UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Signalling in single cell wound healing Liao, Laura 2013

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2013_fall_liao_laura.pdf [ 20.44MB ]
Metadata
JSON: 24-1.0074055.json
JSON-LD: 24-1.0074055-ld.json
RDF/XML (Pretty): 24-1.0074055-rdf.xml
RDF/JSON: 24-1.0074055-rdf.json
Turtle: 24-1.0074055-turtle.txt
N-Triples: 24-1.0074055-rdf-ntriples.txt
Original Record: 24-1.0074055-source.json
Full Text
24-1.0074055-fulltext.txt
Citation
24-1.0074055.ris

Full Text

Signalling in Single Cell WoundHealingThe Role of Protein Kinase CbyLaura LiaoB.Sc., Ryerson University, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2013c? Laura Liao 2013AbstractA single cell, such as a frog egg, is able to repair injuries by orchestratinga localized signalling response on the plasma membrane. Proteins calledRho GTPases are recruited to, and form patterns around, the wound site.Patterning allows testable hypotheses to be made about the structure of thesignalling network. Here, we extend a Rho GTPase signalling model fromSimon et al. (2013) to test how a family of enzymes, protein kinase C (PKC),plays a role in cell repair signalling. Our models let PKCs affect basal RhoGTPase activation and/or inactivation rates, with increasing spatial detail.Ultimately, the model variants do not account for Rho GTPase patterningin all experiments. We suggest a new round of modelling and experimentsto correct these issues.iiPrefaceChapter 2 is work conducted in University of Wisconsin-Madison?s Labora-tory of Cell and Molecular Biology by Prof. W. Bement. The unpublisheddata motivated the modelling work herein.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Wound healing in single cell systems . . . . . . . . . . . . . . 11.2 GTPases: RhoA and Cdc42 . . . . . . . . . . . . . . . . . . . 11.3 GTPases signal to actin and myosin . . . . . . . . . . . . . . 21.4 Protein kinase Cs (PKC) influence Rho GTPases . . . . . . . 21.5 Cell signalling in Xenopus oocyte wound healing . . . . . . . 31.5.1 Signalling components can be probed and quantified . 31.5.2 Patterning hints at network?s structure . . . . . . . . 51.5.3 Thesis focus . . . . . . . . . . . . . . . . . . . . . . . 72 Protein kinase Cs (PKCs), lipids and their roles . . . . . . 102.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Diacylglycerol (DAG) overlaps the RhoA zone . . . . . . . . 132.3 PKCs bind DAG, localizing to regions comprising Rho GT-Pase zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 PKC beta and PKC eta play opposing roles in RhoA andCdc42 activation . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 PKC beta positively influences Rho GTPase activa-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16ivTable of Contents2.4.2 PKC eta negatively influences RhoA and Cdc42 acti-vation . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Main questions to be addressed . . . . . . . . . . . . . . . . 193 Models: definitions, background, and logical framework . 213.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Defining the Simon et al. (2013) Rho GTPase model . . . . . 213.2.1 Model features . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Successes and limitations of the model . . . . . . . . 263.3 The PKCs ? revising the basic model . . . . . . . . . . . . . 283.3.1 Criteria for model validation . . . . . . . . . . . . . . 294 PKCs as spatially constant parameters . . . . . . . . . . . . 304.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 PKCs act on GEFs or GAPs . . . . . . . . . . . . . . . . . . 304.2.1 Bifurcations on the basal rates of RhoA (in)activation,kr0 (k2) . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.2 Bifurcations on the basal rates of Cdc42 (in)activation,kc0 (k7) . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Simulations of PKC manipulations . . . . . . . . . . . . . . . 374.3.1 Overexpression of PKC beta . . . . . . . . . . . . . . 384.3.2 Expression of dominant-negative PKC beta . . . . . . 394.3.3 Overexpression of PKC eta . . . . . . . . . . . . . . . 414.3.4 Expression of dominant-negative PKC eta . . . . . . 424.4 Possible ways to achieve inverted Rho GTPase zones . . . . 434.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 PKCs as spatially dependent parameters: step function rep-resentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Defining PKC activity profiles . . . . . . . . . . . . . . . . . 475.2.1 PKC spatial profiles as step functions . . . . . . . . . 485.2.2 PKC zone localization . . . . . . . . . . . . . . . . . . 485.3 Spatially dependent basal Rho GTPase activation rates . . . 505.3.1 Parameter estimation . . . . . . . . . . . . . . . . . . 515.3.2 Constraints set by control data . . . . . . . . . . . . . 535.4 Simulations of PKC manipulations . . . . . . . . . . . . . . . 545.4.1 Overexpression of PKC beta . . . . . . . . . . . . . . 555.4.2 Expression of dominant-negative PKC beta . . . . . . 565.4.3 Overexpression of PKC eta . . . . . . . . . . . . . . . 58vTable of Contents5.4.4 Expression of dominant-negative PKC eta . . . . . . 595.5 Addressing Cdc42 insensitivity to PKC beta . . . . . . . . . 605.5.1 Spatially dependent basal Cdc42 inactivation rate . . 605.5.2 Parameter estimation and constraints . . . . . . . . . 615.5.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . 615.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 PKCs as spatially dependent parameters: explicit activityprofiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Defining PKC activity profiles . . . . . . . . . . . . . . . . . 676.2.1 Explicit PKC activity profiles from data . . . . . . . 686.2.2 PKC localization to regions overlapping Rho GTPasezones . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Spatially dependent basal Rho GTPase activation rates . . . 716.3.1 Parameter estimation and constraints . . . . . . . . . 716.4 Simulation of the control experiment . . . . . . . . . . . . . 736.4.1 Revising the kr0(x, t), kc0(x, t) relationship with PKCbeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 Full nonlinear parameter fitting of model to data . . . . . 797.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.2 Fitting spatial background activation rates to control RhoGTPase activites . . . . . . . . . . . . . . . . . . . . . . . . . 807.2.1 Matching the simulated wound edge to data . . . . . 807.2.2 Fits to control RhoA data . . . . . . . . . . . . . . . 817.2.3 Fits to control Cdc42 data . . . . . . . . . . . . . . . 827.3 Simulations of PKC manipulations . . . . . . . . . . . . . . . 847.3.1 Overexpression of PKC beta . . . . . . . . . . . . . . 847.3.2 Expression of dominant-negative PKC beta . . . . . . 887.3.3 Overexpression of PKC eta with inverted Rho GTPasezones . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.4 Revisiting fits of background activation rates to Rho GTPaseactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.4.1 Constrained fitting of kr0(x, t) to control RhoA activity 927.4.2 Constrained fitting of kc0(x, t) or k7(x, t) to controlCdc42 activity . . . . . . . . . . . . . . . . . . . . . . 967.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97viTable of Contents8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102AppendicesA Zone amplification factors from box-and-whisker plots . . 104B Amalgamating PKC and Rho GTPase datasets . . . . . . . 106B.1 Overexpression of PKC beta . . . . . . . . . . . . . . . . . . 107B.2 Expression of dominant-negative PKC beta . . . . . . . . . . 108B.3 Overexpression of PKC eta with inverted Rho GTPase zones 109viiList of Tables3.1 Parameters in Model 1 from Simon et al. (2013). . . . . . . . 256.1 Parameterization of kr0(x, t) (Equations 6.1-6.3) by method ofleast-squares. The sum of squared residuals (SSR) is given. . 736.2 Parameterization of kc0(x, t) by method of least-squares. Thesum of squared residuals (SSR) is given. . . . . . . . . . . . . 73A.1 Fold change in RhoA and Cdc42 zone activities, relative tocontrol, when PKC beta is overexpressed. Based on meanintensity data (in arbitrary units) taken from Figure 2.2 D.Fold changes that are not statistically significant are indicatedby ?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104A.2 Fold change in RhoA and Cdc42 zone activities, relative tocontrol, when dominant-negative PKC beta is expressed. Basedon mean intensity data (in arbitrary units) taken from Fig-ure 2.2 E. Fold changes in background activity that are notstatistically significant are indicated by ?. . . . . . . . . . . 104A.3 Fold change in RhoA and Cdc42 zone activities, relative tocontrol, when PKC eta is overexpressed. Based on mean in-tensity data (in arbitrary units) taken from Figure 2.2 F. . . 105A.4 Fold change in RhoA and Cdc42 background activities, rel-ative to control, when dominant-negative PKC eta is ex-pressed. Based on mean intensity data (in arbitrary units)taken from Figure 2.2 G. Fold changes in zones that are notstatistically significant are indicated by ?. . . . . . . . . . . 105viiiList of Figures1.1 Rings of active RhoA (inner ring) and active Cdc42 (outerring) around the wound site (Figure from (Bement et al.,2012)). The wound contracts and closes over the course of afew minutes. Scale bar is 20 ?m. . . . . . . . . . . . . . . . . 21.2 Events in single cell wound healing. Upon wounding, an in-flux of Ca2+ into the cell triggers cell repair. Diacylglycerol(DAG), a lipid, accumulates around the wound. Protein ki-nase Cs (PKC) are lipid targets that are activated by DAG,and influence RhoA and Cdc42 activation. RhoA and Cdc42play instrumental roles in actin assembly and contraction ofthe wound. This thesis is focused on later events involvingPKCs and Rho GTPases. . . . . . . . . . . . . . . . . . . . . 31.3 Spatial patterns of active RhoA (green) and active Cdc42(red) in a wounded frog egg (left). Probe intensities are mea-sured in a line scan drawn outwards from the wound center(right). The wound edge is marked with a vertical black line. 41.4 Time sequence showing a single cell wound experiment. RhoAand Cdc42 zones rise, narrow, and translocate towards thewound center during wound healing. Late stages of the heal-ing response (zone segregation and zone maintenance; 54? 84 s)are depicted in these radial concentration profiles. The woundedge is denoted W. Rho GTPase intensities were scaled toconcentrations using estimates of basal and elevated Rho GT-Pase concentrations found in literature (Simon et al., 2013). . 51.5 The signalling network of RhoA and Cdc42 with interactionsmediated through Abr. Reprinted from ?Pattern formation ofRho GTPases in single cell wound healing,? by C. M. Simon,2013, Molecular Biology of the Cell, 24(3), p. 424. Copyright2013 by the American Society for Cell Biology. Reprintedwith permission. . . . . . . . . . . . . . . . . . . . . . . . . . 6ixList of Figures1.6 This thesis starts from the Rho GTPase signalling networkof Simon et al. (2013) (left), and extends it by investigat-ing the effects of PKCs on background (in)activation rates ofRhoA and Cdc42 (center). PKC activation or inhibition ofbackground rates is investigated (right). Potential PKC betaeffects are shown in the blue arrows/bar-headed lines. PKCeta (not shown) has the opposite effect. Adapted from ?Pat-tern formation of Rho GTPases in single cell wound healing,?by C. M. Simon, 2013, Molecular Biology of the Cell, 24(3),p. 424. Copyright 2013 by the American Society for CellBiology. Adapted with permission. . . . . . . . . . . . . . . . 81.7 A flow chart summarizing model variants and the chaptersin which they are discussed. PKC effects are introduced intothe Rho GTPase background (in)activation rates in the Simonet al. (2013) model. Model variations incorporate increasingspatial detail and additional assumptions. . . . . . . . . . . . 92.1 Five lipids characterized by time to appearance, relative toCdc42 zone formation (15? 20 s) (Figure from Bement et al.(2012)). Lipid domains of phosphoinositide triphosphate (PIP3),phosphoinositide bisphosphate (PIP2) and phosphatidylser-ine (PS) organize around the wound edge at late times (A-C).Lipid domains of phosphatidic acid (PA) and diacylglycerol(DAG) organize around the wound edge at early times (D,E). Lipid distributions at 90 s, relative to the Cdc42 zone(A?-E?). Relative positions of lipid compartments by pairwisecomparison (F-I). Lipid compartment positions in relation tothe wound edge (J). Scale bar = 20 ?m. W = Wound edge.Panels are individually reproduced and discussed in the text. 11xList of Figures2.2 Localization of PKC beta and PKC eta (A) and their distribu-tions at 90 s (Figure from Bement et al. (2012)). Overexpres-sion (OE), or expression of dominant-negative (DN) variants,of PKC beta or PKC eta showing influence on RhoA andCdc42 zone activation (C). Quantification of average RhoA(green) and Cdc42 (red) zone intensities at 30 s and 60 s foreach manipulation (D, F, H). Quantification of average RhoAand Cdc42 zone and background intensities at 60 s for ma-nipulations which show late zone formation (E, H). Top andbottom whiskers indicate maximum and minimum intensities,respectively. Scale bar = 20 ?m. W = Wound edge. Panelsare individually reproduced and discussed in the text. . . . . 122.3 Diacylglycerol appears to accumulate around the wound atearly times, relative to the formation of the Cdc42 zone (E).DAG localizes to regions close to the wound edge, where theRhoA zone is present (E?). The Cdc42 zone is used to marka 5 ?m distance from the wound edge. W denotes the woundedge. Note that the orientation of the axis is reversed fromlater simulations, where the wound is on the left. . . . . . . . 132.4 Localization of PKC beta and PKC eta relative to the Cdc42zone (A, B). PKC beta overlaps regions of both RhoA andCdc42 zones (A?). PKC eta overlaps the region containingonly the RhoA zone (B?). . . . . . . . . . . . . . . . . . . . . 142.5 Time sequence of PKC beta and PKC eta localization showbroad zones which narrow as healing progresses (A). Linescans compare the broad PKC beta localization to the narrowPKC eta localization (B). . . . . . . . . . . . . . . . . . . . . 152.6 Time sequence images of a control wound experiment whereRhoA (green) and Cdc42 (red) are probed. . . . . . . . . . . 162.7 Elevated zone activities of RhoA (green) and Cdc42 (red)when PKC beta is overexpressed. Quantification of zone ac-tivities at 30 s and 60 s (D). . . . . . . . . . . . . . . . . . . . 162.8 Diminished zone activities of RhoA (green) and Cdc42 (red)when dominant-negative PKC beta is expressed. Quantifica-tion of zone activities and background activities at 60 s (E). . 172.9 Diminished zone activities of RhoA (green) and Cdc42 (red)when PKC eta is overexpressed. Quantification of zone ac-tivities at 30 s and 60 s (F). . . . . . . . . . . . . . . . . . . . 18xiList of Figures2.10 Elevated background activities of RhoA (green) and Cdc42(red) when dominant-negative PKC eta is expressed. Quan-tification of zone activities and background activities at 60 s(G). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.11 One-fifth of PKC overepxression experiments result in in-verted zones where Cdc42 encircles the wound, with RhoAoutside the Cdc42 zone. . . . . . . . . . . . . . . . . . . . . . 193.1 RhoA (green) and Cdc42 (red) zones accumulate around thewound on the plasma membrane (left). RhoA and Cdc42concentrations are tracked in the model through time andspace. The two-dimensional spatial domain is represented inradial coordinates where r = 0 (in ?m) corresponds to thewound center (right). . . . . . . . . . . . . . . . . . . . . . . 223.2 The signalling network of RhoA and Cdc42 with interactionsmediated through Abr. Reprinted from ?Pattern formation ofRho GTPases in single cell wound healing,? by C. M. Simon,2013, Molecular Biology of the Cell, 24(3), p. 424. Copyright2013 by the American Society for Cell Biology. Reprintedwith permission. . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 The Simon et al. (2013) model captures basic features inRho GTPase zones such as the speed of amplification andthe steady states. RhoGTPases 54? 84 s post-wounding aresimulated (lines), with data shown in circles for comparison.The simulated wound edge is given by the vertical dashedblack line, to be compared with the vertical solid black line.Notice that the orientation of the axis is reversed (wound edgetowards the left) in comparison to previous line scan data. . . 243.4 The effects of PKCs on background Rho GTPase activationrates kr0, kc0, and background inactivation rates k2, k7. Po-tential PKC beta effects are shown in the blue arrows/bar-headed lines. PKC eta (not shown) has the opposite effect.Adapted from ?Pattern formation of Rho GTPases in singlecell wound healing,? by C. M. Simon, 2013, Molecular Biologyof the Cell, 24(3), p. 424. Copyright 2013 by the AmericanSociety for Cell Biology. Adapted with permission. . . . . . . 28xiiList of Figures4.1 The simulated control (Model 1) possesses two stable steadystates in RhoA and three stable steady states in Cdc42. RhoA(green) and Cdc42 (red) have high steady states within thezone, and low steady states at background. Cdc42 has anadditional steady state in the Cdc42 bump. . . . . . . . . . . 314.2 A bifurcation diagram of kr0 showing RhoA steady states.At the control value (vertical violet line), RhoA possessesstable low and high steady states which correspond to thebackground and zone activities, respectively (violet arrows).The unstable steady state corresponds to the bistable thresh-old. The interval 0.87-1.37 marks the bistable region (verticaldashed lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 A bifurcation diagram of kr0 showing Cdc42 steady states. Asin Figure 4.2, but a third intermediate stable steady stateis shown (violet arrows). Cdc42 is bistable over the inter-val 0-1.37, but only the interval where both GTPases retainbistability is called the bistable region (vertical dashed lines). 334.4 Bifurcation diagrams (space-free Equations 3.2-3.4) with re-spect to RhoA basal activation (top) and inactivation (bot-tom). Stable branches towards the top (bottom) of each di-agram indicate high (low) steady states. Cdc42 possesses anadditional intermediate steady state. Bistable region (bothGTPases) marked by vertical dashed lines. . . . . . . . . . . . 344.5 As in Figure 4.4 but the intermediate Cdc42 steady stateis not shown. RhoA is unaffected by either Cdc42 basal(in)activation rate, and therefore RhoA retains bistabilitythroughout each diagram. . . . . . . . . . . . . . . . . . . . . 364.6 Results for Model 2 where PKC beta is overexpressed. Thesimulated control is shown for comparison (circles/thick curves). 384.7 Results for Model 2 where dominant-negative PKC beta isexpressed. The simulated control is shown for comparison(circles/thick curves). . . . . . . . . . . . . . . . . . . . . . . 404.8 Results for Model 2 where PKC eta is overexpressed. Thesimulated control is shown for comparison (circles/thick curves). 414.9 Results for Model 2 where dominant-negative PKC eta is ex-pressed. The simulated control is shown for comparison (cir-cles/thick curves). . . . . . . . . . . . . . . . . . . . . . . . . 424.10 Results for Model 2 where inverted Rho GTPase zones occurbecause Cdc42 sweeps out in front of the RhoA zone. Thesimulated control is shown for comparison (circles/thick curves). 44xiiiList of Figures4.11 Results for Model 2 where inverted Rho GTPase zones oc-cur because the RhoA zone protrudes into a uniform field ofCdc42. The simulated control is shown for comparison (cir-cles/thick curves). . . . . . . . . . . . . . . . . . . . . . . . . 455.1 Activity profiles of PKC beta, ?(x, t), and PKC eta, ?(x, t),are modelled as normalized step functions with PKC beta cov-ering a greater breadth than PKC eta. The simulated woundedge is marked by the vertical black line, and is towards theleft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 Definition of RhoA zone location at early times (left) and latetimes (right). Top of the thick black line indicates maximumactivity above background; base of the black line indicatesbackground activity. Top violet line indicates 40% of heightspanned by black line; bottom violet line indicates the definedzone location. A similar determination of the Cdc42 zonelocation is used with a threshold of 40 %. . . . . . . . . . . . 495.3 Determination of 40 % activity threshold from control data.Time series of Rho GTPases highlighting the RhoA zone loca-tion (green interval) and Cdc42 zone location (red interval).Chosen activity threshold corresponds to visual estimation ofzone location accuracy. . . . . . . . . . . . . . . . . . . . . . . 495.4 Distinct regions in the simulated control where both PKCsare present (Region 1), only PKC beta is present (Region 2),and neither PKCs are present (Region 3). . . . . . . . . . . . 515.5 Control simulation for Model 3. Note the appearance of theRhoA shoulder. The control data are shown for comparison(circles/thick curves). . . . . . . . . . . . . . . . . . . . . . . 545.6 Results for Model 3 where PKC beta is overexpressed by 5 %.The simulated control is shown for comparison (circles/thickcurves). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.7 Results for Model 3 where dominant-negative PKC beta isexpressed. PKC beta activity is expressed at 80 % of controllevels. Initial conditions have been adjusted to guarantee zonemaintenance. The simulated control is shown for comparison(circles/thick curves). . . . . . . . . . . . . . . . . . . . . . . 575.8 Results for Model 3 where PKC eta is overexpressed by 5 %.The simulated control is shown for comparison (circles/thickcurves). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58xivList of Figures5.9 Results for Model 3 where dominant-negative PKC eta is ex-pressed. PKC eta activity is expressed at 40 % of control lev-els. Initial conditions have been adjusted to guarantee zonemaintenance. The simulated control is shown for comparison(circles/thick curves). . . . . . . . . . . . . . . . . . . . . . . 595.10 Control simulation for Model 4. Note the appearance of theRhoA shoulder. The control data are shown for comparison(circles/thick curves). . . . . . . . . . . . . . . . . . . . . . . 625.11 Results for Model 4 where PKC beta is overexpressed by210 %. The simulated control is shown for comparison (cir-cles/thick curves). . . . . . . . . . . . . . . . . . . . . . . . . 636.1 Normalized intensity profiles of PKC eta and PKC beta ac-tivities. Relevant times 54? 84 s post-wounding are shown. . 686.2 PKC activity profiles and Rho GTPase activities from differ-ent control datasets are compared. PKC zones do not overlapthe RhoA zone. W denotes the wound edge from the RhoGTPase dataset. . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Aligning the PKC profiles with the Cdc42 peak at 90 s post-wounding. A 7 ?m shift towards the wound center is illustrated. 