Global stability of perturbed complex-balanced systems Polly Y. Yu1 1NSFSimons Center for Mathematical and Statistical Analysis of Biology Harvard University

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Global stability of perturbed complex-balanced systems

Polly Y. Yu1

1NSF–Simons Center for Mathematical and Statistical Analysis of Biology, Harvard University

Abstract

A class of polynomial dynamical systems called complex-balanced are locally stable and

conjectured to be globally stable. In general, complex-balancing is not a robust property, i.e.,

small changes in parameter values may result in the loss of the complex-balanced property. We

show that robustly permanent complex-balanced systems are globally stable even after the rate

constants have been perturbed.

1 Introduction

Robustness is vital to biological systems. From cells to ecosystems, the dynamical behaviour

is prescribed by a set of core interactions, which are inﬂuenced by intrinsic and environmental

noise. It is often observed that the system maintains its stable dynamical behaviour despite these

external inﬂuences. In other words, the system is robust with respect to noise. One way to measure

robustness is to ask whether certain qualitative dynamics remains after the parameters have been

perturbed.

Common models for chemical and biological systems assume mass-action kinetics. The dynamics

comes from a system of ODEs built using a list of reactions or interactions, each with a rate

constant. In general, mass-action systems can display a diverse set of dynamics, from multistability,

oscillations, and even chaos. A particularly stable family of mass-action systems is that of complex-

balancing [15], introduced as a generalization of systems at thermodynamic equilibrium. As

such, these systems have exactly one positive steady states (up to conservation laws), which is

asymptotically stable [15] and conjectured to be globally attracting [14].

Not only are complex-balanced systems dynamically stable and algebraically rich, they are

completely characterized by their underlying networks. A mass-action system is complex-balanced

if and only if the underlying network is weakly reversible and the rate constants satisfy some

algebraic equations [5,13]. The number of these equations is dictated by the network’s topology

and geometry.

In the context of biochemical reactions, rate constants are not truly constant; they depend on

temperature, pressure, the presence of solvents and ions, etc., and are subjected to thermal

ﬂuctuation. Thus the algebraic condition for complex-balancing is not satisﬁed in general, i.e.,

complex-balancing is generally not robust. With small perturbations in the rate constants, we can

no longer claim all of complex-balancing’s stable dynamical properties.

arXiv:2210.13633v1 [math.DS] 24 Oct 2022

In this work, we show that robustly permanent complex-balanced systems are globally stable even

with small changes in the rate constants.

Theorem. Let (G, κ∗)be a complex-balanced system that is robustly permanent with respect to

κ∗. Then on every compatibility class U, there exists ε > 0such that for every κ∈B(κ∗, ε), the

mass-action system (G, κ)has a unique globally attracting point within U.

It is worth noting that robust permanence for complex-balanced systems follows immediately from

the Permanence Conjecture for variable-κweakly reversible systems, which has been proved for

several special classes; see for example [1,2,7,10].

This paper is organized as follows. Mathematical notations used throughout this work are covered in

Section 2. We cover the necessary background on mass-action systems, complex-balanced systems,

and permanence in Section 3. Then we prove our main result and give some examples in Section 4.

2 Notations

Throughout this work, we let R≥and R>denote the set of non-negative and positive real numbers

respectively. Accordingly, Rn

≥and Rn

>denote the set of vectors in Rnwith non-negative and positive

entries respectively. We say x>0if x∈Rn

>, and x≥0is deﬁned analogously. Let B(x, ε) be the

ε-ball around x. For any x∈Rn

>and y∈Rn

≥, let xy=xy1

1xy2

2· · · xyn

3 Mass-action systems

We now provide a brief introduction to reaction networks and mass-action systems. For a more

in-depth introduction, see [9,11,21].

Areaction network (or network for short) is a directed graph G= (V, E) with no self-loops

and no isolated vertices, where V⊂Rn

≥and E⊆V×V. An edge (yi,yj)∈Eis denoted (i, j)

or yi→yj. A vertex is also called a complex , while an edge is called a reaction . Throughout

this work, Rnwill be the ambient space; m=|V|,r=|E|, and `is the number of connected

components of G. A network is said to be weakly reversible if every connected component is

strongly connected.

In some classical literature on mass-action systems [9], a reaction network is deﬁned to be a triple

(S,C,R), where Sis the set of species,Cis the set of complexes, and Ris the set of reactions.

The two deﬁnitions are equivalent via a natural identiﬁcation between the species and the standard

orthonormal basis of Rn. For example when n= 3, the complex Xis identiﬁed with (1,0,0) and

2Y +Zwith (0,2,1), and so forth.

Assuming mass-action kinetics, the time-evolution of the concentration vector x(t) is given by the

system of autonomous ODEs

x(t) = X

(i,j)∈E

κij x(t)yi(yj−yi),(1)

where κij >0 is the rate constant of the reaction yi→yj. We say (1) is the associated system

of the mass-action system (G, κ), where κ= (κij )(i,j)∈E.

Remark 3.1. In this work, we restrict the set of vertices to V⊂({0}∪ [1,∞))nor V⊂Zn

≥. Under

this assumption, the state space Rn

>is forward-invariant; moreover, the right-hand side of (1) is

Lipschitz continuous, so solution to (1) with any initial condition in Rn

>is unique [20].

The stoichiometric subspace of a reaction network Gis the linear subspace

S= span{yj−yi: (i, j)∈E},

which contains ˙

x(t). Hence for any initial condition x(0) ∈Rn

>, the solution x(t) is conﬁned to the

compatibility class (x(0) + S)>= (x(0) + S)∩Rn

A generalization of (1) is the variable-κmass-action system

x(t) = X

(i,j)∈E

κij (t)x(t)yi(yj−yi),(2)

where ε≤κij (t)≤ε−1for some uniformly chosen 0 < ε < 1 [7]. One might further assume

the coeﬃcient functions κij (t) to be suﬃciently smooth, for example Lipschitz. Regardless, the

dynamics of (2) is also constrained on compatibility classes. Clearly, the dynamics of (1) is

replicated by (2) when each κij (t) is constant.

3X 3Y

X+Y+Z

κ14

κ43

κ31 κ32

κ21

(a)

κ14

κ43 κ32

κ31

κ21

(b) (c)

Figure 1: (a) A list of reactions that make up (b) the reaction network G. (c)

The dynamics of the mass-action system (G, κ) conﬁned to a compatibility class.

Example 3.2. Consider the ﬁve reactions listed in Figure 1(a). They generate the weakly reversible

reaction network Gin Figure 1(b) embedded in R3. The associated system for any choice of κij >0

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Globalstabilityofperturbedcomplex-balancedsystemsPollyY.Yu11NSF{SimonsCenterforMathematicalandStatisticalAnalysisofBiology,HarvardUniversityAbstractAclassofpolynomialdynamicalsystemscalledcomplex-balancedarelocallystableandconjecturedtobegloballystable.Ingeneral,complex-balancingisnotarobustproperty...

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Global stability of perturbed complex-balanced systems Polly Y. Yu1 1NSFSimons Center for Mathematical and Statistical Analysis of Biology Harvard University

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