A systematic upscaling of nonlinear chemical uptake within a biofilm

When modelling transport of a chemical species to a colony of bacteria in a biofilm, it is computationally expensive4 to treat each bacterium even as a point sink, let alone to capture the finite nature of each bacterium. Instead, models tend to5 treat the bacterial and extracellular matrix domains as a single phase, over which an effective bulk uptake is imposed. In this6 paper, we systematically derive the effective equations that should govern such a system, starting from the microscale problem of a7 chemical diffusing through a colony of finite-sized bacteria, within which the chemical species can also diffuse. The uptake within8 each bacterium is a nonlinear function of the concentration; across the bacterial membrane the concentration flux is conserved and9 the concentration ratio is constant. We upscale this system using homogenization via the method of multiple scales, investigating10 the two distinguished limits for the effective uptake and the effective diffusivity, respectively. This work is a natural sequel to11 Dalwadi et al. 2018 (SIAM J Appl Math, 78(3), 1300-1329), the main difference in this current work being nonlinear uptake within12 the bacteria and a general partition coefficient across the bacterial membrane. The former results in a significantly more involved13 general asymptotic analysis, and the latter results in the merging of two previous distinguished limits. We catalogue the different14 types of microscale behaviour that can occur in this system and the effect they have on the observable macroscale uptake. In15 particular, we show how the nonlinearities in microscale uptake should be modified when upscaled to an effective uptake and how16 different microscale uptake properties and behaviours, such as chemically depleted regions within the bacteria, can lead to the same17 observed uptake.