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  • Technical Review
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Computational models for active matter

Nature Reviews Physics volume 2, pages 181–199 (2020)Cite this article

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Abstract

Active matter, which ranges from molecular motors to groups of animals, exists at different length scales and timescales, and various computational models have been proposed to describe and predict its behaviour. The diversity of the methods and the challenges in modelling active matter primarily originate from the out-of-equilibrium character, lack of detailed balance and of time-reversal symmetry, multiscale nature, nonlinearity and multibody interactions. Models exist for both dry active matter and active matter in fluids, and can be agent-based or continuum-level descriptions. They can be generic, emphasizing universal features, or detailed, capturing specific features. We compare various modelling approaches and numerical techniques to illuminate the innovations and challenges in understanding active matter.

Key points

  • Active matter exhibits a wide range of emergent non-equilibrium phenomena, theoretical studies of which often require computer simulations.

  • Active matter encompasses synthetic and living systems, including active gels and the cytoskeleton, cells and tissues, nanorobots and microrobots, synthetic and biological microswimmers, and animal herds.

  • Active matter is characterized by out-of-equilibrium behaviour, nonlinearity, multibody interactions, lack of detailed balance or time-reversal symmetry and, generically, absence of an equation of state.

  • The wide spectrum of systems and phenomena requires a multitude of models and simulation techniques, from agent-based to continuum-level approaches, and combinations thereof.

  • Active agents can interact in many ways, such as volume exclusion, contact attraction, visual information and hydrodynamics. Hydrodynamic interactions are ubiquitous for self-propelled particles in an aqueous environment, which implies a classification into dry and wet active matter.

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Fig. 1: Modelling active matter.
Fig. 2: Models of dry active matter.

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References

  1. Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143–1189 (2013). Comprehensive overview of the hydrodynamic theories of active matter.

    ADS  Google Scholar 

  2. Ramaswamy, S. The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys. 1, 323–345 (2010).

    ADS  Google Scholar 

  3. Toner, J., Tu, Y. & Ramaswamy, S. Hydrodynamics and phases of flocks. Ann. Phys. 318, 170–244 (2005).

    ADS  MathSciNet  MATH  Google Scholar 

  4. Elgeti, J., Winkler, R. G. & Gompper, G. Physics of microswimmers—single particle motion and collective behavior: a review. Rep. Prog. Phys. 78, 056601 (2015). Review of theories, simulations and experiments on microswimmers.

    ADS  MathSciNet  Google Scholar 

  5. Bechinger, C. et al. Active particles in complex and crowded environments. Rev. Mod. Phys. 88, 045006 (2016). Guided tour of artificial self-propelling microparticles and nanoparticles, and their application to the study of nonequilibrium phenomena.

    ADS  MathSciNet  Google Scholar 

  6. Solon, A. P. et al. Pressure is not a state function for generic active fluids. Nat. Phys. 11, 673–678 (2015).

    Google Scholar 

  7. Bialké, J., Speck, T. & Löwen, H. Crystallization in a dense suspension of self-propelled particles. Phys. Rev. Lett. 108, 168301 (2012).

    ADS  Google Scholar 

  8. Redner, G. S., Hagan, M. F. & Baskaran, A. Structure and dynamics of a phase-separating active colloidal fluid. Phys. Rev. Lett. 110, 055701 (2013).

    ADS  Google Scholar 

  9. Cates, M. E. & Tailleur, J. Motility-induced phase separation. Annu. Rev. Condens. Matter Phys. 6, 219–244 (2015). Review of theoretical descriptions of motility-induced phase separation in active matter.

    ADS  Google Scholar 

  10. Wysocki, A., Winkler, R. G. & Gompper, G. Cooperative motion of active Brownian spheres in three-dimensional dense suspensions. Europhys. Lett. 105, 48004 (2014).

    ADS  Google Scholar 

  11. Stenhammar, J., Marenduzzo, D., Allen, R. J. & Cates, M. E. Phase behaviour of active Brownian particles: the role of dimensionality. Soft Matter 10, 1489–1499 (2014).

    ADS  Google Scholar 

  12. Wysocki, A., Winkler, R. G. & Gompper, G. Propagating interfaces in mixtures of active and passive Brownian particles. New J. Phys. 18, 123030 (2016).

    ADS  Google Scholar 

  13. Stenhammar, J., Wittkowski, R., Marenduzzo, D. & Cates, M. E. Activity-induced phase separation and self-assembly in mixtures of active and passive particles. Phys. Rev. Lett. 114, 018301 (2015).

    ADS  Google Scholar 

  14. Digregorio, P. et al. Full phase diagram of active Brownian disks: from melting to motility-induced phase separation. Phys. Rev. Lett. 121, 098003 (2018).

    ADS  Google Scholar 

  15. Fily, Y., Henkes, S. & Marchetti, M. C. Freezing and phase separation of self-propelled disks. Soft Matter 10, 2132–2140 (2014).

    ADS  Google Scholar 

  16. Elgeti, J. & Gompper, G. Wall accumulation of self-propelled spheres. EPL 101, 48003 (2013).

    ADS  Google Scholar 

  17. Fily, Y., Baskaran, A. & Hagan, M. Dynamics of self-propelled particles under strong confinement. Soft Matter 10, 5609–5617 (2014).

    ADS  Google Scholar 

  18. Das, S., Gompper, G. & Winkler, R. G. Local stress and pressure in an inhomogeneous system of spherical active Brownian particles. Sci. Rep. 9, 6608 (2019).

    ADS  Google Scholar 

  19. Wysocki, A. & Rieger, H. Capillary action in scalar active matter. Phys. Rev. Lett. 124, 048001 (2019).

    ADS  Google Scholar 

  20. Takatori, S. C., Yan, W. & Brady, J. F. Swim pressure: stress generation in active matter. Phys. Rev. Lett. 113, 028103 (2014).

    ADS  Google Scholar 

  21. Winkler, R. G., Wysocki, A. & Gompper, G. Virial pressure in systems of active Brownian particles. Soft Matter 11, 6680–6691 (2015).

    ADS  Google Scholar 

  22. Fily, Y., Kafri, Y., Solon, A. P., Tailleur, J. & Turner, A. Mechanical pressure and momentum conservation in dry active matter. J. Phys. A 51, 044003 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  23. Wysocki, A., Elgeti, J. & Gompper, G. Giant adsorption of microswimmers: duality of shape asymmetry and wall curvature. Phys. Rev. E 91, 050302(R) (2015).

    ADS  Google Scholar 

  24. Romanczuk, P., Bär, M., Ebeling, W., Lindner, B. & Schimansky-Geier, L. Active Brownian particles. Eur. Phys. J. Spec. Top. 202, 1–162 (2012).

    Google Scholar 

  25. Nguyen, N. H. P., Klotsa, D., Engel, M. & Glotzer, S. C. Emergent collective phenomena in a mixture of hard shapes through active rotation. Phys. Rev. Lett. 112, 075701 (2014).

    ADS  Google Scholar 

  26. Löwen, H. Chirality in microswimmer motion: from circle swimmers to active turbulence. Eur. Phys. J. Spec. Top. 225, 2319–2331 (2016).

    Google Scholar 

  27. Peruani, F. Active Brownian rods. Eur. Phys. J. Spec. Top. 225, 2301–2317 (2016).

    Google Scholar 

  28. ten Hagen, B. et al. Can the self-propulsion of anisotropic microswimmers be described by using forces and torques? J. Phys. Condens. Matter 27, 194110 (2015).

    ADS  Google Scholar 

  29. Kaiser, A., Babel, S., ten Hagen, B., von Ferber, C. & Löwen, H. How does a flexible chain of active particles swell? J. Chem. Phys. 142, 124905 (2015).

    ADS  Google Scholar 

  30. Eisenstecken, T., Gompper, G. & Winkler, R. G. Conformational properties of active semiflexible polymers. Polymers 8, 304 (2016).

