A Stochastic Modelling Framework for Single Cell Migration: Coupling Contractility and Focal Adhesions
Abstract
:1. Introduction
2. The Cell Motility Model
2.1. The Cell Representation
 The traction stresses are largely applied on the cell periphery and their magnitude decays rapidly towards the centre [46,47]. Thus, the cell body SF ends are at or near mechanical equilibrium. Since contractile forces are generated by SFs, then a cell body SF end must be balanced by all other SFs (due to the equilibrium). Hence, it is reasonable to have a single connecting node of radial SFs which is either at mechanical equilibrium (for stationary cells) or tends to it.
 Paul et al. [48] demonstrated that application of force, originating from nuclear region, on FAs by starlike SF arrangement, results in cells acquiring morphologies typical for motile cells. Since the forces applied on FAs by SFs determine which one ruptures, then it also influences the motion of a cell (due to retraction). Since I am primarily interested in cell migration, it is justified to assume that this architecture represents a realistic situation. Furthermore, Oakes et al. [49] found that modelling SFs embedded in contractile networks, where only SFs actively contract, yields a behaviour mimicking their experimental results—the cytoskeletal flow was directed along the stress fibres. In the same study, the authors concluded that it is appropriate to treat an SF as a 1D viscoelastic contractile element, which also justifies neglecting inertia in Equation (2).
 Since motile cells assume a wide variety of cell shapes and continuously remodel their actin cytoskeleton, one can view this representation as a cell shape normalization (it is implicitly assumed that a cell volume remains constant). That is, Figure 1b depicts a cell normalized to a circle. Möhl et al. [46] applied the shape normalization technique to a timelapse series data of migrating keratinocytes and demonstrated that this allows consistent analysis of FA dynamics, actin flow and traction forces. In view of their results, a particular cell traction force map and FA configuration normalized to a circle can be effectively captured by our representation.
2.2. The Cell Migration Cycle
2.2.1. Focal Adhesion Binding
2.2.2. Focal Adhesion Unbinding
2.3. Specification of Kinematics
 The spatial and cell length scales are defined by cell radii ${R}_{cell}$.
 The time scale is set by FA disassociation rate ${k}_{off}^{0}$, since FA unbinding of leads to locomotion.
 The force scale is defined by the characteristic force ${F}_{b}$ usually observed at an FA.
3. FA Event Model
3.1. Focal Adhesion Events
3.2. Combining the Cell Motility and the FA Event Model
 The time ${T}_{1}$ of the FA event is chosen such that $\mathbb{P}({T}_{1}>s)=S\left(s\right)$.
 The system evolves according to Equation (7) for $t\in [0,{T}_{1})$.
 At time $t=\tau $, the index j of the FA event is chosen with probability ${K}_{index}\left(j\right{T}_{1},0,{\mathbf{y}}^{0})$ $\mathbf{Y}$ and $\mu $ jump to new values:$$\begin{array}{c}\mathbf{Y}(t=\tau )=\{\begin{array}{c}{\mathbf{y}}^{0}+{\widehat{\mathbf{e}}}_{i},\phantom{a}i=(j+1)/2,\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}j\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{odd}\hfill \\ {\mathbf{y}}^{0}{\widehat{\mathbf{e}}}_{i},\phantom{a}i=j/2,\phantom{\rule{4.pt}{0ex}}\mathrm{else}\hfill \end{array},\hfill \\ \mu (t=\tau )=\{\begin{array}{l}0,\text{}\mathrm{if}j\mathrm{is}\mathrm{odd}\\ 0,\text{}\mathrm{else}\end{array},\hfill \end{array}$$
 The cycle starts anew with initial time $t={T}_{1}$ and initial values of other variables at this time: starting at $t={T}_{1}$ the time ${T}_{2}$ of the FA event is chosen such that$$\begin{array}{c}\mathbb{P}({T}_{2}>s{T}_{1})=exp\left(\right)open="("\; close=")">{\int}_{{T}_{1}}^{{T}_{1}+s}{a}_{0}(\mathbf{y},{\tau}^{\prime})d{\tau}^{\prime}.\hfill \end{array}$$
 The system evolves according to Equation (7) for $t\in [{T}_{1},{T}_{1}+{T}_{2})$ and so on.
4. Piecewise Deterministic Process
4.1. PDMP Overview
 I
 Vector fields $({\mathbf{H}}_{\nu},\nu \in A)$ such that for all $\nu \in A$ there exists a unique global solution ${\mathbf{X}}_{t}\in \mathsf{\Gamma}$ to the following equation:$$\begin{array}{cc}\frac{d}{dt}{\mathbf{X}}_{t}\hfill & ={\mathbf{H}}_{\nu}\left({\mathbf{X}}_{t}\right)\hfill \\ {\mathbf{X}}_{0}\hfill & \in \mathsf{\Gamma}.\hfill \end{array}$$Let ${\varphi}_{\nu}:[0,\infty )\times \mathsf{\Gamma}\to \mathsf{\Gamma}$ denote the flow corresponding to Equation (12), i.e.,$$\begin{array}{c}{\varphi}_{\nu}(t,{\mathbf{X}}_{0})={\mathbf{X}}_{t}.\hfill \end{array}$$
 II
 A measurable function ${a}_{0}:E\to {\mathbb{R}}_{+}$ such that the function $s\mapsto {a}_{0}(\nu ,{\varphi}_{\nu}(s,{\mathbf{X}}_{0}))$ is integrable.
