Stability analysis of uncertain genetic regulatory newtworks

Abstract

Genetic regulatory network (GRN) is a fundamental research area in systems biology.

This thesis studies the stability of a class of GRN models. First, a condition is proposed to ensure the robust stability of uncertain GRNs with SUM regulatory functions. It is assumed that the uncertainties are in the form of a parameter vector that determines the coefficients of the model via given functions. Then, the global asymptotic stability conditions of uncertain GRNs affected by disturbances and time delays are further explored. The conditions are obtained by solving a convex optimization problem by exploring the sum of squares (SOS) of matrix polynomials and by introducing polynomially parameter-dependent Lyapunov-Krasovskii functionals (LKFs). Moreover, based on the uncertain GRNs with guaranteed disturbance attenuation, it is shown that estimates of the sought stable uncertainty sets can be obtained through a recursive strategy based on parameter-dependent Lyapunov functions and the SOS.

Second, the stability conditions of GRNs described by piecewise models are considered. Depending on whether the state partitions and mode transitions are known or unknown as priori, the proposed networks are divided into two categories, i.e., switched GRNs and hybrid GRNs. It is shown that, by using common polynomial Lyapunov functions and piecewise polynomial Lyapunov functions, two conditions are established to ensure the global asymptotic stability for switched and hybrid GRNs, respectively. In addition, it is shown that, by using the SOS techniques, stability conditions in the form of LMIs for both models can be obtained.

Third, the multi-stability of uncertain GRNs with multivariable regulation functions is investigated. It is shown that, by using the Lyapunov functional method and LMI technology, a criterion is established to ensure the robust asymptotical stability of the uncertain GRNs, and such condition can be extended to deal with the multi-stability problem. Moreover, it is shown that by using the square matrix representation (SMR) and by adopting polynomially parameter-dependent Lyapunov functions, a condition in the form of LMIs for robust stability for all admissible uncertainties can be obtained.

Examples with synthetic and real biological models are presented in each section to illustrate the applicability and effectiveness of the theoretical results.