Self-Trapping of Microorganisms Steering Toward their Own Trail

TL;DR

It is argued that, contrary to intuition, this trajectory instability occurs for any chemotactic coupling strength, and arises either through a potential-barrier first-passage problem or from a rare event analysis.

Abstract

Active matter systems comprise self-propelled particles that move on a substrate while leaving chemical trails that influence other particles through chemotaxis (e.g., slime-depositing bacteria). Orientational chemotaxis manifests as a torque that steers the particle toward the chemical gradient. As each particle is coupled to its own trail, the dynamics exhibits an instability: when the particle gently diffuses, it abruptly transitions to trajectories with a radius of curvature comparable to its own size, becoming apparently trapped. We argue that, contrary to intuition, this trajectory instability occurs for any chemotactic coupling strength. Depending on the coupling regime, this arises either through a potential-barrier first-passage problem or from a rare event analysis.

Figures (4)

• Figure 1: Sketch of a bacterium (green circle) diffusing on a substrate and leaving a trail of slime (gray ribbon) while randomly changing direction. The torque $\Gamma \propto \bm{u}_\perp \cdot \bm{\nabla} c$, which couples the angular velocity to the slime concentration gradient, enhances the rotation and can lead to a circling instability of the trajectory kranz_effective_2016.
• Figure 2: Fourth-order polynomial potential $V(\omega)$ governing the diffusion of the particle’s angular velocity in the regime $\mu'\lesssim\mu'_c$ and $\varepsilon\ll1$, shown for $\Delta\hat{\mu}'=0.3$ and $\varepsilon=0.1$. The trapping instability corresponds to the passage of the potential barrier. Inset: Trapping instability observed in our numerical simulation for $\mu'/\mu'_c=0.49$ and $\varepsilon=0.1$. The green line shows the particle’s trajectory, and the grey levels indicate the deposited slime.
• Figure 3: Mean trapping time $T$ for $\mu'<\mu'_c$, obtained from the statistics of our numerical simulations. The blue dashed line in the inset shows the analytical prediction from the mean first-passage time corresponding to Eq. \ref{['eq:T_near_threshold']}. The red dashed line in the main plot represents a one parameter fit to the trapping time estimated with Eq. \ref{['eq:T_asymptotic']}.
• Figure 4: Rotational ($\overline{D}_\theta$) and translational ($\overline{D}_\text{tr}$) diffusion coefficients of the particle for $\varepsilon=0.1$, each normalized by their values at $\mu'=0$. The dashed lines show the analytical predictions in the limit $\varepsilon \to 0$ (Eq. \ref{['Diffusion']}). The graphs end when the trapping time is too short to allow a well-defined diffusive regime. The constant $\mu'_0$ is defined in Eq. \ref{['thresholds']}.