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A Systematic Upscaling of Nonlinear Chemical Uptake Within a Biofilm
SIAM Journal on Applied Mathematics ( IF 1.9 ) Pub Date : 2020-07-23 , DOI: 10.1137/19m130220x Mohit P. Dalwadi , John R. King
SIAM Journal on Applied Mathematics ( IF 1.9 ) Pub Date : 2020-07-23 , DOI: 10.1137/19m130220x Mohit P. Dalwadi , John R. King
SIAM Journal on Applied Mathematics, Volume 80, Issue 4, Page 1723-1750, January 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.
中文翻译：
生物膜内非线性化学吸收的系统放大
SIAM应用数学杂志，第80卷，第4期，第1723-1750页，2020年1月。
当将化学物种转移到生物膜中细菌菌落的模型中时，将每种细菌作为点汇处理在计算上都是昂贵的，更不用说捕获每种细菌的有限性质了。取而代之的是，模型倾向于将细菌和细胞外基质域视为单一相，在该相上进行有效的大量摄取。在本文中，我们系统地推导了控制该系统的有效方程式，从化学物质通过有限大小细菌菌落扩散的微观问题开始，化学物质也可以在其中扩散。每种细菌内的摄取是浓度的非线性函数。在整个细菌膜上，浓度通量是守恒的，并且浓度比是恒定的。我们使用均质化通过多种规模化方法对该系统进行规模升级，分别研究了有效吸收和有效扩散率的两个显着极限。这项工作是Dalwadi等人的自然续集。[SIAM J. Appl。Math。，78（2018），1300--1329]，当前工作的主要区别是细菌内的非线性吸收和跨细菌膜的一般分配系数。前者导致更复杂的一般渐近分析，而后者则导致两个先前的极限合并。我们列出了在该系统中可能发生的不同类型的微观行为，以及它们对可观测的宏观摄取的影响。尤其是，
更新日期：2020-07-28
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.
中文翻译：
生物膜内非线性化学吸收的系统放大
SIAM应用数学杂志，第80卷，第4期，第1723-1750页，2020年1月。
当将化学物种转移到生物膜中细菌菌落的模型中时，将每种细菌作为点汇处理在计算上都是昂贵的，更不用说捕获每种细菌的有限性质了。取而代之的是，模型倾向于将细菌和细胞外基质域视为单一相，在该相上进行有效的大量摄取。在本文中，我们系统地推导了控制该系统的有效方程式，从化学物质通过有限大小细菌菌落扩散的微观问题开始，化学物质也可以在其中扩散。每种细菌内的摄取是浓度的非线性函数。在整个细菌膜上，浓度通量是守恒的，并且浓度比是恒定的。我们使用均质化通过多种规模化方法对该系统进行规模升级，分别研究了有效吸收和有效扩散率的两个显着极限。这项工作是Dalwadi等人的自然续集。[SIAM J. Appl。Math。，78（2018），1300--1329]，当前工作的主要区别是细菌内的非线性吸收和跨细菌膜的一般分配系数。前者导致更复杂的一般渐近分析，而后者则导致两个先前的极限合并。我们列出了在该系统中可能发生的不同类型的微观行为，以及它们对可观测的宏观摄取的影响。尤其是，
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