Quasi-Steady-State Analysis

Analysis title

Quasi-Steady-State Analysis

Provider

Institute of Systems Biology

Class

QuasiSteadyStateAnalysis

Plugin

biouml.plugins.modelreduction (Model reduction plug-in)

Description

In homogeneous chemical kinetics, the dynamic model can be written in the following form of the ODE.

Since a chemical reaction system generally consists of production and loss terms, the ODE can be rewritten as:

or with a matrix-vector notation,

where y ∈ R⁺ⁿ is a concentration vector, P_i: R⁺ⁿ → R⁺ is a production term, and L_i : R⁺ⁿ → R⁺ is a loss terms. S is a stoichiometric matrix and v is a reaction rate vector. The subscripts P and L denote the production and loss, respectively.

Generally, it can be said that if y_i exhibits a quasi-steady state behavior, such behavior is observed after a short period of time for the corresponding P_i and L_i to balance each other. At the moment either P_i or L_i enlarge, the period during which P_i balances with L_i can be evaluated in a simple manner.

By chain rule,

where f = (f₁, f₂,...)^T. Let δt_i be a short time period after which P_i and L_i balance each other. Then, we have following relationship:

where superscript 0 indicates the reference value. Rearranging the equation gives:

(1)

If the denominator on the righthand side of equation (1) is not zero, namely

for some ε_d > 0, we can compute the time scale δt_i from this equation. If the magnitude of the time scale is large, namely, |δt_i| > ε_t for some ε_t > 0, y_i is considered to exhibit slow dynamics.

If 0 ≤ δt_i ≤ ε_t, it will reach a balancing state quickly and QSSA can be applied.

If -ε_t ≤ δt_i ≤ 0, another scale factor should be used to determine whether the i'th variable is balanced or not; the ratio of |f_i| to the larger one of P_i and L_i,

If r_i is large, namely r_i > ε_r for some ε_r > 0, the production and loss are neither balanced nor can be balanced soon, hence QSSA is not applicable to such y_i.

The applicability of QSSA to the ith variable is summarized in table 1.

Table 1: Detection fast/slow dynamics for variable y_i

		0 ≤ δti ≤ εt		-εt < δti ≤ 0		|δti| > εt
ri ≤ εr		Fast		Fast		Slow
ri > εr		Fast		Slow		Slow

References

1. J Choi, KW Yang, TY Lee and SY Lee. "New time-scale criteria for model simplification of bio-reaction systems". BMC Bioinformatics, 9:338, 2008.