Robust Stability Analysis for Cyclic Gene Regulatory Networks: Analytic Necessary and Sufficient Condition

Abstract

This paper is concerned with robust stability conditions of gene regulatory networks with cyclic interconnections, where dynamics of each gene's expression has a certain degree of uncertainty. We first derive a graphical condition for robust stability based on the robustness analysis scheme for large-scale systems presented in the authors' previous work. A key feature of the graphical condition is that robust stability is systematically confirmed by drawing three elementary shapes that are determined by network structure, gene's dynamics, and its uncertainty bounds. Moreover, it is shown that the criterion can be greatly simplified by using a monotonicity of gene's dynamics. This reduction enables us to derive an analytic condition for the robust stability. The developed analytic condition is expressed explicitly in terms of biological parameters and the uncertainty bound. Thus, it can be easily applied to large-scale gene regulatory networks consisting of any number of genes.