Continuum model for linked fibers with alignment interactions

We introduce an individual-based model for fiber elements having the ability to cross-link or unlink each other and to align with each other at the cross links. We first formally derive a kinetic model for the fiber and cross-links distribution functions. We then consider the fast linking/unlinking regime in which the model can be reduced to the fiber distribution function only and investigate its diffusion limit. The resulting macroscopic model consists of a system of nonlinear diffusion equations for the fiber density and mean orientation. In the case of a homogeneous fiber density, we show that the model is elliptic.

[1]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[2]  P. Degond,et al.  A Hierarchy of Heuristic-Based Models of Crowd Dynamics , 2013, 1304.1927.

[3]  Pierre Degond,et al.  Existence of solutions and diffusion approximation for a model Fokker-Planck equation , 1987 .

[4]  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.

[5]  Curtiss,et al.  Dynamics of Polymeric Liquids , .

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

[7]  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.

[8]  Pierre Degond,et al.  HYDRODYNAMIC MODELS OF SELF-ORGANIZED DYNAMICS: DERIVATION AND EXISTENCE THEORY ∗ , 2011, 1108.3160.

[9]  W. Maier,et al.  Eine einfache molekulare Theorie des nematischen kristallinflüssigen Zustandes , 1958 .

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

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

[12]  C. Schmeiser,et al.  Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments , 2008, Cell adhesion & migration.

[13]  L. Palade,et al.  Fragmentation and monomer lengthening of rod-like polymers, a relevant model for prion proliferation , 2011, 1112.4342.

[14]  L. Onsager.  THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES , 1949 .

[15]  S. Mischler,et al.  A new approach to quantitative propagation of chaos for drift, diffusion and jump processes , 2011, 1101.4727.

[16]  M Cristina Marchetti,et al.  Hydrodynamics of self-propelled hard rods. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  A. Czirók,et al.  Collective Motion , 1999, physics/9902023.

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

[19]  E. Bertin,et al.  Mesoscopic theory for fluctuating active nematics , 2013, 1305.0772.

[20]  S. Mischler,et al.  Kac’s program in kinetic theory , 2011, Inventiones mathematicae.

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

[22]  Markus Bär,et al.  Large-scale collective properties of self-propelled rods. , 2009, Physical review letters.

[23]  Amic Frouvelle,et al.  A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters , 2009, 0912.0594.

[24]  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.

[25]  F. Poupaud,et al.  Diffusion approximation of the linear semiconductor Boltzmann equation : analysis of boundary layers , 1991 .

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

[27]  Yoshio Sone,et al.  Kinetic Theory and Fluid Dynamics , 2002 .

[28]  G. Fredrickson.  The theory of polymer dynamics , 1996 .

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

[30]  Pierre Degond,et al.  Kinetic hierarchy and propagation of chaos in biological swarm models , 2013 .

[31]  Andreas Deutsch,et al.  Nonequilibrium clustering of self-propelled rods. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[33]  C. Bardos,et al.  DIFFUSION APPROXIMATION AND COMPUTATION OF THE CRITICAL SIZE , 1984 .

[34]  Pierre Degond,et al.  Kinetic limits for pair-interaction driven master equations and biological swarm models , 2011, 1109.4538.