Realizations of kinetic differential equations

Abstract

The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.2. There is an internal requirement within this branch of science: One should like to know as much as possible about the structure of differential equations that arise from modelling a chemical system.3. Given a system of polynomial differential equations in any field of pure or applied mathematics, one may wish to have statements on stability or oscillations, similar to those offered by the Horn-Jackson Theorem [26], Zero Deficiency Theorem [21], Volpert's theorem [50], or the Global Attractor Conjecture, where several cases have been proven [3,12,23,35] 1 . Then it comes in handy to see that the system of differential equations of interest belongs to a well behaving class.4. Lastly, results of formal reaction kinetics (to use an expression introduced in [5, 6]), e.g. on the existence of stationary points, may offer alternative methods for solving problems in algebraic geometry [17,32]. For instance, one might be able to show the existence of positive roots of a polynomial if the system of polynomial equations is known to be the right hand side of the induced kinetic differential equation of a reversible or weakly reversible reaction network [7].The structure of our paper is as follows. Section 2 introduces the essential concepts of reaction networks and mass action systems. Section 3 formulates the problem we are interested in, that of realizability of kinetic differential equations. Section 4 treats two special cases: finding realizations for compartmental models (defined later) and weakly reversible networks. Section 5 focuses on the general problem of realizability. We first review existing algorithms available. Then we outline several procedures that modify a reaction network while preserving the system of differential equations, including adding and removing vertices from the reaction graph. Section 6 explores the relation between weakly reversible and complex balanced realizations. Here we work with families of symbolic kinetic differential equations. In Section 7, w...

Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems

“…The case for uniqueness when there is a single linkage class follows immediately from Theorem 3.4. This case was first proved in [12] using linear algebraic methods; a more geometric proof can be found in [9].…”

“…Theorem 3.4 ( [9,14]). The deficiency of a reaction network is zero if and only if 1. the network has affinely independent linkage classes, and 2. the stoichiometric spaces of the linkage classes are linearly independent.…”

“…The goal of this paper is to prove that for an associated dynamical system of a mass-action system, there is at most one weakly reversible, deficiency zero realization, which we call a WR 0 realization. This problem has been solved when there is only one linkage class [9,12]. Here, we consider the problem in full generality.…”

Two Nested Limit Cycles in Two-Species Reactions

“…All the research mentioned up to this point solved direct problems: given a reaction or its kinetic differential equation, some properties of it are found. (Note that whereas the route from reactions to differential equations is straightforward [24,25], it is far from being true in the opposite direction, see, e.g., [26].) Our present paper also belongs to this category.…”

“…In this case the point A is an unstable focus. As a next step, keeping c 1 , d 1 , d 2 , e 1 , f 1 , f 2 the same as in ( 21)-( 24), we perturb e 2 as e 2 = 278 1000 (26) and obtain that the system has three singular points in the first quadrant: A(1, 1.23247), B(0.7895, 2.26181) and C(0.414968, 4.09327) and g 1 ≈ −0.0090896 < 0…”