Microorganism Billiards

Recent experiments andnumerical simulations have shown that certain types ofmicroorganisms ‘‘reflect’’ off of a flat surface at a critical angle of departure, independent of the angle of incidence. The nature of the reflection may be active (cell and flagellar contact with the surface) or passive (hydrodynamic) interactions. We explore the billiard-like motion of a body with this empirical reflection law inside a regular polygon and show that the dynamics can settle on a stable periodic orbit or can be chaotic, depending on the swimmer’s departure angle and the domain geometry. The dynamics are often found to be robust to the introduction of weak random fluctuations. The Lyapunov exponent of swimmer trajectories can be positive or negative, can have extremal values, and can have discontinuities depending on the degree of the polygon. A passive sorting device is proposed that traps swimmers of different departure angles into separate bins. We also study the external problem of a microorganism swimming in a patterned environment of square obstacles, where the departure angle dictates the possibility of trapping or diffusive trajectories. © 2016 Elsevier B.V. All rights reserved.

[1]  H. Stark,et al.  Rectification of self-propelled particles by symmetric barriers. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Rothschild,et al.  Non-random Distribution of Bull Spermatozoa in a Drop of Sperm Suspension , 1963, Nature.

[3]  M. V. van Loosdrecht,et al.  Influence of interfaces on microbial activity. , 1990, Microbiological reviews.

[4]  R. Di Leonardo,et al.  Self-starting micromotors in a bacterial bath. , 2008, Physical review letters.

[5]  M. Nishimura,et al.  A fluid-dynamic interpretation of the asymmetric motion of singly flagellated bacteria swimming close to a boundary. , 2005, Biophysical journal.

[6]  G. Volpe,et al.  Active Particles in Complex and Crowded Environments , 2016, 1602.00081.

[7]  V V Moshchalkov,et al.  Geometrical guidance and trapping transition of human sperm cells. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  William S. Ryu,et al.  Real-Time Imaging of Fluorescent Flagellar Filaments , 2000, Journal of bacteriology.

[9]  M. Tasinkevych,et al.  Self-propulsion of a catalytically active particle near a planar wall: from reflection to sliding and hovering. , 2014, Soft matter.

[10]  Gianluigi Del Magno,et al.  Non-Hamiltonian dynamics in optical microcavities resulting from wave-inspired corrections to geometric optics , 2008, 0805.4555.

[11]  T. Powers,et al.  The hydrodynamics of swimming microorganisms , 2008, 0812.2887.

[12]  D. Barkley,et al.  Non-specular reflections in a macroscopic system with wave-particle duality: spiral waves in bounded media. , 2013, Chaos.

[13]  Samuel Sánchez,et al.  Topographical pathways guide chemical microswimmers , 2016, Nature Communications.

[14]  I. Aranson,et al.  Swimming bacteria power microscopic gears , 2009, Proceedings of the National Academy of Sciences.

[15]  Roberto Markarian,et al.  Pinball billiards with dominated splitting , 2009, Ergodic Theory and Dynamical Systems.

[16]  D. J. Smith,et al.  Spermatozoa scattering by a microchannel feature: an elastohydrodynamic model , 2014, Royal Society Open Science.

[17]  Pedro Duarte,et al.  SRB Measures for Polygonal Billiards with Contracting Reflection Laws , 2014 .

[18]  M. Hagan,et al.  Dynamics of self-propelled particles under strong confinement. , 2014, Soft matter.

[19]  L. Fauci,et al.  Sperm motility in the presence of boundaries. , 1995, Bulletin of mathematical biology.

[20]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[21]  D. Crowdy Treadmilling swimmers near a no-slip wall at low Reynolds number , 2011 .

[22]  R Di Leonardo,et al.  Self-Sustained Density Oscillations of Swimming Bacteria Confined in Microchambers. , 2015, Physical review letters.

[23]  Leo R. M. Maas,et al.  Geometric focusing of internal waves , 1995, Journal of Fluid Mechanics.

[24]  Jun Zhang,et al.  Experiments and theory of undulatory locomotion in a simple structured medium , 2012, Journal of The Royal Society Interface.

[25]  K. Stratford,et al.  Hydrodynamic oscillations and variable swimming speed in squirmers close to repulsive walls. , 2015, Soft matter.

[26]  A. Najafi,et al.  Three-sphere low-Reynolds-number swimmer near a wall. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  Raymond E. Goldstein,et al.  Ciliary contact interactions dominate surface scattering of swimming eukaryotes , 2013, Proceedings of the National Academy of Sciences.

[28]  C. A. Condat,et al.  Influence of swimming strategy on microorganism separation by asymmetric obstacles. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  R. Rosenfeld Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[30]  J. Koplik,et al.  Self-diffusiophoretic colloidal propulsion near a solid boundary , 2015, 1505.07172.

[31]  P. Duarte,et al.  Chaos in the square billiard with a modified reflection law. , 2011, Chaos.

[32]  Felix J. H. Hol,et al.  Zooming in to see the bigger picture: Microfluidic and nanofabrication tools to study bacteria , 2014, Science.

[33]  D. Turaev,et al.  Billiards: a singular perturbation limit of smooth Hamiltonian flows. , 2012, Chaos.

[34]  R Di Leonardo,et al.  Hydrodynamic Trapping of Swimming Bacteria by Convex Walls. , 2015, Physical review letters.

