Team:TUDelft/DennisModel

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Overview

With our modeling, we aimed to apply a control systems approach to achieve stability of gene expression across bacterial species. To make expression host-independent, we included an incoherent feedforward loop (iFFL) in our design. An iFFL can be used to make the output of a system independent of the input. Our analytical steady-state solution of this loop showed that expression was completely independent of plasmid copy number and transcriptional-translational rates. We verified this analytical solution by the implementation of a full kinetic model.

The key variables in the design of genetic circuits are plasmid copy number and transcriptional-translational rates. These variables determine the steady-state levels of gene expression. However, when transferring genetic circuits between organisms, these variables change in unpredictable ways.

Promoters have different strengths in different organisms. Some promoters only work in a very narrow range of bacterial species (Yang, Liu et al. 2018). To circumvent host-related changes, we chose our system orthogonal to the host. We do this by using T7 RNA polymerase. However, orthogonal transcription might not behave similarly when applied in varying biological contexts. Through our modeling we show that gene expression levels remain the same in varying biological contexts when using our genetic circuit implementation of an iFFL.

Ribosome binding sites contain the Shine-Dalgarno sequence, where the 16s rRNA of the ribosome binds. However, this sequence varies across species, and often ribosome binding sites are extremely inefficient when applied in phylogenetically distant species (Salis, Mirsky et al. 2009). Our model shows that similar expression levels across organisms can be maintained when all genes in our genetic circuit contain the same ribosome binding site. Assuming translation elongation is similar across different species, our model shows that expression levels in different organisms can be maintained when all genes in our genetic circuit contain the same ribosome binding site. Nevertheless, translation elongation is influenced by codon usage, which differs per organism. We, therefore, developed a software tool that determines a coding sequence similar in codon usage across different species. The similar codon usage minimize the chance of different translation elongation rates between organisms.

The core of our design - Incoherent feed forward loop

We implemented an incoherent feedforward loop (iFFL) in a genetic circuit. In an iFFL, the input signal regulates both the activator and the repressor of the output of the system in the same way. The iFFL results in perfect adaptation to the input when the binding of the repressor is fully non-cooperative (binding of one repressor at a time) (Segall, et al. 2018). In our case, the input is the plasmid copy number of the DNA template, and the output is the steady-state expression of a GOI. In our system, we use a transcription activator-like effector (TALE) protein as a repressor. TALE proteins recognize DNA by a simple DNA-binding mechanism (Doyle 2013) and have been shown to bind fully non-cooperative (Segall-Shapiro, et al. 2018). The promoter controlling the GOI has been engineered to contain a binding site of a TALE protein. When the TALE protein is bound to the promoter, the expression of the GOI is repressed as demonstrated by 2018 iGEM Thessaloniki.

We have modeled the function of the genetics of this system. An analytical steady-state solution of the system showed that the steady-state expression level of a GOI is completely independent to plasmid copy number and can be independent of transcriptional and translational rates when the right design choices are made. After further verification through the implementation of a full ordinary differential equation (ODE) model, we designed experiments to test the independence to these variables. Key design choices were identified by modeling. These consist of:

The need for good insulation of the genes.
The promoter strengths of the TALE protein and the GOI need to maintain the same ratio.
The ribosome binding site strengths of the TALE protein and the GOI need to maintain the same ratio.

The kinetics

In this section, we explain the kinetics of our iFFL and derive a system of ordinary differential equations to describe the interactions within the genetic circuit. We step-wise derive a steady-state solution from the system of equations and describe the properties of the system. In the other sections, we use the steady-state solution and ODE model to describe how our circuit can be used to transfer genetic circuits between prokaryotes. Figure 3, depicts all interactions considered in our system.