    Google Scholar 

  31. Eisenstecken, T., Gompper, G. & Winkler, R. G. Internal dynamics of semiflexible polymers with active noise. J. Chem. Phys. 146, 154903 (2017).

    ADS  Google Scholar 

  32. Kourbane-Houssene, M., Erignoux, C., Bodineau, T. & Tailleur, J. Exact hydrodynamic description of active lattice gases. Phys. Rev. Lett. 120, 268003 (2018).

    ADS  Google Scholar 

  33. Klamser, J. U., Kapfer, S. C. & Krauth, W. Thermodynamic phases in two-dimensional active matter. Nat. Commun. 9, 5045 (2018).

    ADS  Google Scholar 

  34. Sadjadi, Z., Shaebani, M. R., Rieger, H. & Santen, L. Persistent-random-walk approach to anomalous transport of self-propelled particles. Phys. Rev. E 91, 062715 (2015).

    ADS  MathSciNet  Google Scholar 

  35. Shaebani, M. R., Sadjadi, Z., Sokolov, I. M., Rieger, H. & Santen, L. Anomalous diffusion of self-propelled particles in directed random environments. Phys. Rev. E 90, 030701 (2014).

    ADS  Google Scholar 

  36. Levis, D. & Berthier, L. Clustering and heterogeneous dynamics in a kinetic Monte Carlo model of self-propelled hard disks. Phys. Rev. E 89, 062301 (2014).

    ADS  Google Scholar 

  37. Najafi, J. et al. Flagellar number governs bacterial spreading and transport efficiency. Sci. Adv. 4, eaar6425 (2018).

    ADS  Google Scholar 

  38. Hafner, A. E., Santen, L., Rieger, H. & Shaebani, M. R. Run-and-pause dynamics of cytoskeletal motor proteins. Sci. Rep. 6, 37162 (2016).

    ADS  Google Scholar 

  39. Vicsek, T. & Zafeiris, A. Collective motion. Phys. Rep. 517, 71–140 (2012). Review of collective motion of active agents, from macromolecules through metallic rods and robots to groups of animals and people.

    ADS  Google Scholar 

  40. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. & Shochet, O. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995).

    ADS  MathSciNet  Google Scholar 

  41. Solon, A. P., Chaté, H. & Tailleur, J. From phase to microphase separation in flocking models: the essential role of nonequilibrium fluctuations. Phys. Rev. Lett. 114, 068101 (2015).

    ADS  Google Scholar 

  42. Solon, A. P. & Tailleur, J. Revisiting the flocking transition using active spins. Phys. Rev. Lett. 111, 078101 (2013).

    ADS  Google Scholar 

  43. Ginelli, F., Peruani, F., Bär, M. & Chaté, H. Large-scale collective properties of self-propelled rods. Phys. Rev. Lett. 104, 184502 (2010).

    ADS  Google Scholar 

  44. Chaté, H., Ginelli, F., Grégoire, G., Peruani, F. & Raynaud, F. Modeling collective motion: variations on the vicsek model. Eur. Phys. J. B 64, 451–456 (2008).

    ADS  Google Scholar 

  45. Strömbom, D. Collective motion from local attraction. J. Theor. Biol. 283, 145–151 (2011).

    MathSciNet  MATH  Google Scholar 

  46. Aldana, M., Dossetti, V., Huepe, C., Kenkre, V. M. & Larralde, H. Phase transitions in systems of self-propelled agents and related network models. Phys. Rev. Lett. 98, 095702 (2007).

    ADS  Google Scholar 

  47. Aldana, M., Larralde, H. & Vazquez, B. On the emergence of collective order in swarming systems: a recent debate. Int. J. Mod. Phys. B 23, 3661–3685 (2009).

    ADS  MATH  Google Scholar 

  48. Peruani, F. & Aranson, I. S. Cold active motion: how time-independent disorder affects the motion of self-propelled agents. Phys. Rev. Lett. 120, 238101 (2018).

    ADS  Google Scholar 

  49. Grossman, D., Aranson, I. S. & Jacob, E. B. Emergence of agent swarm migration and vortex formation through inelastic collisions. New J. Phys. 10, 023036 (2008).

    ADS  Google Scholar 

  50. Nagy, M., Daruka, I. & Vicsek, T. New aspects of the continuous phase transition in the scalar noise model (snm) of collective motion. Physica A 373, 445–454 (2007).

    ADS  Google Scholar 

  51. Peruani, F., Klauss, T., Deutsch, A. & Voss-Boehme, A. Traffic jams, gliders, and bands in the quest for collective motion of self-propelled particles. Phys. Rev. Lett. 106, 128101 (2011).

    ADS  Google Scholar 

  52. Ginelli, F. & Chaté, H. Relevance of metric-free interactions in flocking phenomena. Phys. Rev. Lett. 105, 168103 (2010).

    ADS  Google Scholar 

  53. Gregoire, G., Chaté, H. & Tu, Y. Moving and staying together without a leader. Physica D 181, 157–170 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  54. Szabó, P., Nagy, M. & Vicsek, T. Transitions in a self-propelled-particles model with coupling of accelerations. Phys. Rev. E 79, 021908 (2009).

    ADS  Google Scholar 

  55. Chaté, H., Ginelli, F. & Montagne, R. Simple model for active nematics: quasi-long-range order and giant fluctuations. Phys. Rev. Lett. 96, 180602 (2006).

    ADS  Google Scholar 

  56. Mahault, B. et al. Self-propelled particles with velocity reversals and ferromagnetic alignment: active matter class with second-order transition to quasi-long-range polar order. Phys. Rev. Lett. 120, 258002 (2018).

    ADS  Google Scholar 

  57. Szabó, B. et al. Phase transition in the collective migration of tissue cells: experiment and model. Phys. Rev. E 74, 061908 (2006).

    ADS  Google Scholar 

  58. Peruani, F., Deutsch, A. & Bär, M. Nonequilibrium clustering of self-propelled rods. Phys. Rev. E 74, 030904 (2006).

    ADS  Google Scholar 

  59. Toner, J. & Tu, Y. Long-range order in a two-dimensional dynamical XY model: how birds fly together. Phys. Rev. Lett. 75, 4326–4329 (1995).

    ADS  Google Scholar 

  60. Toner, J. & Tu, Y. Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E 58, 4828–4858 (1998).

    ADS  MathSciNet  Google Scholar 

  61. Bertin, E., Droz, M. & Grégoire, G. Boltzmann and hydrodynamic description for self-propelled particles. Phys. Rev. E 74, 022101 (2006).

    ADS  Google Scholar 

  62. Ihle, T. Kinetic theory of flocking: derivation of hydrodynamic equations. Phys. Rev. E 83, 030901 (2011).

    ADS  Google Scholar 

  63. Ranft, J. et al. Fluidization of tissues by cell division and apoptosis. Proc. Natl Acad. Sci. USA 107, 20863–20868 (2010).

    ADS  Google Scholar 

  64. Narayan, V., Ramaswamy, S. & Menon, N. Long-lived giant number fluctuations in a swarming granular nematic. Science 317, 105–108 (2007).

    ADS  Google Scholar 

  65. Ramaswamy, S., Simha, R. A. & Toner, J. Active nematics on a substrate: giant number fluctuations and long-time tails. EPL 62, 196–202 (2003).

    ADS  Google Scholar 

  66. Hemingway, E. J. et al. Active viscoelastic matter: from bacterial drag reduction to turbulent solids. Phys. Rev. Lett. 114, 098302 (2015).

    ADS  Google Scholar 

  67. Schwarz, U. S. & Safran, S. A. Physics of adherent cells. Rev. Mod. Phys. 85, 1327–1381 (2013).

    ADS  Google Scholar 

  68. Hohenberg, P. C. & Halperin, B. I. Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977).

    ADS  Google Scholar 

  69. Wittkowski, R. et al. Scalar \({\phi }^{4}\) field theory for active-particle phase separation. Nat. Commun. 5, 4351 (2014).

