György Lipták

Computing zero deficiency realizations of kinetic systems

In the literature, there exist strong results on the qualitative dynamical properties of chemical reaction networks (also called kinetic systems) governed by the mass action law and having zero deficiency. However, it is known that different network structures with different deficiencies may correspond to the same kinetic differential equations. In this paper, an optimization-based approach is presented for the computation of deficiency zero reaction network structures that are linearly conjugate to a given kinetic dynamics. Through establishing an equivalent condition for zero deficiency, the problem is traced back to the solution of an appropriately constructed mixed integer linear programming problem. Furthermore, it is shown that weakly reversible deficiency zero realizations can be determined in polynomial time using standard linear programming. Two examples are given for the illustration of the proposed methods.

Approximation of delayed chemical reaction networks

Chemical reaction networks with arbitrary constant delays assigned to the reactions are studied in this paper. The delayed models are approximated using the chain method known from the theory of differential equations. It is shown that important structural properties (such as reversibility and deficiency) of the approximated models are preserved in the approximating reaction networks. Moreover, the approximation gives rise to a Lyapunov–Krasovskii functional candidate for the original delayed systems that can be efficiently used for stability analysis.

Semistability of complex balanced kinetic systems with arbitrary time delays

Abstract In this letter we introduce a class of delayed kinetic systems derived from mass action type reaction network models. We define the time delayed positive stoichiometric compatibility classes and the notion of complex balanced time delayed kinetic systems. We prove the uniqueness of equilibrium solutions within the time delayed positive stoichiometric compatibility classes for such models. In our main result we prove the semistability of the equilibrium solutions for complex balanced systems with arbitrary time delays using an appropriate Lyapunov–Krasovskii functional and LaSalle’s invariance principle. As a consequence, we obtain that every positive complex balanced equilibrium solution is locally asymptotically stable relative to its positive stoichiometric compatibility class.

On the Parametric Uncertainty of Weakly Reversible Realizations of Kinetic Systems

The existence of weakly reversible realizations within a given convex domain is investigated. It is shown that the domain of weakly reversible realizations is convex in the parameter space. A LP-based method of testing if every element of a convex domain admits weakly reversible realizations is proposed. A linear programming method is also presented to compute a stabilizing kinetic feedback controller for polynomial systems with parametric uncertainty. The proposed methods are illustrated using simple examples.