TALE system — Figure 3: Scheme of genetic circuit interactions developed by *(Segall, et al. (2018))*

Click here to find out more about the kinetics

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

Parameter	Value	Unit	Explanation	Source
$a_T$	1.03	nM/min	Transcription rate TALE	2018 iGEM Thessaloniki
$y_m$	log(2)/5	1/min	degradation rate mRNA	Kushwaha and Salis 2015
$b_T$	0.44	1/min	Translation rate TALE	2018 iGEM Thessaloniki.
$y_T$	0.0347	1/min	Degradation rate TALE
n	1	-	Cooperativity of binding	Segall-Shapiro et al.
$k_{on}$	9.85	1/(nM*min)	Binding of TALE to promoter	2018 iGEM Thessaloniki.
$k_{off}$	2.19	1/min	Unbinding of TALE to promoter	2018 iGEM Thessaloniki.
$a_{Gmax}$	3.78	1/(nM*min)	Maximum transcription of GFP	2018 iGEM Thessaloniki
$a_{Gmin}$	0	1/(nM*min)	Maximum transcription of GFP	Segall-Shapiro, Sontag et al.
$b_G$	3.65	1/min	Translation rate GFP	2018 iGEM Thessaloniki
$y_G$	0.0347	1/min	Degradation rate GFP
$c$	variable	Unitless	Plasmid copy number of plasmid

Variable	Explanation
$m_T$	concentration of TALE mRNA
T	concentration of TALE
$P_G$	Promoter GFP
$P_{G.T$}	Promoter GFP with TALE bound
$m_G$	concentration of mRNA GFP
G	concentration of GFP

Simplification of the system

This system can be simplified by making a few assumptions (Segall-Shapiro, et al. 2018):

Amount of TALE protein is much larger than binding sites for TALE
TALE binding and unbinding occurs much more rapidly than protein production and degradation
When the promoter is repressed the expression level is negligable.

Click here to see the full derivation
This derivation follows the derivation of this system by (Segall-Shapiro, et al. 2018).
By making use of assumption 2, we can assume a quasi-steady state for $\frac{dP_G}{dt}$ and $\frac{dP_{G.T}}{dt}$. Quasi-steady state means we assume it reaches steady-state much quicker than all the other variables in the model and thus the rate of change is zero. This results in the following system of equations.
1. ${dm_T \over dt} = {c \cdot a_T - y_m \cdot m_T}$
2. $\frac{dT}{dt} = b_T \cdot m_T - y_T \cdot T - n \cdot k_{on}\cdot T^n \cdot P_G + n \cdot k_{off} \cdot P_{G.T} + n \cdot (n-1)\cdot y_T \cdot P_{G.T}$
3. $\frac{dP_G}{dt} = k_{off} \cdot P_{G.T} - n \cdot k_{on} \cdot T^n \cdot P_G + n \cdot y_T \cdot P_{G.T} = 0$
4. $\frac{dP_{G.T}}{dt} = n \cdot k_{on} \cdot T^n \cdot P_G - k_{off} \cdot P_{G.T} - n \cdot y_T \cdot P_{G.T} = 0$
5. $\frac{dm_G}{dt} = a_{Gmax} \cdot P_G + a_{Gmin} \cdot P_{G.T} - y_m \cdot m_G $
6. $\frac{dG}{dt} = b_G \cdot m_G - y_G \cdot G $
We can now use equation 3 to simplify the system:
- $\color{red}{k_{off} \cdot P_{G.T} - n \cdot k_{on} \cdot T^n \cdot P_G} + n \cdot y_T \cdot P_{G.T} = 0$
- The red part can be taken to one side of the equation:
- $\color{red}{n \cdot k_{on} \cdot T^n \cdot P_G - k_{off} \cdot P_{G.T}} = n \cdot y_T \cdot P_{G.T}$
- This is pasted in equation 2:
- $\frac{dT}{dt} = b_T \cdot m_T - y_T \cdot T \color{red}{- n \cdot k_{on}\cdot T^n \cdot P_G + n \cdot k_{off} \cdot P_{G.T}}+ n \cdot (n-1)\cdot y_T \cdot P_{G.T}$
- Which becomes:
  $\frac{dT}{dt} = b_T \cdot m_T - y_T \cdot T -(n \cdot y_T \cdot P_{G.T}) + n \cdot (n-1)\cdot y_T \cdot P_{G.T} = b_T \cdot m_T - y_T \cdot T$
Furthermore, we can use equation 3 to solve for $P_G$
- $k_{off} \cdot P_{G.T} - n \cdot k_{on} \cdot T^n \cdot P_G + n \cdot y_T \cdot P_{G.T} = 0$
- $P_G = \frac{k_{off}}{k_{on}} \cdot P_{G.T} + \frac{1}{k_{on}} \cdot n \cdot y_T \cdot P_{G.T}$
Using assumption 1, we can say T >> c. It follows that the amount of free repressor barely changes when some of them bind to $P_G$, meaning $T \approx T + nP_{G.T}$
- $P_G = \frac{k_{off}}{k_{on}} \cdot P_{G.T}$
- Using: $c = P_G + P_{G.T}$ we get the following:
- $P_G = \frac{c}{1 + K_D \cdot R^n}$, Where: $K_D = \frac{k_{off}}{k_{on}}$
- Plugging this into equation 5 and again making use of $c = P_G + P_{G.T}$, equation 5 becomes:
- $\frac{dm_G}{dt} = c \cdot (a_{Gmin} + (a_{Gmax} - a_{Gmin})[\frac{K_D}{K_D + R^n}]) - y_m \cdot m_G$
Assumption 3 tells us that we can assume $a_{Gmin} \approx 0$. Again using assumption 1, we can say $T^n >> K_D$, resulting in $T^n + K_D \approx R^n$ Using these two assumptions equation 5 can be further simplified to:
- $\frac{dm_G}{dt} = c \cdot a_{G} \cdot[\frac{K_D}{R^n}] - y_m \cdot m_G$, where $a_{G} = a_{Gmax}K_D$
Using this reduced system of equations we can now derive the steady-state solution for the GOI.
- $ c \cdot a_T - y_m \cdot m_T = 0$
- $ b_T \cdot m_{T} - y_T \cdot y_T \cdot T = 0$
- $ c \cdot a_{G} \cdot[\frac{K_D}{R^n}] - y_m \cdot m_G = 0$
- $ b_G \cdot m_{G} - \cdot y_G \cdot G = 0$
- $ m_T = c \frac{a_T}{y_m} $
- $ T = \frac{b_T \cdot m_{T}}{y_T}$
- $ m_G = c \frac{a_{G} \cdot[\frac{K_D}{R^n}]}{y_m}$
- $ G = \frac{b_G \cdot m_{G}}{y_G} $

