Model

In order to predict the status of fatty acid production in E.coli, this model takes the red fluorescence and the green fluorescence generated from the bacteria as the qualitative indexes for the preliminary judgment. We then measured the protein expression of RFP and GFP to get a more accurate quantitative relationship and this can be regarded as a reflection for the fatty acid concentration in E.coli.

Multiple regression analysis can be used for studying the relationship between the independent variable and the dependent variable Y. Suppose that there is a linear correlation between Y and , then the data we got can fit this regression model as followed:

In that model, since the error sum of squares: , only when the parameter ’s least squares estimator can make the error sum of squares reach its minimum.

It can prove that ’s least squares estimator is the solution to the m+1th-order linear equations.

In practical problems, the linear correlation between Y and remains a hypothesis before the estimate of regression coefficient is calculated. In order to determine whether Y and do have a linear correlation, statistical test should be carried out including the significance test of regression equation and regression coefficient. When the statistical quantity P is less than a certain (which is 0.05 occasionally) , the test for parameter is significant.

In our model, we set the concentration of fatty acid as Y, the protein expression of RFP and GFP as respectively. Our aim is to figure out in which period of time there is a quite significant linear correlation between the concentration of fatty acid and the protein expression of RFP and GFP. By using the SAS software, we can find that there is a quite significant linear correlation in the interval between 6 hours and 84 hours after we started the experiments.

Calculation Results:

Conclusion:

In the synthesis and degradation process of fatty acid in E.coli, during the time between 6 and 84 hours, the content of fatty acid and the expression of RFP and GFP have a well linear fitting. The linear correlation is below:

Results for other time period is attached:

Data: 12 hours to 108 hours from the beginning

Data: 18 hours to 120 hours from the beginning

Data: 24 hours to 132 hours from the beginning

We can find that there is always some parameter which fail to pass the test in other period. So the content of fatty acid and the expression of RFP and GFP cannot fit the linear correlation well during these time period.From the perspective of biology, the reasons for the nonlinearity of the model after 84 hours could be explained as follows. We think that when bacteria are cultured for over three days and a half, the medium is gradually depleted of nutrients, which has been insufficient to sustain the growth of the bacteria. Therefore, there is not nutrient for fatty acid synthesis.