# American Institute of Mathematical Sciences

December 2017, 12(4): 551-590. doi: 10.3934/nhm.2017023

## Sharp interface limit in a phase field model of cell motility

 1 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA 2 Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering of National Academy of Sciences of Ukraine, 47 Nauky Ave., 61103 Kharkiv, Ukraine

* Corresponding author: Mykhailo Potomkin, mup20@ucs.psu.edu

Received  March 2017 Revised  August 2017 Published  October 2017

Fund Project: The work of LB was supported by NSF grants DMS-1106666 and DMS-1405769. The work of VR and MP was partially supported by NSF grant DMS-1106666

We consider a phase field model of cell motility introduced in [40] which consists of two coupled parabolic PDEs. We study the asymptotic behavior of solutions in the limit of a small parameter related to the width of the interface (sharp interface limit). We formally derive an equation of motion of the interface, which is mean curvature motion with an additional nonlinear term. In a 1D model parabolic problem we rigorously justify the sharp interface limit. To this end, a special representation of solutions is introduced, which reduces analysis of the system to a single nonlinear PDE that describes the interface velocity. Further stability analysis reveals a qualitative change in the behavior of the system for small and large values of the coupling parameter. Using numerical simulations we also show discontinuities of the interface velocity and hysteresis. Also, in the 1D case we establish nontrivial traveling waves when the coupling parameter is large enough.

