September  2020, 15(3): 307-352. doi: 10.3934/nhm.2020021

Modelling pattern formation through differential repulsion

1. 

Institut Denis Poisson, Université d'Orléans, CNRS, Université de Tours, B.P. 6759, 45067 Orléans cedex 2, France, Institut Universitaire de France, Paris, France

2. 

Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

3. 

INRIA Team Mamba, INRIA Paris, 2 rue Simone Iff, CS 42112, 75589 Paris, France, Sorbonne Université, UMR 7598 LJLL, BC187, 4, Place de Jussieu, F-75252 Paris Cedex 5, France

4. 

Department of Mathematics, University College London, Gower Street London WC1E 6BT, UK, United Kingdom

Received  June 2019 Revised  November 2019 Published  September 2020 Early access  September 2020

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.

Citation: Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks and Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021
References:
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[2]

R. AlonsoJ. Young and Y. Cheng, A particle interaction model for the simulation of biological, cross-linked fibers inspired from flocking theory, Cellular and Molecular Bioengineering, 7 (2014), 58-72.  doi: 10.1007/s12195-013-0308-5.

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J. Armero, J. Casademunt, L. Ramírez-Piscina and J. M. Sancho, Ballistic and diffusive corrections to front propagation in the presence of multiplicative noise, Phys. Rev. E., 58 (1998). doi: 10.1103/PhysRevE.58.5494.

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J. A. Åström, P. B. S. Kumar, I. Vattulainen and M. Karttunen, Strain hardening, avalanches, and strain softening in dense cross-linked actin networks, Phys. Rev. E, 77 (2008), 051913.

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C. BardosR. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0.

[7]

J. BarréP. Degond and E. Zatorska, Kinetic theory of particle interactions mediated by dynamical networks, Multiscale Model. Simul., 15 (2017), 1294-1323.  doi: 10.1137/16M1085310.

[8]

J. BarréJ. A CarrilloP. DegondD. Peurichard and E. Zatorska, Particle interactions mediated by dynamical networks: Assessment of macroscopic descriptions, J. Nonlinear Sci., 28 (2018), 235-268.  doi: 10.1007/s00332-017-9408-z.

[9]

A. Baskaran and M. C. Marchetti, Hydrodynamics of self-propelled hard rods, Phys. Rev. E, 77 (2008), 011920, 9 pp. doi: 10.1103/PhysRevE.77.011920.

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E. Bertin, H. Chaté, F. Ginelli, S. Mishra, A. Peshkov and S. Ramaswamy, Mesoscopic theory for fluctuating active nematics, New J. Phys., 15 (2013), 085032. doi: 10.1088/1367-2630/15/8/085032.

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E. CarlenR. ChatelinP. Degond and B. Wennberg, Kinetic hierarchy and propagation of chaos in biological swarm models, Phys. D, 260 (2013), 90-111.  doi: 10.1016/j.physd.2012.05.013.

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E. CarlenP. Degond and B. Wennberg, Kinetic limits for pair-interaction driven master equations and biological swarm models, Math. Models Methods Appl. Sci., 23 (2013), 1339-1376.  doi: 10.1142/S0218202513500115.

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J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis and A. Schlichting, Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus, Arch. Ration. Mech. Anal., 235 (2020), 635–690, arXiv: 1806.01719. doi: 10.1007/s00205-019-01430-4.

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J. A. CarrilloY. Huang and M. Schmidtchen, Zoology of a non-local cross-diffusion model for two species, SIAM J. Appl. Math., 78 (2018), 1078-1104.  doi: 10.1137/17M1128782.

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L. Chayes and V. Panferov, The McKean-Vlasov equation in finite volume, Journal of Statistical Physics, 138 (2010), 351-380.  doi: 10.1007/s10955-009-9913-z.

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I. S. CiupercaE. HingantL. I. Palade and L. Pujo-Menjouet, Fragmentation and monomer lengthening of rod-like polymers, a relevant model for prion proliferation, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 775-799.  doi: 10.3934/dcdsb.2012.17.775.

