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May  2022, 42(5): 2381-2407. doi: 10.3934/dcds.2021194

Self polarization and traveling wave in a model for cell crawling migration

Alessandro Cucchi 1, , Antoine Mellet 2, and  Nicolas Meunier 3,

1. 

Map5, UMR 8145 CNRS, Université de Paris, France
2. 

Department of Mathematics, University of Maryland, College Park MD 20742 USA
3. 

LaMME, UMR 8071 CNRS, Université Évry Val d'Essonne, France

Received  April 2021 Revised  September 2021 Published  May 2022 Early access  December 2021

Fund Project: The second author is supported by NSF grant DMS-2009236

In this paper, we prove the existence of traveling wave solutions for an incompressible Darcy's free boundary problem recently introduced in [6] to describe cell motility. This free boundary problem involves a nonlinear destabilizing term in the boundary condition which describes the active character of the cell cytoskeleton. By using two different methods, a constructive method via a graph analysis and a local bifurcation method, we prove that traveling wave solutions exist when the destabilizing term is strong enough.

Keywords: traveling waves, free boundary, cell migration and cell polarization.
Mathematics Subject Classification: 35R35; 35B32; 35C07; 92C17.
Citation: Alessandro Cucchi, Antoine Mellet, Nicolas Meunier. Self polarization and traveling wave in a model for cell crawling migration. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2381-2407. doi: 10.3934/dcds.2021194
References:
[1]

L. Berlyand, J. Fuhrmann and V. Rybalko, Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility, Commun. Math. Sci., 16 (2018), 735-762.  doi: 10.4310/CMS.2018.v16.n3.a6.

[2]

L. Berlyand, M. Potomkin and V. Rybalko, Phase-field model of cell motility: Traveling waves and sharp interface limit, C. R. Math. Acad. Sci. Paris, 354 (2016), 986-992.  doi: 10.1016/j.crma.2016.09.001.

[3]

L. Berlyand, M. Potomkin and V. Rybalko, Sharp interface limit in a phase field model of cell motility, Netw. Heterog. Media, 12 (2017), 551-590.  doi: 10.3934/nhm.2017023.

[4]

L. Berlyand and V. Rybalko, Stability of steady states and bifurcation to traveling waves in a free boundary model of cell motility, Submitted, available on arXiv.org, 2019.

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[6]

A. Cucchi, A. Mellet and N. Meunier, A Cahn-Hilliard model for cell motility, SIAM J. Math. Anal., 52 (2020), 3843-3880.  doi: 10.1137/19M1267969.

[7]

C. Etchegaray, N. Meunier and R. Voituriez, Analysis of a non-local and non-linear Fokker-Planck model for cell crawling migration, SIAM J. Appl. Math., 77 (2017), 2040-2065.  doi: 10.1137/16M1088715.

[8]

A. Friedman, A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-159.  doi: 10.3934/dcdsb.2004.4.147.

[9]

A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639.  doi: 10.1016/j.jde.2005.09.008.

[10]

A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth, rans. Amer. Math. Soc., 353 (2001), 1587-1634.  doi: 10.1090/S0002-9947-00-02715-X.

[11]

M. Günther and G. Prokert, On travelling wave solutions for a moving boundary problem of Hele-Shaw type, IMA J. Appl. Math., 74 (2009), 107-127.  doi: 10.1093/imamat/hxn029.

[12]

K. Keren, Z. Pincus, G. M. Allen, E. L. Barnhart, G. Marriott, A. Mogilner and J. A. Theriot, Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480. 

[13]

I. Lavi, N. Meunier, R. Voituriez and J. Casademunt, Motility and morphodynamics of confined cells, Phys. Rev. E., 110 (2020), 078102. 

[14]

H. Levine, D. Shao and J. W. Rappel, Coupling actin flow, adhesion, and morphology in a computational cell motility model, Proc Nat Acad Sci, 109 (2015), 6851-6856. 

[15]

M. S. 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.

[16]

M. S. Mizuhara and P. Zhang, Uniqueness and traveling waves in a cell motility model, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2811-2835.  doi: 10.3934/dcdsb.2018315.

