Hermann J. Eberl

Mathematical modelling of biofilm structures

The morphology of biofilms received much attention in the last years. Several concepts to explain the development of biofilm structures have been proposed. We believe that biofilm structure formation depends on physical as well as general and specific biological factors. The physical factors (e.g. governing substrate transport) as well as general biological factors such as growth yield and substrate conversion rates are the basic factors governing structure formation. Specific strain dependent factors will modify these, giving a further variation between different biofilm systems. Biofilm formation seems to be primarily dependent on the interaction between mass transport and conversion processes. When a biofilm is strongly diffusion limited it will tend to become a heterogeneous and porous structure. When the conversion is the rate-limiting step, the biofilm will tend to become homogenous and compact. On top of these two processes, detachment processes play a significant role. In systems with a high detachment (or shear) force, detachment will be in the form of erosion, giving smoother biofilms. Systems with a low detachment force tend to give a more porous biofilm and detachment occurs mainly by sloughing. Biofilm structure results from the interplay between these interactions (mass transfer, conversion rates, detachment forces) making it difficult to study systems taking only one of these factors into account.

A three-dimensional numerical study on the correlation of spatial structure, hydrodynamic conditions, and mass transfer and conversion in biofilms

Abstract A three-dimensional model for convection, diffusion, and reaction in a porous, heterogeneous system has been implemented. It is used to analyse the influence of hydrodynamics and structural heterogeneities on mass transfer and conversion processes of solutes in biofilm systems. The mathematical model comprises the full incompressible Navier–Stokes equations and mass transfer with nonlinear reactions in the biofilm. It is found that increased biofilm surface roughness means decreased mass conversion in the solid biofilm. Secondly, a correlation between bulk flow Reynolds number and the Sherwood number for mass transfer across the irregular liquid/solid interface is formulated. In a further study the contribution of convective transport to overall mass transfer from bulk liquid into the biofilm was analysed. The main result was that the experimental observation of high convective flux of solutes in biofilm channels not necessarily is coupled with an equally high net convective contribution to mass transfer from bulk liquid into biofilm.

A mathematical model of quorum sensing regulated EPS production in biofilm communities

BackgroundBiofilms are microbial communities encased in a layer of extracellular polymeric substances (EPS). The EPS matrix provides several functional purposes for the biofilm, such as protecting bacteria from environmental stresses, and providing mechanical stability. Quorum sensing is a cell-cell communication mechanism used by several bacterial taxa to coordinate gene expression and behaviour in groups, based on population densities.ModelWe mathematically model quorum sensing and EPS production in a growing biofilm under various environmental conditions, to study how a developing biofilm impacts quorum sensing, and conversely, how a biofilm is affected by quorum sensing-regulated EPS production. We investigate circumstances when using quorum-sensing regulated EPS production is a beneficial strategy for biofilm cells.ResultsWe find that biofilms that use quorum sensing to induce increased EPS production do not obtain the high cell populations of low-EPS producers, but can rapidly increase their volume to parallel high-EPS producers. Quorum sensing-induced EPS production allows a biofilm to switch behaviours, from a colonization mode (with an optimized growth rate), to a protection mode.ConclusionsA biofilm will benefit from using quorum sensing-induced EPS production if bacteria cells have the objective of acquiring a thick, protective layer of EPS, or if they wish to clog their environment with biomass as a means of securing nutrient supply and outcompeting other colonies in the channel, of their own or a different species.

Exposure of biofilms to slow flow fields: The convective contribution to growth and disinfection

A previously introduced degenerate diffusion-reaction model of biofilm growth and disinfection is extended to account for convective transport of oxygen and disinfectants in an aqueous environment. To achieve this in a computationally efficient manner we employ a thin-film approximation to the (Navier)-Stokes equations that can be solved analytically. In numerical experiments, we investigate how the convective transport of nutrients and disinfectants due to bulk flow hydrodynamics affects the balance between growth and disinfection processes. It is found that the development of biofilms can be significantly affected by the flow field even at extremely low Reynolds numbers. While it is natural to expect that increased bulk flow velocities imply increased mass transfer of both, nutrients and disinfectants, and hence an acceleration of both, growth and decay of biomass, it is found, furthermore, that in many instances the actual flow conditions, determine the success or failure of disinfection, i.e. persistence or extinction of a biofilm community.

