Spatiotemporal Self-Organization of Fluctuating Bacterial Colonies

We model an enclosed system of bacteria, whose motility-induced phase separation is coupled to slow population dynamics. Without noise, the system shows both static phase separation and a limit cycle, in which a rising global population causes a dense bacterial colony to form, which then declines by local cell death, before dispersing to reinitiate the cycle. Adding fluctuations, we find that static colonies are now metastable, moving between spatial locations via rare and strongly nonequilibrium pathways, whereas the limit cycle becomes almost periodic such that after each redispersion event the next colony forms in a random location. These results, which hint at some aspects of the biofilm-planktonic life cycle, can be explained by combining tools from large deviation theory with a bifurcation analysis in which the global population density plays the role of control parameter.

Figure 1

Left panel: Bifurcation diagram in a two-dimensional projection. The $x$ axis shows the spatial mean of $\rho$, the $y$ axis shows its signed asymmetry $A[\rho ]$ (see text for details). For carrying capacities ${\rho }_S<{\rho }_0<{\rho }_0^c$, no stable fixed point of the dynamics exist. A limit cycle (light blue) and a trajectory with a small amount of noise (dark blue) that exhibits a transition to the lower stable branch are projected into the diagram. Right panels (a),(b), and (c): The time evolution of the limit cycle with a colony located at the left boundary (upper cycle on the left panel) is shown in subplot (a). Fluctuations allow the solution to randomly jump between the limit cycles with colonies located at the left or right of the domain, as shown in subplot (b). Subplot (c) depicts the almost periodic regime in a spatially periodic domain, where the fluctuations randomize the colony location.

Figure 2

Phase diagram as a function of ${\rho }_0$ and $\delta$, showing regimes with homogeneous, almost periodic, and metastable solutions. The dashed line shows ${\rho }_{\mathrm{S}}=2/(1-2{\delta }^2{\pi }^2)$, the dotted line ${\rho }_0^c$.

Figure 3

Transition paths between ${\rho }_L$ and ${\rho }_R$. Here, $x\in [0,1]$ denotes the spatial extend and $s\in [0,1]$ the normalized arc length along each trajectory. Left: Projection into bifurcation diagram; see Fig. 1. Center: MLP from ${\rho }_L$ to ${\rho }_R$. Right: Reversible transition from ${\rho }_L$ to ${\rho }_R$.