Robust stability analysis of gene–protein regulatory networks with cyclic activation–repression interconnections

This paper investigates analytic robust stability criteria for large-scale cyclic gene–protein regulatory network systems consisting of genes with heterogeneous dynamics combined with parametric or unstructured uncertainties from a control-theoretic viewpoint. We first consider a class of gene expressions, which is described as an uncertain linear transcription–translation model (LTTM) with feedback loops from translation products to transcription. Next, we show that such a model belong to a class of large-scale dynamical linear network systems with a generalized frequency variable. Then, we derive robust stability criteria systematically for the following two types of biological uncertain LTTM: (i) LTTMs consisting of heterogeneous gene dynamics with parametric uncertainties and (ii) LTTMs consisting of homogeneous nominal gene dynamics with unstructured uncertainties. These criteria provide fairly simple analysis methodologies, which can be readily applied to the analysis for large-scale genetic regulatory networks and give some biological insight.