  70. Tjhung, E., Nardini, C. & Cates, M. E. Cluster phases and bubbly phase separation in active fluids: reversal of the ostwald process. Phys. Rev. X 8, 031080 (2018).

    Google Scholar 

  71. Stenhammar, J., Tiribocchi, A., Allen, R. J., Marenduzzo, D. & Cates, M. E. Continuum theory of phase separation kinetics for active Brownian particles. Phys. Rev. Lett. 111, 145702 (2013).

    ADS  Google Scholar 

  72. Cates, M. E. & Tailleur, J. When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-induced phase separation. EPL 101, 20010 (2013).

    ADS  Google Scholar 

  73. Tailleur, J. & Cates, M. E. Statistical mechanics of interacting run-and-tumble bacteria. Phys. Rev. Lett. 100, 218103 (2008).

    ADS  Google Scholar 

  74. Purcell, E. M. Life at low Reynolds number. Am. J. Phys. 45, 3–11 (1977).

    ADS  Google Scholar 

  75. Spagnolie, S. E. & Lauga, E. Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J. Fluid Mech. 700, 105–147 (2012).

    ADS  MathSciNet  MATH  Google Scholar 

  76. Winkler, R. G. & Gompper, G. in Handbook of Materials Modeling: Methods: Theory and Modeling (eds Andreoni, W. & Yip, S.) 1–20 (Springer, 2018).

  77. Berke, A. P., Turner, L., Berg, H. C. & Lauga, E. Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101, 038102 (2008).

    ADS  Google Scholar 

  78. Lauga, E. & Powers, T. R. The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601 (2009).

    ADS  MathSciNet  Google Scholar 

  79. Elgeti, J. & Gompper, G. Microswimmers near surfaces. Eur. Phys. J. Spec. Top. 225, 2333–2352 (2016).

    Google Scholar 

  80. Li, G. & Tang, J. X. Accumulation of microswimmers near a surface mediated by collision and rotational Brownian motion. Phys. Rev. Lett. 103, 078101 (2009).

    ADS  Google Scholar 

  81. Elgeti, J. & Gompper, G. Self-propelled rods near surfaces. EPL 85, 38002 (2009).

    ADS  Google Scholar 

  82. Elgeti, J. & Gompper, G. Run-and-tumble dynamics of self-propelled particles in confinement. EPL 109, 58003 (2015).

    ADS  Google Scholar 

  83. Schaar, K., Zöttl, A. & Stark, H. Detention times of microswimmers close to surfaces: influence of hydrodynamic interactions and noise. Phys. Rev. Lett. 115, 038101 (2015).

    ADS  Google Scholar 

  84. Saintillan, D. & Shelley, M. J. Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102 (2007).

    ADS  Google Scholar 

  85. Saintillan, D. & Shelley, M. J. Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys. Rev. Lett. 100, 178103 (2008).

    ADS  Google Scholar 

  86. Sanchez, T., Chen, D. T. N., DeCamp, S. J., Heymann, M. & Dogic, Z. Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–434 (2012).

    ADS  Google Scholar 

  87. Thampi, S. P., Golestanian, R. & Yeomans, J. M. Velocity correlations in an active nematic. Phys. Rev. Lett. 111, 118101 (2013).

    ADS  Google Scholar 

  88. Giomi, L., Bowick, M. J., Ma, X. & Marchetti, M. C. Defect annihilation and proliferation in active nematics. Phys. Rev. Lett. 110, 228101 (2013).

    ADS  Google Scholar 

  89. Keber, F. C. et al. Topology and dynamics of active nematic vesicles. Science 345, 1135–1139 (2014).

    ADS  Google Scholar 

  90. Mathijssen, A. J. T. M., Culver, J., Bhamla, M. S. & Prakash, M. Collective intercellular communication through ultra-fast hydrodynamic trigger waves. Nature 571, 560–564 (2019).

    Google Scholar 

  91. Qiu, T. et al. Swimming by reciprocal motion at low Reynolds number. Nat. Commun. 5, 5119 (2014).

    ADS  Google Scholar 

  92. Qin, B., Gopinath, A., Yang, J., Gollub, J. P. & Arratia, P. E. Flagellar kinematics and swimming of algal cells in viscoelastic fluids. Sci. Rep. 5, 9190 (2015).

    ADS  Google Scholar 

  93. Patteson, A. E., Gopinath, A., Goulian, M. & Arratia, P. E. Running and tumbling with E. coli in polymeric solutions. Sci. Rep. 5, 15761 (2015).

    ADS  Google Scholar 

  94. Li, G. & Ardekani, A. M. Collective motion of microorganisms in a viscoelastic fluid. Phys. Rev. Lett. 117, 118001 (2016).

    ADS  Google Scholar 

  95. Lauga, E. Propulsion in a viscoelastic fluid. Phys. Fluids 19, 083104 (2007).

    ADS  MATH  Google Scholar 

  96. Fu, H. C., Wolgemuth, C. W. & Powers, T. R. Swimming speeds of filaments in nonlinearly viscoelastic fluids. Phys. Fluids 21, 033102 (2009).

    ADS  MATH  Google Scholar 

  97. Spagnolie, S. E., Liu, B. & Powers, T. R. Locomotion of helical bodies in viscoelastic fluids: enhanced swimming at large helical amplitudes. Phys. Rev. Lett. 111, 068101 (2013).

    ADS  Google Scholar 

  98. Man, Y. & Lauga, E. Phase-separation models for swimming enhancement in complex fluids. Phys. Rev. E 92, 023004 (2015).

    ADS  MathSciNet  Google Scholar 

  99. Liu, B., Powers, T. R. & Breuer, K. S. Force-free swimming of a model helical flagellum in viscoelastic fluids. Proc. Natl Acad. Sci. USA 108, 19516–19520 (2011).

    ADS  Google Scholar 

  100. Gagnon, D. A., Keim, N. C. & Arratia, P. E. Undulatory swimming in shear-thinning fluids: experiments with Caenorhabditis elegans. J. Fluid Mech. 758, R3 (2014).

    ADS  MathSciNet  Google Scholar 

  101. Martinez, V. A. et al. Flagellated bacterial motility in polymer solutions. Proc. Natl Acad. Sci. USA 111, 17771–17776 (2014).

    ADS  Google Scholar 

  102. Zöttl, A. & Yeomans, J. M. Enhanced bacterial swimming speeds in macromolecular polymer solutions. Nat. Phys. 15, 554–558 (2019).

    Google Scholar 

  103. McNamara, G. R. & Zanetti, G. Use of the Boltzmann equation to simulate lattice-gas automata. Phys. Rev. Lett. 61, 2332–2335 (1988).

    ADS  Google Scholar 

  104. Dünweg, B. & Ladd, A. J. C. Lattice Boltzmann simulations of soft matter systems. Adv. Polym. Sci. 221, 89–166 (2009).

    Google Scholar 

  105. Español, P. & Warren, P. Statistical mechanics of dissipative particle dynamics. EPL 30, 191–196 (1995).

    ADS  Google Scholar 

  106. Kapral, R. Advances in Chemical Physics (Wiley, 2008).

  107. Gompper, G., Ihle, T., Kroll, D. M. & Winkler, R. G. Multi-particle collision dynamics: a particle-based mesoscale simulation approach to the hydrodynamics of complex fluids. Adv. Polym. Sci. 221, 1–87 (2009).

    Google Scholar 

  108. Ishikawa, T. & Pedley, T. J. Coherent structures in monolayers of swimming particles. Phys. Rev. Lett. 100, 088103 (2008).

    ADS  Google Scholar 

  109. Mathijssen, A. J. T. M., Doostmohammadi, A., Yeomans, J. M. & Shendruk, T. N. Hydrodynamics of micro-swimmers in films. J. Fluid Mech. 806, 35–70 (2016).

    ADS  MathSciNet  MATH  Google Scholar 

  110. Singh, R., Ghose, S. & Adhikari, R. Many-body microhydrodynamics of colloidal particles with active boundary layers. J. Stat. Mech. Theor. Exp. 2015, P06017 (2015).

    MathSciNet  Google Scholar 

  111. Elgeti, J., Kaupp, U. B. & Gompper, G. Hydrodynamics of sperm cells near surfaces. Biophys. J. 99, 1018–1026 (2010).

    ADS  Google Scholar 

  112. Hu, J., Yang, M., Gompper, G. & Winkler, R. G. Modelling the mechanics and hydrodynamics of swimming E. coli. Soft Matter 11, 7867–7876 (2015).

    ADS  Google Scholar 

  113. Watari, N. & Larson, R. G. The hydrodynamics of a run-and-tumble bacterium propelled by polymorphic helical flagella. Biophys. J. 98, 12–17 (2010).

    ADS  Google Scholar 

  114. Shum, H.,Gaffney, E. A. & Smith D. J. Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry. Proc. R. Soc. A 466, 1725–1748 (2010).

    ADS  MathSciNet  MATH  Google Scholar 

  115. Pimponi, D., Chinappi, M., Gualtieri, P. & Casciola, C. M. Hydrodynamics of flagellated microswimmers near free-slip interfaces. J. Fluid Mech. 789, 514–533 (2016).

    ADS  MathSciNet  Google Scholar 

  116. Lighthill, M. J. On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Comm. Pure Appl. Math. 5, 109–118 (1952).

    MathSciNet  MATH  Google Scholar 

  117. Blake, J. R. A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199–208 (1971).

    ADS  MATH  Google Scholar 

  118. Ishikawa, T., Simmonds, M. P. & Pedley, T. J. Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119–160 (2006).

    ADS  MathSciNet  MATH  Google Scholar 

  119. Pedley, T. J. Spherical squirmers: models for swimming micro-organisms. IMA J. Appl. Math. 81, 488–521 (2016).

    MathSciNet  MATH  Google Scholar 

  120. Llopis, I. & Pagonabarraga, I. Hydrodynamic interactions in squirmer motion: swimming with a neighbour and close to a wall. J. Nonnewton. Fluid Mech. 165, 946–952 (2010).

    MATH  Google Scholar 

  121. Götze, I. O. & Gompper, G. Mesoscale simulations of hydrodynamic squirmer interactions. Phys. Rev. E 82, 041921 (2010).

    ADS  Google Scholar 

  122. Evans, A. A., Ishikawa, T., Yamaguchi, T. & Lauga, E. Orientational order in concentrated suspensions of spherical microswimmers. Phys. Fluids 23, 111702 (2011).

    ADS  Google Scholar 

  123. Alarcon, F. & Pagonabarraga, I. Spontaneous aggregation and global polar ordering in squirmer suspensions. J. Mol. Liq. 185, 56–61 (2013).

    Google Scholar 

  124. Molina, J. J., Nakayama, Y. & Yamamoto, R. Hydrodynamic interactions of self-propelled swimmers. Soft Matter 9, 4923–4936 (2013).

    ADS  Google Scholar 

  125. Yoshinaga, N. & Liverpool, T. B. Hydrodynamic interactions in dense active suspensions: from polar order to dynamical clusters. Phys. Rev. E 96, 020603 (2017).

    ADS  Google Scholar 

  126. Ishimoto, K. & Gaffney, E. A. Squirmer dynamics near a boundary. Phys. Rev. E 88, 062702 (2013).

    ADS  Google Scholar 

  127. Lintuvuori, J. S., Brown, A. T., Stratford, K. & Marenduzzo, D. Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls. Soft Matter 12, 7959–7968 (2016).

    ADS  Google Scholar 

  128. Theers, M., Westphal, E., Gompper, G. & Winkler, R. G. Modeling a spheroidal microswimmer and cooperative swimming in a narrow slit. Soft Matter 12, 7372–7385 (2016).

    ADS  Google Scholar 

  129. Theers, M., Westphal, E., Qi, K., Winkler, R. G. & Gompper, G. Clustering of microswimmers: interplay of shape and hydrodynamics. Soft Matter 14, 8590–8603 (2018).

    ADS  Google Scholar 

  130. Keller, S. R. & Wu, T. Y. A porous prolate-spheroidal model for ciliated micro-organisms. J. Fluid Mech. 80, 259–278 (1977).

    ADS  MATH  Google Scholar 

  131. Theers, M., Westphal, E., Gompper, G. & Winkler, R. G. From local to hydrodynamic friction in Brownian motion: a multiparticle collision dynamics simulation study. Phys. Rev. E 93, 032604 (2016).

    ADS  Google Scholar 

  132. Nash, R. W., Adhikari, R., Tailleur, J. & Cates, M. E. Run-and-tumble particles with hydrodynamics: sedimentation, trapping, and upstream swimming. Phys. Rev. Lett. 104, 258101 (2010).

    ADS  Google Scholar 

  133. Hernandez-Ortiz, J. P., Stoltz, C. G. & Graham, M. D. Transport and collective dynamics in suspensions of confined swimming particles. Phys. Rev. Lett. 95, 204501 (2005).

    ADS  Google Scholar 

  134. de Graaf, J. et al. Lattice Boltzmann hydrodynamics of anisotropic active matter. J. Chem. Phys. 144, 134106 (2016).

    ADS  Google Scholar 

  135. Menzel, A. M., Saha, A., Hoell, C. & Löwen, H. Dynamical density functional theory for microswimmers. J. Chem. Phys. 144, 024115 (2016).

    ADS  Google Scholar 

  136. Lighthill, J. Flagellar hydrodynamics. SIAM Rev. 18, 161–230 (1976).

    MathSciNet  MATH  Google Scholar 

  137. Saggiorato, G. et al. Human sperm steer with second harmonics of the flagellar beat. Nat. Commun. 8, 1415 (2017).

    ADS  Google Scholar 

  138. Shum, H. & Gaffney, E. A. Hydrodynamic analysis of flagellated bacteria swimming near one and between two no-slip plane boundaries. Phys. Rev. E 91, 033012 (2015).

    ADS  Google Scholar 

  139. Lauga, E. Bacterial hydrodynamics. Annu. Rev. Fluid Mech. 48, 105–130 (2016). Comprehensive review of the hydrodynamics of bacteria.

    ADS  MathSciNet  MATH  Google Scholar 

  140. Rode, S., Elgeti, J. & Gompper, G. Sperm motility in modulated microchannels. New J. Phys. 21, 013016 (2019).

    ADS  Google Scholar 

  141. Reigh, S. Y., Winkler, R. G. & Gompper, G. Synchronization and bundling of anchored bacterial flagella. Soft Matter 8, 4363–4372 (2012).

    ADS  Google Scholar 

  142. Reichert, M. & Stark, H. Synchronization of rotating helices by hydrodynamic interactions. Eur. Phys. J. E 17, 493–500 (2005).

    Google Scholar 

  143. Vogel, R. & Stark, H. Motor-driven bacterial flagella and buckling instabilities. Eur. Phys. J. E 35, 15 (2012).

    Google Scholar 

  144. Janssen, P. J. A. & Graham, M. D. Coexistence of tight and loose bundled states in a model of bacterial flagellar dynamics. Phys. Rev. E 84, 011910 (2011).

    ADS  Google Scholar 

  145. Hu, J., Wysocki, A., Winkler, R. G. & Gompper, G. Physical sensing of surface properties by microswimmers - directing bacterial motion via wall slip. Sci. Rep. 5, 9586 (2015).

    ADS  Google Scholar 

  146. Lemelle, L., Palierne, J.-F., Chatre, E., Vaillant, C. & Place, C. Curvature reversal of the circular motion of swimming bacteria probes for slip at solid/liquid interfaces. Soft Matter 9, 9759–9762 (2013).

    ADS  Google Scholar 

  147. Lauga, E., DiLuzio, W. R., Whitesides, G. M. & Stone, H. A. Swimming in circles: motion of bacteria near solid boundaries. Biophys. J. 90, 400–412 (2006).

    ADS  Google Scholar 

  148. Di Leonardo, R., Dell Arciprete, D., Angelani, L. & Iebba, V. Swimming with an image. Phys. Rev. Lett. 106, 038101 (2011).

    ADS  Google Scholar 

  149. Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R. E. & Kessler, J. O. Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93, 098103 (2004).

    ADS  Google Scholar 

  150. Matas-Navarro, R., Golestanian, R., Liverpool, T. B. & Fielding, S. M. Hydrodynamic suppression of phase separation in active suspensions. Phys. Rev. E 90, 032304 (2014).

    ADS  Google Scholar 

  151. Gaspard, P. & Kapral, R. Thermodynamics and statistical mechanics of chemically powered synthetic nanomotors. Adv. Phys. X 4, 1602480 (2019).

    Google Scholar 

  152. Bayati, P., Popescu, M. N., Uspal, W. E., Dietrich, S. & Najafi, A. Dynamics near planar walls for various model self-phoretic particles. Soft Matter 15, 5644–5672 (2019).

    ADS  Google Scholar 

  153. Rückner, G. & Kapral, R. Chemically powered nanodimers. Phys. Rev. Lett. 98, 150603 (2007).

    ADS  Google Scholar 

  154. Yang, M. & Ripoll, M. Simulations of thermophoretic nanoswimmers. Phys. Rev. E 84, 061401 (2011).

    ADS  Google Scholar 

  155. Saha, S., Golestanian, R. & Ramaswamy, S. Clusters, asters, and collective oscillations in chemotactic colloids. Phys. Rev. E 89, 062316 (2014).

    ADS  Google Scholar 

  156. Michelin, S. & Lauga, E. Phoretic self-propulsion at finite Péclet numbers. J. Fluid Mech. 747, 572 (2014).

    ADS  MathSciNet  MATH  Google Scholar 

  157. Liebchen, B., Marenduzzo, D., Pagonabarraga, I. & Cates, M. E. Clustering and pattern formation in chemorepulsive active colloids. Phys. Rev. Lett. 115, 258301 (2015).

    ADS  Google Scholar 

  158. Stark, H. Artificial chemotaxis of self-phoretic active colloids: collective behavior. Acc. Chem. Res. 51, 2681–2688 (2018).

    Google Scholar 

  159. Moran, J. L. & Posner, J. D. Phoretic self-propulsion. Annu. Rev. Fluid Mech. 49, 511–540 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  160. Howse, J. R. et al. Self-motile colloidal particles: from directed propulsion to random walk. Phys. Rev. Lett. 99, 048102 (2007).

    ADS  Google Scholar 

  161. Uspal, W. E., Popescu, M. N., Dietrich, S. & Tasinkevych, M. Self-propulsion of a catalytically active particle near a planar wall: from reflection to sliding and hovering. Soft Matter 11, 434 (2015).

    ADS  Google Scholar 

  162. Ishimoto, K. & Gaffney, E. A. Fluid flow and sperm guidance: a simulation study of hydrodynamic sperm rheotaxis. J. Roy. Soc. Interface 12, 20150172 (2015).

    Google Scholar 

  163. Koh, J. B. Y., Shen, X. & Marcos. Theoretical modeling in microscale locomotion. Microfluid. Nanofluid. 20, 98 (2016).

    Google Scholar 

  164. Uspal, W. E., Popescu, M. N., Dietrich, S. & Tasinkevych, M. Rheotaxis of spherical active particles near a planar wall. Soft Matter 11, 6613–6632 (2015).

    ADS  Google Scholar 

  165. Mathijssen, A. et al. Oscillatory surface rheotaxis of swimming E. coli bacteria. Nat. Commun. 20, 3434 (2019).

    ADS  Google Scholar 

  166. Friedrich, B. M. & Jülicher, F. Chemotaxis of sperm cells. Proc. Natl Acad. Sci. USA 104, 13256 (2007).

    ADS  Google Scholar 

  167. Tu, Y. Quantitative modeling of bacterial chemotaxis: signal amplification and accurate adaptation. Annu. Rev. Biophys. 42, 337–359 (2013).

    Google Scholar 

  168. Camley, B. A., Zimmermann, J., Levine, H. & Rappel, W.-J. Emergent collective chemotaxis without single-cell gradient sensing. Phys. Rev. Lett. 116, 098101 (2016).

    ADS  Google Scholar 

  169. ten Hagen, B. et al. Gravitaxis of asymmetric self-propelled colloidal particles. Nat. Commun. 5, 4829 (2014).

    ADS  Google Scholar 

  170. Kuhr, J.-T., Blaschke, J., Rühle, F. & Stark, H. Collective sedimentation of squirmers under gravity. Soft Matter 13, 7548–7555 (2017).

    ADS  Google Scholar 

  171. Cohen, J. A. & Golestanian, R. Emergent cometlike swarming of optically driven thermally active colloids. Phys. Rev. Lett. 112, 068302 (2014).

    ADS  Google Scholar 

  172. Martin, P. C., Parodi, O. & Pershan, P. S. Unified hydrodynamic theory for crystals, liquid crystals, and normal fluids. Phys. Rev. A 6, 2401–2420 (1972).

    ADS  Google Scholar 

  173. Prost, J., Jülicher, F. & Joanny, J.-F. Active gel phys. Nat. Phys. 11, 111–117 (2015). Introduction to active-gel models for actomyosin.

    Google Scholar 

  174. Jülicher, F., Grill, S. W. & Salbreux, G. Hydrodynamic theory of active matter. Rep. Prog. Phys. 81, 076601 (2018).

    ADS  MathSciNet  Google Scholar 

  175. Carenza, L. N., Gonnella, G., Lamura, A., Negro, G. & Tiribocchi, A. Lattice Boltzmann methods and active fluids. Eur. Phys. J. E 42, 81 (2019).

    Google Scholar 

  176. AditiSimha, R. & Ramaswamy, S. Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Phys. Rev. Lett. 89, 058101 (2002).

    ADS  Google Scholar 

  177. Hatwalne, Y., Ramaswamy, S., Rao, M. & Simha, R. A. Rheology of active-particle suspensions. Phys. Rev. Lett. 92, 118101 (2004).

    ADS  Google Scholar 

  178. Kruse, K., Joanny, J. F., Jülicher, F., Prost, J. & Sekimoto, K. Asters, vortices, and rotating spirals in active gels of polar filaments. Phys. Rev. Lett. 92, 078101 (2004).

    ADS  Google Scholar 

  179. Giomi, L., Marchetti, M. C. & Liverpool, T. B. Complex spontaneous flows and concentration banding in active polar films. Phys. Rev. Lett. 101, 198101 (2008).

    ADS  Google Scholar 

  180. Baskaran, A. & Marchetti, M. C. Statistical mechanics and hydrodynamics of bacterial suspensions. Proc. Natl Acad. Sci. USA 106, 15567–15572 (2009).

    ADS  Google Scholar 

  181. Linkmann, M., Marchetti, M. C., Boffetta, G. & Eckhardt, B. Condensate formation and multiscale dynamics in two-dimensional active suspensions. Preprint at arXiv https://arxiv.org/abs/1905.06267 (2019).

  182. Marenduzzo, D., Orlandini, E., Cates, M. E. & Yeomans, J. M. Steady-state hydrodynamic instabilities of active liquid crystals: hybrid lattice Boltzmann simulations. Phys. Rev. E 76, 031921 (2007).

    ADS  Google Scholar 

  183. Marenduzzo, D., Orlandini, E. & Yeomans, J. M. Hydrodynamics and rheology of active liquid crystals: a numerical investigation. Phys. Rev. Lett. 98, 118102 (2007).

    ADS  Google Scholar 

  184. Giomi, L., Mahadevan, L., Chakraborty, B. & Hagan, M. F. Excitable patterns in active nematics. Phys. Rev. Lett. 106, 218101 (2011).

    ADS  Google Scholar 

  185. Doostmohammadi, A., Ignes-Mullol, J., Yeomans, J. M. & Sagues, F. Active nematics. Nat. Commun. 9, 3246 (2018).

    ADS  Google Scholar 

  186. Dunkel, J., Heidenreich, S., Bär, M. & Goldstein, R. E. Minimal continuum theories of structure formation in dense active fluids. New J. Phys. 15, 045016 (2013).

    ADS  Google Scholar 

  187. Slomka, J. & Dunkel, J. Generalized Navier-Stokes equations for active suspensions. Eur. Phys. J. Spec. Top. 224, 1349–1358 (2015).

    Google Scholar 

  188. Dunkel, J. et al. Fluid dynamics of bacterial turbulence. Phys. Rev. Lett. 110, 228102 (2013).

    ADS  Google Scholar 

  189. Wensink, H. H. et al. Meso-scale turbulence in living fluids. Proc. Natl Acad. Sci. USA 109, 14308–14313 (2012).

    ADS  MATH  Google Scholar 

  190. Slomka, J. & Dunkel, J. Geometry-dependent viscosity reduction in sheared active fluids. Phys. Rev. Fluids 2, 043102 (2017).

    ADS  Google Scholar 

  191. Slomka, J. & Dunkel, J. Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids. Proc. Natl Acad. Sci. USA 114, 2119–2124 (2017).

    MathSciNet  MATH  Google Scholar 

  192. Tiribocchi, A., Wittkowski, R., Marenduzzo, D. & Cates, M. E. Active model H: scalar active matter in a momentum-conserving fluid. Phys. Rev. Lett. 115, 188302 (2015).

    ADS  Google Scholar 

  193. Cates, M. E. Active field theories. Preprint at arXiv https://arxiv.org/abs/1904.01330 (2019).

  194. Mogilner, A. & Oster, G. Cell motility driven by actin polymerization. Biophys. J. 71, 3030–3045 (1996).

    ADS  Google Scholar 

  195. Dogterom, M., Kerssemakers, J. W., Romet-Lemonne, G. & Janson, M. E. Force generation by dynamic microtubules. Curr. Opin. Cell Biol. 17, 67–74 (2005).

    Google Scholar 

  196. Mogilner, A. Mathematics of cell motility: have we got its number? J. Math. Biol. 58, 105 (2008).

    MathSciNet  MATH  Google Scholar 

  197. Erlenkämper, C. & Kruse, K. Uncorrelated changes of subunit stability can generate length-dependent disassembly of treadmilling filaments. Phys. Biol. 6, 046016 (2009).

    ADS  Google Scholar 

  198. Howard, J. & Kruse, K. Mechanics of Motor Proteins and the Cytoskeleton (Sinauer Associates, 2001).

  199. Nielsen, S. O., Bulo, R. E., Moore, P. B. & Ensing, B. Recent progress in adaptive multiscale molecular dynamics simulations of soft matter. Phys. Chem. Chem. Phys. 12, 12401–12414 (2010).

    Google Scholar 

  200. Ekimoto, T. & Ikeguchi, M. Multiscale molecular dynamics simulations of rotary motor proteins. Biophys. Rev. 10, 605–615 (2018).

    Google Scholar 

  201. Kolomeisky, A. B. & Fisher, M. E. Molecular motors: a theorist’s perspective. Annu. Rev. Phys. Chem. 58, 675–695 (2007).

    ADS  Google Scholar 

  202. Chowdhury, D. Stochastic mechano-chemical kinetics of molecular motors: a multidisciplinary enterprise from a physicist’s perspective. Phys. Rep. 529, 1–197 (2013). Physicist’s view on molecular motors.

    ADS  Google Scholar 

  203. Klumpp, S. & Lipowsky, R. Cooperative cargo transport by several molecular motors. Proc. Natl Acad. Sci. USA 102, 17284–17289 (2005).

    ADS  Google Scholar 

  204. Appert-Rolland, C., Ebbinghaus, M. & Santen, L. Intracellular transport driven by cytoskeletal motors: general mechanisms and defects. Phys. Rep. 593, 1–59 (2015).

    ADS  MathSciNet  Google Scholar 

  205. Bausch, A. R. & Kroy, K. A bottom-up approach to cell mechanics. Nat. Phys. 2, 231–238 (2006).

    Google Scholar 

  206. Huber, F. et al. Emergent complexity of the cytoskeleton: from single filaments to tissue. Adv. Phys. 62, 1–112 (2013).

    ADS  Google Scholar 

  207. Broedersz, C. P. & MacKintosh, F. C. Modeling semiflexible polymer networks. Rev. Mod. Phys. 86, 995–1036 (2014).

    ADS  Google Scholar 

  208. Mohapatra, L., Goode, B. L., Jelenkovic, P., Phillips, R. & Kondev, J. Design principles of length control of cytoskeletal structures. Annu. Rev. Biophys. 45, 85–116 (2016).

    Google Scholar 

  209. Mogilner, A. & Craig, E. Towards a quantitative understanding of mitotic spindle assembly and mechanics. J. Cell Sci. 123, 3435–3445 (2010).

    Google Scholar 

  210. Pavin, N. & Tolić, I. M. Self-organization and forces in the mitotic spindle. Annu. Rev. Biophys. 45, 279–298 (2016).

    Google Scholar 

  211. Broedersz, C. P. & MacKintosh, F. C. Molecular motors stiffen non-affine semiflexible polymer networks. Soft Matter 7, 3186–3191 (2011).

    ADS  Google Scholar 

  212. Ronceray, P., Broedersz, C. P. & Lenz, M. Fiber networks amplify active stress. Proc. Natl Acad. Sci. USA 113, 2827–2832 (2016).

    ADS  Google Scholar 

  213. Jülicher, F., Kruse, K., Prost, J. & Joanny, J.-F. Active behavior of the cytoskeleton. Phys. Rep. 449, 3–28 (2007).

    ADS  MathSciNet  Google Scholar 

  214. Joanny, J. F. & Prost, J. Active gels as a description of the actin–myosin cytoskeleton. HFSP J. 3, 94–104 (2009).

    Google Scholar 

  215. Nedelec, F. & Foethke, D. Collective langevin dynamics of flexible cytoskeletal fibers. New J. Phys. 9, 427–427 (2007).

    ADS  Google Scholar 

  216. Jilkine, A. & Edelstein-Keshet, L. A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues. PLoS Comput. Biol. 7, e1001121 (2011).

    ADS  MathSciNet  Google Scholar 

  217. Holmes, W. R. & Edelstein-Keshet, L. A comparison of computational models for eukaryotic cell shape and motility. PLoS Comput. Biol. 8, e1002793 (2012).

    ADS  Google Scholar 

  218. Danuser, G., Allard, J. & Mogilner, A. Mathematical modeling of eukaryotic cell migration: insights beyond experiments. Annu. Rev. Cell Dev. Biol. 29, 501–528 (2013). Comprehensive overview of mathematical modelling of cell migration.

    Google Scholar 

  219. te Boekhorst, V., Preziosi, L. & Friedl, P. Plasticity of cell migration in vivo and in silico. Annu. Rev. Cell Dev. Biol. 32, 491–526 (2016).

    Google Scholar 

  220. Doubrovinski, K. & Kruse, K. Cell motility resulting from spontaneous polymerization waves. Phys. Rev. Lett. 107, 258103 (2011).

    ADS  Google Scholar 

  221. Wolgemuth, C. W., Stajic, J. & Mogilner, A. Redundant mechanisms for stable cell locomotion revealed by minimal models. Biophys. J. 101, 545–553 (2011).

    ADS  Google Scholar 

  222. Ziebert, F., Swaminathan, S. & Aranson, I. S. Model for self-polarization and motility of keratocyte fragments. J. R. Soc. Interface 9, 1084–1092 (2012).