[35]  C. Reichhardt,et al.  Dynamics and separation of circularly moving particles in asymmetrically patterned arrays. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  T. Ishikawa,et al.  Hydrodynamic entrapment of bacteria swimming near a solid surface. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Eric Lauga,et al.  Geometric capture and escape of a microswimmer colliding with an obstacle. , 2014, Soft matter.

[38]  He Li,et al.  Asymmetric gear rectifies random robot motion , 2013 .

[39]  Patrick T. Underhill,et al.  Dynamics of confined suspensions of swimming particles , 2008, Journal of physics. Condensed matter : an Institute of Physics journal.

[40]  B. Ai,et al.  Rectification and diffusion of self-propelled particles in a two-dimensional corrugated channel. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  I. Llopis,et al.  Hydrodynamic interactions in squirmer motion: Swimming with a neighbour and close to a wall , 2010 .

[42]  L. Lugiato,et al.  Cavity soliton billiards , 2011 .

[43]  J. Dunkel,et al.  Fluid dynamics and noise in bacterial cell–cell and cell–surface scattering , 2011, Proceedings of the National Academy of Sciences.

[44]  G. Gompper,et al.  Motility-sorting of self-propelled particles in microchannels , 2014, 1409.1417.

[45]  Isaac Klapper,et al.  Mathematical Description of Microbial Biofilms , 2010, SIAM Rev..

[46]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[47]  Eric Lauga,et al.  Hydrodynamic attraction of swimming microorganisms by surfaces. , 2008, Physical review letters.

[48]  M. Cates,et al.  Sedimentation, trapping, and rectification of dilute bacteria , 2009, 0903.3247.

[49]  Jun Zhang,et al.  Hydrodynamic capture of microswimmers into sphere-bound orbits. , 2013, Soft matter.

[50]  J. Feijen,et al.  Bacterial migration along solid surfaces , 1992, Applied and environmental microbiology.

[51]  Enkeleida Lushi,et al.  Fluid flows created by swimming bacteria drive self-organization in confined suspensions , 2014, Proceedings of the National Academy of Sciences.

[52]  Roman Stocker,et al.  Failed escape: solid surfaces prevent tumbling of Escherichia coli. , 2014, Physical review letters.

[53]  F. Frischknecht,et al.  Geometrical model for malaria parasite migration in structured environments. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  H. H. Wensink,et al.  Capturing self-propelled particles in a moving microwedge. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[55]  H. Stark,et al.  Detention Times of Microswimmers Close to Surfaces: Influence of Hydrodynamic Interactions and Noise. , 2014, Physical review letters.

[56]  Cees Dekker,et al.  Bacterial growth and motility in sub-micron constrictions , 2009, Proceedings of the National Academy of Sciences.

[57]  Enkeleida Lushi,et al.  Microalgae Scatter off Solid Surfaces by Hydrodynamic and Contact Forces. , 2015, Physical review letters.

[58]  Z. Nussinov,et al.  Rectification of swimming bacteria and self-driven particle systems by arrays of asymmetric barriers. , 2007, Physical review letters.

[59]  E. Lauga,et al.  Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations , 2012, Journal of Fluid Mechanics.

[60]  Leo R. M. Maas,et al.  Wave attractors: Linear Yet Nonlinear , 2005, Int. J. Bifurc. Chaos.

[61]  H. H. Wensink,et al.  How to capture active particles. , 2012, Physical review letters.

[62]  J. Lintuvuori,et al.  Swimming in A Crystal: Using Colloidal Crystals to Characterise Micro-swimmers , 2014 .

[63]  Anton Zorich Flat Surfaces , 2006 .

[64]  Robert Austin,et al.  A Wall of Funnels Concentrates Swimming Bacteria , 2007, Journal of bacteriology.

[65]  R Di Leonardo,et al.  Bacterial ratchet motors , 2009, Proceedings of the National Academy of Sciences.

[66]  R. Di Leonardo,et al.  Swimming with an image. , 2011, Physical review letters.

[67]  H. Shum,et al.  Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[68]  Ben Parker Chaotic Billiards , 2006 .

[69]  D. Crowdy,et al.  Two-dimensional point singularity model of a low-Reynolds-number swimmer near a wall. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[70]  R. Kolter,et al.  Biofilm formation as microbial development. , 2000, Annual review of microbiology.

[71]  George M Whitesides,et al.  Swimming in circles: motion of bacteria near solid boundaries. , 2005, Biophysical journal.

[72]  G. Gompper,et al.  Giant adsorption of microswimmers: Duality of shape asymmetry and wall curvature. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[73]  D. Smith,et al.  Surface accumulation of spermatozoa: a fluid dynamic phenomenon , 2010, 1007.2153.

[74]  Clemens Bechinger,et al.  Microswimmers in patterned environments , 2011, 1104.3203.

[75]  Aidan T Brown,et al.  Swimming in a crystal. , 2014, Soft matter.

[76]  L. Lemelle,et al.  Counterclockwise Circular Motion of Bacteria Swimming at the Air-Liquid Interface , 2010, Journal of bacteriology.

[77]  Dominique Benielli,et al.  Observation of an internal wave attractor in a confined, stably stratified fluid , 1997, Nature.

[78]  Jörn Dunkel,et al.  Confinement stabilizes a bacterial suspension into a spiral vortex. , 2013, Physical review letters.

[79]  Jackson Kirkman-Brown,et al.  Human sperm accumulation near surfaces: a simulation study , 2009, Journal of Fluid Mechanics.