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| \begin{document} |
|
|
| \title{Stability Analysis in COPASI} |
|
|
| \author{Stefan Hoops\\ |
| Virginia Bioinformatics Institute \\ |
| 1880 Pratt Dr. \\ |
| Blacksburg, VA 24060 \\ |
| USA \\ |
| shoops@vt.edu$ $} |
|
|
|
|
| \date{2004-11-15} |
|
|
| \maketitle |
| \begin{abstract} |
| The stability analysis is an important feature of COPASI. Since its |
| implementation differs from Gepasi's yielding different results an |
| explanation is in order. This document describes the methods used |
| within COPASI in detail and explains the differences in comparison |
| to Gepasi. |
| \end{abstract} |
|
|
| \section{Calculation of the Jacobian} |
|
|
| \subsection{Complete Model} |
| The Jacobian of a vector function $\boldsymbol{f}: \R^{n} \rightarrow \R^{n}$ |
| is defined as: |
| |
| \begin{eqnarray}\label{Jacobian} |
| \boldsymbol{J}_{i,j}(\boldsymbol{x}) = |
| {\left. |
| \frac{\partial f_i}{\partial x_j} |
| \right|}_{\boldsymbol{x}} |
| & \mbox{where} & |
| i = 1, \ldots, n \;\mbox{and}\; j = 1, \ldots, m |
| \end{eqnarray} |
| |
| Let us consider $M$ chemical species participating in $N$ |
| reactions. The kinetics of such a system can then be described by: |
| |
| \begin{eqnarray}\label{FullModel} |
| \dot{\boldsymbol{x}} & = & \boldsymbol{N} \, \boldsymbol{v}(\boldsymbol{x}) |
| \end{eqnarray} |
| |
| where $\boldsymbol{N}$ is the stoichiometry matrix, $\boldsymbol{v}$ is the vector |
| of reaction velocities, and $\dot{\boldsymbol{x}}$ denotes the change |
| of the chemical species over time. |
|
|
| Using the definition of the Jacobian (\ref{Jacobian}) we derive |
| for $\boldsymbol{J}$: |
| |
| \begin{eqnarray} |
| \boldsymbol{J}_{i,j}(\boldsymbol{x}) |
| = |
| {\left. |
| \frac{\partial \dot{x}_i}{\partial x_j} |
| \right|}_{\boldsymbol{x}} |
| = |
| \boldsymbol{N}_{i,k} \, |
| {\left. |
| \frac{\partial v_k}{\partial x_j} |
| \right|}_{\boldsymbol{x}} |
| = |
| \boldsymbol{N}_{i,k} \, \boldsymbol{E}_{k,j}(\boldsymbol{x}) |
| \end{eqnarray} |
| |
| where we used the definition of the Elasticity Matrix: |
| |
| \begin{eqnarray} |
| \boldsymbol{E}_{i,j}(\boldsymbol{x}) = |
| {\left. |
| \frac{\partial v_i}{\partial x_j} |
| \right|}_{\boldsymbol{x}} |
| \end{eqnarray} |
|
|
| \subsection{Reduced Model} |
| Let us assume that the Complete Model (\ref{FullModel}) contains $p$ |
| independent species and $q = M - p$ dependent species. Furthermore, we |
| can assume without loss of generalization that the first $p$ species |
| are independent. We therefore can describe the dependent species |
| through the following equations: |
| |
| \begin{eqnarray}\label{MassConservation} |
| x_i = c_i + \sum_{k=0}^p \alpha_{i,k} \, x_k & \mbox{where} & i = q, \ldots, M, |
| \end{eqnarray} |
| |
| which are also know as Mass Conservations or Moieties. |
| Using this ability to express the dependent species we rewrite the |
| velocity function so that the new velocity function $\boldsymbol{\tilde{v}}$ |
| only depends on the first $p$ species $\boldsymbol{x}_r$. |
| |
| \begin{eqnarray} |
| \boldsymbol{\tilde{v}}(x_r) = \boldsymbol{v}(x_1, \ldots, x_p, |
| c_q + \sum_{i=0}^p \alpha_{q,i} x_i, \ldots, |
| c_M + \sum_{i=0}^M \alpha_{M,i} x_i) |
| \end{eqnarray} |
| |
| This leads to the reduced system: |
| |
| \begin{eqnarray}\label{ReducedModel} |
| \dot{\boldsymbol{x_r}} & = & \boldsymbol{N}_r \, |
| \boldsymbol{\tilde{v}}(\boldsymbol{x}_r) |
| \end{eqnarray} |
| |
| where $\boldsymbol{N}_r$ is the matrix comprised of first $p$ rows of |
| $N$. |
| |
| Using the definition of the Jacobian (\ref{Jacobian}) once again we derive |
| for the Jacobian ($p \mbox{x} p$) of the reduced system: |
| |
| \begin{eqnarray} |
| \boldsymbol{J}_{i,j}(\boldsymbol{x}_r) |
| & = & |
| {\left. |
| \frac{\partial \dot{x}_i}{\partial x_j} |
| \right|}_{\boldsymbol{x}_r} |
| = |
| \boldsymbol{N}_{i,k} \, |
| {\left. |
| \frac{\partial \tilde{v}_k}{\partial x_j} |
| \right|}_{\boldsymbol{x}_r} \\ |
| & = & |
| \boldsymbol{N}_{i,k} \, |
| \left( |
