Collective Behavior of Quorum-Sensing Run-and-Tumble Particles under Confinement

We study a generic model for quorum-sensing bacteria in circular confinement. Every bacterium produces signaling molecules, the local concentration of which triggers a response when a certain threshold is reached. If this response lowers the motility, then an aggregation of bacteria occurs which differs fundamentally from standard motility-induced phase separation due to the long-ranged nature of the concentration of signal molecules. We analyze this phenomenon analytically and by numerical simulations employing two different protocols leading to stationary cluster and ring morphologies, respectively.

Figure 1

Snapshot of a configuration with $N=1000$ quorum-sensing run-and-tumble particles. The color code indicates the local concentration $\widehat{c}$ of signal molecules felt by each particle.

Figure 2

Clustering of slow particles. (a) The particle density $\rho (r)$ for reduced speed $\alpha =0.4$ and persistence length $\ell =0.01$ obtained numerically for $N=1000$ particles (the symbols) and from the mean-field theory (the dashed line). The solid line is a fit to Eq. (11). A sketch of a cluster with radius $r_{*}$ and density ${\rho }_1$ is shown in the inset, where ${\rho }_2$ is the density of the dilute region surrounding the cluster. (b) Densities ${\rho }_1$ (dense, right branch) and ${\rho }_2$ (dilute, left branch) for threshold $\overline{c}=1.45c_0$. The numerical results slightly depend on the persistence length $\ell$; still, the overall agreement with the theoretical prediction is excellent. (c) Numerical phase diagram of reduced speed $\alpha$ vs threshold $\overline{c}$ for $\ell =0.01$. The color bar indicates the density difference $({\rho }_1-{\rho }_2)/{\rho }_0$, whereas open circles indicate that no formation of a cluster has occurred. Also shown is the theoretical prediction (the solid lines).

Figure 3

Formation of a ring for two thresholds ${\overline{c}}_2>{\overline{c}}_1$. Shown is (a) the particle densities $\rho (r)$, (b) the concentration profiles $c(r)$, and (c) the actual average speeds $⟨v(\widehat{c})⟩$ as functions of the radial distance $r/R$ for three different values of the upper threshold ${\overline{c}}_2$, with the lower threshold ${\overline{c}}_1=1.4c_0$ held fixed. Symbols indicate simulation results for $N=1000$ particles, solid lines correspond to the mean-field predictions. Parameters are persistence length $\ell =0.01$ and reduced speed $\alpha =0.3$. The inset in (b) shows a snapshot of the ring (${\overline{c}}_2=1.6c_0$, colors as in Fig. 1), while in the inset of (c) the protocol is sketched.