Communication: Coordinate-dependent diffusivity from single molecule trajectories

Abstract

Single-molecule observations of biomolecular folding are commonly interpreted using the model of one-dimensional diffusion along a reaction coordinate, with a coordinate-independent diffusion coefficient. Recent analysis, however, suggests that more general models are required to account for single-molecule measurements performed with high temporal resolution. Here, we consider one such generalization: a model where the diffusion coefficient can be an arbitrary function of the reaction coordinate. Assuming Brownian dynamics along this coordinate, we derive an exact expression for the coordinate-dependent diffusivity in terms of the splitting probability within an arbitrarily chosen interval and the mean transition path time between the interval boundaries. This formula can be used to estimate the effective diffusion coefficient along a reaction coordinate directly from single-molecule trajectories.

While single-molecule experiments are routinely used to observe large conformational changes in biomolecules (e.g., protein folding), recent studies achieve temporal and spatial resolution that permits observation of finer details of molecular trajectories.^1–7 In single-molecule force spectroscopy, for example, the extension x(t) of a protein or DNA can be monitored with microsecond time resolution, enabling observation of fast conformational fluctuations. The information contained in a trajectory x(t) can be used to develop more accurate dynamical models that go beyond phenomenological theories commonly employed to describe transition rates.

One such phenomenological theory that has been particularly successful in interpreting numerous experimental observations is that of one-dimensional diffusion along a reaction coordinate x in the potential of mean force V(x). In single-molecule studies, the potential of mean force can be directly inferred from the equilibrium probability distribution of x,^8–10

$$p_{eq}\left(x\right)=e^{-\beta V\left(x\right)},$$ (1)

where $\beta ={\left(k_{B}T\right)}^{-1}$, while the value of the coordinate-independent diffusivity D is usually estimated by fitting the transition rates or, more recently, other statistical characteristics of the trajectory x(t) such as the distribution of the transition path time,¹¹ with the corresponding predictions of the one-dimensional diffusion model.

Despite the remarkable success of this model in describing many experimental observations,¹² theoretical studies have highlighted its limitations.^13–17 Recent experimental work has also brought possible shortcomings of this model into spotlight: although its predictions^18,19 seem to fit well the observed distributions of transition path times for biomolecular folding,³ the barrier height estimates obtained from such fits were inconsistent with the measured potentials of mean force and transition rates,³ motivating several theoretical proposals to account for this discrepancy.^20–22

Perhaps the most straightforward improvement of the one-dimensional diffusion model is to allow the diffusion coefficient to be a coordinate-dependent function D(x) (for a justification of the position dependence of the diffusivity, see the discussion in Ref. 23). Estimation of this function from molecular simulations has been the subject of considerable theoretical work^24–31 but has rarely been applied to experimental data (Ref. 32 being the only exception that we know). The purpose of this note is to point out a simple analytical formula for D(x) that can be used to estimate of the coordinate-dependent diffusivity directly from a trajectory x(t) (or a collection of such trajectories). Moreover, this formula highlights the connection between the function D(x) and two key characteristics of stochastic dynamics, the so-called splitting probability³³ (also known as committor probability or P_fold^34–40) and the mean transition path time (also known as mean direct transit time⁴¹ or simply mean transit time). As both of these quantities have already been evaluated from single-molecule trajectories,^3,42 obtaining the function D(x) from the same data should be straightforward (provided that statistical errors are manageable).

Assuming the validity of the one-dimensional diffusion model with a coordinate-dependent diffusivity, we wish to find the function D(x) for $a<x<b$, where the interval boundaries a and b can be chosen at will. We start with the known expression⁴³ for the spitting probability $\mathrm{\Phi }\left(x\to b|a\right)$ that the system, having started from x, will reach the boundary b before reaching a,

$$\mathrm{\Phi }\left(x\to b|a\right)=\frac{\underset{a}{\overset{x}{\int }}{D}^{-1}\left(y\right)e^{\beta V\left(y\right)}dy}{\underset{a}{\overset{b}{\int }}{D}^{-1}\left(y\right)e^{\beta V\left(y\right)}dy}.$$ (2)

The probability of reaching a before reaching b is, obviously, $\mathrm{\Phi }\left(x\to a|b\right)=1-\mathrm{\Phi }\left(x\to b|a\right)$. Differentiating Eq. (2) with respect to x, we obtain

$$\frac{d\mathrm{\Phi }\left(x\to b|a\right)}{dx}=\frac{D^{-1}\left(x\right)e^{\beta V\left(x\right)}}{\underset{a}{\overset{b}{\int }}{D}^{-1}\left(y\right)e^{\beta V\left(y\right)}dy}.$$ (3)

The integral appearing in the denominator of Eq. (3) can be estimated if we consider the mean transition path time from a to b. A transition path from a to b originates from a and ends in b, continuously residing within the interval $a<x\left(t\right)<b$ in between these two events without touching the interval endpoints. We emphasize that the mean transition path time differs from (is shorter than) the mean first passage time from a to b.⁴¹ For one-dimensional diffusion, the mean transition path time can be written in terms of the splitting probability in the form proposed by Szabo,

$$\begin{array}{ccc}\hfill ⟨t_{TP}\left(a\to b\right)⟩& =\hfill & \underset{a}{\overset{b}{\int }}{D}^{-1}\left(y\right)e^{\beta V\left(y\right)}dy\underset{a}{\overset{b}{\int }}dx\mathrm{\Phi }\left(x\to b|a\right)\hfill \\ & & \times \left[1-\mathrm{\Phi }\left(x\to b|a\right)\right]e^{-\beta V\left(x\right)}.\hfill \end{array}$$ (4)