706.4 To correct the PKC localization between datasets, the PKCprofiles are merged with Rho GTPase profiles under a 7 ?mshift towards the wound center. . . . . . . . . . . . . . . . . . 716.5 PKC zones via the zone localization definition. Mean PKCactivity within the zone, and mean PKC activity outside thezone are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . 726.6 Control simulation for Model 5. The control data are shownfor comparison (circles/thick curves). . . . . . . . . . . . . . . 746.7 Control simulation for Model 6. The control data are shownfor comparison (circles/thick curves). . . . . . . . . . . . . . . 766.8 Control simulation for Model 7. The control data are shownfor comparison (circles/thick curves). . . . . . . . . . . . . . . 77xvList of Figures7.1 Simulated wound edge position (left) and velocity vc (right)showing the previous choices by Simon et al. (2013) in red,and our improvement in blue. The simulated wound edgeposition matches data at 54 s and 84 s post-wounding (left,red) when a constant vc(t) =1.12 ?m2/s is used (right, red).The simulated wound edge position matches data at six timespoints (left, blue) when a time-dependent vc(t) is used (right,blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.2 Model 8 fits to RhoA activity with matched wound edge. Fit-ted parameters that resulted in a sum of squared residualsof 0.053 ?M2 were: krbasal =0.0058 /s, krPKC =0.38 /?M/s,?1 =86 /?M, ?r0 =0.60 ?M. Control data are shown as cir-cles/thick curves. . . . . . . . . . . . . . . . . . . . . . . . . . 827.3 Model 9 fits to Cdc42 activity, as in Figure 7.2. However, theparameters in the basal RhoA activation rate are fixed by theprevious RhoA fit. Fitted parameters that resulted in a sumof squared residuals of 0.094 ?M2 were: kcbasal =0.00025 /s,kcPKC =0.38 /?M/s, ?2 =85 /?M, ?c0 =0.043 ?M. An addi-tional parameter, Cshift =?2.47 ?m, was introduced so thatthe offset Cdc42 zone may be fitted. . . . . . . . . . . . . . . 837.4 Results for Model 9 where PKC beta Rho GTPase initial con-ditions are used and PKCs are at control levels. Rho GTPasedata from the corresponding PKC manipulation are shownfor comparison (circles/thick curves). However, RhoA dataare not shown after 72 s since RhoA activity begins to divebelow the focal plane. . . . . . . . . . . . . . . . . . . . . . . 857.5 As in Figure 7.4, showing results for Model 9 where PKC betais overexpressed by two-fold. . . . . . . . . . . . . . . . . . . . 867.6 As in Figure 7.4, showing results for Model 9 where PKC betais overexpressed by 1.5-fold. . . . . . . . . . . . . . . . . . . . 877.7 Results for Model 9 where dominant-negative PKC beta RhoGTPase initial conditions are used and PKCs are at controllevels. Rho GTPase data from the corresponding PKC ma-nipulation are shown for comparison (circles/thick curves). . 887.8 Results for Model 9 where dominant-negative PKC beta isexpressed at half the control activity. Rho GTPase data fromthe corresponding PKC manipulation are shown for compar-ison (circles/thick curves). . . . . . . . . . . . . . . . . . . . . 89xviList of Figures7.9 Results for Model 9 where overexpressed PKC eta Rho GT-Pase initial conditions are used and PKCs are at control lev-els. Rho GTPase data from the corresponding PKC manipu-lation are shown for comparison (circles/thick curves). . . . . 907.10 Results for Model 9 where PKC eta is expressed at twice thecontrol activity. Rho GTPase data from the correspondingPKC manipulation are shown for comparison (circles/thickcurves). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.11 Model 10 fits to control RhoA activity. Fitted parametersthat resulted in a sum of squared residuals of 0.059 ?M2were: krbasal =0.0027 /s, krPKC =0.29 /?M/s, ?1 =24.83 /?M,?r0 =0.46 ?M. Control data are shown for comparison (cir-cles/thick curves). . . . . . . . . . . . . . . . . . . . . . . . . 937.12 Model 11 fits to control RhoA activity. Fitted parametersthat resulted in a sum of squared residuals of 0.071 ?M2 were:krbasal =0.0057 /s, krPKC =0.096 /?M/s, ?1 =17.25 /?M, and?r0 =42.06 ?M. . . . . . . . . . . . . . . . . . . . . . . . . . . 957.13 Results for Model 10 (olive) and Model 11 (blue) where PKCbeta is overexpressed by two-fold. The unconstrained fit fromModel 8 is shown for comparison (green). RhoA data from thecorresponding PKC manipulation are also shown for compar-ison (circles/thick curves). However, RhoA data are removedafter 72 s because RhoA dives below the focal plane. . . . . . 96B.1 Alignment of control PKC profiles with Rho GTPase profilesin PKC beta overexpression. PKC profiles are aligned withRho GTPase data as is (unshifted). The wound edge is notshown but is towards the left. At 72 s, the RhoA zone divesaway from the focal plane. . . . . . . . . . . . . . . . . . . . . 107B.2 Alignment of control PKC profiles with Rho GTPase profilesin dominant-negative PKC beta expression. PKC profiles arealigned with Rho GTPase data when shifted 5 ?m away fromthe wound center. The wound edge is not shown but is to-wards the left. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108B.3 Alignment of control PKC profiles with Rho GTPase profilesin PKC eta overexpression. PKC profiles are aligned withRho GTPase data as is (unshifted). The wound edge is notshown but is towards the left. . . . . . . . . . . . . . . . . . . 109xviiAcknowledgementsI offer my gratitude to my supervisor Prof. Leah Edelstein-Keshet, who wasa fantastic and generous mentor. I thank my readers Profs. William Bementand Dan Coombs for their involvement and encouragement. Special thanksto Raibatak (Dodo) Das who provided invaluable advice on data fittingin this work, and Cory Simon for the inception of this modelling work. Iacknowledge funding from NSERC in the form of the Alexander GrahamBell Canada Graduate Scholarship, as well as the NSERC Discovery Grantto Leah Edelstein-Keshet.I owe thanks to the Keshet research group and all its members. Inparticular, thanks to my funny conspirators Meghan Dutot and MayangMata who worked alongside me throughout my graduate studies at UBC.Lastly, the students who make up the Institute of Applied Mathematicsdeserve acknowledgement for the enduring sense of community and supportthey provided.xviiiTo Justin MartelxixChapter 1Introduction1.1 Wound healing in single cell systemsA single cell is able to repair injuries and reseal part of its torn membrane.In experiments by our collaborator, W. Bement, egg cells of the frog (Xeno-pus) were injured by laser, and the process of repair was studied (Bementet al., 2012). Wounds seal shut by first orchestrating a localized signallingresponse, assembling a local cytoskeleton (actin for structure and myosinfor force production) and finally contracting shut in a purse-string manner(Bement et al., 1999; Mandato and Bement, 2001).How is wound healing in single cells orchestrated? In Xenopus oocytes,the healing response is triggered by an influx of Ca2+ upon wounding. A con-tractile ring consisting of actin and myosin-II assembles around the wound.As the wound closes, myosin-II segregates into a narrow ring in the interiorof the contractile ring, independently of actin, in response to upstream reg-ulators (Bement et al., 1999; Mandato and Bement, 2001). Among possibleupstream regulators are a family of small proteins called Rho GTPases, onwhich Xenopus oocyte healing is dependent (Bement et al., 1999). Theseproteins form radial patterns that motivate the research in this thesis.1.2 GTPases: RhoA and Cdc42Rho GTPases are G proteins (GTP-binding regulatory protein) which aretypically engaged in signal transduction of an extracellular message to intra-cellular message. A prototypical example of G protein function begins withan extracellular ligand binding to a plasma membrane-bound cell receptor,inducing a conformational change in the cell receptor. The cell receptor cou-ples to a G protein and promotes exchange of the G protein?s GDP to GTP.The now active, membrane-bound G protein binds to and activates othertargets until the G protein hydrolyzes its GTP to GDP, thereby terminatingsignalling.Since Rho GTPases have an active state (GTP-bound) and an inac-11.3. GTPases signal to actin and myosintive state (GDP-bound), they are considered molecular switches. The RhoGTPases cycle between an inactive, cytosolic state to an active, membrane-bound state. The toggling between these states is regulated by guanine nu-cleotide exchange factors (GEFS) and GTPase-activating proteins/guanosinenucleotide dissociation inhibitors (GAPS/GDIs) (Jaffe and Hall, 2005). GEFspromote the exchange of GDP to GTP, and thus promote Rho GTPase sig-nalling. GAPs promote hydrolysis of GTP to GDP, and thus inhibit RhoGTPase signalling. GDIs sequester inactive Rho GTPases in the cytosol(DerMardirossian and Bokoch, 2005).1.3 GTPases signal to actin and myosinTwo Rho GTPases, RhoA and Cdc42, are implicated in driving the as-sembly of the actomyosin contractile ring during Xenopus oocyte woundclosure (Benink and Bement, 2005). RhoA promotes phosphorylation of themyosin light chain, inducing contractile forces along actin stress fibers (viaan enzyme Rho kinase) (Amano et al., 1996). Cdc42 enhances the growth ofactin by branching its filaments (mediated by complexes Arp2/3 and WASP)(Miki and Takenawa, 2003). Thus RhoA regulates contractile forces in theinterior domain of the ring, while Cdc42 regulates cytoskeleton remodelingin the exterior domain of the ring (Figure 1.1). The regulation of the con-tractile array by Rho GTPases falls in line with RhoA and Cdc42?s knownroles.Figure 1.1: Rings of active RhoA (inner ring) and active Cdc42 (outer ring)around the wound site (Figure from (Bement et al., 2012)). The woundcontracts and closes over the course of a few minutes. Scale bar is 20 ?m.1.4 Protein kinase Cs (PKC) influence RhoGTPasesThe contractile array occupies discrete regions on the plasma membrane.Subcellular domains of Rho GTPase localization are observed (Figure 1.1),21.5. Cell signalling in Xenopus oocyte wound healingwith dynamic Rho GTPases that are in constant flux on and off the mem-brane (Bement et al., 2006; Benink and Bement, 2005). However, the RhoGTPase-powered contractile array appears late in single cell wound healing.Lipids are involved in the very early response preceding the appearance ofRho GTPases. Rho GTPases, in turn, are influenced by lipids and lipidtargets (Figure 1.2) (Bement et al., 2012).Ca2+Woundactin assemblyactomyosin contractsRhoACdc42? LIPIDLIPID TARGETG PROTEINFigure 1.2: Events in single cell wound healing. Upon wounding, an influxof Ca2+ into the cell triggers cell repair. Diacylglycerol (DAG), a lipid,accumulates around the wound. Protein kinase Cs (PKC) are lipid targetsthat are activated by DAG, and influence RhoA and Cdc42 activation. RhoAand Cdc42 play instrumental roles in actin assembly and contraction of thewound. This thesis is focused on later events involving PKCs and RhoGTPases.As discussed in Chapter 2, the lipid diacylglycerol (DAG) and its targetprotein kinase C (PKC) are important as upstream signalling componentsin wound healing. PKCs are shown to influence Rho GTPase activity, andthe way in which they do so within the context of Xenopus oocyte woundhealing becomes our primary focus.1.5 Cell signalling in Xenopus oocyte woundhealing1.5.1 Signalling components can be probed and quantifiedLipids, lipid targets (protein kinase C), and Rho GTPases orchestrate singlecell wound healing through a signalling network. Xenopus oocytes are auseful experimental system in which to study such cell signalling networks.A key tool are fluorescent probes which visualize processes occurring on theplasma membrane ? e.g., membrane resealing and cytoskeleton remodelling.Fluorescent patterns of small molecules on the plasma membrane, such as31.5. Cell signalling in Xenopus oocyte wound healingactive Rho GTPases, can be quantified in a line scan (Figure 1.3). The linescan measures probe intensities which correspond to concentrations of RhoA(green) and Cdc42 (red).10 ?m1000 ?mr0 5 10 15 20 25 3000.050.160 s post-woundingFigure 1.3: Spatial patterns of active RhoA (green) and active Cdc42 (red)in a wounded frog egg (left). Probe intensities are measured in a line scandrawn outwards from the wound center (right). The wound edge is markedwith a vertical black line.Probes reveal zones of Rho GTPases around the woundProbes reveal spatiotemporal patterning of Rho GTPase activity in imagesand line scans of the wound. The images show the accumulation of RhoAand Cdc42 around the wound at activities two to five times higher thanbasal levels in a resting cell. We refer to these regions of elevated activitiesas discrete zones. The zones circumscribe the wound in concentric circles,with RhoA closest to the wound edge, and Cdc42 outside of the RhoA zone(Figure 1.1).The line scans show the evolution of RhoA and Cdc42 zones (Figure 1.4).Early in wound healing, zones are amplified from an initially shallow RhoGTPase gradient into broad and overlapping zones (zone formation). Ashealing progresses, broad zones narrow into mutually exclusive zones (zonesegregation). Zone segregation suggests that RhoA inhibits Cdc42, ex-cluding Cdc42 from any region that RhoA occupies. As the zones narrow,they reach a characteristic high concentration as they are pulled forwardtowards the wound center during closure (zone maintenance).41.5. Cell signalling in Xenopus oocyte wound healing0 5 10 15 20 25 3000.050.10.15 84 s post-wounding0 5 10 15 20 25 3000.050.10.15 78 s post-wounding0 5 10 15 20 25 3000.050.10.15 72 s post-wounding0 5 10 15 20 25 3000.050.10.15 66 s post-wounding0 5 10 15 20 25 3000.050.10.15 60 s post-wounding0 5 10 15 20 25 3000.050.10.15 RhoACdc42W54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 1.4: Time sequence showing a single cell wound experiment. RhoAand Cdc42 zones rise, narrow, and translocate towards the wound centerduring wound healing. Late stages of the healing response (zone segregationand zone maintenance; 54? 84 s) are depicted in these radial concentra-tion profiles. The wound edge is denoted W. Rho GTPase intensities werescaled to concentrations using estimates of basal and elevated Rho GTPaseconcentrations found in literature (Simon et al., 2013).1.5.2 Patterning hints at network?s structureBy tracking the patterns in time and space, inferences about the signallingnetwork can be made. For instance, the temporal arrangement of each com-ponent in the pathway might be ascertained from observing which probesappear early (upstream) or late (downstream) in the response. The spatialpatterns, too, are telling of component interactions with each other. Ab-sences of one component where another component is concentrated mightindicate inhibition, whereas amplification in the presence of another compo-51.5. Cell signalling in Xenopus oocyte wound healingnent might indicate activation. Moreso, manipulations of the experimentalsystem reinforce our ideas about component interactions. Components canbe overexpressed (OE), or expressed as functionally impaired versions (DN,dominant-negative), with the resultant changes to patterning also suggestiveof regulatory roles. Indeed, the input-output nature of the system allowsfor perturbation of the signalling network so that we might understand thenetwork?s structure.Figure 1.5: The signalling network of RhoA and Cdc42 with interactionsmediated through Abr. Reprinted from ?Pattern formation of Rho GTPasesin single cell wound healing,? by C. M. Simon, 2013, Molecular Biology ofthe Cell, 24(3), p. 424. Copyright 2013 by the American Society for CellBiology. Reprinted with permission.Previous work (Simon et al., 2013) viewed Xenopus oocyte wound heal-ing as a pattern formation system. They focused on Rho GTPases latein the wound healing response, and asked whether the current hypothesesof Rho, Cdc42 and Abr (a dual GEF/GAP of Rho GTPase (Chuang et al.,1995)) interaction could explain the patterning (Vaughan et al., 2011). Mod-elling of a network involving RhoA, Cdc42 and Abr, captured key featuresin Rho GTPase patterns under control conditions, and several experimentalmanipulations (inhibition of RhoA by C3 exotransferase, overexpression ofGEF-dead or GAP-dead Abr, microinjection of Abr) (Figure 1.5). Modelvalidation occurred through predictive experiments involving two wounds atvarious distances apart. The model made subtle predictions on Rho GTPasepatterning in the space between the wounds. The in silico predictions wereconfirmed when two-wound experiments were performed in vivo. The au-thors concluded that Abr, as the chief mediator of Rho GTPase behaviour,was sufficient to account for key aspects of Rho GTPase patterning. Ana-61.5. Cell signalling in Xenopus oocyte wound healinglyzing cell signalling through the lense of pattern formation was shown tobe a valuable perspective.1.5.3 Thesis focusIn this chapter, we have introduced the actomyosin contractile ring in singlecell wound healing and its regulation by RhoA and Cdc42. Both occur latein wound healing, and we have alluded to earlier roles for lipids and lipidtargets. We have mentioned that these signalling components can be con-veniently probed and quantified in line scans. More importantly, line scansreveal spatiotemporal patterns (zones) that hint at the signalling network?sstructure. Recent work has tested a Rho GTPase signalling network and suc-cessfully recapitulated Rho GTPase patterning (Simon et al., 2013). Thisthesis extends the previous model, by Simon et al. (2013), of the signallingnetwork in Figure 1.5.The signalling network describes the following interactions: active RhoAbinds to Abr, Abr-bound active RhoA acts as a GEF for RhoA and as aGAP for Cdc42, active Cdc42 regulates itself (Figure 1.6, left). In addition,the network included background activation and background inactivationrates of both RhoA and Cdc42. This thesis is focused entirely on thesebackground (in)activation rates of Rho GTPases, which are clearly markedin Figure 1.6 (center). This thesis investigates PKC activation or inhibitionof the background (in)activation rates (Figure 1.6, right).71.5. Cell signalling in Xenopus oocyte wound healingModel 2GTPRhoA*GTPCdc42*Model 3AbrGEF GAPModel 1Model 2GTPRhoA*GTPCdc42*Model 3AbrGEF GAPModel 1Model 2GTPRhoA*GTPCdc42*Model 3AbrGEF GAPModel 1Figure 1.6: This thesis starts from the Rho GTPase signalling network ofSimon et al. (2013) (left), and extends it by investigating the effects ofPKCs on background (in)activation rates of RhoA and Cdc42 (center). PKCactivation or inhibition of background rates is investigated (right). PotentialPKC beta effects are shown in the blue arrows/bar-headed lines. PKC eta(not shown) has the opposite effect. Adapted from ?Pattern formation ofRho GTPases in single cell wound healing,? by C. M. Simon, 2013, MolecularBiology of the Cell, 24(3), p. 424. Copyright 2013 by the American Societyfor Cell Biology. Adapted with permission.We use the Rho GTPase network to investigate the roles and effects ofPKCs. In Chapter 2, we present in vivo experiments which motivate the re-search in this thesis. The in vivo experiments demonstrate PKC interactionswith Rho GTPases in the single cell system. In Chapter 3, we define theSimon et al. (2013) Rho GTPase model which is the starting point of thiswork. We introduce PKC effects with increasing spatial detail into the back-ground (in)activation rates of the model. A flow chart summarizes the modelvariants detailed in the indicated chapters (Figure 1.7). We begin with theSimon et al. (2013) model and look at the effect of spatially constant PKCson the background (in)activation rates. We graduate to examining spatiallydistributed PKCs and their effects on the background (in)activation rates.The spatially distributed PKCs are represented as simple step functions, oras explicit distributions given by data. We work with the explicit distribu-tions, revise some assumptions, and perform rigorous fits. As we shall see,we can improve on the control model, but, given current hypotheses, it isnot yet possible to account for all experimental manipulations. In the finalchapter, we suggest how future rounds of modelling can correct this.81.5. Cell signalling in Xenopus oocyte wound healingSimon's model (M1)CH 3 Spatially constant PKCs (M2)CH 4 Spatially distributed PKCsStep functions (M3, M4)CH 5 Explicit profiles (M5)CH 6Explicit profiles M5 w/ Michaelian term (M6, M7)CH 6Explicit profiles M7 w/ fitted wound edge (M8, M9)CH 7Explicit profiles M9 w/ constrained fitting (M10, M11)CH 7Figure 1.7: A flow chart summarizing model variants and the chapters inwhich they are discussed. PKC effects are introduced into the Rho GTPasebackground (in)activation rates in the Simon et al. (2013) model. Modelvariations incorporate increasing spatial detail and additional assumptions.9Chapter 2Protein kinase Cs (PKCs),lipids and their roles2.1 OverviewThis chapter presents in vivo experiments of Bement et al. (2012) that ini-tiated research described in this thesis. Briefly, the in vivo experimentsinvestigated the role of lipids in single cell wound healing. Lipid domainswere observed to converge around the wound. Of the five lipids investigated,diacylglycerol (DAG) was shown to be essential to Rho GTPase activation.Further investigation into DAG targets, protein kinase Cs, also showed pat-terning around the wound edge. PKCs were demonstrated to play opposingroles in wound healing. PKC beta enhanced RhoA and Cdc42 activity, whilePKC eta inhibited RhoA and Cdc42 activity.In these experiments, spherical Xenopus oocytes, approximately 1000 ?min diameter, were wounded by a laser. The circular wounds were consistentin size, at diameters of 10 ?m. Wounds closed under circumferential tensionover the course of a few minutes. Lipids, PKCs and Rho GTPases wereprobed. Three types of data were shown: images of the wound, line scansthrough the wound, and box-and-whisker plots quantifying zone intensities.Figures 2.1 and 2.2 were given to us summarizing the experimental results.For reference, we leave the figures intact but repeat panels throughout aswe highlight important results.102.1. OverviewFigure 2.1: Five lipids characterized by time to appearance, relative toCdc42 zone formation (15? 20 s) (Figure from Bement et al. (2012)). Lipiddomains of phosphoinositide triphosphate (PIP3), phosphoinositide bispho-sphate (PIP2) and phosphatidylserine (PS) organize around the wound edgeat late times (A-C). Lipid domains of phosphatidic acid (PA) and diacyl-glycerol (DAG) organize around the wound edge at early times (D, E). Lipiddistributions at 90 s, relative to the Cdc42 zone (A?