The bistable genetic toggle switch

The synthetic genetic toggle switch consists of two mutually repressing transcrip-tion factors. It was first developed by Gardner, Cantor, & Collins (2000), and con-sists of the following ODEs:

𝑑𝑢 𝑑𝑡 = 𝑎

1 + 𝑣 − 𝑢 (3.1)

𝑑𝑣 𝑑𝑡 = 𝑎

1 + 𝑢 − 𝑣, (3.2)

where u is the concentration of repressor 1, v the concentration of repressor 2,𝑎 and𝑎 denote the effective rates of synthesis of repressors 1 and 2 respectively, β is the cooperativity of repression of promoter 1 and γ of repressor 2. This model is capable of bistable behaviour when𝑎 and𝑎 are balanced and when β, γ are >

1 (Gardner, Cantor, & Collins 2000). This model is derived from the biochemical rate

The bistable genetic toggle switch 53

Figure 3.1 Simple toggle switch model using the Shea-Ackers formalism

equations of gene expression for the two promoters in the system. Using the Shea-Ackers formalism (as described in Section 2.2.2.2), these are shown in Figure 3.1.

The model can thus also be described by the following ODEs:

𝑑𝑢

𝑑𝑡 = 𝑎 𝑙

1 + 𝑙 + 𝑘 𝑣 − 𝑢 (3.3)

𝑑𝑣

𝑑𝑡 = 𝑎 𝑙

1 + 𝑙 + 𝑘 𝑢 − 𝑣, (3.4)

The model shown in Equations 3.1-3.2 is the dimensionless version of the model shown in Equations 3.3-3.4. This is constructed by measuring 𝑎 and𝑎 in units of 𝑘 and𝑘 respectively (Phillips et al. 2013) and setting leakiness to zero. In Section 3.3.2 I will use the more realistic Shea-Ackers version of the model to show that it is also capable of bistable behaviour.

3.3.1 The quasi steady state approximation and the genetic toggle switch

In order for a system to be able to be studied mathematically, a number of assump-tions have to be made. The system under consideration has to be reduced to very few equations and parameters in order to make the system solvable. This requires as-sumptions to be made about the system that cannot always be justified, such as the quasi-steady state approximation (QSSA). The QSSA assumes that the binding/un-binding processes are much faster than any other process (Loinger et al. 2007) thus

54 Positive feedback loops can increase the robustness of a genetic toggle switch

the bound intermediate is assumed to always be in steady state. The QSSA assump-tion is met in vitro but often does not hold in vivo. Its misuse can lead to large errors and incorrectly estimated parameters (Pedersen, Bersani, & Bersani 2007).

Equations 3.1-3.2 of the genetic toggle switch can be derived from the full model by using the quasi-steady state approximation (QSSA). In this section I will discuss how Equations 3.1-3.2 can be derived from the full model constructed under the mass action formalism by using the QSSA. Consider the set of reactions given in Table 3.1 representing the genetic toggle switch.

Table 3.1 Toggle switch model reactions under mass action kinetics

Equation Description

𝑔𝑢 ⟶ 𝑔𝑣 + 𝑢

gene expression 𝑔𝑣 ⟶ 𝑔𝑢 + 𝑣

𝛽𝑢−−⇀↽−−_Km^Kdu

u 𝑢

dimerization 𝛾𝑣 −−⇀↽−− 𝑣

𝑔𝑢 + 𝑣 −−⇀↽−− 𝑣 • 𝑔𝑢 repression 𝑔𝑣 + 𝑢 −−⇀↽−− 𝑢 • 𝑔𝑣

𝑢 ⟶ ∅

degradation 𝑣 ⟶ ∅

Using mass action kinetics, this set of reactions gives us the following ODEs:

𝑑𝑢

𝑑𝑡 = 𝑎 𝑔𝑢 − 𝐾𝑑 𝑢 + 𝐾 𝑢 − 𝐷 𝑢 (3.5)

𝑑𝑣

𝑑𝑡 = 𝑎 𝑔𝑣 − 𝐾𝑑 𝑣 + 𝐾 𝑣 − 𝐷 𝑣 (3.6)

𝑑𝑢

𝑑𝑡 = 𝐾𝑑 𝑢 − 𝐾𝑚 𝑢 − 𝐾𝑓 𝑔𝑣𝑢 (3.7)