[22]

P. DegondC. Appert-RollandM. MoussaidJ. Pettré and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068.  doi: 10.1007/s10955-013-0805-x.

[23]

P. DegondF. Delebecque and D. Peurichard, Continuum model for linked fibers with alignment interactions, Math. Models Methods Appl. Sci., 26 (2016), 269-318.  doi: 10.1142/S0218202516400030.

[24]

P. Degond, G. Dimarco, T. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion, Commun. Math. Sci., 13 (2015), 1615–1638, arXiv: 1404.4886. doi: 10.4310/CMS.2015.v13.n6.a12.

[25]

P. DegondJ.-G. LiuS. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114.  doi: 10.4310/MAA.2013.v20.n2.a1.

[26]

P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation, Transport Theory and Statistical Physics, 16 (1987), 589-636.  doi: 10.1080/00411458708204307.

[27]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), suppl., 1193–1215. doi: 10.1142/S0218202508003005.

[28] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, International Series of Monographs on Physics, Oxford University Press, Vol. 73, 1999. 
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NM Le Douarin, Cell line segregation during peripheral nervous system ontogeny, Science, 231 (1986), 1515-1522.  doi: 10.1126/science.3952494.

[30]

A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011, 40 pp. doi: 10.1142/S021820251250011X.

[31]

F. Ginelli, F. Peruani, M. Bär and H. Chaté, Large-scale collective properties of self-propelled rods, Phys. Rev. Lett., 104 (2010), 184502. doi: 10.1103/PhysRevLett.104.184502.

[32]

J. A. Glazier and F. Graner, Simulation of the differential adhesion driven rearrangement of biological cells, Phys. Rev. E, 47 (1993), 2128-2154.  doi: 10.1103/PhysRevE.47.2128.

[33]

D. A Head, A. J. Levine and F. C MacKintosh, Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks, Phys. Rev. E, 68 (2003), 061907. doi: 10.1103/PhysRevE.68.061907.

[34]

E. Y. C. Hsia, Y. Zhang, H. S. Tran, A. Lim, Y.-H. Chou, G. Lan, P. A. Beachy and X. Zheng, Hedgehog mediated degradation of Ihog adhesion proteins modulates cell segregation in Drosophila wing imaginal discs, Nature Communications, 8 (2017). doi: 10.1038/s41467-017-01364-z.

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J. F. Joanny, F. Jülicher, K. Kruse and J. Prost, Hydrodynamic theory for multi-component active polar gels, New J. Phys., 9 (2007), 422. doi: 10.1088/1367-2630/9/11/422.

[36]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

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H. KarsherJ. LammerdingH. HuangR. T. LeeR. D. Kamm and M. R. Kaazempur-Mofrad, A three-dimensional viscoelastic model for cell deformation with experimental verification, Biophysical Journal, 85 (2003), 3336-3349.  doi: 10.1016/S0006-3495(03)74753-5.

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D. A. Kessler and H. Levine, Fluctuation-induced diffusive instabilities, Nature, 394 (1998), 556-558.  doi: 10.1038/29020.

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J. J. Kupiec, A Darwinian theory for the origin of cellular differentiation, Molecular and General Genetics MGG, 255 (1997), 201-208.  doi: 10.1007/s004380050490.

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W. Maier and A. Saupe, Eine einfache molekulare Theorie des nematischen kristallinflüssigen Zustandes, Z. Naturforsch., 13 (1958), 564-566.  doi: 10.1515/zna-1958-0716.

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S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.

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S. MischlerC. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59.  doi: 10.1007/s00440-013-0542-8.

[43]

S. Nesic, R. Cuerno and E. Moro, Macroscopic response to microscopic intrinsic noise in three-dimensional fisher fronts, Phys. Rev. Lett., 113 (2014), 180602. doi: 10.1103/PhysRevLett.113.180602.

