# Genetic Switch

The field of Synthetic Biology allows to construct synthetic genetic circuits with the tools of Genetic Engineering. One of the advances in this relatively new area was the construction of a genetic toggle switch in Escherichia coli. In this wiki article we study the mathematical model that describes the underlying dynamics of the toggle switch system. The switch was successfully tested experimentally by transducing a plasmid with all necessary circuit components into E. coli.

## Biology a Summary

Toggle switch circuit from "Construction of a genetic toggle switch in Escherichia coli" T. S. Gardner et al.

The bistable synthetic gene-regulatory network consists of two promoters and two repressors. The second operon is composed of the promoter 2 the repressor for promoter 1 called repressor 1 and a reporter GFP. Therefore the GFP reports for the expression level of repressor 1.

In the absence of inducer there are two stable states possible:

1. 'low' state: repressor 2 is transcribed and repressor 1 is repressed
2. 'high' state: repressor 1 (+GFP) is transcribed and repressor 2 is repressed

Assuming the switch is in a stable position, addition of inducer for the repressed repressor causes full activation of transcription, until the originally active repressor is repressed. In this manner the inducers allow to switch between the two stable states.

Toggle switch plasmid from "Construction of a genetic toggle switch in Escherichia coli" T. S. Gardner et al.

Gardner et al used two classes of toggle switch plasmids, the pTAK class which uses a temperature sensitive $\lambda$-repressor (cIts) that is induced by high temperature (42°C) and the pIKE class which uses the Tet Repressor tetR as inducer. They built four pTAK class circuits each with different RBS strength:

CLASS ONE

1. pTAK117 (high promoter strength)
2. pTAK130
3. pTAK131
4. pTAK132 (low promoter strength)

and two pIKE circuits:

CLASS TWO

1. pIKE107
2. pIKE105 (no bistability observed)

moreover monostable controls, with only one promoter were used

1. pTAK102 for the IPTG inducible promoter
2. pTAK106 for the temperature inducible promoter

The following dynamics have been observed experimentally

Toggle switch dynamics from "Construction of a genetic toggle switch in Escherichia coli" T. S. Gardner et al.

Due to the stochastic nature of gene expression only a subpopulation switched to the 'hight' stable steady state, as shown for the pTAK117 strain via single cell FACS measurement. Thus a bimodal cell count was observed in the single cell FACS measurement.

Toggle switch bimodal from "Construction of a genetic toggle switch in Escherichia coli" T. S. Gardner et al.

## Mathematics - the Model

The time evolution of the chemical species is governed by the dimensionless system \begin{align} \dot{u}=\frac{\alpha_1}{1+v^{\beta}}-u\\ \dot{v}=\frac{\alpha_2}{1+u^{\gamma}}-v \end{align} where $u$ is the concentration of repressor 1 and $v$ is the concentration of repressor 2. $\alpha_1$ and $\alpha_2$ are the production rates dictated by the respective promoter, so that $v$ is produced at a maximal rate of $\alpha_2$ and $u$ at a maximal rate of $\alpha_1$. The ratios represent the well known Hill function for $K_m=1$ and $\beta$ and $\gamma$ are cooperativity coefficients. To analyse the model we first observe the nullclines by setting $\dot{x}=\dot{y}=0$, so we get \begin{align} u=\frac{\alpha_1}{1+v^{\beta}}\\ v=\frac{\alpha_2}{1+u^{\gamma}} \end{align} To see how the nullclines change throughout the four-dimensional parameterspace play with the paremeters $\alpha_1,\alpha_2,\beta, \gamma$. You can use Mathematica with the Manipulate command.

$u[\text{a1$\_$},\text{b$\_$}]\text{:=}\text{a1}/(1+v{}^{\wedge}b)$
$\text{uv}[\text{a2$\_$},\text{g$\_$}]\text{:=}((\text{a2}-v)/v){}^{\wedge}(1/g)$
$\text{Manipulate}[\text{Plot}[\{u[c,e],\text{uv}[d,f]\},\{v,0,10\},\text{PlotRange}\to \{\{0,10\},\{0,10\}\},\text{AxesLabel}\to \{v,u\},\text{AspectRatio}\to \text{Automatic}, \text{BaseStyle}\text{-$>$}\{\text{FontWeight}\text{-$>$}\text{Bold},\text{FontSize}\to 20\},\text{Ticks}\to \text{None}],\{c,1,10\},\{d,1,10\},\{e,2,10\},\{f,2,10\}]$

This plots show nullclines and phaseportaits, which allow to study the qualitative behaviour throughout changes of the parameters. plots on the left: $\alpha_1=\alpha_2$,$\beta=\gamma=2$; plots in the middle: $\alpha_1/\alpha_2=4/10$, $\beta=6$ and $\gamma=2$; plots on the right $\alpha_1/\alpha_2=3/10$, $\beta=6$ and $\gamma=2$

For equal parameters $\alpha_1=\alpha_2$ the phaseportrait as well as the nullclines are symmetric accross the median, there exists an unstable fixed point that pushes the trajectory to one of the stable fixed points similar to the Competitive Exclusion Model problem. Also analogous to the Competitive Exclusion Model problem is the stable manifold going through the unstable fixed point, separating two basins of attraction, one for the 'high' fixed point and one for the 'low' fixed point. As $\beta$ is increased the blue nullcline becomes more step-like (in fact for $\beta \rightarrow \infty$ the nullcline becomes a stepfunction). Making $\alpha_1$ smaller and smaller the fixed points coalesce to a semistable fixed point and finally disappear, leaving only the fixed point we called the 'low' state in the Biology summary.

This results explain, why the pIKE105 strain does not exhibit bistable behaviour. Since the strength of the pIKE105 RBS is too weak, $\alpha_1$ is so small that the nullclines don't intersect, except at the 'low' state like displayed in the rightmost nullcline picture above.