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Mathematical Model on Fatty Acid Degradation Device

　　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.

Mathematical Equations

　　We assume that the concentration of fatty acid outside the bacteria remains constant over time and build our ODE-based model as follows:

　　For simplicity

　　① the Complex, or variable x₇, refers to the Fatty acyl-CoA-FadR complex;

　　② FadA, FadB, FadI and FadJ are amalgamated into a single variable x₄, namely, the β-Oxidase, since they have similar functions.

　　The meaning of the parameters in the equations:

　　① a is the affinity of FadR to the promoter PfadR, and V is the background expression rate of related genes. The expression rates of FadL, FadD and β-Oxidase are equivalent since they have the same promoter.

　　② E is the constitutive expression rate of FadR, and R, L, D, B, C is the degradation rate of FadR, FadL, FadD, β-Oxidase and Complex, respectively.

　　③ k₁ and k₂ are 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,

　　while f is the concentration of fatty acid outside the bacteria, K_L is the Michaelis constant of FadL, and k_L is the maximal activity of FadL.