[44]

D. OelzC. Schmeiser and J. V. Small, Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments, Cell Adhesion and Migration, 2 (2008), 117-126.  doi: 10.4161/cam.2.2.6373.

[45]

L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. New York Acad. Sci., 51 (1949), 627-659. 

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F. Peruani, A. Deutsch and M. Bär, Nonequilibrium clustering of self-propelled rods, Phys. Rev. E, 7 (2006), 030904(R). doi: 10.1103/PhysRevE.74.030904.

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D. PeurichardF. DelebecqueA. LorsignolC. BarreauJ. RouquetteX. DescombesL. Casteilla and P. Degond, Simple mechanical cues could explain adipose tissue morphology, J. Theor. Biol., 429 (2017), 61-81.  doi: 10.1016/j.jtbi.2017.06.030.

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M. S. Steinberg, Differential adhesion in morphogenesis: A modern view, Curr. Opin. Genet. Dev., 17 (2007), 281-286.  doi: 10.1016/j.gde.2007.05.002.

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L. A. TaberY. ShiL. Yang and P. V. Bayly, A poroelastic model for cell crawling including mechanical coupling between cytoskeletal contraction and actin polymerization, Journal of Mechanics of Materials and Structures, 6 (2011), 569-589.  doi: 10.2140/jomms.2011.6.569.

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show all references

References:
[1]

R. Aharon, A mathematical model for Eph/Ephrin-directed segregation of intermingled cells, PLoS ONE, 9 (2014), 111-803. 

[2]

R. AlonsoJ. Young and Y. Cheng, A particle interaction model for the simulation of biological, cross-linked fibers inspired from flocking theory, Cellular and Molecular Bioengineering, 7 (2014), 58-72.  doi: 10.1007/s12195-013-0308-5.

[3]

W. Alt and M. Dembo, Cytoplasm dynamics and cell motion: Two phase flow models, Math. Biosci., 156 (1999), 207-228.  doi: 10.1016/S0025-5564(98)10067-6.

[4]

J. Armero, J. Casademunt, L. Ramírez-Piscina and J. M. Sancho, Ballistic and diffusive corrections to front propagation in the presence of multiplicative noise, Phys. Rev. E., 58 (1998). doi: 10.1103/PhysRevE.58.5494.

[5]

J. A. Åström, P. B. S. Kumar, I. Vattulainen and M. Karttunen, Strain hardening, avalanches, and strain softening in dense cross-linked actin networks, Phys. Rev. E, 77 (2008), 051913.

[6]

C. BardosR. Santos and R. Sentis, Diffusion approximation and computation of the critical size, Trans. Amer. Math. Soc., 284 (1984), 617-649.  doi: 10.1090/S0002-9947-1984-0743736-0.

[7]

J. BarréP. Degond and E. Zatorska, Kinetic theory of particle interactions mediated by dynamical networks, Multiscale Model. Simul., 15 (2017), 1294-1323.  doi: 10.1137/16M1085310.

[8]

J. BarréJ. A CarrilloP. DegondD. Peurichard and E. Zatorska, Particle interactions mediated by dynamical networks: Assessment of macroscopic descriptions, J. Nonlinear Sci., 28 (2018), 235-268.  doi: 10.1007/s00332-017-9408-z.

[9]

A. Baskaran and M. C. Marchetti, Hydrodynamics of self-propelled hard rods, Phys. Rev. E, 77 (2008), 011920, 9 pp. doi: 10.1103/PhysRevE.77.011920.

[10]

E. Bertin, H. Chaté, F. Ginelli, S. Mishra, A. Peshkov and S. Ramaswamy, Mesoscopic theory for fluctuating active nematics, New J. Phys., 15 (2013), 085032. doi: 10.1088/1367-2630/15/8/085032.

