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。

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.