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Model for protein expression system

 

1.      Overview

 

Amino acids are the basic building blocks of every protein. The rate of protein translation (V) increases with the concentration of amino acids (C) and bounded when the amount of amino acids is saturated. (i.e. there exists V_max, such that V£V_max no matter how high C is.) Normally, the protein expression rates of different system are compared by the transcription efficiency of the system only because we can provide enough nitrogen sources to make V close to V_max. However, this is not quite applicable to our protein expression system. The transcription process of our system is repressed by nitrogen sources in the cell, so we believe that it is not practical to make C so large that it is saturated. On the other hand, this system allows us to achieve a constant protein expression rate by controlling the duration for expressing T7 polymerase.

 

2.      Analysis and Discussions

 

Let us review our protein expression system. The schematic diagram is shown below.

 

2.1 Assumptions

First, we may assume that:

1. Concentration of σ-RNA polymerase is constant.

2. All proteins being expressed share the same nitrogen source.

 

 

Then, let:

[N] be the concentration of nitrogen source in cell;

[T7^p] be the concentration of T7 polymerase;

[TP] be the concentration of target protein;

K_nc be the rate of nitrogen consumption out of this protein expression system;

V_max([T7^p]) be the maximum rate for target protein expression;

A,B,C,D,E,F,G, I be constant.

 

2.2 Discussions of the two stages

(1) In the first stage:

In this stage, we want to optimize the production of T7 polymerase by setting a relatively low concentration of nitrogen source.

 

We have the following equations:

 
 

Reason for proposing (1): It is expected that at [N]à0 or [N]à¥, =0 and it should be a continuous function. So, this kind of function could be a candidate.

Graph of (1). (y-axis:  x-axis:)

 

Reason for proposing (2): This is similar to the Michaelis–Menten Equation as it is believed that the ribosome would be saturated with nitrogen sources like enzymes saturated with substrate. Also, we expected that V_max is a function of T7 polymerase.

 

Reason for proposing (3): In the right-hand side of the equation, the first term corresponds to the nitrogen consumption rate for T7 polymerase. The second term corresponds to the nitrogen consumption rate for target protein. The third term corresponds to the nitrogen consumption rate for other protein expressed by the bacteria.

 

In this stage, we want to find out the optimized value for [N] to produce polymerase as fast as possible. So, we want to have the maximum of (1).

Solving the equation, we have

 

(2) In the second stage:

In this stage, we want to optimize the production of target protein by setting an excess nitrogen source.

 

In this condition, we have .

So,.

This implies that.

As a result,.

Since, when [N]à¥,,

We have .

The constant that varies with different initial conditions is used.

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