# Detailed balanced mass action systems

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

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

for all ${\displaystyle i,j=1,\ldots ,m}$. Using the short-hand notation ${\displaystyle {\vec {a}}^{\vec {y}}=a_{1}^{y_{1}}\cdots a_{m}^{y_{m}}}$ we can rewrite this as ${\displaystyle k(i,j)({\vec {a}})^{{\vec {y}}_{i}}=k(j,i)({\vec {a}})^{{\vec {y}}_{j}}}$ for all ${\displaystyle i,j=1,\ldots ,m}$. 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[1]).

## Matrix formulation

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

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

where ${\displaystyle \Psi ({\vec {a}}\in \mathbb {R} _{\geq 0}^{m}}$ is the mass action vector, ${\displaystyle A_{k}\in \mathbb {R} ^{m\times m}}$ is the kinetic or Kirchhoff matrix, and ${\displaystyle {\mbox{diag}}\left\{{\vec {v}}\right\}}$ where ${\displaystyle {\vec {v}}=[v_{1},v_{2},\ldots ,v_{m}]^{T}\in \mathbb {R} ^{m}}$ is the ${\displaystyle m\times m}$ matrix with the elements ${\displaystyle v_{1},v_{2},\ldots ,v_{m}}$ along the diagonal and zeroes elsewhere.

## References

1. Fritz Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal., 49:172--186, 1972