Phase-field model of cell motility: Traveling waves and sharp interface limit

Berlyand, Leonid; Potomkin, Mykhailo; Rybalko, Volodymyr

Comptes Rendus. Mathématique, Tome 354 (2016) no. 10, pp. 986-992.

Nous présentons dans cet article des résultats récents sur l'analyse asymptotique d'un modèle EDP pour la migration de cellules eucaryotes. Nous dérivons formellement l'équation limite pour l'interface, qui décrit le mouvement de la membrane cellulaire. Dans le cas unidimensionnel, nous justifions cette limite de façon rigoureuse, et nous observons numériquement quelques propriétés surprenantes, comme par exemple une discontinuité dans les vitesses à l'interface et un phénomène d'hystérésis. Nous montrons l'apparition d'ondes de propagation non triviales quand le paramètre physique clé dépasse un certain seuil.

In this paper, we report our recent results on the asymptotic analysis of a PDE model for the motility of an eukaryotic cell. We formally derive the sharp interface limit, which describes the motion of the cell membrane. In the 1D case, we rigorously justify the limit, and, using numerical simulations, observe some surprising features, such as discontinuity of interface velocities and hysteresis. We show that nontrivial traveling wave solutions appear when the key physical parameter exceeds a critical value.

[1] Physical Models of Cell Motility (Aranson, I.S., ed.), Springer, 2016

[2] Barles, G.; Souganidis, P. A new approach to front propagation problems: theory and applications, Arch. Ration. Mech. Anal., Volume 141 (1998) no. 3, pp. 237-296

[3] Barles, G.; Soner, H.M.; Souganidis, P.E. Front propagation and phase field theory, SIAM J. Control Optim., Volume 31 (1993) no. 2, pp. 439-469

[4] Barnhart, E.; Lee, K.; Keren, K.; Mogilner, A.; Theriot, J. An adhesion-dependent switch between mechanisms that determine motile cell shape, PLoS Biol., Volume 9 (2011) no. 5

[5] Barnhart, E.; Lee, K.; Allen, G.; Theriot, J.; Mogilner, A. Balance between cell-substrate adhesion and myosin contraction determines the frequence of motility initiation in fish keratocytes, Proc. Natl. Acad. Sci. USA, Volume 112 (2015) no. 16, pp. 5045-5050

[6] Berlyand, L.; Potomkin, M.; Rybalko, V. Sharp interface limit in a phase field model of cell motility (submitted for publication, preprint available at) | arXiv

[7] Camley, B.; Zhao, Y.; Li, B.; Levine, H.; Rappel, W. Periodic migration in a physical model of cells on micropatterns, Phys. Rev. Lett., Volume 111 (2013) no. 15

[8] Chen, X. Spectrums for the Allen–Cahn, Cahn–Hilliard, and phase field equations for generic interface, Commun. Partial Differ. Equ., Volume 19 (1994), pp. 1371-1395

[9] Chen, X.; Hilhorst, D.; Logak, E. Mass conserving Allen–Cahn equation and volume preserving mean curvature flow, Interfaces Free Bound., Volume 12 (2010) no. 4, pp. 527-549

[10] Evans, L.C.; Soner, H.M.; Souganidis, P.E. Phase transitions and generalized motion by mean curvature, Commun. Pure Appl. Math., Volume 45 (1991), pp. 1097-1123

[11] Golovaty, D. The volume preserving motion by mean curvature as an asymptotic limit of reaction–diffusion equations, Quart. Appl. Math., Volume 55 (1997), pp. 243-298

[12] Keren, K.; Pincus, Z.; Allen, G.; Barnhart, E.; Marriott, G.; Mogilner, A.; Theriot, J. Mechanism of shape determination in motile cells, Nature, Volume 453 (2008), pp. 475-480

[13] Lio, F.D.; Kim, C.I.; Slepcev, D. Nonlocal front propagation problems in bounded domains with Neumann-type boundary conditions and applications, Asymptot. Anal., Volume 37 (2004) no. 3–4, pp. 257-292

[14] Maiuri, P.; Rupprecht, J.-F.; Wieser, S.; Ruprecht, V.; Bénichou, O.; Carpi, N.; Coppey, M.; Beco, S.; Gov, N.; Heisenberg, C.-F.; Crespo, C.; Lautenschlaeger, F.; Berre, M.; Lennon-Dumenil, A.-M.; Raab, H.-R.; Thiam, M.; Piel, M.; Sixt, M.; Voiteriez, R. Actin flows mediate a universal coupling between cell speed and cell persistence, Cell, Volume 161 (2015) no. 2, pp. 374-386

[15] Majda, A.; Souganidis, P. Large-scale front dynamics for turbulent reaction–diffusion equations with separated velocity scales, Nonlinearity, Volume 7 (1994) no. 1, pp. 1-30

[16] Mizuhara, M.; Berlyand, L.; Rybalko, V.; Zhang, L. On an evolution equation in a cell motility model, Physica D, Volume 318–319 (2015), pp. 12-25 | DOI

[17] Mottoni, P.; Schatzman, M. Geometrical evolution of developed interfaces, Trans. Am. Math. Soc., Volume 347 (1995), pp. 1533-1589

[18] Ölz, D.; Schmeiser, C. Derivation of a model for symmetric lamellipodia with instantaneous cross-link turnover, Arch. Ration. Mech. Anal., Volume 198 (2010), pp. 963-980

[19] Ölz, D.; Schmeiser, C. How do cells move? Mathematical modeling of cytoskeleton dynamics and cell migration, Cell Mechanics: From Single Scale-Based Models to Multiscale Modeling, Chapman and Hall/CRC Press, Boca Raton, FL, USA, 2010, pp. 133-157

[20] Recho, P.; Truskinovsky, L. Asymmetry between pushing and pulling for crawling cells, Phys. Rev. E, Volume 87 (2013)

[21] Recho, P.; Putelat, T.; Truskinovsky, L. Mechanics of motility initiation and motility arrest in crawling cells, J. Mech. Phys. Solids, Volume 84 (2015), pp. 469-505

[22] Rubinstein, B.; Jacobson, K.; Mogilner, A. Multiscale two-dimensional modeling of a motile simple-shaped cell, Multiscale Model. Simul., Volume 3 (2005) no. 2, pp. 413-439

[23] Sandier, E.; Serfaty, S. Gamma-convergence of gradient flows with applications to Ginzburg–Landau, Commun. Pure Appl. Math., Volume 57 (2004) no. 12, pp. 1627-1672

[24] Semplice, M.; Veglio, A.; Naldi, G.; Serini, G.; Gamba, A. A bistable model of cell polarity, PLoS ONE, Volume 7 (2012) no. 2

[25] Serfaty, S. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., Ser. A, Volume 31 (2011) no. 4, pp. 1427-1451

[26] Ziebert, F.; Aranson, I. Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells, PLoS ONE, Volume 8 (2013) no. 5

[27] Ziebert, F.; Swaminathan, S.; Aranson, I. Model for self-polarization and motility of keratocyte fragments, J. R. Soc. Interface, Volume 9 (2011) no. 70, pp. 1084-1092