Stochastic Analysis

Besides the parameter space search, we performed another type of analysis in order to verify that our system is robust and that it is not bistable. We were especially interested in the GFP band, whether it is always present and whether the amount of GFP produced in the cells for different concentrations of Acetaldehyde has large fluctuations. Because gene expression is an intrinsically stochastic process, we performed stochastic simulations to see how our system reacts to noise and how it responds to perturbations.

The following results show that our system is indeed robust and it does not have bifurcations.

Method

For the stochastic simulations we used the [http://magnet.systemsbiology.net/software/Dizzy/ Dizzy] software, to which we gave as an input an SBML file with the description of our system. Moving from deterministic to stochastic simulations we had to change several things in the SBML file. Firstly, we exported the SBML file to a CMDL format.Next, we converted the species values as well as the production rates and repression coefficients from concetrations (in μM) to numbers of molecules. Since our initial system was described in terms of ODEs, we also had to change this by separating the system into distinct reaction channels (i.e. separate degradation from activation/repression) and specifying the reaction rates.

With the new description of the system, we could start the simulations. For analyzing the robustness of the system, we chose 5 strategic Acetaldehyde concentrations (see Figure 1), and for each of them we performed stochastic simulations. We selected the following Acetaldehyde concentrations:

• 0 μM (0 molecules). At this concentration the GFP output should approach 0 μM.
• 25 μM (30110 molecules). At this concentration GFP rises quickly, so it should be very sensitive to noise.
• 46 μM (55402 molecules). This is the concentration that gives the peak GFP value.
• 100 μM (120440 molecules). At this concentration GFP is decreasing, so it should be sensitive to noise.
• 300 μM (361320 molecules). At this concentration the GFP output should again approach 0 μM.

Figure 1: GFP versus Acetaldehyde. The red squares denote the 5 Acetaldehyde concentrations for which we simulated the time course of GFP steady state in stochastic mode.

In order to be sure that we are indeed simulating the time course of the stochastic steady state, and that we are not in the transient regime, we implemented the following approach:

• We ran a deterministic simulation for 5000 minutes (~ 83 hours) to get to the deterministic steady state. We took over the concentration values for the last time point as new initial conditions for the next simulation.
• We ran a stochastic simulation for 10000 minutes (~ 7 days), to make sure that we are somewhere around the “stochastic steady state”. Once again we took over the concentration values for the last time point as new initial conditions for the next simulation.
• We ran a long stochastic simulation for 100000 minutes (~ 70 days) taking samples every minute.

Simulations

Figure 4: Stochastic simulation of GFP steady state for an Acetaldehyde concentration of 46 μM.

Figure 2: Exploring the parameter space of TetR degradation rate	Figure 3: Exploring the parameter space of CI degradation rate
Figure 5: Exploring the parameter space of CI degradation rate	Figure 6: Exploring the parameter space of TetR degradation rate
Figure 7: Exploring the parameter space of CI degradation rate We see in Figure 1 that we obtain a Gaussian distribution of GFP number of molecules at the steady state, with a mean of 23491 molecules and standard deviation of 667 molecules. If we convert this value to concentration we get 19.5046 μM, which is very close to what we obtained deterministically. We see that the system keeps fluctuating very closely to its steady state, with no large jumps away from it. The fact that the Gaussian distribution is unimodal tells us that our system is monostable. From this analysis we confirmed what we concluded before and that is the robustnes of our system, especially of the GFP band. We can now be sure that for a certain acetaldehyde concentration once the band appears, it won't start fading away.

Analysis of Results

Insert overview of page here