but to mobile and dynamic strucutre. A previous study (274) was able to investigate the mobility of TCR towards the contact junction between a T cell and an APC during immune synapse formation and a similar analysis may be performed in our case. The mechanism by which TCRs aggregate to form TCR clusters on the cell membrane remains controversial. There is evidence that protein-protein interactions are required (275) while other evidence suggest that the cy- toskeleton is required (276). Having shown that TCRs can be immobilized by the cytoskeleton in Chapter 5 and knowing that signaling through a TCR can initiate local actin polymerization (271), our work suggests a model whereby productive engagement of TCR by pMHC leads to local actin polymerization. The resulting increase in localized actin increases the probability of TCR binding and hence promotes the formation of a TCR cluster. Indeed, high concentration of polymerized actin has been shown to be confined to TCR clusters (276). 122 References [240] Valitutti, S., S. Mller, M. Cella, E. Padovan, and A. Lanzavecchia. 1995. Serial triggering of many T-cell receptors by a few peptide-MHC complexes. Nature. 375:148–151. [241] Itoh, Y., B. Hemmer, R. Martin, and R. N. Germain. 1999. 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Davis, and M. Krummel. 2002. Quantifying signaling-induced reorientation of T cell receptors during immunological synapse formation. Proc Natl Acad Sci U S A. 99:15024–9. [275] Douglass, A. D., and R. D. Vale. 2005. Single-molecule microscopy reveals plasma membrane mi- crodomains created by protein-protein networks that exclude or trap signaling molecules in T cells. Cell. 121:937–950. [276] Varma, R., G. Campi, T. Yokosuka, T. Saito, and M. L. Dustin. 2006. T cell receptor-proximal signals are sustained in peripheral microclusters and terminated in the central supramolecular activation cluster. Immunity. 25:117–127. 125 Appendix A Matched asymptotic solution for multiple stationary TCR clusters (Chapter 2) In this section we use matched asymptotic analysis to solve for t2 (equation 2.17b) in the limit that b → 0. We begin by rescaling the equation for t2 using u = t2/t∗ and x = r/Rs. After some rearranging we obtain, pikoffDP t ∗ konNT ∇2u = −R 2 s b2 I (A.1) where we have used the fact that Tmc = NT /pib2. We identify the dimensionless quantity b/Rs with a small parameter = b/Rs and define t∗ = konNT /pikoffDP . We obtain the following equation that we wish to solve in the limit → 0: ∇2u = − 1 2 I(x; x1...xN) (A.2) with u(|x| = 1) = 0. The indicator function (equation 2.16) becomes, I(x; x1, ...xN ) = 1 | x− xj |≤ 0 | x− xj |> . Inner solution In the region near the the jth TCR microcluster we introduce the local space variable y = (x− xj)/. Consequently we write uin(y) = u(xj + y) to obtain the governing equation near the j’th cluster, ∇2yuinj = −1 | y |≤ 1 0 | y |> 1 . (A.3) We expand the inner solution in powers of v = −1/log(), uinj = uj,-1v −1 + uj,0 + ∑ i=1 viuj,i (A.4) Appendix A. Matched asymptotic solution for multiple stationary TCR clusters (Chapter 2) 126 where the i subscript indicates the order of the solution. Substituting eqn. A.4 into eqn. A.3 and matching orders of v we obtain, v−1; ∇2yuj,-1 = 0 v0; ∇2yuj,0 = −1 | y |≤ 1 0 | y |> 1 vi; ∇2yuj,i = 0, i ≥ 1 (A.5) The solution to the i = 0 equation is (up to an additive constant), uj,0 = − 12 log | y | | y |> 1 1 4 (1− y2) | y |≤ 1 The solutions to all other orders are uj,i = c1log | y | +Aj,i (i 6= 0). We set c1 = 0 to avoid blow up. In summary, the solution to the inner problem around the jth TCR microcluster is uinj = Aj,-1v −1 + uj,0 +Aj,0 + ∑ i=1 viAj,i (A.6) As | y |→ ∞ the inner solution becomes uinj = Aj,-1v −1 − 1 2 log | x− xj | +1 2 log() +Aj,0 + ∑ i=1 viAj,i. (A.7) The unknown constants Aj,i will be determined by matching this exact inner solution with the outer solution. Outer solution The outer problem is ∇2uout = 0 , x ∈ Ω\ {∑ i=1 | x− xj |≤ } (A.8) with uout(| x |= 1) = 0 and uout → −(1/2)log(| x− xj |) as x → xj. The related Green’s function problem is, ∇2Ĝ = δ(x− xk), Ĝ(| x |= 1) = 0 (A.9) Using the method of images the solution can be written as Ĝ(x, xk) = G(x, xk) +R(x, xk), where (A.10) G(x, xk) = 1 2pi log | x− xk | and R(x, xk) = − 1 2pi log [ | x | xk |2 −xk | | xk | ] . Appendix A. Matched asymptotic solution for multiple stationary TCR clusters (Chapter 2) 127 R is known as the regular part of the Green’s function. We see that we obtain the required log dependence and consequently reformulate the outer problem as ∇2uout = −pi ∑ k=1 δ(x− xk) , x ∈ Ω (A.11) with uout(| x |= 1) = 0. We write the solution in terms of the Green’s function uout = −pi ∑ k=1 Ĝ(x, xk). (A.12) As x → xj the solution becomes uout = −1 2 log | x− xj | −piR(xj, xj)− pi ∑ k=1 k 6=j Ĝ(xj, xk). (A.13) Matching condition We can now match the inner expansion (equation A.7) with the outer expansion (equation A.13) to all orders of v. We obtain Aj,-1 = 1/2, Aj,0 = −pi R(xj, xj)− pi ∑ k=1,k 6=j Ĝ(xj, xk) and Aj,i = 0 for i ≥ 1. Therefore the infinite expansion in the inner region reduces to just two terms. Mean escape time We compute the mean of u by averaging over the whole domain. < u > = 1 |Ω| ∫∫ Ω (1− I)uout + I N∑ j=1 uinj dx where the outer solution, equation A.12, is valid outside clusters, and the inner solution, equation A.6, is valid within clusters. Rearranging and computing the integral we obtain < u > = 1 |Ω| ∫∫ Ω uout + 1 |Ω| ∫∫ Ω I N∑ j=1 uinj − uout dx = 1 4 N∑ k=1 ( 1− |xk|2 ) + ϑ(2) (A.14) We show how to explicitly compute the first integral below. The second integral is of order 2 because of the appearance of the indicator function, I. Finally, we multiply< u > by t∗ to obtain the unscaled quantity< t2 >. Assuming the initial position of the particle is within one of the N microclusters we may compute < u >: < uin > = 1 Npi2 ∫∫ Ω I N∑ j=1 uinj After computing the integral and unscaling the equation we obtain equation 2.20. Appendix A. Matched asymptotic solution for multiple stationary TCR clusters (Chapter 2) 128 Mean escape time integral In this section we evaluate the first integral in equation A.14. After