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Flux

The rate of flow of metabolites through a metabolic pathway, representing the change in metabolite concentration over time.
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The statement of the theorem

Let M={M1,M2,…,MN}\mathbf{M} = \{M_1, M_2, \dots, M_N\} be the set of NN metabolites in a system, and let v={v1,v2,…,vE}\mathbf{v} = \{v_1, v_2, \dots, v_E\} be the set of EE enzymatic reactions. The flux J∈RE\mathbf{J} \in \mathbb{R}^E is defined by the steady-state mass balance equation for each metabolite MiM_i: \begin{equation} \label{eq:mass_balance} \frac{d[M_i]}{dt} = \sum_{j=1}^{E} \nu_{ij} v_j = 0 \end{equation}, where νij\nu_{ij} is the stoichiometric coefficient of MiM_i in reaction jj, and vjv_j is the flux through reaction jj. The flux vector J\mathbf{J} must satisfy the stoichiometry matrix N∈RN×E\mathbf{N} \in \mathbb{R}^{N \times E} such that NJ=0\mathbf{N} \mathbf{J} = \mathbf{0}.