Detailed balanced mass action systems

A detailed balanced equilibrium concentration of a mass action system Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i,j = 1, \ldots, m} . Using the short-hand notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{a}^{\vec{y}} = a_1^{y_1} \cdots a_m^{y_m}} we can rewrite this as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k(i,j) (\vec{a})^{\vec{y}_i} = k(j,i) (\vec{a})^{\vec{y}_j}} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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).

Matrix formulation

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{diag} \left\{ \Psi(\vec{a}) \right\} A_k^T = A_k \mbox{diag} \left\{ \Psi(\vec{a}) \right\} }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi(\vec{a} \in \mathbb{R}_{\geq 0}^m} is the mass action vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_k \in \mathbb{R}^{m \times m}} is the kinetic or Kirchhoff matrix, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{diag} \left\{ \vec{v} \right\}} where ${\vec {v}}=[v_{1},v_{2},\ldots ,v_{m}]^{T}\in \mathbb {R} ^{m}$ is the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times m} matrix with the elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1, v_2, \ldots, v_m} along the diagonal and zeroes elsewhere.