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  • Featured in Physics
  • Open Access

Topological Sound and Flocking on Curved Surfaces

Suraj Shankar1,2, Mark J. Bowick1,2, and M. Cristina Marchetti1,2

  • 1Physics Department and Syracuse Soft Matter Program, Syracuse University, Syracuse, New York 13244, USA
  • 2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA

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Phys. Rev. X 7, 031039 – Published 7 September, 2017

DOI: https://doi.org/10.1103/PhysRevX.7.031039

Abstract

Active systems on curved geometries are ubiquitous in the living world. In the presence of curvature, orientationally ordered polar flocks are forced to be inhomogeneous, often requiring the presence of topological defects even in the steady state because of the constraints imposed by the topology of the underlying surface. In the presence of spontaneous flow, the system additionally supports long-wavelength propagating sound modes that get gapped by the curvature of the underlying substrate. We analytically compute the steady-state profile of an active polar flock on a two-sphere and a catenoid, and show that curvature and active flow together result in symmetry-protected topological modes that get localized to special geodesics on the surface (the equator or the neck, respectively). These modes are the analogue of edge states in electronic quantum Hall systems and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering.

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Synopsis

Even Flocks are Topological

Published 7 September, 2017

A flocking model that describes birds and cells exhibits topological features when the moving entities are confined to a curved surface.

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Popular Summary

Flocking, the self-organized and spontaneous motion of a large collection of self-propelled entities, is ubiquitous. Large murmurations of starlings are a familiar example, but such collective motion can also be found on much smaller scales, such as in groups of cells advancing en masse during growth and development. In many biological processes, from cell repair in the folded intestine to the shaping of embryonic limb buds, the motion of cells takes place on curved surfaces. Understanding the physical aspects of flocking on curved surfaces therefore remains crucial to developing insight into the organizational principles of living matter.

Using a continuum hydrodynamic model, we performed analytical computations that show that flocks on a sphere take the form of a moving band that wraps around an equator. Our calculations show that similar inhomogeneous traveling structures that would be impossible in equilibrium arise in other curved geometries. Furthermore, on the sphere, we find that a finite energy is required to excite traveling sound waves. We show that such sound waves are “topologically protected” and therefore robust to scattering by impurities and disorder. The exotic unidirectional sound modes found when flocks move on a curved surface are akin to edge states in integer quantum Hall systems and to well-known equatorial waves observed in the flows of oceans or in the Earth’s atmosphere.

The existence of unidirectional propagating modes that are robust against disorder appears to be a generic property of flocking systems in curved spaces, suggesting the intriguing possibility that nature may take advantage of this mechanism to control cell motion and force transmission in curved environments.

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References (86)