-E?). Relative positionsof lipid compartments by pairwise comparison (F-I). Lipid compartment po-sitions in relation to the wound edge (J). Scale bar = 20 ?m. W = Woundedge. Panels are individually reproduced and discussed in the text. 112.1. OverviewFigure 2.2: Localization of PKC beta and PKC eta (A) and their distribu-tions at 90 s (Figure from Bement et al. (2012)). Overexpression (OE), orexpression of dominant-negative (DN) variants, of PKC beta or PKC etashowing influence on RhoA and Cdc42 zone activation (C). Quantificationof average RhoA (green) and Cdc42 (red) zone intensities at 30 s and 60 sfor each manipulation (D, F, H). Quantification of average RhoA and Cdc42zone and background intensities at 60 s for manipulations which show latezone formation (E, H). Top and bottom whiskers indicate maximum andminimum intensities, respectively. Scale bar = 20 ?m. W = Wound edge.Panels are individually reproduced and discussed in the text. 122.2. Diacylglycerol (DAG) overlaps the RhoA zone2.2 Diacylglycerol (DAG) overlaps the RhoAzoneOut of five lipids probed in the study, diacylglycerol was established to beessential to Rho GTPase activation. The black and white time sequenceof wound images in Figure 2.3 E depict probes for DAG (white). DAGappeared early, relative to the time of Cdc42 zone formation (at 15? 20 s),suggesting a role for DAG upstream of Rho GTPase activation. Perturbationto the DAG generation pathway supported that DAG is indeed essential toRho GTPase activation (not shown). Since DAG was necessary for woundhealing, the next component investigated was a DAG target.Before considering DAG targets, however, we should also note DAG lo-calization from Figure 2.3 E?. At 90 s, wound images show DAG probes(green) in relation to Cdc42 (red). The Cdc42 zone consistently forms 5 ?mfrom the wound edge, and is used to gauge relative positions. The corre-sponding line scan displays the spatial profile of DAG and Cdc42, with thewound edge towards the right. Here, it is important to notice that DAGlocalizes to the same region where the RhoA zone would appear.Figure 2.3: Diacylglycerol appears to accumulate around the wound at earlytimes, relative to the formation of the Cdc42 zone (E). DAG localizes toregions close to the wound edge, where the RhoA zone is present (E?). TheCdc42 zone is used to mark a 5 ?m distance from the wound edge. Wdenotes the wound edge. Note that the orientation of the axis is reversedfrom later simulations, where the wound is on the left.2.3 PKCs bind DAG, localizing to regionscomprising Rho GTPase zonesDiacylglycerol is a well-known activator of a family of enzymes called proteinkinase C (PKC). As a result, the DAG targets, PKCs, were subsequentlyinvestigated. Members of the PKC family are grouped into three classes ?conventional, novel, and atypical ? based on their activation requirements.132.3. PKCs bind DAG, localizing to regions comprising Rho GTPase zonesConventional PKCs must bind to both Ca2+ and DAG in order for catalyticactivation to occur. Novel PKCs must bind only to DAG. Atypical PKCsrequire neither Ca2+ nor DAG for their activation. The PKCs consideredhere are PKC beta (conventional) and PKC eta (novel). They both requireDAG as an activator, and must compete for DAG in cell wounding.Figure 2.4 shows images and line scans of the PKCs (green) in relationto the Cdc42 zone (red). In Figures 2.4 A and A?, PKC beta localizes toregions which overlap both the RhoA and Cdc42 zones. In Figures 2.4 Band B?, PKC eta localizes to a region which overlaps only the RhoA zone.Since DAG overlaps the RhoA zone, and both PKCs must bind DAG inorder to be activated, it is no surprise that both PKC beta and PKC etaalso overlap the RhoA zone.Figure 2.4: Localization of PKC beta and PKC eta relative to the Cdc42zone (A, B). PKC beta overlaps regions of both RhoA and Cdc42 zones (A?).PKC eta overlaps the region containing only the RhoA zone (B?).Simultaneous probing of PKC beta (green) and PKC eta (red) revealbroad overlapping zones that narrow into well-defined, discrete zones ashealing progresses (Figure 2.5 A). A line scan at 90 s shows that PKC betalocalizes to a broader region than PKC eta (Figure 2.5 B). The takeawaymessage on PKC localization is that PKC beta is targeted to a broad regioncomprising both Rho GTPase zones, while PKC eta is targeted to a narrowregion comprising only the RhoA zone.142.4. PKC beta and PKC eta play opposing roles in RhoA and Cdc42 activationFigure 2.5: Time sequence of PKC beta and PKC eta localization showbroad zones which narrow as healing progresses (A). Line scans comparethe broad PKC beta localization to the narrow PKC eta localization (B).2.4 PKC beta and PKC eta play opposing rolesin RhoA and Cdc42 activationIn the previous sections, we discussed the spatial profiles of DAG and PKCswith respect to RhoA and Cdc42 localization. Since they were probed ina control wounding experiment, we observed the localization of endogenouscomponents. The next panels will show manipulations to the system, andthe effect of exogenous PKCs on Rho GTPase activation.The subsequent experiments probed Rho GTPases and monitored forchanges in zone activity in response to a PKC manipulation. PKCs canbe manipulated through overexpression (OE), or through expression of adominant-negative (DN) version. Overexpression of PKC means more PKCwithin the cell, resulting in increased PKC effect. Conversely, a DN PKC isfunctionally impaired and, when expressed, results in decreased PKC effect.In either manipulation, PKCs retain their capacity to localize. We laterassume that exogenous PKCs (OE or DN manipulation) localize in the sameway as endogenous PKCs (control).The following sections show images of wounds with RhoA in green andCdc42 in red. Each manipulation is labelled as ?OE PKC beta?, for example.The images show qualitative changes to Rho GTPase zone activities that aremeant to be compared against the control (Figure 2.6). An accompanyingbox-and-whisker plot quantifies the changes in zone activities in comparisonto the control.152.4. PKC beta and PKC eta play opposing roles in RhoA and Cdc42 activationFigure 2.6: Time sequence images of a control wound experiment whereRhoA (green) and Cdc42 (red) are probed.2.4.1 PKC beta positively influences Rho GTPaseactivationFigure 2.7: Elevated zone activities of RhoA (green) and Cdc42 (red) whenPKC beta is overexpressed. Quantification of zone activities at 30 s and 60 s(D).When PKC beta is overexpressed, both Rho GTPase zones appear brighter(Figure 2.7). The zone quantification shows higher RhoA zone activity at30 s, compared to control, and higher Cdc42 zone activity at 30 s and 60 s(Figure 2.7 E). We express the difference in zone activities, between themanipulation and control, as fold changes in Table A.1. These are laterused in simulations as zone amplification factors.162.4. PKC beta and PKC eta play opposing roles in RhoA and Cdc42 activationFigure 2.8: Diminished zone activities of RhoA (green) and Cdc42 (red)when dominant-negative PKC beta is expressed. Quantification of zoneactivities and background activities at 60 s (E).When dominant-negative PKC beta is expressed, both Rho GTPasezones appear diminished (Figure 2.8). Since zones are not visible at 30 s,zone quantification at only 60 s can be done. The box-and-whisker plotcompares zone activities, as well as background activities, to the control.Both RhoA and Cdc42 zone activities are lower than control. No signifi-cant changes to background activities are observed. Table A.2 expresses thedifference in zone activities as fold changes.In summary, the overexpression of PKC beta results in upregulation ofboth RhoA and Cdc42 zone activities, while the expression of dominant-negative PKC beta results in downregulation of both RhoA and Cdc42 zoneactivities. PKC beta positively influences both RhoA and Cdc42 zone acti-vation.172.4. PKC beta and PKC eta play opposing roles in RhoA and Cdc42 activation2.4.2 PKC eta negatively influences RhoA and Cdc42activationFigure 2.9: Diminished zone activities of RhoA (green) and Cdc42 (red)when PKC eta is overexpressed. Quantification of zone activities at 30 sand 60 s (F).In comparison to manipulations of PKC beta, manipulations involving PKCeta produce opposite effects on the Rho GTPase zones. In short, overexpres-sion of PKC eta results in qualitatively and quantitatively diminished zones(Figure 2.9). Expression of dominant-negative PKC eta results in qualita-tively and quantitatively elevated background activities (Figure 2.10). Thus,PKC eta negatively influences both RhoA and Cdc42 zone activation.Figure 2.10: Elevated background activities of RhoA (green) and Cdc42(red) when dominant-negative PKC eta is expressed. Quantification of zoneactivities and background activities at 60 s (G).Overall, these results indicate that PKC beta promotes the activation ofRhoA and Cdc42. Conversely, PKC eta inhibits the activation of RhoA andCdc42.182.5. Main questions to be addressedA subset of OE PKC eta manipulations result in inverted RhoAand Cdc42 zonesIn the control and the majority of experiments where PKC eta is overex-pressed, RhoA encircles the wound, and Cdc42 encircles the RhoA zone(Figure 2.6). However, approximately one-fifth of PKC eta overexpressionexperiments result in a surprising swap of RhoA and Cdc42 zones (Fig-ure 2.11). The inverted zones are compelling since they are an unusualresult that might serve as a way to evaluate hypotheses we make about thesignalling network. A successful hypothesis should explain these invertedzones.Figure 2.11: One-fifth of PKC overepxression experiments result in invertedzones where Cdc42 encircles the wound, with RhoA outside the Cdc42 zone.2.5 Main questions to be addressedWhat picture of single cell wound healing do these data paint for us? In acontrol wounding experiment, DAG localizes around the wound, overlappingthe RhoA zone. Both PKC eta and PKC beta bind to DAG in the RhoAzone, though PKC beta broadly extends over the Cdc42 zone as well. PKCbeta and PKC eta exert opposing effects on both zones of RhoA and Cdc42activation. In an experiment where PKCs are manipulated, we assume theexogenous PKCs localize in the same way as endogenous PKCs. Overex-pression of PKC means there are more PKCs within the cell competing forDAG, and exerting more effect on the GTPases. Expression of dominant-negative PKC means that PKC function is impaired, and their effect on RhoGTPases is diminished.We plan to use the Simon et al. (2013) model and test hypotheses abouthow PKCs exert their influence. We formulate questions that will be ad-dressed throughout this thesis.1. Does PKC beta upregulate Rho GTPase activity by enhancing activa-tion (through a GEF), or by depressing inactivation (through a GAP)?A similar question can be asked about PKC eta.192.5. Main questions to be addressed2. Is the spatial distribution of PKC beta and PKC eta important inaccounting for the observed RhoA and Cdc42 zones?3. Why does PKC eta only appear in the RhoA zone? Is is possible thatPKC eta influences RhoA, but that RhoA also influences PKC eta inorder to maintain the discrete eta zone?4. If PKC eta is present only in the RhoA zone, how does it exert influenceover the Cdc42 zone?5. In what ways does DAG modulate PKC effects on Rho GTPases?The following chapters address these questions in the above order. Chap-ter 4 addresses how PKCs affect Rho GTPases, and shows degeneracies be-tween GEF and GAP regulation. Chapter 5 implements spatial distributionsof PKCs which further matches the simulated RhoA zone to the observedRhoA zone. Chapter 5 also implements a feedback from RhoA to PKC etawhich we ultimately reject. Chapter 6 increases the spatial detail in PKCdistributions and also attempts to answer how PKC eta exerts influence overthe Cdc42 zone. Chapter 6 places limits on PKC overexpression due to afinite amount of DAG. See Figure 1.7 for a schematic flow chart of the mod-elling and organization of the thesis. So far, as described in this thesis, theincreasingly detailed model variants are not able to account for the RhoAand Cdc42 zone inversion.20Chapter 3Models: definitions,background, and logicalframework3.1 OverviewIn this chapter, the mathematical model is defined and its parameters are ex-plained. We give background on how it was built, and outline model features(i.e., crosstalk and spatial bistability). We discuss the model?s successes andlimitations. We then discuss our use of the model and the revisions that wemake to it. Finally, we give criteria for model validation.3.2 Defining the Simon et al. (2013) Rho GTPasemodelOur tool for probing the PKC connection to Rho GTPase activation is aset of reaction-diffusion-advection equations (Simon et al., 2013). Theseequations track the concentrations of three main species ? the Rho GTPasesRhoA and Cdc42, and the dual GEF/GAP Abr ? through time and space.The active forms of Rho GTPases are modelled over a two-dimensional cellsurface that is represented in radial coordinates r, centred on the wound(Figure 3.1). Each species in the model reacts, diffuses and advects accordingto?[G?]?t=1r??r????rD?[G?]?r? ?? ?Diffusion+ rv[G?]? ?? ?Advection????+ f([G?], [G])? ?? ?Reaction(3.1)where G denotes the species and an asterisk denotes the active form. Thereaction term f([G?], [G]) describes activation and inactivation of G.213.2. Defining the Simon et al. (2013) Rho GTPase model10 ?m1000 ?mr0 5 10 15 20 25 3000.050.160 s post-woundingFigure 3.1: RhoA (green) and Cdc42 (red) zones accumulate around thewound on the plasma membrane (left). RhoA and Cdc42 concentrationsare tracked in the model through time and space. The two-dimensionalspatial domain is represented in radial coordinates where r = 0 (in ?m)corresponds to the wound center (right).Active Rho GTPases diffuse in the plane of the plasma membrane (withdiffusion coefficient D). They also exchange with inactive GTPases. Sincethe wounding GTPase pattern is very small relative to the cell size, weassume that inactive GTPases are found in excess and at constant level inthe cell. As the wound closes, the membrane advects radially inwards bypurse-string closure, leading to GTPase advection. Advection occurs withvelocity v = vc(t)r which satisfies the continuity equation, and agrees withwound edge data.The Rho GTPase reaction kinetics are summarized in Figure 3.2. Ac-tive RhoA binds to Abr, which acts as a GEF for RhoA, creating a positivefeedback loop between RhoA and Abr (Figure 3.2 Model 1). Abr simul-taneously acts as both GEF and GAP for Cdc42, but with much strongerGAP activity on Cdc42 (Figure 3.2 Model 2). Cdc42 positively feeds backon itself (Figure 3.2 Model 3).223.2. Defining the Simon et al. (2013) Rho GTPase modelFigure 3.2: The signalling network of RhoA and Cdc42 with interactionsmediated through Abr. Reprinted from ?Pattern formation of Rho GTPasesin single cell wound healing,? by C. M. Simon, 2013, Molecular Biology ofthe Cell, 24(3), p. 424. Copyright 2013 by the American Society for CellBiology. Reprinted with permission.The full model from Simon et al. (2013) describing Abr-bound RhoA[A-R?], RhoA [R?], and Cdc42 [C?] is given below.?[A-R?]?t=L[A-R?] + k3[R?]? k4[A-R?] (3.2)?[R?]?t=L[R?] +(kr0 +k1[A-R?]nKnA + [A-R?]n)[R]? (k2 + k3)[R?] (3.3)?[C?]?t=L[C?]+(kc0+k5[A-R?]+k6[C?]nKnC+[C?]n)[C]?(k7+k8[A-R?])[C?](3.4)where L[G?] =1r??r(rD?[G?]?r+ rv[G?])produces the diffusion-advectionterms. We hypothesize that PKCs affect the terms shown in magenta, i.e.,the basal rates of activation or inactivation of the GTPases.Simon et al. (2013) specifically focused on the effect of the GEF/GAPAbr, neglecting other GEF and GAP activity. In Equations 3.3 and 3.4,Abr appears as a GEF for RhoA and Cdc42. Abr activates RhoA at arate k1 in a Michaelian way, and linearly activates Cdc42 at a rate k5. InEquation 3.4, Abr acts as a GAP for Cdc42 and linearly inactivates Cdc42at a rate k8. Equation 3.2 describes Abr itself, which binds to RhoA at ratek3 and unbinds at rate k4. The remaining Michaelian term describes Cdc42activation of itself at a rate k6. The remaining linear terms describe thebasal activation (kr0, kc0) or basal inactivation (k2, k7) of RhoA and Cdc42.233.2. Defining the Simon et al. (2013) Rho GTPase modelHere we explore the hypothesis that PKCs affect the basal rates of activationor basal rates of inactivation of Rho GTPases.A control wound healing experiment is simulated using the model andestimated parameters from Table 3.1 (Figure 3.3). Simulated curves areshown against Rho GTPase data plotted as circles (thick curves). The sim-ulated total amount of RhoA (i.e., the sum of active RhoA and Abr-boundRhoA) is plotted in green. Simulated Cdc42 and Abr correspond to red andblack, respectively. The simulated wound edge is shown as a vertical dashedline, against wound edge data (vertical solid line).0 5 10 15 20 25 3000.050.1 84 s post-wounding0 5 10 15 20 25 3000.050.1 78 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 3.3: The Simon et al. (2013) model captures basic features inRho GTPase zones such as the speed of amplification and the steadystates. RhoGTPases 54? 84 s post-wounding are simulated (lines), withdata shown in circles for comparison. The simulated wound edge is given bythe vertical dashed black line, to be compared with the vertical solid blackline. Notice that the orientation of the axis is reversed (wound edge towardsthe left) in comparison to previous line scan data.243.2. Defining the Simon et al. (2013) Rho GTPase modelSimulations begin with initial conditions specifying each species? con-centration profile. The initial conditions are piecewise linear (triangular)profiles fitted to Rho GTPase data. Simulations are computed over a 50 ?mdomain that begins at the wound edge. Since disrupted plasma membranecannot support Rho GTPases, a no flux boundary condition is imposed atthe wound edge. Far from the wound, Rho GTPases are fixed to basal ac-tivities. The wound edge position is given by w, and the wound edge moveswith a prescribed velocity, dwdt = ?vcw . Data on wound edge position throughtime was used to set the constant vc.Parameter Interpretation Value Unitkr0 Basal RhoA activation 0.0091 /skc0 Basal Cdc42 activation 0.0018 /sk1 Maximum GEF activity of Abr on RhoA 0.0241 /sKA Measure of switch location in Hill eqn. 0.0090 ?MKC Measure of switch location in Hill eqn. 0.0635 ?Mk2 GAP/GDI inactivation rate of RhoA 1 /sk3 Abr binding rate to RhoA 0.1 /sk4 GAP/GDI inactivation rate of Abr-RhoA 6/9 /sk5 Abr GEF activity on Cdc42 2.0802 /sk6 Maximum autocatalysis rate of Cdc42 0.0326 /sk7 Background GAP/GDI inactivation rate of Cdc42 0.6889 /sk8 Abr GAP activity on Cdc42 79.5006 /svc Advection velocity parameter 2.1231 ?m2/sn Hill coefficient 6Table 3.1: Parameters in Model 1 from Simon et al. (2013).In order to conveniently keep track of changes between model variants,we provide model definitions. All model variants start from Model 1 (M 1)base assumptions. Subsequent model definitions note modifications to thebasic assumptions.Model 1. The Simon et al. (2013) model.Equations 3.2-3.4Parameters Table 3.1Initial conditions Piecewise linear (triangular) profiles fittedto control data.253.2. Defining the Simon et al. (2013) Rho GTPase modelBoundary conditions No flux condition at wound edge, basalactivity far from the wound.Advection velocity Constant vc such that initial and final woundedge locations agree with data.3.2.1 Model featuresThe equations in Model 1 were developed by Simon et al. (2013) in a sys-tematic way. Simon began with simple, linear reaction terms which weresuccessively replaced by complex, nonlinear terms until key Rho GTPasefeatures were captured. The two key features were crosstalk and spatialbistability.Crosstalk is the inhibition of one species by another. In the contextof single cell wound healing, crosstalk is evident in zone segregation whereinitially overlapping RhoA and Cdc42 zones narrow into mutually exclusivezones. Through inhibition, RhoA excludes Cdc42 from regions that it occu-pies. This crosstalk feature is present in the model as Abr-mediated RhoAinhibition of Cdc42.Spatial bistability refers to Rho GTPase activities occupying either a sta-ble high steady state or a stable low steady state across the spatial domain.High and low steady states are separated by a threshold. Activities belowthe threshold rapidly fall down to the low steady state, while activities abovethe threshold rapidly rise up to the high steady state. Switching betweenthe two requires significant perturbation. Spatial bistability is present inthe model as the Rho GTPase zones: a peak at high steady state within thezone, and background activity (low steady state) elsewhere.3.2.2 Successes and limitations of the modelModel 1 successfully simulates Rho GTPase zones in a control experiment(Figure 3.3). It captures zone segregation and zone maintenance from54? 84 s post-wounding. In the simulation, the initial profiles narrow intozones that are rapidly amplified. The speed of rise in both zones, as well aspeak and background activities, compare well with data.While the approximate patterns are accounted for, the model is unableto capture several features. For one, the simulated zones are too narrowat the base, and the Cdc42 zone is offset from the RhoA zone (Figure 3.3,84 s). Then, if we look at the amount of Cdc42 within the RhoA zone,the simulated Cdc42 activity shows a ?bump? above background. We knowthis bump should not be present since the data shows Cdc42 at background263.2. Defining the Simon et al. (2013) Rho GTPase modellevels within the RhoA zone. On the other hand, if we look at the amountof RhoA within the Cdc42 zone, the simulated RhoA activity is as low asbackground. The data instead show a ?shoulder? of RhoA within the Cdc42zone, above background. We find that the inclusion of PKCs refines RhoGTPase patterning. In particular, we improve the shape of each zone andcapture the RhoA shoulder. Correcting the Cdc42 zone offset and bumprequire Abr-related changes which we do not consider.It is worth mentioning that the RhoA shoulder is a subtle, yet relevant,detail in patterning. In wounded cells where contraction has been inhibited,a faint rim of RhoA is visible outside the ring of Cdc42. At its most extreme,the rim can be very broad, manifesting as zones on either side of the Cdc42zone. The rim of RhoA is very likely the RhoA shoulder which peeks outfrom underneath the Cdc42 zone.Model 1 is also limited to portraying wound healing during 54? 