Forward Propulsion Force

 As mentioned above, only the cilia on the trapezoid contribute to the rota-tion. Other cilia are responsible for translational propulsion. Considering that the reversed effective strokes on the cathodal side produce a backward force, we can derive the magnitude of the force F as

Equations of Motion of Paramecium Cell

Using the torque estimated in section 2.3, we now discuss the equations of motion of the Paramecium cell. In the micrometer-scale world that paramecia inhabit, the inertial resis-tance of the fluid is small enough to be negligible, and the viscous resistance becomes dominant. Hence, we can apply Stokes' law, derived from the Navier-Stokes equation by ignoring inertial force. Since a rigorous evalua-tion of the viscous resistance around an ellipsoid is quite complicated, here we approximate the viscous force by applying the formula for a sphere as a substitute. According to Stokes' law, the force exerted on a sphere with radius a, moving with velocity v in a viscous fluid is given by F^ — SKfiav, where ji is the viscosity of the fluid. From this equation, the viscous force around the ellipsoidal cell can be obtained by replacing the radius a by the cell radius R. Thus, the equation of motion for the translational motion of the cell can be approximated by MX + DX = F, where X = (X, Y)^ is the  

cell position (the superscript T means the transposition), F = Fe is 3, for-ward propulsive force, e = (cos0,sin(^)^ is a unit vector along the body axis, D = Fs/\X\ = SKJIR is the viscous friction coefficient, M = pF is the cell mass, p is the cell density, and V = 47tLR^/3 is the cell volume. In addition, the equation of motion for the rotation is given by 70 +Z)'0 = Tz{(j)), where / = 7tM{R^ +L^)/5 is the moment of inertia for an ellipsoid, D' = 57tjiL^ is the viscous friction coefficient, and 5 is a coefficient to compensate for errors in the model. Finally, integration of the equations of motion for the translational motion and the rotational motion leads to the following equations with a notation common in systems theory:  

 

 where q = {X, Y,X,7,0,0)^, andP = 2fn\x^\. Practically, acceleration terms can be ignored because of the smallness of the cellsOgawa et al. (2007). In this paper, we included acceleration terms for accurate description.

Numerical Experiments

We performed some numerical experiments to verify the equations of mo-tion using numerical analysis software (MATLAB, MathWorks Inc.). Table 1 shows several physical parameters used in the numerical experi-ments. We obtained the cell size by observing cells incubated in our labo-ratory. The value of j8, the increase in beating frequency with electric field, was estimated from the fact that the frequency increased to around 50 Hz under a stimulation of a few volts per centimeter from the frequency in the regular state of around 15-20 Hz (Naitoh and Sugino 1984). The force yielded by cilia per unit length, fn, is still an unknown pa-rameter. However, our model itself is based on approximation, and a strict evaluation of fn is not so critical. Hence, we estimated the order of fn by