Evolution of Complex Dynamics in Mathematical and Electronic Models of Genetic Networks

Leon Glass, Department of Physiology, McGill University,

Genetic activity is partially regulated by a complicated network of proteins called transcription factors. In the early 1960s Jacob and Monod suggested that genetic networks, in which a transcription factor coded by one gene acts as a regulatory inputs to another gene, could underlie cellular differentiation and and oscillation. I will describe a set on nonlinear and piecewise linear differential equations to model genetic networks. These equations are represented schematically using a directed graph on an hypercube. Because of the discrete representation of the continuous dynamics, the numbers of different networks with N model genes can be counted and classified. The methods are helpful in identifying networks that have certain types of dynamic behaviors such as stable fixed points, stable cycles, and chaotic dynamics. These methods can be used to help design in vitro genetic networks that show oscillation and multistability. They can also be used to determine gene network structure based on the patterns of activation of genes. Finally, the framework offers novel ways to study the evolution of rhythmic patterns in model equations and also in electronic circuits that simulate the differential equations.