Citation: Leonid Berlyand, Mykhailo Potomkin, Volodymyr Rybalko. Sharp interface limit in a phase field model of cell motility. Networks & Heterogeneous Media, 2017, 12 (4) : 551-590. doi: 10.3934/nhm.2017023
##### References:
 [1] M. Alfaro, Generation, motion and thickness of transition layers for a nonlocal Allen-Cahn equation, Nonlinear Analysis, 72 (2010), 3324-3336. doi: 10.1016/j.na.2009.12.013. [2] M. Alfaro and P. Alifrangis, Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local, Interfaces Free Bound., 16 (2014), 243-268. doi: 10.4171/IFB/319. [3] S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2. [4] G. Barles and F. D. Lio, A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions, Interfaces Free Bound., 5 (2016), 239-274. doi: 10.4171/IFB/79. [5] E. Barnhart, K. C. Lee, G. M. Allen, J. A. Theriot and A. Mogilner, Balance between cellsubstrate adhesion and myosin contraction determines the frequency of motility initiation in fish keratocytes, Proceedings of the National Academy of Sciences, 112 (2015), 5045-5050. doi: 10.1073/pnas.1417257112. [6] L. Berlyand, M. Potomkin and V. Rybalko, Phase-field model of cell motility: Traveling waves and sharp interface limit, Comptes Rendus Mathematique, 354 (2016), 986-992. doi: 10.1016/j.crma.2016.09.001. [7] K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Mathematical Notes, 20. Princeton University Press, Princeton, N. J. , 1978. i+252 pp. [8] M. Brassel and E. Bretin, A modified phase field approximation for mean curvature flow with conservation of the volume, Mathematical Methods in the Applied Sciences, 34 (2011), 1157-1180. doi: 10.1002/mma.1426. [9] L. Bronsard and B. Stoth, Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), 769-807. doi: 10.1137/S0036141094279279. [10] X. Chen, D. Hilhorst and E. Logak, Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term, Nonlinear Analysis: Theory, Methods & Applications, 28 (1997), 1283-1298. doi: 10.1016/S0362-546X(97)82875-1. [11] X. Chen, D. Hilhorst and E. Logak, Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces Free Bound., 12 (2010), 527-549. doi: 10.4171/IFB/244. [12] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786. doi: 10.4310/jdg/1214446564. [13] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. Ⅰ, J. Differential Geom., 33 (1991), 635-681. doi: 10.4310/jdg/1214446559. [14] L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. doi: 10.1002/cpa.3160450903. [15] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. vi+93 pp. doi: 10.1137/1.9781611970180. [16] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1. [17] D. Golovaty, The volume preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations, Q. of Appl. Math., 55 (1997), 243-298. doi: 10.1090/qam/1447577. [18] M. Grayson, The heat equation shrinks embedded plane curves to points, J. Differential Geom., 26 (1987), 285-314. doi: 10.4310/jdg/1214441371. [19] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306. doi: 10.4310/jdg/1214436922. [20] M. H. Holmes, Introduction to Perturbation Methods, 2nd edition. Texts in Applied Mathematics, 20. Springer, New York, 2013. xviii+436 pp. doi: 10.1007/978-1-4614-5477-9. [21] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266. doi: 10.4310/jdg/1214438998. [22] K. Keren, Z. Pincus, G. M. Allen, E. Barnhart, G. Marriott, A. Mogilner and J. A. Theriot, Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480. doi: 10.1038/nature06952. [23] R. Kohn, F. Otto, M. Reznikoff and E. Vanden-Eijden, Action minimization and sharpinterface limits for the stochastic Allen-Cahn equation, Comm. Pure Appl. Math., 60 (2007), 393-438. doi: 10.1002/cpa.20144. [24] F. D. Lio, C. I. Kim and D. Slepčev, Nonlocal front propagation problems in bounded domains with Neumann-type boundary conditions and applications, Asymptot. Anal., 37 (2004), 257-292. [25] M. Mizuhara, L. Berlyand, V. Rybalko and L. Zhang, On an evolution equation in a cell motility model, Phys. D, 318/319 (2016), 12-25. doi: 10.1016/j.physd.2015.10.008. [26] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230. [27] A. Mogilner, Mathematics of cell motility: Have we got its number?, J. Math. Biol., 58 (2009), 105-134. doi: 10.1007/s00285-008-0182-2. [28] P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589. doi: 10.1090/S0002-9947-1995-1672406-7. [29] F. Otto, H. Weber and G. Westdickenberg, Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size, Electron. J. Probab., 19 (2014), 1-76. doi: 10.1214/EJP.v19-2813. [30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp. doi: 10.1007/978-1-4612-5561-1. [31] J. Prúss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. [32] P. Recho, T. Putelat and L. Truskinovsky, Mechanics of motility initiation and motility arrest in crawling cells, J. Mech. Phys. Solids, 84 (2015), 469-505. doi: 10.1016/j.jmps.2015.08.006. [33] P. Recho and L. Truskinovsky, Asymmetry between pushing and pulling for crawling cells, Phys. Rev. E, 87 (2013), 022720. doi: 10.1103/PhysRevE.87.022720. [34] B. Rubinstein, K. Jacobson and A. Mogilner, Multiscale two-dimensional modeling of a motile simple-shaped cell, Multiscale Model. Simul., 3 (2005), 413-439. doi: 10.1137/04060370X. [35] J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264. doi: 10.1093/imamat/48.3.249. [36] J. Rubinstein, P. Sternberg and J. B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133. doi: 10.1137/0149007. [37] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. A, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427. [38] D. Shao, W. Rappel and H. Levine, Computational model for cell morphodynamics, Physical Review Letters, 105 (2010), 108104. doi: 10.1103/PhysRevLett.105.108104. [39] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview press, 2014, xiii+513 pp. [40] F. Ziebert, S. Swaminathan and I. S. Aranson, Model for self-polarization and motility of keratocyte fragments, J. R. Soc. Interface, 9 (2011), 1084-1092. doi: 10.1098/rsif.2011.0433.

