Peculiarities of the Mean Transition Path Time Dependence on the Barrier Height in Entropy Potentials

Abstract

A transition path is a part of a one-dimensional trajectory of a diffusing particle, which starts from point a and is terminated as soon as it comes to point b for the first time. It is the trajectory’s final segment that leaves point a and goes to point b without returning to point a. The duration of this segment is called transition path time or, alternatively, direct transit time. We study the mean transition path time in monotonically increasing entropy potentials of the narrowing cones in spaces of different dimensions. We find that this time, normalized to its value in the absence of the potential, monotonically increases with the barrier height for the entropy potential of a narrowing two-dimensional cone, is independent of the barrier height for a narrowing three-dimensional cone, and monotonically decreases with the barrier height for narrowing cones in spaces of higher dimensions. Moreover, we show that as the barrier height tends to infinity, the normalized mean transition path time approaches its universal limiting value n/3, where n = 2, 3, 4, … is the space dimension. This is in sharp contrast to the asymptotic behavior of this quantity in the case of a linear potential of mean force, for which it approaches zero in this limit.

Graphical Abstract

INTRODUCTION

This study was inspired by recent progress in single-molecule experiments, which now allows to observe not only jumps between folded and unfolded states of proteins and nucleic acids but also the actual folding trajectories (see recent review articles¹ and references therein for the detailed discussion), that is, the time-resolved transitions between different metastable states. Another example is time-resolved protein capture by the nanopores of ion channels.² Using a naturally occurring protein with charge distribution along its peptide chain reminiscent of that of a diblock copolymer, it was possible to discriminate between events of polymer translocation, its retraction from the nanopore, and its fluctuations between different positions within the nanopore. In these and other examples, the entropic contributions to the free energy profile along the reaction coordinate are recognized to be crucially important, if not determining, factors. Therefore, theoretical treatment of the reaction dynamics in the presence of entropy barriers³ is currently an emerging field. One of the quantities of interest that can be extracted from trajectories measured with high resolution is the mean transition path time,⁴ also referred to as direct transit time.⁵ This mean time is the focus of the present work.

Consider stochastic trajectories of a particle diffusing along a one-dimensional reaction coordinate x that cannot escape to infinity because of a constraining potential U(x). Trajectories start at point a and are terminated as soon as they come to point b for the first time. Each of these trajectories consists of a transition path and a looping segment.⁶ The former is the final part of the trajectory that starts from point a and goes to point b without touching point a. The rest of the trajectory is its looping part formed by loops that start and end at point a, as shown in Figure 1.

Figure 1.

Example of a one-dimensional trajectory of a diffusing particle, showing trajectory separation into the looping segment (red) and the transition path (green). Trajectory starts from point x = a and is terminated as soon as it comes to point x = b for the first time.

Here, we focus on the mean duration of the transition path (trp) denoted by ${\overline{t}}_{\text{trp}}(a\to b)$. In addition, we consider particle trajectories that start from point b and are terminated as soon as they come to point a for the first time. These trajectories also consist of transition paths and looping segments. The former are direct transitions from point b to point a without touching point b, whereas the latter are loops starting and ending at point b. The mean duration of the transition path from b to a is denoted by ${\overline{t}}_{\text{trp}}(b\to a)$. Because of time-reversibility of diffusion trajectories, distributions of the transition path times between points a and b, and, hence their mean values are direction-independent^5,7

$${\overline{t}}_{\text{trp}}(a\to b)={\overline{t}}_{\text{trp}}(b\to a)={\overline{t}}_{\text{trp}}(a,b)$$ (1)

