Spectral Rate Theory for Two-State Kinetics

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Spectral Rate Theory for Two-State Kinetics

Jan-Hendrik Prinz, John D. Chodera, and Frank Noé

Phys. Rev. X 4, 011020 – Published 21 February 2014

Abstract

Classical rate theories often fail in cases where the observable(s) or order parameter(s) used is a poor reaction coordinate or the observed signal is deteriorated by noise, such that no clear separation between reactants and products is possible. Here, we present a general spectral two-state rate theory for ergodic dynamical systems in thermal equilibrium that explicitly takes into account how the system is observed. The theory allows the systematic estimation errors made by standard rate theories to be understood and quantified. We also elucidate the connection of spectral rate theory with the popular Markov state modeling approach for molecular simulation studies. An optimal rate estimator is formulated that gives robust and unbiased results even for poor reaction coordinates and can be applied to both computer simulations and single-molecule experiments. No definition of a dividing surface is required. Another result of the theory is a model-free definition of the reaction coordinate quality. The reaction coordinate quality can be bounded from below by the directly computable observation quality, thus providing a measure allowing the reaction coordinate quality to be optimized by tuning the experimental setup. Additionally, the respective partial probability distributions can be obtained for the reactant and product states along the observed order parameter, even when these strongly overlap. The effects of both filtering (averaging) and uncorrelated noise are also examined. The approach is demonstrated on numerical examples and experimental single-molecule force-probe data of the p5ab RNA hairpin and the apo-myoglobin protein at low $p H$ , focusing here on the case of two-state kinetics.

Received 18 July 2012

DOI:https://doi.org/10.1103/PhysRevX.4.011020

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Jan-Hendrik Prinz^1,*, John D. Chodera^2,†, and Frank Noé^1,‡

¹Free University Berlin, Arnimallee 6, 14195 Berlin, Germany
²Computational Biology Program, Memorial Sloan-Kettering Cancer Center, New York, New York 10065, USA

^*jan-hendrik.prinz@fu-berlin.de
^†choderaj@mskcc.org
^‡Corresponding author. frank.noe@fu-berlin.de

Popular Summary

How is the rate constant of a chemical process defined, and how can we measure it? These are elementary questions in chemical physics as old as the discipline itself. The traditional paradigm is the so-called classical transition-state theory. It evokes the picture of a barrier transition state clearly separating the initial and the final states and the system jumping back and forth across the barrier aided by thermal fluctuations. Just as fundamentally, it assumes that a “reaction coordinate” that unambiguously represents this state landscape and the system’s dynamics directly corresponds to an experimental observable. It then determines the rate (constant) by counting the number of barrier crossings in a given time of observation.

But contemporary single-molecule experiments of extraordinary precision have revealed fundamental shortcomings of the classical theory and many other theories and rate estimators derived from it. The classical theory, in its most original form, drastically overestimates the rate of many single-molecule processes, and the derived theories designed to address the overcounting issue work well only in situations where the assumption of the direct correspondence between the experimental observable(s) and the reaction coordinate holds. In this paper, we provide an answer to these shortcomings by developing a “spectral rate theory,” whose effectiveness is demonstrated on numerical examples and experimental single-molecule force-probe data on an RNA hairpin and a myoglobin protein molecule.

Our theory allows the quality of an experimental observable at hand as a reaction coordinate and, in turn, the need to repeat the experiment with other accessible observables, to be systematically estimated. In addition, our theory also provides a new optimal rate estimator that gives robust and unbiased results even when poor reaction coordinates are used, even in the case when the initial and final states correspond to a population of many long-living states and show no clear separation in the presentation by the available experimental observable(s).

Our work represents a significant advance on the solution to one of the most fundamental but outstanding problems in chemical and biological physics. It should lead to experimental applications and further theoretical development that will make the approach more accurate, powerful, and more broadly useful.

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Vol. 4, Iss. 1 — January - March 2014

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Figure 1

Illustration of the observed dynamics for which a rate theory is formulated here. Top row: The full-dimensional dynamics $x (t)$ in phase space $Ω$ . These dynamics are assumed to be Markovian, ergodic, and reversible as is often found for physical systems in thermal equilibrium. Furthermore, the theory here is formulated for two-state kinetics, i.e., the system has two metastable states exchanging at rates $k_{A B}$ and $k_{B A}$ , giving rise to a relaxation rate of $κ_{2} = k_{A B} + k_{B A}$ . Middle row: One order parameter $y (x)$ of the system is observed, such as the distance between two groups of a molecule, or the Förster Resonance Energy Transfer (FRET) efficiency between two fluorescent groups. The projection of the full-space dynamics $x (t)$ onto the order parameter $y$ generates a time series $y (t)$ that, however, may not be directly observable. The projection also acts on functions of state space, such as the stationary distribution of configurations in full state space that is projected onto a density in the observable $μ (y)$ . The reaction coordinate quality ${\hat{α}}_{y}$ measures how well the order parameter $y$ resolves the slow transition. It is 1 when $A$ and $B$ are perfectly separated and 0 when they completely overlap. Bottom row: The experimental device used may distort or disperse the signal, for example, by adding noise. The resulting observed signal $o (t)$ is distorted and the observable density $μ^{o} (o)$ is smoothed. ${\hat{α}}_{o}$ measures the observation quality (OQ) of the observed signal, and it is shown in the Supplemental Material [28] that ${\hat{α}}_{o} \leq {\hat{α}}_{y}$ holds.
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Figure 2

