Quasi-Steady-State Analysis - Revision history

WikiBot: Automatic synchronization with BioUML

2020-12-09T11:14:50Z

Automatic synchronization with BioUML

WikiBot: Automatic synchronization with BioUML

2017-02-13T12:00:37Z

Automatic synchronization with BioUML

WikiBot: Protected "Quasi-Steady-State Analysis": Autogenerated page (‎[edit=sysop] (indefinite))

2015-08-25T05:20:22Z

Protected "Quasi-Steady-State Analysis": Autogenerated page (‎[edit=sysop] (indefinite))

WikiBot: Automatic synchronization with BioUML

2015-08-25T05:20:15Z

Automatic synchronization with BioUML

New page

;Analysis title
:[[File:Default-analysis-icon.png]] Quasi-Steady-State Analysis
;Provider
:[[Institute of Systems Biology]]
;Class
:{{Class|biouml.plugins.modelreduction.QuasiSteadyStateAnalysis}}
;Plugin
:[[Biouml.plugins.modelreduction (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.

:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa1.png]]

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

:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa2.png]]

or with a matrix-vector notation,

:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa3.png]]

where ''y'' ∈ ''R''+''n'' is a concentration vector, ''Pi'': ''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 ''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,

:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa4.png]]

where ''f'' = (''f''1, ''f''2,...)''T''. Let ''δti'' be a short time period after which ''Pi'' and ''Li'' balance each other. Then, we have following relationship:

:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa5.png]]

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

:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa6.png]]

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

:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa7.png]]

for some ''εd'' > 0, we can compute the time scale ''δti'' from this equation. If the magnitude of the time scale is large, namely, <nowiki>|</nowiki>''δti''<nowiki>|</nowiki> > ''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''<nowiki>'</nowiki>th variable is balanced or not; the ratio of <nowiki>|</nowiki>''fi''<nowiki>|</nowiki> to the larger one of ''Pi'' and ''Li'',

:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa8.png]]

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 ''i''th variable is summarized in table 1.

'''Table 1: Detection fast/slow dynamics for variable ''yi'''''

{|
|-
|
|
|0 ≤ ''δti'' ≤ ''εt''
|
| -''εt'' < ''δti'' ≤ 0
|
|<nowiki>|</nowiki>''δti''<nowiki>|</nowiki> > ''εt''
|-
|''ri'' ≤ ''εr''
|
|Fast
|
|Fast
|
|Slow
|-
|''ri'' > ''εr''
|
|Fast
|
|Slow
|
|Slow
|}

==== 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.

[[Category:Analyses]]
[[Category:DAE models (analyses group)]]
[[Category:ISB analyses]]
[[Category:Autogenerated pages]]

@@ Line 1: / Line 1: @@
 ;Analysis title
-:[[File:DAE-models-Quasi-Steady-State-Analysis-icon.png]] Quasi-Steady-State Analysis
+:[[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-icon.png]] Quasi-Steady-State Analysis
 ;Provider
 :[[Institute of Systems Biology]]
@@ Line 11: / Line 11: @@
 In homogeneous chemical kinetics, the dynamic model can be written in the following form of the ODE.
-:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa1.png]]
+:: [[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:
-:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa2.png]]
+:: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa2.png]]
 or with a matrix-vector notation,
-:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa3.png]]
+:: [[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.
@@ Line 27: / Line 27: @@
 By chain rule,
-:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa4.png]]
+:: [[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:
-:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa5.png]]
+:: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa5.png]]
 where superscript 0 indicates the reference value. Rearranging the equation gives:
@@ Line 38: / Line 38: @@
+|
-:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa6.png]]
+:: [[File:Differential-algebraic-equations-Quasi-Steady-State-Analysis-qssa6.png]]
 |(1)
@@ Line 44: / Line 44: @@
 If the denominator on the righthand side of equation (1) is not zero, namely
-:: [[File:DAE-models-Quasi-Steady-State-Analysis-qssa7.png]]
+:: [[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.
@@ Line 52: / Line 52: @@
 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:DAE-models-Quasi-Steady-State-Analysis-qssa8.png]]
+:: [[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>''.
@@ Line 92: / Line 92: @@
 [[Category:Analyses]]
-[[Category:DAE models (analyses group)]]
+[[Category:Differential algebraic equations (analyses group)]]
 [[Category:ISB analyses]]
 [[Category:Autogenerated pages]]