First-passage time

Figure C2.5.9. Examples of folding trajectories iT=T derived from the condition = 0.21. (a) Fast folding trajectory as monitored by y/t). It can be seen that sequence reaches the native state very rapidly in a two-state manner without being trapped in intennediates. The first passage time for this trajectory is 277 912 MCS. (b) Slow folding trajectory for the same sequence. The sequence becomes trapped in several intennediate states with large y en route to the native state. The first passage time is 11 442 793 MCS. Notice that the time scales in both panels are dramatically different.

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. 

The first approach to obtain exact time characteristics of Markov processes with nonlinear drift coefficients was proposed in 1933 by Pontryagin, Andronov, and Vitt [19]. This approach allows one to obtain exact values of moments of the first passage time for arbitrary time constant potentials and arbitrary noise intensity moreover, the diffusion coefficient may be nonlinear function of coordinate. The only disadvantage of this method is that it requires an artificial introducing of absorbing boundaries, which change the process of diffusion in real smooth potentials. 

One can obtain an exact analytic solution to the first Pontryagin equation only in a few simple cases. That is why in practice one is restricted by the calculation of moments of the first passage time of absorbing boundaries, and, in particular, by the mean and the variance of the first passage time. 

If the probability density wj(t,x0) of the first passage time of boundaries c and d exists, then by the definition [18] we obtain... 

Moments of the first passage time may be expressed from the probability density wt(U xo) as... 

Equation (4.10) allows to find one-dimensional moments of the first passage time. For this purpose let us use the well-known representation of the characteristic function as the set of moments ... 

Equations (4.11) allow us to sequentially find moments of the first passage time for n — 1,2,3,... (7o = 1). These equations should be solved at the corresponding boundary conditions, and by physical implication all moments Tn(c,xo,d) must have nonnegative values, Tn(c,xo,d) > 0. 

For the case when both boundaries are absorbing, the required moments of the first passage time have more complicated form [18]. 

When the initial probability distribution is not a delta function, but some arbitrary function Wo(xo) where xq C (c,d), then it is possible to calculate moments of the first passage time, averaged over initial probability distribution ... 

As discussed in the previous section, the first passage time approach requires an artificial introduction of absorbing boundaries therefore, the steady-state... 

Moments of Transition Time. Consider the probability Q(t, xo) of a Brownian particle, located at the point xo within the interval (c, d), to be at the time t > 0 outside of the considered interval. We can decompose this probability to the set of moments. On the other hand, if we know all moments, we can in some cases construct a probability as the set of moments. Thus, analogically to moments of the first passage time we can introduce moments of transition time i9 (c,xo, d) taking into account that the set of transition events may be not complete, that is, lim Q(t,xo) < 1 ... 

It is easy to check that the normalization condition is satisfied at such a definition, wT(t.xoj dt = 1. The condition of nonnegativity of the probability density wx(f,x0) > 0 is, actually, the monotonic condition of the probability Q(t, xq). In the case where c and d are absorbing boundaries the probability density of transition time coincides with the probability density of the first passage time wT(t,x0y. 

Finally, for additional support of the correctness and practical usefulness of the above-presented definition of moments of transition time, we would like to mention the duality of MTT and MFPT. If one considers the symmetric potential, such that (—oo) = <1>( I oo) = +oo, and obtains moments of transition time over the point of symmetry, one will see that they absolutely coincide with the corresponding moments of the first passage time if the absorbing boundary is located at the point of symmetry as well (this is what we call the principle of conformity [70]). Therefore, it follows that the probability density (5.2) coincides with the probability density of the first passage time wT(f,xo) w-/(t,xo), but one can easily ensure that it is so, solving the FPE numerically. The proof of the principle of conformity is given in the appendix. 

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). 

In order to illustrate how the Floquet representation of the FPE, Eq. (5.44), may be used to calculate first passage times, we first take the Laplace transform of... 

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... 

Probably, a similar procedure was previously used (see Refs. 1 and 93-95) for summation of the set of moments of the first passage time, when exponential distribution of the first passage time probability density was demonstrated for the case of a high potential barrier in comparison with noise intensity. 

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.

The moments of transition time of a dynamical system driven by noise, described by arbitrary potential cp(x) such that cp( oo) = oo, symmetric relatively to some point x = d, with initial delta-shaped distribution, located at the point xo < d [Fig. A 1(a)], coincides with the corresponding moments of the first passage time for the same potential, having an absorbing boundary at the point of symmetry of the original potential profile [Fig. A 1(b)]. 

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