Arrested phase separation in reproducing bacteria creates a generic route to pattern formation

Abstract

We present a generic mechanism by which reproducing microorganisms, with a diffusivity that depends on the local population density, can form stable patterns. For instance, it is known that a decrease of bacterial motility with density can promote separation into bulk phases of two coexisting densities; this is opposed by the logistic law for birth and death that allows only a single uniform density to be stable. The result of this contest is an arrested nonequilibrium phase separation in which dense droplets or rings become separated by less dense regions, with a characteristic steady-state length scale. Cell division predominates in the dilute regions and cell death in the dense ones, with a continuous flux between these sustained by the diffusivity gradient. We formulate a mathematical model of this in a case involving run-and-tumble bacteria and make connections with a wider class of mechanisms for density-dependent motility. No chemotaxis is assumed in the model, yet it predicts the formation of patterns strikingly similar to some of those believed to result from chemotactic behavior.

Fig. 1.

Growth of the instability in the supercritical (Left) and subcritical cases (Right). The three lines correspond to three successive times. A small perturbation around ρ₀ (red line) growth toward harmonic or anharmonic patterns in the supercritical or subcritical case, respectively. (Left) Supercritical case (α = κ = 0.01, λ = 0.02, ρ₀ = 15, ; times: 10², 10³, 10⁴). (Right) Subcritical case (α = κ = 0.005, λ = 0.02, ρ₀ = 11, ; times: 3.10², 3.10³, 10⁵).

Fig. 2.

Three plots of Λ_q(q) for , 2, 3 (from bottom to top). At the transition, only one critical mode q = q_c is unstable.

Fig. 3.

(Top) Phase diagram in the (R,Φ) plane. The outer region corresponds to stable behavior, whereas within the curve, patterning occurs. The solid line is the theoretical phase boundary—Eq. 7—which accurately fits the numerics (black squares). The blue and red sections correspond to continuous and discontinuous transitions, respectively. The two magenta dots correspond to two 2D simulations that show ordered harmonic patterns close to supercriticality and amorphous patterns otherwise. (Bottom Left) Transition in the supercritical regime. The blues lines correspond to the theory—Eq. 8—whereas the squares come from simulations (Φ = 1.5, 1.35, 1.2 from top to bottom). (Bottom Right) Transition in the subcritical regime for Φ = 1.06 and Φ = 1.7 (bottom to top).

Fig. 4.

Numerical results for a 2D simulation with size equal to (in dimensionless units, see Eq. 5) 28 × 28, R ≃ 316, and Φ = 1.5. Times corresponding to the snapshots are (from left to right) , 0.12, 0.20, and 0.51. The color bar shows values of the dimensionless density u; see Eq. 5.

Fig. 5.

Plot of the characteristic domain size, , as a function of time, (both in dimensionless units), for a system in the inhomogeneous phase, with initially random density fluctuations around ρ₀. Parameters were R ∼ 316 and Φ = 1.35, whereas the system size was 35.5 × 35.5 (in dimensionless units). The solid line corresponds to a single run, whereas the dashed line is an average over six runs. The steps in the single run curve correspond to evaporation-condensation events, highlighted by black squares in the snapshots shown in the figure (before and after one of the steps, respectively, arrows indicate positions on the plot corresponding to the two snapshots). The color bar shows values of the dimensionless density u.

Fig. 6.

Dynamics of formation of patterns in 2D, starting from a single small bacterial droplet in the middle of the simulation sample. (Top) Formation of rings in a system with R = 100 and Φ = 1.65. The simulation box has size 125 × 125 (in dimensionless units). The snapshots correspond to times equal to (from left to right) , 5, 10, and 27. (Bottom) Breakage of rings into dots. The four snapshots correspond to the time evolution of a system with R = 100 and Φ = 1.3. We show a 125 × 125 fraction of the simulation box, with the boundaries far away and not affecting the pattern. The snapshots correspond to times equal to (from left to right) , 7, 29, and 122. For both rows, the color bar shows values of the dimensionless density u.