 III
 A transition measure $Q:\mathcal{E}\times E\to [0,1]$, such that for fixed $C\in \mathcal{E}$, $(\nu ,\mathbf{\xi})\mapsto Q(C;(\nu ,\mathbf{\xi}\left)\right)$ is measurable for $(\nu ,\mathbf{\xi})\in E$ and $Q(\xb7;(\nu ,\mathbf{\xi}\left)\right)$ is a probability measure for all $(\nu ,\mathbf{\xi})$ on $(E,\mathcal{E})$.
4.2. Cell Motility and PDMP
 I
 By Proposition 4 we see that for all $\nu \in A$, there exists a unique global solution to (12).
 II
 Note that in our case the rate function ${a}_{0}$ is given by (recalling Section 3.1):$$\begin{array}{c}{a}_{0}(\nu ,{\mathbf{X}}_{t})={a}_{0}({\mathbf{\alpha}}_{2}\left(\nu \right),{\mathbf{X}}_{t})=\sum _{j=1}^{2M}{a}_{j}({\mathbf{\alpha}}_{2}\left(\nu \right),{\mathbf{X}}_{t}).\hfill \end{array}$$Here, I abuse the notation introduced in Section 3.1: ${a}_{j}({\mathbf{\alpha}}_{2}\left(\nu \right),t)={a}_{j}({\mathbf{\alpha}}_{2}\left(\nu \right),{\mathbf{X}}_{t})={a}_{j}(\nu ,{\mathbf{X}}_{t})$ for $j=0,\dots ,2M$. Thus, for the integrability condition to be satisfied, I assume that each probability rate function ${a}_{j}$ is integrable along the solution of Equation (12). An exact form of the rates ${a}_{j}$ satisfying this condition will be given in the subsequent section. Note that ${a}_{0}$ is nonzero, which follows from Equation (8).
 III
 In our case, the measure $Q(\xb7;(\nu ,\mathbf{\xi}\left)\right)$ is given by (recalling Section 3):$$\begin{array}{cc}Q(\left\{\eta \right\}\times d{\mathbf{\xi}}^{\prime};(\nu ,\mathbf{\xi}))={\delta}_{\mathbf{\xi}}\left(d{\mathbf{\xi}}^{\prime}\right)\sum _{j=1}^{M}\hfill & {\delta}_{{\alpha}_{1}\left(\eta \right),0}\frac{{a}_{j}^{+}({\mathbf{\alpha}}_{2}\left(\nu \right),\mathbf{\xi})}{{a}_{0}({\mathbf{\alpha}}_{2}\left(\nu \right),\mathbf{\xi})}{\delta}_{{\mathbf{\alpha}}_{2}{\left(\eta \right)}_{j},1}\prod _{i\ne j}^{M}{\delta}_{{\mathbf{\alpha}}_{2}{\left(\eta \right)}_{i},{\mathbf{\alpha}}_{2}{\left(\nu \right)}_{i}}\hfill \\ & +{\delta}_{{\alpha}_{1}\left(\eta \right),1}\frac{{a}_{j}^{}({\mathbf{\alpha}}_{2}\left(\nu \right),\mathbf{\xi})}{{a}_{0}({\mathbf{\alpha}}_{2}\left(\nu \right),\mathbf{\xi})}{\delta}_{{\mathbf{\alpha}}_{2}{\left(\eta \right)}_{j},0}\prod _{i\ne j}^{M}{\delta}_{{\mathbf{\alpha}}_{2}{\left(\eta \right)}_{i},{\mathbf{\alpha}}_{2}{\left(\nu \right)}_{i}},\hfill \end{array}$$
 $\mathbb{P}\left(\right)open="("\; close=")">j{\mathbf{\alpha}}_{2}\left(\nu \right),\mathbf{\xi}$ is the probability that the FA event j occurs, given ${\mathbf{\alpha}}_{2}\left(\nu \right)$ and $\mathbf{\xi}$.
 $\mathbb{P}\left(\right)open="("\; close=")">{\alpha}_{1}\left(\eta \right),{\mathbf{\alpha}}_{2}\left(\eta \right)j,{\mathbf{\alpha}}_{2}\left(\nu \right),\mathbf{\xi}$ is the probability of a jump into cell state $\left(\right)$, given ${\mathbf{\alpha}}_{2}\left(\nu \right)$ and $\mathbf{\xi}$, and that the FA event j occurred.
5. Adhesion Kinetics
5.1. Unbinding Rate
5.2. Binding Dynamics
5.2.1. Rac Dependence
5.2.2. Force Dependence
 ${k}_{force}\left(0\right)={k}_{on}^{0}$, i.e., if there is no force applied, the rate is ${k}_{on}^{0}$.
 If the applied force is below the characteristic force ${F}_{b}$, then ${k}_{force}$ is greater than the unbinding rate, i.e., it is more likely that an FA increases in size than that it ruptures.
 If the characteristic force ${F}_{b}$ is applied, the rate ${k}_{force}$ should equal the unbinding rate, i.e., I assume that there is a dynamic equilibrium in some sense.
 If the applied force is larger than ${F}_{b}$, then the unbinding rate dominates the binding rate. Note that as FA increases in size, the force dependence diminishes [59]. Thus, ${k}_{force}$ should plateau around ${F}_{b}$. I also assume that for large applied forces ${k}_{force}$ plateaus at ${k}_{on}^{0}$, since exceeding loads rupture integrin bonds frequently and impede stable maturation.
6. Numerical Simulations
 Uniform environment with no cues.
