Team:SCUT-FSE-CHINA/Model
Modeling
Growth Model
Logistic Growth Model
To verify that the our recombinant E. coli with the recombinant plasmids of the formamidase gene and the phosphite dehydrogenase gene could work efficiently, we cultivated and compared the growth of three strains, including the original BL21(DE3), BL21(DE3)(pGEX-for-ptx) and BL21(DE3)(pETDuet-for-ptx), on MOPS medium with NH3CO (200mM) as the sole nitrogen source and Na2HPO3•5H2O (1.32mM) as the sole phosphorus source. To understand the growth difference better, we modelled them by Logistic equation, of which form is
In the function, $t$ is the culture time, $N(t)$ is the flora number (CFU/ml), $M$ is the maximum flora number (CFU/ml), and $r$ is the relative growth rate coefficient. It means the growth rate decreases with the course of time, and finally tend to be zero. The growth curve shows the model of "S" growth type. Its integral form is
In this function, $N_0$ is the start flora number (CFU/ml).
With the help of logistic function, we will be able to compare the maximum flora number $M$ and the curvature constant $r$ of BL21(DE3)(pGEX-for-ptx) and BL21(DE3)(pETDuet-for-ptx) and BL21(DE3), then to analyze the difference between them.
Figure 1. Growth status of three strains cultivated in MOPS medium containing Na2HPO3•5H2O (1.32mM) and NH4Cl as the phosphorus and nitrogen sources, respectively.
According to Figure 1, the $r$ value of BL21(DE3), BL21(DE3)(pGEX-for-ptx) as well as BL21(DE3)(pETDuet-for-ptx) was $0.238$, $0.497$ and $0.363$ respectively. In MOPS medium, BL21(DE3)(pGEX-for-ptx) showed the highest growth rate, followed by BL21(DE3)(pETDuet-for-ptx) and unmodified BL21(DE3) successively. It also indicated that the original BL21(DE3) had the ability to utilize phosphite and there was no significant difference between their maximal bacterial concentration.
Figure 2. Growth status and fitting results of recombinant E. coli (with two ways of constructed plasmids possessing the formamidase gene and phosphite dehydrogenase gene) and BL21(DE3)(negative control) in a basal MOPS medium (Na2HPO3•5H2O (1.32mM) , NH3CO (200mM))
Theoretically, only one maximal growth rate will be showed, while our results did not show the same curves and two maximal growth rate were measured during the period. This reminds us that logistic equation may not fit them greatly and we should try to find other models to fit them. We will conduct more experiments to find out the cause of this result in our future work. Though the modeling result is not perfectly fit to the theoretical model, our modeling work still proved that the growth of recombinant strains of both construction ways were encouraging, indicating the recombinant strains were able to utilize formamide and phosphite as their nitrogen and phosphorus sources respectively. At the same time, the negative control group was unable to grow normally due to the lack of nitrogen and phosphorus sources.
Co-growth Model of Different Strains
Besides, growth model can not only apply on pure culture of one kind of bacteria, but also suitable for co-culture of different strains. If different strains culture in the same medium and each of them do not produce the substances which will affect the other bacteria, plundering of nutrients will be the only competition among them. Under this circumstance, their growth curves can still be modeled by a modified logistic function.
We assume the curvature constant of strain 1 is $r_1$, the curvature constant of strain 2 is $r_2$, and the maximum flora number of both strain is $M$. Then we use another rate $r_3$ to measure the parameter for describing the mutual inhibition of both strains. Finally, we can get the co-growth model of two strains. Its form is
We have done some experiments to monitor the co-growth situation of strain BL21(pGEX-f-p+pET28a-GFP) and other unexpected species.( Here shows more details.) While in order to complete the co-growth model, we still need data on the single growth status of both co-cultured strains. Further experiments will be conducted to finish this co-growth model.
Metabolic Model
Relative Growth Rate Model
We want to figure out the variation of bacteria growth ability after the modification of the N&P pathways. We have drawn out the growth curves to compare the different growth situation of our engineering strain with unmodified BL21(DE3) and modelled their growth situation with Logistic equation. To achieve further integral modeling, we tried to associate the different growth status with each other using Monod equation.
According to current data, BL21(DE3) itself can partly decompose phosphite. While Monod equation is used to relate microbial growth rates in an aqueous environment to the concentration of a limiting nutrient. If we add the pathway of utilizing phosphite as phosphorus source, our work on model will be quite complex. So we attempted to model the metabolism of formamide in engineering bacteria with modified nitrogen pathways.
Relative growth rate
To examine the growth ability of the bacteria, we use growth model to fit our growth curves and then figure out the growth rate. Logistic equation has been mentioned before, of which image fits the S-shape growth curve very well.
Its differential form is
So the relative growth rate is
If $N \to 0$, $\frac{1}{N}\frac{dN}{dt} \to r$. So at the beginning of culturing, $r$ is the relative growth rate.
Monod equation
Monod equation relates microbial growth rates in an aqueous environment to the concentration of a limiting nutrient. Its form is
In the equation, $\mu$ is the relative growth rate, $\mu_max$ is the maximum relative growth rate, $S$ is the concentration of the restrictive substrate for growth, and $ K_S $ is the "half-velocity constant"—the value of $S$ when $\mu/\mu_{max}=0.5 $.
Our engineering strain will generate formamidase and decompose formamide into ammonium ions, which only adds one step to the initial metabolic pathway. Therefore, when cultivating the engineering bacteria in MOPS medium with formamide as the sole nitrogen source, the environmental ammonium ions comes from the decomposition of formamide by formamidase. So we used the following model to characterize the growth of BL21(DE3)(pGEX-for-ptx) in the medium of which ammonium ions was the only limiting substrate.
