Although we assumed that the experimental data not satisfying the defini-tion of avoiding reaction nor escape reaction were hydrodynamic interac-tions, as shown in figure 2, there was no evidence that two cells did not ac-tively change their swimming motions during the interaction. Thus, we also performed numerical simulations. The model micro-organism introduced in this paper (a squirmer) is used by Ishikawa etal. (2006). A squirmer is assumed to be neutrally buoyant, because the sedimentation velocity for typical aquatic micro-organisms is much less than the swimming speed. The centre of buoyancy
of the micro-organism may not coincide with its geometric centre {bottom-heaviness). The model micro-organism is, therefore, force-free but may not be torque-free. Swimming speeds of micro-organisms such as P. cauda-tum range up to several hundred |Lim/s. However, the Reynolds number based on the swimming speed and the radius of individuals is usually less than 10"^, so the flow field around the cells can be assumed to be Stokes flow with negligible inertia compared to viscous effects. Brownian motion is usually not taken into account, because typical locomotive cells are too large for Brownian effects to be important. The model micro-organism was assumed to propel itself by generating tangential velocities on its surface. In fact, it is a reasonable model to de-scribe the locomotion of ciliates, which propel themselves by beating ar-rays of short hairs (cilia) on their surface in a synchronised way. In particular, the so-called symplectic metachronal wave, in which the cilia tips remain close together at all times, employed by Opalina, for instance, can be modelled simply as the stretching and displacement of the surface formed by the envelope of these tips. When there are N cells in an infinite fluid, the Stokes flow field exter-nal to the cells can be given in integral form as: .,.(x)-(«,.(x)) = --i-^i J,(x-y)^,(yM^, (1) where u is the velocity, J is the Green function, q is the single-layer poten-tial, and A is the surface of a particle. The parentheses <> indicate the sus-pension average. The boundary condition is given by u(x) = U^+Q^x(x-xJ + u,^^ , XE^ (2) where U^ and Q^ are the translational and rotational velocities of squirmer m, respectively, x^ is the centre of squirmer m, and u^,^ is the squirming velocity of squirmer m. In order to obtain u^, the velocity field around a swimming P. caudatum cell was measured by a PIV technique. Velocity vector was calculated for a time series of images, and approximately 80,000 velocity vectors around a swimming cell were obtained. These vec-tors were averaged by assuming that the velocity field is axisymmetric and time-independent. Then the surface velocity, u^, is interpolated from the surrounding flow field. The boundary element method was employed to discretize equation (1). For the computational mesh, a maximum of 590 triangle elements per particle were generated, and the mesh was finer in the near-contact region. Time-marching was performed by 4th-order Runge-Kutta schemes. In simulating hydrodynamic interactions between two squirmers, we assume that the surface velocity is independent of the distance between the cells.
Thus, no biological reaction is modelled, and the interaction is purely hy-drodynamic. Figure 3 shows the interactions between two squirmers, in which one squirmer collides with the anterior end of the other. In this case two cells tend to swim side by side at first, then move away from each other with an acute angle. This tendency is the same as the experimental results shown in figure 2. We also performed simulations with various initial conditions, and found that the hydrodynamic interaction data in the experiments agreed well with the numerical results and that the interaction was purely hydrodynamic. In most of the previous analytical studies on cell-cell interactions (Guell etal 1988; Ramia etal 1993; Nasseri and Phan-Thien, 1997; Jiang etal, 2002), two cells in close contact were not discussed. (Lega and Pas-sot, 2003 included an ad hoc interactive force acting between cells, that in
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