 Nonuniform environment with external cue gradient and uneven myosin motor activity within a cell.
 Striped ECM architecture.
 $\mathbf{x}$ is at the origin, ${\mathbf{x}}_{n}$ is uniformly distributed on a circle with radius ${R}_{cell}$, and $\theta =0$.
 Each FA is in (un)bound state and each cell is in moving state with probability $1/2$.
6.1. Uniform Environment
6.2. External Cue Gradient
6.3. Fibrillar Architecture of ECM
6.4. Asymmetric Contractility
7. Discussion and Outlook
 The migration cycle is reduced to two steps: FA unbinding leads to movement and cytoskeletal reconfiguration, while binding to the latter only (Section 2).
 Cell shape is assumed to be spherical and constant (Section 2.1).
 The number of adhesion sites (occupied and unoccupied) is constant (Section 2.3).
 Adhesion binding and unbinding depend on the force applied by stress fibres and position of the cell, i.e., we omit intracellular biochemical interactions (Section 5).
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
FA  Focal adhesion 
ECM  Extracellular matrix 
SF  Stress fibre 
PDMP  Piecewise deterministic Markov process 
MSD  Meansquared displacement 
Appendix A. Equations of Cell Motion
Appendix B. Parameter Assessment
Parameter  Value  Value  Parameter  Value  Value 

${T}_{i}$  4 nN  0.72  ${R}_{cell}$  25 $\mathsf{\mu}$m  1 
$EA$  15 nN  2.72  ${L}_{0}$  27.5 $\mathsf{\mu}$m  1.1 
${F}_{b}$  5.5 nN  1  ${L}_{c}$  5 $\mathsf{\mu}$m  0.2 
${k}_{off}^{0}$  0.05 s^{−1}  1  ${\beta}_{rot}$  $1.56\times {10}^{11}N\xb7s\xb7m$  5.68 
${k}_{on}^{0}$  0.01 s^{−1}  0.2  ${\beta}_{ECM}$  $5\times {10}^{4}\frac{N\xb7s}{m}$  0.11 
${R}_{nucleus}$  5 $\mathsf{\mu}$m  0.2  ${\beta}_{cell}$  $5\times {10}^{3}\frac{N\xb7s}{m}$  1.14 
Appendix C. Data Analysis
Data Variability
Appendix D. Simulation of the PDMP
Algorithm A1 Simulation of the PDMP. 

Appendix D.1. Simulation of the Next Event Time
 Piecewise constantForward: ${a}_{0}({\nu}_{t},{\varphi}_{{\nu}_{t}}(s,{\mathbf{X}}_{t}))={a}_{0}({\nu}_{t},{\varphi}_{{\nu}_{t}}({\tau}_{i},{\mathbf{X}}_{t}))={a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}})$.$$\begin{array}{c}I\left(\right)open="("\; close=")">{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}}),{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i+1}});{h}_{i},h={h}_{i}{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}}).\hfill \end{array}$$Backward: ${a}_{0}({\nu}_{t},{\varphi}_{{\nu}_{t}}(s,{\mathbf{X}}_{t}))={a}_{0}({\nu}_{t},{\varphi}_{{\nu}_{t}}({\tau}_{i+1},{\mathbf{X}}_{t}))={a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i+1}})$$$\begin{array}{c}I\left(\right)open="("\; close=")">{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}}),{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i+1}});{h}_{i},h={h}_{i}{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i+1}}).\hfill \end{array}$$
 Average: ${a}_{0}({\nu}_{t},{\varphi}_{{\nu}_{t}}(s,{\mathbf{X}}_{t}))=\frac{1}{2}\left(\right)open="("\; close=")">{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}})+{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i+1}})$.$$\begin{array}{c}I\left(\right)open="("\; close=")">{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}}),{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i+1}});{h}_{i},h=\frac{1}{2}{h}_{i}\left(\right)open="("\; close=")">{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}})+{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i+1}})\hfill & .\end{array}$$
 Piecewise linear:$$\begin{array}{c}{a}_{0}({\nu}_{t},{\varphi}_{{\nu}_{t}}(s,{\mathbf{X}}_{t}))=(st{\tau}_{i})\frac{{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i+1}}){a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}})}{h}+{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}}).\hfill \end{array}$$$$\begin{array}{cc}I({a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}}),{a}_{0}({\nu}_{t},\hfill & {\mathbf{X}}_{t+{\tau}_{i+1}});{h}_{i},h).\hfill \\ & ={h}_{i}{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i}})\left(\right)open="("\; close=")">1\frac{{h}_{i}}{h}+\frac{{h}_{i}^{2}}{2h}{a}_{0}({\nu}_{t},{\mathbf{X}}_{t+{\tau}_{i+1}}).\hfill \end{array}$$
Algorithm A2 Event time computation. 