One can find a detailed derivation of this formula in Ref. 44. The mean time $⟨t_{TP}\left(b\to a\right)⟩$ for a transition path that originates from b and ends in a is also given by this expression, since $⟨t_{TP}\left(b\to a\right)⟩$ and $⟨t_{TP}\left(a\to b\right)⟩$ are identical.^45–47 Combining Eqs. (3) and (4) and using Eq. (1), we arrive at the central result of this note,

$$D\left(x\right)=\frac{\underset{a}{\overset{b}{\int }}dz\mathrm{\Phi }\left(z\to b|a\right)\left[1-\mathrm{\Phi }\left(z\to b|a\right)\right]p_{eq}\left(z\right)}{p_{eq}\left(x\right)\left[d\mathrm{\Phi }\left(x\to b|a\right)/dx\right]⟨t_{TP}\left(a\to b\right)⟩}$$ (5)

for any x within the region $a<x<b$.

Equation (5) differs from and is complementary to the formula for D(x) derived by Hinczewski et al.²⁷ Computation of the diffusion coefficient in their approach requires evaluation of the mean round-trip time between two points. The mean round-trip time between arbitrary points $x_1$ and $x_2$ is defined as the sum of the mean first passage times from $x_1$ to $x_2$ and from $x_2$ to $x_1$. Evaluation of these mean times involves considering excursions of the trajectories outside the interval $x_1<x<x_2$. In contrast, Eq. (5) is independent of the dynamics outside the interval $a<x<b$.

Figure 1 illustrates the utility of Eq. (5) with three numerical examples. Two different potentials were considered, the symmetric potential $\beta V\left(x\right)={\left(x^2-1\right)}^2$ (a) and the asymmetric potential $\beta V\left(x\right)={\left(x^2-1\right)}^2-x/2$ (b). Two different functions $D\left(x\right)$ were considered for the symmetric potential [Fig. 1(a)], $D\left(x\right)=1+sin\left(7x\right)exp\left(-3x^2\right)/2$ and $D\left(x\right)=1+e^{-3x^2}/2$ (the latter function being an even function of x). For the second potential, the position dependence of the diffusivity is given by $D\left(x\right)=1+tanh\left(-7x\right)/2-exp\left[-7{\left(x+0.5\right)}^2\right]/2$. The boundaries of the transition region were the same in all cases: $b=-a=1$. The diffusivity dependence on x was reconstructed from a single numerically generated long Brownian trajectory, which contained 7000 - 15 000 transition paths from a to b or b to a. The derivative of the splitting probability was simply estimated using finite differences on an equally spaced grid of 30 - 50 points between a and b.

FIG. 1.

Reconstructing position-dependent diffusivity using Eq. (5). The top of each panel shows the model double well potential used to generate a Brownian dynamics trajectory in each case. The boundaries of the transition region [within which the dependence D(x) was estimated] are shown as dashed lines. The bottom of each panel shows numerically estimated diffusivity values within the transition region (markers) vs the exact diffusivity (solid lines). See text for further details.

To conclude, let us comment on several practical issues arising when Eq. (5) is applied to molecular trajectories. Although splitting probabilities and transition path times are most commonly employed to describe the dynamics of transitions between two basins of attraction (associated with distinct molecular conformations and corresponding to minima of the potential of mean force), no assumption is made here regarding the physical nature of the boundaries a and b or about the shape of the underlying potential of mean force V(x). Equation (5) can be applied to a potential of any shape and to arbitrary interval boundaries a and b. In fact, independence of the estimated D(x) of the interval may be used as a test for the internal consistency of the 1D diffusion model as a description of the system’s dynamics.

As seen from the above numerical examples, rather long trajectories were used, each including $\sim 10^4$ transition paths. Using trajectories that are significantly shorter gives rise to much greater numerical noise in the estimated D(x). Since accurate estimation of the mean transition path time is readily accomplished from $\sim 10^2\text{–}10^3$ transition paths²² and since the numerator of Eq. (5) is a constant, the main source of noise is the numerical differentiation of the splitting probability in Eq. (5). Although it may be possible to reduce the noise by employing a higher order finite difference estimate for the derivative or by using a numerical interpolation/fit of the estimated splitting probability (similarly to the approach of Ref. 27), we anticipate that our method would be most useful for analyzing long experimental trajectories rather than simulation data. In the latter case, long uninterrupted trajectories often come at a premium, and approaches that utilize more efficient sampling of long-time dynamics (such as milestoning²⁸) or employ Bayesian analysis²⁹ may be advantageous.

Finally, we note that experimental kinetic artifacts arising, e.g., from finite time resolution potentially complicate the analysis of experimental data. Having attracted recent attention,^48–50 such artifacts can be accounted for by introducing an additional degree of freedom describing the instrument itself.