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