Modeling

Content

Part 1 Abstract

Part 2 Model

Part 3 Parameters and References

Abstract

In our project, we engineered E.coli cells (BL21) to express SttH-Linker-Fre, MaFMO, and TnaA with the idea of synthetic biology. To help and guide our experiments, we develop a model based on mathematical calculations and the Michaelis-Menten equation.

Hypothesis

(1) The original amount of enzyme (E_0) does not change during the reactions.
(2) 6Br-Trp and 6Br-Indole enter in the E.coli with different rates

Modeling Equations

Process of 6BrIG production:

The whole process can be expressed by the following reactions:

k1 is the bimolecular association rate constant of enzyme-substrate binding of reaction 1; k-1 is the unimolecular rate constant of the C1 complex dissociating to regenerate free enzyme and substrate; k2 is the unimolecular rate constant of the C1 complex dissociating to give free enzyme and product P.
Similarly, k3, k-3, k4, for the second reaction, and k5, k-5, k6 for the last reaction.
S1 represents the substrate Tryptophan, E1 represents enzyme Trp 6-Hal, C1, C2 and C3 represent the enzyme-substrate complex, which forms the P1, or the product, which is 6Br-Trp. The product of the first reaction becomes the substrate for the next reaction, which the E2 or the enzyme is TnaA, and the second reaction results in the generation of 6Br-Indole. Similarly, the third reaction's substrate is the product of the previous one and produces P3, our final product 6BrIG with E3.

The relationship between the rate of change of concentration of each element in these three reactions can be simulated through a set of equations with the reaction rate constants:

The rate of change of S1 is equal to the speed of its production subtracts the speed of its degradation. The rate of its production is equal to the rate of degradation of C1, which is k-1 * [C1], and the rate of its degradation is equal to the rate of production of C1, which is k1 * [S1] * [E1]. Thus, the rate of change of S1 is k-1 * [C1] - k1 * [S1] * [E1].

Likewise, the rate of change of C1 is equal to the speed of its production subtracts the speed of its degradation. The rate of its production is equal to k1 * [S1] * [E1], and the degradation of C1 has two paths. One is from dissociating to regenerate free enzyme and the substrate, k-1 * [C1], and the other is forming the product, k2 * [C1]. Thus, its degradation rate equals to k-1 * [C1] + k2 * [C1]. And the rate of change of C1 equals k1 * [S1] * [E1] - k-1 * [C1] - k2 * [C1].

The rate of change of P1 is equal to the speed of its production subtracts the speed of its degradation. Its production has two paths, which are from the formation from the first reaction k2 * [C1], and the dissociation of the C2 in the second reaction, k-3 * [C2]. Its degradation rate is equal to the production rate of C2 in the second reaction, which is k3 * [P1] * [E2]. Thus, the rate of change of P1 is equal to k2 * [C1] + k-3 * [C2] - k3 * [P1] * [E2].

And the following equations of reactions 2 and 3 follow the same principle.

Among these equations, the rate of change of S1, P1, P2, and P3, representing Tryptophan, 6Br-Trp, 6Br-Indole, and 6BrIG would be graphed to make further analysis.

Result Analysis and Conclusion