Bayesian Uncertainty Quantification for Bond Energies and Mobilities Using Path Integral Analysis

Abstract

Dynamic single-molecule force spectroscopy is often used to distort bonds. The resulting responses, in the form of rupture forces, work applied, and trajectories of displacements, are used to reconstruct bond potentials. Such approaches often rely on simple parameterizations of one-dimensional bond potentials, assumptions on equilibrium starting states, and/or large amounts of trajectory data. Parametric approaches typically fail at inferring complicated bond potentials with multiple minima, while piecewise estimation may not guarantee smooth results with the appropriate behavior at large distances. Existing techniques, particularly those based on work theorems, also do not address spatial variations in the diffusivity that may arise from spatially inhomogeneous coupling to other degrees of freedom in the macromolecule. To address these challenges, we develop a comprehensive empirical Bayesian approach that incorporates data and regularization terms directly into a path integral. All experimental and statistical parameters in our method are estimated directly from the data. Upon testing our method on simulated data, our regularized approach requires less data and allows simultaneous inference of both complex bond potentials and diffusivity profiles. Crucially, we show that the accuracy of the reconstructed bond potential is sensitive to the spatially varying diffusivity and accurate reconstruction can be expected only when both are simultaneously inferred. Moreover, after providing a means for self-consistently choosing regularization parameters from data, we derive posterior probability distributions, allowing for uncertainty quantification.

Figure 1

Dynamic force spectroscopy (DFS) setup and measurement. (a) Schematic of a DFS pulling experiment. A pulling device with spring constant K and reference control position L(t) is attached to one end of a bond. As the device is lifted, it deflects by amount d, but also stretches the observed bond coordinate ξ, which is a measurement of the underlying true bond coordinate x. (b) Schematic of trajectories for L(t), d(t), and ξ(t) ≡ L(t) − d(t). In reconstructions based on rupture forces, the maximum value d_max determines the force at rupture, indicated by the sharp increase in ξ(t). To see this figure in color, go online.

Figure 2

Trajectory data. Simulations using bond force and diffusivity given by (a) Eqs. S1 and S2 in the Supporting Material and (b) Eq. S3 in the Supporting Material. (Solid) Three individual simulated trajectories (out of 10³). Each trajectory represented a different pulling experiment of duration 5 s, sampled at 10 kHz, with V = 20, K = 0.15. (Shaded region) Compactly supported area; it represents the intensity of all 10³ trajectories through each space-time point. While these trajectories are rather featureless, the histogram of positions observed across all trajectories (up to time 5 s) is shown on the right and contains more features. Each point in the histogram represents a single instance in which a position is sampled. Thus, each trajectory can sample a specific position many times. The total number of sample points is 10³ trajectories × 10 kHz × 5 s = 5 × 10⁷. These data can be aggregated across different experimental conditions and contain sufficient information with which to simultaneously reconstruct f(x) and g(x). To see this figure in color, go online.

Figure 3

Failure to account for diffusivity variations. Molecular bond force F^∗(x) = f^∗(x) + F_d(x) derived from unregularized (thin black) and regularized (solid blue) reconstruction data simulated using a given ground truth force field (dashed red). For reconstruction purposes, a constant diffusivity $D_0^{\ast }$ estimated from Eq. S16 in the Supporting Material was assumed. Although regularization allows for smoother and more stable reconstructions, the neglect of spatial structure in D(x) leads to inaccurate results. For example, the reconstructions in (a) cannot accurately determine the position of the minima, while those in (b) miss the minima entirely. The errors are especially apparent in regions where the diffusivity is significantly different from the constant value: (a) $D_0^{\ast }$ = 1.0042, (b) $D_0^{\ast }$ = 0.9995. To see this figure in color, go online.

Figure 4

Regularized reconstruction with a variable number of trajectories. Reconstruction of the bond force and diffusivity is given in Eq. S3 in the Supporting Material. (Shaded yellow) 95% semiclassical posterior confidence interval. (Gray) Unregularized binwise reconstruction. The noising reconstruction here arises from narrow bins and intrinsic sampling variability. (Blue) Regularized reconstructions. Optimal parameters used at trajectories were $D_0^{\ast }$ = 0.9995, β_f = 19,884, β_f = 2.28, and γ_g = 1.02. To see this figure in color, go online.