Overview

Figure 1. The expression system of our proposed mechanism

To synthesize B5R protein, we chose to use BL21(DE3) E. coli strain since it lacks proteases that can degrade our recombinant protein. The BL21(DE3) has a specific T7 RNA Polymerase (T7 RNAP) gene controlled by an IPTG-regulated lacUV5 promoter (Figure 1). This T7 RNAP is derived from λ bacteriophage, and lacUV5 promoter is a leakier promoter than wt lac promoter. The lacUV5 promoter is inhibited by LacI inhibitor when IPTG is absent. It forms dimers in cytoplasm, and LacI dimer can be inhibited by IPTG [1]. Since we use BL21(DE3) strain, we decided to use pET23a plasmid for expression of B5R since it contains a T7 promoter that specifically allows T7 RNAP binding [2]. 

Why do we need a mathematical model?

The high expression of T7 polymerase resulted in high metabolic burden on cells, higher chances of mutations in target gene, and loss of expression plasmid [3]. Therefore, it is important to regulate IPTG concentration and still have optimal recombinant protein expression. Moreover, the modeling of optimal IPTG concentration can be useful in mass production of our protein. You can learn more about our future plans in the Implementation part. 

Expression of T7 polymerase

In our system, T7 polymerase is synthesized from LacUV5 promoter via the following reactions:

And the ODEs that model the change of concentration of and during transcription and translation processes are:

(1)

(2)

Assuming that the translation process is much longer than the transcription, for most of translation, transcription has reached steady state conditions:

(3)

Thus,

(4)

(5)

And the change in concentration of the protein can be described as:

(6)

After the translation process has reached steady-state, the concentration of T7 polymerase can be calculated and the concentration of T7 polymerase depending on the lacUV5 operon can be modeled as:

(7)

Where the constant is:

(8)

The model shows the speed of production and the amount of T7 polymerase produced with the pET23a plasmid. It can be noted that the assumption that the transcription process is at steady state is confirmed (Figure 2 and 3).

Figure 2. Concentration change during T7 expression in 5-hour period

Figure 3. Concentration change during T7 expression in 40-hour period

Parameters used for the modeling:

Expression of B5R

Similar logic is used to model the expression of the B5R protein using the T7 polymerase. The reactions of transcription and translation are:

And the ODEs representing the rate of transcription and translation are:

(9)

(10)

where is the copy number of the plasmid pET-23 that we used. Similarly, assuming that transcription reaches the steady-state conditions for most of the translation process, the equation for the rate of translation is written as:

(11)

And when translation has reached the steady-state conditions, concentration of B5R can be modeled from  the initial concentration of T7 polymerase:

(12)

(13)

Modeling the influence of IPTG on the T7 polymerase expression

Using the same reasoning, the amount of LacI protein, that suppresses the lacUV5 promoter of T7 polymerase production, can be determined as:

(14)

The reactions are:

And adding IPTG into the system, its production theoretically becomes:

(15)

Therefore, reducing the production of LacI, meaning that it represses the repressor of the T7 polymerase production, thus adding IPTG results in a more yielding of the T7 polymerase. 

Modeling the concentration change of LacI, it is noticed that with increasing the concentration of IPTG, the suppressor LacI is depleted faster. This data is useful to pick the concentrations of IPTG to add to obtain the desired concentration of T7 and B5R proteins.

Figure 4. The concentration of free LacI (µM) vs time (second)