23 July 2020
SIAM Journal on Applied Mathematics
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.
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