1. Construction of a genetic toggle
switch in Escherichia coli
2014/15
School of Mathematical Sciences
University of Nottingham
Anna Papanicola, Giulia Del Panta, Safyan Khan,
Philippa Jordan
I have read and understood the School and University guidelines on plagiarism. I confirm
that this work is my own, apart from the acknowledged references.

2. 1 Introduction
This paper presents the construction of a genetic toggle switch in Escherichia Coli and
provides a simple theory that predicts the conditions needed to achieve bistability. Es-
cherichia Coli (E. Coli) is a gram negative, facultatively anaerobic, rod shaped bacterium
that is commonly found in the lower intestine of warm blooded organisms. Its role is very
important in biological engineering and industrial microbiology due to the fact that it has
a long history of laboratory culture and is easy to manipulate. The toggle is constructed
from two repressible Promoters arranged in a mutually inhibitory network and is flipped
between stable states using transient chemical or thermal induction. Bistability is a very
important behaviour of biological systems. It appears in a wide range of systems; from
the λ phage switch in bacteria to cellular signal transduction pathways in mammalian
cells. This experiment was conducted because certain properties, such as multistability
and oscillations, had not yet been demonstrated in networks of non-specialized regulatory
components but had been observed in specialized gene circuits. Such specialized gene
circuits are the bacteriophage λ switch and the Cyanobacteria circadian oscillator.
It has been shown, using a reconstituted enzyme system, that non-linear mathematics
can predict qualitative behaviours of biochemical reaction networks. A practical theory of
enzyme regulatory networks has been developed much more than that of gene regulatory
networks. Many physical and mathematical approaches have been used in the past but,
due to the difficulty of testing predictions, theories have not been verified using experi-
ments. In this paper, a synthetic and bistable gene circuit has been constructed with the
help of a simple mathematical model. This report begins with a description of the toggle
switch design followed by a description of the experiment and its results. After this, the
mathematical model and the analysis of the experiment is given.
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3. 2 The Biological Model
Figure 1 Toggle Switch: Repressor 1 inhibits transcription from Promoter 1 and is in-
duced by Inducer 1 while Repressor 2 inhibits transcription from Promoter 2 and is
induced by Inducer 2.
A toggle switch is a device that moves between two states. Such a system is charac-
terized as bistable, which means that the two states are persistent. This is achieved in
a user-controlled manner. In this particular paper, the toggle switch is composed of two
Repressors and two constitutive Promoters. Each Promoter is inhibited by the Repressor
that is transcribed by the opposing Promoter. Two inducers are included which allow
the system to switch between the two states. The design also has a reporter gene which
is used to observe the system’s activity. This particular design was selected because it
requires the fewest genes and cis-regulatory elements to achieve robust bistable behaviour.
Robust behaviour in this context means that the toggle switch exhibits bistability over a
wide range of parameter values and also will not flip randomly between states. Special-
ized Promoters have not been used as bistability is achievable with any set of Promoters
and Repressors. Having said this, the Promoters and Repressors must satisfy certain
conditions. To be more specific, the bistability of the toggle switch is observed due to
the mutually inhibitory arrangement of the Repressor genes. If the two inducers were not
included in the model, two steady states would be possible. The first one arises when
Promoter 1 transcribes Repressor 2 and the second when Promoter 2 transcribes Repres-
sor 1. The switching occurs by transiently introducing an inducer of the currently active
Repressor.
In the experiment used within the paper, two types of toggle switch were created
(pTAK and pIKE) in E.coli strains, which differed in the second Promoter-Repressor pair
(P1/R1). Both types included the Promoter-Repressor pair of the Prtc-2 Promoter in
conjunction with the Lac Repressor. pTAK can be switched between states by either a
pulse of isopropyl-beta-D-thiogalactopyranoside (IPTG), or a thermal pulse, as the second
2