84 s.It does not recapitulate wound healing from the very beginning, because itwas not intended to model calcium or early events in signalling. It does notrecapitulate zone formation at times slightly before 54 s because the RhoGTPase gradient is too shallow for amplification, and because componentsother than Abr likely play a role. Lastly, the model does not account fortimes after 84 s, due to broadening of the simulated zones. When the simu-lated zones rise up and plateau, the trailing edge bleeds outwards due to aconstant supply of inactive, cytosolic GTPase fuelling activation. As such,discrete zones are no longer retained after 84 s. Biologically, a non-Abr GAPis hypothesized to act on the trailing edge of the RhoA zone (Vaughan et al.,2011). Thus, the model which accounts for Rho GTPase patterning throughAbr, is limited to simulating this window of time. Incorporating PKCs doesnot expand this window, but we suspect that they assist in maintainingdiscrete zones by curbing the outwards spread of the trailing edge.273.3. The PKCs ? revising the basic model3.3 The PKCs ? revising the basic modelModel 2GTPRhoA*GTPCdc42*Model 3AbrGEF GAPModel 1k0rk0ck2k7Figure 3.4: The effects of PKCs on background Rho GTPase activationrates kr0, kc0, and background inactivation rates k2, k7. Potential PKC betaeffects are shown in the blue arrows/bar-headed lines. PKC eta (not shown)has the opposite effect. Adapted from ?Pattern formation of Rho GTPasesin single cell wound healing,? by C. M. Simon, 2013, Molecular Biology ofthe Cell, 24(3), p. 424. Copyright 2013 by the American Society for CellBiology. Adapted with permission.How do we investigate the effects of PKCs on Rho GTPases? We begin withModel 1 and assume that PKCs affect Rho GTPases through backgroundGEFs and GAPs. That is, PKCs affect the background (in)activation ratesof Rho GTPases: kr0, kc0, k2, k7 (Figure 3.4). We progress from model tomodel by increasing the spatial detail of the PKCs, and hence, the spatialdependence of the background (in)activation rates (See flowchart in Fig-ure 1.7). In Chapter 4, we assume that PKCs are spatially uniform acrossthe domain. In Chapter 5, we assume that PKCs localize to the RhoA zone,and represent their spatial profiles as simple step functions. In Chapters6 and 7, we assume that PKCs overlap regions where the RhoA zone ispresent, and that their spatial profiles are given explicitly by data.With each model variant, we simulate the control and fit the model pa-rameters. In order to validate the model variant, we reproduce a battery ofPKC manipulations by making fold changes in parameters kr0, kc0, k2 and283.3. The PKCs ? revising the basic modelk7. The simulated Rho GTPase zones are model outputs evaluated againstquantitative and qualitative criteria listed in the next section. Ideally, in-verted zones in the overexpression of PKC eta would also be reproduced.Under the current assumptions, we do not successfully reproduce thePKC manipulations. In the final chapter, we make suggestions for futuremodelling of PKCs through the basal rates of Rho GTPase activation. Inprinciple, however, the PKCs can eventually be modelled as dynamic vari-ables.3.3.1 Criteria for model validationWith each model variant, we evaluate how well it reproduces Rho GTPasezones in the set of PKC manipulations. A manipulation is recapitulated ifthe following criteria are met.C1 Given appropriate initial Rho GTPase spatial profiles, zones are main-tained.C2 The quantitative fold changes to zone activities, relative to the control,match the extent seen in data (Tables A.1-A.4).C3 Qualitative features such as zone breadth, gaps, inversion and otherwise,match experimental images (Figure 2.2 C).29Chapter 4PKCs as spatially constantparameters4.1 OverviewIn this chapter, we consider a simplified view in which the effect of thePKCs is uniform in space, ignoring their known spatial distribution. UsingModel 1, we first investigate two hypotheses for the effect of PKCs:(i) PKCs modulate the basal Rho GTPase activation rate.(ii) PKCs modulate the basal Rho GTPase inactivation rate.The first possibility implies that PKCs act on GEFs, while the second impliesGAPs. Next, we simulate PKC manipulations (OE, DN) and find that thecurrent assumptions do not explain their effect on the Rho GTPase zones.We conclude that PKC localization is necessary.4.2 PKCs act on GEFs or GAPsWe are interested in whether Rho GTPases are affected by PKCs throughGAPs or GEFs. For instance, PKC beta could upregulate Rho GTPaseactivity either by activating GEFs, or by depressing GAPs. GEF (GAP)regulation is synonymous with the GTPase basal (in)activation rates kr0, kc0(k2, k7). To look at differences between GEF and GAP regulation by PKCs,we examine bifurcation diagrams with respect to each parameter (kr0, kc0, k2,k7) in turn1.If we consider a space-free version of Equations 3.2-3.4, we are left withordinary differential equations (ODE) which describe Rho GTPase reactionkinetics. By carrying out a bifurcation analysis of the ODEs, we find two1Aside from the bifurcation parameter, all other parameters retain values from Ta-ble 3.1.304.2. PKCs act on GEFs or GAPsstable RhoA steady states and three stable Cdc42 steady states (s.s.). Fig-ure 4.1 illustrates that simulated RhoA has a high s.s. within the zone, andlow s.s. at background. Simulated Cdc42 has similar high and low steadystates, but also possesses a third stable s.s. which manifests as the Cdc42bump in Figure 4.1.0 5 10 15 20 25 3000.050.1 84 s post-woundingDistance from wound center (?m)Concentration(?M) 0 5 10 15 20 25 3000.050.1 84 s post-woundingFigure 4.1: The simulated control (Model 1) possesses two stable steadystates in RhoA and three stable steady states in Cdc42. RhoA (green) andCdc42 (red) have high steady states within the zone, and low steady statesat background. Cdc42 has an additional steady state in the Cdc42 bump.314.2. PKCs act on GEFs or GAPsTo ease comprehension of a bifurcation diagram, we show the bifurcationdiagram of RhoA steady states on kr0 (Figure 4.2). The normalized controlparameter value (kr0 = 1) is marked with a vertical violet line which crossesthrough two stable branches (solid green) and an unstable branch (dashedgreen). Arrows mark features in patterning which correspond to the stablesteady states. The RhoA threshold corresponds to an unstable steady state.As the parameter kr0 is increased (right of violet line), the unstable steadystate and low steady state converge and disappear. A fold change in kr0greater than 1.37 (dashed-solid black vertical line) results in a single highsteady state, and thus RhoA monostability. Conversely, a fold change inkr0 less than 0.87 (dashed black vertical line) results in a single low steadystate, and also RhoA monostability. Consequently, the region within a foldchange of 0.87-1.37 in kr0 is a regime where bistability is possible. Bistabilityis necessary for zone maintenance and thus we only consider this range ofpossible variation of kr0 in the next simulations.0 5 10 15 20 25 3000.050.1 84 s post-wounding0 0.25 0.5 0.75 1 1.25 1.5 1.75 2Fold change in background activation rate of RhoA, kR [s-1]00.0250.050.0750.10.1250.15Concentration(?M) Cdc42 stable branchCdc42 unstable branchRhoA stable branchRhoA unstable branchbistable regionFigure 4.2: A bifurcation diagram of kr0 showing RhoA steady states. Atthe control value (vertical violet line), RhoA possesses stable low and highsteady states which correspond to the background and zone activities, re-spectively (violet arrows). The unstable steady state corresponds to thebistable threshold. The interval 0.87-1.37 marks the bistable region (verti-cal dashed lines).324.2. PKCs act on GEFs or GAPsThe bifurcation diagram of Cdc42 steady states on kr0 can be read in asimilar manner (Figure 4.3). Notice the presence of the third stable steadystate that corresponds to the Cdc42 bump. Additionally, the region of Cdc42bistability extends from 0-1.37 fold change in kr0. However, the bistableregion that we indicate in the diagram is narrower, because we indicate theregime where both RhoA and Cdc42 exhibit bistability. The vertical dashed-solid black line marks the loss of both RhoA and Cdc42 bistability, while thevertical dashed black line marks the loss of only one GTPase?s bistability.Subsequent bifurcation diagrams show RhoA and Cdc42 steady states in thesame plot.0 5 10 15 20 25 3000.050.1 84 s post-wounding0 0.25 0.5 0.75 1 1.25 1.5 1.75 2Fold change in background activation rate of RhoA, kR [s-1]00.0250.050.0750.10.1250.15Concentration(?M) Cdc42 stable branchCdc42 unstable branchRhoA stable branchRhoA unstable branchbistable regionFigure 4.3: A bifurcation diagram of kr0 showing Cdc42 steady states. Asin Figure 4.2, but a third intermediate stable steady state is shown (violetarrows). Cdc42 is bistable over the interval 0-1.37, but only the intervalwhere both GTPases retain bistability is called the bistable region (verticaldashed lines).334.2. PKCs act on GEFs or GAPs4.2.1 Bifurcations on the basal rates of RhoA(in)activation, kr0 (k2)0 0.25 0.5 0.75 1 1.25 1.5 1.75 2Fold change in background activation rate of RhoA, kR [s-1]00.0250.050.0750.10.1250.15Concentration (?M)Cdc42 stable branchCdc42 unstable branchRhoA stable branchRhoA unstable branchbistable region0 0.25 0.5 0.75 1 1.25 1.5 1.75 2Fold change in background inactivation rate of RhoA, k2 [s-1]00.0250.050.0750.10.1250.15Concentration (?M)Cdc42 stable branchCdc42 unstable branchRhoA stable branchRhoA unstable branchbistableregionFigure 4.4: Bifurcation diagrams (space-free Equations 3.2-3.4) with respectto RhoA basal activation (top) and inactivation (bottom). Stable branchestowards the top (bottom) of each diagram indicate high (low) steady states.Cdc42 possesses an additional intermediate steady state. Bistable region(both GTPases) marked by vertical dashed lines.344.2. PKCs act on GEFs or GAPsWe first consider the bifurcation diagram on the background activation rateof RhoA (Figure 4.4, top). Within the bistable region, increasing kr0 leadsto higher RhoA zone activity, and higher RhoA and Cdc42 backgroundactivities. With increasing RhoA zone activity also comes a lowering of thethreshold. Lower thresholds tend to result in broad zones, as we shall see inthe simulations. Outside of the bistable region, two scenarios occur. EitherRhoA (monostable) fills the domain at high background activity with Cdc42at low background activity, or RhoA fills the domain at low backgroundactivity with zones of Cdc42 still possible.A similar bifurcation diagram on the background inactivation rate ofRhoA is shown in Figure 4.4 (bottom). The k2 bifurcation diagram appearsas a mirror image of the kr0 diagram, though not all features are exactlyreflected. The k2 diagram possesses a narrower bistable region and a verysteep stable RhoA branch. It appears that RhoA activity can be tuned withsimilar effects through either RhoA basal rate. However, k2 is restricted toa narrower bistable region, which means reduced prospects of zone mainte-nance under manipulations that change k2.4.2.2 Bifurcations on the basal rates of Cdc42(in)activation, kc0 (k7)In the bifurcation diagram on the background activation rate of Cdc42 (Fig-ure 4.5, top), we first notice that RhoA steady states are unaffected by Cdc42parameters. Therefore, RhoA is bistable for all changes in the Cdc42 basalrate. Within the bistable region, increasing kc0 leads to higher Cdc42 zoneand background activities, with an accompanying decline of the thresholdvalue. Outside the bistable region, Cdc42 fills the domain at high back-ground activity, while RhoA zones are possible.Again, a similar mirrored bifurcation diagram on the background inac-tivation rate of Cdc42 occurs (Figure 4.5, bottom). It appears that Cdc42activity can be tuned through either Cdc42 basal rate. However, some valuesof kc0 are inaccessible since they fall below zero. As well, k7 possesses a steepstable upper branch allowing for Cdc42 zones with much higher activity.354.2. PKCs act on GEFs or GAPs0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5Fold change in background activation rate of Cdc42, kC [s-1]00.0250.050.0750.10.1250.15Concentration (?M) RhoA stable branchCdc42 stable branchCdc42 unstable branchbistable region0 0.25 0.5 0.75 1 1.25 1.5 1.75Fold change in background inactivation rate of Cdc42, k7 [s-1]00.0250.050.0750.10.1250.15Concentration (?M) RhoA stable branchCdc42 stable branchCdc42 unstable branchbistable regionFigure 4.5: As in Figure 4.4 but the intermediate Cdc42 steady state is notshown. RhoA is unaffected by either Cdc42 basal (in)activation rate, andtherefore RhoA retains bistability throughout each diagram.364.3. Simulations of PKC manipulationsOverall, these bifurcation diagrams show degeneracies between the basalactivation and inactivation rates. Discriminating between GEF and GAPregulation might entail further interpretation of the narrowness of eachbistable region, or consideration of the steepness of each diagram?s upperstable branch. We cannot immediately rule out that PKCs exclusively acton a GEF or GAP, so we test how changing any of the four parameters kr0,kc0, k2 and k7 affects RhoA and Cdc42 activities.4.3 Simulations of PKC manipulationsIn this section, we simulate the PKC manipulations using information ob-tained from the bifurcation diagrams (Figures 4.4, 4.5). Simulating spatiallyuniform PKCs does not require changes to the equations in Model 1, andonly requires fold changes to control initial conditions and control parametervalues. We can define Model 2 in the following way:Model 2. The spatially uniform PKC model.Parameters Fold changes in the bistable region according tobifurcation diagrams (Figures 4.4, 4.5).Initial conditions Piecewise linear (triangular) profiles fittedto control data, and scaled by amplification factors in Ta-bles A.1-A.4 at 30 s.The Model 2 control simulation is the same as the control from Model 1since no fold changes to control parameter values or initial conditions aremade. Thus, the Model 2 control simulation is shown in Figure 3.3.We reproduce PKC manipulations by specifying initial conditions andthe parameter fold changes which mimic overexpression or expression ofdominant-negative PKC. The initial conditions have the same shape as thecontrol profiles, but are amplified by factors in Tables A.1-A.4 (at 30 s).Recall that the amplification factors were derived from the box-and-whiskerplot zone quantifications (Figure 2.2). Parameters are adjusted according tothe significant experimental outcome we aim to reproduce, e.g., an upregu-lated Cdc42 zone is achieved by decreasing Cdc42 background inactivationrate. In order to guarantee zone maintenance, the corresponding parameterfold changes are chosen from the bistable regions given by the bifurcationdiagrams (Figures 4.4-4.5).To evaluate whether the simulated manipulations are successful, we referto criteria in Section 3.3.1. Zone maintenance (C1) and qualitative aspects374.3. Simulations of PKC manipulationsof the simulated zones (C2) can be visually scrutinized. We quantitativelyassess the fold change in simulated zone activities at 84 s (C3), relative tothe control, and compare them to fold changes in Tables A.1-A.4 at 60 s. Assuch, the simulated GTPases in a PKC manipulation are plotted over thesimulated control so that changes in zone activities can be readily compared.4.3.1 Overexpression of PKC betaSimulated results are shown in Figure 4.6. In the 54 s panel we show theadjusted initial conditions based on Table A.1. In all panels, the controlsimulation is shown for comparison (circles/thick curves). Experimentally,the overexpression of PKC beta results in a significantly upregulated Cdc42zone, so we decrease the background inactivation rate k7 (fold change of0.625).0 5 10 15 20 25 3000.050.184 s post-wounding0 5 10 15 20 25 3000.050.178 s post-wounding 0 5 10 15 20 25 3000.050.172 s post-wounding0 5 10 15 20 25 3000.050.166 s post-wounding 0 5 10 15 20 25 3000.050.160 s post-wounding0 5 10 15 20 25 3000.050.10.150.20.25 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 4.6: Results for Model 2 where PKC beta is overexpressed. Thesimulated control is shown for comparison (circles/thick curves).384.3. Simulations of PKC manipulationsIn the last panel, we observe the RhoA zone with the same level ofactivity as the control, but the zone broadens. The breadth of the RhoAzone is a result of initial conditions with greater activity levels than thethreshold value, allowing faster plateau to high steady state, and outwardsbleeding. We also observe a very broad Cdc42 zone whose activity is elevatedabove the control. The breadth of the Cdc42 zone is a combination of initialconditions with greater activity levels, and a lowering of the threshold dueto the decrease in k7.While zones are maintained (C1), Model 2 fails to satisfy criteria C2and C3. Quantitatively, the Cdc42 zone is elevated by a 1.3 fold changeabove the control. This falls short of what is observed (Table A.1, 60 s).Qualitatively, both zones are much broader than the control zones. This issimply not observed in wounding images (Figure 2.2 C). Consequently, thesimplest model cannot account for experimental observations.4.3.2 Expression of dominant-negative PKC betaSimulated results are shown in Figure 4.7. In the 54 s panel, we show slightlyelevated initial conditions. Experimentally, zones do not form at early times,cannot be quantified, and therefore this leaves us without a 30 s zone am-plification factor in Table A.2. Instead, we adjust the initial conditions bythe smallest amount such that zone maintenance is guaranteed. Experimen-tally, the expression of dominant-negative PKC beta results in downregu-lated zones. To simulate this, we decrease the basal RhoA activation rate kr0(fold change of 0.9) and increase the basal Cdc42 inactivation rate k7 (foldchange of 1.25).394.3. Simulations of PKC manipulations0 5 10 15 20 25 3000.050.1 84 s post-wounding0 5 10 15 20 25 3000.050.1 78 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.10.150.2 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 4.7: Results for Model 2 where dominant-negative PKC beta isexpressed. The simulated control is shown for comparison (circles/thickcurves).At 84 s, we observe both RhoA and Cdc42 zones whose activities are de-pressed. Both zones maintain widths similar to the control. The zone widthsare narrow despite elevated initial conditions because the corresponding pa-rameter manipulations effectively raise the thresholds. Lastly, we observe acurious gap between the RhoA and Cdc42 zones.In the simulation of dominant-negative PKC beta, zone maintenance(C1) was enforced. Regardless, Model 2 fails to satisfy criteria C2 andC3. Quantitatively, we expect to see zones depressed by more than half thecontrol activity (Table A.2). Qualitatively, no such gap between zones isobserved (Figure 2.2 C). Therefore, Model 2 cannot account for experimentalobservations.404.3. Simulations of PKC manipulations4.3.3 Overexpression of PKC etaFigure 4.8 shows the simulated results where PKC eta is overexpressed. Inthe first panel, we show the adjusted initial conditions (Table A.3). Ex-perimentally, overexpression of PKC eta results in the downregulation ofboth RhoA and Cdc42 zones. To simulate this, we decrease the basal RhoAactivation rate kr0 (fold change of 0.89), and increase the basal Cdc42 inac-tivation rate k7 (fold change of 1.25).0 5 10 15 20 25 3000.050.1 84 s post-wounding0 5 10 15 20 25 3000.050.1 78 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 4.8: Results for Model 2 where PKC eta is overexpressed. Thesimulated control is shown for comparison (circles/thick curves).We observe the rapid decay of both RhoA and Cdc42 zones. Zones can-not be maintained due to a combination of diminished initial conditions andincreased thresholds. In this simulation where PKC eta is overexpressed,zone maintenance is not achieved. Therefore, Model 2 fails to satisfy crite-rion C1 and cannot be assessed on the remaining criteria. As a result, our414.3. Simulations of PKC manipulationscurrent assumptions do not account for experimental observations.4.3.4 Expression of dominant-negative PKC etaSimulated results are shown in Figure 4.9. In the first panel, the initial con-ditions are unchanged from the control. Experimentally, zones do not format early times, and we do not have 30 s amplification factors in Table A.4.Instead, we use control initial conditions which guarantee zone maintenance.To simulate the experimental result where dominant-negative PKC eta up-regulates both background activities, we decrease the basal RhoA inactiva-tion rate k2 (fold change of 0.9), and decrease the basal Cdc42 inactivationrate k7 (fold change of 0.8).0 5 10 15 20 25 3000.050.184 s post-wounding0 5 10 15 20 25 3000.050.178 s post-wounding 0 5 10 15 20 25 3000.050.172 s post-wounding0 5 10 15 20 25 3000.050.166 s post-wounding 0 5 10 15 20 25 3000.050.160 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 4.9: Results for Model 2 where dominant-negative PKC eta isexpressed. The simulated control is shown for comparison (circles/thickcurves).424.4. Possible ways to achieve inverted Rho GTPase zonesIn the last panel, we observe RhoA and Cdc42 zones with elevated ac-tivities. The RhoA zone is broad, due to a decreased threshold. Moreimportantly, the background activities are only slightly above control.While zones are maintained (C1), Model 2 fails to satisfy criteria C2and C3. Quantitatively, we expect to see background activities that arealmost twice the control (Table A.4). Qualitatively, the simulated RhoAzone is broader than Cdc42, though experimental wound images show theopposite (Figure 2.2 C). We conclude that this model cannot recapitulateexperimental observations.4.4 Possible ways to achieve inverted RhoGTPase zonesGiven that PKC eta overexpression could not be qualitatively or quantita-tively reproduced by Model 2, we cannot infer how zone inversion occurs inthat circumstance. The simulated manipulation failed because RhoA andCdc42 zones could not be sustained since the initial conditions were belowthe threshold. Now we wonder: outside of PKC eta overexpression, can weever observe inverted Rho GTPase zones? We relax the constraint used toadjust the initial conditions (Table A.3), and find two ways of achievinginverted zones.In the first example, we see the simulation begin with reduced initialconditions (Figure 4.10). The basal RhoA inactivation rate k2 is decreased(0.89 fold change) and the basal Cdc42 inactivation rate k7 is decreased (0.8fold change). We observe the simulated Cdc42 zone spreading close to thewound edge before the RhoA zone can establish. In this way, Cdc42 is ableto sandwich the RhoA zone. Here is one possible way to see the Cdc42 zonein front of the RhoA zone.434.4. Possible ways to achieve inverted Rho GTPase zones0 5 10 15 20 25 3000.050.184 s post-wounding0 5 10 15 20 25 3000.050.178 s post-wounding 0 5 10 15 20 25 3000.050.172 s post-wounding0 5 10 15 20 25 3000.050.166 s post-wounding 0 5 10 15 20 25 3000.050.160 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 4.10: Results for Model 2 where inverted Rho GTPase zones occurbecause Cdc42 sweeps out in front of the RhoA zone. The simulated controlis shown for comparison (circles/thick curves).In the second example, we decrease the basal Cdc42 inactivation rate k7(0.5 fold change). We observe the monostable Cdc42 establish a uniformfield of Cdc42 over the domain (Figure 4.11). The simulated RhoA zoneestablishes, suppressing Cdc42, creating a divot in the uniform field. Again,Cdc42 sandwiches the RhoA zone, but also fills the domain far from thewound.444.5. Conclusions0 5 10 15 20 25 3000.050.184 s post-wounding0 5 10 15 20 25 3000.050.178 s post-wounding 0 5 10 15 20 25 3000.050.172 s post-wounding0 5 10 15 20 25 3000.050.166 s post-wounding 0 5 10 15 20 25 3000.050.160 s post-wounding0 5 10 15 20 25 3000.10.20.30.4 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 4.11: Results for Model 2 where inverted Rho GTPase zones oc-cur because the RhoA zone protrudes into a uniform field of Cdc42. Thesimulated control is shown for comparison (circles/thick curves).It appears that inverted zones can be achieved through the use of pre-carious initial conditions, or by enabling Cdc42 monostability. In both ex-amples, the parameter fold changes resemble a manipulation where PKCbeta is overexpressed. Thus, we are unable to investigate how zone inver-sion occurs within the context of PKC eta overexpression. We are, however,convinced that inverted zones are not impossible using Model 2.4.5 ConclusionsIn this chapter, we showed that using a model with spatially constant PKCsfails to account for quantitative and qualitative observations in experimentswhere PKCs are manipulated. At best, simulations of the overexpression of454.5. ConclusionsPKC beta and the expression of dominant-negative PKC eta result in thecorrect trend of upregulated zones. However, features such as zone breadthand peak zone activity do not match what is observed. Even when zonebreadth is not an issue, like in the simulation of dominant-negative PKCbeta, qualitative features such as a gap between zones works against modelvalidation. At worst, this model can never account for the overexpression ofPKC eta. Zones are simply never sustained due to initial conditions belowthreshold.With respect to zone inversion, we sought any manipulation that wouldproduce the Cdc42 zone in front of the RhoA zone, even if it meant choicesdissimilar to PKC eta overexpression. Manipulations resembling PKC betaoverexpression were found to produce Cdc42 zones which sandwiched theRhoA zone. Though these results are unconvincing in explaining inversionin PKC eta overexpression, they illustrate that Model 2 is capable of suchphenomena.In this chapter, we also showed that PKC impact on Rho GTPases can bethrough either GEFs or GAPs. We examined bifurcation diagrams on fourkey parameters (kr0, kc0, k2, k7) and showed that basal activation rates sharedsimilarities with basal inactivation rates. The bifurcation diagrams identifiedparameter regimes where bistability, and therefore zone maintenance, waspossible.Based on these results, we asked whether taking into account the nonuni-form spatial distribution of PKCs could correct the flawed model predictions.In the next chapter, we explore localized PKC effects modulating the basalRho GTPase activation rates (i.e., through GEFs).46Chapter 5PKCs as spatially dependentparameters: step functionrepresentation5.1 OverviewBased on the failure of the simplest model variant, we revise the approach torepresent spatially distributed PKC activity. To do so, we use experimentaldata to define PKC activity profiles ?