ILDM (Deuflhard)

The ILDM (Deuflhard) method employs the algorithm developed by Deuflhard and Heroth (see Deuflhard96, Zobeley05, Surovtsova09 for details).

Basic concept of decomposition into “slow” and “fast” modes

The block slow–fast decomposition of the Jacobian is performed in two steps:

• First, a real Schur decomposition yields a block upper triangular matrix.
• Next, the desired decoupled structure of the transformed Jacobian is obtained by solving a Sylvester equation.

This basic procedure results in a transformation of the state vectors into new modes, which can then be separated into slow and fast modes. As a result, the dynamics of a full reaction system with $n$ ordinary differential equations (ODEs) is reduced to a differential-algebraic equation (DAE) system consisting of $n_{\text{slow}}$ ODEs and $n - n_{\text{slow}}$ algebraic equations. The number $n_{\text{slow}}$ of slow variables is determined iteratively using the tolerance criterion proposed by Deuflhard and Heroth.

The method implemented in COPASI not only emphasizes the reduction of the mathematical equations, but also places special focus on reducing the underlying biochemical network. To achieve this, COPASI analyzes the transformation matrices obtained after solving the Sylvester equation.

The implementation in COPASI

The Jacobian is calculated by using finite differences. The Schur transformation and the solution of the Sylvester equation are carried out using CLAPACK.

COPASI presents the results in four matrices and four vectors:

• The “Species” matrix shows the contribution of each metabolite to each mode.
• The “Modes” table summarizes the mode distribution for each metabolite.
• The “Slow space” and “Fast space” vectors display the contribution of each concentration variable to all slow or fast modes, respectively.
• The “Reactions contributions to modes” and “Reactions distribution between modes” matrices are the products of the stoichiometric and transformation matrices, normalized by column and row, respectively.
• The “Reactions slow space” and “Reactions fast space” vectors provide the contribution of each reaction to the slow and fast spaces.

Metabolites with the largest contribution to the fast space are considered “fast.” Their ordinary differential equations are replaced by the corresponding algebraic equations (i.e., with the same right-hand side). If a subset of species contributes only to the fast space (and not the slow space), time-scale decomposition results in a separation of the reaction network. Reactions that dominate in the fast modes are classified as fast reactions.