Model

Modeling

Because of the drastic decrease in the number of sharks, which play an important role in the balance of the whole nature, there’s an urgent need for shark’s protection. Considering that the reason why sharks are posed to danger of extinction is the high value of squalene extracted from its liver oil, the solution coming up with is to produce squalene using microorganisms.

Currently we have conducted experiments to measure the squalene yield using microorganisms in lab, but to reach our goal of shark’s protection, we need to calculate the yield of squalene in industrial production and judge its feasibility for the purpose of protecting sharks in a meaningful way. Limited by our lab conditions and time, we currently cannot conduct experiments using 50L fermentor, which is used in industries. Therefore, it’s essential to apply mathematical modeling to predict the squalene yield in a 50L vessel.

It’s not difficult for us to conduct experiments to measure the squalene yield in small vessels. We can change the volume of the experimental vessel (3ml, 20ml, 50ml), and measure time, the number of coli bacillus grown, and yields of squalene in different types of vessels. Therefore, we can set up a relation between the volume of vessel and squalene yield.

We know that there are many other factors to consider during the production in factories. However, the data collected from lab is very limited, so it’s hard to estimate the relationship between squalene yield and other parameters when put into industrial production.

Regarding our model about the relationship between squalene yield and vessel’s volume, we can further conduct an experiment to verify the reliability of the model. On account of the limit of the experimental conditions, a 5L cultivating pot is used to simulate the large volume in industrial production. We will compare the squalene yield in a 5L cultivating pot and that according to our model. It helps the model to be further improved and to reach an ideal consequence which should be close to reality.

Materials and Methods

Since data is not accessible currently due to our experiment cycle, we would expect the model to be as relevant to parameters as follows:

The number of coli bacillus grown in the experimental vessel, with a positive correlation to optical density value (OD), defined as n;

Time, defined as t;

Volume of the experimental vessel, defined as V.

Different engineered coli bacillus using different plasmid will be experimented separately for comparison in order to determine the best engineered coli bacillus acquired in the laboratory.

With these parameters, we would hopefully set a model for yields, defined as y, to predict the situation in industrial manufacture when located in a larger volume. Meanwhile, the number of coli bacillus and the yield are considered to be related. Within an appropriate range, y is supposed to increase with n, otherwise, it falls sharply when vast resources are occupied and contended

among coli bacillus. Thus, the model would be set as . Applying the model can enable further planning in industrial production, which is closed related to our original intention for the project.

Since our project aims to produce squalene in order to protect sharks, we expect it to be used in industrial manufacture with larger reaction vessels. Consequently, we built the model for predicting the condition in industry situation.

The independent variables volume of the experimental vessel and time of the cultivation are respectively simplified as V and t, and the dependent variables yields of the squalene is simplified as y. After the experiment, we got the data of y, OD, V, t as needed (shown in the end), among which the figure of Optical Density can show the concentration of E. coli.

Originally, we considered OD as an independent variable which closely relates to the yields. However, after fitting the data with linear model, it isn’t that fitted as expected. The figure of OD regarding y is plotted and shown in Figure1:

Figure1 Curve Fitting of y(OD)

R-square in this model is 0.1708, which is quite low thus should be considered that there’s no direct or significant connection between them. Thus, this model cannot predict the situation in larger vessels.

After analyzing the conditions in the experiment, we thought that probably the agglomeration of E. coli

Then we turned to

It is considered that OD can be relevant to V and t. With the same processing, a model is achieved and illustrated in Figure2:

Figure2: Curve Fitting of OD(V, t)

A model has been achieved, with an R-square of 0.4616:

OD = f (V, t) = 2.587-0.0153V+3.341t+7.027*10^ (-6) *V^2+0.002283*V*t-0.4636*t^2

Quadratic regression model is applied in the dataset because it is expected that the number of bacteria will have a similar trend to V and t separately in data analysis.

Since y is correlated to OD, while OD is correlated to V and t, it is reasonable considering that y is correlated to V and t in the situation that y cannot fitted well with OD. Thus a better model towards y(V, t) is set, shown in Figure3:

Figure3: Curve Fitting of y(V, t)

In this situation, R-square is 0.5204. However, a problem of the data in day 2 is found that fit the model badly generally. After the abnormal data of day 2 is deleted, the model is better fitted to the factual data point, which is supposed to be a more accurate and precise model thus can better predict the results. The improved model is shown in Figure4:

Figure4: Improved Curve Fitting of y(V, t)

In this situation, R-square is 0.8188, better than the previous model. This model is expressed as:

y = f (V, t) = -799.6 - 0.4308V + 947.7t + 6.794 * 10^ (-4)V^2 - 0.07427Vt—106.6t^2

Verification

Now that a model for the yield of squalene(y) relevant to the volume(V) and time(t) has been built, it will be verified in this section.

R²

A point can be got from the hook surface of the model and compared with experimental data. Suppose V = 1000mL, while in the fifth day of the experiment, a yield of 1221.58 mol/mL is got when V and t are substituted into the model

Significance:

Discussion:

The original data is shown in Table1:

Team:SEFLS Shanghai/Model

Modeling

Materials and Methods