Distinguishable spreading dynamics in microbial communities

TL;DR

The work suggests that cell-level growth limitations can be inferred from population-level dynamics, and it offers a methodology for connecting these two scales.

Abstract

A packed community of exponentially proliferating microbes will spread in size exponentially. However, due to nutrient depletion, mechanical constraints, or other limitations, exponential proliferation is not indefinite, and the spreading slows. Here, we theoretically explore a fundamental question: is it possible to infer the dominant limitation type from the spreading dynamics? Using a continuum active fluid model, we consider three limitations to cell proliferation: intrinsic growth arrest (e.g., due to sporulation), pressure from other cells, and nutrient access. We find that memoryless growth arrest still results in superlinear (accelerating) spreading, but at a reduced rate. In contrast, pressure-limited growth results in linear (constant-speed) spreading in the long-time limit. We characterize how the expansion speed depends on the maximum growth rate, the limiting pressure value, and the effective fluid friction. Interestingly, nutrient-limited growth results in a phase transition: depending on the nutrient supply and how efficiently nutrient is converted to biomass, the spreading can be either superlinear or sublinear (decelerating). We predict the phase boundary in terms of these parameters and confirm with simulations. Thus, our results suggest that when an expansion slowdown is observed, its dominant cause is likely nutrient depletion. More generally, our work suggests that cell-level growth limitations can be inferred from population-level dynamics, and it offers a methodology for connecting these two scales.

Figures (9)

• Figure 1: Setup and three types of growth limitation. (A) We model a biofilm as a radially expanding active fluid and ask how the growth limitation type determines the expansion dynamics. (B) Type 1: temporal growth arrest, e.g. by sporulation at a fixed rate. (C) Type 2: Pressure-limited growth. Pressure is highest in the center and expansion speed is highest at the edge. (D) Type 3: Nutrient-limited growth. Diffusing nutrients are consumed by cells, leading to cell growth and biofilm expansion.
• Figure 2: Temporal growth arrest. (A) Radial expansion for different sporulation rates $k$, from numerical solution and analytic expression (Eq. \ref{['eq:R1']}). We see that growth arrest from sporulation slows expansion, but expansion remains superlinear in time. (B) Snapshot of density fractions $\rho_1(r,t)/c$ (orange) and $\rho_2(r,t)/c$ (purple) over distance $r$ with $k = 0.2$/h. We see that the densities are uniform throughout the biofilm. Parameters are $g=1$/h and $c=1$/$\mu$m$^2$, and the biofilm starts at size $R_0=1$$\mu$m with all growing cells, i.e., $\rho_1(r,0) = c\Theta(R_0-r)$.
• Figure 3: Pressure-limited growth. (A) Radial expansion for different parameters (see legend). We see that expansion is linear in time at long times. (B) Fitted speed vs. $\xi/gp_0$. Black line shows predicted dependence from Eq. \ref{['eq:linear_speed']}. Parameters are $c=1$/$\mu$m$^2$ and $R_0=1$$\mu$m. In A, $g=1$/h. In B, $\xi$ is varied for several choices of $g$ and $p_0$.
• Figure 4: Nutrient-limited growth with a large system size $L\gg\sqrt{D/g}$. (A) Radial expansion from numerical solution for different values of metabolic conversion parameter $\gamma$ (here $n_0=2.4$/$\mu$m$^2$ and $c=1$/$\mu$m$^2$). Color indicates whether curve is superlinear or sublinear at long times (last $0.1$ s of curve). (B) Phase diagram for many values of $\gamma$ and $n_0$ (here $c=1$/$\mu$m$^2$), along with predicted phase boundaries $n_0 = \gamma c$ (dashed, Eq. \ref{['eq:phase']}) and $n_0 = 3\gamma c$ (solid, Eq. \ref{['acc2']}) in the slow- and fast-diffusion regimes, respectively. We see agreement with the slow-diffusion boundary, as expected for the large system size.Other parameters are $L = 6000$$\mu$m, $g=1$/h, $D = 100$$\mu$m$^2$/s, $n_g=1$$\mu$M, and $R_0 = 6$$\mu$m.
• Figure 5: Nutrient-limited growth with a small system size $L\ll\sqrt{D/g}$. (A) Radial expansion from numerical solution for different values of metabolic conversion parameter $\gamma$ (here $n_0=2.4$/$\mu$m$^2$ and $c=1$/$\mu$m$^2$). Color indicates whether curve is superlinear or sublinear at long times (last $0.1$ s of curve). (B) Phase diagram for many values of $\gamma$ and $n_0$ (here $c=1$/$\mu$m$^2$), along with predicted phase boundaries $n_0 = \gamma c$ (dashed, Eq. \ref{['eq:phase']}) and $n_0 = 3\gamma c$ (solid, Eq. \ref{['acc2']}) in the slow- and fast-diffusion regimes, respectively. We see agreement with the fast-diffusion boundary, as expected for the small system size. Other parameters are $L = 180$$\mu$m, $g=1$/h, $D = 100$$\mu$m$^2$/s, $n_g=1$$\mu$M, and $R_0 = 1$$\mu$m.