Appendix D.2. Sampling from the Transition Measure
References
 Lauffenburger, D.A.; Horwitz, A.F. Cell migration: A physically integrated molecular process. Cell 1996, 84, 359–369. [Google Scholar] [CrossRef] [Green Version]
 Abercrombie, M. The Croonian Lecture, 1978—The crawling movement of metazoan cells. Proc. R. Soc. Lond. B Biol. Sci. 1980, 207, 129–147. [Google Scholar]
 Raftopoulou, M.; Hall, A. Cell migration: Rho GTPases lead the way. Dev. Biol. 2004, 265, 23–32. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Ridley, A.J.; Schwartz, M.A.; Burridge, K.; Firtel, R.A.; Ginsberg, M.H.; Borisy, G.; Parsons, J.T.; Horwitz, A.R. Cell migration: Integrating signals from front to back. Science 2003, 302, 1704–1709. [Google Scholar] [CrossRef] [Green Version]
 Holmes, W.R.; EdelsteinKeshet, L. A comparison of computational models for eukaryotic cell shape and motility. PLoS Comput. Biol. 2012, 8, e1002793. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Ziebert, F.; Aranson, I.S. Computational approaches to substratebased cell motility. Npj Comput. Mater. 2016, 2, 16019. [Google Scholar] [CrossRef] [Green Version]
 Mogilner, A. Mathematics of cell motility: Have we got its number? J. Math. Biol. 2009, 58, 105. [Google Scholar] [CrossRef] [Green Version]
 Camley, B.A.; Zhao, Y.; Li, B.; Levine, H.; Rappel, W.J. Crawling and turning in a minimal reactiondiffusion cell motility model: Coupling cell shape and biochemistry. Phys. Rev. E 2017, 95, 012401. [Google Scholar] [CrossRef] [Green Version]
 Oelz, D.; Schmeiser, C. How Do Cells Move? Mathematical Modeling of Cytoskeleton Dynamics and Cell Migration. Cell Mechanics: From Single ScaleBased Models to Multiscale Modeling; Chapman and Hall: New York, NY, USA, 2010. [Google Scholar]
 Manhart, A.; Oelz, D.; Schmeiser, C.; Sfakianakis, N. An extended Filament Based Lamellipodium Model produces various moving cell shapes in the presence of chemotactic signals. J. Theor. Biol. 2015, 382, 244–258. [Google Scholar] [CrossRef] [Green Version]
 Preziosi, L.; Vitale, G. A multiphase model of tumor and tissue growth including cell adhesion and plastic reorganization. Math. Model. Methods Appl. Sci. 2011, 21, 1901–1932. [Google Scholar] [CrossRef]
 Rubinstein, B.; Jacobson, K.; Mogilner, A. Multiscale TwoDimensional Modeling of a Motile SimpleShaped Cell. Multiscale Model. Simul. 2005, 3, 413–439. [Google Scholar] [CrossRef] [PubMed]
 Shao, D.; Rappel, W.J.; Levine, H. Computational model for cell morphodynamics. Phys. Rev. Lett. 2010, 105, 108104. [Google Scholar] [CrossRef] [PubMed]
 Copos, C.A.; Walcott, S.; del Álamo, J.C.; Bastounis, E.; Mogilner, A.; Guy, R.D. Mechanosensitive Adhesion Explains Stepping Motility in Amoeboid Cells. Biophys. J. 2017, 112, 2672–2682. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Zhu, J.; Mogilner, A. Comparison of cell migration mechanical strategies in threedimensional matrices: A computational study. Interface Focus 2016, 6, 20160040. [Google Scholar] [CrossRef] [Green Version]
 Marée, A.F.; Jilkine, A.; Dawes, A.; Grieneisen, V.A.; EdelsteinKeshet, L. Polarization and movement of keratocytes: A multiscale modelling approach. Bull. Math. Biol. 2006, 68, 1169–1211. [Google Scholar] [CrossRef]
 Marée, A.F.M.; Grieneisen, V.A.; EdelsteinKeshet, L. How Cells Integrate Complex Stimuli: The Effect of Feedback from Phosphoinositides and Cell Shape on Cell Polarization and Motility. PLoS Comput. Biol. 2012, 8, 1–20. [Google Scholar] [CrossRef] [Green Version]
 Arrieumerlou, C.; Meyer, T. A local coupling model and compass parameter for eukaryotic chemotaxis. Dev. Cell 2005, 8, 215–227. [Google Scholar] [CrossRef] [Green Version]
 Dickinson, R.B.; Tranquillo, R.T. A stochastic model for adhesionmediated cell random motility and haptotaxis. J. Math. Biol. 1993, 31, 563–600. [Google Scholar] [CrossRef]
 Groh, A.; Louis, A.K. Stochastic modelling of biased cell migration and collagen matrix modification. J. Math. Biol. 2010, 61, 617–647. [Google Scholar] [CrossRef]
 Satulovsky, J.; Lui, R.; Wang, Y.l. Exploring the control circuit of cell migration by mathematical modeling. Biophys. J. 2008, 94, 3671–3683. [Google Scholar] [CrossRef] [Green Version]
 Tranquillo, R.T.; Lauffenburger, D.A.; Zigmond, S. A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations. J. Cell Biol. 1988, 106, 303–309. [Google Scholar] [CrossRef] [PubMed]