In the equation, $\mu_1$ is the relative growth rate of BL21(DE3)(pGEX-for-ptx) when cultivated in MOPS medium (containing phosphite and different concentration of NH4Cl as the phosphorus and nitrogen source, respectively), $\mu_{max_1}$ is the maximum relative growth rate, $S_1$ is the concentration of formamide, and $K_{S_1}$ is the value of $S_1$ when $\mu_1/\mu_{max_1}=0.5$.
Figure 3. The growth status of BL21(DE3)(pGEX-for-ptx) cultured in LB medium containing 1.32mM phosphite and different concentrations of NH4Cl as the phosphorus and nitrogen source, respectively.
The curvature constant turned out to be 0.479, 0.667, 0.543 and 0.677 under 2.38mM, 4.76mM, 9.52mM and 19.04mM NH4Cl condition, respectively. These data showed that $\mu_{max_1}$ was 0.680, and $K_{S_1}$ was 0.884mM, though the results did not showed perfect positive correlation between the microbial growth rates and the concentration of the NH4Cl. We will do more experiments to complete this part.
We use $S_1$ to indicate the concentration of ammonium as limiting substrate, and $S_2$ for the concentration of formamide in the medium.
Figure 4. The growth status of BL21(DE3)(pGEX-for-ptx) cultured in MOPS medium containing 1.32mM phosphite and different concentrations of formamide, as the phosphorus and nitrogen source, respectively.
Figure 4. is the fitting effect of BL21(DE3)(pGEX-for-ptx) under different concentration of formamide, of which curvature constant is 0.0467 for 25mM formamide, 0.102 for 50mM formamide, 0.106 for 100mM formamide and 0.183 for 200mM formamide. Then we calculate out that $\mu_{max_2}$ is 0.290 and $K_{S_2}$ is 125mM. But as formamide can affect the growth of E. coli and high concentration of formamide may inhibit growth of E. coli, these data may need further revise.
Enzyme activity model
When cultivating the engineering bacteria in MOPS medium containing formamide as the sole nitrogen source, its growth situation had been modeled by Monod equation with ammonium as the limiting substrate. Another model had to be established to measure the rate of conversion from formamide to ammonium. So we had done experiments on the variation of enzyme activity of formamidase during the culture period.
Figure 5. The variation of enzyme activity of formamidase during the culture period in a basal MOPS medium (Na2HPO3•5H2O (1.32mM) , NH3CO (200mM)).
Next, we will measure the flora number and the corresponding enzyme activity of formamidase at each sample time to finish the enzyme activity model. We believe that this model can help to calculate the decomposition rate of formamide $V_N$, with particular initial formamide concentration.
Metabolic Model
The ammonium ions generated from the decomposition of formamide, will be eventually utilized by BL21(DE3)( pGEX-for-ptx). We assume that the amount of ammonium being utilized is positively correlated to the flora number $N$, then
In the equation, $k$ is the amount of ammonium ion consumed by a single cell for its growth and metabolism. As $N$ can be written in the form $N(t)$,
With the $S_1(t)$ and Monod equation, we can calculate the relative growth rate of our BL21(DE3)(pGEX-for-ptx) at any culture time. We will then compare the measured relative growth rate with theoretically calculating results of BL21(DE3)(pGEX-for-ptx) in MOPS medium containing 1.32mM phosphate and different concentrations of formamide to analyze the error.
Scale Up
Scale-up Oxygen Dissolve Model
So far, all of our experiments are done in lab level environments. When putting a bioprocess into industrial scale, it usually need to face many problems as the environment may change a lot from a shake flask to a tank reactor. Therefore, we plan to put our design in large scale industrial production, rather than limit it in laboratory level. We have considered about some problems when applying the experimental design to production-scale. So we tried to establish such a model, to lay the foundation for the industrial production in the future.
In aerobic processes, continuous oxygen supplement will be very necessary because the solubility of oxygen in aqueous environment is very low. E. coli is facultative anaerobic that can grow in a hypoxia environment, but for industrial fermentation, we usually need to supply enough oxygen for its growth and production.
To measure the availability of oxygen, people usually use the oxygen transfer rate (OTR). OTR must be known for an optimum design operation and scale-up of bioreactors, and it should be able to be predicted, if possible. It can be described as
In the equation, $k_La$ is the volumetric mass transfer coefficient. $C^*$ is the dissolved oxygen concentration, which is the concentration of oxygen in liquid phase when the partial pressure of oxygen in the gas phase reaches equilibrium. $C_L$ is the dissolved oxygen concentration on gas-liquid interface in liquid phase. The equation shows that OTR can be described proportional to the concentration gradient. Because of low solubility of oxygen in aqueous solutions, it usually costs a lot when we increase the partial pressure of oxygen by increasing the total pressure of air or using purer oxygen as Henry’s Law said. Therefore, measuring the $k_La$ will be quite necessary.
According to previous experiments, we know that many factors can effect OTR, such as the shape of reactors, temperature, pressure, power for mixing, liquid density and so on. There have been many empirical equations to determine $k_La$, and some even try to theoretically predict it recently.
In order to measure the change in oxygen concentration over time, we also need oxygen uptake rate（OUR）. OUR is the oxygen uptake rate of the microorganisms, which stands for the oxygen consumed by microorganisms for growth and production. It can be described as
where $\frac{dC}{dt}$ is the oxygen accumulation rate in liquid phase. After measuring OTR and OUR, we will be able to know the accumulation of oxygen in the liquid phase of our bioreactors, and to regulate the oxygen concentration during the bioprocess, so as to achieve an economical and efficient fermentation process.