A Mathematical Model of Quorum Sensing Induced Biofilm Detachment

Background Cell dispersal (or detachment) is part of the developmental cycle of microbial biofilms. It can be externally or internally induced, and manifests itself in discrete sloughing events, whereby many cells disperse in an instance, or in continuous slower dispersal of single cells. One suggested trigger of cell dispersal is quorum sensing, a cell-cell communication mechanism used to coordinate gene expression and behavior in groups based on population densities. Method To better understand the interplay of colony growth and cell dispersal, we develop a dynamic, spatially extended mathematical model that includes biofilm growth, production of quorum sensing molecules, cell dispersal triggered by quorum sensing molecules, and re-attachment of cells. This is a highly nonlinear system of diffusion-reaction equations that we study in computer simulations. Results Our results show that quorum sensing induced cell dispersal can be an efficient mechanism for bacteria to control the size of a biofilm colony, and at the same time enhance its downstream colonization potential. In fact we find that over the lifetime of a biofilm colony the majority of cells produced are lost into the aqueous phase, supporting the notion of biofilms as cell nurseries. We find that a single quorum sensing based mechanism can explain both, discrete dispersal events and continuous shedding of cells from a colony. Moreover, quorum sensing induced cell dispersal affects the structure and architecture of the biofilm, for example it might lead to the formation of hollow inner regions in a biofilm colony.

Existence and longtime behavior of a biofilm model

Chapter 5 is devoted to biofilm modelling (meso-scale level), analysis and simulation which is one of the most active areas in modern microbiology. To this end it is enough to refer to: “It is the best of times for biofilm research” (Nature 76, vol. 15, pp. 76–81, 2007). In contrast to existing biofilm models, which are based mostly on discrete rules or hybrid models, we are mainly interested in a deterministic and continuous model which is described by PDEs. Chapter 5 consists of five sections. The first two Sects. 5.1 and 5.5 are concerned with the single species/single substrate models. In Sect. 5.1 we derive governing equations which describe spatial spreading mechanisms of biomass.The feature of these equations is that they are highly nonlinear density-dependent degenerate reaction-diffusion systems comprising two kind of degeneracy: porous medium and fast diffusion. We prove the well-posedness of the obtained equations and study the long-time dynamics of their solutions in terms of a global attractor. Moreover we analyze dependence of solutions on boundary conditions. Our numerical simulations of derived equations lead to mushroom patterns which were observed in the experimental studies.

A Nonlinear Master Equation for a Degenerate Diffusion Model of Biofilm Growth

We present a continuous time/discrete space model of biofilm growth, starting from the semi-discrete master equation. The probabilities of biomass movement into neighboring sites depend on the local biomass density and on the biomass density in the target site such that spatial movement only takes place if (i) locally not enough space is available to accommodate newly produced biomass and (ii) the target site has capacity to accommodate new biomass. This mimics the rules employed by Cellular Automata models of biofilms. Grid refinement leads formally to a degenerate parabolic equation. We show that a set of transition rules can be found such that a previously studied ad hoc density-dependent diffusion-reaction model of biofilm formation is approximated well.

OpenMP parallelization of a mickens time-integration scheme for a mixed-culture biofilm model and its performance on multi-core and multi-processor computers

We document and compare the performance of an OpenMP parallelized simulation code for a mixed-culture biofilm model on a desktop workstation with two quad core Xeon processors, and on SGI Altix Systems with single core and dual core Itanium processors. The underlying model is a parabolic system of highly non-linear partial differential equations, which is discretized in time using a non-local Mickens scheme, and in space using a standard finite difference method.

Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones.

Bacterial biofilms are complex microbial depositions on immersed interfaces that form wherever the environmental conditions sustain microbial growth. Despite their name, biofilms can develop in highly irregular structures. Recently several mathematical concepts have been introduced to model these spatially structured microbial populations. Regardless of the type of model, they all have, even for microbially relatively simple systems, many parameters which generally are known at most approximately. We investigate the effect of uncertainties in model parameters on four morphological and four ecological output parameters using a nonlinear diffusion model for a biofilm in which two species compete for a shared nutrient. To this end we conduct an extensive computer simulation experiment for two different levels of data uncertainty, three different hydrodynamic conditions, and two different scenarios of bulk substrate availability. Our results indicate that input model parameter uncertainties have a much larger effect on ecological than on morphological output parameters.

Longtime behavior of one-dimensional biofilm models with shear dependent detachment rates

We investigate the role of non shear stress and shear stressed based detachment rate functions for the longterm behavior of one-dimensional biofilm models. We find that the particular choice of a detachment rate function can affect the model prediction of persistence or washout of the biofilm. Moreover, by comparing biofilms in three settings: (i) Couette flow reactors, (ii) Poiseuille flow with fixed flow rate and (iii) Poiseuille flow with fixed pressure drop, we find that not only the bulk flow Reynolds number but also the particular mechanism driving the flow can play a crucial role for longterm behavior. We treat primarily the single species-case that can be analyzed with elementary ODE techniques. But we show also how the results, to some extent, can be carried over to multi-species biofilm models, and to biofilm models that are embedded in reactor mass balances.