Stoichiometry

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Stoichiometry in the mathematical modeling of chemical reactions is the manner in which the quantities of reactants and products involved in chemical reactions are tracked. Of particular interest is the number of molecules require to catalyze a reaction and the number produced by a reaction. For physical reasons, the stoichiometric quantities are typically assumed to be whole numbers.

In a general reaction of the form \sum_{j=1}^n y_{ij} \mathcal{A}_j \rightarrow \sum_{j=1}^n y_{ij}' \mathcal{A}_j, a vector is assigned to the reactant and product complexes according to \vec{y}_i = [ y_{i1}, y_{i2}, \ldots, y_{in}]^T \in \mathbb{Z}_{\geq 0}^n and \vec{y}_i' = [ y_{i1}', y_{i2}', \ldots, y_{in}']^T \in \mathbb{Z}_{\geq 0}^n. For example, in the reaction network


\begin{array}{ll} \mathcal{R}_1: \; \; \; \; \; & \mathcal{A}_1 + \mathcal{A}_2 \rightarrow \mathcal{A}_3 \\ \mathcal{R}_2: \; \; \; \; \; & 2\mathcal{A}_3 \rightarrow \mathcal{A}_4 \end{array}

the following vectors are assigned


\vec{y}_1 = \left[ \begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \end{array} \right], \; \; \; \vec{y}_1' = \left[ \begin{array}{c} 0 \\ 0 \\ 1 \\ 0  \end{array} \right], \; \; \; \vec{y}_2 = \left[ \begin{array}{c} 0 \\ 0 \\ 2 \\ 0 \end{array} \right],\vec{y}_2' = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right]

to indicate that the reactant complex of the first reaction consists of one element of the first species and second species, the product complex of the first reaction consists of one element of the third species, and so on.

Complex matrix

The complex matrix Y \in \mathbb{R}^{n \times m} is the matrix with columns given by the stoichiometrically distinct complexes. For the reaction network given above we have


Y = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right].

Stoichiometric subspace

The stoichiometric subspace S is given by the span of the reaction vectors \vec{y}_i'-\vec{y}_i, i = 1, \ldots, r. In other words, we define


S = \mbox{ span } \left\{ (\vec{y}_i' - \vec{y}_i) \; | \; (\mathcal{C}_i, \mathcal{C}_i') \in \mathcal{R} \right\}.

The dimension of the stoichiometric subspace S is commonly denoted s = dim(S). For the network above we have


\vec{y}_1'-\vec{y}_1 = \left[ \begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array} \right] - \left[ \begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \end{array} \right] = \left[ \begin{array}{c} -1 \\ -1 \\ 1 \\ 0 \end{array} \right] \; \; \; \mbox{and} \; \; \; 
\vec{y}_2'-\vec{y}_2 = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right] - \left[ \begin{array}{c} 0 \\ 0 \\ 2 \\ 0 \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ -2 \\ 1 \end{array} \right].

In other words, there is a net loss of one molecule of species one and two and a net gain of one molecule of species three as a result of reaction one, and a loss of two molecules of species three and gain of one molecule of species four as a result of reaction two. It follows that


S = \mbox{span} \left\{ \left[ \begin{array}{c} -1 \\ -1 \\ 1 \\ 0 \end{array} \right], \left[ \begin{array}{c} 0 \\ 0 \\ -2 \\ 1 \end{array} \right] \right\}

and the dimension is s = 2.

Stoichiometric matrix

The stoichiometric matrix \Gamma \in \mathbb{Z}^{m \times r} is the matrix with the i^{th} column given by the i^{th} reaction vector \vec{y}_i' - \vec{y}_i. For the network above we have


\Gamma = \left[ \begin{array}{cc} -1 & 0 \\ -1 & 0 \\ 1 & -2 \\ 0 & 1 \end{array} \right].

Stoichiometric compatibility classes

The stoichiometric compatibility classes of a chemical reaction network are the sets (\vec{c}_0 + S) \cap \mathbb{R}^n_{\geq 0}. They are affine translations of the stoichiometric subspaces S and their interiors are locally homeomorphic to the Euclidean space of dimension s.

Stoichiometric compatibility classes arise from the general kinetic form of a chemical reaction network. Integrating directly gives


\vec{c}(t)= \vec{c}(0) + \sum_{i = 1}^r (\vec{y}_i' - \vec{y}_i) \int_0^t R(s,\vec{c}(s)) \; ds \in \vec{c}(0) + S

for all t \geq 0. When  s < n, it follows that solutions \vec{c}(t) may not wander freely about the positive orthant \mathbb{R}^n_{\geq 0}; rather, they are restricted to a translation (determined by the initial condition) of stoichiometric subspace. With modest assumptions of the form of the rate functions R_i(t,\vec{c}), it follows that \vec{c}(t) \in (\vec{c}_0 + S) \cap \mathbb{R}^n_{\geq 0} for all t \geq 0 [1]. For the above network, even though the system is five-dimensional, trajectories \vec{c}(t) remain in two-dimensional translations of S.

References

  1. Aizik I. Vol'pert and Sergei I. Hudjaev, Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics, Martinus Nijhoff Publishers, Dordrecht, Netherlands, 1985.