4. Repressor is temperature sensitive. The second type of plasmids can be switched between
states by a pulse of IPTG or a pulse of anhydrotetracycline (aTc).
The high state is defined as when P2 (Prtc-2) is transcribed/P1 repressed, and the
opposing state (low state) when P1 is transcribed/P2 repressed. When the switch is in a
stable position, addition of the inducer for the repressed Repressor causes transcription
of the Repressor, until the originally active Repressor is repressed. In this manner the
inducers allow the toggle to switch between the two stable states.
All toggles used the gfpmut3 gene, which was induced when the toggle was in the high
state, expressing a green fluorescent colour.
For example, the pTAK plasmids were grown for 6 hours with IPTG, thus repressing
P1, entering the high state and inducing gfpmut3. Cells were then grown for further time
without the presence of IPTG, and continued to remain in the same state. When grown at
a higher temperature, P1 was transcribed and Prtc-2 repressed, thus the toggles switched
to the low state, and gfpmut3 expression was stopped. The pIKE plasmids again induced
gfpmut3 expression when grown with IPTG, but were switched back to the low state
after been grown with aTc, rather than at a high temperature, due to the difference in
the second Repressor.
The experiment uses 6 variants of the toggle switch, 4 pTAK and 2 pIKE plasmids. The
plasmids within each class were varied by inserting RBS sequences of different strengths
into the RBS1 site (see Figure 2). All plasmids demonstrated bistability except for one
of the pIKE plasmids (as shown in Figure 3). This was concluded as being most likely
due to the reduced efficiency of the Tet Repressor (R1 within pIKE plasmids), and could
be fixed by reducing the strength of the P1 Promoter.
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5. Figure 2 Toggle switch plasmid: The diagram shows the layout of the toggle switch
plasmid. The purple outlined boxes represent ribosome binding sites. Switching these
binding sites is what changes the rate of synthesis for the Repressors (changes the values
of α1 and α2 within the mathematical model). T1T2 are terminators of the system. The
experiment uses different P1 Promoters, R1 Repressors and RBS1 binding sites for differ-
ent toggle switches, for example the differences between the pTAK and pIKE plasmids,
as explained previously. The diagram shows in more detail the placement of gfpmut3
within the system, demonstrating why it is only activated when the Prtc-2 Promoter is
transcribed.
Figure 3 The demonstration of bistability: The grey shaded areas correspond to the
periods of chemical or thermal induction. The lines are approximations of the switching
dynamics. All four pTak plasmids exhibited bistability, whereas only one pIKE plasmid
(pIKE107) exhibited bistability. Figure 3a corresponds to the pTAK toggle plasmids
and controls. Figure 3b corresponds to the pIKE plasmids and controls. (This figure
was taken from the original paper: Construction of a genetic toggle switch in Escherichia
Coli.)
4

6. 3 The Mathematical Model
The toggle switch and the conditions required for bistability can be understood using the
following dimensionless model for the network, which is composed of coupled ordinary
differential equations (ODEs):
du
dt
=
α1
1 + vβ
− u
dv
dt
=
α2
1 + uγ
− v
where u and v are the concentrations of Repressor 1 and Repressor 2 respectively, α1
and α2 are the effective rates of synthesis of Repressor 1 and Repressor 2 respectively,
β is the cooperativity of repression of Promoter 2 and γ is the cooperativity of repres-
sion of Promoter 1.The model is derived from a biochemical rate equation formulation
of gene expression. The first term in each equation represents the cooperative repression
of constitutively transcribed Promoters and the second term in each equation the degra-
dation/dilution of the Repressors. The parameters α1 and α2 describe the total effect
of RNA polymerase binding, open-complex formation, transcript elongation, transcript
termination, Repressor binding, ribosome binding and polypeptide elongation. β and γ,
which describe cooperativity, can appear from the multimerization of the Repressor pro-
teins and the cooperative binding of Repressor multimers to multiple operator sites in the
Promoter.
The nullclines of the model are calculated by setting du
dt
and dv
dt
equal to zero:
α1
1 + vβ
− u = 0
α2
1 + uγ
− v = 0
This gives:
u =
α1
1 + vβ
, v =
α2
1 + uγ
5

7. a b c
Figure 4
Figure 4a: α1 = 7, α2 = 10, β = γ = 1.5
Figure 4b: α1 = 10, α2 = 10, β = γ = 1.5
Figure 4c: α1 = 13, α2 = 10, β = γ = 1.5
In these graphs, the nullclines were plotted for three different values of α1, with con-
stant values for β, γ and α2. If α1 = 7, the nullclines intersect only once. Since the
concentration of u is lower than that of v, we would believe that at the steady state the
amount of u would be lower than that of v as well which is confirmed in Figure 4a. If α1
and α2 are balanced there are three steady states, as depicted in Figure 4b. When α1 is
greater than α2 there is only one steady state, which is shown in Figure 4c. Hence, we
conclude that the values of α1 and α2 have to be balanced in order for there to be three
steady states.
a b
Figure 5 Bifurcation diagram: The two figures are bifurcation diagrams where β was
used as the bifurcation parameter. The figures depict how the value of u of the steady
state changes when varying β. The red dots represent the stable steady states while the
blue line represents the unstable steady states.
Figure 5a shows that there is a saddle node bifurcation for β = 1.3. This means
that for β greater than 1.3 there exist three steady states. Two stable ones, which can
be seen in Figure 5a, and an unstable one, which is between the two steady states. The
6

8. unstable steady state is not a long term state of the model since almost every trajectory
diverges away from it. Very few initial conditions would lead to the unstable steady state
and since the initial conditions are randomly generated, it is highly unlikely that this will
occur. This is why the blue line was drawn in to represent the unstable steady states.
In Figure 5b γ is set to one. It can be seen from the graph that no matter what
value β is, there will always be just one steady state.
The bifurcation diagram also shows how robust the system is, i.e. how much the
model is affected by random fluctuations in the model. The closer the system gets to the
bifurcation point, the less robust the system is. Thus, increasing β and γ increases the
tolerance of the fluctuations of the steady states.
Hence, by analysing the bifurcation diagram, an important aspect of the model was
identified: at least one of the two parameters β and γ has to be greater than 1.3 in order
for there to be two stable steady states.
The Jacobian can be used to evaluate the stability of the steady states depending on
our choices for the parameters. The general form of the Jacobian is the following:
J =