  1. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel Type of Phase Transition in a System of Self-Driven Particles, Phys. Rev. Lett. 75, 1226 (1995).
  2. J. Toner, Y. Tu, and S. Ramaswamy, Hydrodynamics and Phases of Flocks, Ann. Phys. (Amsterdam) 318, 170 (2005).
  3. S. Ramaswamy, The Mechanics and Statistics of Active Matter, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  4. M. Cristina Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of Soft Active Matter, Rev. Mod. Phys. 85, 1143 (2013).
  5. A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini, and M. Viale, Scale-Free Correlations in Starling Flocks, Proc. Natl. Acad. Sci. U.S.A. 107, 11865 (2010).
  6. A. Sokolov, I. S. Aranson, J. O. Kessler, and R. E. Goldstein, Concentration Dependence of the Collective Dynamics of Swimming Bacteria, Phys. Rev. Lett. 98, 158102 (2007).
  7. B. Szabo, G. J. Szöllösi, B. Gönci, Z. Jurányi, D. Selmeczi, and T. Vicsek, Phase Transition in the Collective Migration of Tissue Cells: Experiment and Model, Phys. Rev. E 74, 061908 (2006).
  8. V. Schaller, C. Weber, C. Semmrich, E. Frey, and A. R. Bausch, Polar Patterns of Driven Filaments, Nature (London) 467, 73 (2010).
  9. T. Sanchez, D. T. N. Chen, S. J. DeCamp, M. Heymann, and Z. Dogic, Spontaneous Motion in Hierarchically Assembled Active Matter, Nature (London) 491, 431 (2012).
  10. F. C. Keber, E. Loiseau, T. Sanchez, S. J. DeCamp, L. Giomi, M. J. Bowick, M. C. Marchetti, Z. Dogic, and A. R. Bausch, Topology and Dynamics of Active Nematic Vesicles, Science 345, 1135 (2014).
  11. J. Deseigne, O. Dauchot, and H. Chaté, Collective Motion of Vibrated Polar Disks, Phys. Rev. Lett. 105, 098001 (2010).
  12. V. Narayan, S. Ramaswamy, and N. Menon, Long-Lived Giant Number Fluctuations in a Swarming Granular Nematic, Science 317, 105 (2007).
  13. J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, and P. M. Chaikin, Living Crystals of Light-Activated Colloidal Surfers, Science 339, 936 (2013).
  14. S. J. Streichan, M. F. Lefebvre, N. Noll, E. F. Wieschaus, and B. I. Shraiman, Quantification of Myosin Distribution Predicts Global Morphogenetic Flow in the Fly Embryo, arXiv:1701.07100.
  15. B. Vasiev, A. Balter, M. Chaplain, J. A. Glazier, and C. J. Weijer, Modeling Gastrulation in the Chick Embryo: Formation of the Primitive Streak, PLoS One 5, e10571 (2010).
  16. A. J. Ewald, A. Brenot, M. Duong, B. S. Chan, and Z. Werb, Collective Epithelial Migration and Cell Rearrangements Drive Mammary Branching Morphogenesis, Dev. Cell 14, 570 (2008).
  17. A. Fatehullah, P. L. Appleton, and I. S. Näthke, Cell and Tissue Polarity in the Intestinal Tract During Tumourigenesis: Cells Still Know the Right Way up, But Tissue Organization Is Lost, Phil. Trans. R. Soc. B 368, 20130014 (2013).
  18. L. Ritsma, S. I. J. Ellenbroek, A. Zomer, H. J. Snippert, F. J. de Sauvage, B. D. Simons, H. Clevers, and J. van Rheenen, Intestinal Crypt Homeostasis Revealed at Single-Stem-Cell Level by In Vivo Live Imaging, Nature (London) 507, 362 (2014).
  19. J. M. Collinson, L. Morris, A. I. Reid, T. Ramaesh, M. A. Keighren, J. H. Flockhart, R. E. Hill, S.-S. Tan, K. Ramaesh, B. Dhillon, and J. D. West, Clonal Analysis of Patterns of Growth, Stem Cell Activity, and Cell Movement During the Development and Maintenance of the Murine Corneal Epithelium, Dev. Dyn. 224, 432 (2002).
  20. M. Mishra, Y. Huang, P. Srivastava, R. Srinivasan, M. Sevugan, R. Shlomovitz, N. Gov, M. Rao, and M. Balasubramanian, Cylindrical Cellular Geometry Ensures Fidelity of Division Site Placement in Fission Yeast, J. Cell Sci. 125, 3850 (2012).
  21. H. G. Yevick, G. Duclos, I. Bonnet, and P. Silberzan, Architecture and Migration of an Epithelium on a Cylindrical Wire, Proc. Natl. Acad. Sci. U.S.A. 112, 5944 (2015).
  22. Y. Fily, A. Baskaran, and M. F. Hagan, Dynamics of Self-Propelled Particles under Strong Confinement, Soft Matter 10, 5609 (2014).
  23. Y. Fily, A. Baskaran, and M. F. Hagan, Dynamics and Density Distribution of Strongly Confined Noninteracting Nonaligning Self-Propelled Particles in a Nonconvex Boundary, Phys. Rev. E 91, 012125 (2015).
  24. Y. Fily, A. Baskaran, and M. F. Hagan, Equilibrium Mappings in Polar-Isotropic Confined Active Particles, arXiv:1612.08719.
  25. N. Nikola, A. P. Solon, Y. Kafri, M. Kardar, J. Tailleur, and R. Voituriez, Active Particles with Soft and Curved Walls: Equation of State, Ratchets, and Instabilities, Phys. Rev. Lett. 117, 098001 (2016).
