On the relation between $\omega$-limit set and boundaries of mass-action chemical reaction networks

Abstract

$\omega$-limit set can be used to understand the long term behavior of a dynamical system. In this paper, we use the Lyapunov Function PDEs method, developed in our previous work, to study the relation between $\omega$-limit points and boundaries for chemical reaction networks equipped with mass-action kinetics. Using the solution of the PDEs, some new checkable criteria are proposed to diagnose non $\omega$-limit points of the network system. These criteria are successfully applied to verify that non-semilocking boundary points and some semilocking boundary points are not $\omega$-limit points. Further, we derive the $\omega$-limit theorem that precludes the limit cycle of some biochemical network systems. The validity of the results are demonstrated through some abstract and practical examples of chemical reaction networks.

Introduction

Chemical reaction networks (CRNs) are widely existing in the fields of chemistry, biology, and medicine, etc. The related studies have been applied to many fields, besides the above mentioned ones, even including those not seemly related to chemical reactions, such as electricity (Samardzija et al., 1989). Mass-action kinetics is one of the most commonly used kinetics to describe the reaction rate of chemical reactions, and the induced systems, termed as mass-action systems (MASs), often give rise to a family of nonequilibrium ordinary differential equations (ODEs). The ODEs are generally highly nonlinear, and also contain many parameters, such as the rate constants of all reactions. Due to difficult/inaccurate measurement problem to these parameters, it often requires to study the dynamical behaviors of MASs only depending on the topological structure of networks, but independent of the system parameters. Here, the dynamical behaviors mainly relate to stability, persistence, etc. The pioneering work on this topic was made by Feinberg (1979), and Horn and Jackson (1972). They defined a class of complex balanced MASs with weakly reversible network structure, and addressed well the issues on number of equilibria, locally asymptotic stability, etc. However, the globally asymptotic stability is still far from being solved. In practice, it is unnecessary to directly tackle this issue itself, but may address another issue of persistence instead. A complex balanced MAS will be globally asymptotically stable if it has persistence. Hence, the persistence study has become quite active in recent decades (Anderson, 2008, Angeli et al., 2007), but the related issues are still open problems in the field of CRNs. Certainly, persistence, as a kind of dynamical behavior, is often used to model the reaction phenomenon that no species will be used up in the course of reaction if it is present at the reaction start. The study on this topic is also interesting from the viewpoint of dynamic analysis.

In this work, we will not follow the line of studying persistence of complex balanced MASs, but launch some preparatory work about persistence of CRNs with any structure. As one might know, $\omega$-limit set is a positively invariable set of the trajectory, and may be used to reflect the dynamical property of system. A MAS is persistence if the intersection of its $\omega$-limit set and boundaries is empty. The studies on the relation between $\omega$-limit set and boundaries of MASs are thus rather significant from the viewpoint of control theory as well as of dynamic analysis. As early as 1970s, Vasil’ev et al. (1974) proved the $\omega$-limit set of each trajectory of a detailed balanced system to be only a single positive equilibrium or a set of boundary equilibria. Siegel and MacLean (2000) generalized this result to complex balanced MASs and use this property to derive the global asymptotic stability of a subclass of complex balanced MASs. In characterizing dynamical behaviors, Lyapunov function is a powerful tool (Blanchini & Giordano, 2014). When the networks are complex balanced, there is a canonical choice, i.e., the well-known pseudo-Helmholtz free energy function as the Lyapunov function (Horn & Jackson, 1972). However, in other cases, there is no general method to obtain a Lyapunov function, and no understanding of the space of possible Lyapunov functions for a given reaction network. Although Anderson et al. (2015) showed that the pseudo-Helmholtz free energy function can be derived from an appropriate limit of the stationary distributions for the stochastic mass-action systems, this is not a practical construction method since for almost all reaction network systems, the stationary distributions are prohibitively hard to obtain. Our early work (Fang & Gao, 2019) followed up this issue, and by connecting the microscopic and the macroscopic levels, thermodynamics and potential theory, proposed a PDE and a boundary condition for a mass-action system, referred to as the Lyapunov Function PDEs. When the PDEs are solved, the solutions are conjectured as Lyapunov functions for the network attached. This conjecture has been proved true when the network is complex balanced, of $1$-dimensional stoichiometric subspace, and a combination of the other cases, respectively. Some special networks with dimension of the stoichiometric subspace more than $2$ also support the conjecture.

In this paper, we continue to utilize the Lyapunov Function PDEs, but to study the relation between $\omega$-limit set and boundaries of each trajectory of MAS. The main contributions are listed below:

• For a MAS, based on the solution of its Lyapunov Function PDEs, two sufficient conditions are proposed to say any boundary point of system to be not $\omega$-limit point.

• Any non-semilocking boundary point and some semilocking boundary non-equilibrium points are proved not $\omega$-limit point.

• The $\omega$-limit set of some MASs, especially of complex balanced MASs, is proved a set of positive equilibria or a set of boundary equilibria.

• Some practical biochemical systems are proved to have no limit cycles.

The remainder of this paper is organized as follows. Section 2 introduces some preliminaries about CRN theory and Lyapunov Function PDEs. Section 3 presents the checkable technical criteria for the $\omega$-limit set under the framework of the Lyapunov Function PDEs, and some concrete examples are also illustrated. In Section 4, we apply our results to some kinds of real biochemical networks. Finally, Section 5 concludes the paper.

Notations: ${\mathbb{R}}^n,{\mathbb{R}}_{\ge 0}^n,{\mathbb{R}}_{>0}^n$ denote $n$-dimensional real space, non -negative real space, positive real space, respectively; ${\mathcal{C}}^i$ denotes the function set whose elements are $i$th continuously differentiable; $\text{su}x$ denotes the support set of a vector $x$ defined as $\text{su}x=\left\{S_j\right|x_j\ne 0\},\forall x\in {\mathbb{R}}_{\ge 0}^n$; $\text{Ln}\left(x\right)$ is defined as a vector whose $j$th element is $ln\left(x_j\right)$.