Stochastic modeling of some aspects of biofilm behavior

Abstract

A unified stochastic description of the effects of internal and external fluctuations on the thickness and roughness of a biofilm is given in terms of linear and nonlinear master equations ($ME$). In the absence of detachment the $ME$ is linear, while erosion renders it to be nonlinear. For the linear case the influence of the environment is modeled through an external noise in one of the transition probabilities per unit time and the $% ME$ is solved analytically. For the nonlinear case we only consider internal fluctuations and use van Kampen's systematic expansion to solve the $ME$. In both cases the thickness and roughness dependence on time is calculated and expressed in terms of the first two moments of the probability distribution function. An analytical expression for roughness as a function of thickness is also obtained in both cases. For both cases we compare our analytical results with reported experimental measurements of these quantities for $P$. Aeruginosa. The best fitting values of the transition probabilities and external noise parameters are determined, so that the relative error $% \delta $ between the calculated and the experimentally measured values of the thickness and roughness is minimized. We find that for the linear case the mean relative error $<\delta >$ is relatively small, 1.8 %-6.2 %, while in the presence of detachment is slightly higher, 6.7 %- 9.3 %. We close the paper by discussing the advantages, scope and limitations of our approach.