Modelling populations of swimming micro-organisms

Bioconvection patterns are observed in shallow suspensions of randomly, but on average upwardly, swimming micro-organisms which are a little denser than water. The basic mechanism is analogous to that of Rayleigh-Benard convection, in which an overturning instability develops when the upper regions of fluid become denser than the lower regions. The reason for the upswimming however depends on the species of micro-organism: certain biflagellate algae are bottom-heavy, and therefore experience a gravitational torque when they are not vertical; certain oxytactic bacteria swim up oxygen gradients that they generate by their consumption of oxygen. Rational continuum models can be formulated and analysed in each of these cases, as long as the cell volume fraction $n$ is low enough for hydrodynamic or other cell-cell interactions to be neglected ($n$ \leq 0.1%). The key mathematical step is the calculation of the probability density function for the cells' swimming velocity from a suitable Fokker-Planck equation, when that is justifiable. Both examples will be discussed from this point of view.

Another sort of pattern-formation ("whorls and jets") is observed in very concentrated, very shallow cultures of swimming bacteria on agar plates. Here cell-cell interactions are crucial, but it is not clear how to derive an appropriate macroscopic model that is consistent with the laws of mechanics at the cellular level. A recent attempt (J Lega & T Passot, Phys Rev E, 67, 2003) succeeds in generating patterns on the correct scale, but appears not to be rationally justified. Here we examine the deterministic swimming of model organisms which interact hydrodynamically but do not exhibit intrinsic randomness except in their initial positions and orientations. A micro-organism is modelled as a squirming, inertia-free sphere with prescribed tangential surface velocity. Pairwise interactions have been computed using the boundary element method, supplemented by lubrication theory, and the results stored in a database. The movement of 27 identical squirmers is computed by the Stokesian Dynamics method, with the help of the database of interactions (the restriction to pairwise interactions requires that the suspension be semi-dilute, with particle volume fraction less than about 0.1). It is found that the spreading is correctly described as a diffusive process a sufficiently long time after the motion is initiated, although all cell movements are deterministic. The effective translational and rotational diffusivities depend strongly on volume fraction and mode of squirming.