Overview：
After cell transformation, pyruvate in cells can not only be converted into acetyl CoA through PFOR enzyme system, but also can be converted into the same product through PDH enzyme system. At the same time, the first pathway PFOR enzyme system is anaerobic, and the PDH enzyme system (aceE) is lactose inducible. Therefore, we use ordinary differential equations to simulate the conversion rate of pyruvate to acetyl CoA under different oxygen concentrations and lactose concentrations. In addition, our model also refers to Michaelis-Menten equation.
Model design and solution
First of all, we use differential equations to simulate two reactions respectively.
where [Actyl - CoA]1 is concentration of Acetyl-CoA in reaction v1
is maximum synthesis rate of Acetyl-CoA in reaction
[Osub2] : concentration of oxygen
K2 : concentration of oxygen required to achieve half of Maximum synthesis rate of Acetyl-CoA in reaction v1
： degradation rate of Acetyl CoA in reaction v1
Path model under lactose inducible
where
[P D H] ：concentration of PDH enzyme system
synPDH ： maximum synthesis rate of PDH enzyme system
[lactose] : concentration of lactose
Klactose : concentration of lactose required to achieve half of maximum synthesis rate of PDH enzyme
deglactose ：degradation rate of lactose
[Acety - CoA]2 ：concentration of Acetyl-CoA in reaction v2
： is the maximum synthesis rate of Acetyl-CoA
KPDH：is the concentration of PDH enzyme required to achieve half of maximum synthesis rate of Acetyl-CoA
degAcetyl - CoA ： is the degradation rate of Acetyl-CoA
degAcetyl - CoA ：is the sum of degradation rate of Acetyl-CoA
Model solution for [PDH]
Let y = [PDH], a = syn phd × `( [lac\t\ose] / {K _{lac\t\ose}+_{[lac\t\ose]}} )`, b = deg lactose Equation (2) can be transformed into the following variable separable differential equation `dy/dt = a -by`
The solution of Equation (4) is `y = {alpha - C_e ^{-bt}} / b`
[PDH] = ` {synPDH \ast ([lac\t\ose]/ {K _{lac\t\ose}+_{[lac\t\ose]})) - (synPDH \ast [lac\t\ose] / {K _{lac\t\ose}+_{[lac\t\ose]}} - deg_{lac\t\ose} \ast y0) \ast e ^{-deg_{lac\t\ose} xt}} / {deg_{lac\t\ose}}`
Since the products of two reactions are the same, it is not easy to distinguish which way the products are generated in the experiments. Hence, we will consider the two models comprehensively, and then combine two equations in a linear fashion.
Joint model of two paths under aerobic
`{d[Acetyl-CoA]} / dt` = `w_1 \ast ( {\[O_2\]} / K_O_2 + \[O_2\]) + W_2 \ast ( [PDH]/K_{PDH} = [PDH])` `-deg_{Acetyl-CoA} \ast [Acetyl - CoA]`
where `w_1 - syn_{Acetyl - CoA_{max1}}` `w_2 = syn_{Acety1-CoA_{max2}}`
Some assumptions are made here to set the boundary of our system.
Assumption for model solution
(1) Initial concentration of PDH enzyme (aceE) is assumed to be 0
(2) Initial concentration of Acetyl-CoA is assumed to be 0.
(3) For the reaction , assume that only the concentration of PDH enzyme system (aceE) affects the reaction rate
(4) All of the established equations follow the characteristics of Michaelis-Menten equation.
Model solution for joint model
To solve ordinary differential equation (7), we use SymPy to speed up the process. And from here we further simplify the symbols in the above system as follows
`{dA} / {dt} = W_1(S_1/{K_1+S_1}) + W_2({P(t)}/{K_2 + P(t)}) - D_1 * A(t)`
`{dp}/{dt} = W_3({S_2}/{K_3 + S_2}) - D_2 * P(t)`
where is the concentration of Acetyl-CoA over time and is the concentration of PDH over time with the following variables and constants
We can first solve as it is not relevant to Acetyl-CoA. Denoting the time point we add the lactose into the system, we have
Solving the formula for PDH is rather easy, but solving for that of Acetyl-CoA requires some tricks. As we can now plug into Equation (8), and then it becomes
This is not solvable either by hand or by computers, mainly due to the term that includes in the denominator of fraction. Therefore, we would need a way to tackle this problem. We can first simply this term as
By replacing `K_4 = {D_2K_2(K_3+S_2)}/{W_3S_2}`, we have
And this would look like the blue line in the following figure.
To eliminate the fraction term, we have to use a polynomial with no negative power terms. Furthermore, to further keep things simple and tidy, we just use a linear term here. We pick the first derivative at , and make it cap at the limit of the original term, which means that we take
where `t1 = t0 + K_4 /{D_2(K_4+1)}`
With such simplification, we can finally solve for . After refactoring the result from SymPy, we have the following solution
Simulation Experiment and Result Analysis
Our models are mainly used to simulate the conversion rate of pyruvate to Acetyl-CoA under different oxygen concentration and lactose concentration.
Experiment assumption
The following parameter setting are used for simulation experiments.
| Parameter | Value |
|---|---|
| `syn_{PDH}` | 18.26 μg/h |
| `K_{lac\t\ose}` | 54.94 μg/h |
| deg_{PDH} | 0.22 μg/h |
| syn_{Acetyl-CoA_{max1}} | 19.56 μg/h |
| syn_{Acetyl-CoA_{max2}} | 6.73 μg/h |
| K_{PDH} | 37.81 μg/h |
| K_{O_2} | 1.24 μg/h |
| deg_{Acetly-CoA} | 0.31 μg/h |
Result analysis
We have the following experiment results.
Experiment 1: Simulate the expression of PDH enzyme with respect to lactose induction.
According to the solution of Equation (2), the upper limit of concentration of PDH increases with the amount of inducer.
Experiment 2:Simulate Acetyl-CoA production under different oxygen and lactose conditions.
From Figure 4, we can clearly see that with the addition of lactose, the system can promisingly produce even more Acetyl-CoA, which means that our second path has the ability to compensate for the loss in the production of the original metabolic path
Experiment 3:Simulate Acetyl-CoA production under different conditions of oxygen, lactose, and the time of adding lactose
From Figure 5, we can see that upon adding lactose, the production of Acetyl-CoA drastically increases, while adding lactose too late will hurt the total production over the same time span. This agrees with the feedback from experiments that adding lactose late will decrease the final concentration of Acetyl-CoA.