substituting the outer solution, equation A.12, we obtain < u > = − pi|Ω| N∑ k=1 ∫∫ Ω Ĝ (x, xk) = − pi|Ω| N∑ k=1 [ L1k + L 2 k ] (A.15) where L1k = 1 2pi ∫∫ Ω log|x− xk|dx and L2k = − 1 2pi ∫∫ Ω log ∣∣∣∣|xk|x− xk|xk| ∣∣∣∣ dx. (A.16) We first calculate L2k . Without loss of generality we assume that xk lies on the positive x-axis and therefore we let x = reiθ r̂ and xk = ρkr̂. With these we find, L2k = − 1 2pi ∫ 2pi 0 dθ ∫ 1 0 drlog|ρkreiθ − 1| (A.17) Next we let z = eiθ to obtain a computable contour integral, L2k = − 1 2pi ∫ 1 0 rdr ∫ |z|=1 dz −ilog|ρkrz − 1| z = − 1 2pi ∫ 1 0 rdr (−i2piiRes(z = 0)) = − 1 2pi ∫ 1 0 rdr (2pilog| − 1|) = 0 The only singularity for |z| ≤ 1 occured where z = 0 and consequently we find L2k = 0. To evaluate L1k , we make the same substitutions for x and xk to obtain L1k = 1 2pi ∫ 2pi 0 dθ ∫ 1 0 drlog|reiθ − ρk|. Here we cannot make the complex substitution directly since the log term will have a singularity within the contour integral. Instead, we rearrange the integrand to obtain log|reiθ − ρk| = 1 2 log(r2 + ρ2k) + 1 2 log(1− µ cos θ) (A.18) where µ = 2rρk/(r2 + ρ2k). Substituting and performing the integration over θ we obtain L1k = 1 2pi ∫ 1 0 rdr [ pilog(r2 + ρ2k) + pilog ( 1 + √ 1− µ2 2 )] Appendix A. Matched asymptotic solution for multiple stationary TCR clusters (Chapter 2) 129 where we have used the identity ∫ 2pi 0 log (1− µ cos θ) dθ = 2pilog ( 1 + √ 1− µ2 2 ) . Next we note that 1 + √ 1− µ2 2 = r2 r2+ρ2k r > ρk ρ2k r2+ρ2k r < ρk so we can establish L1k = 1 2pi [ pilog(ρ2k) ∫ ρk 0 rdr + pi ∫ 1 ρk rlog(r2)dr ] = 1 4 (ρ2k − 1). (A.19) Finally, we combine the results for L1k and L2k to obtain < u > = −1 4 N∑ k=1 [|xk|2 − 1] (A.20) where we have used |Ω| = pi and ρk = |xk|. 130 Appendix B Monte Carlo simulation of pMHC dynamics with multiple mobile TCR clusters (Chapter 2) In this appendix we develop an agent-based Monte Carlo simulation that models the formation and mobility of multiple TCR clusters, the diffusion of pMHC, and the interaction between pMHC and clustered TCR in the immune synapse (IS). Although we do not explicitly simulate individual binding events between pMHC and TCR/coreceptors in clusters, the model is fully stochastic (details below). B.1 Model of pMHC and TCR cluster dynamics Diffusion module We model the diffusion of a single pMHC on a lattice where the rate of hopping between the four nearest neighbour sites is D/h2 (see Isaacson and Peskin (277)), where D is the diffusion coefficient and h is the lattice spacing. We take h = 0.01µm, approximately the size of pMHC or TCR in the membrane, in order to ensure that we capture both reaction limited and diffusion limited regimes. Our domain is the immune synapse which is a disc of radius Rs = 6µm, see figure B.1. However, diffusing pMHC on the APC or a supported planar bilayer can diffuse beyondRs and subsequently re-enter the synapse. To allow for synapse re-entry, we terminate the simulation once the pMHC has reached a radius of Re = 7µm. The accumulation of pMHC in the cSMAC suggests that pMHC become trapped in the cSMAC upon entry (278; 279; 280). We therefore also terminate the simulation if the pMHC has reached the cSMAC, which has a radius of Rc = 1µm (278). In summary, we simulate the diffusion of pMHC in a large annulus and terminate the simulation once the pMHC has diffused out of the annulus at Rc or Re, see figure B.1. TCR cluster module We model multiple TCR clusters (of radius b) that are stochastically advecting along actin filaments towards the cSMAC. In the immune synapse, actin filaments have been shown to be radially organized and to be undergoing treadmilling, presumably moving TCR clusters to the cSMAC (281). The stochasticity in Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 131 V b 1. 2. 3. 4.5. θ y xRc Rp Rs Re Figure B.1: Illustration of model geometry and the interaction between diffusing pMHC and advecting TCR clusters (not to scale). TCR clusters are continually generated in a random location between Rp and Rs and advect with velocity V towards the cSMAC center (the origin). 1. Simulations begin with the pMHC at a radius of 5 µm. 2. Diffusion of pMHC and/or advection of TCR clusters may result in the pMHC entering a cluster. Once inside a cluster, the pMHC may bind to TCR/coreceptors. Upon binding, the pMHC remains in the cluster for a period of time determined by ψ during which it is carried by the cluster towards the cSMAC. 3. Upon unbinindg, the pMHC exits the cluster at an angle determined from χ and resumes free diffusion. 4. It may subsequently enter and bind to TCR/coreceptors in a different cluster. 5. The simulation is terminated once the pMHC has entered the cSMAC. TCR cluster velocity in our model implicitly captures the stochasticity in actin treadmilling and binding events between a TCR cluster and actin. The rate of advection in our model is given by V/h, where V is the effective TCR cluster velocity and h is the lattice spacing. TCR clusters exit our simulation once they fully enter the cSMAC. The formation of new TCR clusters is observed in the periphery of the immune synapse (279; 280; 282) and we therefore simulate the formation of new TCR clusters, with a rate kmc, in a random location between the synapse boundary Rs = 6µm and Rp = 5µm. We denote the total number of TCR clusters in the synapse at any given time as N . In summary, this module generates new TCR clusters in an annulus in the periphery of the immune synapse (between Rp and Rs) that undergo stochastic directed motion towards the cSMAC where they exit from our simulation. Full simulation Combining both the pMHC diffusion module and the TCR cluster module we can now consider reactions whenever the pMHC diffuses into a TCR cluster. Contained within the TCR cluster is some number of Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 132 TCR and coreceptors that are able to bind pMHC. Since these reactions have very fast kinetics we do not resolve individual binding events in order to create efficient simulations. Instead, once the pMHC has bound a molecule in the TCR cluster, we solve the relevant partial differential equation (PDE) to determine the cumulative