Team:WHU-China/Project/FattyAcidModel

From 2012.igem.org

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　　The fatty acid degradation device is of particular importance in our project because of its bistability: gene fadL, fadD, etc, are highly-expressed when the concentration of fatty acid in the gut is high, and vice versa. So it is interesting to investigate when and how the device will be bistable, and thus we build a mathematical model based on ordinary differential equations (ODE) to describe the metabolism of fatty acid in E.coslim.
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The Fatty Acid Degradation Device may be the most complicated part in our project, along with its great importance. The antagonistic relationship between gene fadR and those related to β oxidation -- the fadL, fadD, etc, -- makes it regulatable to the concentration of fatty acid in the environment. Thus, it is necessary to explore the quantitative response corresponding to the concentration change of fatty acid. We build an ordinary differential equations-based mathematical model to describe the device and find a proper set of parameters under which the proportion of the steady expression level of fadL to fadR changes broadly from 0.02 to 50. The model mathematically demonstrates the effectiveness of the Fatty Acid Degradation Device and also provides meaningful clues for the optimization of the device in experiments.  </p><p>
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<h2>The Ordinary Differential Equations of the Model</h2>
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<h2>Mathematical Equations</h2>
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　　We assume that the concentration of fatty acid outside the bacteria remains constant over time and build our ODE-based model as follows:
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We conduct an evaluation by mathematical modeling and build the ordinary differential equations (ODE) as follows: </p><p>
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<img src="https://static.igem.org/mediawiki/2012/7/7c/Fatty_Fml_1.png" width="529" height="317" hspace="2" vspace="1" border="2" align="top" />…………………………①</p><p>
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<img src="https://static.igem.org/mediawiki/2012/3/3d/Fatty_equation.png" width="476" height="340" hspace="2" vspace="1" border="2" align="top" />
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　　<strong>For simplicity</strong> </p><p>
 
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　　① the <i>Complex</i>, or variable <i>x<sub>7</sub></i>, refers to the Fatty acyl-CoA-FadR complex;  </p><p>
 
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　　② FadA, FadB, FadI and FadJ are amalgamated into a single variable <i>x<sub>4</sub></i>, namely, the <i>&beta;-Oxidase</i>, since they have similar functions.  </p><p>
 
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　　<strong>The meaning of the parameters in the equations:</strong>  </p><p>
 
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　　① <i>a</i> is the affinity of <i>FadR</i> to the promoter PfadR, and <i>V</i> is the background expression rate of related genes. The expression rates of <i>FadL, FadD and &beta;-Oxidase</i> are equivalent since they have the same promoter.  </p><p>
 
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　　② <i>E</i> is the constitutive expression rate of <i>FadR</i>, and <i>R, L, D, B, C</i> is the degradation rate of <i>FadR, FadL, FadD, &beta;-Oxidase</i> and <i>Complex</i>, respectively.  </p><p>
 
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　　③ <i>k<sub>1</sub></i> and <i>k<sub>2</sub></i> are the forward and reverse reaction rate coefficients, respectively. <i>k<sub>3</sub></i> to <i>k<sub>6</sub></i> are parameters related to enzyme-catalyzed reactions based on the Michaelis-Menten Equation. Specially,  </p><p>
 
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<img src="https://static.igem.org/mediawiki/2012/b/bc/K3Relation.png" width="96" height="82" hspace="2" vspace="1" border="2" align="top" /></p></center>
 
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　　while <i>f</i> is the concentration of fatty acid outside the bacteria, <i>K<sub>L</sub></i> is the Michaelis constant of <i>FadL</i>, and <i>k<sub>L</sub></i> is the maximal activity of <i>FadL</i>.
 
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Revision as of 22:55, 23 September 2012

Mathematical Model on Fatty Acid Degradation Device

The Fatty Acid Degradation Device may be the most complicated part in our project, along with its great importance. The antagonistic relationship between gene fadR and those related to β oxidation -- the fadL, fadD, etc, -- makes it regulatable to the concentration of fatty acid in the environment. Thus, it is necessary to explore the quantitative response corresponding to the concentration change of fatty acid. We build an ordinary differential equations-based mathematical model to describe the device and find a proper set of parameters under which the proportion of the steady expression level of fadL to fadR changes broadly from 0.02 to 50. The model mathematically demonstrates the effectiveness of the Fatty Acid Degradation Device and also provides meaningful clues for the optimization of the device in experiments.

The Ordinary Differential Equations of the Model

We conduct an evaluation by mathematical modeling and build the ordinary differential equations (ODE) as follows:

…………………………①

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