A hybrid particle/continuum model for micro-organism motility

Abstract

We describe a method for simulating the dynamics of a flexible filament immersed in a fluid. The model treats the filament as a set of rigidly connected beads. The forces acting on these beads are characteristic of the instantaneous bending, tension, viscous and active forces operating on an elemental length. We show that, with an appropriate choice of parameters, the effects of inertia can be minimised to the point where the “inertialess” equations of motion are essentially satisfied. Typically, this regime is appropriate for micron sized filaments. The flagella that propel micro-organisms are one such example. With this in mind, we apply the model to simulate the planar swimming motion characteristic of simple spermatozoa. Assuming a relatively naive bending mechanism, a force quadrupole in the form of a travelling wave, the model generates waveforms in very good agreement with experiment. This is only true, however, if the bending forces acting on the filament are large compared to the viscous forces. Experimental measurements of the tail stiffness imply this should not be the case. We discuss the implications of this observation in the context of the sperm’s swimming mechanism.