Polynomial relaxation-based conditions for global asymptotic stability of equilibrium points of genetic regulatory networks

An important problem in systems biology consists of establishing whether an equilibrium point of a genetic regulatory network (GRN) is stable or not. This article investigates this problem for GRNs with SUM or PROD regulatory functions. It is shown that sufficient conditions for global asymptotical stability of an equilibrium point of these networks can be derived in terms of convex optimisations with linear matrix inequality constraints. These conditions are obtained by looking for a Lyapunov function through the use of suitable polynomial relaxations, and do not introduce approximations of the nonlinearities present in the GRNs. The benefit of these conditions is that their conservatism can be decreased by increasing the degree of the introduced polynomial relaxations. Numerical examples illustrate the usefulness of the proposed conditions.postprin

Network analysis of complex biological systems : boundedness of weakly reversible chemical reaction networks and conditions for synchronisation of coupled oscillators

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Reduction of dynamical biochemical reaction networks in computational biology

Biochemical networks are used in computational biology, to model the static and dynamical details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as combinatorial explosion are strong obstacles against analyzing the dynamics of large models of this type. Multi-scaleness is another property of these networks, that can be used to get past some of these obstacles. Networks with many well separated time scales, can be reduced to simpler networks, in a way that depends only on the orders of magnitude and not on the exact values of the kinetic parameters. The main idea used for such robust simplifications of networks is the concept of dominance among model elements, allowing hierarchical organization of these elements according to their effects on the network dynamics. This concept finds a natural formulation in tropical geometry. We revisit, in the light of these new ideas, the main approaches to model reduction of reaction networks, such as quasi-steady state and quasi-equilibrium approximations, and provide practical recipes for model reduction of linear and nonlinear networks. We also discuss the application of model reduction to backward pruning machine learning techniques

Investigating stochastic stability of uncertain genetic networks via LMIs

Part of 2010 IEEE Multi-Conference on Systems and ControlThis paper addresses the problem of investigating stochastic stability of uncertain genetic networks with SUM regulatory functions. Specifically, the genetic network is assumed to be affected by Wiener processes, and its coefficients are parametrized by an unknown vector constrained in a hypercube. By using the square matricial representation (SMR) of matrix polynomials, it is shown that a condition for stochastic stability of the uncertain genetic network with disturbance attenuation guaranteed for all admissible values of the parameter can be derived in terms of linear matrix inequalities (LMIs). Some examples illustrate the proposed condition. © 2010 IEEE.published_or_final_versionThe 2010 IEEE International Symposium on Computer-Aided Control System Design (CACSD), Yokohama, Japan, 8-10 September 2010. In Proceedings of CACSD, 2010, p. 731-73

On the Steady States of Uncertain Genetic Regulatory Networks

This correspondence addresses the analysis of the steady states of uncertain genetic regulatory networks (GRNs). The uncertainty is represented as a vector constrained in a given set that affects the coefficients of the mathematical model of the GRN. It is shown how regions containing all possible steady states can be estimated via an iterative strategy that progressively splits the concentration space into smaller sets, discarding those that are guaranteed not to contain equilibrium points of the considered model. This strategy is based on worst case evaluations of some appropriate functions of the uncertainty via linear matrix inequality optimization.published_or_final_versio

Control Theory: On the Way to New Application Fields

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently, deep interactions are emerging with new application areas, such as systems biology, quantum control and information technology. In order to address the new challenges posed by the new application disciplines, a special focus of this workshop has been on the interaction between control theory and mathematical systems biology. To complement these more biology oriented focus, a series of lectures in this workshop was devoted to the control of networks of systems, fundamentals of nonlinear control systems, model reduction and identification, algorithmic aspects in control, as well as open problems in control