There is an exact quadrature expression for the mean transition path time^4,5,8

$${\overline{t}}_{\text{trp}}(a,b)=\frac{{\int }_a^b\left({\int }_a^x\left({\text{e}}^{\beta U(y)}/D(y)\right)\text{d}y\right)\left({\int }_x^b\left({\text{e}}^{\beta U(y)}/D(y)\right)\text{d}y\right){\text{e}}^{-\beta U(x)}\text{d}x}{{\int }_a^b\left({\text{e}}^{\beta U(y)}/D(y)\right)\text{d}y}$$ (2)

where D(x) is the particle position-dependent diffusivity, β = 1/(k_BT), k_B is the Boltzmann constant, and T is the absolute temperature. When the diffusivity is position-independent, D(x) = D = const, the above expression simplifies and takes the form

$${\overline{t}}_{\text{trp}}(a,b)=\frac{{\int }_a^b\left({\int }_a^x{\text{e}}^{\beta U(y)}\text{d}y\right)\left({\int }_x^b{\text{e}}^{\beta U(y)}\text{d}y\right){\text{e}}^{-\beta U(x)}\text{d}x}{D{\int }_a^b{\text{e}}^{\beta U(y)}\text{d}y}$$ (3)

Performing the integrations, one can find that in the absence of the potential, U(x) = 0, the mean transition path time between two points separated by distance L is

$${\overline{t}}_{\text{trp}}(0,L)=\frac{L^2}{6D}$$ (4)

Note that the mean first-passage time from a reflecting wall to a point separated from the wall by distance L is L²/(2D). This mean time is longer than the mean transition path time (eq 4). The difference between the two mean times, L²/(3D), is the mean looping time of the particle near the reflecting wall.

This paper focuses on the mean transition path time in the entropy potential of mean force, which is a conditional free energy and hence a thermodynamic function. In the analysis of dynamic problems, the entropy potential, as any potential of mean force, should be used with caution because averaging resulting in this potential may fail.⁹ This is especially true in the case of the mean transition path time because here only a minor subset of all diffusive trajectories is involved in the averaging.

We study the mean transition path time between points 0 and L in the presence of a monotonically increasing entropy potential U(x), that is, when the force F(x) = −dU(x)/dx pushes the particle in the negative x-direction, F(x) < 0, back to the origin (Figure 2), so that the particle has to climb a potential barrier to reach x = L. Our focus is on the dependence of ${\overline{t}}_{\text{trp}}(0,L)$ on the barrier height, ΔU = U(L) − U(0), for potentials of different shapes. For the sake of comparison, we begin with the case of a linear potential, U(x) = Fx (Figure 2A), and show that here ${\overline{t}}_{\text{trp}}(0,L)$, scaled by its value in the absence of the potential, given in eq 4, monotonically decreases with the barrier height, approaching zero as the barrier height tends to infinity. Then, we proceed to entropy potentials and consider particle diffusion in narrowing cones in three and two dimensions, as shown schematically in Figure 2B,C. Particle motion in these cases can be described as diffusion in corresponding one-dimensional entropy potentials of mean force, which monotonically increase with the coordinate x measured along the cone axis. We will see that for diffusion in the three-dimensional cone, the mean transition path time scaled by ${\overline{t}}_{\text{trp}}(0,L)$ is independent of the barrier height. In contrast, its two-dimensional counterpart monotonically increases with the barrier height, approaching a finite limiting value as the barrier height tends to infinity. This demonstrates that the dependence of the scaled mean transition path time on the barrier height may change its qualitative behavior as the shape of the potential U(x) changes. Finally, we consider diffusion in high-dimensional spaces and derive an expression giving the scaled mean transition path time for n-dimensional narrowing cones with n = 4, 5, …. We show that, as the barrier height tends to infinity, the asymptotic behavior of the scaled mean transition path times becomes universal and is given by 3/n for cones of all dimensions starting from n = 2. In contrast to the linear potential, the scaled mean transition path time in the entropy potential remains finite as the barrier height tends to infinity for any n, approaching zero as n tends to infinity as 1/n.

Figure 2.

Schematics of three systems considered: (A) one-dimensional diffusion in the presence of a constant force, (B) diffusion in a three-dimensional narrowing cone, and (C) diffusion in a two-dimensional narrowing cone. Particles start from the reflecting boundary at x = 0 and are trapped by the absorbing boundary at x = L. Particle initial distribution over the boundary is uniform.