−1
−α1βv(β−1)
(1 + vβ)2
−α2γu(γ−1)
(1 + uγ)2
−1




This is then calculated for specific values of α1, α2, β and γ.
For the remainder of the analysis we will be using α1 = α2 = 10, β = γ = 4 as these
values give the three steady states necessary for the toggle switch.
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9. Figure 6
The intersections of the nullclines represent the steady states of the model. We can
see three steady states in the graph. For our chosen parameter values these steady states
are calculated as (0.001,10), (1.533,1.533) and (10, 0.001). The values of 10 are indicative
of our choice of 10 for α1 and α2, and this would vary accordingly when α1 and α2 are
varied.
The stability of these three steady states is determined by substituting the steady
state values into the Jacobian matrix, and calculating the corresponding eigenvalues.
For our chosen parameter values:
J =




−1
−40v3
(1 + v4)2
−40u3
(1 + u4)2
−1




By evaluating the Jacobian matrix at each of the steady states, we can discover the
stability of each state.
J(0.001,10) =
−1 −0.0004
0 −1
This yields eigenvalues of -1 and -1, showing a stable node. In relevance to the toggle
switch, this state is known as the high state.
J(1.553,1.553) =
−1 −3.3869
−3.3869 −1
This yields eigenvalues of -4.3860 and 2.3869, showing a saddle point.
8

10. J(10,0.001) =
−1 0
−0.0004 −1
This yields eigenvalues of -1 and -1, showing again a stable node. In relevance to the
toggle switch, this state is known as the low state.
There are two steady states which attract neighbouring trajectories, whereas the un-
stable steady state, which is between the two stable steady states on the diagonal (sepa-
ratrix), does not.
Figure 7 Phase Portrait:
This graph depicts the phase portrait of the model. The graph shows how the con-
centrations of u and v, starting at a random initial point, converge to a steady state.
Since α1 and α2 are balanced and also β and γ are balanced, the phase portrait displays
a symmetric behaviour. Every trajectory converges to a steady state, and the basins of
attraction represent the areas where the trajectories converge to the same steady state.
As the direction arrows show, there are two basins of attraction which are divided by the
separatrix. As a result, if the initial conditions are above the separatrix, the trajectory
corresponding to these initial conditions converges to the stable steady state in the upper
basin. On the other hand, if the initial conditions are below the separatrix, the trajectory
converges to the stable steady state in the lower basin. The only way the trajectory will
converge to the unstable steady state (saddle point) is if the initial conditions lie on the
separatrix.
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11. 4 Conclusions
The genetic toggle switch that was constructed represents a very important departure
from conventional genetic engineering. This is because this design mainly relied on the
manipulation of network architecture instead of on the engineering of proteins and other
regulatory elements. The toggle theory and the experiment that was carried out seem
to be in agreement. This suggests that theoretical design of complex and practical gene
networks is a realistic and achievable goal. The toggle switch, as a practical device, may
find applications in gene therapy and biotechnology.
The Repressors and Promoters within the toggle switch need to be selected carefully
in order for the values of α1, α2, β, and γ to be the correct values. When α1 and α2 are
balanced, with either β and γ at a value of over 1.3, the model will have the bistability
necessary for the toggle switch to function as intended.
5 Appendix
Included are MATLAB script files used for plotting the nullclines, phase plane, and bi-
furcation plots.
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14. References
[1] Timothy S. Gardner, Charles R.Cantor and James J.Collins, “Construction of a
genetic toggle swith in Escherichia coli”, Nature 403, 339–342 (2000).
[2] Enakshi Guru, Sanghamitra Chatterjee, “Study of Synthetic Biomolecular Network
in Escherichia Coli”, International Journal of Biophysics 3(1), 38–50 (2013).
[3] Tianhai Tian and Kevin Burrage, “Stochastic Models for Regulatory Networks of
the Genetic Toggle Switch”, PNAS 103), 8372–8377 (2006).
[4] Brian Ingalls, “Mathematical Modelling in Systems Biology: An Introduction”,
(2012), URL : http://www.math.uwaterloo.ca/ bingalls/MMSB/Notes.pdf
(cited on 30 November 2014).
[5] Website http://www.bio-physics.at/wiki/index.php?title=Genetic Switch,
visited 23 November 2014.
[6] Website http://http://2013.igem.org/Team:Duke/Modeling/Kinetic Model,
visited 23 November 2014.
[7] Website http://en.wikipedia.org/wiki/Escherichia coli, visited 25 November
2014.
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