Team:Paris/Modeling/FromMolReactToNLOde

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{{Paris/Header|From Molecular Reactions to Non-Linear ODEs}}
{{Paris/Section_contents_characterization}}

== Idea and Assumptions ==

We will here propose a Mathematical Modeling of the [[Team:Paris/Modeling/Molecular_Reactions|elementary molecular reactions]]. The idea of the ''Characterization Approach'' is that this modelization must both accounts for [[Team:Paris/Modeling/Protocol_Of_Characterization|every small steps]] of the system and allow the [[Team:Paris/Modeling/Protocol_Of_Characterization|experimental characterizations]].


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Therefore, the following equations do not describe properly what ''really'' happens in the cells. For exemple, we know that the transcription factor FlhD-FlhC is actually an ''hexamere FlhD<sub>4</sub>C<sub>2</sub>''. But, as we will surely not get access to the ''dissociation constant'' of the ''hexamerisation'', we just treat it as an ''abstract'' inducer protein "''FlhDC''", with an order (''n'') in its ''complexation caracterization'' probably around 2*4 = 8 (but perhaps completly different ! ; the estimation of the error by the [[Team:Paris/Modeling/Implementation#Parameters_Finder_Programs|Parameters Finder Programs]] would tell us if it stays consistent).

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== Mathematical Interpretation and Simulation of the Molecular Reactions ==

First, we consider the '''complexation phenomenon'''. We show, under the ''quasi steady-state hypothesis'', how it leads to ''Non-Linear interactions'' like "Hill functions".

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Then, we apply these results to the ''Molecular Reactions'' of our system. That gives all the theoretical calculation of the complexations, and finally, with the ''previous assumptions'', we get the ''equations of the full system of ODEs''.

{| border="1" width="100%" style="text-align: center"
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| width="75%" | '''Complexation Steady-State''' <br>
The double triangle ('''><''') is a symbol for complexations : <br>
[[Image:compl.jpg|110px]] means ''n<sub>1</sub>'' molecules of ''FlhDC'' complexed with a ''pFliA'' promoter<br><br>

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{{Paris/Toggle2|Complexation Steady-State|Team:Paris/Modeling/Compl_SS}}
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| width="25%" | '''On the Example :''' <br>
<small> How given amounts of ''FlhDC'' and ''FliA'' would produce ''FP1'' ? </small><br>
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* ''FlhDC'' and ''FliA'' bind to ''pFliL''<br>
following "Hill functions" : <br>
[[Image:DCiLcomplEq.jpg|270px]]

[[Image:AiLcomplEq.jpg|270px]]

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| width="75%" | '''Equations of the Full System''' <br>
The double triangle ('''><''') is a symbol for complexations : <br>
[[Image:compl.jpg|110px]] means ''n<sub>1</sub>'' molecules of ''FlhDC'' complexed with a ''pFliA'' promoter<br><br>

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| width="25%" | '''On the Example :''' <br>
<small> How given amounts of ''FlhDC'' and ''FliA'' would produce ''FP1'' ? </small>
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* ''FlhDC><pFliL'' and ''FliA><pFliL'' causes "production" of ''FP1'', diluted along times :
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[[Image:bFP1dot.jpg|bFP1dot.jpg]]
the outlined parameters are necessary and enough to fully characterize &#131;5
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<div style="text-align: right">For complete PDF with all Reactions and Numbered References : [https://static.igem.org/mediawiki/2008/6/61/Equa_Core_System.pdf complete list of reactions and equations] </div>

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