Part 1 Summary

The expression vector pET-28a-BFD-M7 was expressed in E. coli BL21 in our project. E. coli must grow to a certain concentration to induce expression of the interest protein (pET-28a-BFD-M7). If it is induced in advance, the expressed protein will inhibit the growth of the cells, thereby affecting not only the growth of E. coli, but also the final expression yield of the protein. However if the E. coli concentration is too high when they are induced, the induced expression will not be high at that time, because the nutrients in the medium have been consumed a lot, and in the late stage, the bacteria will grow slowly due to lack of nutrients and high density. So this is a dilemma when inducing expression proteins. In order to solve this problem, we use mathematical models to try to find the optimal growth concentration of bacteria, and then induce the expression when the bacteria growth to reach this level, which can achieve the maximum protein production and the highest economic benefit. (The concentration of E. coli is measured by OD600, and the protein product concentration is expressed in mg/L units.)

When the OD600 is the initial value, the expression of the target protein is detected every 6 hours, and the growth density of E. col is measured.

Part 2 General assumption and parameter

2.1 Assumption

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

2.1.2 Additionally, since there is no formula for explaining the relationship between the growing time, the concentration with different initial value of OD600 which is well studied, we assume that they are in a polynomial relation.

2.2 Parameter

**//*: Iin the experiment in each times, we use different initial value of OD₆₀₀ . details are in here:

Part 3 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. I 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(hour).

As is shown in the diagram, most of trend of each set of data has one turning-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 b(mg/L) with t(hour).

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 b at each t their own.

The essence of the increase speed is △bn/△t, which is also the first derivative of bn. Then we find the derivative of each b:

The graph of them produced by MATLAB can intuitively reflect some useful information:

Comparison table of the graph:

This can also be the speed vs t graph.

In terms of choosing a best environment for E.coil to grow, we need to see which line can reach a higher level speed. thus, b4(with 0.8 OD600 initially), b6(with 1.2OD600 initially)and b7(with 1.5OD600 initially). Then, with we thinking about the economized way, b4(with 0.8 OD600 initially)is the best in three without any doubt.

Part 4 Conclusion and self-reflection

4.1 Conclusion

In the experiment of our project, OD600 need to be kept around 0.8, because it is not only effective but also economized.

4.2 Disadvantages of the model

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

② In fact the value of each b at the time 0h must be 0; however, result of the fitting still has a constant, though it is small enough for us to ignore them.

Team:JNFLS/Model