SDE simulation is stochastic in nature. To evaluate the convergence and stability of a simulation, we rely on statistical analysis of the stochastic trajectories, e.g., by mean levels and/or their standard deviations. Using the Ornstein–Uhlenbeck process (see Part 6C) as an example, design a simulation experiment to evaluate the convergence of simulation statistics with respect to the computational cost (here, we consider that the computational cost is proportional to the length of the simulation).
Consider a geometric Brownian motion governed by the following stochastic differential equation (SDE):
\[dX = \mu X dt+ \sigma XdW \tag{1}\] , where \(\mu\) and \(\sigma\) are positive constants. Show using SDE simulations with the Euler–Maruyama method that
(a) Mean \(<X> = X_0 e^{\mu t}\)
(b) Variance \(Var(X) = X_0^2 e^{2\mu t} (e^{\sigma^2 t} - 1)\)
In this exercise, we set \(\mu = 0.1\), \(\sigma = 0.2\), and the same initial condition \(X_0 = 1\).
Hint: Simulate multiple time trajectories for the same system using the same parameters and initial condition. The only differences between these simulations are random numbers. From these simulations, compute the mean and variance for different \(t\) values. You will need to perform at least 100 simulations and evaluate the statistics for 10 different time points \(t\) (choose small \(t\)s and short total simulation time to reduce the computational cost). To answer the question, evaluate whether the statistics from the simulations are consistent with the above formula.
We revisit the flip-flop circuit that we have experienced in Part 3H. Now we consider gene expression noise. Write down the corresponding SDEs for the circuit and simulate its stochastic dynamics. What types of dynamics do the circuit exhibit with the presence of noise?
We had multiple discusses of effective potential for non-linear dynamical systems (see Parts 2D and 3G). Over there, we defined the effective potential by the analogy of over-damped dynamics. However, this definition has issues for a high dimensional system. Here, instead, we define a new effective potential by
\[ U(X, Y) = - log(P(X,Y)) \tag{2}\] , where \(P\) is the probability of the states near (\(X\), \(Y\)) in the phase plane.
Use the toggle switch circuit as an example (see Parts 3AB and 3EFG), perform SDE simulations, and compute the effective potentials. Plot the potential curves along each of the nullclines.
We revisit the transition rate calculations in Part 6D. Now we vary the noise levels. Identify the relationship between the transition rates and the noise levels. Check if your calculations are consistent with Arrhenius equation, i.e.,
\[\kappa \propto e^{-\frac{\Delta U}{D}} \tag{3}\]
, where \(D\) represents the noise level.