Modeling

In Reconby we transform bacteria with a plasmid (circular DNA with a promoter which indicates to the DNA-polymerase where to start transcripting to RNA) that has our nanobodies's expression system cloned in it.
This process require energy and resources from bacteria, which might directly affect its resource allocation and growth (1)*.
Knowing the main characteristics of the bacteria (E. coli UT5600) and the system we are going to use, we aim to describe mathematically this process and its consequences to the bacteria and to its expression system.
For the model design we got inspiration from previous work of King College of London team from 2019 (2) and an from the paper "Deconvolution of Gene Expression Noise into Spatial Dynamics of Transcription Factor–Promoter Interplay" *(3)

Steps in which we divide the modelling

Step 1: Activation of our promoter

Part 1

Promoters are activated by transmission factors, in this case, the protein we are interested in has two forms, desactivated or activated.
To activate our promotor, we need to have our XylS in its activated form XylS². The molecule 3-methyl-benzoate (3mB) is the inductor that binds to XylS activating it. We assume that 3mB is in excess compared to XylS and that, at the working pH, the 3mB is fully protonated being able to enter to the cell.

The union and dissociation of 3mB to XylS is regulated by the constant of union and dissociation (4)*:

The total concentration of XylS is:

The concentration of XylS activated depends on the the concentration of the inductor and the constant of binding (k_x) and can be described as (4)*:

Where the power n is the probability of finding n copies of 3mB at the same place at the same time.
To understand the actuation of the constant n, we ran a simulation of this equation were we assumed the concentration of XylS to be (2)*:

We show the plot of the results on how the concentration of XylS ativated changes with the concentration of 3mB and how the curve depends on the value of n:

The maximum value that the constant n we are going to considered is supposed to be 4. Larger values than 1, are associated with cooperative reactions. It is known that the lac promotor is associated with a value of n=2. In our project we are going to consider that n=1 for simplicity.

Part 2

Afterwards, two activated XylS form join the distal and proximal parts of our promotor so that the promoter is activated.

This reaction takes place twice for each promoter and is regulated by the constant of union and dissociation (3)*

In order to simplify the model, we are going to assume that the constants that regulate the union and dissociation are the same.
We can describe the concentration of the promoter in the activated form as:

In the stationary state, the equations (4.1) and (4.2):

So that:

We define the dissociation constant:

The larger the dissociation constant, the higher the rate of dissociation of the complex, that is, the weaker the binding of XylS activated and promoter.
If we reorganise:

So we can deduce:

We describe the conservation equation, where the total Pm is given by:

And from the previous equations (5.1) and (5.3) we obtain:

So that we can study what concentrations of Pm we have when change the concentration of XylS:

We conclude that:

Transcription

Now that we have our promoter activated, it is ready to be recognized by the RNA-polymerase that will transcript the sequence into mRNA. This is a irreversible process but mRNA degradation has to be considered:

Where the constant union that regulate the first equation and the degradation constant are : (3)*, (2)*

For optimizated models: (3)*

For determining the levels of mRNA transcribed from the induced transcription we define the next equation.

An activator protein increases the rate of transcription when it binds to its DNA site in the promoter. The rate of transcription is thus proportional to the probability thatthe activator XylS is bound to Pm. So this binding is described by a Michaclis-Menten function (4)*:

For the first union we would have that:

But as we have two unions:

Using equation (2) and (10):

We show the plot of the results on how the concentration of mRNA changes with the concentration of 3mB, equation 2) and again we how the curve depends on the value of n:

For this plot we had set the value parameter of Kd with the values of the constants as (3)*

We are going to use this values for the constants and to assume that the concentration [mRNA] converges to the value of:

Nevertheless, in the next plot we are going to visualice for the fixed value of n=1 the results of (9) for the value of kd we used in the past simulation and the other two different values of kd given by:

So we can see how for smaller values of kd, we obtain less concentration of mRNA.

Translation

This is the process of translating the sequence of a messenger RNA (mRNA) molecule to a sequence of amino acids in order to synthesise the Intimin protein, which is generated at the cytoplasm.

With a constant of:

In optimized models (3)*:

Transport

Once intimin is translated, it is continuously transported to the surface of the cell, therefore intimins' concentration in the cell also decreases.

We also have keep in mind the degradation of the Intimine at the membrane.

Where the constants are (2)* (3)*:

In this step of our model ,we are going to describe the production levels of the surface protein from translation.
The rate of intimin production is subject to changes in the rate of mRNA translation.
To account for fluctuation, we created a function that takes the transport rate of intimin to the surface to the cells into consideration as well.(alpha)

Our final differential equation simulates the change in concentration of intimin expressed on the cell surface over time. This equation takes both the rate at which intimin is being transported to the cell surface and the intimins’ degradation rate into consideration.

The transport of the intimin to the surface is defined, similarly to a Michaelis Menten equation, by,

Where:

With:

This equation was modified from KCL model, they defined it as (2):

However, for us, it is more realistic to change the fraction in order to keep it real as:

If the concentration of intimin in the cytoplasm increases, there would be more transport towards the membrane as φ (phi) increases.
If the concentration of intimin in the membrane increases, there would be less room for more intimin and transport would slow down as φ (phi) is reduced.

With the initial condition that Intc at time=0 is 0, we simulate the ODE (14) so that we obtain Intc(t):

References

(1)* Borkowski O, Ceroni F, Stan GB, Ellis T. Overloaded and stressed: whole-cell considerations for bacterial synthetic biology. Curr Opin Microbiol. 2016 Oct 1;33:123–30.
(2)* Team:UCL/Model4 - 2017.igem.org [Internet]. [cited 2021 Oct 20]. Available from: https://2017.igem.org/Team:UCL/Model4
(3)* Goñi-Moreno Á, Benedetti I, Kim J, Lorenzo V de. Deconvolution of Gene Expression Noise into Spatial Dynamics of Transcription Factor–Promoter Interplay. ACS Synth Biol. 2017 Jul 21 ;6(7):1359–69.
(4)* Uri Alon-An introduction to systems biology_ design principles of biological circuits-Chapman & Hall_CRC (2007)

GitHub

Notebook in Python used for obtaining the plots: https://github.com/Halvus/IGEM2021UNIZAR_RECONBY

Team:UNIZAR/Model