Mean, first passage time

To unbind from a protein the ligand has to move from a, the minimum of the potential U x), to 6, the maximum of U x). The mean first passage time t F) of such motion is (Izrailev et ah, 1997)... 

Calculation of Mean First Passage Times from Differential Recurrence Relations... 

For the solution of real tasks, depending on the concrete setup of the problem, either the forward or the backward Kolmogorov equation may be used. If the one-dimensional probability density with known initial distribution deserves needs to be determined, then it is natural to use the forward Kolmogorov equation. Contrariwise, if it is necessary to calculate the distribution of the mean first passage time as a function of initial state xo, then one should use the backward Kolmogorov equation. Let us now focus at the time on Eq. (2.6) as much widely used than (2.7) and discuss boundary conditions and methods of solution of this equation. 

We also remark that Eq. (5.44) may be decomposed into separate sets of equations for the odd and even ap(t) which are decoupled from each other. Essentially similar differential recurrence relations for a variety of relaxation problems may be derived as described in Refs. 4, 36, and 73-76, where the frequency response and correlation times were determined exactly using scalar or matrix continued fraction methods. Our purpose now is to demonstrate how such differential recurrence relations may be used to calculate mean first passage times by referring to the particular case of Eq. (5.44). 

The use of the differential recurrence relations to calculate the mean first passage time is based on the observation that if in Eq. (5.48) one ignores the term sY(x, s) (which is tantamount to assuming that the process is quasi-stationary, i.e., all characteristic frequencies associated with it are very small), then one has... 

Figure 10. Evolution of the survival probability for the potential Tf.v) — ax2 - for3 for different values of noise intensity the dashed curve denoted as MFPT (mean first passage time) represents exponential approximation with MFPT substituted into the factor of exponent.

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 standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann distribution of reactants is postulated to persist during a reaction.68 The equilibrium theory first passage time is the TV -> oo limit in Eq. (6), Corrections to it then are to be expected when the second term in this equation is no longer negligible, i.e., when N is not much greater than e — e- )-1. The mean first passage time and rate of activation deviate from their equilibrium value by more than 10% when... 

Using Eq. (8) for P(t), mean first passage times may then be calculated and compared to either the equilibrium result (N- oo) to test the range of validity of this assumption or to experiment. The agreement with experiment is not very good3 4 but then again the model outlined here is only a first step. The obvious refinement is the use of a more realistic... 

One may ask the following question. Suppose the random walker starts out at site m at t = 0 how long does it take him to reach a given site R for the first time This first-passage time is, of course, different for the different realizations of his walk and is therefore a random quantity. Our purpose is to find its probability distribution, and in particular the average or mean first-passage time ]... 

If this is equal to unity, then fR,m(t) is the probability density for the first-passage time at R. In that case the mean first-passage time is [compare (VI.7.5)]... 

Exercise. Write the corresponding equations for the case of a left exit point L < m. Exercise. Solve the first-passage problem for the simple symmetric random walk. Show that any site R is reached with probability nR m= 1, but that the mean first-passage time is infinite. 

Exercise. A Brownian particle obeys the diffusion equation (VIII.3.1) in the interval Lsplitting probabilities nL(X0) and (Xq) as functions of its starting point X0. Also the conditional mean first-passage times. 

The conditional mean first-passage time for those exits that take place between l and / + dl is... 

Secondly consider the mean first-passage time xRytn. We suppose that on the left there is a reflecting boundary L, so that (2.4) holds and R is reached with probability 1. At t = 0 the particle sets out at m. In the next At it jumps to the right with probability gm At or to the left with probability rm At or it remains at m with probability 1 — gm At — rm At. One therefore has (fig. 33) ... 

Our derivation holds for L + 1 m < R — 2. One sees in the same way as before that (2.5) remains valid for m = R — 1 if one puts xRtR = 0. Physically this condition expresses the obvious fact that the mean first-passage time for a particle starting at the boundary R itself is zero. Also, a similar argument applied to m = L gives... 

This is identical with (2.5) because at the reflecting boundary rL = 0. The conclusion is When boundary L is reflecting the mean first-passage time at R is obtained by solving (2.5) for L m R — 1 with the boundary condition xr,r = 0- The solution is given in (2.10). 

Thirdly we take a one-step process with two exits L and R. There are two splitting probabilities nR m and nLtm obeying (2.2) with the appropriate boundary conditions for each. There are also two conditional mean first-passage times xRm, iL m. In order to compute them we introduce the products R,m = nR,m R,m and L,m = L,m L,m- In the same way as before one argues that... 

Exercise. For the random walk, nR m has been computed in (2.3). Find SRftn and subsequently the conditional mean first-passage time ... 

If it happens that L is a reflecting boundary one has in analogy with (2.4) the boundary condition dnR/dy = 0 at L. It then follows trivially that nR(y) = 1 for all y, as expected. In this case one has a mean first-passage time Tfl(y). It obeys... 

When L is not a reflecting boundary there is a conditional mean first-passage time My). It is obtained from the equation510 for the product quantity 3 (y) = nR(y)TR(y) ... 

Exercise. A particle obeys the ordinary diffusion equation in the space between two concentric spheres. Find the splitting probability and the conditional mean first-passage times. 

Exercise. For the Smoluchowski equation in one dimension show that the conditional mean first-passage time r (y) obeys... 

Exercise. For the random walk it is not true that m(0) is finite. Accordingly the mean first-passage time is infinite (as found before). 

Type (ii). The particle has a non-zero probability to reach the wall, but we show that it takes an infinite time to arrive. The conditional mean first-passage time at R — s is, according to (3.11),... 

Exercise. For the same case derive the expressions for the two conditional mean first-passage times tl(X0, V0) and tr X0, V0). 

Exercise. Write the analogous equations for the mean first-passage times through L and R. 

The less ambitious approach in 2 through the adjoint equation aims only at the splitting probabilities and the mean first-passage times. We construct an equation analogous to (2.2) and (3.1). If at time t the value of Y equals x, it will have at t + At another value x with probability At JT(x x), or it has the same value x. The probability 7rK(x) that Y, starting at x, will exit through R obeys therefore the identity... 

Exercise. Find the mean first-passage time at R for the M-equation (7.5) with W given by (7.6). ... 

The mean first-passage time is given by (XII.3.9) replace R with c y with a and take L = — oo ... 

Exercise. For the discrete case show that the escape time over a barrier as given by the mean first-passage time (XII.2.11) may be approximated by... 

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