Cellular Automata and Discrete Particle Systems

Applied to Biology

Modeling in biology requires the development of new techniques which reflect the uniqueness of biological systems. Discrete individual-based models (discrete particle systems) are easily applied to a wide range of biological problems since they reflect the intrinsic individuality of particles and are straight-forward to build from a microscopic level understanding of particle-particle interactions. They afford the ability to simulate stochastic mechanisms in a natural way (see the simulation of FRAP, below) and the possibility of including relationships and behaviors which are difficult to formulate as continuum equations. Biological problems are characterized by complex individual-based interactions that naturally lend themselves to investigation through individual-based models.

Particle Diffusion

Particle diffusion is modeled as particles randomly switch between velocity channels (4 directions on a square lattice, 6 directions on a hexagonal lattice). Slower diffusion is modeled by adding an increased probability that the cell rests during a time step.

 Fast: p(resting) = 0 Slow: p(resting) = 0.9

Application: Simulation of FRAP

Modeling the diffusion of fluorescent molecules and “photobleaching” a region of the lattice to look at fluorescence recovery.

Mathematics of Biological Applications

In addition to their usefulness as a tool to investigate biological problems, discrete particle systems and cellular automata models yield fascinating mathematics. This mathematics is quite challenging and requires new techniques since cellular automata models do not generally lend themselves to analytical analysis. Indeed, it is shown by equivalence with Universal Turing Machines that the only way of determining the properties of most cellular automata is to forward-evaluate the cellular automaton [12]. Thus there is interesting and well-developed cellular automata research in the field of recursion theory. Viewed as dynamical systems, cellular automata yield waves and oscillations, periodic and chaotic behavior, stable attractors, and bifurcations. Analytic analysis of deterministic 1D cellular automata has yielded results regarding their reversibility, invariants, criticality, fractal dimension, and computational power [13]. When a rigorous connection is made between a set of macroscopic and microscopic equations, there may be surprising differences between the corresponding models. For example, the canonical lattice gas model of [14] known as the FHP model has been shown to approximate the Navier-Stokes equation at low velocities. However, for higher velocities the lattice gas model does not conserve Galilean invariance: fluid regions in which particles have a higher particle velocity have a higher particle density, even under the assumption of constant pressure.

Last updated August 13th, 2013.