Estimation results using overdamped Langevin dynamics in a two-dimensional two-well potential that is projected onto different observables: (I) perfect projection, (II) average-quality projection, (III) poor projection. Results are compared without noise (left half of panels) and with additional measurement noise (right half of panels). (1) Full state space with indicated direction of the used order parameter. (2) Top: Stationary density $μ^{y} (y)$ in the observable of the two partial densities of states $A$ (orange) and $B$ (gray). Results with noise are shaded lighter and are more spread out. Bottom: Second eigenvector without noise (solid, blue), with noise (solid, red), and dividing surface (black, dashed line). (3) Estimation quality $α$ from spectral estimation (OQ, red line), and from exponential fitting to the number correlation function using a diving surface at $y = 0$ (green line). (4) Estimated relaxation rate $κ_{2}$ : TST with averaging window of size $W$ (indicated in the $x$ axis). Dividing surface at $o = 0$ with single- $τ$ (dashed green line) and multi- $τ$ (solid green line) estimators. Estimates from a MSM-derived second eigenvector ${\tilde{ψ}}_{2}$ with a single- $τ$ estimate (normal MSM, dashed red line) and multi- $τ$ estimate (spectral estimation, solid red line). The black line is the reference solution, obtained from a direct MSM estimate for $τ = 50$ in row 1. (5) The transition rates $k_{A B}$ from state $A$ to $B$ . The coloring is identical to panels (4).
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Figure 3

Probed systems by optical tweezer experiments. Top: RNA hairpin p5ab. Bottom: H36Q mutant of sperm whale apo-myoglobin. Panels (1) and (4) show schematic views of the probed system in its native secondary or tertiary structure including the direction of the pulling force (green and black arrows). Panels (2) and (5) show the traces used for analysis. Row (a) reports the results when directly analyzing the measured 50 kHz data, while row (b) reports the results when analyzing data that has been binned to 1 kHz to reduce the noise amplitude. Panels (3) and (6) show the estimated phenomenological rates ${\hat{κ}}_{2}$ for TPT (blue line) using different averaging window sizes $W$ ( $x$ axis) and spectral estimation (red line) for different lag times $τ$ ( $x$ axis). For apo-myoglobin, the inset displays the behavior of TST at large window sizes $W$ , where the rate is systematically underestimated.
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Figure 4

Estimates for rates and estimation qualities from passive-mode single-molecule force-probe experiments of the p5ab RNA hairpin. All panels report the estimation results, showing the direct MSM estimate (black line), a fit to the fluctuation autocorrelation function using a dividing surface at $o = 12.80 pN$ (green line), and spectral estimation (red line). (1a),(1b) The stationary probability of observing a given force value (solid black line). The partial probabilities of states $A$ (gray) and $B$ (orange) obtained by spectral estimation show that there is very little overlap between the states. (2a),(2b) The estimation quality $α_{o}$ , coinciding with the OQ $\hat{α_{o}}$ for spectral estimation. (3a),(3b) Estimated relaxation rate $κ_{2}$ . (4a),(4b) Estimated microscopic transition rates: the folding rate $k_{A B}$ (dashed line) and the unfolding rate $k_{B A}$ (solid line).
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Figure 5

Estimates for rates and estimation qualities from passive-mode single-molecule force-probe experiments of apo-myoglobin. All panels report the estimation results, showing the direct Markov model estimate (black line), a fit to the fluctuation autocorrelation function using a dividing surface at the histogram maximum (minimum between two maxima for filtering) $o = 4.6 pN$ (green line), and spectral estimation (red line). (1a),(1b) The stationary probability of observing a given force value (solid black line). The partial probabilities of states $A$ (gray) and $B$ (orange) obtained by spectral estimation show that there is very little overlap between the states. (2a),(2b) The estimation quality $α_{o}$ , coinciding with the OQ ${\hat{α}}_{o}$ , for spectral estimation. (3a),(3b) Estimated relaxation rate $κ_{2}$ . (4a),(4b) Estimated microscopic transition rates $k_{A B}$ (dashed line) and $k_{B A}$ (solid line).
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