Modelling pattern formation through differential repulsion

Motivated by experiments on cell segregation, we present a two-species model of interacting particles, aiming at a quantitative description of this phenomenon. Under precise scaling hypothesis, we derive from the microscopic model a macroscopic one and we analyze it. In particular, we determine the range of parameters for which segregation is expected. We compare our analytical results and numerical simulations of the macroscopic model to direct simulations of the particles, and comment on possible links with experiments.

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

[2]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[3]  Lincoln Chayes,et al.  The McKean–Vlasov Equation in Finite Volume , 2009, 0910.4615.

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

[5]  J. M. Sancho,et al.  Ballistic and diffusive corrections to front propagation in the presence of multiplicative noise , 1998 .

[6]  N. L. Le Douarin.  Cell line segregation during peripheral nervous system ontogeny. , 1986, Science.

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

[8]  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.

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

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

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

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

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

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

[15]  Markus Schmidtchen,et al.  Zoology of a Nonlocal Cross-Diffusion Model for Two Species , 2017, SIAM J. Appl. Math..

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

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

[18]  M. S. Steinberg,et al.  Differential adhesion in morphogenesis: a modern view. , 2007, Current opinion in genetics & development.

[19]  W. Taylor,et al.  Cell segregation and border sharpening by Eph receptor–ephrin-mediated heterotypic repulsion , 2017, Journal of The Royal Society Interface.

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

[21]  William R. Taylor,et al.  Simulation of Cell Movement and Interaction , 2011, J. Bioinform. Comput. Biol..

[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. Beachy,et al.  Hedgehog mediated degradation of Ihog adhesion proteins modulates cell segregation in Drosophila wing imaginal discs , 2017, Nature Communications.

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

[26]  Uwe C. Täuber,et al.  Field Theory of Branching and Annihilating Random Walks , 1997 .

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

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

[29]  W. Taylor,et al.  A Mechanical Model of Cell Segregation Driven by Differential Adhesion , 2012, PloS one.

[30]  Herbert Levine,et al.  Fluctuation-induced diffusive instabilities , 1998, Nature.

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

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

[33]  G. A. Pavliotis,et al.  Long-Time Behaviour and Phase Transitions for the Mckean–Vlasov Equation on the Torus , 2018, Archive for Rational Mechanics and Analysis.

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

[35]  P. Degond,et al.  Macroscopic models of collective motion with repulsion , 2014, 1404.4886.

[36]  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.

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

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

[39]  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.

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

[41]  Pierre Degond,et al.  Kinetic Theory of Particle Interactions Mediated by Dynamical Networks , 2016, Multiscale Model. Simul..

[42]  K. Hamza,et al.  A Mathematical Model for Eph/Ephrin-Directed Segregation of Intermingled Cells , 2014, PloS one.

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

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

[45]  Pierre Degond,et al.  Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions , 2017, Journal of Nonlinear Science.

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

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

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

[49]  J. Carrillo,et al.  A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure , 2014, 1402.4252.

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

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

[52]  Glazier,et al.  Simulation of the differential adhesion driven rearrangement of biological cells. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[54]  S. Nesic,et al.  Macroscopic response to microscopic intrinsic noise in three-dimensional Fisher fronts. , 2014, Physical review letters.

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