Phototaxis

Phototaxis is directed organism motion in response to light. An ideal spherical swimmer subject to light torque $\mathbf{p}\times \mathbf{L}$ and rotational resistance $\alpha$ aligns towards the light while ambient vorticity rotates it. Its deterministic orientation satisfies $\dot{\mathbf{p}}=\mathbf{\omega }\times \mathbf{p}/2+[\mathbf{L}-(\mathbf{p}\cdot \mathbf{L})\mathbf{p}]/\alpha$. The right-hand side is perpendicular to $\mathbf{p}$ and preserves its unit length.

In a long channel $-a<x<a$, uniform horizontal mean swimming $V_s{K}_0{\mathbf{e}}_x$ and isotropic translational diffusion coefficient $D$ give zero transverse cell flux when $n^{\prime }=\beta n$, $\beta =V_s{K}_0/D$. Normalizing the transverse mean concentration to $n_0$ gives the displayed exponential. For $\beta a\gg 1$, cells form a layer of thickness ${\beta }^{-1}$ at the illuminated wall. This is a bulk approximation; exact closed-end cell conservation equation also requires the integrated axial cell flux to vanish.

In the Boussinesq approximation, downward excess cell weight gives $\mu w^{\prime \prime }=G+g^{\prime }n(x)$. The no-slip boundary condition at both walls and zero total fluid flux determine the excess axial pressure gradient $G$, with $b=a\beta$. In particular, $w=C[e^{\beta x}-\cosh b-(x/a)\sinh b]+G(x^2-a^2)/(2\mu )$, $C=g^{\prime }{n}_{0}a\beta /(\mu {\beta }^2\sinh b)$. For large $b$, writing $\xi =x/a$ and $W=g^{\prime }{n}_0{a}^2/(\mu b)$ gives the outer profile $w/W=(1+\xi )(1-3\xi )/2$. A thin layer restores no slip at the illuminated wall. Cells descend there and fluid returns upward on the other side. At $b=0$ the limiting solution is $G=-g^{\prime }{n}_0$, $w=0$.

For isotropic rotational diffusion, orientation drift and probability density obey the displayed Fokker-Planck equation, normalized with respect to solid angle. Without flow, constant light $L{\mathbf{e}}_x$ gives the zero-current density $f_0=\lambda e^{\lambda p_x}/(4\pi \sinh \lambda )$, $\lambda =L/(\alpha D_r)$. Integration over the sphere gives unit mass; its mean projection is the Langevin function. The zero-light limit is the uniform density $1/(4\pi )$.

For light along ${\mathbf{e}}_x$, vorticity along ${\mathbf{e}}_y$ and $\epsilon =\omega /(2D_r)$, spherical coordinates $\mathbf{p}=(\cos \theta ,\sin \theta \cos \varphi ,\sin \theta \sin \varphi )$ give a first orientation-density correction proportional to $\sin \varphi g(\theta )$. The linearized alignment-diffusion operator preserves azimuthal harmonics and the rotation of the axisymmetric equilibrium forces only this sine harmonic. Angular integration makes the first correction to the mean purely vertical. Uniqueness follows from the weighted identity $\mathcal{A}(f_{0}h)={\nabla }_p\cdot (f_0{\nabla }_{p}h)$: integration by parts shows the only homogeneous smooth solution is a multiple of $f_0$, removed by normalization.