Modeling

Introduction

Since the effectuality of our genetic-modified parts are highly depended on the copper concentration of the outside environment, so it is significant to test the optimal copper concentration for the use of our product. With the consideration that using waste water collected from a factory will bring unnecessary factors other than the copper concentration and disable us to conclude the relation to the parts. We rather undergo complex mathematical process to calculate the predicted optimal concentration with the experimental data.

We have carried out several testing experiments to investigate the maximum capacity of both our medium, the e.coli, and e.coli with PcopA encoded in a comparably severe environment. We measured the fluorescence every two hours of E.coli in different concentration of copper solution (Cupric Chloride) , then the data sheets of E.coli growth against time against Copper(II) ion concentration have been achieved.

The following analysis is highly based on the sheets that we have attached, aiming to analyze the correlations between these two independent variables and how much these variables could cause the e.coli to be more active and effective at working.

Variable justification

Model development

1. The correlation of E.coli fluorescence against time against copper concentration

The data sheet we could obtain is shown below, which the side column represents the concentration of Cupric ion in a certain solution and the side row represents the time for e.coli bacteria placing in the solution. We have recorded 5 groups with one of them being the control group. And the result in this sheet will be analyzed in the following section.

Based on a common knowledge, both time and concentration will affect the bacterium’s growth, we had the first empirical equation, the two-variable linear function is then applied to test the simulation of these real-world data.

E = a × C + b × H + d (a, b, d are constants)

By the regression in the Matlab, we have obtained the function of E = 17.4 × C + 68.23 × H + 1181 where R² = 0.782

To increase the accuracy of this simulation, the trial with the assertion that the relations between these three factors may not be linear has been tested and applied.

In this equation, the R² has increased to 0.888 compared to the previous 0.782.

E = 13.02 × C² + 1.504 × H² + 37.23 × C × H - 237.2 × C + 23.51 × H +1500

By seting the upper bound of time lasting (which is 10 hours), the best activation of the bacteria occurs when there is no copper ion in the solution. This is reasonable that the bacteria works best at the cultivating solution and this result also proves that there is no obvious abnormality in this model.

Compared to the average activation of e.coli which is 1630 fluorescence, generally, most experimental groups could stand their corresponding Cu concentration. And some bacteria would develop their suffering capacity as time lasts longer to against the harm of permeating in a dense copper solution. Therefore, we could conclude that e.coli itself could stand the copper concentration of 2.4 mg/L (which is the comparably high copper concentration (only solution counts) in agriculture factory or daily waste dump) and have certain degree of self-adjusting and self-repairing capacity.

2. The correlation of PcopA1 and PcopA2

We have investigated two general groups of E.coli with PcopA1 and PcopA2 respectively. To test the accuracy of the data and avoid any abnormal data recorded to impact on the final result, we measured the correlation of these two groups.

Since they are encoded with the same gene with same effect towards the copper concentration, ideally, the result obtained should be linear. With linear regression in Matlab, the equation 3 could be acquired with a high R^2 which is 0.9916. It proves that data of these two experimental groups could be used in combination but qualified by the linear relationship. This solves the problem of having a few data for regression which may affect the overall accuracy.

PcopA2 = 1.07 × PcopA1 - 187.8

3. The correlation of PcopA against time against copper concentration

By regression of binary linear function, we got the equation

PcopA = 933.2 × C + 814.7 × H - 1951

With non-linear regression, we got the regression function 5, and the graph has shown below. R²=0.9626 which shows a high improvement compared to the previous one.

PcopA1 = -233.8 × C² + 99.05 × H² + 274.4 × C × H - 152.1 × C - 703.1 × H +2629

4. The correlation of E.coli GFP fluorescence against time against copper concentration

The further prediction and regression should be carried out when it comes to the GFP which is a type of protein that could be activated by certain effectual parts in the e.coli plasmid.

With binary non-linear regression, we obtained

E = 0.000005209 × C² - 0.001433 × H² - 0.0001952 × C × H -0.004257 × C + 0.1132 × H + 0.6607

Overall, the R²=0.71 which has improvement but still not high enough to conclude the correlation between these three factors when it applied its effect on GFP.

Team:ASTWS-China/Model