THEORY

Diffusion in a Linear Potential.

For U(x) = Fx, F > 0, eq 3 leads to¹⁰

$${\overline{t}}_{\text{trp }}(0,L)=\frac{\beta FL-2\text{tanh}(\beta FL/2)}{D{(\beta F)}^2\text{tanh}(\beta FL/2)}$$ (5)

Normalizing this mean transition path time to t₀ = L²/(6D), that is, its value in the absence of the potential (F = 0), and introducing the barrier height ΔU = U(L) − U(0) = FL, we find that the dimensionless mean transition path time, considered as a function of the dimensionless barrier height measured in units of thermal energy, βΔU = βFL, is

$$\frac{{\overline{t}}_{\text{trp}}(0,L)}{t_0}=\frac{12}{{(\beta \Delta U)}^2}\left[\frac{\beta \Delta U/2}{\text{tanh}(\beta \Delta U/2)}-1\right]$$ (6)

This shows that the dimensionless mean transition path time decreases monotonically from unity to zero, as the barrier height increases from zero to infinity. Its asymptotic behavior at low and high barriers is given by

$$\frac{{\overline{t}}_{\text{trp}}(0,L)}{t_0}=\{\begin{array}{ll}1-{(\beta \Delta U)}^2/60,\hfill & \beta \Delta U\to 0\hfill \\ 6/(\beta \Delta U),\hfill & \beta \Delta U\to \infty \hfill \end{array}$$ (7)

Diffusion in a Three-Dimensional Conical Channel.

Consider diffusion in a narrowing three-dimensional cone (Figure 2B) of radius r(x) = a + λ(L − x), 0 < x < L, λ > 0, with the x-coordinate measured along the cone axis, in the presence of a reflecting boundary at the origin, where the cone radius is a + λL. When the rate of the cone radius change λ is smaller than one, λ < 1, particle motion along the channel axis can be described by the generalized Fick–Jacobs equation^9a–h as effective one-dimensional diffusion along the channel axis in the entropy potential of mean force, U(x), given by

$$\begin{gathered} U(x)=-{\beta }^{-1}\text{ln}(A(x)/A(0)) \\ =-2{\beta }^{-1}\text{ln}[1-\lambda x/(a+\lambda L)] \end{gathered}$$ (8)

where A(x) = πr²(x) is the cone cross-section area. Reduction of diffusion in a tube of varying radius, r(x), to effective one-dimensional diffusion in the entropy potential of mean force is accompanied by the replacement of the initial diffusivity by an effective one. The latter is a function of the radius variation rate, dr(x)/dx, and hence is, in general, position dependent. However, in the particular case of the conical channel dr(x)/dx = λ = const, and, therefore, the effective diffusivity is position independent. Various approximate expressions for this diffusivity, denoted by D_λ, are discussed in the references cited above.

Substituting U(x), eq 8, into eq 3 with D replaced by D_λ and performing the integrations, we arrive at^3b

$${\overline{t}}_{\text{trp}}(0,L)=\frac{L^2}{6D_{\lambda }}$$ (9)

This shows that the mean transition path time in the narrowing three-dimensional cone is independent of the height of the entropy potential barrier ΔU = U(L) − U(0) = 2β⁻¹ ln(1 + λL/a). One can see that the mean transition path time in the entropy potential, eq 8, is the same as its counterpart for free diffusion (U(x) = 0) in eq 4, with D replaced by D_λ. Thus, in three dimensions, the conical geometric constraint manifests itself only in the renormalization of the particle diffusivity.

For further comparison, we need a dimensionless mean transition path time, which in this case is independent of the barrier height as its dimensional counterpart. Here, we choose the time in eq 9 as the scaling time with the goal to make the dimensionless mean transition path time equal to unity

$$\frac{{\overline{t}}_{\text{trp}}(0,L)}{t_0}=1$$ (10)

where t₀ = L²/(6D_λ).

Diffusion in a Two-Dimensional Conical Channel.

