Part 1. Summary

The yield of amylose produced by E. coli is related to many culture conditions, such as culture temperature, medium component (glucose concentration), pH value, fermentation time and inoculum size, etc., and these factors also influence each other. We use the relationship of glucose concentration and fermentation time as an example to illustrate this problem.

When the glucose concentration of medium component is low, the bacteria grow slowly and synthesize amylose slowly. However, when the glucose concentration is too high, the cell grows vigorously and the primary metabolism is too strong. However, the cell will have a "glucose" effect, inhibit the cell growth and inhibit the synthesis of secondary metabolites. Therefore, maintaining the appropriate glucose concentration in the fermentation process can effectively reduce the accumulation of by-products and improve the synthesis efficiency of amylose. The two conditions are not conducive to the production of amylose. In order to acquire the maximum economic benefit, we tried to use mathematical model to find the optimal glucose concentration of medium component and the optimal fermentation time, so as to acquire the maximum yield of amylose.

The glucose concentration was expressed as a percentage of the culture medium volume; the bacterial density was expressed as OD600 value, and the unit of amylose yield was ug/L.

Part 2. Fitting function between culture conditions and yield

Fitting function is a data processing method that approximately describes or compares the functional relationship between the coordinates represented by the discrete point group on the plane with the continuous curve. In our experiments, the sets of data pairs (xi, yi) (i = 1, 2,... n) of quantities x and y are obtained through experiments, in which each xi is from culture temperature, pH value, fermentation time and inoculum size, and yi. represents the different yield of amylose. Usually, the kind of analytical expression, y = f (x, c), is suitable for the law of experimental data to reflect the dependence between quantity x and y, that is, to "best" fit the known data in a certain sense for amylose yield. The fitting function of culture temperature, pH value, fermentation time and inoculum size with amylose yield were shown as follows.

Part 3. General assumption and parameter

3.1 Assumption

3.1.1 First, the bacteria are growing in an ideal environment (with a suitable growth temperature, humidity and pH value).

3.1.2 Additionally, since there is no formula for explaining the relationship between the fermentation time and the medium component with different value of OD600, which is well studied, we assume that they are in a polynomial relation.

3.2 Parameter

**//*: In the experiment in each times, we use different initial different glucose concentration in medium components. Details are in here:

Part 4. Mathematic analysis

In this topic, the speed of growing is obviously not keeping increasing all the time, because of some limitation in real environment. Therefore, bn/t will no keep increasing. We try to use such a unique information as a breakthrough point to find a better situation for E. coli to grow.

Such is the diagram of bn/t vs t (hours).

As is shown in the diagram, most of trend of each set of data has one peak-point, thus we can assume that quadric expression can represent the relation of them. Then, if bn/t and t has a quadric relation, the bn and t must have a cubic relation. According to this, we can do the cubic fitting for each set of bn (ug/L) with t (hours).

Result of fitting:

However, it is not the eventual result that we want, because what we want is finding the best option after considering the speed of increasing in bn and the amount of OD600 we need to use instead of the value of bn at each t their own.

The graph of them produced by MATLAB can intuitively reflect some useful information of day, OD600, amylose and initial glucose concentrations.

Comparison table of the graph:

In terms of choosing a best environment for E. coil to grow, we need to see which plane can reach a higher amylose amounts. The models show the 1% initial glucose concentration amount is the best in all.

Part 5. Conclusion and self-reflection

5.1 Conclusion

In the experiment of our project, 37°C, pH7.6, fermentation 22h, 14% inoculum size, and 1% initial glucose concentration amount are the best, because they are effective in all.

5.2 Disadvantages of the model

① The assumption that the relation of them can be represented as a polynomial is a little cursory.

② The result of the fitting still has a constant, though it is small enough for us to ignore them.

Team:ICJFLS/Model