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

TL;DR

This paper introduces an analytical model showing a transition from traveling front solutions to diffusion-limited, sublinear growth in expanding populations, explaining experimental observations of microbial colony spreading.

Contribution

The study presents a new model supporting both traveling fronts and sublinear growth, elucidating the transition mechanism and its biological implications.

Findings

Sublinear fronts preserve shape with diverging effective diffusion.

Nutrient depletion causes biomass redistribution slowing.

Model explains linear colony growth observed in 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.