Stoichiometry

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 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 \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 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{y}_i = [ y_{i1}, y_{i2}, \ldots, y_{in}]^T \in \mathbb{Z}_{\geq 0}^n} 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 \vec{y}_i' = [ y_{i1}', y_{i2}', \ldots, y_{in}']^T \in \mathbb{Z}_{\geq 0}^n} . For example, in the reaction network

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 \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

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{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 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 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

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 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 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{y}_i'-\vec{y}_i} , 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 = 1, \ldots, r} . In other words, we define

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 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 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 s = dim(S)} . For the network above we have

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{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

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 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 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 s = 2} .

Stoichiometric matrix

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

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 \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 $({\vec {c}}_{0}+S)\cap \mathbb {R} _{\geq 0}^{n}$. They are affine translations of the stoichiometric subspaces 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 S} and their interiors are locally homeomorphic to the Euclidean space of dimension 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 s} .

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

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{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 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 t \geq 0} . When 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 s < n} , it follows that solutions 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{c}(t)} may not wander freely about the positive orthant 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 \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 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 R_i(t,\vec{c})} , it follows that 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{c}(t) \in (\vec{c}_0 + S) \cap \mathbb{R}^n_{\geq 0}} 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 t \geq 0} . For the above network, even though the system is five-dimensional, trajectories ${\vec {c}}(t)$ remain in two-dimensional translations of 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 S} .