(x, t) and ?(x, t). Here we considersimple representations of that spatial dependence, namely piecewise constantfunctions of space. As a result the quantities kr0 and kc0 are in turn functions.In this chapter, we implement PKC activity profiles without using ex-plicit data on their temporal evolution2. Instead, we use observations fromChapter 2 to inform their shape and localization relative to the Rho GTPasezones. For simplicity, we let PKCs modulate the basal Rho GTPase acti-vation rates. With this model variant, we calibrate the control simulation.PKC manipulations are then simulated, but we find that the current as-sumptions do not explain the experimental observations. We conclude thatmore detailed PKC activity profiles are necessary.5.2 Defining PKC activity profilesIn order to make a simplified approximation of the PKC activity profiles,several features demand attention:(i) PKC beta and PKC eta accumulate around wounds as zones of elevatedactivity (Figure 2.5). Both PKC beta and PKC eta overlap the RhoAzone, though PKC beta also encompasses the Cdc42 zone (Figures 2.4).2We do eventually receive explicit data on PKC activity profiles, and use them in thenext chapter.475.2. Defining PKC activity profiles(ii) The PKC zones appear to exhibit zone segregation and zone mainte-nance in concert with the Rho GTPase zones (Figure 2.5 A).5.2.1 PKC spatial profiles as step functionsThe spatial profile of PKC activity (Feature (i)) can be approximated as astep function (Figure 5.1). Both PKC step functions are normalized to 1,representing the elevated activity within zones. Outside of each step func-tion, the corresponding PKC activity is assumed to be essentially zero. ThePKC beta step function ?(x, t) is broader than the PKC eta step function?(x, t).0 5 10 15 20 25 30Distance from wound center (?m)00.250.50.7511.251.51.752Intensity (A.U.)PKC betaPKC etaFigure 5.1: Activity profiles of PKC beta, ?(x, t), and PKC eta, ?(x, t), aremodelled as normalized step functions with PKC beta covering a greaterbreadth than PKC eta. The simulated wound edge is marked by the verticalblack line, and is towards the left.5.2.2 PKC zone localizationWe have picked our simplified representation of the PKC activity profiles,but how do we determine where these PKC zones should localize? Based onFeature (ii), they should track the Rho GTPase zones. For us to recognizewhere a Rho GTPase zone is located, we adopt the following quantitativedefinition.Zone Localization The RhoA (Cdc42) zone localization is defined as theregion of the domain with RhoA (Cdc42) activity exceeding 40 % themaximum RhoA (Cdc42) activity above background.485.2. Defining PKC activity profilesThe zone localization definition is illustrated in Figure 5.2. The 40 % activitythreshold was chosen by applying various thresholds to Rho GTPase controldata and visually estimating the accuracy of zone locations (Figure 5.3). Welet PKC beta follow the union of RhoA and Cdc42 zone locations, and PKCeta follow the RhoA zone location.0 5 10 15 20 25 30Distance from wound center (?m)00.050.10.150.2Concentration (?M) Simulated RhoA54 s post-wounding0 5 10 15 20 25 30Distance from wound center (?m)00.050.10.150.2Concentration (?M)84 s post-woundingFigure 5.2: Definition of RhoA zone location at early times (left) and latetimes (right). Top of the thick black line indicates maximum activity abovebackground; base of the black line indicates background activity. Top violetline indicates 40% of height spanned by black line; bottom violet line indi-cates the defined zone location. A similar determination of the Cdc42 zonelocation is used with a threshold of 40 %.0 5 10 15 20 25 30Distance from wound center (?m)00.050.10.15Concentration (?M) RhoACdc4254 s post-wounding0 5 10 15 20 25 30Distance from wound center (?m)00.050.10.15Concentration (?M)84 s post-woundingFigure 5.3: Determination of 40 % activity threshold from control data.Time series of Rho GTPases highlighting the RhoA zone location (greeninterval) and Cdc42 zone location (red interval). Chosen activity thresholdcorresponds to visual estimation of zone location accuracy.495.3. Spatially dependent basal Rho GTPase activation ratesBy defining Rho GTPase zone locations which dictate the localizationof PKCs, we have imposed implicit feedbacks between the two. That is,PKCs localize to the GTPase zones, while simultaneously regulating GTPaseactivity. In turn, PKCs impact Rho GTPase zone widths and locations,which feeds back into dictating PKC recruitment. Without yet consideringthe role of the lipid DAG, this is one way to implement the coordinatedmovement of PKC zones with Rho GTPase zones.5.3 Spatially dependent basal Rho GTPaseactivation ratesThe above arguments defined the functions ?(x, t) and ?(x, t). With this, werevise the background activation rates kr0 and kc0. The simplest assumptionis that these functions take a form:kr0(x, t) = krbasal + krPKC?(x, t)1 + ?1?(x, t)(5.1)kc0(x, t) = kcbasal + kcPKC?(x, t)1 + ?2?(x, t)(5.2)where kbasal is the basal rate of Rho GTPase activation, kPKC is the contri-bution of Rho GTPase activation from PKCs, and ? is the strength of PKCeta relative to PKC beta. ?(x, t) is 1 on the spatial domain wherever bothRho GTPase zones are located, and 0 elsewhere. ?(x, t) is 1 on the domainwhere the RhoA zone is located, and 0 elsewhere.We chose these forms for the following reasons:1. In the absence of PKCs (?(x, t) = ?(x, t) = 0), we obtain a basalactivation rate kr,c0 (x, t) = kr,cbasal.2. PKC beta enhances the basal activation rate.3. PKC eta counteracts PKC beta and reduces the basal activation rate.Based on the forms 5.1 and 5.2, we must identify six parameters values,namely kr,cPKC, kr,cbasal and ?1,2.We consider three separate regions, each with its own PKC beta andPKC eta level (Figure 5.4).? Region 1 where both PKCs are present and influence RhoA zone ac-tivity.505.3. Spatially dependent basal Rho GTPase activation rates? Region 2 where only PKC beta is present and influences Cdc42 zoneactivity.? Region 3 where neither PKCs are present.We apply constraints in each region to estimate the parameter values. Wefollow a sequence of steps each time, first considering Region 3 (PKCsabsent), then Region 1 (PKCs present), and finally Region 2 (PKC betapresent).0 5 10 15 20 25 30Distance from wound center (?m)00.050.10.15Concentration (?M)Sim total RhoASim Cdc42Sim PKC betaSim PKC etaRegion 1 Region 2 Region 3Figure 5.4: Distinct regions in the simulated control where both PKCs arepresent (Region 1), only PKC beta is present (Region 2), and neither PKCsare present (Region 3).5.3.1 Parameter estimationIn Region 3, both PKCs are absent (?(x, t) = ?(x, t) = 0) and the RHS ofEquation 5.1 is now krbasal. We constrain the basal rate tokrbasal = ?1krctrl (5.3)515.3. Spatially dependent basal Rho GTPase activation rateswhere ?1 (Region 3 constraint) is a fold change of the base parameter value,kr0 (from Table 3.1), which we have called krctrl. The first parameter krbasal isnow determined.In Region 1, both PKCs are present (?(x, t) = ?(x, t) = 1) and weconstrain the RHS of Equation 5.1 to the base value of the RhoA basalactivation rate krctrlkrbasal +krPKC1 + ?1= krctrl. (5.4)This constraint ensures that the simulated control RhoA zone behaves thesame as the control RhoA zone from Model 1. The above allows us todetermine the second parameter krPKC in terms of the remaining parameter?1 and the base value krctrl? krPKC = (1? ?1)(1 + ?1)krctrl. (5.5)In Region 2, only PKC beta is present (?(x, t) = 1, ?(x, t) = 0). Weconstrain the RHS of Equation 5.1 to ?1krctrl, and substitute determinedparameters from 5.3 and 5.5. The last parameter ?1 is now determined:krbasal + krPKC = ?1krctrl (5.6)krctrl(?1 + (1? ?1)(1 + ?1)) = ?1krctrl? ?1 =?1 ? ?11? ?1? 1. (5.7)The prefactor ?1 (Region 2 constraint) describes the fold change necessaryfor setting the RhoA activation rate in Region 2 to match the activity of theRhoA shoulder.Altogether, with the three determined parameters, the basal RhoA acti-vation rate (Equation 5.1) can be expressed in terms of constraints ?1 and?1:kr0(x, t) = krctrl(?1 + (1? ?1)(1 + ?1)?(x, t)1 + ?1?(x, t))(5.8)where ?1 =?1??11??1? 1.Repeating the same process for Cdc42, we havekc0(x, t) = kcctrl(?2 + (1? ?2)(1 + ?2)?(x, t)1 + ?2?(x, t)). (5.9)So far, we have defined the PKC activity profiles as step functions whichlocalize to Rho GTPase zones. We have further assumed that the back-ground RhoA and Cdc42 activation rates are spatially dependent on the525.3. Spatially dependent basal Rho GTPase activation ratesPKC activity profiles. The basal activation rates take on a particular form(Equations 5.1, 5.2) which can be parameterized under several constraints.Next, we specify how we choose the exact constraints to simulate the controlexperiment.5.3.2 Constraints set by control dataIn this section, we detail how we set constraints ?1,2 and ?1,2 to faithfullysimulate the control. We consult the bifurcation diagrams (Figures 4.4,4.5) when matching control data on Rho GTPase activity (Figure 1.4) toapproximate kr0 and kc0 values. We discuss the constraints for each region inturn.What do we expect to happen when PKCs are absent? We expect thatzone formation and maintenance would not be possible, and that Rho GT-Pases are essentially monostable. Furthermore, we expect background levelsof Rho GTPase activities. For these reasons, we set the constraint on krbasalto ?1 = 0.75. This is outside the RhoA bistable region, and also resultsin background activity comparable to data. Similarly, we set ?2 = 0.01,which is inside the bistable region3 with a high threshold, and still resultsin background activity comparable to data.In Region 2, we match the control RhoA shoulder activity (0.05 ?M) tothe bifurcation diagram. According to Figure 4.4, ?1 = 1.25 will produceactivities close to what is observed. We also match the control Cdc42 zoneactivity ([C?] =0.09 ?M). To do this, we consider Cdc42 steady state kinetics(Equation 3.4) which sets ?2 = 2.7 for us:(k7 + k8)[C?][C]? k5[A-R?]?k6[C?]nKnC + [C?]n= 2.7kcctrl (5.10)where [A-R?] is the stable low steady state determined from Equations 3.2and 3.3, with kr0 = 1.25krctrl.With these constraints, we initiate a control simulation using backgroundactivation rates of RhoA and Cdc42 with the following parameterizations:kr0(x, t) = krctrl(0.75 + 0.5?(x, t)1 + ?(x, t))(5.11)kc0(x, t) = kcctrl(0.01 + 2.71?(x, t)1 + 1.73?(x, t))(5.12)3Recall that regions of low monostability are inaccessible when modulating the param-eter kc0.535.4. Simulations of PKC manipulations5.4 Simulations of PKC manipulationsIn this section, we simulate the PKC manipulations by modifying PKC betaand PKC eta activities. We can define Model 3 in the following way:Model 3. The spatially distributed PKC (as step functions) model.Equations 5.11, 5.12Initial conditions Piecewise linear (triangular) profiles fittedto control data, and scaled by amplification factors in Ta-bles A.1-A.4 at 30 s.0 5 10 15 20 25 3000.050.1 84 s post-wounding0 5 10 15 20 25 3000.050.1 78 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 5.5: Control simulation for Model 3. Note the appearance of theRhoA shoulder. The control data are shown for comparison (circles/thickcurves).The Model 3 control simulation is shown in Figure 5.5. The simulatedcontrol captures the speed of rise in the RhoA and Cdc42 zone, as well as545.4. Simulations of PKC manipulationspeak GTPase zone activities. A few improvements are visible in the RhoAactivity. The first notable feature is the appearance of the RhoA shoulderwhich was emphasized in Section 3.2.2. The second improvement followsfrom the RhoA shoulder, and is a widening in the base of the zone, withclose coupling between the GTPase zones.To simulate a PKC manipulation, we specify initial conditions as inSection 4.3, and modify the relevant PKC activity. When a particularPKC is overexpressed, its activity is increased from control levels. Whena dominant-negative PKC is expressed, its activity is decreased from con-trol levels. The simulated manipulation is evaluated against criteria in Sec-tion 3.3.1, just as in the previous chapter. The following results show simu-lated GTPases plotted over the Model 3 simulated control for comparison.5.4.1 Overexpression of PKC betaSimulations of PKC beta overexpression is shown in Figure 5.6. The firstpanel shows the adjusted initial conditions (Table A.1). The simulated ma-nipulation overexpresses PKC beta activity by 5 %. Further overexpressionresults in a very broad RhoA zone annihilating the Cdc42 zone.Throughout the simulation, we observe an extremely broad RhoA zone,trailed by the Cdc42 zone which has a width comparable to the control. Thebroad RhoA zone is a result of initial conditions with greater activity thancontrol, in conjunction with lowering of the threshold by the manipulation.The Cdc42 zone does not broaden as extremely because the RhoA zoneexpands at the Cdc42 zone?s expense.At 84 s, both zones have activities that are comparable to the con-trol. The RhoA zone appears slightly elevated, while the Cdc42 appearsunchanged from control. Cdc42 insensitivity to PKC beta overexpression,and thereby kc0, can be simply explained by the bifurcation diagram in Fig-ure 4.5. The diagram shows that the Cdc42 upper stable branch has arange between 0.093? 0.1 ?M over a fold change of 0-3 in kc0 (bistable re-gion). Thus, the very narrow range in Cdc42 stable high steady states overthe bistable region is responsible for the Cdc42 zone?s insensitivity to PKCmodulation. The PKC beta overexpression might have been very slight, butdoes indeed exert effects on both zones.In this manipulation, the simulated results exhibit zone maintenance,satisfying the first criterion C1. Qualitatively, we expect upregulation of thezones, though the simulations do not strongly reflect this. Furthermore, weexpect similar zone widths, but the simulated RhoA zone becomes extremelybroad (Figure 2.2 C). Since the zones are not significantly upregulated, this555.4. Simulations of PKC manipulationssimulation does not satisfy criteria C2 and C3 (Table A.1). Model 3 isunable to recapitulate experimental observations.0 5 10 15 20 25 30 3500.050.10.15 84 s post-wounding0 5 10 15 20 25 30 3500.050.10.15 78 s post-wounding0 5 10 15 20 25 30 3500.050.10.15 72 s post-wounding0 5 10 15 20 25 30 3500.050.10.15 66 s post-wounding0 5 10 15 20 25 30 3500.050.10.15 60 s post-wounding0 5 10 15 20 25 30 3500.050.10.150.20.25 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 5.6: Results for Model 3 where PKC beta is overexpressed by 5 %.The simulated control is shown for comparison (circles/thick curves).5.4.2 Expression of dominant-negative PKC betaSimulated results are shown in Figure 5.7. The first panel shows slightlyelevated initial conditions. Experimentally, zones do not form early, so wedo not have zone amplification factors at 30 s (Table A.2). We asked whetheradjusting the initial conditions could improve this result. Consequently, weadjusted the initial conditions by the minimum amount needed to guaranteezone formation. To simulate expression of dominant-negative PKC beta,the PKC beta activity is expressed at 80 % of the control activity. Furtherunderexpression does not lead to very different results.In the simulation, we observe zones with heights and widths very similar565.4. Simulations of PKC manipulationsto the control. We observe the RhoA zone slightly above control activity,and the (insensitive) Cdc42 very slightly below the control activity. Thecounterintuitive result where RhoA is slightly elevated can be explainedsimply. We raised the initial conditions, guaranteeing zone maintenance,because the PKC manipulation raised the bistability threshold. The ampli-fied RhoA initial conditions rose to high steady state faster than the controlRhoA zone, thus appearing slightly elevated at 84 s. In fact, seconds afterthe last panel, the RhoA zone downregulation becomes fully apparent.0 5 10 15 20 25 3000.050.184 s post-wounding0 5 10 15 20 25 3000.050.178 s post-wounding 0 5 10 15 20 25 3000.050.172 s post-wounding0 5 10 15 20 25 3000.050.166 s post-wounding 0 5 10 15 20 25 3000.050.160 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 5.7: Results for Model 3 where dominant-negative PKC beta is ex-pressed. PKC beta activity is expressed at 80 % of control levels. Initialconditions have been adjusted to guarantee zone maintenance. The simu-lated control is shown for comparison (circles/thick curves).While the simulation exhibits zone maintenance (C1), it fails to satisfycriteria C2 and C3. Under this manipulation, we expect to observe sig-nificantly downregulated zone activities (Table A.2). Since the simulated575.4. Simulations of PKC manipulationszones are not qualitatively and quantitatively downregulated to the extentwe expect, Model 3 cannot reproduce experimental observations.5.4.3 Overexpression of PKC etaSimulated results are shown in Figure 5.8. The first panel shows the adjustedinitial conditions (Table A.3). PKC eta is overexpressed by 5 %, and furtheroverexpression does not change the results.0 5 10 15 20 25 3000.050.184 s post-wounding0 5 10 15 20 25 3000.050.178 s post-wounding 0 5 10 15 20 25 3000.050.172 s post-wounding0 5 10 15 20 25 3000.050.166 s post-wounding 0 5 10 15 20 25 3000.050.160 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 5.8: Results for Model 3 where PKC eta is overexpressed by 5 %.The simulated control is shown for comparison (circles/thick curves).The simulation shows rapid decay of both RhoA and Cdc42 zones. Thezones cannot be sustained because the initial conditions are set below thethreshold, and the threshold is increased by the manipulation. This simula-tion fails to satisfy the criterion on zone maintenance (C1), and thus cannotbe evaluated on remaining criteria. Under the current assumptions, Model 3585.4. Simulations of PKC manipulationscannot reproduce the experimental observations.5.4.4 Expression of dominant-negative PKC etaSimulation results are shown in Figure 5.9. The first panel shows initialconditions exactly as in the control. Experimentally, zones do not form early,so we do not have zone amplification factors at 30 s (Table A.4). Instead, wetested initial conditions which guarantee zone maintenance. The expressionof dominant-negative PKC eta was simulated by expressing PKC eta at40 % of PKC eta control activity. Further underexpression produces similarresults.0 5 10 15 20 25 3000.050.184 s post-wounding0 5 10 15 20 25 3000.050.178 s post-wounding 0 5 10 15 20 25 3000.050.172 s post-wounding0 5 10 15 20 25 3000.050.166 s post-wounding 0 5 10 15 20 25 3000.050.160 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 5.9: Results for Model 3 where dominant-negative PKC eta is ex-pressed. PKC eta activity is expressed at 40 % of control levels. Initialconditions have been adjusted to guarantee zone maintenance. The simu-lated control is shown for comparison (circles/thick curves).595.5. Addressing Cdc42 insensitivity to PKC betaAt 84 s, we observe a RhoA zone with elevated activity. The RhoA zoneis slightly broader than control. We also observe the Cdc42 zone whoseactivity and width remains comparable to control. In fact, the Cdc42 zoneis completely unaffected by changes in PKC eta. By definition, PKC etalocalizes solely to the RhoA zone, and is absent from the Cdc42 zone. More-over, because PKCs were assumed to be essentially zero outside the GTPasezones, the background activities are completely unaffected by the PKCs.In this simulation, the first criterion on zone maintenance is satisfied(C1). Qualitatively, we expect upregulation of both RhoA and Cdc42 zoneactivity, but only observe RhoA upregulation. Experimental images alsoshow that the Cdc42 zone is slightly broader than the RhoA zone, but weobserve the opposite (does not satisfy C2; Figure 2.2 C). Quantitatively, thedata show that background activities should be significantly elevated, butno change is observed in the simulation (does not satisfy C3; Table A.4).Therefore, Model 3 and the current assumptions are unable reproduce ex-perimental observations.5.5 Addressing Cdc42 insensitivity to PKC betaIn Sections 5.4.1 and 5.4.2, we simulated PKC beta manipulations. Sim-ulation results showed that Cdc42 activity was insensitive to PKC beta.Furthermore, we discussed that Cdc42 insensitivity was primarily due toPKC effects via the basal Cdc42 activation rate kc0. Instead, we attemptto address this issue by allowing PKC beta effects through the basal Cdc42inactivation rate k7. If we recall the bifurcation diagram in Figure 4.5, weexpect dramatic changes in Cdc42 activity from modest fold changes in k7(due to the steep Cdc42 upper stable branch).In this section, we allow the PKCs to affect the Rho GTPases throughkr0 and k7. We simulate the control and overexpression of PKC beta. Wedemonstrate that the Cdc42 zone can be tuned more effectively through thebasal Cdc42 inactivation rate k7.5.5.1 Spatially dependent basal Cdc42 inactivation rateWe assume that k7 takes on the form:k7(x, t) = k7basal ? k7PKC?