[11]

R. Bird, C. Curtiss, R. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory, John Wiley & Sons, New York, 1987.

[12]

C. P. Broedersz, M. Depken, N. Y. Yao, M. R. Pollak, D. A. Weitz and F. C. MacKintosh, Cross-link-governed dynamics of biopolymer networks, Phys. Rev. Lett., 105 (2010), 238101. doi: 10.1103/PhysRevLett.105.238101.

[13]

G. A. BuxtonN. Clarke and P. J. Hussey, Actin dynamics and the elasticity of cytoskeletal networks, Express Polymer Letters, 3 (2009), 579-587.  doi: 10.3144/expresspolymlett.2009.72.

[14]

J. L. Cardy and U. C. Täuber, Field theory of branching and annihilating random walks, Journal of Statistical Physics, 90 (1998), 1-56.  doi: 10.1023/A:1023233431588.

[15]

E. CarlenR. ChatelinP. Degond and B. Wennberg, Kinetic hierarchy and propagation of chaos in biological swarm models, Phys. D, 260 (2013), 90-111.  doi: 10.1016/j.physd.2012.05.013.

[16]

E. CarlenP. Degond and B. Wennberg, Kinetic limits for pair-interaction driven master equations and biological swarm models, Math. Models Methods Appl. Sci., 23 (2013), 1339-1376.  doi: 10.1142/S0218202513500115.

[17]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.

[18]

J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis and A. Schlichting, Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus, Arch. Ration. Mech. Anal., 235 (2020), 635–690, arXiv: 1806.01719. doi: 10.1007/s00205-019-01430-4.

[19]

J. A. CarrilloY. Huang and M. Schmidtchen, Zoology of a non-local cross-diffusion model for two species, SIAM J. Appl. Math., 78 (2018), 1078-1104.  doi: 10.1137/17M1128782.

[20]

L. Chayes and V. Panferov, The McKean-Vlasov equation in finite volume, Journal of Statistical Physics, 138 (2010), 351-380.  doi: 10.1007/s10955-009-9913-z.

[21]

I. S. CiupercaE. HingantL. I. Palade and L. Pujo-Menjouet, Fragmentation and monomer lengthening of rod-like polymers, a relevant model for prion proliferation, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 775-799.  doi: 10.3934/dcdsb.2012.17.775.

[22]

P. DegondC. Appert-RollandM. MoussaidJ. Pettré and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068.  doi: 10.1007/s10955-013-0805-x.

[23]

P. DegondF. Delebecque and D. Peurichard, Continuum model for linked fibers with alignment interactions, Math. Models Methods Appl. Sci., 26 (2016), 269-318.  doi: 10.1142/S0218202516400030.

[24]

P. Degond, G. Dimarco, T. B. N. Mac and N. Wang, Macroscopic models of collective motion with repulsion, Commun. Math. Sci., 13 (2015), 1615–1638, arXiv: 1404.4886. doi: 10.4310/CMS.2015.v13.n6.a12.

[25]

P. DegondJ.-G. LiuS. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114.  doi: 10.4310/MAA.2013.v20.n2.a1.

[26]

P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation, Transport Theory and Statistical Physics, 16 (1987), 589-636.  doi: 10.1080/00411458708204307.

[27]

P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), suppl., 1193–1215. doi: 10.1142/S0218202508003005.

[28] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, International Series of Monographs on Physics, Oxford University Press, Vol. 73, 1999. 
[29]

NM Le Douarin, Cell line segregation during peripheral nervous system ontogeny, Science, 231 (1986), 1515-1522.  doi: 10.1126/science.3952494.

[30]

A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011, 40 pp. doi: 10.1142/S021820251250011X.

[31]

F. Ginelli, F. Peruani, M. Bär and H. Chaté, Large-scale collective properties of self-propelled rods, Phys. Rev. Lett., 104 (2010), 184502. doi: 10.1103/PhysRevLett.104.184502.

[32]

J. A. Glazier and F. Graner, Simulation of the differential adhesion driven rearrangement of biological cells, Phys. Rev. E, 47 (1993), 2128-2154.  doi: 10.1103/PhysRevE.47.2128.