𝑑𝑣

𝑑𝑡 = 𝐾𝑑 𝑣 − 𝐾𝑚 𝑣 − 𝐾𝑓 𝑔𝑢𝑣 (3.8)

𝑑𝑔𝑢 • 𝑣

𝑑𝑡 = 𝐾𝑓 𝑔𝑣𝑣 − 𝐾𝑟 𝑔𝑢 • 𝑣 (3.9)

𝑑𝑔𝑣 • 𝑢

𝑑𝑡 = 𝐾𝑓 𝑔𝑢𝑢 − 𝐾𝑟 𝑔𝑣 • 𝑢 . (3.10)

The principal quasi steady state assumption being made is that the rate of binding and unbinding of the repressor to the promoter happens very fast. We assume that it

The bistable genetic toggle switch 55 happens so much faster than any other reaction in the system that we can assume that ^• and ^• are constant, i.e. in equilibrium. This assumption is also known as the separation of timescales in transcriptional regulation. In order for

• and ^• to be in equilibrium, we must assume that

𝑑𝑔𝑢 • 𝑣

𝑑𝑡 = 𝐾𝑓 𝑔𝑣𝑣 − 𝐾𝑟 𝑔𝑢 • 𝑣 = 0 (3.11)

𝑑𝑔𝑣 • 𝑢

𝑑𝑡 = 𝐾𝑓 𝑔𝑢𝑢 − 𝐾𝑟 𝑔𝑣 • 𝑢 = 0, (3.12)

therefore,

𝐾𝑓 𝑔𝑣𝑣 − 𝐾𝑟 𝑔𝑢 • 𝑣 = 0 (3.13)

𝐾𝑓 𝑔𝑢𝑢 − 𝐾𝑟 𝑔𝑣 • 𝑢 = 0. (3.14)

Now we have a set of algebraic equations rather than differential equations. Solving for ^• and ^• respectively we get:

𝑔𝑢 • 𝑣 = 𝐾𝑓 𝑔𝑣𝑣

𝐾𝑟 (3.15)

𝑔𝑣 • 𝑢 = 𝐾𝑓 𝑔𝑢𝑢

𝐾𝑟 . (3.16)

This can now be substituted into the set of Equations 3.5-3.10. The second assump-tion that is made in this system is that the rate of formaassump-tion and dissociaassump-tion of the polymerised transcription factor is in steady state. This allows us to solve the system for𝑢in a similar way as shown above and substitute it in Equations 3.5-3.10.

This results in a simplified model, with fewer differential equations and parameters.

3.3.2 Phase space and bifurcation analysis

First, I study the model given in Equations 3.3-3.4 by conducting a bifurcation lysis in order to confirm that it is capable of bistable behaviour. A bifurcation ana-lysis is used to determine the properties of a system in parameter space (Alon 2007).

Here I used the PyDSTool (Clewley 2012), a python package used for the analysis of dynamical systems.

The parameters chosen here for the phase space analysis are within the range suggested by Gardner, Cantor, & Collins (2000). Parameters𝑎 and𝑎 are set to 10, and β, γ set to 2. A vector plot shows that the system has two steady states as shown

56 Positive feedback loops can increase the robustness of a genetic toggle switch

in Figure 3.2. Both states were found to be stable by examining the eigenvalues of the system at each steady state using Mathematica (Mathematica 2016).

I further study the system by conducting a bifurcation analysis, where all para-meters remain constant to the values shown above, and only one parameter (𝑎 ) is varied. The bifurcation diagram, given in Figure 3.2C shows a saddle-node bifurca-tion. We observe that by varying the parameter for the effective rate of synthesis of repressor 1 while all other parameters remain constant, the system is bistable when 7 ≥𝑎 ≤ 17.

0 2 4 6 8 10

U 0

2 4 6 8 10

V

1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5

(A) (B)

(C)

v

Figure 3.2 The Gardner, Cantor, & Collins (2000) toggle switch is capable of bistable behaviour given𝑎 and 𝑎 = 10, and β, γ = 2. (A) The time course of the simulated model using multiple initial conditions for𝑣. (B) The vector plot of the Gardner switch shows there are two stable steady states at (u, v) = (4.791, 0.208) and (u, v) = (0.208, 4.791). (C) A bifurcation diagram shows that the system is bistable when 7 ≥𝑎 ≤ 17.

Designing a simple synthetic switch 57