    Google Scholar 

  223. Ziebert, F. & Aranson, I. S. Computational approaches to substrate-based cell motility. NPJ Comput. Mater. 2, 16019 (2016).

    ADS  Google Scholar 

  224. Linsmeier, I. et al. Disordered actomyosin networks are sufficient to produce cooperative and telescopic contractility. Nat. Commun. 7, 12615 (2016).

    ADS  Google Scholar 

  225. Singer-Loginova, I. & Singer, H. M. The phase field technique for modeling multiphase materials. Rep. Prog. Phys. 71, 106501 (2008).

    ADS  Google Scholar 

  226. Nonomura, M. Study on multicellular systems using a phase field model. PLoS One 7, 1–9 (2012).

    Google Scholar 

  227. Camley, B. A. et al. Polarity mechanisms such as contact inhibition of locomotion regulate persistent rotational motion of mammalian cells on micropatterns. Proc. Natl Acad. Sci. USA 111, 14770–14775 (2014).

    ADS  Google Scholar 

  228. Najem, S. & Grant, M. Phase-field model for collective cell migration. Phys. Rev. E 93, 052405 (2016).

    ADS  MathSciNet  Google Scholar 

  229. Camley, B. A. & Rappel, W.-J. Physical models of collective cell motility: from cell to tissue. J. Phys. D 50, 113002 (2017). Overview of physical models describing collective motion of cells and tissues.

    ADS  Google Scholar 

  230. Mueller, R., Yeomans, J. M. & Doostmohammadi, A. Emergence of active nematic behavior in monolayers of isotropic cells. Phys. Rev. Lett. 122, 048004 (2019).

    ADS  Google Scholar 

  231. Wenzel, D., Praetorius, S. & Voigt, A. Topological and geometrical quantities in active cellular structures. J. Chem. Phys. 150, 164108 (2019).

    ADS  Google Scholar 

  232. Abaurrea-Velasco, C., Ghahnaviyeh, S. D., Pishkenari, H. N., Auth, T. & Gompper, G. Complex self-propelled rings: a minimal model for cell motility. Soft Matter 13, 5865–5876 (2017).

    ADS  Google Scholar 

  233. Abaurrea-Velasco, C., Auth, T. & Gompper, G. Vesicles with internal active filaments: self-organized propulsion controls shape, motility, and dynamical response. New J. Phys. https://doi.org/10.1088/1367-2630/ab5c70 (2019).

  234. Preziosi, L., Ambrosi, D. & Verdier, C. An elasto-visco-plastic model of cell aggregates. J. Theor. Biol. 262, 35–47 (2010).

    MathSciNet  MATH  Google Scholar 

  235. Rodriguez, E. K., Hoger, A. & McCulloch, A. D. Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27, 455–467 (1994).

    Google Scholar 

  236. Drasdo, D. & Höhme, S. A single-cell-based model of tumor growth in vitro: monolayers and spheroids. Phys. Biol. 2, 133–147 (2005).

    ADS  Google Scholar 

  237. Basan, M., Prost, J., Joanny, J.-F. & Elgeti, J. Dissipative particle dynamics simulations for biological tissues: rheology and competition. Phys. Biol. 8, 026014 (2011).

    ADS  Google Scholar 

  238. Malmi-Kakkada, A. N., Li, X., Samanta, H. S., Sinha, S. & Thirumalai, D. Cell growth rate dictates the onset of glass to fluidlike transition and long time superdiffusion in an evolving cell colony. Phys. Rev. X 8, 021025 (2018).

    Google Scholar 

  239. Matoz-Fernandez, D. A., Martens, K., Sknepnek, R., Barrat, J. L. & Henkes, S. Cell division and death inhibit glassy behaviour of confluent tissues. Soft Matter 13, 3205–3212 (2017).

    ADS  Google Scholar 

  240. Graner, F. & Glazier, J. A. Simulation of biological cell sorting using a two-dimensional extended potts model. Phys. Rev. Lett. 69, 2013–2016 (1992).

    ADS  Google Scholar 

  241. Glazier, J. A. & Graner, F. Simulation of the differential adhesion driven rearrangement of biological cells. Phys. Rev. E 47, 2128–2154 (1993).

    ADS  Google Scholar 

  242. Chen, N., Glazier, J. A., Izaguirre, J. A. & Alber, M. S. A parallel implementation of the cellular Potts model for simulation of cell-based morphogenesis. Comput. Phys. Commun. 176, 670–681 (2007).

    ADS  Google Scholar 

  243. Maree, A. F. M. & Hogeweg, P. How amoeboids self-organize into a fruiting body: multicellular coordination in dictyostelium discoideum. Proc. Natl Acad. Sci. USA 98, 3879–3883 (2001).

    ADS  Google Scholar 

  244. Merks, R. M. H., Perryn, E. D., Shirinifard, A. & Glazier, J. A. Contact-inhibited chemotaxis in de novo and sprouting blood-vessel growth. PLoS Comp. Biol. 4, e1000163 (2008).

    ADS  MathSciNet  Google Scholar 

  245. Maree, A. F. M., Jilkine, A., Dawes, A., Grieneisen, V. A. & Edelstein-Keshet, L. Polarization and movement of keratocytes: a multiscale modelling approach. Bull. Math. Biol. 68, 1169–1211 (2006).

    MATH  Google Scholar 

  246. Albert, P. J. & Schwarz, U. S. Dynamics of cell shape and forces on micropatterned substrates predicted by a cellular potts model. Biophys. J. 106, 2340–2352 (2014).

    ADS  Google Scholar 

  247. Segerer, F. J., Thöroff, F., PieraAlberola, A., Frey, E. & Rädler, J. O. Emergence and persistence of collective cell migration on small circular micropatterns. Phys. Rev. Lett. 114, 228102 (2015).

    ADS  Google Scholar 

  248. Hufnagel, L., Teleman, A. A., Rouault, H., Cohen, S. M. & Shraiman, B. I. On the mechanism of wing size determination in fly development. Proc. Natl Acad. Sci. USA 104, 3835–3840 (2007).

    ADS  Google Scholar 

  249. Fletcher, A. G., Osterfield, M., Baker, R. E. & Shvartsman, S. Y. Vertex models of epithelial morphogenesis. Biophys. J. 106, 2291–2304 (2014).

    ADS  Google Scholar 

  250. Sussman, D. M., Schwarz, J. M., Marchetti, M. C. & Manning, M. L. Soft yet sharp interfaces in a vertex model of confluent tissue. Phys. Rev. Lett. 120, 058001 (2018).

    ADS  Google Scholar 

  251. Barton, D. L., Henkes, S., Weijer, C. J. & Sknepnek, R. Active vertex model for cell-resolution description of epithelial tissue mechanics. PLoS Comput. Biol. 13, e1005569 (2017).

    ADS  Google Scholar 

  252. Chiang, M. & Marenduzzo, D. Glass transitions in the cellular Potts model. Europhys. Lett. 116, 28009 (2016).

    ADS  Google Scholar 

  253. Bi, D., Yang, X., Marchetti, M. C. & Manning, M. L. Motility-driven glass and jamming transitions in biological tissues. Phys. Rev. X 6, 021011 (2016).

    Google Scholar 

  254. Oswald, L., Grosser, S., Smith, D. M. & Käs, J. A. Jamming transitions in cancer. J. Phys. D 50, 483001 (2017). Review of tissue dynamics and liquid-like versus solid-like behaviour.

    ADS  Google Scholar 

  255. Fung, Y.-C. Biomechanics: Mechanical Properties of Living Tissues 2nd edn (Springer, 2010).

  256. Dunlop, J. W. C., Fischer, F. D., Gamsjäger, E. & Fratzl, P. A theoretical model for tissue growth in confined geometries. J. Mech. Phys. Solids 58, 1073–1087 (2010).

    ADS  MathSciNet  MATH  Google Scholar 

  257. Ambrosi, D., Preziosi, L. & Vitale, G. The interplay between stress and growth in solid tumors. Mech. Res. Commun. 42, 87–91 (2012).

