Conceived and designed the experiments: MPC. Performed the experiments: IC MAR MPC. Analyzed the data: lm DAWB IC. Contributed reagents/materials/analysis tools: LM DAWB. Wrote the paper: LM SS DDB MdB. Critically revised the manuscript: LM DAWB SS DDB MdB.
The authors have declared that no competing interests exist.
Systems and Synthetic Biology use computational models of biological pathways in order to study
The emerging field of Synthetic Biology aims at constructing novel biological circuits in the cell
Comparison between the topology of the actual version of the network (A) and the reengineered topologies (B)–(E); in all the cases we only consider the galactose growing condition. A thicker line corresponds to an increase of the strength of the corresponding interaction; the strength is intended in terms of MichaelisMenten coefficient and/or Hill coefficient and/or maximal transcriptional velocity. The parameters in red are the ones that we are varying from the nominal value. (A) Topology of IRMA. In galactose growing conditions, the topology consists of one delayed positive and one negative feedback loop, since the proteinprotein interaction between Gal4 and Gal80 is switched off. (B) Reengineering of IRMA in order to turn it into an autonomous oscillator, Scenario 1. Tuning the parameters v_{2}, k_{6}, h_{2} and h_{6} we increase the strength of the following interactions: Cbf1 on Gal4, Swi5 on Ash1 and Ash1 on Cbf1. Both the original positive and the negative feedback lops are present. (C) Reengineering of IRMA in order to turn it into an autonomous oscillator, Scenario 2. Tuning the parameters v_{2}, k_{1}, k_{2}, k_{6}, h_{3} and h_{6} we increase the strength of the following interactions: Cbf1 on Gal4, Swi5 on Ash1 and Ash1 on Cbf1. The original positive feedback loop is removed. (D) Reengineering of IRMA in order to turn it into an autonomous oscillator, Scenario 3. The topology is identical to the one in Scenario 2 with the addition of a positive autofeedbackloop on Swi5. The tuned parameters are: v_{2}, k_{1}, k_{2}, k_{6}, h_{3} and h_{6}. (E) Reengineering of IRMA in order to turn it into a bistable switch, Scenario 4. Properly tuning the parameters v_{2}, k_{1}, k_{2}, h_{1} and h_{3} we increase the strength of the following interactions: Cbf1 on Gal4, Swi5 on Cbf1. The negative feedback loop is removed.
The main aim of this paper is to show how to use novel tools from numerical bifurcation theory (e.g. DDEBIFTOOL
In particular, the aim is to understand if and how IRMA can be turned into a robust and tunable synthetic oscillator or a bistable switch. Oscillations have a crucial role in cell behaviour: the circadian clock and the cell cycle are common examples
In the case of IRMA, the goal is challenging, both in terms of the mathematical analysis and in terms of the
In addition, we discovered that by reducing the topology of the network to a single positive feedback loop, IRMA can be turned into a bistable system (a “toggle switch”, that toggles between two discrete, alternative stable steady states). Hysteretic examples have been observed in several natural examples, including the control of lactose utilization in
In
With the aim of tuning the dynamics of IRMA and turning it into an autonomous biochemical oscillator, we shall seek to achieve the desired dynamic behaviour by appropriately varying the model parameters. In so doing it is obviously fundamental both to remain inside the physically feasible range and to minimize the number of changes to the existing network topology and nominal parameter values, in order to speed up the experimental implementation.
In our specific case, the number of physical parameters is quite high (33), thus an exhaustive exploration of the parameter space would be excessively complicated and time consuming. On the other hand, from the analytical view point it is cumbersome to get any results about the structural stability of equilibria under parameters variations since the system is timedelayed and highly nonlinear, due to the large value that the Hill coefficients can assume. For the case of our multiparametric delayed gene network, it is then crucial to restrict the number of parameters to be changed to induce sustained oscillations. For the selection of the parameter subset to be used to carry out the bifurcation analysis, we use as guidelines the links between the topology and the occurrence of autonomous oscillations presented in the recent literature
In the analytical studies of simple twocomponents networks modelled by differential equations
For the analysis of the IRMA network, we decide to consider only the galactose growing condition, since in such a condition the network is “switched on” and the genes are significantly expressed. Note that, in such condition the proteinprotein interaction between Gal4 and Gal80 is switched off (see the section
In what follows, we analyse 3 possible reengineering scenarios in order both to compare the oscillator tunability and robustness due to different network topologies and to explore different experimental strategies for their implementation.
By looking at the values of the kinetic parameters estimated from
Parameter  Nominal Value  Scenario 1 (A, B)  Scenarios 2, 3  Scenario 4 (A, B) 
1  1  –  0.0477 

0.035  0.035  0.00035 
–  
0.037  0.037  0.037  0.037  
0.010  0.010  0.010  0.010  
1.884  1.884  1.884  1.884  
1.884  0.0477 
0.0477 
1.884  
0  0  0  0  
1.49 
1.49 
1.49 
1.49 

3 
3 
3 
3 

7.4 
7.4 
7.4 
7.4 

6.1 
6.1 
6.1 
6.1 

0.040  0.040  0.040  0.040  
8.82 
0.026 
0.026 
8.82 

0.020  0.020  0.020  0.020  
0.014  0.014  0.014  0.014  
0.018  0.018  0.018  0.018  
0.022  0.022  0.022  0.022  
0.047  0.047  0.047  0.047  
0.421  0.421  0.421  0.421  
0.098  0.098  0.098  0.098  
0.050  0.050  0.050  0.050  
1  1  –  4 