  26. F. Smallenburg and H. Löwen, Swim Pressure on Walls with Curves and Corners, Phys. Rev. E 92, 032304 (2015).
  27. R. Green, J. Toner, and V. Vitelli, The Geometry of Threshold-less Active Flow in Nematic Microfluidics, arXiv:1602.00561.
  28. Y. Fily, A. Baskaran, and M. F. Hagan, Active Particles on Curved Surfaces, arXiv:1601.00324.
  29. R. Sknepnek and S. Henkes, Active Swarms on a Sphere, Phys. Rev. E 91, 022306 (2015).
  30. D. R. Nelson, Defects and Geometry in Condensed Matter Physics (Cambridge University Press, Cambridge, England, 2002).
  31. A. Maitra, P. Srivastava, M. Rao, and S. Ramaswamy, Activating Membranes, Phys. Rev. Lett. 112, 258101 (2014).
  32. S. A. Mallory, C. Valeriani, and A. Cacciuto, Curvature-Induced Activation of a Passive Tracer in an Active Bath, Phys. Rev. E 90, 032309 (2014).
  33. A. Souslov, B. C. van Zuiden, D. Bartolo, and V. Vitelli, Topological Sound in Active-Liquid Metamaterials, arXiv:1610.06873.
  34. Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, Topological Acoustics, Phys. Rev. Lett. 114, 114301 (2015).
  35. E. Prodan and C. Prodan, Topological Phonon Modes and Their Role in Dynamic Instability of Microtubules, Phys. Rev. Lett. 103, 248101 (2009).
  36. C. L. Kane and T. C. Lubensky, Topological Boundary Modes in Isostatic Lattices, Nat. Phys. 10, 39 (2014).
  37. S. Raghu and F. D. M. Haldane, Analogs of Quantum-Hall-Effect Edge States in Photonic Crystals, Phys. Rev. A 78, 033834 (2008).
  38. M. Z. Hasan and C. L. Kane, Colloquium: Topological Insulators, Rev. Mod. Phys. 82, 3045 (2010).
  39. P. Delplace, J. B. Marston, and A. Venaille, Topological Origin of Geophysical Waves, arXiv:1702.07583.
  40. P. Srivastava, R. Shlomovitz, N. S. Gov, and M. Rao, Patterning of Polar Active Filaments on a Tense Cylindrical Membrane, Phys. Rev. Lett. 110, 168104 (2013).
  41. M. Eisenberg and R. Guy, A Proof of the Hairy Ball Theorem, Am. Math. Mon. 86, 571 (1979).
  42. Y. Tu, J. Toner, and M. Ulm, Sound Waves and the Absence of Galilean Invariance in Flocks, Phys. Rev. Lett. 80, 4819 (1998).
  43. J. Toner and Y. Tu, Long-Range Order in a Two-Dimensional Dynamical XY Model: How Birds Fly Together, Phys. Rev. Lett. 75, 4326 (1995).
  44. J. Toner and Y. Tu, Flocks, Herds, and Schools: A Quantitative Theory of Flocking, Phys. Rev. E 58, 4828 (1998).
  45. Manfredo Perdigao do Carmo Valero, Riemannian Geometry (Prentice-Hall, Englewood Cliffs, NJ, 1992).
  46. X. Yang, M. L. Manning, and M. C. Marchetti, Aggregation and Segregation of Confined Active Particles, Soft Matter 10, 6477 (2014).
  47. S. C. Takatori, W. Yan, and J. F. Brady, Swim Pressure: Stress Generation in Active Matter, Phys. Rev. Lett. 113, 028103 (2014).
  48. J. Toner, Reanalysis of the Hydrodynamic Theory of Fluid and Polar-Ordered Flocks, Phys. Rev. E 86, 031918 (2012).
  49. A. Gopinath, M. F. Hagan, M. C. Marchetti, and A. Baskaran, Dynamical Self-Regulation in Self-Propelled Particle Flows, Phys. Rev. E 85, 061903 (2012).
  50. E. Bertin, M. Droz, and G. Grégoire, Hydrodynamic Equations for Self-Propelled Particles: Microscopic Derivation and Stability Analysis, J. Phys. A 42, 445001 (2009).
  51. A. P. Solon, J.-B. Caussin, D. Bartolo, H. Chaté, and J. Tailleur, Pattern Formation in Flocking Models: A Hydrodynamic Description, Phys. Rev. E 92, 062111 (2015).
  52. J.-B. Caussin, A. Solon, A. Peshkov, H. Chaté, T. Dauxois, J. Tailleur, V. Vitelli, and D. Bartolo, Emergent Spatial Structures in Flocking Models: A Dynamical System Insight, Phys. Rev. Lett. 112, 148102 (2014).
  53. S. Mishra, A. Baskaran, and M. C. Marchetti, Fluctuations and Pattern Formation in Self-Propelled Particles, Phys. Rev. E 81, 061916 (2010).
  54. E. Bertin, M. Droz, and G. Grégoire, Boltzmann and Hydrodynamic Description for Self-Propelled Particles, Phys. Rev. E 74, 022101 (2006).
  55. P. Gilles De Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1993).
  56. W. Kung, M. C. Marchetti, and K. Saunders, Hydrodynamics of Polar Liquid Crystals, Phys. Rev. E 73, 031708 (2006).
  57. A. Baskaran and M. C. Marchetti, Hydrodynamics of Self-Propelled Hard Rods, Phys. Rev. E 77, 011920 (2008).
  58. P. C. Martin, O. Parodi, and P. S. Pershan, Unified Hydrodynamic Theory for Crystals, Liquid Crystals, and Normal Fluids, Phys. Rev. A 6, 2401 (1972).
  59. A. E. Gill, Atmosphere–Ocean Dynamics (Elsevier, New York, 2016).