Next, we consider diffusion in a narrowing two-dimensional channel (Figure 2C) of width w(x) = 2[a + λ(L − x)], L > x > 0, and λ > 0, with the x-coordinate measured along the channel axis, in the presence of a reflecting boundary at the origin, where the channel width is 2(a + λL). When one-half of the rate of the channel width changes, λ, is smaller than one, λ < 1, particle motion along the channel axis can again be described by the generalized Fick−Jacobs equation^{9a–d,f–h,11} as effective one-dimensional diffusion along the channel axis in the entropy potential of mean force, U(x), which in this case is

$$\begin{gathered} U(x)=-{\beta }^{-1}\text{ln}(w(x)/w(0)) \\ =-{\beta }^{-1}\text{ln}[1-\lambda x/(a+\lambda L)] \end{gathered}$$ (11)

This effective diffusion, as in the three-dimensional case, is also characterized by the λ-dependent effective diffusivity, D_λ, which differs from its three-dimensional counterpart. One can see that with the same a and λ, U(x) in eq 8 increases with x twice faster than U(x) in eq 11. It turns out that this surprisingly results in a qualitative change in the mean transition path time dependence on the barrier height.

Substituting U(x), eq 11, into eq 3 with D replaced by D_λ and performing the integrations, we arrive at

$${\overline{t}}_{\text{trp}}(0,L)=\frac{\left(2a^2+2a\lambda L+{\lambda }^2{L}^2\right)\text{ln}(1+\lambda L/a)-\lambda L(2+\lambda L)}{4D_{\lambda }{\lambda }^2\text{ln}(1+\lambda L/a)}$$ (12)

Introducing the barrier height ΔU = U(L) − U(0) = β⁻¹ ln(1 + λL/a) and the time scale t₀ = L²/(6D_λ), which we use to make the mean transition path time in eq 12 dimensionless, we obtain

$$\frac{{\overline{t}}_{\text{trp}}(0,L)}{t_0}=\frac{3\left[\beta \Delta U\left(1+{\text{e}}^{2\beta \Delta U}\right)+1-{\text{e}}^{2\beta \Delta U}\right]}{2\beta \Delta U{\left({\text{e}}^{\beta \Delta U}-1\right)}^2}$$ (13)

One can check that the dimensionless mean transition path time monotonically increases from 1 to 1.5, as the barrier height increases from zero to infinity. The asymptotic behavior of the ratio ${\overline{t}}_{\text{trp}}(0,L)/t_0$ is given by

$$\frac{{\overline{t}}_{\text{trp}}(0,L)}{t_0}=\{\begin{array}{ll}1+{(\beta \Delta U)}^2/60,\hfill & \beta \Delta U\to 0\hfill \\ 1.5(1-1/(\beta \Delta U)),\hfill & \beta \Delta U\to \infty \hfill \end{array}$$ (14)

Comparison.

As shown in Figure 3, we compare the three dimensionless mean transition path times, given in eqs 6, 10, and 13, considered as functions of the dimensionless barrier height, βΔU, obtained for the three potentials corresponding to the three systems, shown in Figure 2. One can see that the slowdown of the increase rate of the potential results in the increase of the dimensionless mean transition path time. Moreover, the functional dependence of the dimensionless mean transition path time on the barrier height may change its qualitative behavior. Although this mean time monotonically decreases with the barrier height for the linear potential, it is independent of the barrier height for the entropy potential in eq 8 (narrowing the three-dimensional cone) and monotonically increases with the barrier height for the entropy potential in eq 11 (narrowing the two-dimensional cone). This qualitative difference in the barrier height dependence of the dimensionless mean transition path time is one of the main results of this work.

Figure 3.

Dimensionless mean transition path times as functions of the dimensionless barrier height. The three curves represent, from bottom to top, the dimensionless mean transition path times in the linear potential (eq 6), in the narrowing three-dimensional cone, eq 10, and the narrowing two-dimensional cone (eq 13.

Numerical Test.

