Model Introduction

Modeling always plays an important role in every subject, even beyond science. In our project, it comes up with real data, and thus make biological theories easier to be realized and observed. Johann Karl Friedrich Gauß said that “Mathematics is the queen of the science.” A proposition of mathematics is reliable and indisputable, whereas other science theories have always been in a risk of being overthrown. The reason why modeling has good reputation and a certain status is that it theorems the scientific phenomenon, and makes them more trustworthy. By conducting modeling, we can have a reasonable embryonic form to estimate a possible solution of a difficult problem. However, the reaction series or the operation mechanism of an unknown equation needs to be reasonably presumed, and this is the most difficult part in the whole process. Inappropriate assumption can lead to a result with large error. After the right theories come out, we can amend our hypothetical surmise, and remake another model. In the modeling process we’ve done, the main technique we used is DE (differential equation). We use derivative to describe the difference of any variables that vary within a very short time. But we’ve met some very complicated equations when solving the problem, so we use the program MATLAB to help calculate the results.

What are we modeling?

- The growth of E. coli

- The Expression of Different Color

- The Concentration Function f:(substance,time)→concentration

- Math Is Long, Life Is Short: Math in Our Life

Model

I. The growth of E. coli (Click to see more)

At first, we assume that E. coli proliferate and die at the same ratio over time, and the value difference is the birth rate (μ_g). So, we do derivative with this assumption.

Substituting the boundary condition, t = 0, N = N₀, we then have ∴ e^C₂-C₁=N₀ Thus, the equation that expresses the relation between bacteria and time is:

N = N₀∙e^μ_gt

What’s more, it is useless to say that E. coli consumes their “food”, LB, all the time. Thus, if E. coli consumes their food steadily, the LB consuming rate will be proportional to N, then we can write down the equation:

By substituting the boundary condition, we then have
𝐶= − 𝑛_𝐿𝐵0/𝑘_𝑐𝑜𝑛−𝑁₀/𝜇_𝑔

So the relation between n_LB and _t is:

II. The Expression of Different Color (Click to see more)

Assumption

1. In order to write the equations down simply, we assume that all the chemical reaction rates are proportional to the concentration of each reagent (e.g. for the reaction: A+B+C→D+E,the forward rate r₊=k₊[A][B][C]).

2.For every substances produced by biobricks, we assume that their production rate =φ[mRNA],
[mRNA]= the concentration of the promoted biobrick

φ= the result of multiplication of rate constant, coefficient of correction (since a biobrick is different from a reagtant), a dimension T^-1

Equations & Solutions

According to the picture, we can write down 3 equations as follows:

P.S. φ= the result of multiplication of rate constant, coefficient of correction (since a promoter is different from a reagtant), a dimension 𝑇⁻¹

By solving these 3 equations, the solution expressed by φ、k and [P^a] are as follows:

When the concentration of each activated promoter reaches to each of their steady state, then we can simplify the equations as follows:

Besides, since lim_t→∞⁡(1-1/e^kt) = 1, satisfying the definition of the horizontal asymptotes. And d(1-1/e^kt)/dt=ke^-kt>0 (t∈[0,∞)), so it is a strictly increasing function.
So, this is a strictly increasing and convergent function with an upper bound 1.

Then the result is that the extremum of the concentration is:

Degradation Rate Constant Calculation

As for the other variable written in the solutions, the degradation rate constant, can also be solved with differential equations. Since the degradation rate is an “order one” reaction, the equation can be written as follow:

dM/dt= -k_dM

Then, after solving the equation and substituting the boundary conditions
(t = 0⇒M = M₀), the the solution is:

According to the project 2008 iGEM KULeuven and 2014 iGEM Edinburgh had done, both GFP-LVA and RFP-LVA degrades to half of the amount within 50 to 60 minutes, so we assume that cjblue is the same. The RFP and BFP reference are as follow (the latter degrades to half of the amount about 50 minutes while the former does about 3 hours). So we can get

From these degradation rate constants and the relation between concentration and time, the “[cjblue],[RFP],[BFP]-t Diagram” is as follow:

According to this simulation diagram, we can know that cjblue and BFP increase faster—coming to 90% of maximum only takes about 170 minutes, while RFP takes about 600 minutes, which is also acceptable.

Through mathematical modeling, when observing the sicker changing to a specific color, we can calculate the ratio of each kind of chromoprotein by quantifying it.

III. The Concentration Function (Click to see more)

Equations

Accroding to their feedback mechanism, we can write down the simultaneous equations as follows.

Since the designations are too complex to be written, we change these deignations to simple ones. Meanwhile, we’ll explain all the individual meanings of every designations. (See the following tables)

Designation Description Table

Concentration

Constant

Here comes the script.

IV. Math is Long, Life is Short: Math in Our Life (Click to see more)

Since it requires complicate and large quantity of computing, you might think mathematics as an unreasonable tool. All it can do is endlessly derivation, and not being able to utilize in the real world. But in fact, mathematics are around us everywhere, while we have not notice them. The following math stuffs will be approachable, including offering formulas, for companies to decide whether they want to use our project; calculate the number of samples, so you can know how much surveys you need to do; offering possible data, give some reference for the team after, etc.

The minimum Number of Cargo Packed in a Box

After having a meeting with Professor Cheng-Ming Chang, we learned that the companies would only like to spend less than 2‰ of the price of the item to guarantee the quality of those item. According to this matter of fact, we can list the following equation:

Sample Size Estimation

Assume that the data of people‘s habits and opinions roughly obey the form of normal distribution. Then, according to the 68-95-99.7 rule, we can know that at 95% confident level, if we allow a deviation (E), the number of samples we should grab is…

Conclusion

- Through mathematical modelling, we can estimate how much LB filled in the sticker is adequate.

- Cjblue and BFP comes to 90% of maximum only takes about 3 hours. As for RFP, it takes about 5 hours to reach 70% of maximum, which is also acceptable.

- When observing the sicker changing to a specific color, we can calculate the ratio of each kind of chromoprotein/fluorescent by quantifying it, even know what time the sticker is activated; when it is exposed to UV or sunlight!

- If you aren’t sure whether the sticker for your product is cost-effective or not, mathematical modelling will be your best solution!