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

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

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.

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. BerlyandJ. 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. BerlyandM. 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. BerlyandM. 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. CucchiA. 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. EtchegarayN. 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. KerenZ. PincusG. M. AllenE. L. BarnhartG. MarriottA. Mogilner and J. A. Theriot, Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480. 

[13]

I. LaviN. MeunierR. Voituriez and J. Casademunt, Motility and morphodynamics of confined cells, Phys. Rev. E., 110 (2020), 078102. 

[14]

H. LevineD. 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. MizuharaL. BerlyandV. 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. PutelatP. 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. BerlyandJ. 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. BerlyandM. 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. BerlyandM. 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. CucchiA. 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. EtchegarayN. 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. KerenZ. PincusG. M. AllenE. L. BarnhartG. MarriottA. Mogilner and J. A. Theriot, Mechanism of shape determination in motile cells, Nature, 453 (2008), 475-480. 

[13]

I. LaviN. MeunierR. Voituriez and J. Casademunt, Motility and morphodynamics of confined cells, Phys. Rev. E., 110 (2020), 078102. 

[14]

H. LevineD. 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. MizuharaL. BerlyandV. 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. PutelatP. 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
[1]

Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 789-799. doi: 10.3934/dcdsb.2014.19.789

[2]

Julien Dambrine, Nicolas Meunier, Bertrand Maury, Aude Roudneff-Chupin. A congestion model for cell migration. Communications on Pure and Applied Analysis, 2012, 11 (1) : 243-260. doi: 10.3934/cpaa.2012.11.243

[3]

Matthew S. Mizuhara, Peng Zhang. Uniqueness and traveling waves in a cell motility model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2811-2835. doi: 10.3934/dcdsb.2018315

[4]

Harunori Monobe. Behavior of radially symmetric solutions for a free boundary problem related to cell motility. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 989-997. doi: 10.3934/dcdss.2015.8.989

[5]

Yangjin Kim, Soyeon Roh. A hybrid model for cell proliferation and migration in glioblastoma. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 969-1015. doi: 10.3934/dcdsb.2013.18.969

[6]

Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Exact multiplicity of stationary limiting problems of a cell polarization model. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5627-5655. doi: 10.3934/dcds.2016047

[7]

Jan Kelkel, Christina Surulescu. On some models for cancer cell migration through tissue networks. Mathematical Biosciences & Engineering, 2011, 8 (2) : 575-589. doi: 10.3934/mbe.2011.8.575

[8]

Marco Scianna, Luigi Preziosi, Katarina Wolf. A Cellular Potts model simulating cell migration on and in matrix environments. Mathematical Biosciences & Engineering, 2013, 10 (1) : 235-261. doi: 10.3934/mbe.2013.10.235

[9]

Jian-Guo Liu, Min Tang, Li Wang, Zhennan Zhou. Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3011-3035. doi: 10.3934/dcdsb.2018297

[10]

Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa, Shoji Yotsutani. Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization. Conference Publications, 2015, 2015 (special) : 861-877. doi: 10.3934/proc.2015.0861

[11]

Zhenzhen Zheng, Ching-Shan Chou, Tau-Mu Yi, Qing Nie. Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1135-1168. doi: 10.3934/mbe.2011.8.1135

[12]

Lianzhang Bao, Zhengfang Zhou. Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 395-412. doi: 10.3934/dcdss.2017019

[13]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

[14]

Nadia Loy, Luigi Preziosi. Stability of a non-local kinetic model for cell migration with density dependent orientation bias. Kinetic and Related Models, 2020, 13 (5) : 1007-1027. doi: 10.3934/krm.2020035

[15]

Tracy L. Stepien, Hal L. Smith. Existence and uniqueness of similarity solutions of a generalized heat equation arising in a model of cell migration. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3203-3216. doi: 10.3934/dcds.2015.35.3203

[16]

Caleb Mayer, Eric Stachura. Traveling wave solutions for a cancer stem cell invasion model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 5067-5093. doi: 10.3934/dcdsb.2020333

[17]

Youngmok Jeon, Eun-Jae Park. Cell boundary element methods for convection-diffusion equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 309-319. doi: 10.3934/cpaa.2006.5.309

[18]

Yuan Wu, Jin Liang, Bei Hu. A free boundary problem for defaultable corporate bond with credit rating migration risk and its asymptotic behavior. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1043-1058. doi: 10.3934/dcdsb.2019207

[19]

Ali Ashher Zaidi, Bruce Van Brunt, Graeme Charles Wake. A model for asymmetrical cell division. Mathematical Biosciences & Engineering, 2015, 12 (3) : 491-501. doi: 10.3934/mbe.2015.12.491

[20]

Deborah C. Markham, Ruth E. Baker, Philip K. Maini. Modelling collective cell behaviour. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5123-5133. doi: 10.3934/dcds.2014.34.5123

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (210)
  • HTML views (166)
  • Cited by (0)

[Back to Top]