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.
When non-motile cells grow, they form dense aggregates such as healthy tissues, tumors, biofilms, microbial mats, and colonies. The growth dynamics of such aggregates influence diverse phenomena, including disease onset and progression, agricultural productivity, geochemical cycles, and the integrity of human-built infrastructure [1][2][3][4][5][6][7][8]. Consequently, understanding these dynamics has been a focus of extensive research, employing both detailed application-specific models and simpler phenomenological frameworks aimed at uncovering general principles of population growth [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].
Among these approaches, reaction-diffusion equations have emerged as the dominant modeling paradigm, because they effectively incorporate nutrient diffusion, cellular growth and motility, mechanical interactions, and other key processes. Theoretical predictions has been most thoroughly tested in the context of microbial colonies due to their accessibility for quantitative measurement and manipulation. In particular, reactiondiffusion models have successfully explained complex pattern formations [10][11][12][13]15] and-perhaps most notablythe observed nearly constant expansion velocity of microbial colonies [24][25][26][27]. This constant front velocity is a striking prediction resulting from the interplay of diffusive transport and exponential growth.
Recent theoretical work has focused on how various biophysical processes, especially mechanical interactions, influence expansion velocity [17][18][19][20][21][22][23]. However, an increasing number of experiments suggest that the commonly assumed linear growth is not universal. In particular, many organisms under diverse growth conditions exhibit sublinear, power-law growth with an exponent close to one-half [26,[28][29][30]. Here, we demonstrate that these experimental observations can be reconciled within the standard reaction-diffusion framework by incorporating the experimentally motivated dependence of biomass redistribution on nutrient concentration-a factor largely overlooked in previous models.
Although there are a great number of reactiondiffusion models of colony growth, they typically fall into one of three classes. The first class includes various generalizations of the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equation [31][32][33]:
Here, the growth rate of biomass b is approximated by the standard logistic curve, which consists of exponential growth at low b and saturation at carrying capacity K.
The value of K is set by the initial nutrient concentration, which is not modeled explicitly. The motility is assumed to follow a random-walk-like pattern, with the effective diffusion constant given by D s . This classic equation was the first model of reaction-diffusion waves in population biology and motivated numerous subsequent studies in various fields [33,34]. It predicts invariant traveling fronts moving with velocity v = 2 √ D s r and an exponentially decreasing population density ahead of the wave. These predictions have been confirmed in many experimental and observational studies [25,33,[35][36][37][38], but only with motile organisms, e.g., bacteria swimming in very thin agar. In dense microbial colonies, the outward motion of cells is not diffusive, and population density abruptly drops to zero instead of showing a more grad-ual exponential decrease [9,[17][18][19][20][21][22][23]26].
To capture the sharp drop of the biomass at the front, density dependence was introduced in the diffusion term of the FKPP equation [33,34]:
where the new parameter D p quantifies the emergent cooperative motility of the cells and could depend on many factors such as the agar concentration, surfactant production, and cell rigidity. Phenomenologically, the nonlinear diffusion could be explained by collective motion due to the repeated rearrangements of cells within the colony as they push against each other. Alternatively, the nonlinear diffusion can be derived from a hydrodynamic model that involves mechanical compression due to growth, friction with the substrate, and the flow of the biomass in response to mechanical forces (see the Supplemental Material [39]). The front velocity in this model equals D p r/2, and the population density vanishes linearly near the colony edge [34,[40][41][42]; power law decay is also possible for slightly different models [39]. Although Eq. (2) recapitulated the growth of circular colonies reasonably well, it could not reproduce two essential aspects of colony growth. First, colonies stop growing well before reaching the edge of the Petri dish, and, second, colonies exhibit non-circular (rough or branched) morphologies at low nutrient and high agar concentrations [10][11][12]29]. Both of these observations can be explained by nutrient limitation [9][10][11][12], which is introduced in the third class of models:
Here, n is the concentration of the growth-limiting nutrient, D n
This content is AI-processed based on open access ArXiv data.