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.