According to our analytical solution, the level of the protein of interest is only dependents on plasmid copy number, and the ratios of transcription and translation rates of the genes in the circuit. In the next sections, we use this steady-state solution to demonstrate how it can be used to transfer genetic circuits between organisms. Furthermore, since this steady-state solution is based on assumptions, we solve the full system of ordinary differential equations in Matlab to gain insight in the kinetics of the system. In each segment, a link to our corresponding wet lab page is given.

Plasmid copy number

The expression levels in a genetic circuit are strongly correlated to the plasmid copy number of the DNA template Segall-Shapiro et al.. The amount of gene plasmid copy number can change when transferred between organisms. Therefore there is a need for consistent expression at a wide range of plasmid copy numbers if the same genetic circuit is used in different organisms. The steady-state solution of our model tells us that when our repressor binding is fully non-cooperative, n = 1, we have complete independence of plasmid copy number:

$$G = \left(\frac{\color{red}c}{\color{red}c^\color{red}n}\right)_{\color{red}n\color{red}=\color{red}1} \left(\frac{a_Gb_Gy_T^ny_m^n}{a_Tb_Ty_Gy_m}\right)$$

This formula is however based on a few assumptions. To see how the system would behave without making these assumptions we implemented the full system of ordinary differential equations (Figure 4).

Figure 4: Steady-state GOI production for gene plasmid copy number 1 to 600 (genome integration to high plasmid copy number plasmid).

The model without assumptions has the same expression level independent of plasmid copy number (figure 4). We therefore can transfer our circuit between organisms and expect the expression of the GOI to be independent of the changes in plasmid copy number of our orthogonal plasmid.

Wet lab

We tested the prediction of plasmid copy number independence by implementing the iFFL system, with GFP as the output. We cloned the system in backbones containing different origins of replication. As a control, we also cloned GFP into the same backbones to demonstrate different levels of expression. More info can be found here.