 Dieterich, P.; Klages, R.; Preuss, R.; Schwab, A. Anomalous dynamics of cell migration. Proc. Natl. Acad. Sci. USA 2008, 105, 459–463. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Li, L.; Nørrelykke, S.F.; Cox, E.C. Persistent Cell Motion in the Absence of External Signals: A Search Strategy for Eukaryotic Cells. PLoS ONE 2008, 3, e2093. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Selmeczi, D.; Mosler, S.; Hagedorn, P.H.; Larsen, N.B.; Flyvbjerg, H. Cell Motility as Persistent Random Motion: Theories from Experiments. Biophys. J. 2005, 89, 912–931. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Othmer, H.G.; Dunbar, S.R.; Alt, W. Models of dispersal in biological systems. J. Math. Biol. 1988, 26, 263–298. [Google Scholar] [CrossRef]
 Engwer, C.; Hillen, T.; Knappitsch, M.; Surulescu, C. Glioma follow white matter tracts: A multiscale DTIbased model. J. Math. Biol. 2015, 71, 551–582. [Google Scholar] [CrossRef]
 Kelkel, J.; Surulescu, C. A multiscale approach to cell migration in tissue networks. Math. Model. Methods Appl. Sci. 2012, 22, 1150017. [Google Scholar] [CrossRef]
 Davis, M.H. Piecewisedeterministic Markov processes: A general class of nondiffusion stochastic models. J. R. Stat. Soc. Ser. B Methodol. 1984, 46, 353–388. [Google Scholar] [CrossRef]
 Davis, M.H.A. Markov Models and Optimization; Chapman and Hall: New York, NY, USA, 1993. [Google Scholar]
 Liu, Y.J.; Le Berre, M.; Lautenschlaeger, F.; Maiuri, P.; CallanJones, A.; Heuzé, M.; Takaki, T.; Voituriez, R.; Piel, M. Confinement and low adhesion induce fast amoeboid migration of slow mesenchymal cells. Cell 2015, 160, 659–672. [Google Scholar] [CrossRef] [Green Version]
 RamirezSan Juan, G.R.; Oakes, P.W.; Gardel, M.L. Contact guidance requires spatial control of leadingedge protrusion. Mol. Biol. Cell 2017, 28, 1043–1053. [Google Scholar] [CrossRef]
 Verkhovsky, A.B.; Svitkina, T.M.; Borisy, G.G. Selfpolarization and directional motility of cytoplasm. Curr. Biol. 1999, 9, 11S1. [Google Scholar] [CrossRef] [Green Version]
 Yam, P.T.; Wilson, C.A.; Ji, L.; Hebert, B.; Barnhart, E.L.; Dye, N.A.; Wiseman, P.W.; Danuser, G.; Theriot, J.A. Actin–myosin network reorganization breaks symmetry at the cell rear to spontaneously initiate polarized cell motility. J. Cell Biol. 2007, 178, 1207–1221. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Uatay, A. A mathematical model of contact inhibition of locomotion: Coupling contractility and focal adhesions. arXiv 2019, arXiv:1901.02386. [Google Scholar]
 Uatay, A. Multiscale Mathematical Modeling of Cell Migration: From Single Cells to Populations. Ph.D. Thesis, Technische Universität Kaiserslautern, Kaiserslautern, Germany, 2019. [Google Scholar]
 Bellomo, N.; Knopoff, D.; Soler, J. On the difficult interplay between life, ”complexity”, and mathematical sciences. Math. Model. Methods Appl. Sci. 2013, 23, 1861–1913. [Google Scholar] [CrossRef]
 Knopoff, D.A.; Nieto, J.; Urrutia, L. Numerical simulation of a multiscale cell motility model based on the kinetic theory of active particles. Symmetry 2019, 11, 1003. [Google Scholar] [CrossRef] [Green Version]
 Gardel, M.L.; Schneider, I.C.; AratynSchaus, Y.; Waterman, C.M. Mechanical integration of actin and adhesion dynamics in cell migration. Annu. Rev. Cell Dev. Biol. 2010, 26, 315–333. [Google Scholar] [CrossRef] [Green Version]
 Murrell, M.; Oakes, P.W.; Lenz, M.; Gardel, M.L. Forcing cells into shape: The mechanics of actomyosin contractility. Nat. Rev. Mol. Cell Biol. 2015, 16, 486–498. [Google Scholar] [CrossRef]
 Guthardt Torres, P.; Bischofs, I.B.; Schwarz, U.S. Contractile network models for adherent cells. Phys. Rev. Lett. E 2012, 85, 011913. [Google Scholar] [CrossRef] [Green Version]
 Kassianidou, E.; Brand, A.C.; Schwarz, U.S.; Kumar, S. Geometry and network connectivity govern the mechanics of stress fibers. Proc. Natl. Acad. Sci. USA 2017, 114, 2622–2627. [Google Scholar] [CrossRef] [Green Version]
 Kumar, S.; Maxwell, I.Z.; Heisterkamp, A.; Polte, T.R.; Lele, T.P.; Salanga, M.; Mazur, E.; Ingber, D.E. Viscoelastic Retraction of Single Living Stress Fibers and Its Impact on Cell Shape, Cytoskeletal Organization, and Extracellular Matrix Mechanics. Biophys. J. 2006, 90, 3762–3773. [Google Scholar] [CrossRef] [Green Version]
