COPASI: Hidden/Intranet/Fast Reactions/Mathematical Analysis

We are dealing with the following system:

(1) $\mathbf{\dot{x}} = \mathbf{N} \, \mathbf{v}(\mathbf{x})$

Where $\mathbf{x}$ is the vector of $m$ species, $\mathbf{v}(\mathbf{x})$$ is the vector of $n$ reaction velocities , and $\mathbf{N}$ is the $m\, x \, n$ stoichiometry matrix.

Let $n_f$ be the number of fast reactions and $m_f$ be the number of species participating in those. Without loss of generality we can assume that the reactions and species are ordered in such a way that the first $n_f$ reactions are fast and the first $m_f$ species participate in them. Therefore the stoichiometry matrix can be written as: $\begin{pmatrix} \mathbf{F} & \mathbf{S_1} \\ \mathbf{0} & \mathbf{S_2} \end{pmatrix}$ . This leads to the following system:

(2) $\mathbf{\dot{x}} = \begin{pmatrix} \mathbf{F} & \mathbf{S_1} \\ \mathbf{0} & \mathbf{S_2} \end{pmatrix} \, \begin{pmatrix} \mathbf{v_f}(\mathbf{x}) \\ \mathbf{v_s}(\mathbf{x}) \end{pmatrix}= \begin{pmatrix} \mathbf{S_1} \\ \mathbf{S_2} \end{pmatrix} \, \mathbf{v_s}(\mathbf{x})$ .

Where we used the quasi steady-state assumtion $\mathbf{F} \, \mathbf{v_f}(\mathbf{x}) = 0$ . Please note: It is important that the initial state fulfills: $\mathbf{F} \, \mathbf{v_f}(\mathbf{x_0}) = 0$

Existence and Uniqueness of Solution of the Quasi Steady-State Assumption

Let $\mathbf{U}\,\mathbf{\Sigma}\,\mathbf{V}$ be a singular value decomposition of $\mathbf{F}$ . We define $\mathbf{\hat{U}} = \begin{pmatrix} \mathbf{U} & \mathbf{0} \\ \mathbf{0} & \mathbf{I} \end{pmatrix}$ , which yields:

(3) $\mathbf{\hat{U}^{T}}\,\mathbf{\dot{x}} = \begin{pmatrix} \mathbf{U^{T}} \, \mathbf{F} & \mathbf{U^{T}}\,\mathbf{S_1} \\ \mathbf{0} & \mathbf{S_2} \end{pmatrix} \, \begin{pmatrix} \mathbf{v_f}(\mathbf{x}) \\ \mathbf{v_s}(\mathbf{x}) \end{pmatrix} = \begin{pmatrix} \mathbf{U^{T}}\,\mathbf{S_1} \\ \mathbf{S_2} \end{pmatrix} \, \mathbf{v_s}(\mathbf{x})$ .

Let $\mathbf{U} = \begin{pmatrix} \mathbf{U_{1}} & \mathbf{U_{2}} \end{pmatrix}$ where $\mathbf{U_{2}}$ are the column vectors of $\mathbf{U}$ corresponding to zero singular values. Thus we have $\mathbf{U^{T}_{2}} \, \mathbf{F} = 0$ independently from $\mathbf{v_f}(\mathbf{x})$ , i.e., $\mathbf{U^{T}_{2}} \, \mathbf{\dot{x}} = 0$ for the fast reactions system. Thus $\mathbf{U^{T}_{2}} \, \mathbf{x} = \mathbf{C}$ represents the mass conservation relationships of the fast reaction system. Any solution to the equation derived from the quasi steady-state assumption must fulfill these mass conservation relationships and thus can be represented as $$ \mathbf{x_{f}} = \begin{pmatrix} \mathbf{U_{1}} & \mathbf{U_{2}} \end{pmatrix} \begin{pmatrix} \mathbf{q} \\ \mathbf{C} \end{pmatrix}$ , where $\mathbf{q} \in \mathbb{R}^{r}$ with $r$ being the number of non zero singular values. Note, $\mathbf{C}$ can still be changed by the slow reactions. Similarly to $\mathbf{U}$ let $\mathbf{V} = \begin{pmatrix} \mathbf{V_{1}} \\ \mathbf{V_{2}} \end{pmatrix}$ . Thus the quasy steady-state assumption can be represented by the following equation:

(4) $\mathbf{V_{1}} \, \, \mathbf{v_f} (\mathbf{U_{1}} \, \mathbf{q} + \mathbf{U_{2}} \, \mathbf{C}, \, \mathbf{x_{s}} ) = 0$

In this equation $\mathbf{x_{s}}$ are the species changed only by the slow reactions and thus are constant. We therefore have $r$ equations for $r$ unknowns $\mathbf{q_{i}}$ , which means the equation is generally uniquely solvable.

Fast Reaction System

To create the complete DAE system we add $m_{f}$ variables $\lambda_{i}$ to our system, i.e., $\mathbf{\tilde{x}} = \begin{pmatrix} \mathbf{x} \\ \mathbf{\lambda} \end{pmatrix}$ . With $\mathbf{A} = \begin{pmatrix} \mathbf{I}_{m \, x \, m} & 0 \\ 0 & 0 \end{pmatrix}$ we derive:

(5) $\mathbf{A^{T}}\,\mathbf{\dot{\tilde{x}}} = \begin{pmatrix} \begin{pmatrix} \mathbf{S_1} \\ \mathbf{S_2} \end{pmatrix} \, \mathbf{v_s}(\mathbf{x}) \\ \\ \mathbf{F} \, \mathbf{v_f}(\mathbf{x})\end{pmatrix}$

We can reduce the number of algebraic equations to $r \le m_{f}$ by making use of the singular value decomposition of $\mathbf{F}$

(5a) $\mathbf{A^{T}}\,\mathbf{\dot{\tilde{x}}} = \begin{pmatrix} \begin{pmatrix} \mathbf{S_1} \\ \mathbf{S_2} \end{pmatrix} \, \mathbf{v_s}(\mathbf{x}) \\ \\ \mathbf{V_1} \, \mathbf{v_f}(\mathbf{x})\end{pmatrix}$