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A Systematic Upscaling of Nonlinear Chemical Uptake Within a Biofilm

Dalwadi, Mohit P.; King, John R.

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Authors

Mohit P. Dalwadi

JOHN KING JOHN.KING@NOTTINGHAM.AC.UK
Professor of Theoretical Mechanics



Abstract

When modelling transport of a chemical species to a colony of bacteria in a biofilm, it is computationally expensive 4 to treat each bacterium even as a point sink, let alone to capture the finite nature of each bacterium. Instead, models tend to 5 treat the bacterial and extracellular matrix domains as a single phase, over which an effective bulk uptake is imposed. In this 6 paper, we systematically derive the effective equations that should govern such a system, starting from the microscale problem of a 7 chemical diffusing through a colony of finite-sized bacteria, within which the chemical species can also diffuse. The uptake within 8 each bacterium is a nonlinear function of the concentration; across the bacterial membrane the concentration flux is conserved and 9 the concentration ratio is constant. We upscale this system using homogenization via the method of multiple scales, investigating 10 the two distinguished limits for the effective uptake and the effective diffusivity, respectively. This work is a natural sequel to 11 Dalwadi et al. 2018 (SIAM J Appl Math, 78(3), 1300-1329), the main difference in this current work being nonlinear uptake within 12 the bacteria and a general partition coefficient across the bacterial membrane. The former results in a significantly more involved 13 general asymptotic analysis, and the latter results in the merging of two previous distinguished limits. We catalogue the different 14 types of microscale behaviour that can occur in this system and the effect they have on the observable macroscale uptake. In 15 particular, we show how the nonlinearities in microscale uptake should be modified when upscaled to an effective uptake and how 16 different microscale uptake properties and behaviours, such as chemically depleted regions within the bacteria, can lead to the same 17 observed uptake. 18

Citation

Dalwadi, M. P., & King, J. R. (2020). A Systematic Upscaling of Nonlinear Chemical Uptake Within a Biofilm. SIAM Journal on Applied Mathematics, 80(4), 1723-1750. https://doi.org/10.1137/19m130220x

Journal Article Type Article
Acceptance Date Apr 8, 2020
Online Publication Date Jul 23, 2020
Publication Date Jul 23, 2020
Deposit Date Jun 21, 2020
Publicly Available Date Mar 29, 2024
Journal SIAM Journal on Applied Mathematics
Print ISSN 0036-1399
Electronic ISSN 1095-712X
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 80
Issue 4
Pages 1723-1750
DOI https://doi.org/10.1137/19m130220x
Keywords homogenization; multiscale; distinguished limits; effective uptake; quiescent core; depleted core 19 AMS subject classifications 35B25; 35B27; 35B40; 92C45 20
Public URL https://nottingham-repository.worktribe.com/output/4696759
Publisher URL https://epubs.siam.org/doi/10.1137/19M130220X

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