# Team:CAU China/Model

Due to the influence of COVID-19 pandemic, our team doesn’t do the whole experiments, without some crucial data existing. So we refer lots of excellent project and use some other projects’ data. There are two models, which mean the population model of pests and transcriptional regulation model, combining closely our project.

## Transcriptional regulation

introduction

Our team applicates the CarH system into the production of dsRNA. The goal is to analyze transcriptional regulation. We then turn to and investigate an alternative protein that might be better suited for real-time control.
Our regulatory part called CarH protein is controlled by green light. In the condition of light, CarH tetramers repress the expression of dsRNA. When we use only green light, the tetramers dissociate, thus activating the expression of dsRNA. However, without the light, there is not any product, dsRNA, in the system we designed in Saccharomvces cerevisiae. Saccharomvces cerevisiae enters the rapid growth phase. The mechanics are described in Figure 1.

Figure 1: the mechanism of different regulation. The expression of dsRNA is regulated both in the light and in the darkness.

light-regulation

CarH tetramer is a dimer of dimers. The tetramer is a repressor of the CarH promoter with ≈ 5th order cooperativity, whereas the individual monomers and dimers do not bind significantly[1]. For simplicity, we consider that the tetramer accords with the Hill function of mRNA transcription. We think that CarH tetramer consists of four CarH monomer with VB12. A photon of light is absorbed by cobalamin in the CarH light-sensing domain and changes its conformation. This reaction of disassociating the tetramer is irreversible, with the subsequently light-state inert CarH domains forming.
We express CarH constitutively and put a dsRNA and isocitrate dehydrogenase under the control of CarH
The model equation are
(1)$${dr1\over dt}={ β1-γ1·r1}.$$
(2) $${dr2\over dt}={ {β2*C1^n\over C1^n+K^n}-γ2·r2}.$$
(3) $${dC1\over dt}={ α1·r1-δ1·C1+φuC4}.$$
(4)$${dC4\over dt}={ {1\over 4}α1·r1-δ2·C4+φuC4}.$$

Where r1 and r2 represent concentration of CarH RNA and dsRNA respectively, C1 and C4 represent concentration of CarH (Mono) and CarH (Ter) respectively.

Model solving

We use MATLAB 2014a to solve the ordinary equations above. The parameters are determined from both others' works and our experiments and listed as follows.

parameter description value
γ1 degradation rate of CarH RNA 1.68*10^−1
δ1 degradation rate of CarH(mono) 5.02*10^+1
δ2 degradation rate of CarH(ter) 1.00*10^+5
β2 the max reaction rate of dsRNA Austria
K Michaelis constant 2.75*10^-1
n Hill constant 67±2 nM
φ quantum yield 0.084
u input light 525 nm 5μmol m^-2 s^-1

The simulated results in Figure 2 where (a) is the curve of the concentration of intermediate products in the light, (b) is the curve concentration of dsRNA

(a) Concentration of intermediate products both in the light and in the darkness

(b) Concentration of dsRNA and isocitrate dehydrogenase

Figure 2 Expression of intermediate and terminal product in the light

## Population model of locust

Introduction

We developed a mathematical population model to predict the amount of locust migratoria manilensis infected by our dsRNA product since we can not do field experiment. We then performed a small-size experiment of locust and attained useful data about locust growing with infecting dsRNA-CHS dsRNA-ATPase and dsRNA-TSP. We expected the mortality rate of these dsRNA helping us better predict the influence when the product will be applied in killing locust.

natural condition

In natural condition outdoors, due to environmental resistance, the population of locust is more likely to follow a S-shaped growth curve, which can be formalized mathematically by logistic function$${dN\over dt}={rN(1-{N\over K})}.$$. After we research statistical yearbook of Yunnan province and paper, we use data fitting method to verify it. However, while the break of locust plague around the world, the population of locust meet the J-shaped growth curve (exponential growth) (Figure 1). The exponential growth is formalized mathematically by function. $$Y={A*e^{BT}}.$$

Figure1 Comparison between logistic growth model and exponential

After injection

After we using dsRNA-CHS dsRNA-ATPase and dsRNA-TSP to infect the locust (see mortality of locust experiment), we develop the population model of locust. These are some assumption listed below. One assumption is that dsRNA has relatively constant toxicity. According to the paper and our experiment, we know the lethality rate of dsRNA. As shown in the equation, the constant toxicity is added.

$$Y={A*e^{BT-C}}.$$

The second assumption is that the toxicity of dsRNA decreases over time. Considering the character of chemical, the decrease is in line with exponential decline. The decrease of toxicity constant is shown in the equation.

$$Y={A*e^{BT-C(1-e^{-DT})}}.$$

According to the analysis of mathematical method using Matlab, there is a figure of model shown in the Figure 2. When Y'(t) is 0, (B=CD·e-D), we have the minimal point: the density of locust reaches the minimal.

$$Tmin={{1\over D}*A*ln{CD\over B}}.$$

Figure 2 Population density curve of locusts

result

parameter description value
B The rate of growth 0.22 d-1
C Toxicity parameter 1.5
D Decrease of toxicity parameter 2.0
A the initial population 10
Y The population of specific time Variable
T Time Variable
Table 1 Parameters in population model of locust

From the above analysis, two discussions can be drawn:
when shRNA concentration increases, locust population density decreases rapidly. When the toxicity decreases fast, the time to reach the minimum locust population density will delay.

## References

1.https://2019.igem.org/Team:Peking/Model#Regulatory
2.https://2019.igem.org/Team:CAU_China/Dynamics_Model