Mathematical Modeling, Analysis and Simulation of Biofilm Processes

细菌生物膜是微生物沉积在被淹没的表面上（又称基质）。在生物膜形成的最初可逆步骤中，细菌附着在表面。保持黏附的细胞开始产生一种细胞外聚合物质，使它们自身嵌入其中，保护它们免受机械冲刷和抗菌剂的侵害。在这个保护层中，生动的微生物群落发展起来。生物膜很重要，例如对废水处理或土壤修复技术的发展。另一方面，生物膜在医学环境中是有害的，它们可能导致难以根除的细菌感染或卫生问题。尽管它们的名字，生物膜往往不是均匀的膜，但可以在高度不规则的结构中发展。生物膜群落中的生命与悬浮或浮游生物群落中的生命有着本质上的不同，而悬浮或浮游生物群落是实验和数学微生物学传统研究的重点。这在很大程度上是由于生物膜的空间组织，导致底物梯度，因此，空间异质性的生长条件。因此，许多传统的微生物生态学模型，通常被制定为批量或连续培养的ode，不能应用，但必须开发一种完全不同的模型。生物膜的微生物和物理复杂性通常反映在这些模型的数学复杂性中。*文献中提出了几种生物膜的数学模型，采用了非常不同的数学概念和方法。我们的重点将放在密度依赖的扩散反应系统上，我们已经证明，它既可以解释为空间结构的微生物种群，也可以解释为作为复杂流体的生物膜的描述。在其最简单的原型形式中，该模型包含因变量消失时的多孔介质简并，同时包含因变量达到最大细胞密度时的超扩散奇异。我们之前已经为这个原型系统开发了一个解理论，以及它们的数值模拟方法。在此资助期间，生物膜的新方面将被纳入该模型框架，这将导致额外的数学挑战，并需要对这些技术进行实质性的扩展和重新思考。*一个重点将是多物种系统的空间混合。我们对以前的模型进行了修正，以显示问题如何导致我们迄今为止忽略的额外交叉扩散项。另一个重点将是我们模糊地称之为化学诱导的分离（以区别于剪切诱导的分离），包括由细胞间信号传导控制的分离，或由酶分解EPS。这将要求我们同时考虑活动和不动的细菌阶段以及这两种生长模式之间的交换。我们想要包括的第三个方面是细菌降解它们生长的基质的情况，这需要我们考虑反应边界条件。这是在生产生物膜的某些生物燃料中观察到的现象。*我们将研究的生物膜的一些方面是基本的，其他方面与特定系统密切相关。在所有情况下，它们都将受到特定生物膜应用的激励。我们考虑的应用将包括通过纤维素水解生物膜生产生物燃料；废水处理工艺；基于信号的生物膜控制策略地下水保护与土壤修复；食品安全和工业中有害生物膜的生物控制。

Bacterial biofilms are microbial depositions on submerged surfaces (a.k.a substratum). In the initial reversible step of biofilm formation bacteria attach to the surface. Cells that stay adhered start producing an extracellular polymeric substance in which they are themselves embedded and that protects them against mechanical washout and antimicrobials. In this protective layer, vivid microbial communities develop. Biofilms are important, e.g for the development of technologies for wastewater treatment or soil remediation. On the other hand biofilms are detrimental in a medical context, where they can lead to difficult to eradicate bacterial infections or hygienic problems. Despite their name, biofilms are often not homogeneous films but can develop in highly irregular architectures. Life in biofilm communities is substantially different from life in suspended or planktonic populations, on which experimental and mathematical microbiology have traditionally focused. This is largely due to the spatial organisation of biofilms, which leads to substrate gradients, and, hence, to spatially heterogeneous growth conditions. Therefore, many of the traditional models of microbial ecology, typically formulated as ODEs for batch or continuous cultures, cannot be applied, but an entirely different class of models must be developed. The microbial and physical complexity of biofilms is often reflected in the mathematical complexity of these models.*Several mathematical models of biofilms have been proposed in the literature, drawing on very different mathematical concepts and approaches. Our focus will be on density-dependent diffusion-reaction systems, which we have shown can be interpreted both as a spatially structured microbial populations and as a description of biofilms as complex fluids. In its simplest prototype form this model comprises a porous medium degeneracy when the dependent variable vanishes and simultaneously a super-diffusion singularity when the dependent variable reaches maximum cell density. We have developed a solution theory for this protoype system previously, as well as numerical methods for their simulation. Over the duration of this grant new aspects of biofilms will be incorporated in this model framework which lead to additional mathematical challenges and require a substantial extension and re-thinking of these techniques.*One focus will be on spatial mixing in multi-species systems. We revise our previous model to show how the problem leads to additional cross-diffusion terms which we have so far neglected. Another emphasis will be on what we vaguely call chemically induced detachment (to distinguish it from shear induced detachment), including detachment controlled by cell-to-cell signaling, or breakdown of the EPS by enzymes. This will require us to consider concurrently motile and sessile bacterial phases and the exchange between these two modes of growth. A third aspect we want to include is the situation where bacteria degrade the substratum on which they grow, which requires us to consider reactive boundary conditions. This is a phenomenon observed for certain biofuel producing biofilms.*Some of the biofilm aspects that we will study are of fundamental nature, others are closely tied to specific systems. In all cases they will be motivated by particular biofilm applications. Applications that we consider will include biofuel production by cellulolytic biofilms; wastewater treatment processes; signal based biofilm control strategies; groundwater protection and soil remediation; (bio)control of detrimental biofilms in food safety and industry.