Stability and dynamics of anisotropically tumbling chemotactic swimmers

Microswimmers such as bacteria perform random walks known as run-and-tumbles to move up chemoattractant gradients and as a result aggregate with others. It is also known that such micro-swimmers can self-organize into macroscopic patterns due to interactions with neighboring cells through the fluidic environment they live in. While the pattern formation resulting from chemotactic and hydrodynamic interactions separately and together have been previously investigated, the effect of the anisotropy in the tumbles of microswimmers has been unexplored. Here we show through linear analysis and full nonlinear simulations that the slight anisotropy in the individual swimmer tumbles can alter the collective pattern formation in nontrivial ways. We show that tumbling anisotropy diminishes the magnitude of the chemotactic aggregates but may result in more such aggregation peaks.

Figure 1

(a) Numerical solution for ${\sigma }_C\left(k\right)$ of the autochemotactic relation Eq. (24) for $\chi =20, D_c=1/20, \lambda =0.25$, and a variety of tumbling anisotropy parameters $\delta =0,0.25,0.5$. (b) Numerical solution for ${\sigma }_H\left(k\right)$ of the hydrodynamic relation Eq. (23) for pushers $\alpha =-1$ for ${\lambda }_0=0,0.05$ and tumbling anisotropy parameters $\delta =0,0.5$. Red arrows show the effect of an increasing parameter.

Figure 2

Phase space of various regimes for autochemotactic and/or hydrodynamic instabilities in suspensions of (a) pullers or neutral swimmers, (b) pushers as a function of the basal tumbling frequency ${\lambda }_0$, correlation of tumbles parameter $\delta$, and chemotactic parameters $\chi ,{\beta }_1,{\beta }_2$. The curve $1/\left[{\lambda }_0(1-\delta /3)\right]$ is shown for $\delta =0$ and $\delta =1$.

Figure 3

Snapshot of the dynamics of microswimmers concentration field $\mathrm{\Phi }$ at long times. Shown are the cases of “neutral” swimmers ($\alpha =0$), pullers ($\alpha =+1$), and pushers ($\alpha =-1$) with isotropic tumbling ($\delta =0$) or with tumbling direction correlation parameters ($\delta =0.25,0.5$). The chemoattractant dynamics follows closely that of the swimmer concentration.

Figure 4

Evolution of the maximum of the swimmer concentration $\mathrm{\Phi }$ and the system configurational entropy $\mathit{S}\left(t\right)$ in time. The cases shown are for neutral (black line), puller (red line), and pusher swimmers (blue line) for isotropic suspensions (solid lines) and anisotropic suspensions with $\delta =0.25$ (dashed lines).