 Russell, R.J.; Xia, S.L.; Dickinson, R.B.; Lele, T.P. Sarcomere mechanics in capillary endothelial cells. Biophys. J. 2009, 97, 1578–1585. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Deguchi, S.; Ohashi, T.; Sato, M. Tensile properties of single stress fiber isolated from cultured vascular smooth muscle cells. J. Biomech. 2006, 39, 2603–2610. [Google Scholar] [CrossRef] [PubMed]
 Möhl, C.; Kirchgessner, N.; Schäfer, C.; Hoffmann, B.; Merkel, R. Quantitative mapping of averaged focal adhesion dynamics in migrating cells by shape normalization. J. Cell Sci. 2012, 125, 155–165. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Schwarz, U.S.; Gardel, M.L. United we stand integrating the actin cytoskeleton and cell matrix adhesions in cellular mechanotransduction. J. Cell Sci. 2012, 125, 3051–3060. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Paul, R.; Heil, P.; Spatz, J.P.; Schwarz, U.S. Propagation of mechanical stress through the actin cytoskeleton toward focal adhesions: Model and experiment. Biophys. J. 2008, 94, 1470–1482. [Google Scholar] [CrossRef] [Green Version]
 Oakes, P.W.; Wagner, E.; Brand, C.A.; Probst, D.; Linke, M.; Schwarz, U.S.; Glotzer, M.; Gardel, M.L. Optogenetic Control of RhoA Reveals Zyxinmediated Elasticity of Stress Fibers. Nat. Commun. 2017, 8, 15817. [Google Scholar] [CrossRef] [Green Version]
 Gillespie, D.T. A rigorous derivation of the chemical master Equation. Phys. A Stat. Mech. Its Appl. 1992, 188, 404–425. [Google Scholar] [CrossRef]
 Ananthakrishnan, R.; Ehrlicher, A. The Forces Behind Cell Movement. Int. J. Biol. Sci. 2007, 3, 303–317. [Google Scholar] [CrossRef]
 Cramer, L.P. Forming the cell rear first: Breaking cell symmetry to trigger directed cell migration. Nat. Cell Biol. 2010, 12, 628–632. [Google Scholar] [CrossRef]
 Bell, G.I. Models for the specific adhesion of cells to cells. Science 1978, 200, 618–627. [Google Scholar] [CrossRef]
 Ridley, A.J. Rho GTPases and cell migration. J. Cell Sci. 2001, 114, 2713–2722. [Google Scholar] [PubMed]
 Giverso, C.; Preziosi, L. Mechanical perspective on chemotaxis. Phys. Rev. E 2018, 98, 062402. [Google Scholar] [CrossRef]
 Holmes, W.R.; Lin, B.; Levchenko, A.; EdelsteinKeshet, L. Modelling cell polarization driven by synthetic spatially graded Rac activation. PLoS Comput. Biol. 2012, 8, e1002366. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Narang, A. Spontaneous polarization in eukaryotic gradient sensing: A mathematical model based on mutual inhibition of frontness and backness pathways. J. Theor. Biol. 2006, 240, 538–553. [Google Scholar] [CrossRef] [Green Version]
 Choi, C.K.; VicenteManzanares, M.; Zareno, J.; Whitmore, L.A.; Mogilner, A.; Horwitz, A.R. Actin and αactinin orchestrate the assembly and maturation of nascent adhesions in a myosin II motorindependent manner. Nat. Cell Biol. 2008, 10, 1039–1050. [Google Scholar] [CrossRef]
 Stricker, J.; AratynSchaus, Y.; Oakes, P.W.; Gardel, M.L. Spatiotemporal constraints on the forcedependent growth of focal adhesions. Biophys. J. 2011, 100, 2883–2893. [Google Scholar] [CrossRef] [Green Version]
 ZaidelBar, R.; Cohen, M.; Addadi, L.; Geiger, B. Hierarchical assembly of cell–matrix adhesion complexes. Biochem. Soc. Trans. 2004, 32, 416–420. [Google Scholar] [CrossRef]
 Balaban, N.Q.; Schwarz, U.S.; Riveline, D.; Goichberg, P.; Tzur, G.; Sabanay, I.; Mahalu, D.; Safran, S.; Bershadsky, A.; Addadi, L.; et al. Force and focal adhesion assembly: A close relationship studied using elastic micropatterned substrates. Nat. Cell Biol. 2001, 3, 466–472. [Google Scholar] [CrossRef]
 Soiné, J.R.D.; Brand, C.A.; Stricker, J.; Oakes, P.W.; Gardel, M.L.; Schwarz, U.S. Modelbased Traction Force Microscopy Reveals Differential Tension in Cellular Actin Bundles. PLoS Comput. Biol. 2015, 11, e1004076. [Google Scholar] [CrossRef] [Green Version]
 Alexandrova, A.Y.; Arnold, K.; Schaub, S.; Vasiliev, J.M.; Meister, J.J.; Bershadsky, A.D.; Verkhovsky, A.B. Comparative dynamics of retrograde actin flow and focal adhesions: formation of nascent adhesions triggers transition from fast to slow flow. PLoS ONE 2008, 3, e3234. [Google Scholar] [CrossRef] [Green Version]
 VicenteManzanares, M.; Zareno, J.; Whitmore, L.; Choi, C.K.; Horwitz, A.F. Regulation of protrusion, adhesion dynamics, and polarity by myosins IIA and IIB in migrating cells. J. Cell Biol. 2007, 176, 573–580. [Google Scholar] [CrossRef] [PubMed]
