In this expression model, we will calculate the exact number of β-galactosidase protein. We will quantify the data by building a mathematical model. The result of the model will visually show the relationship between time and the number of molecules of the target protein. Hence, through this graph, we will be able to identify the proper time to check the protein production and extract it. In order to simplify the model, we will choose cell-free environment first. In the end, we will apply the model back to the cell environment.
Fig. 1 |The whole process
Design
Fundamental gene expression includes two steps: transcription and translation. The product of transcription is mRNA, while the product of translation is protein. After a certain time, the protein will be degraded into fraction by particular enzymes. Fig.1 below shows the basic mechanism.Hence, we establish the following equations which represent different biological reactions. The condition of these equation is that all the reaction rate is proportional to the concentration of reactants and products. Table 1 shows all the equations. Table 2 explains the meaning of different variables.
Fig. 2 |All equations
Fig. 3 |Definition
Based on the equation above, we describe two relationships: time and the amount of mRNA, time and protein. The functions are listed below:
Fig. 4 |Two functions
The values of the constants used in the functions are listed in table 3.
(all the constants are calculated from ratio. The protein size and its molecular mass are used in the calculation.)
Fig. 3 |Definition
Assumption:
To make sure the result is reliable, we need to make several prerequisites to our model:
1. Before induction, the target protein and mRNA level is zero
2. The transcription speed of E. coli RNA polymerase is 50 nt/s
3. The synthesis and degradation rate of the protein is always constant
4. No other substance will affect the reaction
Solving equations:
We choose to use MATLAB as our calculating and demonstrating tool. The result of the expression model in MATLAB uses two files: f1.m defines the system of differential equation; run1.m solve the model. The final result is shown in Fig.2