Our analysis is based on general expression for the mean transition path time, eq 3, which is the exact result when the particle diffusive dynamics is one-dimensional, as in the case of the linear potential. However, in the cases of the narrowing three- and two-dimensional conical channels, we use the potentials of mean force, given in eqs 8 and 11, which are of entropic origin.^9i,12 Because these potentials of mean force are thermodynamic functions, they should be used with caution in analyzing the kinetics. This is especially true with respect to the conditional mean first-passage times because they are obtained by averaging over very restricted subsets of diffusion trajectories formed by their transition paths.

In view of these circumstances, to test the accuracy of our theoretical predictions, given in eqs 1, 10, and 13, we determine the mean transition path times for the 0-to-L and L-to-0 transitions in the three- and two-dimensional cones, shown in Figure 2, in Brownian dynamics simulations. Each mean transition path time was obtained by averaging over 25,000 realizations of the particle trajectory. This was performed for cones of several lengths up to L = 20a with λ = 0.4 and λ = 0.6. The mean transition path times obtained from the simulations are used (i) to examine the independence of these times of the transition direction, as claimed in eq 1, and (ii) to check the accuracy of the relations given in eqs 10 and 13.

We found that the identity in eq 1 with a = 0 and b = L is fulfilled with high precision, and that our theoretically predicted mean transition path times are in perfect agreement with those obtained from our simulations. This is shown in Figure 4, where solid curves are the theoretical dimensionless mean transition path times, ${\overline{t}}_{\text{trp}}(0,L)/t_0$, for the narrowing three- and two-dimensional conical channels shown as functions of the dimensionless length, $\tilde{L}=\lambda L/a$, and symbols are the simulation results (see the figure caption for more details). As follows from eqs 8 and 11, the relations between $\tilde{L}$ and the dimensionless barrier heights βΔU for the three- and two-dimensional cones are given by

$$\tilde{L}=\lambda L/a=\{\begin{array}{ll}{\text{e}}^{\beta \Delta U/2}-1,\hfill & \text{in three dimensions}\hfill \\ {\text{e}}^{\beta \Delta U}-1,\hfill & \text{in two dimensions}\hfill \end{array}$$ (15)

This allows us to write the dimensionless mean transition path time, given in 13 as a function of $\tilde{L}$ as

$$\frac{{\overline{t}}_{\text{trp }}(0,L)}{t_0}=\frac{3\left[\left(2+2\tilde{L}+{\tilde{L}}^2\right)\text{ln}(1+\tilde{L})-\tilde{L}(2+\tilde{L})\right]}{2{\tilde{L}}^2\text{ln}(1+\tilde{L})}$$ (16)

which can also be obtained using eq 12.

Figure 4.

Comparison of the theoretically predicted dimensionless mean transition path times (curves) with the simulation results (symbols). The curves are, from bottom to top, the dependences for the narrowing three-dimensional cone, shown in Figure 2B, eq 10, and the narrowing two-dimensional cone, shown in Figure 2C, eq 16, plotted as functions of the dimensionless length $\tilde{L}=\lambda L/a$, which is related to the dimensionless barrier height by eq 15. Different symbols show simulation results for 0-to-L and L-to-0 transitions at λ = 0.4 and λ = 0.6, respectively.

Diffusion in a High-Dimensional Conical Channel.

The question naturally arises what happens with the mean transition path time dependence on the barrier height in the entropy potentials of narrowing cones in spaces of dimensions n > 3. The x-dependence of the entropy potential U_n(x) in this case is given by (cf. eq 8)

$$U_n(x)=-{\beta }^{-1}\text{ln}\frac{{\Sigma }_n(x)}{{\Sigma }_n(0)}=-{\beta }^{-1}(n-1)\text{ln}\frac{a+\lambda (L-x)}{a+\lambda L}$$ (17)