 Chan, C.E.; Odde, D.J. Traction Dynamics of Filopodia on Compliant Substrates. Science 2008, 322, 1687–1691. [Google Scholar] [CrossRef] [PubMed]
 Paňková, K.; Rösel, D.; Novotnỳ, M.; Brábek, J. The molecular mechanisms of transition between mesenchymal and amoeboid invasiveness in tumor cells. Cell. Mol. Life Sci. 2010, 67, 63–71. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Friedl, P.; Noble, P.; Shields, E.; Zänker, K. Locomotor phenotypes of unstimulated CD45RAhigh and CD45ROhigh CD4+ and CD8+ lymphocytes in threedimensional collagen lattices. Immunology 1994, 82, 617. [Google Scholar]
 Gorelik, R.; Gautreau, A. Quantitative and unbiased analysis of directional persistence in cell migration. Nat. Protoc. 2014, 9, 1931. [Google Scholar] [CrossRef]
 Petrie, R.J.; Doyle, A.D.; Yamada, K.M. Random versus directionally persistent cell migration. Nat. Rev. Mol. Cell Biol. 2009, 10, 538–549. [Google Scholar] [CrossRef] [Green Version]
 Kubow, K.E.; Shuklis, V.D.; Sales, D.J.; Horwitz, A.R. Contact guidance persists under myosin inhibition due to the local alignment of adhesions and individual protrusions. Sci. Rep. 2017, 7, 14380. [Google Scholar] [CrossRef] [Green Version]
 Shellard, A.; Szabó, A.; Trepat, X.; Mayor, R. Supracellular contraction at the rear of neural crest cell groups drives collective chemotaxis. Science 2018, 362, 339–343. [Google Scholar] [CrossRef] [Green Version]
 Fletcher, A.; Osterfield, M.; Baker, R.; Shvartsman, S. Vertex Models of Epithelial Morphogenesis. Biophys. J. 2014, 106, 2291–2304. [Google Scholar] [CrossRef] [Green Version]
 Mori, Y.; Jilkine, A.; EdelsteinKeshet, L. Wavepinning and cell polarity from a bistable reactiondiffusion system. Biophys. J. 2008, 94, 3684–3697. [Google Scholar] [CrossRef] [Green Version]
 Roycroft, A.; Mayor, R. Molecular basis of contact inhibition of locomotion. Cell. Mol. Life Sci. 2016, 73, 1119–1130. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Svitkina, T.M.; Surguchova, I.G.; Verkhovsky, A.B.; Gelfand, V.I.; Moeremans, M.; De Mey, J. Direct visualization of bipolar myosin filaments in stress fibers of cultured fibroblasts. Cell Motil. Cytoskelet. 1989, 12, 150–156. [Google Scholar] [CrossRef] [PubMed]
 AratynSchaus, Y.; Oakes, P.W.; Gardel, M.L. Dynamic and structural signatures of lamellar actomyosin force generation. Mol. Biol. Cell 2011, 22, 1330–1339. [Google Scholar] [CrossRef] [PubMed]
 Verkhovsky, A.B.; Borisy, G.G. Nonsarcomeric mode of myosin II organization in the fibroblast lamellum. J. Cell Biol. 1993, 123, 637–652. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Finer, J.T.; Simmons, R.M.; Spudich, J.A. Single myosin molecule mechanics: Piconewton forces and nanometre steps. Nature 1994, 368, 113–119. [Google Scholar] [CrossRef]
 Molloy, J.E.; Burns, J.E.; KendrickJones, J.; Tregear, R.T.; White, D.C. Movement and force produced by a single myosin head. Nature 1995, 378, 209–212. [Google Scholar] [CrossRef]
 Gardel, M.L.; Valentine, M.T.; Crocker, J.C.; Bausch, A.R.; Weitz, D.A. Microrheology of Entangled FActin Solutions. Phys. Rev. Lett. 2003, 91, 158302. [Google Scholar] [CrossRef] [Green Version]
 Kalwarczyk, T.; Ziȩbacz, N.; Bielejewska, A.; Zaboklicka, E.; Koynov, K.; Szymański, J.; Wilk, A.; Patkowski, A.; Gapiński, J.; Butt, H.J.; et al. Comparative analysis of viscosity of complex liquids and cytoplasm of mammalian cells at the nanoscale. Nano Lett. 2011, 11, 2157–2163. [Google Scholar] [CrossRef]
 Mastro, A.M.; Babich, M.A.; Taylor, W.D.; Keith, A.D. Diffusion of a small molecule in the cytoplasm of mammalian cells. Proc. Natl. Acad. Sci. USA 1984, 81, 3414–3418. [Google Scholar] [CrossRef] [Green Version]
 Arcizet, D.; Meier, B.; Sackmann, E.; Rädler, J.O.; Heinrich, D. Temporal Analysis of Active and Passive Transport in Living Cells. Phys. Rev. Lett. 2008, 101, 248103. [Google Scholar] [CrossRef] [Green Version]
 Lammerding, J. Mechanics of the Nucleus. In Comprehensive Physiology; American Cancer Society: Atlanta, GA, USA, 2011; pp. 783–807. [Google Scholar]
 Burridge, K.; ChrzanowskaWodnicka, M. Focal adhesions, contractility, and signaling. Annu. Rev. Cell Dev. Biol. 1996, 12, 463–519. [Google Scholar] [CrossRef] [Green Version]
 Geiger, B.; Spatz, J.P.; Bershadsky, A.D. Environmental sensing through focal adhesions. Nat. Rev. Mol. Cell Biol. 2009, 10, 21. [Google Scholar] [CrossRef]
