Persistence and the Global Attractor Conjecture

Persistence is a concept from population dynamics. In words, a model of species interactions (e.g. predator-prey) is persistent if, given an initial condition where all species have non-zero population, none of the species can become extinct. The meaning in chemical reaction network theory is analogous: a CRN model is called persistent if, given an initial condition where all chemical species have non-zero concentration, none of the species can be completely "used up", i.e. none of their concentrations can tend to zero. In mathematical terms, a CRN model is persistent if

$\omega (x)\cap \partial \mathbb {R} _{\geq 0}^{n}=\emptyset \ \forall x\in \mathrm {int} (\mathbb {R} _{\geq 0}^{n})$

where $\omega (x)$ denotes the omega limit set of $x$.

The persistence conjecture

Feinberg - 1987
Given a chemical reaction network system, assume:

1. All reactions follow mass action kinetics,
2. The network is weakly reversible and
3. Each trajectory remains bounded.

Then the system is persistent.

Counterexample for power law kinetics

The assumption that all reactions obey the law of mass action is an essential part of the conjecture. Consider the following simple example network:

$2A{\begin{array}{c}k_{1}\\\rightleftharpoons \\k_{2}\end{array}}A+B$

Let $a,b$ represent the concentrations of $A$ and $B$ respectively. It is easy to demonstrate that $b$ cannot tend to zero under standard kinetic assumptions. Note also that $a+b=c$, where $c$ is a constant, and that all trajectories are bounded as a result. Under mass action kinetics,

${\dot {a}}=-k_{1}a^{2}+k_{2}ab=-(k_{1}+k_{2})a^{2}+k_{2}ac$

For $a\ll 1$, the second term $k_{2}ac$ dominates, and so ${\dot {a}}>0$, implying that the system is persistent. Now consider the same system under more general kinetics, where the right-to-left reaction rate has been replaced by more general power law kinetics:

${\dot {a}}=-k_{1}a^{2}+k_{2}a^{3}b=-k_{1}a^{2}-k_{2}a^{4}+k_{2}a^{3}c$

In this case, for $a\ll 1$ the term $-k_{1}a^{2}$ dominates, and so ${\dot {a}}<0$, meaning that the system is not persistent.