On robust stability of stochastic genetic regulatory networks with time delays: A delay fractioning approach

Copyright [2009] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.Robust stability serves as an important regulation mechanism in system biology and synthetic biology. In this paper, the robust stability analysis problem is investigated for a class of nonlinear delayed genetic regulatory networks with parameter uncertainties and stochastic perturbations. The nonlinear function describing the feedback regulation satisfies the sector condition, the time delays exist in both translation and feedback regulation processes, and the state-dependent Brownian motions are introduced to reflect the inherent intrinsic and extrinsic noise perturbations. The purpose of the addressed stability analysis problem is to establish some easy-to-verify conditions under which the dynamics of the true concentrations of the messenger ribonucleic acid (mRNA) and protein is asymptotically stable irrespective of the norm-bounded modeling errors. By utilizing a new Lyapunov functional based on the idea of “delay fractioning”, we employ the linear matrix inequality (LMI) technique to derive delay-dependent sufficient conditions ensuring the robust stability of the gene regulatory networks. Note that the obtained results are formulated in terms of LMIs that can easily be solved using standard software packages. Simulation examples are exploited to illustrate the effectiveness of the proposed design procedures

Synchronization of stochastic genetic oscillator networks with time delays and Markovian jumping parameters

The official published version of the article can be found at the link below.Genetic oscillator networks (GONs) are inherently coupled complex systems where the nodes indicate the biochemicals and the couplings represent the biochemical interactions. This paper is concerned with the synchronization problem of a general class of stochastic GONs with time delays and Markovian jumping parameters, where the GONs are subject to both the stochastic disturbances and the Markovian parameter switching. The regulatory functions of the addressed GONs are described by the sector-like nonlinear functions. By applying up-to-date ‘delay-fractioning’ approach for achieving delay-dependent conditions, we construct novel matrix functional to derive the synchronization criteria for the GONs that are formulated in terms of linear matrix inequalities (LMIs). Note that LMIs are easily solvable by the Matlab toolbox. A simulation example is used to demonstrate the synchronization phenomena within biological organisms of a given GON and therefore shows the applicability of the obtained results.This work was supported in part by the Biotechnology and Biological Sciences Research Council (BBSRC) of the UK under Grants BB/C506264/1 and 100/EGM17735, the Royal Society of the UK, the National Natural Science Foundation of China under Grant 60804028, the Teaching and Research Fund for Excellent Young Teachers at Southeast University of China, the International Science and Technology Cooperation Project of China under Grant 2009DFA32050, and the Alexander von Humboldt Foundation of Germany

Monostability and multistability of genetic regulatory networks with different types of regulation functions

The official published version of the article can be found at the link below.Monostability and multistability are proven to be two important topics in synthesis biology and system biology. In this paper, both monostability and multistability are analyzed in a unified framework by applying control theory and mathematical tools. The genetic regulatory networks (GRNs) with multiple time-varying delays and different types of regulation functions are considered. By putting forward a general sector-like regulation function and utilizing up-to-date techniques, a novel Lyapunov–Krasovskii functional is introduced for achieving delay dependence to ensure less conservatism. A new condition is then proposed for the general stability of a GRN in the form of linear matrix inequalities (LMIs) that are dependent on the upper and lower bounds of the delays. Our general stability conditions are applicable to several frequently used regulation functions. It is shown that the existing results for monostability of GRNs are special cases of our main results. Five examples are employed to illustrate the applicability and usefulness of the developed theoretical results.This work was supported in part by the Biotechnology and Biological Sciences Research Council (BBSRC) of the U.K. under Grant BB/C506264/1, the Royal Society of the U.K., the National Natural Science Foundation of China under Grants 60504008 and 60804028, the Program for New Century Excellent Talents in Universities of China, and the Alexander von Humboldt Foundation of Germany

Robustness analysis of genetic regulatory networks affected by model uncertainty

A fundamental problem in systems biology consists of investigating robustness properties of genetic regulatory networks (GRNs) with respect to model uncertainty. This paper addresses this problem for GRNs where the coefficients are rationally affected by polytopic uncertainty, and where the saturation functions are not exactly known. First, it is shown that a condition for ensuring that the GRN has a globally asymptotically stable equilibrium point for all admissible uncertainties can be obtained in terms of a convex optimization problem with linear matrix inequalities (LMIs), hence generalizing existing results that mainly consider only the case of GRNs where the coefficients are linearly affected by the uncertainty and the regulatory functions are in SUM form. Second, the problem of estimating the worst-case convergence rate of the trajectories to the equilibrium point over all admissible uncertainties is considered, and it is shown that a lower bound of this rate can be computed by solving a quasi-convex optimization problem with LMIs. Third, the paper considers the problem of estimating the set of uncertainties for which the GRN has a globally asymptotically stable equilibrium point. This problem is addressed, first, by showing how one can compute estimates with fixed shape by solving a quasi-convex optimization problem with LMIs, and second, by deriving a procedure for computing estimates with variable shape. Numerical examples illustrate the use of the proposed techniques. © 2010 Elsevier Ltd. All rights reserved.postprin