Team:NJMU-China/Modeling of Hardware

Our team’s project involves in a test strip for domestic test, so it’s important to demonstrate its plausibility—we need to know whether the growth of our bacteria meets up our expectations,which is the best temperature to cultivate our bacteria and whether the color intensity difference can be observed by eyes at home, etc. That’s where mathematical model comes in.

The growth model for bacteria

We first put forward the concept of $\tau$ , the average life of a bacterium, as bacteria will not disintegrate immediately when there is no nutrition support, its vital signs can maintain for some time. The same bacteria community can be described by the same $\tau$ , for it is objective, and is only related with the varieties of bacteria and the surrounding environment of the bacteria.

It can also be explained in this way: suppose there is some bacteria of the total number which loses nutrition support at the time t , then at the time t+$\tau$, there will be some bacteria of the total number of n₂ which disintegrate. Then there will be:

n₁ $\approx$ n₂

In the culture dish, as the growth of bacteria is limited by the environment, it can only grow outwards outside the surface of culture dish. Since only the outside part of the bacteria can get the nutrition support from the culture solution, the inside part of bacteria will not growth. Thus according to this feature, together with the average life of a bacteria,

Suppose N(t) as the function of the number of bacteria, depicts the relationship between time and bacteria number. Suppose culture solution is used up when t=T, which means there is no ntrition support, and N, the maximum capacity of bacteria in the culture dish.

There will be only three kinds of relationship between t and $\tau$ : $\tau$ < T , $\tau$=T ,$\tau$> T.
So we can get the initial expression of N(t):

when $\tau$ < T , $N_{0}(t)=\left\{\begin{array}{c}N(t), 0 \leq t \leq T \\ N-N(t-\tau), T \leq t \leq \tau \\ N-N(t-\tau), \tau \leq t \leq \tau+T \\ 0, \tau+T \leq t\end{array}\right.$

when $\tau$ = T , $N_{0}(t)=\left\{\begin{array}{c}N(t), 0 \leq t \leq T \\ N-N(t-\tau), \tau \leq t \leq \tau+T \\ 0, \tau+T \leq t\end{array}\right.$

when $\tau$ > T , $N_{0}(t)=\left\{\begin{array}{c}N(t), 0 \leq t \leq T \\ N, T \leq t \leq \tau \\ N-N(t-\tau), \tau \leq t \leq \tau+T \\ 0, \tau+T \leq t\end{array}\right.$

The following is the solving of N(t)

Suppose that the growth rate of bacteria varies with time:

$$\frac{d n}{d t}=a n$$

But the bacteria are not growing unlimited, so we introduce the limit factor b:

$$\frac{d n}{d t}=a n-b n^{2}$$

Suppose that the number of bacteria is n0 at the initial time t0:

$$N(t 0)=n 0$$

According to the feature of bacteria growth on the two-dimensional surface, the number of bacteria in growth is in proportion to the current radius of the bacteria colony:

n = (2$\pi$R)$\rho$, $\rho$ as the linear density of the bacteria clolony

According to the previous differential equation, we get:

2$\pi$$\rho$dR = 2$\pi$$\rho$R(a-b2$\rho$R)dt

According to the previous differential equation, we get:

$$d R=R(a-b \cdot 2 \pi \rho R) d t$$

Solving this dR by getting the qua, so

$$R(t)=\frac{a R_{0} e^{a t}}{a-2 \pi \rho R_{0} b+2 \pi \rho R_{0} b e^{a t}}$$

We also know that the linear circle of bacteria generated bacteria of the total number

d_n= $\rho$(2$\pi$RdR)

Finally, we can get:

N(t) = n₀ + $\pi$$\rho$R²,

According to the previous expression of dR

$$N(t)=n_{0}+\pi \rho \cdot\left(\frac{a R_{0} e^{a t}}{a-2 \pi \rho R_{0} b+2 \pi \rho R_{0} b e^{a t}}\right)^{2}$$

Finding the optimized temperature

Now that we have the growth model of our bacteria, we still need to find out which is the best temperature to cultivate our bacteria and what is the optimistic temperature range, as domestic tests requires both convenience and sensitivity, and most importantly, this test strip can be used in the environment of a common family.

We carried out experiments to find out the relationship between the temperature and the number of bacteria. Since bacteria cannot grow in either too low or too high temperature and the visibility of fluorescence can only be observed on the basis of a certain number of bacteria, we can use these criteria to find out the temperature range and the optimized temperature.

Based on Gompertz model, we used experimental data to optimize the parameters. The result is shown in the graph below.

From the graph, we can see that the best temperature range is [26℃,30℃], which means that the test strip can be used in domestic environment.

The RNA transcription model

After building models on a macroscopic scale, now we can focus on a much microscopic scale—the cellular level. The bacteria RNA will start to transcribe when there is specific biomarker in the tested urine and the product will react and then generate a blue fluorece.

The reaction process can be described as follows. Some relevant parameters can be found in the documentation^2-11.

So it’s possible to observe the blue fluorescence in the domestic environment.

The Solution diffusion on test script model

We expected to react with the engineered bacteria at different gradient concentrations and take photos of the final results. ImageJ was used to analyze the photos to obtain the chromogenic brightness under the conditions of different gradient concentrations of serotonin.

$$y=A_{1}+\frac{A_{2}-A_{1}}{1+10^{p\left(\log x_{0}-x\right)}}$$

We predict that the final color intensity will be higher than the range that human eyes can recognize, which further confirms the feasibility of home testing of this project.In the future, the concentration of engineering bacteria will be gradually adjusted, and the two indexes of engineering bacteria quantity and color brightness maximization will be weighed, so as to serve as the basis for the production of Test Script.

References

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2. Picó, J., Vignoni, A., Picó-Marco, E., & Boada, Y. (2015). Modelado de sistemas bioquímicos: De la ley de acción de masas a la aproximación lineal del ruido. Revista Iberoamericana de Automática e Informática Industrial RIAI, 12(3), 241-252.
3. Y. Boada, A. Vignoni, and J. Picó. (2017). Engineered control of genetic variability reveals interplay among quorum sensing, feedback regulation, and biochemical noise. ACS Synthetic Biology, 6(10):1903–1912, 2017a.
4. Segel, L. A., & Slemrod, M. (1989). The quasi-steady-state assumption: a case study in perturbation. SIAM review, 31(3), 446-477.
5. Boada, Y., Vignoni, A., Reynoso-Meza, G., & Picó, J. (2016). Parameter identification in synthetic biological circuits using multi-objective optimization. Ifac-Papersonline, 49(26), 77-82.
6. R. Milo and R. Phillips. Cell Biology by the Numbers. First edition, 2015. ISBN9780815345374. 7. Boada, Y., Vignoni, A., & Picó, J. (2017). Reduction of population variability in protein expression: A control engineering approach. Actas de las XXXVIII Jornadas de Automática.
8. U. Alon. An Introduction to Systems Biology. Desing Principles of Biological Circuits. Champan and Hall/CRC, Edition, 2007.
9. Schleif, R. (2000). Regulation of the L-arabinose operon of Escherichia coli. Trends in Genetics, 16(12), 559-565.
10. Boada, Y. (2018). A systems engineering approach to model, tune and test synthetic gene circuits. PhD. Thesis, Universitat Politècnica de València.
11. N. E. Buchler, U. Gerland, and T. Hwa. Nonlinear protein degradation and the function of genetic circuits. Proceedings of the National Academy of Sciences of the United States of America, 102(27):9559–9564, 2005.