1  4 
1  –  
1  1  4 
4 

1  1  1  1  
1  4 
4 
1  
4  4  4  4  
0.6  0.6  0.6  0.6  
100  100  100  100 
Then, we evaluate the effect of the nonlinearity of the reaction kinetics generated by the Hill functions on the network behavior. Since the stiffness of such sigmoidal function is determined by the Hill coefficients, which describe the cooperativity of the promoters, we perform our numerical investigation increasing the Hill coefficients
Once oscillations are obtained, a fundamental step in the theoretical analysis is the investigation of the robustness and the tunability of the oscillator. To this aim we use numerical continuation techniques
The limit cycle can be continued on each of the
Furthermore, continuation allows us to investigate the tunability of the oscillator in terms of amplitude and period (
Finally, it is useful to test for the robustness of the oscillator under initial conditions variations. To this aim, we perform a significant number of time simulations (5000) fixing the parameters to the values in
At this point, it is crucial to address the feasibility of reengineering IRMA
In order to increase the maximal transcription velocity
In
In
Expression levels of IRMA genes at different methionine concentrations in glucose (white bars) or in galactose/raffinose (grey bars). The control is the standard complete medium, YEP, which contains 140 mM of methionine. Data represent the 2^{−DCt} (mean of two experiments±Standard Error).
Regarding the changes to the
The last parameter to be tuned is the Hill coefficient
The positive loop in Scenario 1 seems difficult to implement in vivo. Therefore, we consider a second scenario, in
Again, we tune both the strength of the negative loop (by decreasing
Such a scenario can be analyzed in terms of robustness to parameters variations and tunability by using the continuation tool DDEBIFTOOL with no delayed variable. The most relevant continuation results, reported in
Furthermore, through continuation we investigate the tunability of the oscillator, discovering that in Scenario 2, contrary to what found for Scenario 1, it is not possible to tune the amplitude independently of the period. The unique parameter that allows to tune the dynamics of oscillations is
Testing through simulations the network dynamics under varying initial conditions within the range [0 1] [a.u], we observe again that robustness is guaranteed. All the trajectories converge to limit cycles of period 1 (results not shown).
The critical parameters which have to be tuned to implement scenario 2
The tuning of parameters
The topology proposed in Scenario 2 appears feasible for
In Scenario 3, the topology of the network is the same as in Scenario 2 with the addition of an autoactivation reaction on
We can compare the robustness to parameter variations of Scenarios 2 and 3 by continuing the Hopf bifurcation on the same pairs of parameters considered previously. By comparing
For the
As our investigation confirms the flexibility of IRMA, we further explore the possibility of turning the network also into a bistable switch. A bistable system is one that toggles between two discrete, alternative stable steady states, in contrast to a monostable system. In biology, bistability has long been established in control of the cell cycle and other oscillations
In our setting, the idea is to reduce the actual version of the topology to a 3 gene positive feedback loop between the genes
The ODEs model can be analyzed by continuing the steady state on the critical parameters.
For the
Again, we can increase the strength of the activation of the
In this work, using numerical and continuation techniques, we showed how IRMA can be turned into a robust and tunable oscillator, or a bistable genetic switch. The deterministic mathematical model, previously formulated and identified to allow data interpretation and experiment planning, is here analysed to guide the reengineering of the network with predictable functions.
IRMA showed great flexibility. Its topology can be reengineered in a number of ways in order to achieve the desired dynamical behaviour. Of note, all the proposed changes are viable
The major conclusion we can draw from our results is that, aiming at constructing a robust and tunable oscillator, the best option is to include in the topology both a delayed negative feedback loop and a fast positive one. This is the case explicitly analyzed in Scenario 3 that results to be most robust and tunable as compared to Scenario 2, in which the topology of the network is reduced to a single negative feedback loop.
In the context of Synthetic Biology, our model guided reengineering framework can be applied to existing topologies with the aim of turning them into oscillators or switches. We analyzed three topologies for the oscillator case and one for the switch case. A crucial point was to minimize the number of experiments needed to modify the synthetic network. Surely, other possible ways to reengineering IRMA can give rise to other oscillatory, switchlike and maybe more complex dynamical behaviours. Of note, once the best performing scenario has been chosen from our deterministic approach, it will be crucial to resort to stochastic simulations in order to estimate the impact of noise on the network dynamics
The mathematical model we used is made up of of five nonlinear Delay Differential Equations that describe the production rates of the five mRNA concentrations, assuming Hill kinetics and proportionality between protein and mRNA levels. The time delay, describing the delayed activation of the
For this work, the mediumdependent parameters in the equation of
Letting
The model analyzed in
To describe
To describe
To obtain
Numerical simulations were run using Matlab 2008b (The MathWorks). For Scenario 1, we adopted the
All the numerical continuation experiments were performed using DDEBIFTOOL package
Continuation results for Scenario 1 A using DDEBIFTOOL software. (A) Two parameters continuation of the Hopf bifurcation on parameters k_{1} (MichealisMenten coefficient of the
(0.26 MB TIF)
Expression of
(0.47 MB TIF)
Continuation results for Scenario 2. Continuation results for Scenario 2 using DDEBIFTOOL software. (A) Two parameters continuation of the Hopf bifurcation on parameters k_{2} (MichealisMenten coefficient of the
(0.30 MB TIF)
Continuation results for Scenario 3. Continuation results for Scenario 3 using DDEBIFTOOL software. (A) Two parameters continuation of the Hopf bifurcation on parameters k_{2} (MichealisMenten coefficient of the
(0.26 MB TIF)
Continuation results for Scenario 4. (A) Scenario 4 A. One parameter continuation of the steady state on k_{1} (MichealisMenten coefficient of the
(0.27 MB TIF)
L.M. acknowledges Marina Scopano, Mariagrazia Marucci and Jolanda Varley for general support.