Multiscale Modelling of De Novo Anaerobic Granulation.

A multiscale mathematical model is presented to describe de novo granulation, and the evolution of multispecies granular biofilms, in a continuously fed bioreactor. The granule is modelled as a spherical free boundary domain with radial symmetry. The equation governing the free boundary is derived from global mass balance considerations and takes into account the growth of sessile biomass as well as exchange fluxes with the bulk liquid. Starting from a vanishing initial value, the expansion of the free boundary is initiated by the attachment process, which depends on the microbial species concentrations within the bulk liquid and their specific attachment velocity. Nonlinear hyperbolic PDEs model the growth of the sessile microbial species, while quasi-linear parabolic PDEs govern the dynamics of substrates and invading species within the granular biofilm. Nonlinear ODEs govern the evolution of soluble substrates and planktonic biomass within the bulk liquid. The model is applied to an anaerobic, granular-based bioreactor system, and solved numerically to test its qualitative behaviour and explore the main aspects of de novo anaerobic granulation: ecology, biomass distribution, relative abundance, dimensional evolution of the granules and soluble substrates, and planktonic biomass dynamics within the bioreactor. The numerical results confirm that the model accurately describes the ecology and the concentrically layered structure of anaerobic granules observed experimentally, and that it can predict the effects on the process of significant factors, such as influent wastewater composition; granulation properties of planktonic biomass; biomass density; detachment intensity; and number of granules.

Continuum and discrete approach in modeling biofilm development and structure: a review.

The scientific community has recognized that almost 99% of the microbial life on earth is represented by biofilms. Considering the impacts of their sessile lifestyle on both natural and human activities, extensive experimental activity has been carried out to understand how biofilms grow and interact with the environment. Many mathematical models have also been developed to simulate and elucidate the main processes characterizing the biofilm growth. Two main mathematical approaches for biomass representation can be distinguished: continuum and discrete. This review is aimed at exploring the main characteristics of each approach. Continuum models can simulate the biofilm processes in a quantitative and deterministic way. However, they require a multidimensional formulation to take into account the biofilm spatial heterogeneity, which makes the models quite complicated, requiring significant computational effort. Discrete models are more recent and can represent the typical multidimensional structural heterogeneity of biofilm reflecting the experimental expectations, but they generate computational results including elements of randomness and introduce stochastic effects into the solutions.

Mathematical modeling of dispersal phenomenon in biofilms.

A mathematical model for dispersal phenomenon in multispecies biofilm based on a continuum approach and mass conservation principles is presented. The formation of dispersed cells is modeled by considering a mass balance for the bulk liquid and the biofilm. Diffusion of these cells within the biofilm and in the bulk liquid is described using a diffusion-reaction equation. Diffusion supposes a random character of mobility. Notably, biofilm growth is modeled by a hyperbolic partial differential equation while the diffusion process of dispersed cells by a parabolic partial differential equation. The two are mutually connected but governed by different equations that are coupled by two growth rate terms. Three biological processes are discussed. The first is related to experimental observations on starvation induced dispersal [1]. The second considers diffusion of a non-lethal antibiofilm agent which induces dispersal of free cells. The third example considers dispersal induced by a self-produced biocide agent.

Modeling multispecies biofilms including new bacterial species invasion.

A mathematical model for multispecies biofilm evolution based on continuum approach and mass conservation principles is presented. The model can describe biofilm growth dynamics including spatial distribution of microbial species, substrate concentrations, attachment, and detachment, and, in particular, is able to predict the biological process of colonization of new species and transport from bulk liquid to biofilm (or vice-versa). From a mathematical point of view, a significant feature is the boundary condition related to biofilm species concentrations on the biofilm free boundary. These data, either for new or for already existing species, are not required by this model, but rather can be predicted as results. Numerical solutions for representative examples are obtained by the method of characteristics. Results indicate that colonizing bacteria diffuse into biofilm and grow only where favorable environmental conditions exist for their development.