# Detailed balanced mass action systems

A **detailed balanced equilibrium concentration** of a mass action system is a point which satisfies

for all . Using the short-hand notation we can rewrite this as for all . 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

where is the mass action vector, is the kinetic or Kirchhoff matrix, and where is the matrix with the elements along the diagonal and zeroes elsewhere.

## Properties of detailed balanced mass action systems

## References

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