Difference between revisions of "Team:TUDelft/DennisModelFull"

Overview - The Model

Many variables play a role in the behavior of synthetic circuits. Many (if not all) of these variables can dramatically change when transferring a circuit between organisms. We adopted a system developed by (Sontag et al. (2018)) which has been shown to result in independence of copy number of the genetic circuit. However, the full potential of this system has not been explored yet. Closer analyzation of this system revealed it would maintain the same output when transferring genetic circuits between organisms.

In our system we exploit a commonly applied control system motif known as an Incoherent Feed-Forward Loop (iFFL), in which an input signal regulates both the activator and the respressor of the output of the system. An iFFL results in perfect adaptation when the negative regulation is uncooperative (Sontag et al. (2018)).
The input in our case is the copy number of the DNA template and the output is the steady-state expression of a gene of interest. This control system is established through the expression of a Transcription activator-like effector (TALE) protein. TALE proteins recognize DNA by a simple DNA-binding mechanism and have been shown to bind fully uncooperative (source). The TALE protein has been engineered to bind to the promoter of a gene of interest and thus repress the expression of it.

The kinetics

In order to model our system we have to identify all interactions and subsequently define the rate equations. Figure 2 depicts a scheme of all interactions.

From these interactions we can derive the following system of ordinary differential equations:

${dm_T \over dt} = {c \cdot a_T - y_m \cdot m_T}$

$\frac{dT}{dt} = b_T \cdot m_T - y_T \cdot T - n \cdot k_{on}\cdot T^n \cdot P_G + n \cdot k_{off} \cdot P_{G.T}$

$\frac{dP_G}{dt} = k_{off} \cdot P_{G.T} - n \cdot k_{on} \cdot T^n \cdot P_G + n \cdot y_T \cdot P_{G.T}$

$\frac{dP_{G.T}}{dt} = n \cdot k_{on} \cdot T^n \cdot P_G - k_{off} \cdot P_{G.T} - n \cdot y_T \cdot P_{G.T} $

$\frac{dm_G}{dt} = a_{Gmax} \cdot P_G + a_{Gmin} \cdot P_{G.T} - y_m \cdot m_G $

$\frac{dm_G}{dt} = b_G \cdot m_G - y_G \cdot G $

Parameter	Explanation
$a_T$	Transcription rate TALE
c	copy number plasmid
$y_m$	degradation rate mRNA
$b_T$	Translation rate TALE
$y_T$	Degradation rate TALE
n	Cooperativity of binding
$k_{on} /k_{off}$	(un)Binding of TALE to promoter
$a_{Gmax}/a_{Gmin}$	Maximum and minimum transcription of GFP
$b_G$	Translation rate GFP
$y_G$	Degradation rate GFP

Simplification of the system

This system can be simplified by making a few assumptions:

1. Amount of TALE protein is much larger than binding sites for TALE
2. TALE binding and unbinding occurs much more rapidly than protein production and degradation
3. There is negligable expression when the promoter is repressed

This results in the following steady-state solution:

$G = (\frac{c}{c^n}) (\frac{a_Gb_Gy_T^ny_m^n}{a_Tb_Ty_Gy_m})$

• Full derivation

In the case of $n = 1$, this system is independent of copy number. However, maybe more notably the system also appears to be dependent only on the ratios of the remaining variables. The dependence on the ratio of these variables has not been previously explored. This dependency on the ratio of only a few variables will be further elaborated in the section "Sensitivity to different environments" (maybe put a link so someone automatically scrolls to this section?).

Copy number independence

Broad host range plasmids are often used when using different hosts however, they do not guarantee the same copy number in every organism (Jain and Srivastava 2013). In some organisms, it is not common practice to use a plasmid but rather insert into the genome. The need for consistent expression at a wide range of copy number is thus a desirable feature of any synthetic circuit. We, therefore, have simulated this system from a range of 1 to 600 (genome integration to high-copy number plasmid).

As predicted by our model simplification the system has no variation in steady-state protein levels.

• Link to wet lab

  To validate our predictions we have

Sensitivity to different environments

As stated before, there are many variables that influence the expression of a gene of interest. However, according to the steady-state model solution, the final level of the gene of interest is only dependent on the ratio of transcription, translation and degradation rates. We can use this to ensure stable expression across different bacterial species if we make the following assumptions:

1. Transcription rates will change in a similar way for both the TALE gene as the gene of interest.
2. Translation initiation rate will change in a similar way for both the TALE gene as the gene of interest if the same ribosome binding site is used for both.
3. Translation elongation rate will change in a similar way for both the TALE gene as the gene of interest if both proteins are similarly codon-optimized.

If these assumptions are correct our steady-state model solution always yields in the same ratio and thus the same steady-state solution.

• Validation of steady-state solution - Simulation of transcription and translation variations using full model

  Our steady-state solution is based off of a couple of assumptions. We therefore simulate our full model in the case of transcription and translation variation to see whether we get the same result:

  

In our system we are using the same promoter for each gene and thus assume similar variation in transcription for each gene when changing between organisms. However, even though we use the same ribosome binding site (RBS) for each gene, we can't assume similar variation in translation rate. Translation rate is dependent both on translation initiation and translation elongation. Translation elongation is largely dependent on codon usage (Frumkin, Lajoie et al. 2018). Both genes in our circuit don't have the same codon adaptation index in E. coli and their adaptation will be irregular to different bacterial species.

• Link to wet lab