Team:SZU-China/Model
Model
To improve our project this year, we built a series of models to achieve better results. At the beginning of the project, we established a population growth model to illustrate the harmfulness of Mikania micrantha. When we presented the third-generation product, we modeled the fermentation process and determined when we could best harvest the Micrancide. In the exploration of the use method of Micrancide, we carried out two fitting models to obtain the fastest time of Mikania micrantha removal and the optimal concentration of Micrancide, respectively.
To explain why we insisted on developing effective Mikania micrantha herbicides, we build a model to describe the biomass of Mikania micrantha population. This model is based on the population growth model and the plant individual growth model. It can be terrific how Mikania micrantha as an invasive species in a land which have no predators.
M. micrantha is a perennial liana plant. Its population generations overlap, and population number change is continuous, satisfying the differential equation model We let the population size be N, and its growth rate be r
Assumption 1: M. micrantha grows in a place without survival pressure. It is very consistent with the characteristics of an invasive species, and there is no limit to the nutrition of M. micrantha, so it meets the exponential growth:
Population Growth Model: Equation 1
We can get:
Population Growth Model: Equation 2
Alternatively
Population Growth Model: Equation 3
Assumption 2: Since the stem of M. micrantha grows roots during its growth, its ability to absorb nutrients is not affected by the distance between the upper and bottom ends of the plant.
We set mass as w, volume as v, nutrient absorption and volume as proportional, proportional coefficient as k, plant density as ρ:
Population Growth Model: Equation 4
Then we can get:
Population Growth Model: Equation 5
Thus, from above equations, we can obtain simultaneously:
Population Growth Model: Equation 6
In mentioned equation, we can find N_0, w_0, e^r, ρ^(k/ρ) are constants, so the equation 1.6 can be rewritten as:
Population Growth Model: Equation 7
The following figure is the parameters and values of this model:
Tabel 1. Parameters and Values in the Population Growth Model
Although some parameters in the model are still unclear, the equation is an exponential function. So if we allow M. micrantha to grow wildly, it will inevitably cause massive damage to the local ecology. We assumed some parameters and got a simulation:
Population Growth Model: Simulated Result
Moreover, in equation 6, we can find that the number of r has the most crucial impact on the weight of M. micrantha. However, if the number of r decreases to 0, the weight will not stop because the plant will keep growing.
The advantage of our model: 1. It can reflect the relationship between population size and mass and time. However, the other population model can only show size or mass. 2. Although we still do not know some parameters in the model, the model reflects a dangerous situation about the local ecosystem all the same.
In the third generation of our products, the most critical link in controlling the bacteria, is the time of cell death. To get the best shRNA yield, we designed a system to control the bacteria death by cracking the cells, but the best harvest time is still unknown. So we set up a model of gene expression to explore the best time to harvest.
Gene expression can be divided into two steps: transcription and translation. The products of transcription and translation are mRNA and protein. After a specific time, those products will degrade by specific enzymes. The approximate mechanism is shown in the figure below.
Expression Model: Gene Expression in E. coli
To explore when shRNA and r-body reach the maximum value, we established an ODE model to describe the biochemical process in the figure above. R-body is composed of three proteins, namely rebA,rebB, and rebC.
1. Let us assume these three proteins come together and immediately become R-body, so the model becomes a model that expresses three proteins and one shRNA.
2. Since shRNA is difficult to degrade in a short time after synthesis, we assume that it will not degrade in bacteria (short time).
3. Since the three proteins have similar molecular weight sizes, we assume that their degradation rates are the same, as are the mRNAs
4. Protein, mRNA, and shRNA levels were 0 before induction.
The process in figure 1 is symbolized as follows:
Tabel 2. Biochemical reaction with reaction rate in the Expression model.