| {\left. \frac{\partial v_k}{\partial x_j} \right|}_{\boldsymbol{x}_r} + |
| \sum_{l=q}^{M} |
| {\left. \frac{\partial v_k}{\partial x_l} \right|}_{\boldsymbol{x}_r} |
| \frac{\partial x_l}{\partial x_j} |
| \right) \\ |
| & = & |
| \boldsymbol{N}_{i,k} \, |
| \left( |
| \sum_{l=0}^p |
| {\left. \frac{\partial v_k}{\partial x_l} \right|}_{\boldsymbol{x}_r} |
| \delta_{l,j} |
| + \sum_{l=q}^{M} |
| {\left. \frac{\partial v_k}{\partial x_l} \right|}_{\boldsymbol{x}_r} |
| \alpha_{l,j} |
| \right) \\ |
| & = & \label{LinkMatrix} |
| \boldsymbol{N}_{i,k} \, \boldsymbol{E}_{k,l}(\boldsymbol{x}) |
| \, \boldsymbol{L}_{l,j} |
| \end{eqnarray} |
| |
| where $\delta_{l,j} = 0$ for $l \neq j$ and $1$ otherwise. The link |
| matrix $\boldsymbol{L}$ in (\ref{LinkMatrix}) is defined as: |
| \begin{eqnarray} |
| \boldsymbol{L}_{l,j} & = & |
| \left\{ |
| \begin{array}{ll} |
| \delta_{l,j} & \mbox{if $l \leq p$} \\ |
| \alpha_{l,j} & \mbox{if $l > p$} |
| \end{array} |
| \right. |
| \end{eqnarray} |
|
|
| \subsection{Comparison to Gepasi} |
| Gepasi's calculation of the Jacobian is based on the reduced |
| system. However it uses the following slightly different formula |
| |
| \begin{eqnarray}\label{GepasiModel} |
| \dot{\boldsymbol{x_r}} & = & \boldsymbol{N}_r \, |
| \boldsymbol{v}(\boldsymbol{x}_r) |
| \end{eqnarray} |
| |
| This yields to the Jacobian ($N \mbox{x} p$) of the reduced system: |
| |
| \begin{eqnarray} |
| \boldsymbol{J}_{i,j}(\boldsymbol{x}_r) |
| & = & |
| \boldsymbol{N}_{i,k} \, \boldsymbol{E}_{k,l}(\boldsymbol{x}_r) |
| \end{eqnarray} |
| |
| The omission to express the kinetic functions in the new reduced |
| variables leads to a different Jacobian in the case that the model |
| contains dependent species. |
|
|
| {\bf Note:} It is important to mention that the method to calculate the Jacobian |
| is not effecting the calculation of the dynamics of a system in |
| Gepasi. The reason for this is that Gepasi uses the full |
| Differential and Algebraic Equation (DAE) system. |
| |
| \section{Stability Analysis} |
| The stability analysis in both COPASI and Gepasi is based on the |
| Jacobian of the reduced system since the complete system has a |
| singular Jacobian whenever it contains dependent species. The |
| interpretation for the reduced system differs between COPASI and |
| Gepasi as shown in (\ref{ReducedModel}) and (\ref{GepasiModel}). We |
| therefore achieve different results for the stability analysis |
| especially the eigenvalues of the Jacobians differ. In most cases this |
| does not effect the classification of the steady-state. |
|
|
| We oberved a second cause for different results for the stability |
| analysis, which is numerical instability. This behviour can be |
| observed in the Brusellator model ({\tt brusselator.gps}), which is |
| distributed with Gepasi. \\ |
| \\ |
| COPASI's result:\\ |
| {\tt \small |
| KINETIC STABILITY ANALYSIS \\ |
| The linear stability analysis based on the eigenvalues \\ |
| of the Jacobian matrix is only valid for steady states. \\ |
| \\ |
| Summary: \\ |
| This steady state is unstable, \\ |
| transient states in its vicinity have oscillatory components \\ |
| \\ |
| Eigenvalue statistics: \\ |
| Largest real part: 2.5 \\ |
| Largest absolute imaginary part: 1.65831 \\ |
| 0 are purely real \\ |
| 0 are purely imaginary \\ |
| 2 are complex \\ |
| 0 are equal to zero \\ |
| 2 have positive real part \\ |
| 0 have negative real part \\ |
| stiffness = 1.00000 \\ |
| time hierarchy = 0.000000 \\ |
| } |
| \\ |
| Gepasi's Result: \\ |
| {\tt \small |
| KINETIC STABILITY ANALYSIS \\ |
| \\ |
| Summary: \\ |
| This steady state is unstable. \\ |
| \\ |
| Eigenvalue statistics: \\ |
| Largest real part: +1.593070e+000 \\ |
| Largest absolute imaginary part: 0.000000e+000 \\ |
| 2 are purely real \\ |
| 0 are purely imaginary \\ |
| 0 are complex \\ |
| 0 are equal to zero \\ |
| 2 have positive real part \\ |
| 0 have negative real part \\ |
| stiffness = 1.72068e+305 \\ |
| time hierarchy = 1 \\ |
| } |
| \end{document} |
|
|