  60. The term involving λ¯(η+2)mv is not relevant as its contribution to the dispersion is subdominant, as can be seen from its absence in Eq. (25).

  61. M. V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes, Proc. R. Soc. A 392, 45 (1984).
  62. M. Nakahara, Geometry, Topology and Physics (Taylor & Francis, London, 2003).
  63. F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the “Parity Anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  64. S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological Insulators and Superconductors: Tenfold Way and Dimensional Hierarchy, New J. Phys. 12, 065010 (2010).
  65. D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems, Phys. Rev. Lett. 118, 040401 (2017).
  66. R. Jackiw and C. Rebbi, Solitons with Fermion Number 1/2, Phys. Rev. D 13, 3398 (1976).

  67. More precisely, one needs a Killing field on the surface dictating a symmetry direction. One can always have the Christoffel symbols vanish at a given point in Riemann normal coordinates, but in the presence of a Killing field, this extends to an integral curve along which the Christoffel symbols vanish, changing sign as it is crossed. This provides the necessary structure for the existence of such topological modes.

  68. M. J. Bowick and L. Giomi, Two-Dimensional Matter: Order, Curvature and Defects, Adv. Phys. 58, 449 (2009).
  69. A. M. Turner, V. Vitelli, and D. R. Nelson, Vortices on Curved Surfaces, Rev. Mod. Phys. 82, 1301 (2010).
  70. M. J. Bowick, D. R. Nelson, and A. Travesset, Interacting Topological Defects on Frozen Topographies, Phys. Rev. B 62, 8738 (2000).
  71. A. R. Bausch, M. J. Bowick, A. Cacciuto, A. D. Dinsmore, M. F. Hsu, D. R. Nelson, M. G. Nikolaides, A. Travesset, and D. A. Weitz, Grain Boundary Scars and Spherical Crystallography, Science 299, 1716 (2003).
  72. W. T. M. Irvine, V. Vitelli, and P. M. Chaikin, Pleats in Crystals on Curved Surfaces, Nature (London) 468, 947 (2010).
  73. M. J. Bowick and Z. Yao, Crystalline Order on Catenoidal Capillary Bridges, Europhys. Lett. 93, 36001 (2011).
  74. E. Bendito, M. J. Bowick, A. Medina, and Z. Yao, Crystalline Particle Packings on Constant Mean Curvature (Delaunay) Surfaces, Phys. Rev. E 88, 012405 (2013).
  75. D. R. Nelson and L. Peliti, Fluctuations in Membranes with Crystalline and Hexatic Order, J. Phys. II (France) 48, 1085 (1987).
  76. H. S. Seung and D. R. Nelson, Defects in Flexible Membranes with Crystalline Order, Phys. Rev. A 38, 1005 (1988).
  77. X. Xing, H. Shin, M. J. Bowick, Z. Yao, L. Jia, and M.-H. Li, Morphology of Nematic and Smectic Vesicles, Proc. Natl. Acad. Sci. U.S.A. 109, 5202 (2012).
  78. N. Ramakrishnan, J. H. Ipsen, and P. B. Sunil Kumar, Role of Disclinations in Determining the Morphology of Deformable Fluid Interfaces, Soft Matter 8, 3058 (2012).
  79. L. Mesarec, P. Kurioz, and A. Iglič, W. Góźdź, and S. Kralj, Curvature-Controlled Topological Defects, Crystals 7, 153 (2017).
  80. D. R. Nelson, Liquids and Glasses in Spaces of Incommensurate Curvature, Phys. Rev. Lett. 50, 982 (1983).
  81. C. D. Modes and R. D. Kamien, Hard Disks on the Hyperbolic Plane, Phys. Rev. Lett. 99, 235701 (2007).
  82. V. Vitelli and A. M. Turner, Anomalous Coupling Between Topological Defects and Curvature, Phys. Rev. Lett. 93, 215301 (2004).
  83. L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, Topological Mechanics of Gyroscopic Metamaterials, Proc. Natl. Acad. Sci. U.S.A. 112, 14495 (2015).
  84. X. Serra-Picamal, V. Conte, R. Vincent, E. Anon, D. T. Tambe, E. Bazellieres, J. P. Butler, J. J. Fredberg, and X. Trepat, Mechanical Waves During Tissue Expansion, Nat. Phys. 8, 628 (2012).
  85. F. E. O. Holmberg, J. B. Seidelin, X. Yin, B. E. Mead, Z. Tong, Y. Li, J. M. Karp, and O. H. Nielsen, Culturing Human Intestinal Stem Cells for Regenerative Applications in the Treatment of Inflammatory Bowel Disease, EMBO Mol. Med. 9, 558 (2017).
  86. S. Hopyan, J. Sharpe, and Y.Yang, Budding Behaviors: Growth of the Limb as a Model of Morphogenesis, Dev. Dyn. 240, 1054 (2011).

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Phys. Rev. X 7, 031039– Published 7 September, 2017
  • Received 27 April 2017
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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