Team:XJTU-China/Model

Model

Description

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Model

Our modeling includes five steps:

• Establish the model of population dynamics, which displays the population change of E. coli;
• Establish the model of toggle switch, where the production of red fluorescent protein (RFP) and green fluorescent protein (GFP) shows the effect of toggle switch;
• Establish the model of genetic circuits based on the model of toggle switch;
• Establish the model of synthesis of tryptophan based on Michaelis-Menten equation;
• Finally, integrate the above models to establish the model of production.

(a) The model of population dynamics

(b) The model of toggle switch

(c) The model of genetic circuits

(d) The model of synthesis of tryptophan

(e) The model of production

The model of population dynamics

First, we establish the model of population dynamics to study the variation of E. coli population density.

Theory

Let $N$ $N$ be the population density of E. coli. With the Logistic equation, we know that

$$\begin{array}{}\text{(1.1)}& \frac{\mathrm{d}N}{\mathrm{d}t}=rN(1-\frac{N}{K}),\end{array}$$

where $r$ $r$ and $K$ $K$ are the growth rate and the environmental capacity of E. coli respectively. Equation $(1.1)$ $(1.1)$ shows that:

• When $N<\frac{K}{2}$ $N<\dfrac{K}{2}$, the population density grows exponentially;
• When $N>\frac{K}{2}$ $N>\dfrac{K}{2}$, the environmental resources have a restrictive effect on E. coli, and finally the population density approaches $K$ $K$.

Parameter

The parameters are shown in the table below.

Parameter	Value	Reference
$k$ $k$	$6.08\times {10}^9\mathrm{CFU}/\mathrm{ml}$ $6.08×10^9\mathrm{CFU/ml}$	https://2018.igem.org/Team:Lund/Model/GrowthCurves/Results
$r$ $r$	$0.0073\sim 0.01{\min }^{-1}$ $0.0073\sim0.01\mathrm{\mathrm{min}^{-1}}$	From experiment.

Result

Let the initial value of population density be $0.01\mathrm{\%}$ $0.01\%$ of the environmental capacity. From the figure, we obtain that the population density reach balance after about $33\mathrm{h}$ $33\mathrm{h}$.

Code

main.m

​x
% initialization
                          
                          
clc;
                          
close all;
                          
clear all;
                          

                      
                          
% numerical solution
                          
[t, N] = ode45(@(t, N) odefcn(t, N)', [0 3000], 6.08*10^9*0.0001);
                          

                      
                          
% draw figure
                          
figure();
                          
plot(t, N, 'LineWidth', 2);
                          
xlabel('Time/min');
                          
ylabel('Population Density')
                          
grid on;
                        
                        
                        
                        
                        
                        

                        

                        
                        
                        

odefun.m

xxxxxxxxxx
                        
                        

                        

                        

                          

                            

                            

                            
function dN = odefcn(t, N)
                          
                          
    r = 0.0073;
                          
    K = 6.08*10^9;
                          
    dN = r*N*(1-N/K);
                        
                        
                        
                        
                        
                        

                        

                        
                        
                        

 

contact us

Xi'an Jiaotong University
28 Xianning West Road
Xi'an, Shaanxi, China, 710049
xjtu_igem@xjtu.edu.cn

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