Team:Qdai/Model
Modeling
We plan to recover phosphorus using a phosphorus accumulation pathway present in E. coli, but simply increasing the amount of phosphorus-accumulating proteins will not solve the problem. We will discuss increasing the total amount of final phosphorus recovery while also considering the negative effects of polyphosphate accumulation and stringent reactions on E. coli growth.
We did not have access to the lab this season because of COVID-19, so we did not have actual experimental data, but we performed the modeling to get a qualitative trend.
1. A model of polyphosphate accumulation and E. coli populations
The simplified pathway for phosphorus accumulation in E. coli is shown in Fig. 1.
Fig.1: The simplified pathway for phosphorus accumulation in E. coli
Applying this reaction to the Michaelis-Menten equation, the concentration of polyphosphoric acid is expressed by the following equation.
\[\frac{dX_{ pp }}{dt}=\frac{b_1\ast V_1\ast X_p}{K_1+X_p}-\frac{b_2\ast V_2\ast X_{ pp }}{K_2\ast(1+\frac{X_{relA}}{K_i})+X_{pp}}\ldots①\]
\(X_{pp}\) is the concentration of polyphosphate. In this equation, it reflects the competitive inhibition of PPX by ppGpp synthesized by relA. The concentration of phosphoric acid in the cell body is assumed to be constant.
\[\frac{dX_p}{dt}=0\]
Next, the population of E. coli is expressed in the next equation. E. coli increases logistically, and the negative effects of growth occur from the accumulation of polyphosphate and the stringent reaction caused by the presence of relA.
\[\frac{dN}{dt}=(1-\frac{N}{K})\ast N \ast r \ast f(X_{pp})\ast g(X_{relA} )…②\]
N is E. coli population. The details of the formulas for negative effects are as follows.
\[g₁(X_{pp} )=\frac{1}{(1+a_1 \ast X_{pp} )}\]
\[g₂(X_{relA} )=\frac{1}{(1+a_2 \ast X_{relA} )}\]
Equations g₁ and g₂ are only hypothetical and can be substituted by any formula as long as the three properties are the same: the equation does not diverge even if the variable is zero, it approaches asymptotically to 1, and it is monotonically decreasing.
The parameters are listed in the following Table1.
| Constants | Definition | Units | Value |
|---|---|---|---|
| \(K\) | Environmental capacity | Unit | 109 |
| \(r\) | Biotic potential | (-) | 0.012 |
| \(X_p\) | Concentration of phosphorous | μM | 10 |
| \(V_1\) | Maximum rate achieved by PPK | s-1 | 1 |
| \(V_2\) | Maximum rate achieved by PPX | s-1 | 1 |
| \(K_1\) | The value of the Michaelis constant | μM | 1 |
| \(K_2\) | The value of the Michaelis constant | μM | 1 |
| \(K_i\) | ppGpp dissociation constant | μM | 10 |
| \(a_1, a_2, b_1, b_2\) | Arbitrary constant | (-) | 1, 1, 1, 1 |
Table1: List of parameters
Solve ① equation for \(X_{pp}\). To make the equation easy to see, \(r_1=b_1 V_1, r_2=b_2 V_2\). Let \(\frac{dX_pp}{dt}=0\), because the polyphosphate takes equilibrium and is stable in E. coli. As a result, the following equations are obtained.
\[X_{pp}=\frac{K_2 r_1 X_p (1+\frac{X_{relA}}{K_i})}{r_2 (K_1+X_p )-r_1 X_p }\]
Next, Define \(N(t^*)=N_1\) and solve ② equation for N.
\[N(t)=\frac{KN_1}{N_1+(K-N_1)e^\frac{-rt}{(1+a_1 )(1+a_2)} }\]
In this study, we can regulate two proteins, relA and ppk, where relA is represented by \(X_{relA}\) and ppk is represented by \(r_1\) in the equation. We look for the value of the variable with the highest phosphorus recovery when these two variables are changed. Since we are evaluating the total amount of phosphorus recovered in the end, we show the product of these two equations as \(F(r_1, z)\).
\[F(r_1,z)=\frac{KK_2 N_1 r_1 X_p (1+\frac{X_{relA}}{K_i })}{N_1+(K-N_1)e^\frac{-rt}{(1+a_1)(1+a_2)}(r_2 (K_1+X_p )-r_1 X_p)}\]
Since the purpose is to see the qualitative data, we have added appropriate provisional values to the parameters. The time for taking the final concentration
is t=86400(s) (1 day later). In this case, we plot the 3D model on the three axes of \((r_1, z,F)\) in the range of \(0<r_1<\frac{r_2 (K_1+X_p)}{X_p}\) , \(0<z\) because \(N(t)>0 \ and \ X_{pp}>0\).
Fig.2: 3D model of total amount of recovered polyphosphoric acid
From Fig. 2 , we can say that the closer \(r_1\) gets to \(\frac{r_2 (K_1+X_p)}{X_p}\) and the closer z gets to positive infinity, the larger the value of \(F(r_1,z)\) becomes. F was also taking a local maximum value when z was near zero.
2. Optimization of relA induction timing
In the above discussion, the relA was calculated to be constant all the time. However, this means that the growth rate of E. coli is negatively affected from the beginning. Since relA is not expressed in our E. coli until the addition of an inducer, we are planning to induce relA after E. coli has entered stationary phase. Then, we also simulate the model when relA is induced in the middle of the proliferation. The initial value of N is 10⁶ and the initial value of \(X_{pp}\) is 10. The relA expression paradigm assumes that the amount of relA expression increases after the administration of IPTG and it takes one hour to reach a concentration z=3600. The values of other constants and variables are those in Table 1. The relationship between the start time of induction and final concentration are shown in Fig. 3. The graphs for growth without relA expression are also shown for comparison in Fig. 4.
The graph shows that if relA is expressed before E. coli finishes growing, the yield will be less than when relA is added after E. coli growth is finished. Also, unless \(X_{pp}\) reaches equilibrium after E. coli finishes growing, the earlier the inducer is added, the longer it increases to the equilibrium point in the presence of relA, so the yield will be higher.
Therefore, it can be said that the best time to induce relA expression is when E. coli populations reach to environmental carrying capacity, and as inducing relA is delayed from there, the final yield of \(X_{pp}\) will decrease.
(Reference) Kuroda, A. et al. (1997). Guanosine Tetra- and Pentaphosphate Promote Accumulation of Inorganic Polyphosphate in Escherichia coli. The Journal of Biological Chemistry. 272(34). 21240-21243. Zhang, Y. et al. (2018). Novel (p)ppGpp Binding and Metabolizing Proteins of Escherichia coli. American Society for Microbiology. 9(2). 1-20.