probability distribution of escape times, ψ(t) (i.e. how long the pMHC has spent in the cluster) and escape angles, χ(t, θ) (i.e. where does it pop out of the cluster). These distributions depend on the initial state of the pMHC (bound to TCR or coreceptor), the initial position within the cluster, the concentration of TCR (Tmc) and coreceptors (Cmc), the TCR cluster radius (b), and the reaction rates between pMHC and TCR/coreceptors. Once initially bound in a TCR cluster, we generate two random numbers which determine the escape time, τesc, and escape angle, θesc, based on the distributions ψ and χ. In the sections below we show to evaluate ψ and χ. Summarizing, the diffusing pMHC may enter and bind a molecule in a TCR cluster. Once bound the pMHC advects with the cluster towards the cSMAC for a certain period of time, stochastically determined by the distribution ψ. In figure B.1 we illustrate a sequence events that a diffusing pMHC may undergo. By allowing the pMHC to interact with multiple TCR clusters, the purely diffusive motion of the pMHC is biased towards cSMAC arrival by the motion of TCR clusters. B.1.1 Simulation Engine The main engine of our simulation is the next reaction method first proposed by Gillespie (283). We begin the main loop by first examining the state of the pMHC to determine all possible reactions, see table B.1, from which we determine the overall rate a. Based on a we determine the time for the next reaction as, τnext = 1 a ln ( 1 r1 ) where r1 is a uniformly distributed random number between 0 and 1. If the pMHC is unbound, we proceed to use a second random number, r2, to determine which of the possible reactions took place. We simulate the reaction, advance the simulation time, tc, by τnext, and return to the start of the loop. A flow chart of the simulation is shown in figure B.2. If a binding event takes place we compute τesc using ψ and store the time at which the pMHC is expected to exit the cluster, te = tc+ τesc. After each computation of τnext we check if the pMHC has escaped the TCR cluster (i.e. tc + τnext > te). If pMHC escape has occured, we simply set tc = te and return to the start of the loop. In each realization of the simulation we collect statistics on the number of clusters the pMHC has visited, the number of TCR engagements, and whether the pMHC has arrived in the cSMAC. The number of TCR engage- ments is calculated using equations 2.7, where < t > is replaced by the stochastically determined escape time, τesc. In order to collect sufficient statistics, we run the simulation for a given parameter set at least 500 times. For a discussion of the parameters see main text. Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 133 Table B.1: The state of pMHC determines the possible reactions pMHC state unbound unbound within a cluster bound within a cluster Possible reactions diffusion diffusion cluster advection cluster advection cluster advection cluster formation cluster formation cluster formation unbinding binding Overall rate a = 4D/h2 +NV/h a = 4D/h2 +NV/h a = NV/h+ kmc +kmc +kmc + λ1 + λ−4 pMHC boundpMHC unbound tc + τnext < te tc + τnext > te simulate reaction tc = tc + τnext tc = te Determine pMHC State (compute τnext) Determine pMHC State (compute τnext) Figure B.2: Flow chart illustrating the main loop in the simulation B.2 Determining ψ and χ B.2.1 PDE governing pMHC dynamics in a TCR cluster In order to carry out our simulations we need to compute the cumulative distributions for pMHC escape time, ψ(t), and escape angle, χ(t, θ). The PDE governing pMHC dynamics within a TCR cluster is, ∂fA ∂t = λ4fD − λ−4fA − λ1fA + λ−1fB +D∇2fA, ∂fB ∂t = λ1fA − λ−1fB − λ2fB + λ−2fC, ∂fC ∂t = λ2fB − λ−2fC − λ3fC + λ−3fD, ∂fD ∂t = λ3fC − λ−3fD − λ4fD + λ−4fA, (B.1) where the fi are the probability density functions for each state, λi are the transition rates, and D is the pMHC diffusion coefficient. We assume a disc domain of radius r = a and take the boundary condition to be fA(r = Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 134 b) = 0. The initial conditions for the fi are, fA(t = 0) = g(r, θ)ΓA fB(t = 0) = g(r, θ)ΓB fC(t = 0) = g(r, θ)ΓC fD(t = 0) = g(r, θ)ΓD (B.2) where, g(r, θ) = δ(r − r0)δ(θ) r (B.3) We take the initial position of the particle (i.e. pMHC) to be (r, θ) = (r0, 0) without loss of generality. The values of Γi indicate the initial state of the particle. For example, if Γ = (ΓA,ΓB,ΓC,ΓD) = (0, 1, 0, 0) then the particle begins bound to TCR in state i = B. The probability that the particle has escaped the cluser is ψ(t) = 1− P (t), where P (t) is the probability that the particle is in the domain. This latter quantity is obtained by simply integrating the solution of equations B.2 over the whole domain, P (t) = ∫ a 0 rdr ∫ 2pi 0 dθ (fA + fB + fC + fD) (B.4) In the last section of this appendix we show how to efficiently evaluate P (t). Given an escape time, we now wish to determine the probability distribution for escape angles, χ(t, θ). We define the escape angle, θ, relative to the initial angle which we take to be 0, see step 3 in figure B.1. We define p(t, θ) to be the flux through the boundary, p(t, θ) = −D (∇fA · dr̂) |r=b (B.5) Note that only fA appears in this equation because in all other states the particle is bound and hence immobile. The probability that the pMHC has escaped at an angle θ at time t is then given by, χ(t, θ) = ∫ θ 0 p(t, θ ′ )dθ ′ ∫ 2pi 0 p(t, θ′)dθ′ (B.6) In the last section of this appendix we show how to evaluate this quantity. Once the pMHC has bound a molecule in a TCR cluster we computeψ(t) using information about the binding event (e.g. location within the cluster, reaction rates of the simulation, etc). We generate a random number, r3, Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 135 and solve the equation r3 = ψ(τesc) for τesc. Next we generate another random number, r4, and solve the equation r4 = χ(τesc, θesc) for θesc. In the sections that follow we show to to solve equations B.1 and how this solution is used to efficiently evaluate ψ and χ. Using this evaluation method we use the Matlab function fzero to solve for τesc and θesc in the above equations. B.2.2 