[33]

D. A Head, A. J. Levine and F. C MacKintosh, Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks, Phys. Rev. E, 68 (2003), 061907. doi: 10.1103/PhysRevE.68.061907.

[34]

E. Y. C. Hsia, Y. Zhang, H. S. Tran, A. Lim, Y.-H. Chou, G. Lan, P. A. Beachy and X. Zheng, Hedgehog mediated degradation of Ihog adhesion proteins modulates cell segregation in Drosophila wing imaginal discs, Nature Communications, 8 (2017). doi: 10.1038/s41467-017-01364-z.

[35]

J. F. Joanny, F. Jülicher, K. Kruse and J. Prost, Hydrodynamic theory for multi-component active polar gels, New J. Phys., 9 (2007), 422. doi: 10.1088/1367-2630/9/11/422.

[36]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[37]

H. KarsherJ. LammerdingH. HuangR. T. LeeR. D. Kamm and M. R. Kaazempur-Mofrad, A three-dimensional viscoelastic model for cell deformation with experimental verification, Biophysical Journal, 85 (2003), 3336-3349.  doi: 10.1016/S0006-3495(03)74753-5.

[38]

D. A. Kessler and H. Levine, Fluctuation-induced diffusive instabilities, Nature, 394 (1998), 556-558.  doi: 10.1038/29020.

[39]

J. J. Kupiec, A Darwinian theory for the origin of cellular differentiation, Molecular and General Genetics MGG, 255 (1997), 201-208.  doi: 10.1007/s004380050490.

[40]

W. Maier and A. Saupe, Eine einfache molekulare Theorie des nematischen kristallinflüssigen Zustandes, Z. Naturforsch., 13 (1958), 564-566.  doi: 10.1515/zna-1958-0716.

[41]

S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., 193 (2013), 1-147.  doi: 10.1007/s00222-012-0422-3.

[42]

S. MischlerC. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59.  doi: 10.1007/s00440-013-0542-8.

[43]

S. Nesic, R. Cuerno and E. Moro, Macroscopic response to microscopic intrinsic noise in three-dimensional fisher fronts, Phys. Rev. Lett., 113 (2014), 180602. doi: 10.1103/PhysRevLett.113.180602.

[44]

D. OelzC. Schmeiser and J. V. Small, Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments, Cell Adhesion and Migration, 2 (2008), 117-126.  doi: 10.4161/cam.2.2.6373.

[45]

L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. New York Acad. Sci., 51 (1949), 627-659. 

[46]

F. Peruani, A. Deutsch and M. Bär, Nonequilibrium clustering of self-propelled rods, Phys. Rev. E, 7 (2006), 030904(R). doi: 10.1103/PhysRevE.74.030904.

[47]

D. PeurichardF. DelebecqueA. LorsignolC. BarreauJ. RouquetteX. DescombesL. Casteilla and P. Degond, Simple mechanical cues could explain adipose tissue morphology, J. Theor. Biol., 429 (2017), 61-81.  doi: 10.1016/j.jtbi.2017.06.030.

[48]

F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers, Asymptot. Anal., 4 (1991), 293-317.  doi: 10.3233/ASY-1991-4402.

[49]

Y. Sone, Kinetic Theory and Fluid Dynamics, Birkhäuser, Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.

[50]

M. S. Steinberg, Differential adhesion in morphogenesis: A modern view, Curr. Opin. Genet. Dev., 17 (2007), 281-286.  doi: 10.1016/j.gde.2007.05.002.

[51]

L. A. TaberY. ShiL. Yang and P. V. Bayly, A poroelastic model for cell crawling including mechanical coupling between cytoskeletal contraction and actin polymerization, Journal of Mechanics of Materials and Structures, 6 (2011), 569-589.  doi: 10.2140/jomms.2011.6.569.

