Modeling biodegradation and growth of microorganisms via particle tracking

Biologically mediated degradation of organic compounds is heavily non-linear. When an organic compound is degraded part of the carbon is present in the form of metabolite while a fraction of it is used to increase the biomass, capable then to enhance the degradation process. The rate of biomass growth is usually modeled with the experimentally derived Monod equation, so that it is proportional to the actual existing biomass multiplied by a non-linear factor in terms of available organic matter. The non-linearity in the degradation equation implies a strong difficulty in directly implementing a numerical solution within a Lagrangian framework. Thus, numerical solutions have traditionally been sought in an Eulerian framework.

Here we pursue a fully Lagrangian solution to the problem. First, the Monod empirical equation is derived from a two-step reaction ( B+C → ^k1 BC → ^k2 B +  ΔB + P); while the approach is less general to other derivations existing in the literature, it allows two things: (1) providing some physical meaning to the actual parameters in Monod equation, and more interestingly (2) formulate a methodology for the solution of the degradation equation incorporating Monod kinetics by means of a particle tracking formulation. For the latter purpose, reactants and biomass are represented by particles, and their location at any given time is represented by a kernel that includes the uncertainty in the actual physical location. By solving the reaction equation in a kernel framework, we can reproduce the Monod kinetics and, as a particular result in the case of no biomass growth is allowed, the Michaelis–Menten kinetics. We show how the method is successfully applied to reproduce two studies of microbially induced degradation. First, the observed kinetics of Pseudomonas putida F1 in batch reactors while growing on benzene, toluene and phenol, and second, the column study of carbon tetrachloride biodegradation by the denitrifying bacterium Pseudomonas Stutzeri KC.