Laplace transform solution of the PDE system We begin by finding a solution to equations B.1 with initial conditions B.2 and boundary condition fA(r = b) = 0. An explicit solution will not be possible and instead we will find a solution in the transformed space which we can numerically invert. We begin by taking the Laplace transform of equations B.1 and denoting the Laplace variable as s and the transformed concentrations as Fi, sFA − fA(t = 0) = λ4FD − λ−4FA − λ1FA + λ−1FB +D∇2FA, sFB − fB(t = 0) = λ1FA − λ−1FB − λ2FB + λ−2FC, sFC − fC(t = 0) = λ2FB − λ−2FC − λ3FC + λ−3FD, sFD − fD(t = 0) = λ3FC − λ−3FD − λ4FD + λ−4FA, (B.7) We can use the last three equations to solve for FB,FC, and FD in terms of FA. We obtain, FB = Λ 1 BFA + Λ 2 Bg(r, θ), FC = Λ 1 CFC + Λ 2 Cg(r, θ), FD = Λ 1 DFD + Λ 2 Dg(r, θ), (B.8) where Λ variables are rational functions of the Laplace variable s. They are explicitly given in the end of this appendix. We can now subsitute FB and FD (from equations B.8) into the equation for FA (first equation in B.7) to obtain (after some rearrangements), ∇2FA − µ2FA = −σδ(r − r0)δ(θ) r (B.9) where, σ = (ΓA + λ4Λ 2 D + λ−1Λ 2 B)/D (B.10) µ2 = (s+ λ1 + λ−4 − λ4Λ1D − λ−1Λ1B)/D (B.11) Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 136 We write the Laplacian for polar coordinates and use the variable subsitution z = µr to obtain, z2 ∂2FA ∂z2 + z ∂FA ∂z + ∂2FA ∂θ2 − z2FA = −σzδ(z− z0)δ(θ) (B.12) where z0 = µr0. We use the method of eigenfunction expansion to write down a plausible solution for FA, FA = ∑ n [Vn(z) sin(nθ) + Un(z) cos(nθ)] (B.13) Subsituting this equation into equation B.12 and using the orthogonality property of the eigenfunctions we obtain two equations, z2V ′′n + zV ′ n − (z2 + n2)Vn = 0 (B.14) z2U ′′n + zU ′ n − (z2 + n2)Un = − σzδ(z− z0) αn (B.15) where, αn = 2pi n = 0 pi n > 0 . The two conditions on equations B.14 and B.15 are (1) solutions remain finite on the domain and (2) solutions are zero on the domain boundary (r = b), za = µb. The solutions to equation B.14 are modified Bessel functions of the first and second kind. After applying the two conditions we find that Vn = 0 for all n. The homogeneous solutions of equation B.15 are also modified Bessel functions,CnIn(z) + DnKn(z). To obtain the particular solution of equation B.15 we use the method of variation of parameters, Un(z) = Xn(z)In(z) + Yn(z)Kn(z) (B.16) where Xn(z) and Yn(z) are determined by substituting the above expression into equation B.15. The ingredients required for this substitution are, U ′n = XnI ′ n + YnK ′ n U ′′n = X ′ nI ′ n + XnI ′′ n + Y ′ nK ′ n + YnK ′′ n and as usual we have set, X′nIn + Y ′ nKn = 0 (B.17) Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 137 Carrying out the substitution followed by some rearrangements we find, X′nI ′ n + Y ′ nK ′ n = − σδ(z− z0) zαn (B.18) Together, equations B.17 and B.18 form a system of equations for the derivatives of our unknown functions. We integrate to find, Xn(z) = −σKn(z0) αn H(z− z0) (B.19) Yn(z) = σIn(z0) αn H(z− z0) (B.20) where H is the Heaviside step function. We can now write the full solution to equation B.15, Un(z) = CnIn(z) + DnKn(z) + Xn(z)In(z) + Yn(z)Kn(z) We set Dn = 0 for all n to obtain a finite solution. Applying the condition that Un(za) = 0 we find, Cn = σ [In(za)Kn(z0)− In(z0)Kn(za)] αnIn(za) (B.21) Finally, we have all necessary quantities to write the solution for FA, FA = ∑ n (CnIn(z) + Xn(z)In(z) + Yn(z)Kn(z)) cos(nθ) (B.22) We obtain FB, FC, and FD using equation B.8. B.2.3 Rapid evaluation of ψ and χ Recall that ψ(t) = 1 − P (t) and we therefore begin by evaluating P (t) as defined in equation B.4. Substituting in the transformed quantities and using the change of variables z = µr we obtain, P (t) = ∫ a 0 rdr ∫ 2pi 0 dθ (fA + fB + fC + fD) = L−1 { (1/µ2) ∫ za 0 zdz ∫ 2pi 0 dθ (FA + FB + FC + FD) } where L−1 is the inverse Laplace transform. Using equations B.8 we substitute all transformed varialbes in terms of FA and subsequently use the solution for FA (equation B.22) to simplify the expression. This simplification Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 138 requires some tedious algebra along with the use of the following identities, zI1 = ∫ zI0dz −zK1 = ∫ zK0dz InKn+1 + In+1Kn = 1/z We finally arrive at the following simple expression, P (t) = L−1 {( Λ2B + Λ 2 C + Λ 2 D ) + σ µ2 ( 1 + Λ1B + Λ 1 C + Λ 1 D )( 1− I0(z0) I0(za) )} (B.23) Using a numerical inverse Laplace transform in Matlab we can invert this quantity in under 1 millisecond (AMD Athlon) and find ψ(t) = 1− P (t). We now wish to evaluate χ(t, θ). We begin by evaluating p(t, θ) using equation B.5. Substituting in the transformed quantities and using the change of variables z = µr we obtain, p(t, θ) = −D (∇fA · dr̂) |r=b = L−1 { −D ∂FA ∂r ∣∣∣∣ r=b } = L−1 { −Dµ ∂FA ∂z ∣∣∣∣ z=za } . (B.24) Taking the derivative of equation B.22, using the Bessel function identities quoted above, along with the following ones, I′n = In+1 + n z In K′n = −Kn+1 + n z Kn , (B.25) we can obtain, after some tedious algebra, the following simple expression p(t, θ) = L−1 { Dσµ za ∑ n In(z0) αnIn(za) cos(nθ) } . (B.26) Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 139 Subsituting this expression into equation B.6 we obtain, χ(t, θ) = [∑ n ( L−1 { Dσµ a In(z0) In(za) })( sin(nθ) nαn )] upslope [( L−1 { Dσµ b I0(z0) I0(za) })( 2pi α0 )] (B.27) Once again, using a numerical inverse Laplace transform routine we can invert this expression in less than 1 ms. B.2.4 Definitions of Λ Λ1B = [ λ1s 2 + (λ4λ1 + λ1λ−3 + λ3λ1 + λ−2λ1)s+ λ−3λ−2λ−4 + λ3λ1λ4 + λ1λ−2λ4 + λ1λ−2λ−3 ] /Λ Λ2B = [ ΓBs 2 + (λ3ΓB + ΓBλ−3 + λ−2ΓC + λ−2ΓB + ΓBλ4)s+ λ3ΓBλ4 + ΓBλ−2λ−3 + λ−3λ−2ΓD+ λ−2λ−3ΓC + λ−2λ4ΓC + ΓBλ−2λ4] /Λ Λ1C = [(λ−3λ−4 + λ1λ2)s+ λ−3λ−1λ−4 + λ−3λ1λ2 + λ−3λ2λ−4 + λ4λ1λ2] /Λ Λ2C = [ ΓCs 2 + (λ−3ΓD + ΓCλ−1 + λ−3ΓC + ΓCλ2 + λ4ΓC + ΓBλ2)s+ λ4ΓBλ2 + λ−3ΓBλ2 + λ−3λ−1ΓD+ λ4ΓCλ−1 + λ4ΓCλ2 + λ−3ΓCλ2 + λ−3ΓCλ−1 + λ−3λ2ΓD] /Λ Λ1D = [ λ−4s2 + (λ−4λ−1 + λ−4λ3 + λ−2λ−4 + λ−4λ2)s+ λ3λ−1λ−4 + λ3λ2λ−4 + λ−2λ−1λ−4+ λ1λ3λ2] /Λ Λ2D = [ ΓDs 2 + (λ2ΓD + λ−2ΓD + ΓDλ−1 + λ3ΓD + ΓCλ3)s+ λ3λ2ΓD + ΓCλ3λ2 + ΓCλ3λ−1+ λ−2λ−1ΓD + ΓBλ3λ2 + λ3λ−1ΓD] /Λ where, Λ = s3 + (λ3 + λ−2 + λ−3 + λ4 + λ2 + λ−1)s2 + (λ−2λ−3 + λ−2λ4 + λ4λ3 + λ2λ−3+ λ4λ2 + λ4λ−1 + λ−3λ−1 + λ3λ2 + λ3λ−1 + λ−2λ−1)s+ λ3λ−1λ4 + λ−2λ−1λ−3+ λ−2λ−1λ4 + λ3λ2λ4 (B.28) B.3 Connection between P (t), p(t, θ), and MFPT Given that earlier sections in Chapter 2 were concerned with calculating the mean escape time from a TCR cluster it is worth pointing out the connection between the quantities derived above and the mean escape time. Consider a particle released at a specific location within our domain and that we measure how long it takes for the particle to first reach the boundary. Repeating this exact experiment many times will produce a distribution, p̄(t), of escape Appendix B. Simulation of pMHC dynamics with mobile TCR clusters (Chapter 2) 140 times (or first passage times) due to stochasticity in the diffusion process. The mean escape time (or mean first passage time (MFPT)) can be obtained from p̄(t) as follows, τ = ∫ ∞ 0 tp̄(t)dt (B.29) where τ is the mean escape time and is equal to tA, as defined in the main text, when there are no reactions. As defined above, P (t), is the probability that the particle has not escaped the domain at time t. The distribution of escape times, p̄(t), is related to P (t) as follows, p̄(t) = −∂P (t) ∂t (B.30) By multiplying equation B.30 by t and integrating from 0 to ∞ we find, τ = ∫ ∞ 0 P (t)dt where we have used the fact that p̄(0) = 0 (i.e. probability of escape is zero at t = 0) and P (0) = 1 (i.e at t = 0 the particle is in the domain). Lastly, we can obtain p̄ by integrating p(t, θ) over θ from 0 to 2pi. 141 References [277] Isaacson, S. A., and C. S. Peskin. 2006. Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations. SIAM J. Sci. Comp. 28:47–74. [278] Grakoui, A., S. K. Bromley, C. Sumen, M. M. Davis, A. S. Shaw, P. M. Allen, and M. L. Dustin. 1999. The immunological synapse: a molecular machine controlling t cell activation. Science. 285:221–227. [279] Yokosuka, T., K. Sakata-Sogawa, W. Kobayashi, M. Hiroshima, A. Hashimoto-Tane, M. Tokunaga, M. L. Dustin, and T. Saito. 2005. Newly generated t cell receptor microclusters initiate and sustain t cell activation by recruitment of zap70 and slp-76. Nat Immunol. 6:1253–1262. [280] Varma, R., G. Campi, T. Yokosuka, T. Saito, and M. L. Dustin. 2006. T cell receptor-proximal signals are sustained in peripheral microclusters and terminated in the central supramolecular activation cluster. Immu- nity. 25:117–127. [281] Kaizuka, Y., A. D. Douglass, R. Varma, M. L. Dustin, and R. D. Vale. 2007. Mechanisms for segregating t cell receptor and adhesion molecules during immunological synapse formation in jurkat t cells. Proc Natl Acad Sci U S A. 104:20296–20301. [282] Campi, G., R. Varma, and M. L. Dustin. 2005. Actin and agonist mhc-peptide complex-dependent t cell receptor microclusters as scaffolds for signaling. J Exp Med. 202:1031–1036. [283] Gillespie, D. 1977. Exact stochastic simulations of coupled chemical reactions. J. Phys. Chem. 81:2340– 2361. 142 Appendix C Analysis of errors due to geometric approximations (Chapter 4) There are two subtleties with the application of equation 4.6 to the geometry of protocol 2. First, the cell radius must be determined. This is a trivial task since, as shown in figure 4.1, a ring of fluorescence from the labeled surface molecule is observed at the equatorial plane of the cell. The cell diameter can be determined from this ring using most image editing software. Since equations 4.7 vary slowly withR, we find that a 10% error in estimating R gives an error of 3% in the diffusion coefficient when using equation 4.6, see figure C.1a. Therefore negligible error is expected to arise in D from inaccuracies in determiningR. Figure C.1: Potential source of error in using the 2D model. Percent differences (defined in figure 4.3) in D as a result of estimating the cell radius are shown as contours. Data was generated using equation 5.9 at various cell radii (Actual R) and were fit using a different cell radius (Estimated R). The bleached and monitoring region parameters, L, were calculated in each case using equations 4.7 with Wx = 2µm, Wy = 2µm, and Wz = 2µm. The second subtlety has due to with approximating the bleached and monitoring regions as perfect rectangles on the surface of the cell. Protocol 2 assumes that a perfect rectangle (of area Ar = LxbLyb) is bleached on the surface of the cell. However, although Lxb represents the curved length at the middle of the stripe, this distance changes throughout the stripe width, Lyb. It maximally differs from Lxb on the edges of the bleached stripe. Appendix C. Analysis of errors due to geometric approximations (Chapter 4) 143 Taking into account this curvature, we calculate the area of the bleached stripe as, Ac = ∫ pi 2 +arcsin ( Wy 2R ) pi 2 −arcsin ( Wy 2R ) 2R sin(φ) arccos ( 1− Wx R sin(φ) ) Rdφ (C.1) The total area of the bleached stripe when we assume a perfect rectangular stripe (neglecting curvature effects) on the surface of the cell is, Ar = LxbLyb = 4R 2 arccos(1−Wx/R) arcsin(Wy/2R) (C.2) We compare these values for a large range of Wx, Wy, andR = 5µm. We find that the difference betweenAr and Ac is negligible (< 0.1%) for all experimental conditions with a large cushion and that increasing R reduces this difference. 