Steering Chiral Swimmers along Noisy Helical Paths

Chemotaxis along helical paths towards a target releasing a chemoattractant is found in sperm cells and many microorganisms. We discuss the stochastic differential geometry of the noisy helical swimming path of a chiral swimmer. A chiral swimmer equipped with a simple feedback system can navigate in a concentration gradient of chemoattractant. We derive an effective equation for the alignment of helical paths with a concentration gradient which is related to the alignment of a dipole in an external field and discuss the chemotaxis index.

Figure 1

(a) A helical path $\mathbf{r}$ can be described as the trajectory of a point on the circumference of a disk which rotates and translates; see text for details. (b) Helical swimming path $\mathbf{r}$ of a chemotactic chiral swimmer in a linear concentration field: The helix vector ${\mathbf{h}}_3$ fluctuates around the direction of the concentration gradient $\mathbf{\nabla }c$. Parameters were $v_0=r_0/\sigma$, ${\tau }_0=0.2/r_0$, $-{\kappa }_1={\tau }_1=0.5/r_0$, $\mu =\sigma$, $\lambda c_0={10}^2/\sigma$, $|{\mathbf{c}}_1|={10}^{-2}{c}_0/r_0$. (c) Histograms of $z=\cos \psi$ where $\psi$ equals the angle enclosed by the helix axis and the gradient direction for a simulated ensemble of helical swimming paths as in (b) with initial distribution $P(z,0)=\delta (z)$ at times $t=10T$, $100T$. Also shown is the analytical solution $P(z,t)$ (red). Approximately, $P(z,100T)$ equals the steady-state distribution $P_0(z)\sim \exp (\mathrm{Pe}z)$.