Through the above biochemical reactions, we can obtain our related variables and ODE models:
Tabel 3. Variables, Biochemical species and Units in the Expression model
Expression Model: ODE model
Among them, the average number of copies of pET-28a (+) plasmid in E. coli cells was 40[1]. The size of rebA and rebB genes were 345bp and 318bp[2], respectivelv. The rebC gene size was 171bp. So the rebA, rebB, and rebC protein sizes were 114aa, 105aa, and 56aa, respectively. The T7 polymerase transcription speed is 60nt/s[3]
The parameter values can be obtained from the above data[4]:
Tabel 4. Parameter and values in the Expression model
We used Mathematica to solve the ODEs mentioned above and obtained the following image through simulation:
Expression Model: Simulated result
In this result, we can find the time for the protein to reach its peak is about 44min. Because shRNA growth is linear, we need to induce the protein expression 44 minutes before harvesting the shRNA to get the best result.
After getting best production time, we worked out a plan to clean M. micrantha. However, in the process of formulating the scheme, we found that although we could successfully produce Micrancide, the best usage of this drug was unknown, so we began to explore how to use Micrancide better
Etiolation rate is an indicator to measure the healthy condition of a plant. To explore the spray intervals and test whether the surfactant used along with siRNA spraying will influence the plant, we carried out the etiolation rate experiment. We selected four M. micrantha plants to spray Micrancide every day(2mL liquid in total for every plant) and take photos for the same leaves. We designed a Matlab program to measure the etiolation rate.
In this experiment, we set up a control group and three experimental groups. All groups have a 10-day test cycle and n=3. The following table is the detailed information of the experimental group and the control group:
Tabel 5. Groups explanation
s+ means siRNA with surfactant; s- means siRNA without surfactant; w+ means water with surfacant; w- means water without surfacant
Ten days after initial of the experiment, we carried out statistical analysis on the average etiolation rate of each group, and the figure below is the result of the statistical analysis:
Etiolation Curve Bar figure
As show in the figure above, there is a significant statistical difference between the s+ group and w+ group (p<0.01). And there is a significant statistical difference between the s- group and w- group (p<0.01). Furthermore, the s+ group and s- group, have no statistical difference. And there is no statistical difference between the w+ group and w- group.
Therefore, it can be inferred that the surfactant does not affect plant etiolation.
We used Matlab to fit the mean values of etiolation rate of w- and s+ groups for ten days, and the results are as follows:
Etiolation Curve: Fitting result
In the figure, the y-coordinate is the percentage of etiolation rate, and the x-coordinate is days after experiment, The equations obtained by fitting are as follows:
Etiolation Curve: Fitting equation
According to the equations above, etiolation rate reaches 80% is predicted on the 13.4 days.
Although we obtain how long we need to spraying to clean the M. micrantha, the best amount of Micrancide is still unknown. An optimum concentration and dosage, may help completely removal of M. micrantha. To explore the potential of Micrancide in clearing M. micrantha, we conducted a fluorescence detection experiments, to detect the optimum concentration of siRNA.
To explore the optimum dosage of Micrancide, we designed an experiment to detect the concentration of shRNA in plant cells. Typically, there are several methods to detect the shRNA level, such as nanodrop, qubit, and fluorescence detection[5]. However, the error of the first two methods is significant. We choose Thioflavin T(ThT) as fluorochrome to detect the in vivo shRNA level of the plant cells after one day spraying. And we fitted our data to get the fitting curve.
We set the shRNA concentration gradient as 50ng/μL, 0ng/μL for the control group, and 500ng/μL for the maximum test. Nine experiment groups
In this experiment, the actual concentration of shRNA is unknown, but the fluorescence intensity has a linear relationship with the exact level. So we directly substitute the shRNA fluorescence intensity for the actual concentration.
We drew and fitted the experimental data, and the results were as follows:
Concentration Curve: Fitting Sesult
The fitting coefficient (R2) is 0.9862, and this imitative effect is good.
Because it is impossible to get a stable number from a logarithmic function. To get the optimum dosage, we divide the former result of the concentration prediction by the latter, and when the number larger than 0.999, we think it is the optimum dosage.
Using the method above, we get the optimum dosage is 274ng/μL. We then set up five experiment groups: 170ng/μL, 220ng/μL,270ng/μL, 320ng/μL, and 370ng/μL to test the hypothetical optimum concertration.
Concentration Curve: Experimental Sesult
As shown in the figure, the fluorescence intensity was much better when the concentration was around 220ng/μL. Marginal increase of fluorescence intensity became less when the concentration level was further increased.
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