144 Appendix D Asymptotic reductions of the full model (Chapter 5) D.1 Full model The full model, described by equations 5.3 and 5.4, has four relevant timescales; two diffusive timescales (τf = L2/Df, τc = L 2/Dc) and two reaction timescales (1/k∗on, 1/koff) . Using formal asymptotic analysis, we demon- strate below, that when there exists a separation of timescales the full model can be reduced to a simpler model consisting of only simple diffusion equations. We non-dimensionalize the full model with the transformations: f̂ = f/Feq, ĉ = c/Ceq, x̂ = x/L, t̂ = t/τ , resulting in the following nondimensional form of the full model: ∂f̂/∂t̂ = τDf L2 ∇2f̂ + τk∗on(ĉ− f̂), (D.1a) ∂ĉ/∂t̂ = τDc L2 ∇2ĉ+ τkoff(f̂ − ĉ). (D.1b) In this form, all timescales are explicit and all terms are of order one. The length scale L is determined from the experimental setup; for example, an appropriate choice is the dimension of the bleaching region. Intuitively, at equilibrium, we expect the labeled ligand to be in a free state (corresponding to the diffusion coefficient Df) for a fraction koff/(k∗on + koff) of the time, and in the bound complex (with diffusion coefficient Dc) for a fraction k∗on/(k ∗ on + koff). Therefore, an appropriate choice of timescale for the full system is: τ = L2/ ( koff k∗on + k∗on Df + k∗on k∗on + koff Dc ) . = L2/ ( 1 1 +K∗ Df + K∗ 1 +K∗ Dc ) (D.2) where K∗ = k∗on/koff is a pseudo-affinity constant that will be the parameter of interest in the model reductions described below. Note that, by definition, Feq = 1/(1 +K∗), and Ceq = K∗/(1 +K∗) Appendix D. Asymptotic reductions of the full model (Chapter 5) 145 D.2 Dimensionless form of full model We now examine the dimensionless system (equation D.1) under different limiting conditions. To do so, we define a small parameter, as the ratio of two timescales, and expand the solutions in powers of this parameter, f̂ ∼ f̂0 + f̂1 + . . . (D.3a) ĉ ∼ ĉ0 + ĉ1 + . . . (D.3b) to obtain leading order equations for f̂0 and ĉ0. As described below, each reduction leads to a simple diffusion equation of the form ∂u ∂t = D∇2u (D.4) with a different definition of u, and the parameter D for each limiting case. This diffusion eqution can be solved using standard techniques, and for each limit, we sum the dimensional forms of the leading order solutions to obtain an expression for fluorescence intensity of the form: g(x, t,D) = f0(x, t) + c0(x, t) = koff k∗on + koff f̂0(x/L, t/τ) + k∗on k∗on + koff ĉ0(x/L, t/τ) (D.5) Integrating (and normalizing) g over the the monitoring region gives rise to a functionG that can be fit directly to FRAP data. D.3 Pure diffusion There are two parameter regimes where we obtain “pure diffusion” limits. The first is when the free pool is large, τfk ∗ on = , τfkoff = 1/ (D.6) In this case the timescale of the FRAP experiment reduces to τ = τf and equation D.1 reduces to, ∂(f̂0 + f̂1) ∂t̂ = ∇2(f̂0 + f̂1) + (ĉ0 + ĉ1 − (f̂0 + f̂1)) (D.7a) ∂(ĉ0 + ĉ1) ∂t̂ = τfτc∇2(ĉ0 + ĉ1) + (1/)(f̂0 + f̂1 − (ĉ0 + ĉ1)), (D.7b) Appendix D. Asymptotic reductions of the full model (Chapter 5) 146 whose first order terms are: 0 : ∂f̂0 ∂t̂ = ∇2f̂0 (D.8a) f̂0 = ĉ0. (D.8b) Transforming these equations back to their dimensional form we find, ∂f0 ∂t = Df∇2f0 (D.9) and, therefore, in this case, g = gpf = u(x, t,Df). The second “pure diffusion” regime occurs when the bound pool is large, τfk ∗ on = 1/, τfkoff = (D.10) and by analogy with the previous case, g = gpc = u(x, t,Dc). D.4 Weighted diffusion When the reaction timescales are fast compared to the diffusive timescales we obtain another reduction which we label as “weighted diffusion”. The appropriate definition for in this case is, τfk ∗ on = 1/ (D.11) We note that there is no separation of scale between the on- and off-rate (i.e. k∗on/koff = constant) and the two diffusion timescales (τf/τc = constant). The timescale τ does not simplify in this case. We rearrange the terms in the full system D.1 to obtain, ∂f̂ ∂t̂ = 1 +K∗ 1 +K∗(Dc/Df) ∇2f̂ + k∗onτf 1 +K∗ 1 +K∗(Dc/Df) (ĉ− f̂) (D.12a) ∂ĉ ∂t̂ = 1 +K∗ K∗ + (Df/Dc) ∇2ĉ+ k∗onτf 1 +K∗ K∗ + (Df/Dc) (f̂ − ĉ) (D.12b) where we can clearly identify the small parameter in the second term of each equation. We expand f̂ and ĉ to Appendix D. Asymptotic reductions of the full model (Chapter 5) 147 obtain, ∂(f̂0 + f̂1) ∂t̂ = 1 +K∗ 1 +K∗(Dc/Df) ∇2(f̂0 + f̂1) + k∗onτf 1 +K∗ 1 +K∗(Dc/Df) (ĉ0 + ĉ1 − (f̂0 + f̂1)) (D.13a) ∂(ĉ0 + ĉ1) ∂t̂ = 1 +K∗ K∗ + (Df/Dc) ∇2(ĉ0 + ĉ1) + k∗onτf 1 +K∗ K∗ + (Df/Dc) (f̂0 + f̂1 − (ĉ0 + ĉ1)) (D.13b) Equating terms with like powers of we obtain the following equations, 0 : f̂0 = ĉ0 (D.14a) 1 : ∂f̂0 ∂t̂ = 1 +K∗ 1 +K∗(Dc/Df) ( ∇2f̂0 + ĉ1 − f̂1 ) (D.14b) ∂ĉ0 ∂t̂ = 1 +K∗ K∗ + (Df/Dc) ∇2ĉ0 + 1 +K ∗ K∗ + (Df/Dc) (f̂1 − ĉ1) (D.14c) Eliminating f̂1 − ĉ1 and rearranging terms we obtain, ∂f̂0 ∂t̂ = ∇2f̂0 (D.15) recast in the dimensional form to, ∂f0 ∂t = ( 1 1 +K∗ Df + K∗ 1 +K∗ Dc ) ∇2f0 = Dwd∇2f0 (D.16) where we have defined an effective diffusion coefficient Dwd = ( 1 1+K∗Df + K∗ 1+K∗Dc ) , that is a weighted av- erage of the individual diffusion coefficients. Since f̂0 = ĉ0 we determine that g = gwd = u(x, t,Dwd). Thus, in this parameter regime we ultimately obtain a simple diffusion equation with the effective diffusion coefficient, Dwd. This regime is equivalent to the “pseudo-effective diffusion” regime of Braga et al. (? ). D.5 Independent diffusion Finally, when the reaction timescales are slow compared to the diffusive timescales we obtain another reduction which we label as “independent diffusion”. The appropriate definition for in this case is, τfk ∗ on = (D.17) As in the weighted diffusion reduction, we do not assume any scale separation within the reaction or diffusion timescales and no simplification to τ is possible. We identify the smaller parameter in equation D.12 and subsitute Appendix D. Asymptotic reductions of the full model (Chapter 5) 148 equation D.3 to obtain, ∂(f̂0 + f̂1) ∂t̂ = 1 +K∗ 1 +K∗(Dc/Df) ∇2(f̂0 + f̂1) + 1 +K ∗ 1 +K∗(Dc/Df) (ĉ0 + ĉ1 − (f̂0 + f̂1)) (D.18a) ∂(ĉ0 + ĉ1) ∂t̂ = 1 +K∗ K∗ + (Df/Dc) ∇2(ĉ0 + ĉ1) + 1 +K ∗ K∗ + (Df/Dc) (f̂0 + f̂1 − (ĉ0 + ĉ1)) (D.18b) The leading order equations are simply, 0 : ∂f̂0 ∂t̂ = τDf∇2f̂0 (D.19a) ∂ĉ0 ∂t̂ = τDc∇2ĉ0 (D.19b) rewritten in their dimensional form as the following system of uncoupled diffusion equations, ∂f0 ∂t = Df∇2f0 (D.20a) ∂c0 ∂t = Dc∇2c0 (D.20b) and the function g is the weighted sum of the solutions, g = gid = (1/(1 + K∗))u(x, t,Df) + (K∗/(1 + K∗))u(x, t,Dc). In other words, in this parameter regime there is effectively no exchange between the free and bound pool on the timescale of the FRAP experiment, and therefore the two pools diffuse independently. D.6 Reaction dominant In the above asymptotic reductions we have not assumed any scale separation betweenDf andDc. We can recover the results of Sprague et al. (284), who analyzed the reaction-diffusion system in equation D.1 withDc = 0, when we assume a large scale separation between Df and Dc. Defining as, τfk ∗ on = , τf/τc = 2 (D.21) and taking the timescale to be, τ = 1/koff, we can reduce