How Quorum Sensing Shapes Clustering in Active Matter

Abstract

Self-propelled particles undergoing persistent motion can accumulate either through excluded-volume interactions or through quorum sensing, where self-propulsion decreases at high local density. Using kinetic balance theory and simulations, we show that the interplay of these two mechanisms produces a reentrant, non-monotonic behavior in which clustering passes through a pronounced minimum as quorum-sensing strength or persistence time varies. Beyond a threshold quorum-sensing strength, we find long-lived transient states that retain memory of initial conditions, including kinetically arrested active gels. Although quorum sensing can mimic attractive interactions, it also acts strongly in dilute regions, producing an effective cluster bistability that is captured by our theory. Our results explain collective states observed experimentally in synthetic and biological active systems.

Figures (4)

• Figure 1: Quorum sensing gives rise to active gels, reentrant clustering, and long-lived transients. (a) Disk-like particles form an arrested cluster and a dilute phase. Each particle self-propels in a direction defined by $\theta$ (gray arrows), with a speed $v$ that decreases with the weighted density $\tilde{\rho}$ of particles within a radius $R_{\mathrm{QS}}$. (b) $v(\tilde{\rho})$ [Eq. \ref{['eq:speedQS']}] drops from one plateau to another (c) Zoom-out of panel (a) reveals a gel pattern caused by low directional persistence and speeds. Here, ${\tau_\mathrm{R}=4}$, ${\alpha=3}$, $\phi=0.4$, ${N=2\times 10^4}$, and ${t=2 \times 10^6}$. (d) Phase diagram in the $\tau_R$–$\alpha$ plane, based on snapshots at ${t = 2 \times 10^6}$ from a homogeneous initial condition. The color bar shows $f_c$, averaged between ${t = 1.7 \times 10^6}$ and ${2\times 10^6}$. A reentrant $f_c$ reveals three regions. Region I shows gels as in panel (c). Region II features motility-induced phase separation due to contact forces. In Region III, slowed particles have just enough persistence to escape without forming clusters. (e) Same as panel (d), but the initial condition is a single cluster with inward-pointing particles at the interface. There is a long-lived transient due to the memory of the initial condition. The diagram in panel (e) converges slowly to that in panel (d); see Supplementary Material SM.
• Figure 2: QS effects on the cluster surface and in the gas explain reentrant clustering and initial-condition dependence. (a) Schematics showing effective escape cone: due to an effective attraction, interface particles pointing outside the cone lack sufficient horizontal self-propulsion to escape. (b) $k_\mathrm{in}$ (blue) and $k_\mathrm{out}$ (red) vs. $\rho_{\rm g}$ [Eqs. \ref{['eq:kin']} and \ref{['eq:kout']}] for ${\alpha = 2}$ and $\tau_\mathrm{R} = 2$, $128$, and $1024$. Inset: $k_\mathrm{out}$ vs. $\tau_\mathrm{R}$. Quorum sensing makes $k_\mathrm{in}$ nontrivial, enabling multiple fixed points $\rho_{\rm g}$. For $\tau_\mathrm{R}=2$, one has $k_\mathrm{out} \approx 0$, giving $\rho_{\rm g} \approx 0$ (Region I clusters). For $\tau_\mathrm{R}=128$, if ${\rho_{\rm g} \approx \rho_{0}=0.49}$ initially, then $k_\mathrm{out}>k_\mathrm{in}$, maintaining ${\rho_{\rm g} \approx \rho_0}$ (Region III, homogeneous). For $\tau_\mathrm{R}=1024$, the system flows to a stable fixed point with low $\rho_{\rm g}$ (Region II clusters). Fitting parameters $C = 1.32$ and $\kappa = 2$ were set by visual comparison of the full phase diagrams.
• Figure 3: Effective kinetic balance theory captures key features of the clustering phase diagram. Fraction of particles in clusters $f_\mathrm{c}$ from our theory [Eq. \ref{['eq:fc_eff']}] is shown for (a) homogeneous and (b) clustered initial conditions. The theory reproduces the reentrant clustering and the initial-condition dependence of the diagram, while finer features such as cluster morphology and the pronounced curvature of the transition lines are not resolved.
• Figure 4: Quorum-sensing kinetic arrest manifests as suppressed mixing and strong memory of initial conditions. Particles are colored by their initial positions, grouped in vertical stripes with alternating colors. The first, second, and third columns show time evolution for $\alpha = 2.5$ and $\tau_\mathrm{R} = 1$, $32$, and $1024$, respectively. The fourth and fifth columns correspond to $\alpha = 5$ and $\tau_\mathrm{R} = 1024$, starting from homogeneous and clustered initial conditions. Preservation of colored stripes over time indicates minimal rearrangement. Mixing slows down with increasing $\alpha$, decreasing $\tau_\mathrm{R}$, or stronger initial clustering.