Cellular automaton modeling of biological pattern formation

Abstract

Examples of biological pattern formation are life cycles of bacteria and social amoebae, embryonic tissue formation, wound healing or tumour growth. Thereby, development of a particular spatio‐temporal "multi‐cellular" pattern may be interpreted as cooperative phenomenon emerging from an intricate interplay of local (e.g. by adhesion) and non‐local (e.g. via diffusing signals) cell interactions. Mathematical models are required for the analysis of cooperative phenomena. Typical modeling attempts focus on a macroscopic perspective, i.e. the models (e.g. partial differential equations) describe the spatio‐temporal dynamics of cell concentrations. More recently, cell‐based models have been suggested in which the fate of each individual cell can be tracked. Cellular automata are discrete dynamical systems and may be utilized as cell‐based models. Here, we analyze spatio‐temporal pattern formation in cellular automaton models of interacting discrete cells. In particular, we introduce lattice‐gas cellular automata. In the same spirit as cellular automata, lattice‐gas cellular automaton (LGCA) and lattice Boltzmann (LB) models are promising tools for studying transport and interaction processes in biological systems. In these models the update rule is split into two parts which are called collision (interaction) and propagation, respectively. The collision rule of LGCA can be compared with the update rule for CA in that it assigns new states to each cell based on the states of the sites in a local neighborhood. After the collision step the state of each node is propagated to a neighboring node. This split of the update rule guarantees propagation of quantities while keeping the rules simple. The desired behaviour, e.g. spatio‐temporal pattern formation of a LGCA shows up in the macroscopic limit which can be derived from a theory of statistical mechanics on a lattice. Instead of discrete particles, LB models deal with continuous distribution functions which interact locally and which propagate after "collision" to the next neighbour node. LB models can be interpreted as mean‐field approximations of LGCA. Model applications are bacterial pattern formation and tumour growth.Ref.:A. Deutsch, S. Dormann (2005) Cellular automaton modelling of biological pattern formation, Birkhauser, Boston