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

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Leon Glass, Department of Physiology, McGill University,
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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.