Here, the x-coordinate is measured along the cone axis, Σ_n(x) is the cone cross-section area, Σ_n(x) ∝ r(x)ⁿ⁻¹, where r(x) is the x-dependent cone radius given by r(x) = a + λ(L − x), 0 ≤ x ≤ L. Using this entropy potential, one can derive general expression for the mean transition path time ${\overline{t}}_{\text{trp}}^{(n)}(0,L)$, starting from eq 3 with D replaced by $D_{\lambda }^{(n)}$, where $D_{\lambda }^{(n)}$ is the particle effective diffusivity in the n-dimensional cone. This leads to the following expression for the dimensionless mean transition path time, ${\overline{t}}_{\text{trp}}^{(n)}(0,L)/t_0^{(n)}$, where $t_0^{(n)}=L^2/\left(6D_{\lambda }^{(n)}\right)$ is the scaling time analogous to that given in eq 4

$$\begin{gathered} \frac{{\overline{t}}_{\text{trp}}^{(n)}(0,L)}{t_0^{(n)}}=\frac{6}{(n-2)\gamma \left[1-{(1-\gamma )}^{n-2}\right]}{I}_n, \\ n=3,4,5,\dots \end{gathered}$$ (18)

where

$$I_n=\left[1+{(1-\gamma )}^{n-2}\right]\left(1-\frac{\gamma }{2}\right)-\frac{1}{n\gamma }\left[1-{(1-\gamma )}^n\right]-{(1-\gamma )}^{n-2}{J}_n$$ (19)

and

$$\begin{gathered} J_n={\int }_0^1\frac{\text{d}x}{{(1-\gamma x)}^{n-3}} \\ =\{\begin{array}{ll}1,\hfill & n=3\hfill \\ \frac{1}{\gamma }\text{ln}\frac{1}{1-\gamma },\hfill & n=4\hfill \\ \frac{1}{(n-4)\gamma }\left[\frac{1}{{(1-\gamma )}^{n-4}}-1\right],\hfill & n=5,6,\dots \hfill \end{array} \end{gathered}$$ (20)

Parameter γ in the above expressions is

$$\gamma =\frac{\tilde{L}}{1+\tilde{L}},\tilde{L}=\frac{\lambda L}{a}$$ (21)

This dimensionless parameter, varying between zero and unity, 0 ≤ γ ≤ 1, determines the barrier height of the entropy potential ΔU_n which is defined as

$$\Delta U_n=U_n(L)-U_n(0)={\beta }^{-1}(n-1)\text{ln}(1+\tilde{L})$$ (22)

The relation in eq 21 allows one to express the barrier height in terms of γ

$$\Delta U_n={\beta }^{-1}(n-1)\text{ln}\frac{1}{1-\gamma }$$ (23)

As γ increases from zero to unity, the barrier height increases from zero to infinity. The above expressions show that in high-dimensional spaces (n > 3), the dimensionless mean transition path time monotonically decreases with the barrier height (parameter γ) from unity to its asymptotic value at infinitely high barrier (γ = 1). In contrast to the linear potential, where the mean transition path time vanishes as ΔU → ∞, see eq 7 and Figure 3, in the case of the entropy potential this time remains finite

$$\underset{\Delta U\to \infty }{\text{lim}}\frac{{\overline{t}}_{\text{trp}}^{(n)}(0,L)}{t_0^{(n)}}=\frac{3}{n}$$ (24)

Note that although this expression for the asymptotic value of the dimensionless mean transition path time is obtained for high-dimensional spaces, n = 4, 5, 6, …, in fact, it is universal. Indeed, eq 24 correctly predicts the asymptotic values in two and three dimensions as 3/2 and 1, respectively.

