Algorithms and complexity in biological pattern formation problems

In this paper we develop a new mathematical approach to the pattern formation problem in biology. This problem was first posed mathematically by A.M. Turing, however some principal questions were left open (for example, whether there exists a “universal” mathematical model that allows one to obtain any spatio-temporal patterns).Here we consider the pattern formation ability of some class of genetic circuits. First, we show that the genetic circuits are capable of generating arbitrary spatio-temporal patterns. Second, we give upper and lower bounds on the number of genes in a circuit generating a given pattern. A connection between the complexity of gene interaction and the pattern complexity is found. We investigate the stochastic stability of patterning algorithms. Results are consistent with experimental data.