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##### References:
 [1] M. Alfaro, Generation, motion and thickness of transition layers for a nonlocal Allen-Cahn equation, Nonlinear Analysis, 72 (2010), 3324-3336. doi: 10.1016/j.na.2009.12.013. [2] M. Alfaro and P. Alifrangis, Convergence of a mass conserving Allen-Cahn equation whose Lagrange multiplier is nonlocal and local, Interfaces Free Bound., 16 (2014), 243-268. doi: 10.4171/IFB/319. [3] S. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2. [4] G. Barles and F. D. Lio, A geometrical approach to front propagation problems in bounded domains with Neumann-type boundary conditions, Interfaces Free Bound., 5 (2016), 239-274. doi: 10.4171/IFB/79. [5] E. Barnhart, K. C. Lee, G. M. Allen, J. A. Theriot and A. Mogilner, Balance between cellsubstrate adhesion and myosin contraction determines the frequency of motility initiation in fish keratocytes, Proceedings of the National Academy of Sciences, 112 (2015), 5045-5050. doi: 10.1073/pnas.1417257112. [6] L. Berlyand, M. Potomkin and V. Rybalko, Phase-field model of cell motility: Traveling waves and sharp interface limit, Comptes Rendus Mathematique, 354 (2016), 986-992. doi: 10.1016/j.crma.2016.09.001. [7] K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Mathematical Notes, 20. Princeton University Press, Princeton, N. J. , 1978. i+252 pp. [8] M. Brassel and E. Bretin, A modified phase field approximation for mean curvature flow with conservation of the volume, Mathematical Methods in the Applied Sciences, 34 (2011), 1157-1180. doi: 10.1002/mma.1426. [9] L. Bronsard and B. Stoth, Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM J. Math. Anal., 28 (1997), 769-807. doi: 10.1137/S0036141094279279. [10] X. Chen, D. Hilhorst and E. Logak, Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term, Nonlinear Analysis: Theory, Methods & Applications, 28 (1997), 1283-1298. doi: 10.1016/S0362-546X(97)82875-1. [11] X. Chen, D. Hilhorst and E. Logak, Mass conserving Allen-Cahn equation and volume preserving mean curvature flow, Interfaces Free Bound., 12 (2010), 527-549. doi: 10.4171/IFB/244. [12] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786. doi: 10.4310/jdg/1214446564. [13] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. Ⅰ, J. Differential Geom., 33 (1991), 635-681. doi: 10.4310/jdg/1214446559. [14] L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. doi: 10.1002/cpa.3160450903. [15] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. vi+93 pp. doi: 10.1137/1.9781611970180. [16] L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1. [17] D. Golovaty, The volume preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations, Q. of Appl. Math., 55 (1997), 243-298. doi: 10.1090/qam/1447577. [18] M. Grayson, The heat equation shrinks embedded plane curves to points, J. Differential Geom., 26 (1987), 285-314. doi: 10.4310/jdg/1214441371. [19] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), 255-306. doi: 10.4310/jdg/1214436922. [20] M. H. Holmes, Introduction to Perturbation Methods, 2nd edition. Texts in Applied Mathematics, 20. Springer, New York, 2013. xviii+436 pp. doi: 10.1007/978-1-4614-5477-9. [21] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266. doi: 10.4310/jdg/1214438998. [22] K. Keren, Z. Pincus, G. M. Allen, E. Barnhart, G. Marriott, A. Mogilner and J. A. Theriot, Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480. doi: 10.1038/nature06952. [23] R. Kohn, F. Otto, M. Reznikoff and E. Vanden-Eijden, Action minimization and sharpinterface limits for the stochastic Allen-Cahn equation, Comm. Pure Appl. Math., 60 (2007), 393-438. doi: 10.1002/cpa.20144. [24] F. D. Lio, C. I. Kim and D. Slepčev, Nonlocal front propagation problems in bounded domains with Neumann-type boundary conditions and applications, Asymptot. Anal., 37 (2004), 257-292. [25] M. Mizuhara, L. Berlyand, V. Rybalko and L. Zhang, On an evolution equation in a cell motility model, Phys. D, 318/319 (2016), 12-25. doi: 10.1016/j.physd.2015.10.008. [26] L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 98 (1987), 123-142. doi: 10.1007/BF00251230. [27] A. Mogilner, Mathematics of cell motility: Have we got its number?, J. Math. Biol., 58 (2009), 105-134. doi: 10.1007/s00285-008-0182-2. [28] P. de Mottoni and M. Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc., 347 (1995), 1533-1589. doi: 10.1090/S0002-9947-1995-1672406-7. [29] F. Otto, H. Weber and G. Westdickenberg, Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size, Electron. J. Probab., 19 (2014), 1-76. doi: 10.1214/EJP.v19-2813. [30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp. doi: 10.1007/978-1-4612-5561-1. [31] J. Prúss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. [32] P. Recho, T. Putelat and L. Truskinovsky, Mechanics of motility initiation and motility arrest in crawling cells, J. Mech. Phys. Solids, 84 (2015), 469-505. doi: 10.1016/j.jmps.2015.08.006. [33] P. Recho and L. Truskinovsky, Asymmetry between pushing and pulling for crawling cells, Phys. Rev. E, 87 (2013), 022720. doi: 10.1103/PhysRevE.87.022720. [34] B. Rubinstein, K. Jacobson and A. Mogilner, Multiscale two-dimensional modeling of a motile simple-shaped cell, Multiscale Model. Simul., 3 (2005), 413-439. doi: 10.1137/04060370X. [35] J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math., 48 (1992), 249-264. doi: 10.1093/imamat/48.3.249. [36] J. Rubinstein, P. Sternberg and J. B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133. doi: 10.1137/0149007. [37] S. Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. A, 31 (2011), 1427-1451. doi: 10.3934/dcds.2011.31.1427. [38] D. Shao, W. Rappel and H. Levine, Computational model for cell morphodynamics, Physical Review Letters, 105 (2010), 108104. doi: 10.1103/PhysRevLett.105.108104. [39] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview press, 2014, xiii+513 pp. [40] F. Ziebert, S. Swaminathan and I. S. Aranson, Model for self-polarization and motility of keratocyte fragments, J. R. Soc. Interface, 9 (2011), 1084-1092. doi: 10.1098/rsif.2011.0433.
Illustration of the ansatz (34). Function $\rho_\varepsilon$ decays to a non-zero constant of the order $\varepsilon$ for $x\to\pm\infty$ and to a constant slightly different from 1 for $-a\leq x\leq a$ (solid). Dashed line represents the limiting profile, which is the characteristic function of $(-a, a)$
Left: $\Phi_\beta(V)$ for $\beta=150$ and $W_{\text{as}}(\rho)=\frac{1}{4}\rho^2(1+\rho^2)(\rho-1)^2$, positive slope illustrates $\Phi'_\beta(0)>0$; Right: $\theta_0$, standing wave for the Allen-Cahn equation for $W_{\text{sym}}(\rho)=\frac{1}{4}\rho^2(\rho-1)^2$ (dashed) and $W_{\text{as}}(\rho)=\frac{1}{4}\rho^2(1+\rho^2)(\rho-1)^2$ (solid)
Left: Plot of function $\Phi_\beta(V)$ for $\beta =150>\beta_{\text{cr}}$; {\it Right}: Plot $c_0V- \Phi_\beta(V)$ for $\beta=150$ vs $F$. For $-F=1.5$ there is one intersection ((61) has one root). For each $-F=1.762$ and $-F=2.264$ there are two intersections ((61) has two roots). For $-F=2$ there are three intersections ((61) has three roots)
Sketch of the function $F(V)=-c_0V+\Phi_\beta (V)$; $F(V)$ has one local minimum, $F_{\text{min}}=F(V_{\text{min}})$, and one local maximum, $F_{\text{max}}=F(V_{\text{max}})$. Left: Until $F<F_{\max}$ we stay on the left branch. When $F$ exceeds $F_{\max}$ we jump on the right branch; Right: Until $F>F_{\min}$ we stay on the right branch; When $F$ becomes less than $F_{\min}$ we jump on the left branch. Red arrows on both figures illustrate jumps in velocities.
Hysteresis loop in the problem of cell motility. Simulations of $V=V(F)$ Left: (61) Jumping from the left to the right branches and back; Right: PDE system (57)-(58). On both figures arrows show in what direction the system $(V(t), F(t))$ evolves as time $t$ grows; red curve is for $F_{\uparrow}(t)$, blue curve is for $F_{\downarrow}(t)$
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