A Systematic Upscaling of Nonlinear Chemical Uptake Within a Biofilm
Mohit P. Dalwadi, John R. King · SIAM Journal on Applied Mathematics · 2020
When modeling transport of a chemical species to a colony of bacteria in a biofilm, it is computationally expensive to treat each bacterium even as a point sink, let alone to capture the finite nature of each bacterium. Instead, models tend to treat the bacterial and extracellular matrix domains as a single phase, over which an effective bulk uptake is imposed. In this paper, we systematically derive the effective equations that should govern such a system, starting from the microscale problem of a chemical diffusing through a colony of finite-sized bacteria, within which the chemical species can also diffuse. The uptake within each bacterium is a nonlinear function of the concentration; across the bacterial membrane the concentration flux is conserved and the concentration ratio is constant. We upscale this system using homogenization via the method of multiple scales, investigating the two distinguished limits for the effective uptake and the effective diffusivity, respectively. This work is a natural sequel to Dalwadi et al. [ SIAM J. Appl. Math., 78 (2018), 1300--1329], the main difference in this current work being nonlinear uptake within the bacteria and a general partition coefficient across the bacterial membrane. The former results in a significantly more involved general asymptotic analysis, and the latter results in the merging of two previous distinguished limits. We catalogue the different types of microscale behavior that can occur in this system and the effect they have on the observable macroscale uptake. In particular, we show how the nonlinearities in microscale uptake should be modified when upscaled to an effective uptake and how different microscale uptake properties and behaviors, such as chemically depleted regions within the bacteria, can lead to the same observed uptake.