Bifurcation Analysis of Gene Regulatory Circuits Subject to Copy Number Variation

Gene regulatory networks are comprised of many small gene circuits. Understanding expression dynamics of gene circuits for broad ranges of parameter space may provide insight into the behavior of larger regulatory networks as well as facilitate the use of circuits as autonomous units performing specific regulatory tasks. In this paper, we consider three common gene circuits and investigate the dependence of gene expression dynamics on the circuit copy number. In particular, we perform a detailed bifurcation analysis of the circuits' corresponding nonlinear gene regulatory models restricted to protein-only dynamics. Employing a geometric approach to bifurcation theory, we are able to derive closed form expressions for conditions which guarantee existence of saddle-node bifurcations caused by variation in the circuit copy number or copy number concentration. This result shows that the drastic effect of copy number variation on equilibrium behavior of gene circuits is highly robust to variation in other parameters in the circuits. We discuss a possibility of extending the current results to higher dimensional models which incorporate more details of the gene regulatory process.