The asymptotical behavior of cyclic genetic regulatory networks

In this paper, the cyclic genetic regulatory networks with repression and inducement is studied. Based on the assumption that one gene is repressed or induced by another single gene, the cyclic genetic regulatory networks consisting of nn genes are classified, and their corresponding models are built to describe the biochemical feedback loops. By the theory of KK-competitive and KK-cooperative system, the asymptotical behavior of KK-inducible or KK-repressive systems is analyzed. For the models consisting of 3 genes, sufficient conditions are presented for the existence of periodic solution and for the global asymptotic stability of the positive equilibrium. As for three models consisting of 4 genes, the results indicate that the limit set is a closed orbit if it contains no equilibria.