Mean first passage time Master equation

In a realistic simulation, one initiates trajectories from the reactant well, which are thermally distributed and follows the evolution in time of the population. If the phenomenological master equations are correct, then one may readily extract the rate constants from this time evolution. This procedure has been implemented successfully for example, in Refs. 93,94. Alternatively, one can compute the mean first passage time for all trajectories initiated at reactants and thus obtain the rate, cf. Ref 95. 

The master equation is solved numerically from which the mean-first passage time is extracted. Analysis of the mean-first passage time indicates that even a moderate increase in the critical number Nc of the beta process leads to entropic slowdown of the dynamics [96]. Furthermore, the fragility of the system is controlled by the ratio of the critical number Nc to total number Ap of beta process [96]. 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

Fig. 11.2. Mean first passage time for C, = 50nM (left) and C(, = 80nM (right) computed from the master equation (solid), the fl expansion (dashed) and the nonlinear Fokker-Planck equation (dotted) for di = 0.13pM,d2 = 3pM,d3 = 0.9434pM, 4 = 0.4133pM, ds = 0.24/iM, 02 = 0.4 = 0.2 (pMs), as = 5 (pMs) , N = 25. The dots in the left panel represent the variance of the Cl expansion. The inset in the right panel shows a blow up of the plot for large IPs concentration.

The master equation and the two Fokker-Planck equations exhibit an increase of the mean first passage time with decreasing IP3 concentration. This increment diverges for lower values of the IP3 concentration. 

Fig. 11.3. Mean first passage time for the master equation (left) and the relative difference r = TyK — Tmb)/Tme of the mean first passage time between van Kampen s method Tyn and the master equation Tme (right) in dependence on the base level for different values of the IP3 concentration I — 0.4/rM (solid), 0.5 /xM (dotted), 0.6 /rM (dashed). Parameter values as in figure 11.2 and as = 1 (/rMs) . ...

---