Modeling and Simulation of a Bacterial Biofilm That Is Controlled by pH and Protonated Lactic Acids

Abstract

We present a mathematical model for growth and control of facultative anaerobic bacterial biofilms in nutrient rich environments. The growth of the microbial population is limited by protonated lactic acids and the local pH value, which in return are altered as the microbial population changes. The process is described by a non-linear parabolic system of three coupled equations for the dependent variables biomass density, acid concentration and pH. While the equations for the dissolved substrates are semi-linear, the equation for bacterial biomass shows two non-linear diffusion effects, a power law degeneracy as the dependent variable vanishes and a singularity in the diffusion coefficient as the dependent variable approaches its a priori known threshold. The interaction of both effects describes the spatial spreading of the biofilm. The interface between regions where the solution is positive and where it vanishes is the biofilm/bulk interface. We adapt a numerical method to explicitly track this interface in x -t space, based on the weak formulation of the biofilm model in a moving frame. We present numerical simulations of the spatio-temporal biofilm model, applied to a probiotic biofilm control scenario. It is shown that in the biofilm neighbouring regions co-exist in which pathogenic bacterial biomass is produced or killed, respectively. Furthermore, it is illustrated how the augmentation of the bulk with probiotic bacteria leads to an accelerated decay of the pathogenic biofilm.

A Mixed‐Culture Model of a Probiotic Biofilm Control System

“…observed in Eberl and Sudarsan [14]. This confirms the findings of the simpler onedimensional study [22], where the growth limiting substrates C and P were controlled from the liquid phase above the biofilm. Mathematically this can be explained by the maximum principle.…”

“…The mathematical model that we study here is a major extension of a biofilm control model that was introduced previously in Khassehkhan and Eberl [22], based on the biocontrol model for suspended cultures [6] and the biofilm model [15]. We allow for mixed culture biofilms, formed by pathogen and control agent.…”

“…As in Khassehkhan and Eberl [22], we assume that nutrients and other required substances are available to the bacteria in abundance. Biofilm growth and decay are assumed to be controlled only by protonated lactic acids and local pH.…”

“…Note that with these assumptions, if the initial colonies are separated by species, the biofilm model reduces locally to the single species model that was introduced in Khassehkhan and Eberl [22], as long as the local growth conditions are positive or neutral.…”

“…More specifically we describe a model for the modulation of lactic acid concentration and pH by a bio-control agent that is more tolerant than the pathogens. In a previous modelling study [22], we described a similar system, consisting of a biofilm forming pathogen and a bio-control agent that was assumed to be suspended in the bulk liquid. Here we propose a mathematical model for a mixed culture biofilm system, consisting of a pathogenic biofilm former and a probiotic biofilm former.…”