Polynomial Superlevel Set Representation of the Multistationarity Region of Chemical Reaction Networks

In this paper we introduce a new representation for the multistationarity region of a reaction network, using polynomial superlevel sets. The advantages of using this polynomial superlevel set representation over the already existing representations (cylindrical algebraic decompositions, numeric sampling, rectangular divisions) is discussed, and algorithms to compute this new representation are provided. The results are given for the general mathematical formalism of a parametric system of equations and so may be applied to other application domains.Comment: 27 pages, 9 figure

On Darboux integrability of Edelstein's reaction system in R³

We consider Edelstein's dynamical system of three reversible reactions in R³ and show that it is not Darboux integrable. To do so we characterize its polynomial first integrals, Darboux polynomials and exponential factors

Reduction of chemical reaction networks with approximate conservation laws

Model reduction of fast-slow chemical reaction networks based on the quasi-steady state approximation fails when the fast subsystem has first integrals. We call these first integrals approximate conservation laws. In order to define fast subsystems and identify approximate conservation laws, we use ideas from tropical geometry. We prove that any approximate conservation law evolves more slowly than all the species involved in it and therefore represents a supplementary slow variable in an extended system. By elimination of some variables of the extended system, we obtain networks without approximate conservation laws, which can be reduced by standard singular perturbation methods. The field of applications of approximate conservation laws covers the quasi-equilibrium approximation, which is well known in biochemistry. We discuss reductions of slow-fast as well as multiple timescale systems. Networks with multiple timescales have hierarchical relaxation. At a given timescale, our multiple timescale reduction method defines three subsystems composed of (i) slaved fast variables satisfying algebraic equations, (ii) slow driving variables satisfying reduced ordinary differential equations, and (iii) quenched much slower variables that are constant. The algebraic equations satisfied by fast variables define chains of nested normally hyperbolic invariant manifolds. In such chains, faster manifolds are of higher dimension and contain the slower manifolds. Our reduction methods are introduced algorithmically for networks with monomial reaction rates and linear, monomial, or polynomial approximate conservation laws. We propose symbolic algorithms to reshape and rescale the networks such that geometric singular perturbation theory can be applied to them, test the applicability of the theory, and finally reduce the networks. As a proof of concept, we apply this method to a model of the TGF-beta signaling pathway

A computational approach to polynomial conservation laws

For polynomial ODE models, we introduce and discuss the concepts of exact and approximate conservation laws, which are the first integrals of the full and truncated sets of ODEs. For fast-slow systems, truncated ODEs describe the fast dynamics. We define compatibility classes as subsets of the state space, obtained by equating the conservation laws to constants. A set of conservation laws is complete when the corresponding compatibility classes contain a finite number of steady states. Complete sets of conservation laws can be used for model order reduction and for studying the multistationarity of the model. We provide algorithmic methods for computing linear, monomial, and polynomial conservation laws of polynomial ODE models and for testing their completeness. The resulting conservation laws and their completeness are either independent or dependent on the parameters. In the latter case, we provide parametric case distinctions. In particular, we propose a new method to compute polynomial conservation laws by comprehensive Gröbner systems and syzygies

Conservation Laws in Biochemical Reaction Networks

We study the existence of linear and non-linear conservation laws in biochemical reaction networks with mass-action kinetics. It is straightforward to compute the linear conservation laws as they are related to the left null-space of the stoichiometry matrix. The non-linear conservation laws are difficult to identify and have rarely been considered in the context of mass-action reaction networks. Here, using Darboux theory of integrability we provide necessary structural (i.e. parameter independent) conditions on a reaction network to guarantee the existence of non-linear conservation laws of certain type. We give necessary and sufficient structural conditions for the existence of exponential factors with linear exponents and univariate linear Darboux polynomials. This allows us to conclude that a non-linear first integrals (similar to Lotka-Volterra system) only exists under the same structural condition. We finally show that the existence of such a first integral generally implies that the system is persistent and has stable steady states. We illustrate our results by examples

Conservation Laws in Biochemical Reaction Networks

Altres ajuts: Universitat Jaume I grant P1-1B2015-16We study the existence of linear and nonlinear conservation laws in biochemical reaction networks with mass-action kinetics. It is straightforward to compute the linear conservation laws as they are related to the left null-space of the stoichiometry matrix. The nonlinear conservation laws are difficult to identify and have rarely been considered in the context of mass-action reaction networks. Here, using the Darboux theory of integrability, we provide necessary structural (i.e., parameter-independent) conditions on a reaction network to guarantee the existence of nonlinear conservation laws of a certain type. We give necessary and sufficient structural conditions for the existence of exponential factors with linear exponents and univariate linear Darboux polynomials. This allows us to conclude that nonlinear first integrals only exist under the same structural condition (as in the case of the Lotka-Volterra system). We finally show that the existence of such a first integral generally implies that the system is persistent and has stable steady states. We illustrate our results by examples

Conservation laws in biochemical reaction networks

Altres ajuts: Universitat Jaume I grant P1-1B2015-16We study the existence of linear and nonlinear conservation laws in biochemical reaction networks with mass-action kinetics. It is straightforward to compute the linear conservation laws as they are related to the left null-space of the stoichiometry matrix. The nonlinear conservation laws are difficult to identify and have rarely been considered in the context of mass-action reaction networks. Here, using the Darboux theory of integrability, we provide necessary structural (i.e., parameter-independent) conditions on a reaction network to guarantee the existence of nonlinear conservation laws of a certain type. We give necessary and sufficient structural conditions for the existence of exponential factors with linear exponents and univariate linear Darboux polynomials. This allows us to conclude that nonlinear first integrals only exist under the same structural condition (as in the case of the Lotka-Volterra system). We finally show that the existence of such a first integral generally implies that the system is persistent and has stable steady states. We illustrate our results by examples