The barrier height dependence of the dimensionless mean transition path time and their asymptotes are shown in Figure 5 for n = 2, 3, 4, 5, and 6. One can see that this dimensionless mean time is independent of the barrier height for n = 3 and monotonically decreases with βΔU in spaces of higher dimensions, whereas in two dimensions this mean time increases with the barrier height. To draw this dependence in two dimensions, we used the result given in eq 12 in which we expressed $\tilde{L}$ in terms of γ. As follows from eq 21, $\tilde{L}=\gamma /(1-\gamma )$ and the dimensionless mean transition path time, eq 12, is

$$\frac{{\overline{t}}_{\text{trp}}^{(2)}(0,L)}{t_0^{(2)}}=\frac{3\left\{\left[1+{(1-\gamma )}^2\right]\text{ln}(1/(1-\gamma ))-\gamma (2-\gamma )\right\}}{2{\gamma }^2\text{ln}(1/(1-\gamma ))}$$ (25)

This can be converted into the dependence on the barrier height using eq 23. Figure 5 shows that the three-dimensional space (n = 3) plays the role of a conditional boundary: the dimensionless mean transition path time decreases with the entropy barrier height in spaces of higher dimensions n = 4, 5, 6, … and increases with the barrier height in two dimensions.

Figure 5.

Dimensionless mean transition path times as functions of the entropy barrier height for narrowing cones in spaces of different dimensions, solid curves (eqs 18–23 and 25) and their asymptotic values, dashed lines (eq 24. Qualitative behavior of the dependences changes from that of monotonically increasing one for n = 2 to that of monotonically decreasing for n ≥ 4, with a special case of n = 3, where the dimensionless mean transition path time is independent of the entropy barrier height.

CONCLUSIONS

Recent progress in single-molecule and single-channel experiments^1,2b makes it possible to study the “fine structure” of the escape trajectories in different biologically relevant systems. Specifically, not only the mean lifetimes of the systems in different long-living metastable states can be measured, but individual transitions between these states are now becoming time resolvable. Such transitions are often described in terms of one-dimensional diffusion along a reaction coordinate, which occurs in the potential of mean force that may contain an entropy contribution. One of the quantities characterizing these transitions is the mean transition path time. This brings about an obvious need to study the mean transition path time in the entropy potential theoretically.

A step in this direction is made in the present study where we derived analytical expressions for the dimensionless mean transition path times of particles diffusing in n-dimensional narrowing cones, n = 2, 3, 4, …. These expressions are obtained using eq 3 in which U(x) is the entropy potential of mean force given in eqs 8, 11, and 17 for n = 3, n = 2, and n = 4, 5, …, respectively. For n = 2 and 3, we compare our analytical results with those obtained from Brownian dynamics simulations, as shown in Figure 4. The comparison shows excellent agreement between the theoretical predictions and their simulation counterparts.

Not surprisingly, the dimensionless mean transition path times are functions of the barrier height, however, it turns out that these functions are qualitatively different. The scaled mean transition path time monotonically increases with the barrier height for diffusion in a two-dimensional cone, is independent of the barrier height for diffusion in a three-dimensional cone, and monotonically decreases with the barrier height in cones of higher dimensions, n = 4, 5, … (see Figure 5). As the barrier height tends to infinity, all these dimensionless mean times approach their finite asymptotic values 3/n, n = 2, 3, 4, …. This is in contrast to the behavior of the dimensionless mean transition path time in the case of the linear potential U(x) = Fx, where it vanishes as the barrier height tends to infinity (see eq 7).

In conclusion, it is necessary to mention that one can think of different ways to vary the height of the entropy barrier, which is proportional to ln(1 + λL/a). Here, we choose to vary the ratio L/a, specifically by varying L at fixed a. This leads to the increase of the distance between the two cone ends and hence results in the increase of the mean transition path time even in the absence of the entropy barrier. With this in mind, we scaled the mean transition path time in the presence of the barrier by the value of this time when the barrier is zero. Alternatively, one can change the entropy barrier by varying the rate of the cone radius change λ, or combining variations of both L/a and λ. In all cases, including those of arbitrarily high entropy barriers, the obtained results hold true as long as the modified Fick–Jacobs equation is applicable. This question was carefully analyzed in our previous studies.⁹ⁱ In particular, Brownian dynamics simulations performed for 3D and 2D cones^9e,11 demonstrated that the requirement of effective averaging that validates the one-dimensional description used here, is fulfilled when λ ≤ 1. The results of our simulations with λ = 0.4 and λ = 0.6, shown in Figure 4, support this claim.