On the dynamics of a gene regulatory network

We examine the dynamics of a network of genes focusing on a periodic chain of genes, of arbitrary length. We show that within a given class of sigmoids representing the equilibrium probability of the binding of the RNA polymerase to the core promoter, the system possesses a single stable fixed point. By slightly modifying the sigmoid, introducing 'stiffer' forms, we show that it is possible to find network configurations exhibiting bistable behaviour. Our results do not depend crucially on the length of the chain considered: calculations with finite chains lead to similar results. However, a realistic study of regulatory genetic networks would require the consideration of more complex topologies and interactions.

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