[52]

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Figure 1.  Scheme of the predictions of the linear stability analysis by acting on the interspecies repulsion force
Figure 2.  Form of the potential $ \tilde{\Phi}^{ST}(x) $ for $ R = 2 $ and $ \frac{\nu^{ST}_c}{\nu^{ST}_d} \frac{\kappa^{ST}}{2} = 1 $. The potential is repulsive on its support
Figure 3.  Values of $ \lambda_1 $ (blue curve), $ \lambda_2 $ (orange curve) and their mean (yellow dotted line) near $ z = 0 $ (z = 0.1), plotted as functions of the scaling parameter $ s $ for $ R = 1 $, $ D^A = D^B = 1, c'^{AA} = c'^{AB} = c'^{BA} = 1, c'^{BB} = 10 $
Figure 4.  (I): Values of $ \lambda_1(z) $ as functions of $ z $ for $ R = 1 $, $ D^A = D^B = 1, \, c^{AA} = c^{AB} = c^{BA} = 1, \, c^{BB} = 10 $ and for different values of $ s $ in the instability regime: $ s = 30 $ (blue curve), $ s = 50 $ (orange curve), $ s = 70 $ (yellow curve), and $ s = 90 $ (red curve). (II): same plots for $ \lambda_2(z) $. (III) Plot of $ z^* $ defined in (36) as a function of parameter $ s $
Figure 5.  Microscopic simulations for Cases 1-4 for parameters described in Table 1. $ A $-cells are represented as red disks, $ B $-cells as green disks. For each subsection, the left figure is obtained starting from a homogeneous distribution of particles, the right one from a segregated initial distribution ($ B $-cells on the half-left of the domain, $ A $-cells on the right)
Figure 6.  Macroscopic simulations for Cases 1-4 for parameters described in table 1 for the final time of simulations equal to $ T = 8000 $
Figure 7.  (I) Simulations of the microscopic and macroscopic models for $ \kappa^{AA} = 4, \kappa^{BB} = \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA} = 1 $ for which $ s^* \approx 2.1 $. Simulations of the microscopic model are performed with $ N_A = N_B = 500 $ (IA) and $ N_A = N_B = 2000 $ (IB) particles. Simulations of the macroscopic model correspond to (IC). We consider 7 values of the interspecies repulsion intensity, from left to right: for $ 2.05 = s<s^* $, for $ s^*<s = \{2.15, 2.2, 2.5, 4, 6, 10 \} $. Type B cells are represented in green, type A cells in red. (II) In (IIA-C), we show the values of the quantifiers at time equilibrium as functions of $ s $. Fig. (IA) shows the mean elongation of the green clusters, (IIB) shows the number of green clusters and (IIC) shows the overlapping amount $ Q $ described by (73). Black curves are obtained with the microscopic model for $ N_A = N_B = 500 $ (corresponding to Figures (IA)), yellow curves are for $ N_A = N_B = 2000 $ and correspond to Figures (IB) and red curves are obtained with the macroscopic model (Figures (IC)). The two bottom figures correspond to a zoom of the corresponding curves close to the transition region for $ s $
