Abstract

PopQC, which is the abbreviation for population quality control, is a new approach designed for biosynthesis production enhancement based on the non-genetic cell-to-cell variation. Because of some nongenetic differences, different cells in a single colony will have considerable variations in protein and metabolite concentrations. Based on this, PopQC was designed as a plasmid-based gene circuit, which continuously selects high-producers to increase production.

We first use ODEs to explain why the concentration of glutamate can vary in a wide range and stay stable. We then use biophysical model to explain the biosynthetic performance of our system based on the inhibition effect of promoter's occupation by RNA polymerase. Finally, we use our model to predict initial condition to get the maximum production, and test our systems' efficiency and accuracy.

Glutamate and GltC Concentration

Figure 1: Pattern diagram of PopQC based on our design

In our project, Bacillus amyloliquefaciens LL3 was selected as the engineered strain introduced into the PopQC plasmid. In Bacillus amyloliquefaciens LL3, the intracellular glutamate concentration varies from 20 mM to 200 mM based on some nongenetic differences, which is from the result of our lab's previous work. We hope in the presence of PopQC, high glutamate strains can stay alive while low glutamate strains are unable to survive.

GltC combines with glutamate to form GltC-Glu complex. And it 's suggested that in the case of Bacillus amyloliquefaciens RNAP, glutamate inhibits GltC-dependent transcription of gltAB mostly by GltC-Glu complex 's additional non-specific inhibitory effect on RNAP for binding on gltAB promoter^[1]. gltAB operon encodes glutamate synthase^[2], so it forms a feedback suppression network by GltC-Glu complex. Meanwhile, GltC-Glu complex also has an inhibitory effect on gltC promoter, which forms a self-negative feedback network. These feedback suppression networks work together to keep intracellular glutamate pool stable. See Figure 2:

Figure 2: Feedback suppression networks based on GltC-Glu complex

We use ordinary differential equations (Table 1) to describe this progress, the fitting result is shown as Figure 3:

Table 1: Ordinary differential equations that describes feedback suppression networks based on GltC-Glu complex. We have estimated the parameters and the result are: k1=0.81, k2=2.15×10⁴, k3=49, k4=0.96, k5=0.84, k6=1.6×10^-4, k7=5.0×10⁶, k8=0.47, k9=1.7×10⁶,k10=9.7×10⁴. β represents gene leakage rate, const=1.

Figure 3: (a) Simulated glutamate production process. (b) Simulated GltC production process

Accroding to Figure 3, it can be seen that the time evolutions are smooth and that no oscillations are present, and eventually the concentration of Glutamate and GltC are stable, which is necessary for bacteria. We have also found that even small changes in mRNA degradation rates or mRNA copy numbers may result in large differences in intracellular glutamate concentrations. This result demonstrates why the concentration of glutamate can vary in a wide range and stay stable, which provides a theoretical basis for our experiment.

However, we find that the intracellular GltC concentration is basically equivalent in different cells because of its self-negative feedback network. So is reasonable to assume that the concentration of GltC-Glu complex is only affected by different intracellular glutamate concentrations. The reaction equilibrium constant in Equation (9) is , so the concentration of GltC-Glu complex is proportional to glutamate concentration. Define its proportional coefficient .

Biophysical Model

We assume that the intracellular glutamate concentration is normally distributed between 20'200 mM, which is from the result of our lab's previous work. See Figure 4:

Figure 4: Using 50,000 cells to simulate the normally distribution

The complex GltC-Glu achieves transcriptional regulation by altering the probability of RNA polymerase binding to the P_gltAB promoter. We model this process and assume that the probability of binding of RNA polymerase to the tetA promoter is linear to the tetA mRNA transcription level.

Total partition function:

(16)

where Z(P, R; N_NS) refers to the partition function of P polymerase and R complex GltC-Glu binding to N_NS non-specific sites (Dividing DNA sites into promoters and non-specific binding sites,which is approximately equal to 4×10⁶ in Bacillus amyloliquefaciens.) β=K_BT. Here we assume that RNA polymerase is randomly collided onto DNA. Therefore, the probability of a promoter occupied by RNA polymerase is:

(17)

Because the number of polymerase and GltC-Glu complexes is negligible relative to the number of effective binding sites, so:

(18)

Substituting Equation (3) into Equation (4):

(19)

We define the degree of deterrence(D) as the ratio of the probability that the promoter is occupied by RNA polymerase in the absence of a complex GltC-Glu to the probability of having a complex GltC-Glu:

(20)

For weak promotors:

(21)

Define the expression level of P_gltAB as 1 when glutamate concentration is 0, so the relative expression level of P_gltAB is:

(22)

According to the in vitro experimental data ^[1] as in Figure 5(a), we can find the parameter in equation (8) as: and get a fitting result, which is shown in Figure 5(b):

Figure 5: Use in vitro experimental data to describe the relationship between P_gltAB transcriptional level and glutamate concentration (a) The in vitro experimental data ^[1]. (b) Fitting result of the relationship between glutamate concentration and gltAB expression level

According to the above formulas and glutamate distribution, we can get the distribution of gltAB expression in the cells, and since the P_gltAB promoter is linked to LacI protein, which can also be considered as the distribution of intracellular LacI in Figure 6: