Approach: Model Analysis by means of Time Scale
Separation
In order to explain the basic notions of time scale decomposition we start with a first-order kinetics system. Then,
the differential equations (ODE) describing the system dynamics ${\bf y}$ are linear:
$$ \frac{d\,{\bf y}}{d\, t} = {\bf J} \cdot {\bf y}$$
with constant Jacobian ${\bf J}$. By using the set of right eigenvectors ${\bf A}$ of ${\bf J}$ as the new basis we
can decompose the Jacobian [
Golub96]
$${\bf x} = {\bf A}^{-1} \cdot {\bf y}, \quad {\bf \Lambda} = {\bf A}^{-1} \cdot {\bf J} \cdot {\bf A}.$$
The components of the transformed concentration vector ${\bf x}$ are called
modes. Because ${\bf \Lambda}$ is
a diagonal matrix of real or complex
eigenvalues $\lambda_i$ of ${\bf J}$, the transformed ODE system is fully decoupled:
$$\frac{d\, x^i}{d\, t}=\lambda^i x^i, \ i=1,...,N.$$
Thus, the modes $x^i \sim e^{\lambda^i t}$ evolve independently of each other.
The reciprocals of $\Re(\lambda^i):$
$$\tau_i = -\frac{1}{\Re(\lambda^i)}$$
have a dimension of time and are called time scales (TS). Ordering them w.r.t. magnitudes $\tau_1 < \tau_2 <
... < \tau_N$ leads to approximate speed
ranking of the modes [
Lam93]. The modes corresponding to fast time
scales (eigenvalues with large negative real part) approach $0$ 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. Its eigenvalues and eigenvectors change with
time.
Hence, in order to obtain a reasonable characterization of the systems dynamics the time scale decomposition has to
be applied repeatedly at many time points through the evaluation time of the reaction system.
The methods presented here relies on the presence of a wide range of characteristic time-scales in biological
systems and are based on the local analysis of the Jacobian, which is partitioned into fast and slow components at
the initial point of a user chosen interval.
All the three methods involves numerical integration using the LSODA solver [
Petzold83],
deterministic
simulation. The TSSA methods uses Schur transformation and the solution of Sylvester equation [
Golub96] (ILDM and Modified ILDM) performed
by
CLAPACK.