    Google Scholar 

  258. Shraiman, B. I. Mechanical feedback as a possible regulator of tissue growth. Proc. Natl Acad. Sci. USA 102, 3318–3323 (2005).

    ADS  Google Scholar 

  259. Byrne, H. & Preziosi, L. Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20, 341–366 (2003).

    MATH  Google Scholar 

  260. Tracqui, P. Biophysical models of tumour growth. Rep. Prog. Phys. 72, 056701 (2009).

    ADS  Google Scholar 

  261. Rieger, H. & Welter, M. Integrative models of vascular remodeling during tumor growth. WIREs Syst. Biol. Med. 7, 113–129 (2015).

    Google Scholar 

  262. Fredrich, T., Rieger, H., Chignola, R. & Milotti, E. Fine-grained simulations of the microenvironment of vascularized tumours. Sci. Rep. 9, 11698 (2019).

    ADS  Google Scholar 

  263. Romanczuk, P., Couzin, I. D. & Schimansky-Geier, L. Collective motion due to individual escape and pursuit response. Phys. Rev. Lett. 102, 010602 (2009).

    ADS  Google Scholar 

  264. Simpson, S. J., Sword, G. A., Lorch, P. D. & Couzin, I. D. Cannibal crickets on a forced march for protein and salt. Proc. Natl Acad. Sci. USA 103, 4152–4156 (2006).

    ADS  Google Scholar 

  265. Agudo-Canalejo, J. & Golestanian, R. Active phase separation in mixtures of chemically interacting particles. Phys. Rev. Lett. 123, 018101 (2019).

    ADS  Google Scholar 

  266. Pearce, D. J. G., Miller, A. M., Rowlands, G. & Turner, M. S. Role of projection in the control of bird flocks. Proc. Natl Acad. Sci. USA 111, 10422–10426 (2014).

    ADS  Google Scholar 

  267. Lavergne, F. A., Wendehenne, H., Bäuerle, T. & Bechinger, C. Group formation and cohesion of active particles with visual perception-dependent motility. Science 364, 70–74 (2019).

    ADS  Google Scholar 

  268. Ballerini, M. et al. Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. Natl Acad. Sci. USA 105, 1232–1237 (2008).

    ADS  Google Scholar 

  269. Bajec, I. L. & Heppner, F. H. Organized flight in birds. Animal Behav. 78, 777–789 (2009).

    Google Scholar 

  270. Mijalkov, M., McDaniel, A., Wehr, J. & Volpe, G. Engineering sensorial delay to control phototaxis and emergent collective behaviors. Phys. Rev. X 6, 011008 (2016).

    Google Scholar 

  271. Charlesworth, H. J. & Turner, M. S. Intrinsically motivated collective motion. Proc. Natl Acad. Sci. USA 116, 15362–15367 (2019).

    ADS  Google Scholar 

  272. Khadka, U., Holubec, V., Yang, H. & Cichos, F. Active particles bound by information flows. Nat. Commun. 9, 3864 (2018).

    ADS  Google Scholar 

  273. Mann, R. P. & Garnett, R. The entropic basis of collective behaviour. J. R. Soc. Interface 12, 20150037 (2015).

    Google Scholar 

  274. Ward, A. J. W., Sumpter, D. J. T., Couzin, I. D., Hart, P. J. B. & Krause, J. Quorum decision-making facilitates information transfer in fish shoals. Proc. Natl Acad. Sci. USA 105, 6948–6953 (2008).

    ADS  Google Scholar 

  275. Abaurrea Velasco, C., Abkenar, M., Gompper, G. & Auth, T. Collective behavior of self-propelled rods with quorum sensing. Phys. Rev. E 98, 022605 (2018).

    ADS  Google Scholar 

  276. Castellano, C., Fortunato, S. & Loreto, V. Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009).

    ADS  Google Scholar 

  277. King, A. J., Douglas, C. M., Huchard, E., Isaac, N. J. & Cowlishaw, G. Dominance and affiliation mediate despotism in a social primate. Curr. Biol. 18, 1833–1838 (2008).

    Google Scholar 

  278. Couzin, I. D., Krause, J., Franks, N. R. & Levin, S. A. Effective leadership and decision-making in animal groups on the move. Nature 433, 513–516 (2005).

    ADS  Google Scholar 

  279. Freeman, R. & Biro, D. Modelling group navigation: dominance and democracy in homing pigeons. J. Navigat. 62, 33–40 (2009).

    Google Scholar 

  280. Bäuerle, T., Fischer, A., Speck, T. & Bechinger, C. Self-organization of active particles by quorum sensing rules. Nat. Commun. 9, 3232 (2018).

    ADS  Google Scholar 

  281. Moussaïd, M., Helbing, D. & Theraulaz, G. How simple rules determine pedestrian behavior and crowd disasters. Proc. Natl Acad. Sci. USA 108, 6884–6888 (2011).

    ADS  Google Scholar 

  282. Faria, J. J., Dyer, J. R., Tosh, C. R. & Krause, J. Leadership and social information use in human crowds. Animal Behav. 79, 895–901 (2010).

    Google Scholar 

  283. Bain, N. & Bartolo, D. Dynamic response and hydrodynamics of polarized crowds. Science 363, 46–49 (2019).

    ADS  MathSciNet  MATH  Google Scholar 

  284. Kim, M.-C., Silberberg, Y. R., Abeyaratne, R., Kamm, R. D. & Asada, H. H. Computational modeling of three-dimensional ECM-rigidity sensing to guide directed cell migration. Proc. Natl Acad. Sci. USA 115, E390–E399 (2018).

    Google Scholar 

  285. Paluch, E. K. & Raz, E. The role and regulation of blebs in cell migration. Curr. Opin. Cell Biol. 25, 582–590 (2013).

    Google Scholar 

  286. Tozluoglu, M. et al. Matrix geometry determines optimal cancer cell migration strategy and modulates response to interventions. Nat. Cell Biol. 15, 751–762 (2013).

    Google Scholar 

  287. Moure, A. & Gomez, H. Three-dimensional simulation of obstacle-mediated chemotaxis. Biomech. Model. Mechanobiol. 17, 1243–1268 (2018).

    Google Scholar 

  288. Besser, A. & Schwarz, U. S. Coupling biochemistry and mechanics in cell adhesion: a model for inhomogeneous stress fiber contraction. New J. Phys. 9, 425–425 (2007).

    ADS  Google Scholar 

  289. Nishikawa, M., Naganathan, S. R., Jülicher, F. & Grill, S. W. Controlling contractile instabilities in the actomyosin cortex. eLife 6, e19595 (2017).

    Google Scholar 

  290. Gross, P. et al. Guiding self-organized pattern formation in cell polarity establishment. Nat. Phys. 15, 293–300 (2019).

    Google Scholar 

  291. Bratanov, V., Jenko, F. & Frey, E. New class of turbulence in active fluids. Proc. Natl Acad. Sci. USA 112, 15048–15053 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  292. Weber, C. A. et al. Long-range ordering of vibrated polar disks. Phys. Rev. Lett. 110, 208001 (2013).

    ADS  Google Scholar 

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Acknowledgements

M.R.S., A.W. and H.R. acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) within SFB 1027 (A3, A7). R.G.W. and G.G. acknowledge funding by DFG within the priority programme SPP 1726 “Microswimmers — from Single Particle Motion to Collective Behaviour”.

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  1. Department of Theoretical Physics and Center for Biophysics, Saarland University, Saarbrücken, Germany

    M. Reza Shaebani, Adam Wysocki & Heiko Rieger

  2. Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany

    Roland G. Winkler & Gerhard Gompper

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Shaebani, M.R., Wysocki, A., Winkler, R.G. et al. Computational models for active matter. Nat Rev Phys 2, 181–199 (2020). https://doi.org/10.1038/s42254-020-0152-1

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