Tetrazines are important bioorthogonal reactive tools due to their rapid ligation with trans-cyclooctene. However, their application is limited by the reactivity-stability paradox in biological environments. Thus, finding a suitable tetrazine is of great importance for the experiment.

The reaction between tetrazine and trans-cyclooctene can be separated into two steps.

Figure 1: Two steps of the reaction between tetrazine and trans-cyclooctene1

Suppose the trans-cyclooctene is A, tetrazine is B, intermediary product is C, and the terminal product is D. The reaction between A and B is Second order reaction, which means the speed of the reaction is linearly dependent to the concentration of A and B.

The Decomposition reaction of the intermediary product C is One order reaction, which means the speed of the reaction is only linearly dependent to the concentration of C.

We should also consider the relationship between A,B,C and D.

In addition we should consider the solvent effect, because only if the numerator enter the Solvent, then it can react with other numerator.

With the help of python, we can get such fit curve.

We can consider the reaction in another view(inspired by ZJU-China 2023): The extent to which the reaction proceeds can be described by the dissociation constant:

Since the probability of binding is occupied by the receptor rate, the probability of the complex occupying the whole is:

Considering the solution lattice model, i.e., θ ligands are randomly distributed in ω lattices, according to the Boltzmann distribution function:

The probabilities can be deduced as:

where is the binding energy and is the energy of the ligand in solution, comparing (10)(12) two equations:

Taking k_b= 1.38×10^-23J/​(K·T), electing the unit volume of 1mm3, ε_b-ε_sol≈-10k_bT in the lattice model.

Figure 2: The probability varies with concentration

Through this model, we can view the first step in such way:

A and B have have the potential to bind which can be described by the model above. And C will decompose into D.

With the help of python, we find:

The Lambert W Function, also known as the "omega function" or "product log function", is the inverse function of f(w)=w.exp(w), where exp(w) is an exponential function and w is an arbitrary number. exp(w), where exp(w) is an exponential function and w is an arbitrary number.

The Lambert W Function's Taylor expansion is:

In our work, we want to achieve protein presenting with CsgA. Thus the production of CsgA is of great importance. In this part, we will build the model to achieve better production of CsgA.

Figure 3: The secretory steps of CsgA

Firstly, the activation of L-Arabinose promoter.

Figure 4: The activation of L-Arabinose promoter2

The PBAD promoter regulates the araBAD operon, which encodes three enzymes that convert L-arabinose to D-xylulose-5-phosphate. In the absence of L-arabinose, PBAD is repressed by the transcriptional regulator AraC, which binds to the distal I1 and O2 half-sites to form a DNA loop that sterically blocks the entry of RNA polymerase into the promoter. In the presence of L-arabinose, transcription of the PBAD promoter is activated by AraC, which also exerts negative feedback on its own promoter, the PC. AraC binds to the neighbouring I1 and I2 halves and recruits RNA polymerase, thus activating the promoter.

The activation of the promoter:

When considering the transporting rate of CsgG,we import the T function, according to the published articles, the transporting rate of one protein has a threshold, before which the transporting rate is linearly dependent to the concentration.

Figure 5: T Function

Here we provide several Ordinary differential equation to describe the process of the expression of CsgA.

Transcription:

Translation:

Transport:

α_m is the transcription rate costant of DNA, β_m is the degration rate constant of mRNA. α_c is the translation rate constant of mRNA, β_c is the degration rate constant of CsgA. T(CsgA_in) describes the rate of transporting CsgA into periplasm. β_pen is the degration rate constant of CsgA in the periplasm. T(CsgA_pen) describes the transporting rate of CsgA from periplasm to outer space.

The Nucleation reactions of CsgA in the membrane can be described in such way:

The nuclear reaction of CsgA:

Thus:

Through the model above, we can get such conclusion:

The yield of CsgA is related to the insert efficiency, reactivity and stability of the noncanonical amino acids, thus it is of great importance to find one noncanonical amino acid which is of all these qualities. Then we find Tet 2.0 can meet our requirement. Tet-v2.0 reacts selectively with cyclopropane-fused trans-cyclooctene (sTCO) with a bimolecular rate constant of 72,500 ± 1660 M-1 s-1 without reacting with other cellular components. It can also achieve high insert efficiency. Thus we choose this as our noncanonical amino acid.

The expression of protein with noncanonical amino acids can be separated into such steps:

Figure 6: The expression of protein

The number of noncanonical amino acids on the membrane surface determines the reaction rate which determines the efficiency of the project. Thus we need to describe the process with the help of ODE tools for better effectiveness.

The number of protein on the membrane surface has been described in previous work, here we will try to describe the exact number of noncanonical amino acid.

We assume that the constants for the binding of CsgA, CsgAtet, and CsgB are respectively: ε_csgA,ε_csgAtet,ε_csgB

Thus: