Complex balanced mass action systems

A complex balanced equilibrium concentration of a mass action system ${\vec {a}}=[a_{1},a_{2},\ldots ,a_{n}]^{T}\in \mathbb {R} _{>0}^{n}$ is a point which satisfies

$\sum _{i=1}^{m}k(i,j)a_{1}^{y_{i1}}\cdots a_{m}^{y_{im}}=a_{1}^{y_{j1}}\cdots a_{m}^{y_{jm}}\sum _{i=1}^{m}k(j,i)$

for $j=1,\ldots ,m$. Using the short-hand notation ${\vec {a}}^{\vec {y}}=a_{1}^{y_{1}}\cdots a_{m}^{y_{m}}$ we can rewrite this as $\sum _{i=1}^{m}k(i,j)({\vec {a}})^{{\vec {y}}_{i}}=({\vec {a}})^{{\vec {y}}_{j}}\sum _{i=1}^{m}k(j,i)$ for $j=1,\ldots ,m$. A mass action system is said to be complex balanced if every equilibrium concentration permitted by the system is a complex balanced equilibrium concentration.

The notion of complex balancing of equilibrium concentrations was introduced in 1972 by Fritz Horn and Roy Jackson as a generalization of detailed balancing. Is is a weaker than detailed balancing in that it does not require each elementary step in the reaction to be balanced by a reverse step, rather it requires that the net input into each complex balances the net flow out of the complex.

Matrix formulation

A complex balanced equilibrium concentration may be alternatively characterized by

$A_{k}\Psi ({\vec {a}})={\vec {0}}$

where $\Psi ({\vec {a}})\in \mathbb {R} _{\geq 0}^{m}$ is the mass action vector, $A_{k}\in \mathbb {R} ^{m\times m}$ is the kinetic or Kirchhoff matrix, and ${\vec {0}}\in \mathbb {R} ^{m}$ is the $m$-dimensional vector with all entries zero.

Properties of complex balanced mass action systems

A number of strong dynamical properties are known to hold for mass action systems with complex balanced equilibrium concentrations. Fritz Horn and Roy Jackson proved the follow results:

• any mass action system permitting a complex balanced equilibrium concentration only permits complex balanced equilibrium concentrations (Lemma 5B);
• each stoichiometric compatibility class $({\vec {c}}+S)\cap \mathbb {R} _{>0}^{m}$ of a complex balanced system has exactly one equilibrium concentration (Theorem 6A and Lemma 4B); and
• that equilibrium concentration is locally asymptotically stable relative to its compatibility class (Theorem 6A and Lemma 4C).

The argument is made by use of the pseudo-Helmholtz function

$H({\vec {c}})=\sum _{i=1}^{n}c_{i}(\ln c_{i}-\ln a_{i}-1)$

where ${\vec {a}}=[a_{1},\ldots ,a_{n}]^{T}\in \mathbb {R} _{>0}^{n}$ is any complex balanced equilibrium concentration permitted by the system. They were able to show that the time-derivative of the $H({\vec {c}})$ along solutions ${\vec {c}}(t)$ of a complex balanced mass action system satisfy

${\frac {d}{dt}}H({\vec {c}}(t))<0$

for all $t>0$. This result is sufficient to prove trajectories are bounded, non-periodic and non-chaotic, and converge to the stoichiometrically compatible equilibrium concentration within a neighbourhood. It is unable, however, to preclude the possibility of trajectories approaching the boundary $\partial [({\vec {c}}_{0}+S)\cap \mathbb {R} _{>0}^{n}]$ because $H({\vec {c}})$ is not radially unbounded relative to $({\vec {c}}_{0}+S)\cap \mathbb {R} _{>0}^{n}$.

Fritz Horn was also able to show that complex balanced mass action systems share many connections with the reaction graph of the underlying chemical reaction network. He was able to show that every chemical reaction network which permits a complex balanced mass action system is weakly reversible (Theorem 2B), that every chemical reaction network which is weakly reversible permits complex balanced mass action systems (discussion after Theorem 3B), and that a mass action system is unconditionally complex balanced (complex balanced for choices of its positive rate constants) if and only if it is weakly reversible and deficiency zero (Theorem 3B).

Global Attractor Conjecture

The Global Attractor Conjecture is the conjecture that the unique equilibrium concentration ${\vec {c}}\in \mathbb {R} _{>0}^{m}$ known to exist in each positive stoichiometric compatibility class $({\vec {c}}_{0}+S)\cap \mathbb {R} _{>0}^{m}$ of a complex balanced mass action system is a global attractor of the compatibility class. That is to say, every trajectory originating in $({\vec {c}}_{0}+S)\cap \mathbb {R} _{>0}^{m}$ converges asymptotically toward a unique point in the compatibility class (the stoichiometrically compatible complex balanced equilibrium concentration).

The belief that complex balanced mass action systems satisfy the conjecture has been widespread through chemical reaction network theory since complex balanced systems were introduced by Fritz Horn and Roy Jackson in 1972 . The conjecture that the convergence to complex balanced equilibria extends globally to $({\vec {c}}+S)\cap \mathbb {R} _{>0}^{n}$ was first made by Fritz Horn in 1974 . The problem has received significant attention since. It is known that in order to prove the conjecture, it is sufficient to prove that trajectories are persistent . Consequently, there is a significant overlap between questions of persistence and attempts to prove the conjecture.

The conjecture was retitled the Global Attractor Conjecture in 2009 by Gheorghe Craciun, Alicia Dickenstein, Anne Shiu and Bernd Sturmfels . The conjecture is currently one of the best known and widely studied problems in chemical reaction network theory .