To introduce the basic ideas of time scale decomposition, we begin by considering a linear system with first-order kinetics. The ordinary differential equations (ODEs) describing the evolution of the state vector $\mathbf{y}$ can be written as:
\[\frac{d\,\mathbf{y}}{dt} = \mathbf{J} \cdot \mathbf{y}\]where $\mathbf{J}$ is a constant Jacobian matrix. By using the set of right eigenvectors $\mathbf{A}$ of $\mathbf{J}$ as a new basis, we can decompose the Jacobian [Golub96]:
\[\mathbf{x} = \mathbf{A}^{-1} \cdot \mathbf{y}, \quad \mathbf{\Lambda} = \mathbf{A}^{-1} \cdot \mathbf{J} \cdot \mathbf{A}.\]The components of the transformed concentration vector $\mathbf{x}$ are called modes. Since $\mathbf{\Lambda}$ is a diagonal matrix composed of the real or complex eigenvalues $\lambda_i$ of $\mathbf{J}$, the transformed ODE system is fully decoupled:
\[\frac{d\, x^i}{dt} = \lambda^i x^i, \quad i = 1, \dots, N.\]Thus, the modes $x^i \sim e^{\lambda^i t}$ evolve independently of each other. The reciprocals of the real parts of the eigenvalues give the time scales:
\[\tau_i = -\frac{1}{\mathrm{Re}(\lambda^i)}\]Each $\tau_i$ has units of time and is referred to as a time scale (TS). Ordering the time scales such that $\tau_1 < \tau_2 < \ldots < \tau_N$ yields an approximate ranking of the speed of the modes [Lam93]. Modes associated with fast time scales (i.e., eigenvalues with large negative real parts) decay toward zero very quickly and can be eliminated from the system for $t \gg \tau_M$, where $\tau_M$ is a fast time scale.
For general nonlinear problems, the Jacobian is time-dependent, and its eigenvalues and eigenvectors change over time. Therefore, to obtain a reasonable characterization of the system’s dynamics, time scale decomposition must be performed repeatedly at various time points throughout the evaluation of the reaction system.
The methods described here take advantage of the wide range of characteristic time scales present in biological systems. They are based on local analysis of the Jacobian, partitioning it into fast and slow components at the initial point of a user-chosen interval.
All three methods involve numerical integration using the LSODA solver [Petzold83], (see deterministic simulation). The TSSA methods use the Schur transformation and solution of the Sylvester equation [Golub96] (as in the ILDM and Modified ILDMapproaches), which are performed by CLAPACK.