 Erdmann, T.; Schwarz, U.S. Stability of Adhesion Clusters under Constant Force. Phys. Rev. Lett. 2004, 92, 108102. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Li, F.; Redick, S.D.; Erickson, H.P.; Moy, V.T. Force measurements of the α5β1 integrin–fibronectin interaction. Biophys. J. 2003, 84, 1252–1262. [Google Scholar] [CrossRef] [Green Version]
 Kassianidou, E.; Kumar, S. A biomechanical perspective on stress fiber structure and function. Biochim. Biophys. Acta BBA Mol. Cell Res. 2015, 1853, 3065–3074. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Chen, W.T. Mechanism of retraction of the trailing edge during fibroblast movement. J. Cell Biol. 1981, 90, 187–200. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Hotulainen, P.; Lappalainen, P. Stress fibers are generated by two distinct actin assembly mechanisms in motile cells. J. Cell Biol. 2006, 173, 383–394. [Google Scholar] [CrossRef] [Green Version]
 Kuragano, M.; Uyeda, T.Q.; Kamijo, K.; Murakami, Y.; Takahashi, M. Different contributions of nonmuscle myosin IIA and IIB to the organization of stress fiber subtypes in fibroblasts. Mol. Biol. Cell 2018, 29, 911–922. [Google Scholar] [CrossRef]
 Lewis, P.A.; Shedler, G.S. Simulation of nonhomogeneous Poisson processes by thinning. Nav. Res. Logist. 1979, 26, 403–413. [Google Scholar] [CrossRef]
 BouchardCôté, A.; Vollmer, S.J.; Doucet, A. The bouncy particle sampler: A nonreversible rejectionfree Markov chain Monte Carlo method. arXiv 2015, arXiv:1510.02451. [Google Scholar] [CrossRef] [Green Version]
 Vose, M.D. A linear algorithm for generating random numbers with a given distribution. IEEE Trans. Softw. Eng. 1991, 17, 972–975. [Google Scholar] [CrossRef] [Green Version]
M  8  16  32 

${s}_{av}$, $\mathsf{\mu}$m/min  1.7595  2.4845  3.6047 
${s}_{IQR}$, $\mathsf{\mu}$m/min  1.4557  2.217  3.1787 
$\overline{\beta}$, 1  1.0683  1.0035  1.0552 
${\beta}_{0}$, $\mathsf{\mu}$m^{2}/${\mathrm{min}}^{\overline{\beta}}$  2.1493  3.3971  4.6308 
$\overline{r}$, 1  0.0483  0.0408  0.0497 
M  8  16  32  

$\delta $  0.05  0.1  0.15  0.05  0.1  0.15  0.05  0.1  0.15 
$\overline{\beta}$, 1  0.9859  1.3184  1.4084  1.0086  1.3505  1.5581  1.1014  1.4299  1.5639 
${s}_{av}$, $\mathsf{\mu}$m/min  1.7656  1.7768  1.7557  2.4918  2.5009  2.5021  3.5818  3.5735  3.6104 
${s}_{IQR}$, $\mathsf{\mu}$m/min  1.3964  1.4319  1.472  2.24  2.2322  2.2355  3.2136  3.1108  3.0269 
${\beta}_{0}$, $\mathsf{\mu}$m^{2}/${\mathrm{min}}^{\overline{\beta}}$  3.0846  0.6934  0.5534  3.4517  0.7103  0.3543  4.1257  0.9536  0.5716 
$\overline{r}$, 1  0.0452  0.0519  0.0597  0.0440  0.0513  0.0587  0.0522  0.05  0.0623 
M  8  16  32  

${\delta}_{E}$  0.05  0.1  0.15  0.05  0.1  0.15  0.05  0.1  0.15 
$\overline{\beta}$, 1  1.2551  1.5051  1.52  1.3405  1.6963  1.7545  1.5427  1.7722  1.8569 
${s}_{av}$, $\mathsf{\mu}$m/min  1.8136  1.9133  2.0365  2.5235  2.5972  2.6089  3.5819  3.4218  3.3074 
${s}_{IQR}$, $\mathsf{\mu}$m/min  1.7384  1.5461  1.6434  2.0398  2.1999  2.1180  2.8058  2.9067  3.0175 
${\beta}_{0}$, $\mathsf{\mu}$m^{2}/${\mathrm{min}}^{\overline{\beta}}$  1.0697  0.4625  0.6845  1.0312  0.3496  0.3654  0.8319  0.3412  0.2242 
$\overline{r}$, 1  0.0523  0.0607  0.0693  0.053  0.08  0.097  0.0726  0.1  0.1223 
M  8  16  32  

${\delta}_{myo}$  0.35  0.40  0.45  0.35  0.40  0.45  0.35  0.4  0.45 
$\overline{\beta}$, 1  1.3072  1.6325  1.7759  1.1311  1.7905  1.8524  1.4650  1.8892  1.9353 
${s}_{av}$, $\mathsf{\mu}$m/min  1.6230  1.5639  1.5103  2.3742  2.3798  2.3516  3.5861  3.7414  3.7701 
${s}_{IQR}$, $\mathsf{\mu}$m/min  1.3827  1.4534  1.4607  2.0273  2.0186  2.2693  3.1370  3.2738  3.1152 
${\beta}_{0}$, $\mathsf{\mu}$m^{2}/${\mathrm{min}}^{\overline{\beta}}$  0.7767  0.3659  0.2493  2.3088  0.2893  0.3037  1.0713  0.9894  1.2787 
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Uatay, A. A Stochastic Modelling Framework for Single Cell Migration: Coupling Contractility and Focal Adhesions. Symmetry 2020, 12, 1348. https://doi.org/10.3390/sym12081348
Uatay A. A Stochastic Modelling Framework for Single Cell Migration: Coupling Contractility and Focal Adhesions. Symmetry. 2020; 12(8):1348. https://doi.org/10.3390/sym12081348
Chicago/Turabian StyleUatay, Aydar. 2020. "A Stochastic Modelling Framework for Single Cell Migration: Coupling Contractility and Focal Adhesions" Symmetry 12, no. 8: 1348. https://doi.org/10.3390/sym12081348