equation D.1 to, ∂f̂/∂t̂ = 1koffτf∇2f̂ + k∗on/koff(ĉ− f̂) (D.22) ∂ĉ/∂t̂ = 1koffτc∇2ĉ+ (f̂ − ĉ) (D.23) Appendix D. Asymptotic reductions of the full model (Chapter 5) 149 Identifying the small parameter, we arrive at the following equations, ∂(f̂0 + f̂1) ∂t̂ = 1∇2(f̂0 + f̂1) + k∗on/koff(ĉ− (f̂0 + f̂1)) (D.24) ∂(ĉ0 + ĉ1) ∂t̂ = ∇2(ĉ0 + ĉ1) + ((f̂0 + f̂1)− (ĉ0 + ĉ1)) (D.25) The leading order equations are simply, 0 : ∇2f̂0 = 0 (D.26a) ∂ĉ0 ∂t̂ = f̂0 − ĉ0 (D.26b) Solving these equations we recover the reaction dominant solution presented in (284), GRD(t) = 1− K ∗ 1 +K∗ e−kofft (D.27) Note: This suggests that for the approximationDc = 0 to be valid, Dc/Df must be O(2). Table D.1 below summarizes the small parameters for each model reduction. Table D.1: Summary of Asymptotic Limits Reduction Small Parameter Definition Pure Diffusion (free pool) τfk∗on = , τfkoff = 1/ Pure Diffusion (bound pool) τfk∗on = 1/, τfkoff = Weighted Diffusion τfk∗on = 1/ Independent Diffusion τfk∗on = Reaction Dominant τfk∗on = , τf/τc = 2 150 References [284] Sprague, B. L., R. L. Pego, D. A. Stavreva, and J. G. McNally. 2004. Analysis of binding reactions by fluorescence recovery after photobleaching. Biophys J. 86:3473–3495. 151 Appendix E Details of model fitting to simulated data (Chapter 5) Table E.1: Model Fitting to Simulated Experiment A Model SSR AIC ∆AIC p Fitted Parameters ST = 1µm −2 WD 0.02469 -828.64 2.93 0.24 Dwd = 0.4795µm2/s ID 0.02398 -831.57 0.00 0.63 K∗ = 0.0160 FM 0.02450 -827.41 4.15 0.14 k∗on = 3.162× 10−5s−1 koff = 1.774× 10−3s−1 ST = 40µm −2 WD 0.1441 -652.25 181.28 0.00 Dwd = 0.1600µm2/s ID 0.02352 -833.53 0.00 0.52 K∗ = 0.6539 FM 0.02313 -833.20 0.33 0.48 k∗on = 7.107× 10−4s−1 koff = 1.045× 10−3s−1 ST = 80µm −2 WD 0.07733 -714.48 113.12 0.00 Dwd = 0.0698µm2/s ID 0.02495 -827.60 0.00 0.64 K∗ = 3.873 FM 0.02492 -825.73 1.87 0.36 k∗on = 3.420× 10−4s−1 koff = 8.806× 10−5s−1 ST = 200µm −2 WD 0.02123 -843.77 0.84 0.37 Dwd = 0.0503µm2/s ID 0.02105 -844.61 0.00 0.46 K∗ = 198.0 FM 0.02127 -841.56 3.05 0.17 k∗on = 75.92s−1 koff = 4.856× 10−2s−1 WD - Weighted Diffusion, ID - Independent Diffusion, FM - Full Model, p - Probability (equation 5.19 in main text) Appendix E. Details of model fitting to simulated data (Chapter 5) 152 Table E.2: Model Fitting to Simulated Experiment B Model SSR AIC ∆AIC p Fitted Parameters ST = 1µm −2 WD 0.02512 -826.92 0.00 0.62 Dwd = 0.4733µm2/s ID 0.02646 -821.75 5.18 0.09 K∗ = 0.0129 FM 0.02521 -824.56 2.36 0.29 k∗on = 3.374× 10−2s−1 koff = 5.965× 10−1s−1 ST = 40µm −2 WD 0.06892 -726.00 98.69 0.00 Dwd = 0.1838µm2/s ID 0.05215 -753.88 70.81 0.00 K∗ = 0.5105 FM 0.02518 -824.69 0.00 1.00 k∗on = 7.674× 10−3s−1 koff = 1.109× 10−2s−1 ST = 80µm −2 WD 0.03887 -783.26 54.52 0.00 Dwd = 0.0911µm2/s ID 0.03887 -783.26 54.52 0.00 K∗ = 2.004 FM 0.02209 -837.78 0.00 1.00 k∗on = 3.089× 10−2s−1 koff = 9.335× 10−3s−1 ST = 200µm −2 WD 0.02198 -840.22 0.00 0.71 Dwd = 0.0535µm2/s ID 0.02337 -834.13 6.09 0.06 K∗ = 32.15 FM 0.02228 -836.81 3.41 0.22 k∗on = 2.358× 10−2s−1 koff = 5.331× 10−4s−1 WD - Weighted Diffusion, ID - Independent Diffusion, FM - Full Model, p - Probability (equation 5.19 in main text) Appendix E. Details of model fitting to simulated data (Chapter 5) 153 Table E.3: Model Fitting to Simulated Experiment C Model SSR AIC ∆AIC p Fitted Parameters ST = 1µm −2 WD 0.02221 -839.26 0.00 0.80 Dwd = 0.5083µm2/s ID 0.02334 -834.29 4.97 0.12 K∗ = 4.978× 106 FM 0.02310 -833.33 5.93 0.08 k∗on = 3.162× 10−5s−1 koff = 1.000× 103s−1 ST = 50µm −2 WD 0.02786 -816.56 0.00 0.80 Dwd = 0.3978µm2/s ID 0.03644 -789.73 26.83 0.00 K∗ = 0.06496 FM 0.02840 -812.67 3.89 0.20 k∗on = 3.580× 10−1s−1 koff = 1.401s −1 ST = 200µm −2 WD 0.02315 -835.09 0.00 0.53 Dwd = 0.2297µm2/s ID 0.01010 -687.79 147.30 0.00 K∗ = 0.3564 FM 0.02278 -834.70 0.39 0.47 k∗on = 8.363× 10−1s−1 koff = 6.030× 10−1s−1 ST = 1000µm −2 WD 0.02090 -845.33 0.00 0.58 Dwd = 0.0940µm2/s ID 0.07804 -713.57 131.76 0.00 K∗ = 2.542 FM 0.02072 -844.17 1.17 0.42 k∗on = 3.024s−1 koff = 2.812× 10−1s−1 WD - Weighted Diffusion, ID - Independent Diffusion, FM - Full Model, p - Probability (equation 5.19 in main text) 154 Appendix F Materials and methods (Chapter 6) F.1 Cell isolation and culture Purification of CD4+ human PBL T cells were isolated from the whole blood of healthy donors (Centre de Trans- fusion Sanguine, CHU Purpan, Toulouse) as described previously (285). Briefly, PBMC were purified from blood by centrifugation through Ficoll-Hypaque (Pharmacia Biotech, Sweden). CD4+ T lymphocytes were purified using RosetteSep Kit (StemCell Technologies, Vancouver, Canada). Cell purity was assessed by FACS analysis (Facscan, Becton Dickinson) using FITC-labelled anti-CD4 mAb (clone RPA-T4, BD Pharmingen). CD4+ frac- tions were routinely ∼ 90% pure. Before use, purified T cells were cultured in complete RPMI 1640 (Gibco, Paisley, Scotland) containing 5% human serum. In parallel experiments, Cord Blood T Lymphocytes (CBTL) were purified from whole cord blood (CHU Purpan, Toulouse) as described above. In some cases CBTL were cul- tured and expanded for 14 days in RPMI 1640, 5% human serum, IL-2 (150 IU/ml) using Dynabeads CD3/CD28 T Cell Expander (Dynalbiotech, Oslo, Norway). Bead to cell ratio was 1:1. F.2 Fluorescence recovery after photobleaching Anti-CD3 mAbs (TR66, IgG1, (286)) were digested by using the IgG1 Fab and F(ab)’2 kit (Pierce biotech- nologyTM, Rockford). Fab fragments (5 mg/ml in 0.1 M sodium carbonate, pH = 9.3) were labelled with Cy5 Monoreactive dye pack (Amersham BioscienceTM, Piscataway). The labelled Fab fragments were separated from non-conjugated dye by gel chromatography by the use of Nap-10 Columns (Amersham Bioscience). This step was followed by an overnight dialysis in 1×PBS by using a Lyser Dialysis cassettes (Pierce biotechnology) (287). PBMC derived CD4+ T cells were washed and re-suspended in RPMI, 5% FCS, 10 mM Hepes. Cells were loaded with 2 µM Fluo4 AM (Molecular Probes, Leiden, The Netherlands) for 30 minutes at 37 ◦C. After washing, cells were stained with 30 µg/mL CD3/Cy5 Fab in RPMI, 1% FCS at 4 ◦C for 30 minutes. Cells were washed and kept in ice until used for FRAP experiments. FRAP was performed in