Figure 8.  Quantifiers computed on the simulation images as functions of the logarithm of the simulation time for $ \kappa^{AA} = 4, \kappa^{BB} = \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA} = 1 $ for the microscopic model with $ N_A = N_B = 500 $ (green curves), $ N_A = N_B = 2000 $ (blue curves), $ N_A = N_B = 4000 $ (yellow curves) and for the macro model (red curve). (I) Green cluster elongation, (II) Number of cell clusters and (III) Overlapping amount $ Q $. On Figure (I), we superimpose linear fits (dotted lines) for short times and large times, showing the two timescales (two slopes) of the micro model compared to the unique timescale of the macro dynamics (single slope)
Table 1.  Model parameters for the inter- and intra- species forces. The value of parameter $ s^* $ has been computed numerically from the formula (43)
value of $ s $ comment
Case I: $ \kappa^{AA}=\kappa^{BB}= {2} $, $ \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA}= {2} $.
IA 0.5 Stable regime ($ s< s^* \approx 1.01 $)
IB 4 Unstable regime ($ s>s^* \approx 1.01 $)
Case II: $ \kappa^{AA}=\kappa^{BB}= 2 $, $ 1=\tilde{\kappa}^{AB}< \tilde{\kappa}^{BA}= 2 $.
IIA 0.5 Stable regime ($ s< s^* \approx 1.43 $)
IIB 4 Unstable regime ($ s>s^* \approx 1.43 $)
Case III: $ 2=\kappa^{AA}>\kappa^{BB}= 1 $, $ \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA}=2 $.
IIIA 0.5 Stable regime ($ s< s^* \approx 0.72 $)
IIIB 4 Unstable regime ($ s>s^* \approx 0.72 $)
Case IV: $ 2=\kappa^{AA}>\kappa^{BB}= 1 $, $ 1=\tilde{\kappa}^{AB}< \tilde{\kappa}^{BA}= 2 $.
IVA 0.5 Stable regime ($ s< s^* \approx 1.02 $)
IVB 4 Unstable regime ($ s>s^* \approx 1.02 $)
value of $ s $ comment
Case I: $ \kappa^{AA}=\kappa^{BB}= {2} $, $ \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA}= {2} $.
IA 0.5 Stable regime ($ s< s^* \approx 1.01 $)
IB 4 Unstable regime ($ s>s^* \approx 1.01 $)
Case II: $ \kappa^{AA}=\kappa^{BB}= 2 $, $ 1=\tilde{\kappa}^{AB}< \tilde{\kappa}^{BA}= 2 $.
IIA 0.5 Stable regime ($ s< s^* \approx 1.43 $)
IIB 4 Unstable regime ($ s>s^* \approx 1.43 $)
Case III: $ 2=\kappa^{AA}>\kappa^{BB}= 1 $, $ \tilde{\kappa}^{AB} = \tilde{\kappa}^{BA}=2 $.
IIIA 0.5 Stable regime ($ s< s^* \approx 0.72 $)
IIIB 4 Unstable regime ($ s>s^* \approx 0.72 $)
Case IV: $ 2=\kappa^{AA}>\kappa^{BB}= 1 $, $ 1=\tilde{\kappa}^{AB}< \tilde{\kappa}^{BA}= 2 $.
IVA 0.5 Stable regime ($ s< s^* \approx 1.02 $)
IVB 4 Unstable regime ($ s>s^* \approx 1.02 $)
Table 2.  Volume fraction of the green family computed on the simulations of FigS. 5-6 at equilibrium for the microscopic model (left column) and for the macroscopic model (right column)
Case VF green cells (microscopic model) VF green cells (macroscopic model)
Homogeneous IC Front-like IC Homogeneous IC Front-like IC
(IB) $ 48.2 \% $ $ 50.0 \% $ $ 49.7\% $ $ 50.0\% $
(IIB) $ 40.2 \% $ $ 42.3 \% $ $ 38.5\% $ $ 42.0\% $
(IIIB) $ 40.6 \% $ $ 42.4 \% $ $ 44.0\% $ $ 46.0\% $
(IVB) $ 34.8 \% $ $ 35.0 \% $ $ 35.9\% $ $ 38.0\% $
Case VF green cells (microscopic model) VF green cells (macroscopic model)
Homogeneous IC Front-like IC Homogeneous IC Front-like IC
(IB) $ 48.2 \% $ $ 50.0 \% $ $ 49.7\% $ $ 50.0\% $
(IIB) $ 40.2 \% $ $ 42.3 \% $ $ 38.5\% $ $ 42.0\% $
(IIIB) $ 40.6 \% $ $ 42.4 \% $ $ 44.0\% $ $ 46.0\% $
(IVB) $ 34.8 \% $ $ 35.0 \% $ $ 35.9\% $ $ 38.0\% $
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