(x, t)1 + ?3?(x, t). (5.13)Here, we have a negative term for the contribution of PKC effects. Thenegative sign is necessary because k7, a GAP term, has the opposite effect605.5. Addressing Cdc42 insensitivity to PKC betaof kc0, a GEF term. The above assumption allows the inactivation rate todecrease with PKC beta, and approach the background level with increasesin PKC eta.5.5.2 Parameter estimation and constraintsTo estimate the parameters k7basal, k7PKC and ?3, we apply constraints toeach region, as previously done. In terms of the constraints, the parameterscan be expressed as:k7basal = ?3k7ctrl (5.14)k7PKC = (1 + ?2)(?3 ? 1)k7ctrl (5.15)?3 =?3 ? ?3?3 ? 1? 1 (5.16)where ?3 is the Region 3 constraint, ?3 is the Region 2 constraint, andk7ctrl is the base value of the basal Cdc42 inactivation rate (Table 3.1). Thebackground Cdc42 inactivation rate expressed in terms of constraints is thenk7(x, t) = k7ctrl(?3 ? (?3 ? 1)(1 + ?3)?(x, t)1 + ?3?(x, t)). (5.17)To set the constraints, we match the control data on GTPase activitiesto fold changes in parameters suggested by the bifurcation diagram (Fig-ure 4.5). Since we expect monostable Cdc42 in the absence of PKCs, we set?3 = 1.6, outside the bistable region. The constraint in Region 2, ?3 = 0.85,is set in a similar way as in Section 5.3.2.5.5.3 SimulationsWe now define Model 4 in the following way:Model 4. The spatially distributed PKC (as step functions) model.Equations 5.11, 5.17Initial conditions Piecewise linear (triangular) profiles fittedto control data, and scaled by amplification factors in Ta-bles A.1-A.4 at 30 s.The Model 4 control simulation is shown in Figure 5.10. The Rho GT-Pase zones look very similar to the Model 3 control simulation (Figure 5.5),and the RhoA shoulder still appears. The only difference between the sim-ulated controls is a narrower Cdc42 zone when Model 4 is used.615.5. Addressing Cdc42 insensitivity to PKC beta0 5 10 15 20 25 3000.050.1 84 s post-wounding0 5 10 15 20 25 3000.050.1 78 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 5.10: Control simulation for Model 4. Note the appearance of theRhoA shoulder. The control data are shown for comparison (circles/thickcurves).Overexpression of PKC betaFigure 5.11 shows the simulation results for the overexpression of PKC beta.In the first panel, the adjusted initial conditions are shown. PKC beta isoverexpressed by 210 %. Further overexpression is not possible since thiswould result in non-positive values of k7 which is inconsistent with its defi-nition.We observe that the initial zones quickly plateau to high steady stateand broaden outwards. Both RhoA and Cdc42 zones are upregulated abovecontrol activities. At 84 s, the Cdc42 zone is more elevated than in previoussimulations where the basal rate of Cdc42 activation (kc0) was affected byPKCs. Here, we demonstrate that varying k7 is more effective in upregulat-ing Cdc42 zone activity.625.5. Addressing Cdc42 insensitivity to PKC beta0 5 10 15 20 25 30 3500.050.10.15 84 s post-wounding0 5 10 15 20 25 30 3500.050.10.15 78 s post-wounding0 5 10 15 20 25 30 3500.050.10.15 72 s post-wounding0 5 10 15 20 25 30 3500.050.10.15 66 s post-wounding0 5 10 15 20 25 30 3500.050.10.15 60 s post-wounding0 5 10 15 20 25 30 3500.050.10.150.20.25 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 5.11: Results for Model 4 where PKC beta is overexpressed by 210 %.The simulated control is shown for comparison (circles/thick curves).In this simulation, zone maintenance is achieved (C1), as well as a qual-itative upregulation of both zones. However, the simulated RhoA zonebreadth is still too broad, so this simulation does not satsify criterion C2(Figure 2.2 C). Quantitatively, the upregulation of either zone does notmatch the extent observed in data (Table A.1, C3 not satisfied). Therefore,under these assumptions, Model 4 is unable to reproduce experimental ob-servations. From this, we conclude that the basal Cdc42 inactivation ratek7 might improve our simulation results, but only with respect to the Cdc42zone?s peak activity.635.6. Conclusions5.6 ConclusionsIn this chapter, we constructed a model (Model 3) with the simplest rep-resentation of PKC activity profiles. We used step functions to representthe PKC activity profiles, and allowed them to localize to the Rho GTPasezones. Simulated PKC beta overlapped both RhoA and Cdc42 zones, whilesimulated PKC eta overlapped the RhoA zone. The PKCs affected the RhoGTPase basal rates of activation (kr0(x, t), kc0(x, t)). By consulting bifurca-tion diagrams and experimental GTPase control activities, we parameterizedthe basal rates? dependence on PKCs.Model 3 reproduced a notable feature in the Rho GTPase control data.It captured the prominent RhoA shoulder and improved the zone width.While Model 3 was able to simulate the control quite well, it did not ac-count for any of the PKC manipulations. In particular, we could not accountfor the overexpression of PKC eta, meaning we also could not account for theinverted Rho GTPase zones. Generally, trends in upregulation or downreg-ulation of RhoA activity were qualitatively captured. However, the modelhad trouble with PKC modulation of Cdc42 activity. The failure of themodel in each manipulation highlights its shortcomings.In simulations where PKC beta is manipulated, we noticed that Cdc42activity was insensitive to modulations in PKC beta activity. We foundthat the insensitivity was due to PKC effects through the basal Cdc42 rateof activation (kc0). In contrast, PKC effects through the basal Cdc42 rate ofinactivation (k7) produced more noticeable changes to Cdc42 zone activity.It is possible that PKCs regulate GAP activity, though we were unable tosuccessfully reproduce a PKC manipulation even with this modification.In simulations where PKC eta is manipulated and zones were maintained,we noticed that Cdc42 activity was completely unaffected by modulationsin PKC eta activity. By definition, simulated PKC eta is absent from theCdc42 zone, and thus does not exert its influence there. This is one of severalmajor issues we will address in turn.Model 3 is fraught with many problems. We need to reconsider thePKC activity profiles, PKC localization, and the Rho GTPase initial condi-tions. Our first observation is that the simple representation of PKC activityprofiles is insufficient in accounting for the manipulations. In simulated ma-nipulations, PKC eta is zero within the Cdc42 zone, rendering the zoneunaffected by PKC eta. We can think of two possible ways to correct this.First, we are inclined to increase the realism and spatial detail of the PKCactivity profiles. If they were peak-like (Figure 2.5), with a gradual transi-tion from peak to baseline, a non-zero amount of PKC eta would be present645.6. Conclusionswithin the Cdc42 zone. A small amount of PKC eta in the Cdc42 zone might?cure? the Cdc42 insensitivity. Second, we might need to implement PKCcompetition for DAG. Our model variants assumed that PKC eta and PKCbeta were independent of each other. However, if they competed for DAG,PKC eta would affect PKC beta in the RhoA zone. And, since PKC beta isbroad and extends over the Cdc42 zone, PKC eta indirectly affects Cdc42zone activity. In the next chapter, we implement the first possibility.Our second observation concerns our implementation of PKC localiza-tion. Biologically, we know that PKCs localize to DAG. There is no reasonto believe that PKC localization is Rho GTPase-dependent, as we have as-sumed in our models. Allowing PKCs to localize to the Rho GTPase zonesis therefore unjustified. Since we also made this erroneous assumption in thecontrol, as well as the manipulations, we further presumed that exogenousPKCs localize differently than endogenous PKCs. Seeing that we want totest very simple assumptions, our next models assume that PKCs are notRho GTPase-dependent in the control. And, we use control PKC localiza-tion (endogenous) in the simulated PKC manipulations (exogenous).Our third observation concerns the appropriateness of the Rho GTPaseinitial conditions. We used initial conditions that were fitted to the RhoGTPase control data, and scaled them appropriately in order to simulatePKC manipulations. This strategy preceded our receipt of explicit dataand might have confounded the simulation results. For example, simula-tions of PKC beta overexpression began with very broad initial conditionswhich resulted in very broad Rho GTPase zones. Thus, our results werein qualitative disagreement with experimental observations. As another ex-ample, simulations of PKC eta overexpression began with very small zoneswhich negated zone maintenance. Moreover, we had to guess at the initialconditions in both the simulations where dominant-negative PKCs were ex-pressed. To remove further ambiguity, in the next chapter we use explicitdata on Rho GTPase activities, under specific PKC manipulations, as ourinitial conditions.Lastly, we would like to note one more interesting possibility. When weparameterized the basal activation rates, we constrained kbasal to a monos-table regime. This means that in regions of zero (or low) PKC activity, simu-lated Rho GTPase zones simply cannot be sustained. Due to the constraint,extreme broadening of simulated Rho GTPase zones, beyond regions of highPKC activity, should not be observed. In this chapter, however, extremelybroad simulated Rho GTPase zones were observed because of another rea-son: our artificial implementation of PKC localization. In the next chapter,we implement this monostable constraint again, in regions of low PKC ac-655.6. Conclusionstivity, and expect focused, discrete Rho GTPase zones. We propose thatPKC activity profiles are partly responsible for the discrete Rho GTPasezones. They might also have the effect of curbing the backwards spread ofthe Rho GTPase zone?s trailing edge. Therefore, no additional GAP at thetrailing edge needs to be hypothesized.Based on these results, in the next chapter we replace the step functionswith data on the explicit PKC activity profiles, and ask whether modelpredictions could be corrected.66Chapter 6PKCs as spatially dependentparameters: explicit activityprofiles6.1 OverviewBased on the failure to explain experimental data using step function rep-resentations of the PKC activities, we asked whether more detailed spa-tiotemporal PKC activity profiles would improve the results. Consequently,we requested detailed profiles of both PKC beta and PKC eta from W.Bement. We received a PKC dataset from a control experiment, as wellas Rho GTPase data from each manipulation. In addition, we also hadprevious control Rho GTPase data at hand (Figure 1.4).In this chapter, we implement explicit spatiotemporal PKC activity pro-files given by control data. For proper PKC localization, we merge the twodifferent control PKC and control Rho GTPase datasets. Given differencesbetween datasets, we must horizontally shift the PKC data so that the RhoAzone and PKC peaks overlap. As in Chapter 5, we let PKCs modulate thebasal Rho GTPase activation rates. We calibrate the model variant by firstapproximating the detailed PKC activity profiles with a step function pro-file, and then apply constraints to estimate parameters. With this modelvariant, the simulated control experiment did not account for the controldata. As such, we revise a major assumption, namely that PKC beta ex-pression should saturate. We conclude that this revision is appropriate, andthat we should continue with a rigorous fit of the control simulation, beforesimulating PKC manipulations.6.2 Defining PKC activity profilesWe received data on PKC activity profiles in the control wound experi-ment. These data remove ambiguity about the shape of the profiles, and676.2. Defining PKC activity profilesalso the temporal evolution of PKC localization. We discuss how our func-tions ?(x, t) and ?(x, t) are defined by the shape and localization of PKCactivity profiles.6.2.1 Explicit PKC activity profiles from dataWe use PKC activity profiles from 54? 84 s, at every 6 s interval. The activ-ity profiles begin at the wound center and end 50 ?m away. Here, we shownormalized PKC activity profiles of PKC beta and PKC eta (Figure 6.1).PKC beta spans a broader region than PKC eta, though both their peakactivities overlap in the same place. The two activity peaks move towardsthe wound center, while amplifying between 54? 78 s, until subsiding at84 s.0 10 20 30 40 5000.20.40.60.81 84 s post-wounding0 10 20 30 40 5000.20.40.60.81 78 s post-wounding0 10 20 30 40 5000.20.40.60.81 72 s post-wounding0 10 20 30 40 5000.20.40.60.81 66 s post-wounding0 10 20 30 40 5000.20.40.60.81PKC betaPKC eta60 s post-wounding0 10 20 30 40 5000.20.40.60.81 54 s post-woundingDistance from wound center (?m)Intensity (A.U.)Figure 6.1: Normalized intensity profiles of PKC eta and PKC beta activi-ties. Relevant times 54? 84 s post-wounding are shown.According to Figure 2.4, we assume that we can scale the PKC activity686.2. Defining PKC activity profilesprofiles to units of concentration in the same way that Cdc42 activity wasscaled at 90 s. The scaled PKC activity profiles are shown in Figure 6.2.Next, we make one more adjustment before defining ?(x, t) and ?(x, t).6.2.2 PKC localization to regions overlapping Rho GTPasezonesThe PKC activity profiles were obtained from a different control experimentthan the Rho GTPase profiles. Because of this, the two different datasetscannot be directly amalgamated. If we merge the two profiles, we see thatthe PKCs do not overlap the RhoA zone as we expect (Figure 6.2).0 5 10 15 20 25 3000.050.10.15 84 s post-wounding0 5 10 15 20 25 3000.050.10.15 78 s post-wounding0 5 10 15 20 25 3000.050.10.15 72 s post-wounding0 5 10 15 20 25 3000.050.10.15 66 s post-wounding0 5 10 15 20 25 3000.050.10.15 PKC betaPKC eta60 s post-wounding0 5 10 15 20 25 3000.050.10.15 RhoACdc42W54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 6.2: PKC activity profiles and Rho GTPase activities from differentcontrol datasets are compared. PKC zones do not overlap the RhoA zone.W denotes the wound edge from the Rho GTPase dataset.In order to resolve this issue, we translate the PKC profiles by a fixed696.2. Defining PKC activity profilesamount. A clue is provided in Figure 2.4 B?, where we observe a gradient ofPKC eta crossing through the Cdc42 zone at 90 s. Accordingly, we make a7 ?m shift to the PKC profile relative to Cdc42 activity, at 90 s (Figure 6.3).The final merged datasets show appropriate PKC localization to the sameregions as the RhoA zone (Figure 6.4).0 5 10 15 20 25 30Cdc4200.0250.050.0750.1PKC betaPKC eta90 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 6.3: Aligning the PKC profiles with the Cdc42 peak at 90 s post-wounding. A 7 ?m shift towards the wound center is illustrated.We emphasize that we are merely correcting for the differences betweendatasets, so that PKCs localize to regions where the RhoA zone manifests.We do not implement any sort of PKC dependence on GTPases, as in Chap-ter 5. With this correction, we can now define our PKC profile functions. Todefine ?(x, t) and ?(x, t), we interpolate the explicit PKC activity profilesover the spatial domain, as well as between time points.706.3. Spatially dependent basal Rho GTPase activation rates0 5 10 15 20 25 3000.050.10.15 84 s post-wounding0 5 10 15 20 25 3000.050.10.15 78 s post-wounding0 5 10 15 20 25 3000.050.10.15 72 s post-wounding0 5 10 15 20 25 3000.050.10.15 66 s post-wounding0 5 10 15 20 25 3000.050.10.15 PKC betaPKC eta60 s post-wounding0 5 10 15 20 25 3000.050.10.15 RhoACdc42W54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 6.4: To correct the PKC localization between datasets, the PKCprofiles are merged with Rho GTPase profiles under a 7 ?m shift towardsthe wound center.6.3 Spatially dependent basal Rho GTPaseactivation ratesThe PKC activity profiles ?(x, t) and ?(x, t), explicitly given by data, influ-ence the Rho GTPase basal activation rates. The rates kr0(x, t) and kc0(x, t)assume the forms in Equations 5.1 and 5.2. Again, six parameter valuesmust be identified.6.3.1 Parameter estimation and constraintsWe would like to take advantage of constraining kr0(x, t) and kc0(x, t) indistinct regions of PKC activity, as we have done in the previous chapter.716.3. Spatially dependent basal Rho GTPase activation ratesTo do so, we apply the zone localization definition to the PKCs, to delineateregions of high and low PKC activity. The mean activity within a PKCzone, and the mean activity outside the zone are shown in Figure 6.5.0 10 20 30 40 50Distance from wound center (?m)00.050.10.15Concentration (?M) PKC betaPKC etaPKC beta zonePKC eta zoneAvg PKC beta conc.Avg PKC eta conc.54 s post-woundingFigure 6.5: PKC zones via the zone localization definition. Mean PKCactivity within the zone, and mean PKC activity outside the zone are shown.As in Section 5.3, we consider three separate regions, each with its ownPKC beta and PKC eta level of activity. In Region 1, PKC beta activityis high (?H) and PKC eta activity is high (?H). In Region 2, PKC betaactivity is high and PKC eta activity is low (?L). In Region 3, PKC betaactivity is low (?L) and PKC eta activity is low. Parameter estimation isset up in the same way as in Section 5.3.1, however ?H,L and ?H,L are used.Equations 6.1-6.3 show the constraints applied in each region (LHS) and thekr0(x, t) parameters we must determine.Region 1 krctrl = krbasal + krPKC?H1 + ?1?H(6.1)Region 2 1.25krctrl = krbasal + krPKC?H1 + ?1?L(6.2)Region 3 0.75krctrl = krbasal + krPKC?L1 + ?1?L(6.3)726.4. Simulation of the control experimentUsing MATLAB?s leasqr, we determine the parameters by minimizingthe sum of squared residuals (SSR) of the LHS of Equations 6.1-6.3. TheCdc42 rate parameter kc0(x, t) is treated similarly. The resulting parametersare shown in Tables 6.1 and 6.2. In the parameterization of kc0(x, t), thebasal rate was fixed to zero because fitting leads to a negative value on theorder of 10?3.krbasal ( /s) 0.0034krPKC ( /?M/s) 0.12?1 ( /?M) 9.1SSR 1.4? 10?25Table 6.1: Parameterization of kr0(x, t) (Equations 6.1-6.3) by method ofleast-squares. The sum of squared residuals (SSR) is given.kcbasal ( /s) 0kcPKC ( /?M/s) 0.15?2 ( /?M) 85SSR 2.53? 10?6Table 6.2: Parameterization of kc0(x, t) by method of least-squares. The sumof squared residuals (SSR) is given.6.4 Simulation of the control experimentIn this section, we simulate the control experiment using detailed PKC activ-ity profiles and the parameterized basal activation rates. We define Model 5in the following way:Model 5. The spatially distributed PKC (as detailed profiles) model.Equations 5.1, 5.2, where ?(x, t) and ?(x, t) are detailed PKCactivity profiles from control data.Parameters Tables 6.1 and 6.2The Model 5 simulation is shown in Figure 6.6. The RhoA zone appearsto rise as the data does, while the Cdc42 zone prematurely rises before thedata. At 84 s post-wounding, the RhoA zone is significantly wider than thedata but still centered correctly. The Cdc42 zone matches the width of thedata but is offset from the data. We also observe the loss of a prominent736.4. Simulation of the control experimentRhoA shoulder. Due to discrepancies in RhoA activity, Model 5 cannotaccount for the experimental control.0 5 10 15 20 25 3000.050.1 54 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1Sim total RhoASim Cdc42Sim AbrSim wound edge78 s post-wounding0 5 10 15 20 25 3000.050.1 84 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 6.6: Control simulation for Model 5. The control data are shown forcomparison (circles/thick curves).6.4.1 Revising the kr0(x, t), kc0(x, t) relationship with PKCbetaThe Model 5 control simulation does not successfully recapitulate the RhoAzone?s width (Figure 6.6). To correct this, we re-examine the relationshipbetween kr0(x, t) and PKC beta. The current assumption is that kr0(x, t) isproportional to ?(x, t) (Equation 5.1). Our current parameterization of thebackground activation rate produces a RhoA zone that is too broad (Fig-ure 6.6). This suggests that PKC beta upregulation of the basal activationrate is too strong. This also suggests revising the PKC contribution term inEquation 5.1 which is currently linear and unbounded.746.4. Simulation of the control experimentInstead, we expect that PKC beta?s effect should saturate, and we revisethe relationship to a Michaelian term in ?(x, t),kr0(x, t) = krbasal + krPKC11 + ?1?(x, t)?r0?(x, t)?r0 + ?(x, t)(6.4)where ?r0 is the concentration of PKC beta at which the added rate is halfof its maximal value krPKC?r01+?1?(x,t). For low PKC activity (small ?(x, t)),the Michaelian term is well approximated by a linear relationship, reducingto our previous approximation in Equation 5.1.We want to estimate ?r0 so that this linearity gradually tapers off forhigher PKC beta activity and ultimately reduces the broad RhoA zone inthe control simulation. Other parameters need no revision and are retainedfrom the fits in the previous section. We sample a variety of ?r0 to find that?r0 = 2.26 ?M sufficiently reduces the width of the RhoA zone and improvesits height (Figure 6.7, Model 6). We define the model variant in the followingway:Model 6. The model with PKC beta saturation in kr0(x, t).Equations 6.4, 5.2, where ?(x, t) and ?