Transcriptional variations

Every promoter might have a different strength when used in different organisms (Yang, Liu et al. 2018). Thus, when using the same promoter in different organisms, you can get unpredictable behavior. The steady-state solution of our model tells us that the steady-state expression level of the GOI is only dependent on the ratio of transcription rates of the GOI and TALE.

$$G = \left(\frac{c}{c^n}\right) \left(\frac{\color{red}a_\color{red}G \color{black} b_Gy_T^ny_m^n}{\color{red}a_\color{red}T \color{black} b_Ty_Gy_m}\right)$$
The formula is, however, based on a few assumptions. We, therefore, implemented the full system of ordinary differential equations to validate our assumptions. We vary both transcription parameters. In figure 5 we plot the resulting steady-state solutions as a function of the transcription rate of TALE and GFP. TALE system

Figure 5: Steady-state GFP production while translation rates of both TALE and GOI are changed. The lines indicate constant ratio of transcription rates

The full kinetic model shows that the expression level of GFP is the same when the transcription rate of TALE and of GFP remain constant (figure 5). In order to achieve constant ratio of transcription rates in our genetic circuit we use the T7 orthogonal transcription system which is transcribed by its own RNA polymerase. We implemented T7 promoters with varying strengths compared to the wild-type, developed by Ryo Komura, et al. (2018). More information can be found here (link to design page).

Wet lab

We demonstrated the prediction of transcription rate independence when the same ratio in transcription rate of both genes is maintained.

[graph]
We made variations of the system where we changed both promoters in the same way. As a control, we also cloned GFP without repression under the control of these same promoters.

We also tested the functioning of the system when different IPTG induction is used. Again, as a control, we also cloned GFP without repression under the control of these same promoters.

[graph]
More info can be found here [link to result page].

Translational variations

Similarly, as in transcription, our model steady-state solution tells us that the steady-state expression level of the GOI is only dependent on the rate of translation of the GOI and TALE,

$$G = \left(\frac{c}{c^n}\right) \left(\frac{a_G \color{red}b_\color{red}Gy_T^ny_m^n}{a_T \color{red}b_\color{red}Ty_Gy_m}\right)$$

In figure 6 we plot the resulting steady-state solutions as a function of the translation rate of TALE and GOI using the full kinetic model to see how the system without assumptions behaves.

Figure 6: Steady-state GOI production while translation rates of both TALE and GOI are changed. The lines indicate constant rate of the translation rates.

As can be seen in figure 6 the full kinetic model maintains the same level of GFP expression when the translation rates for both genes remain in a constant ratio. To keep the same ratio in translation rates across organisms we used the same ribosome binding site (RBS) for both genes. Using the same RBS ensures that translation initiation for both genes change in a similar manner (Salis, H. M., et al. 2009), more on the design choices can be found here (link to design page).

Wet lab

We tested the prediction of translational rate independence by implementing the iFFL system, where the output is GFP. We made variations of the system where we change both ribosome binding sites in the same way. As a control, we also cloned GFP without repression into under control of these ribosome binding sites. More info can be found here.

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Importance of insulation

In our model solutions so far we assumed the promoter of the GOI to be completely insulated from expression from the TALE protein. However, in reality when two transcription units are placed in series leaky expression of the second gene can occur. This is due to the efficiency of the terminator of the first gene (Cheng, Ying-Ja et al. 2018). The iFFL system originally developed by Segall-Shapiro et al, uses the ECK120029600 terminator for the TALE protein. This terminator has a reported efficiency of 1/612, meaning that for every 612 TALE proteins produced, 1 protein of the GOI is made (Cheng, Ying-Ja et al. 2018). We incorporate this efficiency into our model and solve again for steady-state GOI expression levels to see the effect of terminator efficiency on plasmid copy number independence. TALE system

Figure 7: Comparison of a perfect terminator and a leaky terminator on the expression level at different plasmid copy number.

The model shows that the leaky expression negatively impacts the system's ability to adapt to gene plasmid copy number. We therefore designed our system to have the transcriptional unit of TALE in a different orientation than the transcriptional unit of the GOI.

TALE system

Figure 8: Design of genetic circuit, to circumvent the issue of leaky expression

Team:TUDelft/DennisModel

Overview

The core of our design - Incoherent feed forward loop

The kinetics

Simplification of the system

Plasmid copy number

Wet lab

Transcriptional variations

Wet lab

Translational variations

Wet lab

Importance of insulation

References