Synchronization of Swimming Microorganisms

Abstract

Some microorganisms, such as spermatozoa, have been observed to synchronize their flagella when swimming in close proximity. Using a two-dimensional model we show that phase-locking can arise from hydrodynamic forces alone. In a Newtonian fluid, as a consequence of the kinematic reversibility of the field equations, there must exist front-back asymmetry in the geometry of their flagellar waveform. The time-evolution of the phase difference between co-swimming cells depends only on the nature of this geometrical asymmetry, and microorganisms can phase-lock into conformations which minimize or maximize energy dissipation. In a viscoelastic fluid the presence of polymeric stresses removes the geometrical asymmetry constraint, and therefore even symmetric swimmers synchronize. Such synchronization occurs on asymptotically faster time scales than in a Newtonian fluid, and the swimmers evolve into a stable in-phase conformation minimizing the energy dissipated in the surrounding fluid. Furthermore, we show that if we consider flexible sheets, with internal symmetric forcing instead of prescribed kinematics, they deform when interacting with each other through the fluid in such a way as to systematically break the overall waveform symmetry, thereby always evolving to an in-phase conformation where energy dissipation is minimized. These dynamics are shown to be mathematically equivalent to those obtained for prescribed waveforms in viscoelastic fluids, emphasizing the crucial role of flexibility in symmetry-breaking and synchronization - be it that of the fluid, or the swimmers themselves.