(x, t) are detailed PKCactivity profiles from control data.Parameters ?r0 = 2.26 ?M, Tables 6.1 and 6.2756.4. Simulation of the control experiment0 5 10 15 20 25 3000.050.1 54 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1Sim total RhoASim Cdc42Sim AbrSim wound edge78 s post-wounding0 5 10 15 20 25 3000.050.1 84 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 6.7: Control simulation for Model 6. The control data are shown forcomparison (circles/thick curves).We similarly treat the basal Cdc42 activation rate and find that a Michaelianexpression with ?c0 = 1.02 ?M improves the Cdc42 zone?s height, width andspeed of rise (Figure 6.8, Model 7). This model variant is defined as:Model 7. The model with PKC beta saturation in kr0(x, t) and kc0(x, t).Equations 6.4 and analogous kc0(x, t) equation, where ?(x, t)and ?(x, t) are detailed PKC activity profiles from controldata.Parameters ?r0 = 2.26 ?M, ?c0 = 1.02 ?M, Tables 6.1 and 6.2766.5. Conclusions0 5 10 15 20 25 3000.050.1 54 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1Sim total RhoASim Cdc42Sim AbrSim wound edge78 s post-wounding0 5 10 15 20 25 3000.050.1 84 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 6.8: Control simulation for Model 7. The control data are shown forcomparison (circles/thick curves).6.5 ConclusionsIn this chapter, we used control data of PKC activity profiles to definethe functions ?(x.t) and ?(x, t). We had to combine two different controldatasets, and ensured that PKCs localized to regions overlapping the RhoAzone. The PKCs acted on the basal rates of activation kr0(x, t) and kc0(x, t).With the current linear dependence of the basal activation rate on PKCbeta, we saw the model fail to explain control data. The simulated controllost the RhoA shoulder, and the simulated RhoA zone grew too broad.We revised a major assumption in the basal activation rate?s depen-dence on PKC beta. We let the basal activation rate saturate with highPKC beta activity. The saturation is a reasonable assumption because PKCbeta expression should be limited. In experiments, PKC beta can only beoverexpressed by a certain amount before the Rho GTPase zones cease up-776.5. Conclusionsregulation. This makes sense when we consider that the lipid DAG, forwhich PKCs compete, is finite, and that PKCs depend on DAG for acti-vation. Thus, PKC activity should be bounded, as well as the activationrates.So far, increasing the spatial detail in the PKC activity profiles has de-manded a major revision in our assumptions on kr0(x, t) and kc0(x, t). Wehave only simulated the control experiments in this chapter, but see pre-liminary improvements. Before we simulate PKC manipulations, we furthermodify the latest model variant so that a rigorous fit of the control simula-tion can be done.78Chapter 7Full nonlinear parameterfitting of model to data7.1 OverviewIn the previous chapter, we used explicit PKC activity profiles and foundthat the basal activation rates had to saturate with PKC beta expression.After this modification, the control was simulated and improvements weremade in the simulated RhoA and Cdc42 zones. In this chapter, we use thesame modification but rigorously fit the control simulation to determine theparameters in the basal activation rates. With the parameters, we thensimulate PKC manipulations.To perform a fit to control data, we modify the advection velocity so thesimulated wound edge is pulled forwards at the same speed as the observedwound edge. Matching the wound edge aligns the simulated RhoA zone withthe data, making a fit possible. We proceed by decoupling the RhoA fit fromthe Cdc42 fit. We can do so since RhoA activity affects Cdc42 activity, butnot vice versa. Thus, we can first fit simulated RhoA to data, and determineparameters in kr0(x, t). Then with kr0(x, t) fixed, we fit simulated Cdc42 todata, determining parameters in kc0(x, t).With the calibrated model, we simulate PKC manipulations. Recall thatwe requested and received data on the PKC manipulations. In addition tothe control PKC profiles previously discussed, we received Rho GTPaseactivity profiles for each manipulation. This allows us to initiate each sim-ulated manipulation with the corresponding Rho GTPase initial conditionsfrom data. Additionally, we assume that PKCs localize in the same way asPKCs in the control simulation. Under the current assumptions, the modelis unable to account for experimental observations.We therefore revisit the fitted basal rates and notice that GTPase zonesare possible even at background PKC activity. At low PKC activity, bothfitted basal activation rates lie within the bistable region. For consistencyto previous model variants, we give the fitting routine a nonlinear constraint797.2. Fitting spatial background activation rates to control Rho GTPase activiteswhich enforces the basal activation rates to lie in a monostable regime wher-ever PKC activity is low. Under this constraint, we find fits to control RhoAdata successful, while fits to Cdc42 are not. Cdc42 fits invariably lead toloss of the Cdc42 zone. We further show that the PKC activity profiles arepartly responsible for discrete RhoA zones. However, we conclude that weneed to assess the validity of this assumption by way of experiment.7.2 Fitting spatial background activation rates tocontrol Rho GTPase activites7.2.1 Matching the simulated wound edge to dataIn order to easily fit Rho GTPase data at all six time points, we match thesimulated velocity of the wound edge to data. By solving the ODE describingwound edge velocity, dwdt = ?vc(t)w , the constant vc can be determined usingdata on wound edge position.54 60 66 72 78 84Time post-wounding (s)23456789Distance from wound center (?m) vc = 1.12 ?m2/s, constantvc(t), time-dependentReal wound edge54 60 66 72 78 84Time post-wounding (s)00.20.40.60.811.21.41.61.82v c(t) (?m2 /s)match wound edge position at 2 time ptsmatch wound edge position at 6 time ptsFigure 7.1: Simulated wound edge position (left) and velocity vc (right)showing the previous choices by Simon et al. (2013) in red, and our im-provement in blue. The simulated wound edge position matches data at54 s and 84 s post-wounding (left, red) when a constant vc(t) =1.12 ?m2/sis used (right, red). The simulated wound edge position matches data at sixtimes points (left, blue) when a time-dependent vc(t) is used (right, blue).Previously, vc was constant in time and was determined using woundedge positions at 54 s and 84 s (Figure 7.1, right). This resulted in the woundedge being pulled at a constant speed early in the simulation, eventuallyspeeding up to meet the exact final location observed in data (Figure 7.1,left). Instead, we choose to allow a time-dependent vc(t) which is determined807.2. Fitting spatial background activation rates to control Rho GTPase activitesusing wound edge position data at each six second interval (Figure 7.1,right). This results in the wound edge being pulled at a velocity whichmatches the wound edge position at all six time points (Figure 7.1, left).7.2.2 Fits to control RhoA dataWe strategically decouple our fitting procedure by first fitting kr0(x, t) toRhoA activities, since Cdc42 activity has no impact on RhoA (Figure 3.2).The model is still governed by Equations 3.2-3.4, but the basal activationrate has been revised to the function kr0(x, t) (Equation 6.4). We solve anonlinear curve-fitting problem using MATLAB?s lsqcurvefit. The fit toRhoA activity is shown in Figure 7.2, and we define Model 8 as follows:Model 8. The RhoA fitted model with PKC beta saturation in kr0(x, t).Equations 6.4, where ?(x, t) and ?(x, t) are detailed PKC ac-tivity profiles from control data.Parameters Listed in Figure 7.2.Advection velocity vc(t) such that wound edge locations agreewith control data throughout (Figure 7.1).Model 8 simulates the control well, but does not reproduce the RhoAshoulder. It captures the rise in the RhoA zone adequately, as well as thewidth of the zone base. Compared to Model 7 results, the backgroundactivities are also much improved.817.2. Fitting spatial background activation rates to control Rho GTPase activites0 5 10 15 20 25 3000.050.1 84 s post-wounding0 5 10 15 20 25 3000.050.1 78 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.2: Model 8 fits to RhoA activity with matched wound edge. Fittedparameters that resulted in a sum of squared residuals of 0.053 ?M2 were:krbasal =0.0058 /s, krPKC =0.38 /?M/s, ?1 =86 /?M, ?r0 =0.60 ?M. Controldata are shown as circles/thick curves.7.2.3 Fits to control Cdc42 dataRecall that Model 1 simulated a control Cdc42 zone that was slightly offsetfrom data (Figure 3.3). In Model 1, and all model variants presented, theoffset remains uncorrected. In order to fit the Cdc42 activity, we introducean additional parameter to adjust for the Cdc42 offset. The subsequent fitto Cdc42 activity is shown in Figure 7.3, using Model 9 which is defined as:Model 9. The fitted model with PKC beta saturation in kr0(x, t) andkc0(x, t).Equations 6.4 and analogous kc0(x, t) equation, where ?(x, t)and ?(x, t) are detailed PKC activity profiles from control827.2. Fitting spatial background activation rates to control Rho GTPase activitesdata.Parameters Listed in Figures 7.2, 7.3.Initial conditions Rho GTPase activity profiles from data.Advection velocity vc(t) such that wound edge locations agreewith control data throughout (Figure 7.1).The fully fitted model (Model 9) is successful in reproducing the con-trol Cdc42 activity. The rise in the simulated Cdc42 zone is slightly fasterthan data, but the shape and width at the final time point matches well.Simulated Cdc42 background activities match well too.0 5 10 15 20 25 3000.050.1 84 s post-wounding0 5 10 15 20 25 3000.050.1 78 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim Cdc42Sim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.3: Model 9 fits to Cdc42 activity, as in Figure 7.2. However,the parameters in the basal RhoA activation rate are fixed by the previousRhoA fit. Fitted parameters that resulted in a sum of squared residualsof 0.094 ?M2 were: kcbasal =0.00025 /s, kcPKC =0.38 /?M/s, ?2 =85 /?M,?c0 =0.043 ?M. An additional parameter, Cshift =?2.47 ?m, was introducedso that the offset Cdc42 zone may be fitted.837.3. Simulations of PKC manipulations7.3 Simulations of PKC manipulationsAfter the full nonlinear parameter fits, we simulate the PKC manipulationsusing Model 9. We reproduce PKC manipulations by specifying Rho GTPaseinitial conditions and changes in PKC activities. Furthermore, we use thecontrol PKC localization in the PKC manipulations. This assumes thatexogenous PKCs localize in the same way as endogenous PKCs.For each manipulation, the corresponding Rho GTPase initial conditionsare specified by data. To gauge the effects of the Rho GTPase initial con-ditions, we perform a simulation where PKC activities are at their controllevels. We then mimic overexpression, or expression of dominant-negativePKC, by modifying the relevant PKC activity by two-fold, or by half. Thesimulated manipulation is evaluated against criteria in Section 3.3.1, as inprevious chapters. In contrast to previous model validation, we have RhoGTPase data from each experimental PKC manipulation. We use these datafor direct qualitative and quantitative comparison to simulated Rho GTPaseresults. The following results show the simulated Rho GTPases plotted overRho GTPase data.7.3.1 Overexpression of PKC betaTo gauge the effect of the Rho GTPase initial conditions, Figure 7.4 showssimulation results where PKC beta activity remains at control levels. Inthe first panel, the Rho GTPase initial conditions are given by data. Inall panels, the Rho GTPase data are shown for comparison (circles/thickcurves).Throughout the simulation, we observe the Cdc42 zone broaden andsandwich the RhoA zone. The height of the Cdc42 zone matches data at54? 60 s and 84 s. The width of the Cdc42 zone is too broad, and the sand-wich effect is largely due to the initial profile. When comparing simulatedRhoA to data, we are limited to 54? 66 s. After this time, the RhoA dataare not shown because the RhoA band begins to dive below the focal plane.For early times, simulated RhoA peak zone activity is higher than data,though the background activities match well.Since this simulation is not a PKC manipulation where we have modifiedthe PKC activities from control levels, we do not evaluate these resultsagainst criteria for model validation. These results are to be contrastedwith the following simulations where the PKC activities are modified.847.3. Simulations of PKC manipulations0 10 20 30 40 5000.050.10.15 84 s post-wounding0 10 20 30 40 5000.050.10.15 78 s post-wounding0 10 20 30 40 5000.050.10.15 72 s post-wounding0 10 20 30 40 5000.050.10.15 66 s post-wounding0 10 20 30 40 5000.050.10.15 60 s post-wounding0 10 20 30 40 5000.050.10.150.20.25 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.4: Results for Model 9 where PKC beta Rho GTPase initial condi-tions are used and PKCs are at control levels. Rho GTPase data from thecorresponding PKC manipulation are shown for comparison (circles/thickcurves). However, RhoA data are not shown after 72 s since RhoA activitybegins to dive below the focal plane.Figure 7.5 shows the simulation where PKC beta is overexpressed bytwo-fold. Again, the simulation begins with initial conditions given by data.After 60 s, the simulated RhoA zone begins to overtake the broad Cdc42zone. By 72 s, RhoA has completely suppressed Cdc42 and filled the domain.In this simulation, we observe the annihilation of the Cdc42 zone. Thecriterion for zone maintenance is not satisfied (C1) since PKC beta overex-pression is too strong. As such, we perform another simulation where PKCbeta activity is overexpressed by less than two-fold.857.3. Simulations of PKC manipulations0 10 20 30 40 5000.050.10.15 84 s post-wounding0 10 20 30 40 5000.050.10.15 78 s post-wounding0 10 20 30 40 5000.050.10.15 72 s post-wounding0 10 20 30 40 5000.050.10.15 66 s post-wounding0 10 20 30 40 5000.050.10.15 60 s post-wounding0 10 20 30 40 5000.050.10.150.20.25 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.5: As in Figure 7.4, showing results for Model 9 where PKC betais overexpressed by two-fold.Figure 7.6 shows the simulation results where PKC beta is overexpressedby 1.5-fold. The first panel shows the initial conditions given by data.At 60 s, simulated Cdc42 broadens and still sandwiches the RhoA zone.Comparing the RhoA zone to the RhoA zones in Figures 7.4 and 7.5, wesee that overexpressed PKC beta impacts the zone intensity, as well as thewidth. Greater PKC beta activity leads to increased RhoA zone intensity,as well as breadth. The simulated RhoA zone widens at the expense of theCdc42 zone. Comparing the Cdc42 zone to Figure 7.4, we see that PKCbeta activity does not impact the Cdc42 zone intensity significantly. Wehave already discussed this issue as Cdc42 insensitivity to PKC beta whenmodulated through a GEF.While zones are maintained (C1), Model 9 does not satisfy the remainingcriteria C2 and C3. By direct comparison to data, the Cdc42 zone heightand width do not quantitatively match. As well, qualitatively, the data does867.3. Simulations of PKC manipulationsnot indicate that Cdc42 sandwiches the RhoA zone. Model 9 is unable toaccount for experimental observations.0 10 20 30 40 5000.050.10.15 84 s post-wounding0 10 20 30 40 5000.050.10.15 78 s post-wounding0 10 20 30 40 5000.050.10.15 72 s post-wounding0 10 20 30 40 5000.050.10.15 66 s post-wounding0 10 20 30 40 5000.050.10.15 60 s post-wounding0 10 20 30 40 5000.050.10.150.20.25 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.6: As in Figure 7.4, showing results for Model 9 where PKC betais overexpressed by 1.5-fold.877.3. Simulations of PKC manipulations7.3.2 Expression of dominant-negative PKC betaFigure 7.7 shows the simulation results where PKC beta activity remainsat control levels. In the first panel, we show the initial conditions whichare given by data. Without modifying PKC beta activities, we observethe simulated Rho GTPases at background activities throughout. At 84 s,simulated RhoA and Cdc42 features above background are visible. We donot evaluate these results against model validation criteria since we have notmodified PKC beta activities yet.0 10 20 30 40 5000.0584 s post-wounding0 10 20 30 40 5000.0578 s post-wounding 0 10 20 30 40 5000.0572 s post-wounding0 10 20 30 40 5000.0566 s post-wounding 0 10 20 30 40 5000.0560 s post-wounding0 10 20 30 40 5000.050.1 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.7: Results for Model 9 where dominant-negative PKC beta RhoGTPase initial conditions are used and PKCs are at control levels. RhoGTPase data from the corresponding PKC manipulation are shown for com-parison (circles/thick curves).Figure 7.8 shows the simulation results where we express dominant-negative PKC beta by modifying PKC beta activities to half the control887.3. Simulations of PKC manipulationslevels. The first panel shows initial conditions given by data. Throughoutthe panels, we observe background activities of simulated RhoA and Cdc42because the manipulation serves to increase the Rho GTPase thresholds.Even when Rho GTPase thresholds were at control level, zones could notbe maintained. Additionally, in comparison to Figure 7.7, the above back-ground features of RhoA and Cdc42 are smoothed out. Since zones are notmaintained (C1), Model 9 cannot account for experimental observations.0 10 20 30 40 5000.0584 s post-wounding0 10 20 30 40 5000.0578 s post-wounding 0 10 20 30 40 5000.0572 s post-wounding0 10 20 30 40 5000.0566 s post-wounding 0 10 20 30 40 5000.0560 s post-wounding0 10 20 30 40 5000.050.1 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.8: Results for Model 9 where dominant-negative PKC beta is ex-pressed at half the control activity. Rho GTPase data from the correspond-ing PKC manipulation are shown for comparison (circles/thick curves).7.3.3 Overexpression of PKC eta with inverted RhoGTPase zonesHere, we simulate PKC eta overexpression where Rho GTPase zone inversionhas occurred. From 54? 84 s post-wounding, the data already show inverted897.3. Simulations of PKC manipulationsRho GTPase zones (Figure 7.9).Figure 7.9 shows simulation results where PKC eta activities remain atcontrol levels. In the first panel, we show initial conditions where the Cdc42zone is closer to the wound edge than the RhoA zone. Throughout thesimulation, Rho GTPase activities are at background levels. We comparethese results to the next simulation where PKC eta activity is modified.0 10 20 30 40 5000.0584 s post-wounding0 10 20 30 40 5000.0578 s post-wounding 0 10 20 30 40 5000.0572 s post-wounding0 10 20 30 40 5000.0566 s post-wounding 0 10 20 30 40 5000.0560 s post-wounding0 10 20 30 40 5000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.9: Results for Model 9 where overexpressed PKC eta Rho GTPaseinitial conditions are used and PKCs are at control levels. Rho GTPasedata from the corresponding PKC manipulation are shown for comparison(circles/thick curves).Figure 7.10 shows simulation results where PKC eta activity is overex-pressed by two-fold. Again, Rho GTPase activities remain at backgroundlevels throughout. In comparison to Figure 7.9, the simulated Rho GTPaseactivities appear further suppressed. This manipulation lowers the Rho GT-Pase thresholds and background activities.907.4. Revisiting fits of background activation rates to Rho GTPase activityIn this simulation, zones are not maintained (C1). Therefore, Model 9cannot account for experimental observations.0 10 20 30 40 5000.0584 s post-wounding0 10 20 30 40 5000.0578 s post-wounding 0 10 20 30 40 5000.0572 s post-wounding0 10 20 30 40 5000.0566 s post-wounding 0 10 20 30 40 5000.0560 s post-wounding0 10 20 30 40 5000.050.10.15 Sim total RhoASim Cdc42Sim AbrSim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.10: Results for Model 9 where PKC eta is expressed at twice thecontrol activity. Rho GTPase data from the corresponding PKC manipula-tion are shown for comparison (circles/thick curves).7.4 Revisiting fits of background activation ratesto Rho GTPase activitySimulations with Model 9 fail to capture the expression of dominant-negativePKC beta and the overexpression of PKC eta. Both simulated manipulationscould not sustain zones. In the overexpression of PKC beta, however, zonemaintenance was possible in the simulation. Qualitative and quantitativeimprovements to the overexpressed PKC beta simulation results are needed.We can think of two immediate improvements to the simulation of PKC917.4. Revisiting fits of background activation rates to Rho GTPase activitybeta overexpression. First, in Figure 7.5, a very broad RhoA zone is ob-served. The broad RhoA zone extends over regions of background PKCactivities. This indicates that RhoA is bistable even when PKC activitiesare low. However, we argued that zone maintenance should not be possibleat basal PKC activities, meaning that Rho GTPases should be monostable(Section 5.3.2). To enforce this assumption of monostability, we specify non-linear constraints on the basal RhoGTPase activation rates kr0(x, t), kc0(x, t).We test whether PKC activities are partly responsible for regulating the dis-creteness of Rho GTPase zones. If this assumption is valid, we expect oursimulation results to show narrower Rho GTPase zones, where zone expan-sion has been curbed.The second improvement addresses the Cdc42 zone insensitivity to PKCbeta. As in Section 5.5, we allow PKCs to modulate the background inac-tivation rate of Cdc42 (k7(x, t), Equation 5.13), instead of the backgroundactivation rate of Cdc42 (kc0(x, t)). Recall that the bifurcation diagram (Fig-ure 4.5) indicates Cdc42 sensitivity to changes in k7.7.4.1 Constrained fitting of kr0(x, t) to control RhoA activityIn this section, we implement the first improvement to simulate the over-expression of PKC beta. We fit the basal activation rates to control RhoGTPase activities. Fits are performed with MATLAB?s fmincon where wehave supplied a nonlinear constraint on kr0(x, t). We contrast these con-strained fits with the previous unconstrained fits (Model 9).We decouple the RhoA fit from the Cdc42 fit. Two RhoA fits are per-formed, with each considering a different nonlinear constraint. One RhoAfit is subject to the constraint:Region 2 0.75krctrl ? krbasal + krPKC?H1 + ?1?L(7.1)where ?H and ?L are from Section 6.3.1. This constraint imposes RhoA in asingle, low background level wherever PKC eta activity is low (i.e., outsideof the RhoA zone; Region 2). Recall that a fold change of 0.75 in the basalRhoA activation rate results in monostable RhoA activity. Model 10 isdefined in the following way:Model 10. The RhoA fitted model with a nonlinear constraint (Equa-tion 7.1) on kr0(x, t).Equations 6.4, where ?(x, t) and ?