Transition from traveling fronts to diffusion-limited growth in expanding populations

TL;DR

The model applies to dense cellular aggregates of nonmotile cells consuming a diffusible nutrient and provides a potential explanation for the linear rather than quadratic increase of colony area with time, which has been observed for many microbes.

Abstract

Reaction-diffusion equations describe various spatially extended processes that unfold as traveling fronts moving at constant velocity. We introduce and solve analytically a model that, besides such fronts, supports solutions advancing as the square root of time. These sublinear fronts preserve an invariant shape, with an effective diffusion constant that diverges at the transition to linear spreading. The model applies to dense cellular aggregates of nonmotile cells consuming a diffusible nutrient. The sublinear spread results from biomass redistribution slowing due to nutrient depletion, a phenomenon supported experimentally but often neglected. Our results provide a potential explanation for the linear rather than quadratic increase of colony area with time, which has been observed for many microbes.

Figures (11)

• Figure 1: (color online). Transition between traveling fronts and diffusion-limited growth. (a) Equation \ref{['linear_model']} supports two types of solutions with the colony radius $R$ increasing either linearly with time or as a square root of time. (b) The coefficient of determination ($R^2$) indicates that $R\propto \sqrt{t}$ for $D<D_c$, and $R\propto t$ for $D>D_c$. Examples of biomass and nutrient profiles in the radial direction, at 10%, 50%, and 90% of the time preceding nutrient depletion at the end of the simulation box are shown in (c) for $D<D_c$ and in (d) for $D>D_c$.
• Figure 2: (color online). Velocity of the traveling front solutions. Circles show the velocities obtained by performing a linear fit on $R(t)$, determined by solving Eq. \ref{['linear_model']} (PDE). These calculations were performed at two systems sizes. The disagreement between them indicates that a traveling front solution does not exist. The dashed line shows the results of the shooting method (ODE), and the solid line is the analytical approximation given by Eq. \ref{['v_analytic']}.
• Figure 3: Biomass diffusivity controls the height and expansion rate of microbial colonies in the regime of square-root growth, as demonstrated for simulations in one and two dimensions. (a) The colony height, defined as the maximal concentration near the edge of the colony, diverges for $D\to0$, as expected for a stationary colony, and it approaches $1$ for $D \to D_c$, as expected for a traveling front solution. The solid line shows the analytical prediction given by Eq. \ref{['H_prediction']}. (b) The rate of colony expansion, quantified by $\varkappa=r_e/(2\sqrt{t})$, increases with $D$. It approaches zero for $D=0$ and diverges at $D\to D_c$. Theoretical predictions based on Eq. \ref{['height']} are shown with a solid line for 1d and a dashed line for 2d, using numerically obtained values of $H$.
• Figure S1: Transition from traveling fronts to diffusion-limited growth is a general feature of nutrient-dependent diffusion. For different models, in 1 dimension, we show the increase of biomass over time at low and high rates of biomass redistribution (left column), and the corresponding biomass and nutrient profiles at 10%, 50%, and 90% of the time preceding nutrient depletion at the end of the simulation box. Models without a maintenance cost (rows (a) and (b)) exhibit both a linear and a square-root increase of the total biomass with time. Introducing a maintenance cost (row (c)) drastically changes the behavior at low dispersal. The nutrient concentration at the edge of the colony decreases over time until it drops below the maintenance cost threshold, at which point the biomass stops growing. For the traveling-front growth, the biomass never reaches the maximum value set by the initial concentration of nutrients because of the maintenance cost.
• Figure S2: The model introduced in Eq. \ref{['mvs_model']} in the main text, with the motility independent of nutrient concentration, does not exhibit the transition seen in our model. There is a traveling wave solution at low $D$ with a velocity that scales linearly with $D$ for $D\ll1$muller:Db_model. The uptick seen at low $D$ for the square-root fitting is due to rapidly increasing transient times as $D \to 0$.