Stoichiometric Analysis


A network of m chemical species and n reactions can be described by the m by n stochiometry matrix \mathbf{N}. \mathbf{N}_{i,j} is the net number of species i produced or consumed in reaction j. The dynamics of the network are described by

\frac{d}{dt}\mathbf{s}(t) = \mathbf{N} \mathbf{v}(\mathbf{s}(t),\mathbf{p},t),

where \mathbf{s} is the vector of species concentrations, \mathbf{p} is a vector of time independent parameters, and t is time.

Each structural conservation, or interchangably, conserved sum (e.g. conserved moiety) in the network coresponds to a lineraly dependent row in the stoichiometry matrix \mathbf{N}.

If there are conserved sums, then the row rank, r of N is < m, and the stochiometry matrix N may first be re-ordered such that the first r are linearly independent, and the remaining m-r rows are linear combinations of the first r rows. The reduced stochiometry matrix \mathbf{N_R} is then formed from the first r rows of N. Finally, N may be expressed as a product of the m \times r link matrix \mathbf{L} and the r \times n \mathbf{N_R} matrix:

\mathbf{N} = \mathbf{L}\mathbf{N_R}.

The link matrix \mathbf{L} has the form

\mathbf{L} = \left[ \begin{array}{c}
                    \mathbf{I}_{r} \\
                    \mathbf{L}_0    \end{array} \right],

where \mathbf{I}_{r} is the r \times r identity matrix and \mathbf{L}_0 is a (m-r) \times r matrix.