(x, t) are detailed PKC ac-tivity profiles from control data.927.4. Revisiting fits of background activation rates to Rho GTPase activityParameters Listed in Figure 7.11, satisfying the nonlinear con-straint in Equation 7.1.Advection velocity vc(t) such that wound edge locations agreewith control data throughout (Figure 7.1).Figure 7.11 shows the RhoA fit using Model 10. The simulation resultscapture the control RhoA zone?s height, width and speed of amplificationadequately. If we contrast this to the unconstrained fit (Figure 7.2), weobserve a minor difference in the simulated background activities. The dif-ference produces a slightly higher sum of squared residuals (SSR) in theModel 10 fit (0.059 ?M2) than the Model 8 fit (0.053 ?M2).0 5 10 15 20 25 3000.050.1 84 s post-wounding0 5 10 15 20 25 3000.050.1 78 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.11: Model 10 fits to control RhoA activity. Fitted parameters thatresulted in a sum of squared residuals of 0.059 ?M2 were: krbasal =0.0027 /s,krPKC =0.29 /?M/s, ?1 =24.83 /?M, ?r0 =0.46 ?M. Control data are shownfor comparison (circles/thick curves).937.4. Revisiting fits of background activation rates to Rho GTPase activityThe second RhoA fit is subject to multiple nonlinear constraints:Region 1 krctrl ? krbasal + krPKC?H1 + ?1?H(7.2)Region 2 1.25krctrl ? krbasal + krPKC?H1 + ?1?L(7.3)Region 3 0.75krctrl ? krbasal + krPKC?L1 + ?1?L(7.4)where ?H,L and ?H,L are from Section 6.3.1. These constraints impose RhoAin a single, low background level wherever both PKC beta and PKC etaare low (i.e., outside both GTPase zones; Region 3). The constraints wereexplained in detail in Section 5.3.2.Model 11 is defined in the following way:Model 11. The RhoA fitted model with nonlinear constraints (Equations 7.2-7.4) on kr0(x, t).Equations 6.4, where ?(x, t) and ?(x, t) are detailed PKC ac-tivity profiles from control data.Parameters Listed in Figure 7.12, satisfying the nonlinear con-straints in Equations 7.2-7.4.Advection velocity vc(t) such that wound edge locations agreewith control data throughout (Figure 7.1).Figure 7.12 shows the RhoA fit using Model 11. The simulation resultscapture the RhoA zone just as well as Model 8 and Model 10. However,the background activities are visibly worse, increasing the sum of squaredresiduals to 0.071 ?M2.947.4. Revisiting fits of background activation rates to Rho GTPase activity0 5 10 15 20 25 3000.050.1 84 s post-wounding0 5 10 15 20 25 3000.050.1 78 s post-wounding0 5 10 15 20 25 3000.050.1 72 s post-wounding0 5 10 15 20 25 3000.050.1 66 s post-wounding0 5 10 15 20 25 3000.050.1 60 s post-wounding0 5 10 15 20 25 3000.050.10.15 Sim total RhoASim wound edge54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.12: Model 11 fits to control RhoA activity. Fitted parameters thatresulted in a sum of squared residuals of 0.071 ?M2 were: krbasal =0.0057 /s,krPKC =0.096 /?M/s, ?1 =17.25 /?M, and ?r0 =42.06 ?M.Simulating overexpression of PKC betaUsing the calibrated Models 10 and 11, we simulate the overexpression ofPKC beta, ignoring Cdc42 for now. Figure 7.13 shows the simulation resultswhere PKC beta is overexpressed by two-fold. Results from Model 9 (green),Model 10 (olive), and Model 11 (blue) are compared. In the first panel, wesee that each simulation begins with initial conditions given by data.As the simulation progresses, we observe that Model 11 produces a nar-rower and more focused RhoA zone. If we consult Figure B.1, we see thatthe RhoA zone breadth is limited to regions where PKC activities are high.Therefore, under the assumption that GTPase zones should not form atbasal PKC activities, we have shown that PKC activity profiles are partlyresponsible for discrete GTPase zones. To implement this assumption, we957.4. Revisiting fits of background activation rates to Rho GTPase activityapplied two different nonlinear constraints to the RhoA fits, but found thatonly Model 11 was successful.0 10 20 30 40 5000.050.10.15 84 s post-wounding0 10 20 30 40 5000.050.10.15 78 s post-wounding0 10 20 30 40 5000.050.10.15 72 s post-wounding0 10 20 30 40 5000.050.10.15 66 s post-wounding0 10 20 30 40 5000.050.10.15 60 s post-wounding0 10 20 30 40 5000.050.10.150.20.25 RhoA, unconstrained fitRegion 2 constrainedRegion 1-3 constrained54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure 7.13: Results for Model 10 (olive) and Model 11 (blue) where PKCbeta is overexpressed by two-fold. The unconstrained fit from Model 8is shown for comparison (green). RhoA data from the corresponding PKCmanipulation are also shown for comparison (circles/thick curves). However,RhoA data are removed after 72 s because RhoA dives below the focal plane.7.4.2 Constrained fitting of kc0(x, t) or k7(x, t) to controlCdc42 activityApplying the nonlinear constraints (Equations 7.2-7.3) resulted in a restric-tion of the extreme RhoA broadening that was previously observed. Weadopt these constraints to fit either kc0(x, t) or k7(x, t). The latter is ex-pected to address the Cdc42 zone insensitivity to PKC beta.When we attempt a constrained fit of either basal Cdc42 (in)activation967.5. Conclusionsrate, the fits result in a loss of the Cdc42 zone, and background Cdc42activity throughout the simulation. Since we cannot calibrate the Cdc42control simulation under these constraints, we do not proceed further andrefrain from simulating PKC manipulations.7.5 ConclusionsIn this chapter, we performed a full nonlinear parameter fit of the modelto control data. We first adjusted the advection velocity of the woundedge to match data at all time points. Then we fitted RhoA in isolationsuccessfully. Next we attempted to fit the full system, including Cdc42. Themodel made assumptions about PKC localization and Rho GTPase initialconditions. We assumed that PKC localization in each manipulation wasthe same as the control. The Rho GTPase initial conditions were assumedto be given by data. The model was fitted and not subject to any nonlinearconstraints. The calibrated model could account for the control (withoutreproducing the RhoA shoulder), but was unable to account for the observedPKC manipulations.We took a second look at model assumptions and the fitting procedure.We assumed that Rho GTPase zones are unable to form when PKC activitiesare at background, thereby assuming that Rho GTPases are monostable overregions of low PKC activity. As such, we imposed a nonlinear constraint onthe parameter fits, enforcing GTPase monostability in regions of low PKCactivity. With these constraints, we fitted the model to RhoA control data.We simulated PKC beta overexpression, and compared the simulations fromconstrained fits to simulations from unconstrained fits. We observed thatthe assumption on zone formation at PKC resting levels, which imposed thenonlinear constraint, was effective at curbing broad RhoA zones. Lastly, wefound that fits to Cdc42 were thoroughly unsuccessful, and therefore themodel could not account for the control data. We were unable to proceedin reproducing PKC manipulations.Based on these results, we ask whether Rho GTPases should be monos-table in regions of low PKC activities. Experimentally, this would mean thata resting cell is unable to sustain GTPase zones. Moreover, if a resting cellcould be locally perturbed by Rho GTPases, without a zone forming, theassumption would be valid. If the assumption is warranted, we would be con-fident in applying the nonlinear constraints in Equations 7.2-7.4. However,we require additional experimental evidence regarding this assumption.97Chapter 8ConclusionsIn this thesis, we tested whether PKCs could account for Rho GTPase pat-terning in single cell wound healing. We started with a Rho GTPase modelfrom Simon et al. (2013), and allowed PKCs to affect the Rho GTPase basal(in)activation rates with increasing spatial detail. We tested models withspatially constant PKCs, as well as spatially distributed PKCs. The spa-tially distributed PKCs were either represented as simple step functions,or were given by explicit PKC activity profiles from data. While our modelvariants could account for the control, they were unable to recapitulate PKCmanipulations so far.We addressed five questions about how PKCs exert their influence onRho GTPases (Section 2.5). We will review each question in turn. The firstquestion asked whether PKC beta upregulated Rho GTPase activity by en-hancing activation (through a GEF) or by depressing inactivation (througha GAP). A similar question applied to PKC eta. To answer this question,we performed a bifurcation analysis on the Rho GTPase basal (in)activationrates in Chapter 4. The bifurcation diagrams identified parameter regimes ofGTPase bistability, and hence regimes where zone maintenance was possible.The diagrams were used primarily for parameter estimation, but they alsoshowed that nearly equivalent effects on GTPase activity occurred througheither increasing the basal activation rate, or decreasing the basal inacti-vation rate, and vice versa. Correspondingly, the bifurcation diagrams ofbasal activation rates were reflections of the bifurcation diagrams of basalinactivation rates. The bifurcation diagrams did not immediately suggestthat PKC modulation of either basal GTPase rate (i.e., through GEFs orGAPs) was more likely.It is possible, however, that PKC modulation of basal Rho GTPase inac-tivation rates is more likely to occur. In Chapters 5-7, we noticed that Cdc42was insensitive to changes in PKC beta activity when the basal Cdc42 activa-tion rate was modulated. Allowing PKC beta to affect the basal Cdc42 inac-tivation rate increased Cdc42?s sensitivity. From this, we can conclude thatPKC regulation of Rho GTPase GAPs is more likely to quantitatively ac-count for observed GTPase zone intensities. Our future model (Model FM)98Chapter 8. Conclusionswill allow PKCs to affect the basal inactivation rates.The next question we asked was whether the spatial distribution of PKCswas important in accounting for the observed GTPase zones. By increasingspatial detail in the PKC activity profiles, we showed that the spatial distri-bution was indeed important in accounting for GTPase patterning. The firstmodel with spatially constant PKCs failed to explain PKC manipulations,allowing us to conclude that PKC spatial detail was necessary. Furthermore,when spatially distributed PKCs were implemented as step functions, we re-produced the RhoA shoulder, a subtle feature in Rho GTPase patterning.The RhoA shoulder within the Cdc42 zone appeared as a result of totalPKC eta absence from the Cdc42 zone.For the next model, we requested and received detailed PKC activityprofiles. But the PKC data was from another experiment, and was notsynchronized with the RhoA and Cdc42 data. We compensated for the dif-ferences in datasets by aligning PKC activity peaks to RhoA activity peaks.Unfortunately, this was not enough to resolve issues between datasets. Asa result, the detailed PKC activity profiles did not recapitulate the RhoAshoulder in GTPase patterning. The loss of the RhoA shoulder was dueto mismatched datasets which allowed the presence of PKC eta within theCdc42 zone. This motivates Model FM to eschew the detailed PKC activityprofiles, and adopt step function representations of the PKCs.The third question we asked was why PKC eta only appeared in theRhoA zone, and was absent from the Cdc42 zone. To mimic this, we im-plemented PKC localization that was Rho GTPase-dependent. In the end,this dependency was rejected because it was unjustified. Instead, we knowthat PKCs are activated by the lipid DAG, so DAG-dependence is likelythe cause of PKC localization. We did not explicitly consider DAG in thisthesis, so we took the spatially detailed PKC activity profiles at face value.Still, we wonder why PKC eta localizes to a narrower region than PKCbeta, even though DAG presence is broad. We speculate that it is possiblethat PKC eta weakly competes for DAG, and is therefore more likely to bepresent in regions abundant in DAG. Since DAG is mostly concentrated inthe RhoA zone, this would explain why PKC eta is present only in thatnarrow region.The fourth question we asked also concerned PKC eta localization. Weasked how PKC eta could be absent from the Cdc42 zone and still exertinfluence over it. Our first simple spatial model defined PKC eta as essen-tially zero within the Cdc42 zone, hence the Cdc42 zone was completelyunaffected by PKC eta. We tried to cure the Cdc42 zone insensitivity toPKC eta by using explicit PKC activity profiles. The PKC profiles had a99Chapter 8. Conclusionsgradient of PKC eta running through the Cdc42 zone. We were unable totest whether this was the correct improvement because we only had datato simulate PKC eta overexpression with inverted GTPase zones. Due toinverted GTPase zones, PKCs did not encompass the Cdc42 zone in theusual way (Figure B.3). A better test would be to run a simulation withcontrol Rho GTPase initial conditions and PKC eta overexpression. Suffi-cient changes to the Cdc42 zone intensity would validate the use of detailedPKC profiles. Unfortunately, we did not try this due to time constraints.Another way to address PKC eta?s influence on the Cdc42 zone is throughthe implementation of PKC competition for DAG. In this scenario, PKC etacompetes with PKC beta for DAG, only in the RhoA zone. Indirectly, PKCeta can affect the Cdc42 zone, without being present there, by modifyingPKC beta which does extend over the Cdc42 zone. We envision implement-ing PKC competition for DAG in Model FM.Our last question asked whether PKCs affected activation rates linearly,or not. This can be related to a question about how DAG modulates PKCeffects on Rho GTPases. We found that the basal activation rates mustsaturate with PKC beta, in order to successfully reproduce a control ex-periment. The saturation possibly reflects the finite amount of DAG, andthe finite expression of PKC attainable. Model FM will also employ PKCsaturation.In the end, our main challenges have been uncertainties in data and theassumptions we have made. In particular, we suggest that PKCs and RhoGTPases be measured simultaneously in a control experiment. This wouldresolve our issues with merging datasets.We would also like clarification on an assumption we have used. Weassumed that at background PKC activities, as in a resting cell, Rho GTPasezones are impossible to sustain (i.e., are monostable). This led us to imposea monostable constraint on Rho GTPases wherever PKC activity was low.We saw that PKC activity profiles were partly responsible for the discreteRho GTPase zones. In order to validate this assumption, we would likean experiment which locally perturbed Rho GTPase activities in a resting(unwounded) cell. The perturbation could be an injection of constitutivelyactive RhoA or Cdc42. If zones are unable to form, our assumption wouldbe justified.We are optimistic that PKCs can account for Rho GTPase patterning,and will ultimately allow us to investigate Rho GTPase zone inversion. Weare optimistic that it would be fruitful to take what we have learned here tomotivate the next model. In the absence of detailed PKC activity profileswithout dataset issues, Model FM should make the following assumptions:100Chapter 8. Conclusions? PKC activity profiles can be represented as simple step functions, asin Models 3, 4.? PKC localization should be inferred from one dataset ? the Rho GT-Pase control. That is, the zone localization definition applied to RhoGTPase control activities should dictate control PKC zones and re-cruitment. In simulated manipulations, we would use the same con-trol PKC zones but adjust the PKC activity levels. This is a newassumption that has not been attempted.? PKCs should affect the basal inactivation rates, as in Model 4.? The basal rates should saturate with PKC activity, as in Models 6, 7.? Regions of background PKC activity should not be able to sustainzones, and GTPase activities should be monostable there, as in Mod-els 3-7, 10, 11.? PKC competition for DAG should be implemented. This is a newassumption.Again, we would use the model to simulate PKC manipulations, and testwhether the experimental observations can be accounted for.101BibliographyMutsuki Amano, Masaaki Ito, Kazushi Kimura, Yuko Fukata, KazuyasuChihara, Takeshi Nakano, Yoshiharu Matsuura, and Kozo Kaibuchi.Phosphorylation and activation of myosin by Rho-associated kinase (Rho-kinase). Journal of Biological Chemistry, 271(34):20246?20249, 1996.William M Bement, Craig A Mandato, and Mary N Kirsch. Wound-inducedassembly and closure of an actomyosin purse string in Xenopus oocytes.Current Biology, 9(11):579?587, 1999.William M Bement, Ann L Miller, and George von Dassow. Rho GTPaseactivity zones and transient contractile arrays. Bioessays, 28(10):983?993,2006.William M Bement, Emily M Vaughan, Hoi-Ying E Yu, Amber Lasek, NickVitale, and Troy A Hornberger. Lipid domain-dependent regulation ofsingle cell wound repair. Unpublished manuscript, 2012.He?le`ne A Benink and William M Bement. Concentric zones of active RhoAand Cdc42 around single cell wounds. The Journal of Cell Biology, 168(3):429?439, 2005.TH Chuang, X Xu, V Kaartinen, N Heisterkamp, J Groffen, andGM Bokoch. Abr and Bcr are multifunctional regulators of the Rho GTP-binding protein family. Proceedings of the National Academy of Sciences,92(22):10282?10286, 1995.Ce?line DerMardirossian and Gary M Bokoch. GDIs: central regulatorymolecules in Rho GTPase activation. Trends in Cell Biology, 15(7):356?363, 2005.Aron B Jaffe and Alan Hall. Rho GTPases: biochemistry and biology.Annual Review of Cell and Developmental Biology, 21:247?269, 2005.Craig A Mandato and William M Bement. Contraction and polymerizationcooperate to assemble and close actomyosin rings around Xenopus oocytewounds. The Journal of Cell Biology, 154(4):785?798, 2001.102Hiroaki Miki and Tadaomi Takenawa. Regulation of actin dynamics byWASP family proteins. Journal of Biochemistry, 134(3):309?313, 2003.Cory M Simon, Emily M Vaughan, William M Bement, and Leah Edelstein-Keshet. Pattern formation of Rho GTPases in single cell wound healing.Molecular Biology of the Cell, 24(3):421?432, 2013.Emily M Vaughan, Ann L Miller, Hoi-Ying E Yu, and William M Bement.Control of local Rho GTPase crosstalk by Abr. Current Biology, 21(4):270?277, 2011.103Appendix AZone amplification factorsfrom box-and-whisker plotsIn Chapter 2, we present Rho GTPase data in PKC manipulations. Changesin Rho GTPase zone intensity are quantified in box-and-whisker plots, andcompared to the control Rho GTPase zones. We express the changes in RhoGTPase zone intensity as a fold change relative to the control. These foldchanges are used as zone amplification factors in simulated PKC manipula-tions to scale the initial conditions.Intensity (A.U.)fold change relative to controlControl OEPKCbeta30 sRhoA 6.0 12.0 2Cdc42 8.0 21.0 2.660 sRhoA 13.2 14.3 ?Cdc42 20.4 35.6 1.7Table A.1: Fold change in RhoA and Cdc42 zone activities, relative tocontrol, when PKC beta is overexpressed. Based on mean intensity data(in arbitrary units) taken from Figure 2.2 D. Fold changes that are notstatistically significant are indicated by ?.Intensity (A.U.)fold change relative to controlControl DNPKCbetaRhoAZone 39.4 19.2 0.49Background 24.1 25.0 ?CdcZone 49.1 19.7 0.40Background 29.0 27.1 ?Table A.2: Fold change in RhoA and Cdc42 zone activities, relative tocontrol, when dominant-negative PKC beta is expressed. Based on meanintensity data (in arbitrary units) taken from Figure 2.2 E. Fold changes inbackground activity that are not statistically significant are indicated by ?.104Appendix A. Zone amplification factors from box-and-whisker plotsIntensity (A.U.)fold change relative to controlControl OEPKCeta30 sRhoA 8.0 1.6 0.2Cdc42 6.8 3.0 0.4460 sRhoA 20.4 2.4 0.12Cdc42 15.4 7.2 0.46Table A.3: Fold change in RhoA and Cdc42 zone activities, relative tocontrol, when PKC eta is overexpressed. Based on mean intensity data (inarbitrary units) taken from Figure 2.2 F.Intensity (A.U.)fold change relative to controlControl DNPKCetaRhoAZone 37.7 35.1 ?Background 24.0 57.2 2.4CdcZone 49.3 40.3 ?Background 31.2 45.5 1.46Table A.4: Fold change in RhoA and Cdc42 background activities, relativeto control, when dominant-negative PKC eta is expressed. Based on meanintensity data (in arbitrary units) taken from Figure 2.2 G. Fold changes inzones that are not statistically significant are indicated by ?.105Appendix BAmalgamating PKC andRho GTPase datasetsIn Chapter 6, we used explicit PKC activity profiles given by control data.Due to differences in datasets, the PKC profiles had to be shifted to correctfor localization to the control Rho GTPase zones. In Chapter 7, we haveRho GTPase data for each PKC manipulation. In each PKC manipulation,we must correct the PKC activity profile and shift it to overlap the RhoAzone. The following figures illustrate these corrections.106B.1. Overexpression of PKC betaB.1 Overexpression of PKC beta0 10 20 30 40 5000.050.10.15 84 s post-wounding0 10 20 30 40 5000.050.10.15RhoACdc42PKC etaPKC beta78 s post-wounding 0 10 20 30 40 5000.050.10.15 72 s post-wounding0 10 20 30 40 5000.050.10.15 66 s post-wounding0 10 20 30 40 5000.050.10.15 60 s post-wounding0 10 20 30 40 5000.050.10.15 54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure B.1: Alignment of control PKC profiles with Rho GTPase profiles inPKC beta overexpression. PKC profiles are aligned with Rho GTPase dataas is (unshifted). The wound edge is not shown but is towards the left. At72 s, the RhoA zone dives away from the focal plane.107B.2. Expression of dominant-negative PKC betaB.2 Expression of dominant-negative PKC beta0 10 20 30 40 5000.050.10.15 84 s post-wounding0 10 20 30 40 5000.050.10.15 78 s post-wounding0 10 20 30 40 5000.050.10.15 72 s post-wounding0 10 20 30 40 5000.050.10.15 66 s post-wounding0 10 20 30 40 5000.050.10.15 60 s post-wounding0 10 20 30 40 5000.050.10.15 RhoACdc42PKC betaPKC eta54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure B.2: Alignment of control PKC profiles with Rho GTPase profilesin dominant-negative PKC beta expression. PKC profiles are aligned withRho GTPase data when shifted 5 ?m away from the wound center. Thewound edge is not shown but is towards the left.108B.3. Overexpression of PKC eta with inverted Rho GTPase zonesB.3 Overexpression of PKC eta with invertedRho GTPase zones0 10 20 30 40 5000.050.184 s post-wounding0 10 20 30 40 5000.050.178 s post-wounding 0 10 20 30 40 5000.050.172 s post-wounding0 10 20 30 40 5000.050.166 s post-wounding 0 10 20 30 40 5000.050.160 s post-wounding0 10 20 30 40 5000.050.1 RhoACdc42PKC betaPKC eta54 s post-woundingDistance from wound center (?m)Concentration (?M)Figure B.3: Alignment of control PKC profiles with Rho GTPase profiles inPKC eta overexpression. PKC profiles are aligned with Rho GTPase dataas is (unshifted). The wound edge is not shown but is towards the left.109

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0074055/manifest

Comment

Related Items