Team:Warwick/Modelling/System

Results
Once the initial models for the experiments had been realised it was clear that some of the dynamics were not being fully captured. Our first ODEs had provided great insight into the way our system operated but the deterministic models did not demonstrate the stability of the system. Simply calculating a trajectory or equilibrium point could easily be very sensitive to initial conditions or changes in parameters. We therefore opted for stochastic DEs. It is a common misconception that the average trajectory of solutions to SDEs will tend towards their ODE counterparts. This will only happen under certain assumptions of stability and so as well as more realistic trajectories, we also got stability analysis.

Next we realised that for any experiment involving RdRp, there would be time when the polymerase was on the mRNA and thus restricting translation. This is particularly relevant to the 3'UTR experiment where the length of the RNA means the polymerase is on for a considerable time. This might be tackled by a new 'combined' RNA and RdRp molecule, however the RdRp is attached to the RNA for a very specific length of time. This lead us to the conclusion that a delay DE would best deal with the dynamics. This is where rate of change is subject not only the current state of the system but also the state of the system in the past. Introducing time delay also brings competition to our model without having to add in probability functions. Our main issue was of combining SDEs and DDEs into one model. Matlab has built in DDE solvers, and there is an well known library that will deal with SDEs. However we could find no m files which would deal with both. So we created it. Our solver runs a modified 4th order Runge-Kutta method which is more accurate than the current tools for SDEs. The dynamics offered by stochastic time delay are much richer than ODEs. These files are available for download .

Our 3'UTR experiment formulated with time delay now looked like; \[ \frac{dR_{+}}{dt} = \tau(t) - \alpha R_{+}(t)E(t) + \alpha R_{+}(t-s)E(t-s) - \mu_{R} R_{+}(t)\] \[ \frac{d[R_{+}E]}{dt} = \alpha R_{+}(t)E(t) - \alpha R_{+}(t-s)E(t-s) \] \[ \frac{dR_{-}}{dt} = \alpha R_{+}(t-s)E(t-s) - \mu_{R_{-}} R(t) \] \[ \frac{dE}{dt} = \beta_{1}R_{+}(t) - \alpha R_{+}(t)E(t) + \alpha R_{+}(t-s)E(t-s) - \mu_{E} E(t) \] \[ \frac{dG}{dt} = \beta_{2}R_{-}(t) - \mu_{G} G(t) \] where the time delays corresponds to the time RdRp takes to copy a strand of RNA.

We then took these equations and adding Weiner noise were able to study how our system behaved in a much more realistic setting. Putting this into our solvers and increasing the number of repetitions to build a more accurate expected value, we found the stochastic model followed the deterministic predictions as one would expect when conducting more repeats; as is the case with experimental data in that the more repeats conducted the smaller the error, as illustrated by the final figure.

The above figures show GFP Output (vs.Time) over 1, 10 and 100 repeats respectively.

The figure above shows a comparison of GFP Output (vs. Time) as given by the deterministic system of equations (in red) vs. that produced from the stochastic system averaged over 100 repeats (in blue).