Detailed balanced mass action systems

A detailed 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

$k(i,j) a_1^{y_{i1}} \cdots a_m^{y_{im}} = k(j,i) a_1^{y_{j1}} \cdots a_m^{y_{jm}}$

for all $i,j = 1, \ldots, m$. Using the short-hand notation $\vec{a}^{\vec{y}} = a_1^{y_1} \cdots a_m^{y_m}$ we can rewrite this as $k(i,j) (\vec{a})^{\vec{y}_i} = k(j,i) (\vec{a})^{\vec{y}_j}$ for all $i,j = 1, \ldots, m$. A mass-action system is said to be detailed balanced if every equilibrium concentration permitted by the system is a detailed balanced equilibrium concentration.

Detailed balancing guarantees that every elementary step in the reaction mechanism is balanced by a reverse elementary step at equilibrium. Every chemical reaction network which permits a detailed balanced mass action system is reversible (Theorem 2A).

Matrix formulation

Detailed balancing at equilibrium can be alternatively characterized by the matrix expression

$\mbox{diag} \left\{ \Psi(\vec{a}) \right\} A_k^T = A_k \mbox{diag} \left\{ \Psi(\vec{a}) \right\}$

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 $\mbox{diag} \left\{ \vec{v} \right\}$ where $\vec{v} = [ v_1, v_2, \ldots, v_m ]^T \in \mathbb{R}^m$ is the $m \times m$ matrix with the elements $v_1, v_2, \ldots, v_m$ along the diagonal and zeroes elsewhere.