"

From 2012.igem.org

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

　　For simplicity, all genes with a promoter PfadR and equally regulated by FadR are deemed as a whole and represented as FadX, i.e., FadX refers to FadL, FadD, FadE, FadA, FadB, FadI, FadJ. And the Complex, or variable x7, refers to the Fatty Acyl-CoA-FadR Complex.

　　Parameters in the ODEs:

　　① E denotes the constitutive expression rate of FadR, and D the degradation rates of FadR, FadX and Complex, which is assumed equal.

　　② a denotes the affinity of FadR to the promoter PfadR, and V denotes the background expression rate of related genes.

　　③ k1 and k2 denote the forward and reverse reaction rate coefficients, respectively. k3 to k6 are parameters related to enzyme-catalyzed reactions based on the Michaelis-Menten Equation. Specially,

…………………………②

　　while f denotes the concentration of fatty acid outside the bacteria, KL the Michaelis constant of FadL, and kL the maximal activity of FadL. Details for the ODE can be illustrated in Fig 1.

Fig 1 Illustration of the meaning of the ODE

Analysis on the Steady State of the ODE