Detailed balanced mass action systems

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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[1]).

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.

Properties of detailed balanced mass action systems

References

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