Modeling

Introduction

In our project, we want to calculate the bacteria concentration in the testers.

However, our devices can only detect GFP intensity, so we can only utilize GFP intensity to calculate bacteria concentration.

What’s more, with the view to making sure our system works successfully, we need to make sure that testers can detect GFP in our devices. Since the GFP in mosquitoes take some time to be synthesized, we can detect the green fluorescence only few hours after the mosquitoes take in the tester’s blood. To prevent from the misleading of our devices and system, we should calculate the very beginning time that the testers can detect the green fluorescence in the devices.

Guiding Questions

1. How many bacteria can be tested in our model ? (Model 1)

2. How long do our devices take to send out signal ? (Model 2)

Focus on Our Model

Since our devices can only detect the GFP intensity, we can only utilize GFP intensity to calculate E.coli concentration. After obtaining E.coli concentration, we will utilize it to calculate the very beginning time that testers can detect GFP. Finally, the two parameters will be demonstrated on our devices for the testers to take as reference.

Model 1: Calculating E.coli Concentration by GFP Intensity

Method

To find the mathematical relationship between GFP and E.coli concentration, we measure the GFP growing curve with different MOI value every two hours. Then, perform a series of calculation and finally arrive at the mathematical relationship between GFP and E.coli concentration.

Obtaining the Mathematical Relationship

Table 1.0 shows the corresponding Fluorescence intensity with different MOI value. However, the units of E.coli we want should be transformed into another form.

Conversion of MOI to E.coli density

The equation of E.coli density is shown below:

Since the MOI value refers to the ratio of E.coli cells to mosquito cells, we can use the density of mosquito cells to calculate the E.coli density. Plus, the mosquito cells are seeded at the density of 1.8×〖10〗^5 cells/well, and the volume of each well is 100μL.

Thus, the equation become

Then, we will turn the MOI in Table 1.0 into E.coli density to form Table 1.1. Next, we will use Table 1.1 to keep figuring out the mathematical relationship between E.coli concentration and GFP intensity.

Forming the mathematical expression

Before we keep working on our calculation, it’s worth noticed that the GFP intensity has already existed while there is no E.coli . Thus, the intensity of GFP with no E.coli should be eliminated as discussing the relationship between E.coli concentration and GFP intensity, which means [GFP] should minus the [〖GFP〗_0] with no E.coli, and add it back in our final result.

With that in mind, we form the Table 1.2

Now we can begin with our data analyzing.

Data Analyzing

Figure 2.0 shows the graphic expression between the [E.coli] and GFP intensity, the Exponential Function is shown below:

Next, we will bring in that [〖GFP〗_0 ]=813 to the Exponential Function and obtain the final graphic expression and function.

Combining the constants, we arrive at

Conclusion

Application

With the formula, we can now build a calculator to calculate the [E.coli] by GFP Intensity, and apply the formula to our devices to from a well-designed prototype. The devices will calculate the [E.coli] automatically based on the GFP intensity they detect. As a result, the testers will be able to know the [E.coli] in their blood through our devices.

Limitation

However, there are some limitations to our Model 1. Not knowing when all the E.coli cells bind to the GAM 1 promoter, we can’t make sure when we can utilize the formula to conduct calculation.

Consequently, we will conduct Model 2 to figure out the limitation.

Model 2: Number of E.coli Cells Binding to The GAM 1 Promoter Increase With Time

Method

To know when all the E.coli cells bind to GAM 1 promoter, we measure the GFP growing curves with different MOI value every two hours, and differentiate the growing curves to find the time that the instant GFP transcription rate reaches the maximum, which is the time all the E.coli cells bind to GAM 1 promoter.

After the differentiation, we will be able to obtain the graphic expression between the binding time and the [E.coli]. By analyzing the graphic expression, the very beginning time that the testers can detect GFP can be calculated.

In addition, the limitation mentioned in Model 1 can also be quantified via the graphic expression we obtained in Model 2.

Standardization of GFP Growing Curve

Since we need to differentiate the GFP growing curve, the standardization is essential. Thus, we will digitize the GFP growing curve so that we can conduct the differentiation successfully.

Raw Data

We perform two experiments respectively with different purposes.

The first one is to made for the blank so there is no DNA inside the well and no inducement of GAM 1 promoter, since the cells itself contribute to some absorbance, too.

The second one consists of E.coli with induced GAM 1 promoter. Both of the experiments are measured by our plate reader every two hours.

The Table 2.0 and Table 2.1 are shown below

Absorbance of green fluorescence protein

The actual absorbance of GFP is the absorbance of the E.coli cells with induced GAM 1 promoter minus the absorbance of E.coli cells without DNA and no inducement of GAM 1 promoter. After the calculation, the result are shown as Table 2.2

Standardization

Then, we will illustrate the growing curves and find the formula of each growing curve so that we can differentiate them. It’s noticed that we don’t adopt the GFP growing curve without E.coli, since its GAM 1 promoter isn’t induced.

After analyzing, we find that cubic equation perfectly fits to our experiment data. Those cubic equations are shown as Figure 3.0 to Figure 3.4

Also, the mathematical expressions of these cubic equations are shown as Table 3.0 and the graphic expressions are shown as Figure 4.0

Number of E.coli Cells Binding to GAM 1 Promoter

Derivative of the green fluorescence growing curve

When all the E.coli cells bind to GAM 1 promoter, the instant GFP transcription rate will reach the maximum value. As a consequence, what we need to do is to differentiate the GFP growing curves at different MOI value, and find the maximum of the differentiated formulas as well as the corresponding time.

With that in mind, we conduct the derivative of the mathematical expressions in Table 3.0 and form Table 3.1

Obtaining The Maximum and The Corresponding Time

To calculate the maximum of the derivative of the green fluorescence growing curve, we need to conduct the second derivative and find the maximum and corresponding time. The result is shown as Table 3.2

Then, we will turn the MOI value into E.coli density to form Table 3.3 Also, the graphic expression of the relationship between time and E.coli density is shown as Figure 5.0

We also arrive at the equation between time and E.coli concentration

Conclusion

Application

With the formula, we can utilize the [E.coli] calculated in Model 1 to calculate how long the testers should wait to detect green fluorescence in our devices and demonstrate it on our devices to inform the testers. Then, the formula will also be applied to our calculator as well.

Conclusion

Our model not only help to build the formulaic system which applied to our devices, but also make us better understand our project. Because our devices can only detect the GFP intensity, our model is required to build a well-designed devices and system. In our model 1, we obtain the formula which allows us to calculate [E.coli] from GFP intensity. While in model 2, we obtain the formula which allows us to calculate how long the testers should wait to get the result of the test based on the [E.coli] calculated in mode 1. For [E.coli] and the time interval, they will be demonstrated on our devices to show them to the testers. To sum up, Our model act as a bridge between our devices and the testers, and quantifies the significant parameters in our project, which allow the masses to simply get the result of the test without complex calculations.