Team:Hainan China/Model

Model

1 Introduction

Our mathematical modeling tries to describe performance of vitamin B₁₂ (VB₁₂) production of Pseudomonas denitrificans in response to the synthetic biology of this probiotic bacterium. The mathematical modeling was programed with MATLAB, a MathWorks® product, which is a numerical computing application with access to symbolic computing abilities. The model provides simulation of the VB₁₂ production by genetically engineered Pseudomonas denitrificans under varying growth conditions.
The modeling idea came from our wet-lab experimentation in which both wild type and genetically engineered P. denitrificans were cultivated separately for 140 hours. Starting from 60 hours, the yields of VB₁₂ were tested every 20 hours.
We did the experiments three times and the results are shown in Figure 1:

We did function fitting for the yields of VB₁₂ VS fermentation time based on Fig. 1. Then we created a mathematical model to predict yields of VB₁₂ of P. denitrificans with fermentation time.

2 Model descriptions and assumptions

We set the following modeling variables:

The amount of VB₁₂ needed can be represented by:

With the fitting function:

Since we know the amount of VB₁₂ needed, we are able to predict the optimal fermentation time for the genetically engineered P. denitrificans.
In this model, we made the following assumptions which all the following modeling process based on:
i. The advantage in yields of the probiotic bacteria in the vgb group over the bacteria in the control groups always exists.
ii. The probiotic bacteria will not die as they are placed in the seawater. Instead, they will keep dividing.
iii. The data get from the experiments are reliable for they all have small standard deviations.
iv. No presence of multiple turning points in the yields.
v. Since the yields data we get from the experiment didn’t show any monotonicity, and according to our third assumption, there should be no turning point within the range covered by our experiment (60-140 hours).

3 Modeling Methods

The modeling process is illustrated in Figure 2.:

We divided the problem into two parts: estimating the yields within and out of the range covered by our experiments in the lab. The range of fermentation time here is from 60 hours to 140 hours. If the yields of fermentation time we want to estimate is within the range, and according to the fifth assumption we’ve mentioned above, we just need to make sure that the fitting function has monotonicity within the range. However, if we are estimating yields out of the range, we have to think about two possibilities: there is a turning point, or there isn’t a turning point. From the experimental data we got we are not able to know the presence of a turning point, and this may be confirmed if we do a long-term fermentation experiment. For the situation of no turning point, curves with monotonicity are preferred, while the others without monotonicity are preferred in the situation with the presence of a turning point.
After choosing the suitable curve, we can move to the next step of finding the fitting function. Here we used Matlab for it has a helpful curve fitting tool. We first did the curve fitting using all kinds of curves that meet the requirements mentioned in the first step. Then, we determined the best fitting function according to their R-square values. In this model, we abandoned the fitting function with an R-square value less than 0.8, and we wanted this value to be as close as possible to one.
After we got the fitting function, we established a predictive model for fermentation time needed to produce the appropriate amount of VB₁₂ for the healthy coral symbiosis.

4 Curve fitting results

We used the Curve Fitting tool in Matlab to do curve fitting of the data we get from the experiment.

The linear fitting result after adjustments with an R-square value of 0.9899 is shown in Figure 3:

The power fitting result after adjustments with an R-square value of 0.9820 is shown in Figure 4:

The rational fitting result after adjustments with an R-square value of 0.9889 is shown in Figure 5:

The smoothing spline fitting result after adjustments with an R-square value of 0.9949 is shown in Figure 6:

The sum of sine fitting result after adjustments with an R-square value of 0.9906 is shown in Figure 7:

The polynomial fitting result after adjustments with an R-square value of 1 is shown in Figure 8:

It can be seen from these figures above that all the fitting curves we got have monotonicity within the range of 60-140 hours.

5 Simulation results

Based on previous curve fitting results, we have established a function for the modeling:

With this function, we can estimate the yields of within the fermentation time of 60-140 hours and out of this fermentation time with an assumption of a turning point.
We are then able to predict the yields out of the fermentation time of 60-140 with the assumption of no turning points as shown in figure 9:

However, if the R-square value of the smoothing spline function and its quadratic fitting function are 0.9949 and 0.9889 respectively, it means that only 0.9449*0.9889=0.9344=93.44% of the data can be explained by the function, which is now lower than the linear fitting function in figure 3 which has an R-square value of 09889. Thus, we finally picked the linear fitting function to make the prediction, whose analytic function is:

6 Conclusions

This model provides us with insights on how the probiotic bacterium we developed with the aid of synthetic biology will function in the real circumstances. It is a rigorous simulation of the actual situations and a reliable prediction. The modeling results enable us to know how much P. denitrificans is needed for how long time for fermentation to reach the VB₁₂ concentration we want. This model uses scientific and precise ways to bring the project more practicality within limited time and budgets.