[17]

F. Otto, Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141 (1998), 63-103.  doi: 10.1007/s002050050073.

[18]

T. Putelat, P. Recho and L. Truskinovsky, Contraction-driven cell motility, Phys Rev Lett, 111 (2013), 108102. 

[19]

F. Ziebert and I. Aronson, Computational Approaches to Substrate-based Cell Motility, Nature Phys. Journal, 2016.

show all references

References:
[1]

L. Berlyand, J. Fuhrmann and V. Rybalko, Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility, Commun. Math. Sci., 16 (2018), 735-762.  doi: 10.4310/CMS.2018.v16.n3.a6.

[2]

L. Berlyand, M. Potomkin and V. Rybalko, Phase-field model of cell motility: Traveling waves and sharp interface limit, C. R. Math. Acad. Sci. Paris, 354 (2016), 986-992.  doi: 10.1016/j.crma.2016.09.001.

[3]

L. Berlyand, M. Potomkin and V. Rybalko, Sharp interface limit in a phase field model of cell motility, Netw. Heterog. Media, 12 (2017), 551-590.  doi: 10.3934/nhm.2017023.

[4]

L. Berlyand and V. Rybalko, Stability of steady states and bifurcation to traveling waves in a free boundary model of cell motility, Submitted, available on arXiv.org, 2019.

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.

[6]

A. Cucchi, A. Mellet and N. Meunier, A Cahn-Hilliard model for cell motility, SIAM J. Math. Anal., 52 (2020), 3843-3880.  doi: 10.1137/19M1267969.

[7]

C. Etchegaray, N. Meunier and R. Voituriez, Analysis of a non-local and non-linear Fokker-Planck model for cell crawling migration, SIAM J. Appl. Math., 77 (2017), 2040-2065.  doi: 10.1137/16M1088715.

[8]

A. Friedman, A hierarchy of cancer models and their mathematical challenges, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 147-159.  doi: 10.3934/dcdsb.2004.4.147.

[9]

A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Differential Equations, 227 (2006), 598-639.  doi: 10.1016/j.jde.2005.09.008.

[10]

A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth, rans. Amer. Math. Soc., 353 (2001), 1587-1634.  doi: 10.1090/S0002-9947-00-02715-X.

[11]

M. Günther and G. Prokert, On travelling wave solutions for a moving boundary problem of Hele-Shaw type, IMA J. Appl. Math., 74 (2009), 107-127.  doi: 10.1093/imamat/hxn029.

[12]

K. Keren, Z. Pincus, G. M. Allen, E. L. Barnhart, G. Marriott, A. Mogilner and J. A. Theriot, Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480. 

[13]

I. Lavi, N. Meunier, R. Voituriez and J. Casademunt, Motility and morphodynamics of confined cells, Phys. Rev. E., 110 (2020), 078102. 

[14]

H. Levine, D. Shao and J. W. Rappel, Coupling actin flow, adhesion, and morphology in a computational cell motility model, Proc Nat Acad Sci, 109 (2015), 6851-6856. 

[15]

M. S. 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.

[16]

M. S. Mizuhara and P. Zhang, Uniqueness and traveling waves in a cell motility model, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2811-2835.  doi: 10.3934/dcdsb.2018315.

[17]

F. Otto, Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141 (1998), 63-103.  doi: 10.1007/s002050050073.

[18]

T. Putelat, P. Recho and L. Truskinovsky, Contraction-driven cell motility, Phys Rev Lett, 111 (2013), 108102. 

[19]

F. Ziebert and I. Aronson, Computational Approaches to Substrate-based Cell Motility, Nature Phys. Journal, 2016.

Figure 1.  A shape of traveling wave solution $ \Omega_0 $ defined by (3.1) for $ \beta/\lambda = 4 $ and $ \gamma = 1 $. The red dots indicate the point $ x_L<0 $ on the left and $ x_R>0 $ on the right. The function $ h(x) $ is defined for $ x \in [x_L,x_R] $ such that conditions (3.2) – (3.3) hold. The graph of $ h $ lies on the $ y $ -positve part of the plane, while the graph of $ -h $ on the $ y $ -negative part
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