Lab-Tek chambers (Nalgene Nunc, Rochester, NY) in pre-warmed 5% FCS/10/mM HEPES at 37 ◦C, 5% CO2 on a confocal microscope (LSM 510 ; Carl Zeiss, Jena, Germany) using a Plan-Apochromate 63x objective and the 633 nm laser. A rectangular region (2 µm by Appendix F. Materials and methods (Chapter 6) 155 1.4 µm) was defined on the surface of T cells. This region was irradiated using the 633 nm, 543 nm, and 488 nm lasers with 100% intensity for 3 s. The aperture of the pinhole was adjusted to obtain optical slices of 2 µm depth. Before and after bleaching, the whole field was visualized by irradiating with 488 nm laser with 4% intensity and 633 nm laser with 17% intensity. Images were taken at 1 s intervals. In some experiments, cells were treated either with ionomycin (0.5 µg/ml) at the time of the time-lapse recording. In some experiments cells were treated with cytochalasin D (10 µM) or latrunculin B (50 nM) before the beginning of the assay. F.3 Fitting procedures We used the method for FRAP measurements of surface diffusion described in (288). Briefly, a small region of the cell membrane was bleached and the fluorescence recovery curve was fit using a two-dimensional recovery equation to estimate the TCR diffusion constant and the mobile fraction of TCR. The Matlab function lsqcurvefit was used for fitting. We also fit a simple exponential to every FRAP experiment. Using the time-constant of the recovery as an indicator of TCR mobility gave the same conclusions as using the diffusion coefficient (analysis not shown). F.4 Quantification of F-actin by phalloidin staining and flow cytometry A method described by Downey et al (289) was used with some modifications (290). Briefly, PBL were fixed with 3% PFA, permeabilized with 0.1% saponin and stained with 160 nM Alexa 488-labelled phalloidin. Cells were analyzed on a FACSCalibur (Becton Dickinson). Gating was done to exclude by forward and side scatter criteria cell debris and cell clumps. In some samples T cells were treated with 2.5 µg/ml ionomycin or with 50 to 500 nM latrunculin B for 15 minutes at 37 ◦C, 5%, before fixation. 156 References [285] Zaru, R., T. Cameron, L. Stern, S. Muller, and S. Valitutti. 2002. Cutting edge: TCR engagement and triggering in the absence of large-scale molecular segregation at the T cell-APC contact site. J Immunol. 168:4287–91. [286] Lanzavecchia, A., and D. Scheidegger. 1987. The use of hybrid hybridomas to target human cytotoxic T lymphocytes. Eur J Immunol. 17:105–11. [287] Depoil, D., R. Zaru, M. Guiraud, A. Chauveau, J. Harriague, G. Bismuth, C. Utzny, S. Muller, and S. Vali- tutti. 2005. Immunological synapses are versatile structures enabling selective T cell polarization. Immunity. 22:185–94. [288] Dushek, O., and D. Coombs. 2008. Improving parameter estimation for cell surface FRAP data. J Biochem Biophys Methods. 70:1224–31. [289] Downey, G., C. Chan, S. Trudel, and S. Grinstein. 1990. Actin assembly in electropermeabilized neu- trophils: role of intracellular calcium. J Cell Biol. 110:1975–82. [290] Valitutti, S., M. Dessing, and A. Lanzavecchia. 1993. Role of cAMP in regulating cytotoxic T lymphocyte adhesion and motility. Eur J Immunol. 23:790–5.
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Mathematical modeling in cellular immunology: T cell activation and parameter estimation Dushek, Omer 2008
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Title | Mathematical modeling in cellular immunology: T cell activation and parameter estimation |
Creator |
Dushek, Omer |
Publisher | University of British Columbia |
Date Issued | 2008 |
Description | A critical step in mounting an immune response is antigen recognition by T cells. This step proceeds by productive interactions between T cell receptors (TCR) on the surface of T cells and foreign antigen, in the form of peptide-major-histocompatibility-complexes (pMHC), on the surface of antigen-presenting-cells (APC). Antigen recognition is exceedingly difficult to understand because the vast majority of pMHC on APCs are derived from self-proteins. Nevertheless, T cells have been shown to be exquisitely sensitive, responding to as few as 10 antigenic pMHC in an ocean of tens of thousands of self pMHC. In addition, T cells are extremely specific and respond only to a small subset of pMHC by virtue of their specific TCR. To explain the sensitivity of T cells to pMHC it has been proposed that a single pMHC may serially bind multiple TCRs. Integrating present knowledge on the spatial-temporal dynamics of TCR/pMHC in the T cell-APC contact interface, we have constructed mathematical models to investigate the degree of TCR serial engagements by pMHC. In addition to reactions within clusters, the models capture the formation and mobility of TCR clusters. We find that a single pMHC serially binds a substantial number of TCRs in a TCR cluster only if the TCR/pMHC bond is stabilized by coreceptors and/or pMHC dimerization. In a separate study we propose that serial engagements can explain T cell specificity. Using Monte Carlo simulations, we show that the stochastic nature of TCR/pMHC interactions means that multiple binding events are needed for accurate detection of foreign pMHC. Critical to our studies are estimates of TCR/pMHC reaction rates and mobilities. In the second half of the thesis, we show that Fluorescence Recovery After Photobleaching (FRAP) experiments can reveal effective diffusion coefficients. We then show, using asymptotic analysis and model fitting, that FRAP experiments can be used to estimate reaction rates between cell surface proteins, like TCR/pMHC. Lastly, we use FRAP experiments to investigate how the actin cytoskeleton modulates TCR mobility and report effective reaction rates between TCR and the cytoskeleton. |
Extent | 3507557 bytes |
Subject |
T cell receptor Mathematical modeling |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-12-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0066848 |
URI | http://hdl.handle.net/2429/2894 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2009-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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