Cellular automata bio lm with included quorum
sensing
Erik Alpkvist, Anders Heyden, Niels Chr. Overgaard
Applied Mathematics Group
School of Technology and Society
Malmo Univeristy
SE-20506 Malmo
email: erik.alpkvist@ts.mah.se
Jens Nilsson
Centre for Mathematical Sciences
Lund University
Box 118
SE-22100 Lund
email: jensn@maths.lth.se
Abstract
The spatial heterogeneous structure in bio lm raises the following question: What rules govern the development of such structures? Some likely candidates are: (a) The interaction with the surrounding liquid. (b) Quorum sensing. (c) The dynamics of transport and consumption of nutrients. Most models consider the bio lm divided into two domains, the biomass domain occupied with biomass and the bulk liquid domain being the surrounding liquid. In this paper we adopt a discrete cellular automata model, where in addition to the nutrient eld we model a con-centration eld of signalling substances. Cellular automata bio lm models are discrete models where the biomass occupies space in an ordered way and nutrient components are represented as a continuous eld. The whole space is subdivided into a point grid. At each point in the biomass domain a quantity of biomass is assumed to be located in a gel bed, represented as a matrix of biomass density. The density matrix is determined from the uptake of nutrient from a relaxed nutrient eld modelled as a continuum and rules for division and shoving of each element. The production of cell-to-cell signalling substances such as quorum sensing molecules is as-sumed to be dependent on the density of the bio lm colony. Thus a model capable of describing a varying density for the colony is required for a re-alistic model of quorum sensing. To be more precise the proposed model contains three dierent steps: (i) the quorum sensing external concen-tration is described by a partial dierential equation in the biomass and bulk liquid region coupled with an ordinary dierential equation for the intracellular concentration describing the intracellular dynamics, (ii)the nutrient concentration is modelled using an ordinary diusion equation
and (iii) the bio lm growth is modelled as a function of nutrient concen-tration and the sensing concenconcen-tration. The cellular densities are necessary to account for dierences in volume fraction for intracellular and extra-cellular space in the model. Together with boundary condition for the partial dierential equation the system is well de ned. Simulations of this model are presented in this paper. The simulations are in two space dimensions in a rectangular domain. Moreover the quorum sensing con-centration is believed to be linked the production of exopolysaccharide and consequently bacterial density. This aects the diusion of nutrients and thus the structure of the bio lm. We present the relaxed quorum sensing eld for dierent spatial structures of the biomass region in an evolving bio lm. Other models have shown the concentration of quorum sensing to switch between dierent steady-states of high and low concen-tration of signalling substance with varying density. These observations are studied for this model and evaluated.
