Comptes Rendus
Partial differential equations
Phase-field model of cell motility: Traveling waves and sharp interface limit
Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 986-992.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.09.001

Leonid Berlyand 1; Mykhailo Potomkin 1; Volodymyr Rybalko 2

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 Nauka Ave., 61103, Kharkiv, Ukraine
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Leonid Berlyand; Mykhailo Potomkin; Volodymyr Rybalko. Phase-field model of cell motility: Traveling waves and sharp interface limit. Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 986-992. doi : 10.1016/j.crma.2016.09.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.09.001/

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

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

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

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

[5] E. Barnhart; K. Lee; G. Allen; J. Theriot; A. Mogilner 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] L. Berlyand; M. Potomkin; V. Rybalko Sharp interface limit in a phase field model of cell motility (submitted for publication, preprint available at) | arXiv

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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