Difference between revisions of "Quasi-Steady-State Analysis"
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In homogeneous chemical kinetics, the dynamic model can be written in the following form of the ODE. | In homogeneous chemical kinetics, the dynamic model can be written in the following form of the ODE. | ||
| − | :: [[File: | + | :: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa1.png]] |
Since a chemical reaction system generally consists of production and loss terms, the ODE can be rewritten as: | Since a chemical reaction system generally consists of production and loss terms, the ODE can be rewritten as: | ||
| − | :: [[File: | + | :: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa2.png]] |
or with a matrix-vector notation, | or with a matrix-vector notation, | ||
| − | :: [[File: | + | :: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa3.png]] |
where ''y'' ∈ ''R''<sup>+''n''</sup> is a concentration vector, ''P<sub>i</sub>'': ''R''<sup>+''n''</sup> → ''R''<sup>+</sup> is a production term, and ''L''<sub>''i''</sub> : ''R''<sup>+''n''</sup> → ''R''<sup>+</sup> 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. | where ''y'' ∈ ''R''<sup>+''n''</sup> is a concentration vector, ''P<sub>i</sub>'': ''R''<sup>+''n''</sup> → ''R''<sup>+</sup> is a production term, and ''L''<sub>''i''</sub> : ''R''<sup>+''n''</sup> → ''R''<sup>+</sup> 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. | ||
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By chain rule, | By chain rule, | ||
| − | :: [[File: | + | :: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa4.png]] |
where ''f'' = (''f''<sub>1</sub>, ''f''<sub>2</sub>,...)<sup>''T''</sup>. Let ''δt<sub>i</sub>'' be a short time period after which ''P<sub>i</sub>'' and ''L<sub>i</sub>'' balance each other. Then, we have following relationship: | where ''f'' = (''f''<sub>1</sub>, ''f''<sub>2</sub>,...)<sup>''T''</sup>. Let ''δt<sub>i</sub>'' be a short time period after which ''P<sub>i</sub>'' and ''L<sub>i</sub>'' balance each other. Then, we have following relationship: | ||
| − | :: [[File: | + | :: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa5.png]] |
where superscript 0 indicates the reference value. Rearranging the equation gives: | where superscript 0 indicates the reference value. Rearranging the equation gives: | ||
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|(1) | |(1) | ||
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If the denominator on the righthand side of equation (1) is not zero, namely | If the denominator on the righthand side of equation (1) is not zero, namely | ||
| − | :: [[File: | + | :: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa7.png]] |
for some ''ε<sub>d</sub>'' > 0, we can compute the time scale ''δt<sub>i</sub>'' from this equation. If the magnitude of the time scale is large, namely, <nowiki>|</nowiki>''δt<sub>i</sub>''<nowiki>|</nowiki> > ''ε<sub>t</sub>'' for some ''ε<sub>t</sub>'' > 0, ''y<sub>i</sub>'' is considered to exhibit slow dynamics. | for some ''ε<sub>d</sub>'' > 0, we can compute the time scale ''δt<sub>i</sub>'' from this equation. If the magnitude of the time scale is large, namely, <nowiki>|</nowiki>''δt<sub>i</sub>''<nowiki>|</nowiki> > ''ε<sub>t</sub>'' for some ''ε<sub>t</sub>'' > 0, ''y<sub>i</sub>'' is considered to exhibit slow dynamics. | ||
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If -''ε<sub>t</sub>'' ≤ ''δt<sub>i</sub>'' ≤ 0, another scale factor should be used to determine whether the ''i''<nowiki>'</nowiki>th variable is balanced or not; the ratio of <nowiki>|</nowiki>''f<sub>i</sub>''<nowiki>|</nowiki> to the larger one of ''P<sub>i</sub>'' and ''L<sub>i</sub>'', | If -''ε<sub>t</sub>'' ≤ ''δt<sub>i</sub>'' ≤ 0, another scale factor should be used to determine whether the ''i''<nowiki>'</nowiki>th variable is balanced or not; the ratio of <nowiki>|</nowiki>''f<sub>i</sub>''<nowiki>|</nowiki> to the larger one of ''P<sub>i</sub>'' and ''L<sub>i</sub>'', | ||
| − | :: [[File: | + | :: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa8.png]] |
If ''r<sub>i</sub>'' is large, namely ''r<sub>i</sub>'' > ''ε<sub>r</sub>'' for some ''ε<sub>r</sub>'' > 0, the production and loss are neither balanced nor can be balanced soon, hence QSSA is not applicable to such ''y<sub>i</sub>''. | If ''r<sub>i</sub>'' is large, namely ''r<sub>i</sub>'' > ''ε<sub>r</sub>'' for some ''ε<sub>r</sub>'' > 0, the production and loss are neither balanced nor can be balanced soon, hence QSSA is not applicable to such ''y<sub>i</sub>''. | ||
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[[Category:ISB analyses]] | [[Category:ISB analyses]] | ||
[[Category:Autogenerated pages]] | [[Category:Autogenerated pages]] | ||
Latest revision as of 18:14, 9 December 2020
- Analysis title
Quasi-Steady-State Analysis
- Provider
- Institute of Systems Biology
- Class
QuasiSteadyStateAnalysis- Plugin
- biouml.plugins.modelreduction (Model reduction plug-in)
[edit] 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, Pi: R+n → R+ is a production term, and Li : 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 yi exhibits a quasi-steady state behavior, such behavior is observed after a short period of time for the corresponding Pi and Li to balance each other. At the moment either Pi or Li enlarge, the period during which Pi balances with Li can be evaluated in a simple manner.
By chain rule,
where f = (f1, f2,...)T. Let δti be a short time period after which Pi and Li 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 εd > 0, we can compute the time scale δti from this equation. If the magnitude of the time scale is large, namely, |δti| > εt for some εt > 0, yi is considered to exhibit slow dynamics.
If 0 ≤ δti ≤ εt, it will reach a balancing state quickly and QSSA can be applied.
If -εt ≤ δti ≤ 0, another scale factor should be used to determine whether the i'th variable is balanced or not; the ratio of |fi| to the larger one of Pi and Li,
If ri is large, namely ri > εr for some εr > 0, the production and loss are neither balanced nor can be balanced soon, hence QSSA is not applicable to such yi.
The applicability of QSSA to the ith variable is summarized in table 1.
Table 1: Detection fast/slow dynamics for variable yi
| 0 ≤ δti ≤ εt | -εt < δti ≤ 0 | |δti| > εt | ||||
| ri ≤ εr | Fast | Fast | Slow | |||
| ri > εr | Fast | Slow | Slow |
[edit] References
- 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.
