Macroscopic Model for Cross-Linked Fibers with Alignment Interactions: Existence Theory and Numerical Simulations

In this paper, we study the macroscopic model of [P. Degond, F. Delebecque, and D. Peurichard, Math. Models Methods Appl. Sci., 26 (2016), pp. 269--318] for fiber elements having the ability to cross-link, or unlink each other, to align with each other at the cross-links, and subjected to an external potential. We first aim to gain insight into the properties of the solutions of the macroscopic model: we provide an existence result under structural conditions for the external potential and perform numerical simulations. The numerical study of the macroscopic model reveals interesting features such as the emergence of a buckling phenomenon. Physical properties of the macroscopic fiber network are then deduced. We finally propose a numerical comparison between the macroscopic model and the microscopic one, a starting point for the derivation of the macroscopic equations. The numerical simulations reveal a good agreement between both models, providing we adapt the regime of study according to the model parameters.

[1]  David A Weitz,et al.  Cross-link-governed dynamics of biopolymer networks. , 2010, Physical review letters.

[2]  R. Galatzer-Levy.  Emergence , 2002 .

[3]  Jian-Guo Liu,et al.  Dynamics in a Kinetic Model of Oriented Particles with Phase Transition , 2011, SIAM J. Math. Anal..

[4]  Alex J Levine,et al.  Filamin cross-linked semiflexible networks: fragility under strain. , 2006, Physical review letters.

[5]  T. Hillen,et al.  Mathematical Analysis of a Kinetic Model for Cell Movement in Network Tissues , 2008, 0807.2249.

[6]  P. Hussey,et al.  Actin dynamics and the elasticity of cytoskeletal networks , 2009 .

[7]  R T Tranquillo,et al.  A methodology for the systematic and quantitative study of cell contact guidance in oriented collagen gels. Correlation of fibroblast orientation and gel birefringence. , 1993, Journal of cell science.

[8]  Richard T. Lee,et al.  A three-dimensional viscoelastic model for cell deformation with experimental verification. , 2003, Biophysical journal.

[9]  Luigi Preziosi,et al.  Modeling cell movement in anisotropic and heterogeneous network tissues , 2007, Networks Heterog. Media.

[10]  R. Dobrescu,et al.  Emergence, self–organization and morphogenesis in biological structures , 2011, Journal of medicine and life.

[11]  P. Degond,et al.  Simple mechanical cues could explain adipose tissue morphology. , 2017, Journal of theoretical biology.

[12]  D. Boal.  Mechanics of the Cell: Membranes , 2012 .

[13]  J. Galle,et al.  From single cells to tissue architecture—a bottom-up approach to modelling the spatio-temporal organisation of complex multi-cellular systems , 2009, Journal of mathematical biology.

[14]  Ilpo Vattulainen,et al.  Strain hardening, avalanches, and strain softening in dense cross-linked actin networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  M. Ben Amar,et al.  A finite dissipative theory of temporary interfibrillar bridges in the extracellular matrix of ligaments and tendons , 2009, Journal of The Royal Society Interface.

[16]  Lorenzo Pareschi,et al.  Modelling and numerical methods for granular gases , 2004 .

[17]  Daniel A. Fletcher,et al.  Reversible stress softening of actin networks , 2007, Nature.

[18]  L Cipelletti,et al.  Slow dynamics and internal stress relaxation in bundled cytoskeletal networks. , 2011, Nature materials.

[19]  Michael E. Taylor,et al.  Partial Differential Equations III , 1996 .

[20]  L. Preziosi,et al.  Mechanics and Chemotaxis in the Morphogenesis of Vascular Networks , 2006, Bulletin of mathematical biology.

[21]  P. Friedl,et al.  The biology of cell locomotion within three-dimensional extracellular matrix , 2000, Cellular and Molecular Life Sciences CMLS.

[22]  T. Pollard,et al.  Actin dynamics. , 2001, Journal of cell science.

[23]  Yingda Cheng,et al.  A Particle Interaction Model for the Simulation of Biological, Cross-Linked Fiber Networks Inspired From flocking Theory , 2014 .

[24]  C. Poignard,et al.  Modeling of the migration of endothelial cells on bioactive micropatterned polymers. , 2013, Mathematical biosciences and engineering : MBE.

[25]  Giacomo Dimarco,et al.  HYDRODYNAMICS OF THE KURAMOTO-VICSEK MODEL OF ROTATING SELF-PROPELLED PARTICLES , 2013, 1306.3372.

[26]  Pierre Degond,et al.  Continuum limit of self-driven particles with orientation interaction , 2007, 0710.0293.

[27]  E. Tadmor,et al.  From particle to kinetic and hydrodynamic descriptions of flocking , 2008, 0806.2182.

[28]  H M Byrne,et al.  The influence of growth-induced stress from the surrounding medium on the development of multicell spheroids , 2001, Journal of mathematical biology.

[29]  R. Carpick,et al.  A General Equation for Fitting Contact Area and Friction vs Load Measurements. , 1999, Journal of colloid and interface science.

[30]  Frank Jülicher,et al.  Hydrodynamic theory for multi-component active polar gels , 2007 .

[31]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[32]  George Em Karniadakis,et al.  Accurate coarse-grained modeling of red blood cells. , 2008, Physical review letters.

[33]  Macroscopic models of collective motion with repulsion , 2014, 1404.4886.

[34]  Pierre Degond,et al.  DIFFUSION IN A CONTINUUM MODEL OF SELF-PROPELLED PARTICLES WITH ALIGNMENT INTERACTION , 2010, 1002.2716.

[35]  F C MacKintosh,et al.  Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  W Alt,et al.  Cytoplasm dynamics and cell motion: two-phase flow models. , 1999, Mathematical biosciences.

[37]  Pierre Degond,et al.  Continuum model for linked fibers with alignment interactions , 2015, 1505.05027.

[38]  T. Mckeown.  Mechanics , 1970, The Mathematics of Fluid Flow Through Porous Media.

[39]  M. G. RUSBRIDGE,et al.  Kinetic Theory , 1969, Nature.

[40]  Pierre Degond,et al.  A Macroscopic Model for a System of Swarming Agents Using Curvature Control , 2010, 1010.5405.

[41]  A. McCulloch,et al.  Stress-dependent finite growth in soft elastic tissues. , 1994, Journal of biomechanics.

[42]  Emmanuel Boissard,et al.  Trail formation based on directed pheromone deposition , 2011, Journal of mathematical biology.

[43]  J. Murray,et al.  A mechanical model for the formation of vascular networks in vitro , 1996, Acta biotheoretica.

[44]  R. Dickinson,et al.  A generalized transport model for biased cell migration in an anisotropic environment , 2000, Journal of mathematical biology.

[45]  N. Balaban,et al.  Calculation of forces at focal adhesions from elastic substrate data: the effect of localized force and the need for regularization. , 2002, Biophysical journal.

[46]  P V Bayly,et al.  A POROELASTIC MODEL FOR CELL CRAWLING INCLUDING MECHANICAL COUPLING BETWEEN CYTOSKELETAL CONTRACTION AND ACTIN POLYMERIZATION. , 2011, Journal of mechanics of materials and structures.

[47]  B. Shraiman,et al.  Mechanical feedback as a possible regulator of tissue growth. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[48]  T. Hillen.  M5 mesoscopic and macroscopic models for mesenchymal motion , 2006, Journal of mathematical biology.