Calculation of the potential of mean force from nonequilibrium measurements via maximum likelihood estimators

Chelli, Riccardo; Marsili, Simone; Procacci, Piero

doi:10.1103/physreve.77.031104

Cited by 31 publications

(44 citation statements)

References 21 publications

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“…Unidirectional estimates of ΔF t based on Jarzynski's equality are markedly biased as the states are further perturbed from the starting equilibrium state. Chelli and coworkers have also developed an asymptotically correct bidirectional estimator that reduces to BAR at the end states [26]. However, the derivation offered by these authors is limited to deterministic systems, and although we have empirical evidence that their estimator approaches the correct ΔF t for Brownian simulations in the limit of a large number of trajectories (data not shown), it leads to a more pronounced bias than Eq.…”

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confidence: 82%

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Optimized Free Energies from Bidirectional Single-Molecule Force Spectroscopy

2008

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An optimized method for estimating path-ensemble averages using data from processes driven in opposite directions is presented. Based on this estimator, bidirectional expressions for reconstructing free energies and potentials of mean force from single-molecule force spectroscopy-valid for biasing potentials of arbitrary stiffness-are developed. Numerical simulations on a model potential indicate that these methods perform better than unidirectional strategies.Crooks' path-ensemble average theorem (Eq. 1) encompasses a set of exact results in nonequilibrium statistical mechanics pertinent to systems driven from thermal equilibrium by a time-dependent external potential [1]. These include Jarzynski's equality [2] and the Crooks fluctuation theorem [3], which relate equilibrium free energy differences to the nonequilibrium work distribution, as well as reweighting relations that allow one to recover arbitrary equilibrium ensemble averages from measurements of driven nonequilibrium processes [1]. Because of the intimate connection between such processes and molecular force spectroscopy, these theorems have been widely invoked to extract free energies and potentials of mean force (PMFs) from single-molecule pulling experiments [4][5][6][7][8].While formally correct, the practical utility of these relations is limited by the presence of exponential averages of the work, which are dominated by rare events and therefore have notoriously slow convergence properties [9]. In order to improve their convergence, strategies such as work-weighted trajectory sampling [10][11][12] have been proposed. Here we suggest another method to accelerate the convergence of these averages: including trajectories from the reverse process in the forward path-ensemble. This is motivated in part by the observation that the exponential average of the work in the forward process is dominated by those rare trajectories that resemble time-reversed counterparts ("conjugate twins") of typical trajectories generated by the reverse protocol [13]. Thus, our goals are to construct optimized forward pathensemble average estimators that explicitly include such trajectories, and apply them to the problem of estimating free energies and potentials of mean force from single-molecule pulling experiments.The starting point of our analysis is Crooks' path-ensemble average theorem, which relates the forward average of an arbitrary functional of the phase space trajectory Γ = {q(t), p (t)} to its work-weighted average in the reverse process, namely [1] (1)In the above, the forward average 〈…〉 F is an average over all trajectories (path-ensemble average) generated in the forward process, wherein an external parameter (e.g. the position of * Electronic Address:

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confidence: 82%

“…Since a general derivation of the results of Ref. [26] is not yet available, at present it is difficult to identify the source of discrepancy between these two estimators, and we leave this question for future investigations.…”

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confidence: 99%

Optimized Free Energies from Bidirectional Single-Molecule Force Spectroscopy

2008

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“…[49][50][51] As we said, the free energy difference between two states can be evaluated through the WFT as F = W * , where W * is the work value such that P F (W * ) = P B (−W * ). More generally, it has been shown that a bidirectional approach, in which the number of available experiments is partitioned between the forward and the backward transformations is statistically more founded.…”

Section: Steered Molecular Dynamicsmentioning

confidence: 99%

ORAC: A molecular dynamics simulation program to explore free energy surfaces in biomolecular systems at the atomistic level

et al. 2009

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Abstract:We present the new release of the ORAC engine (Procacci et al., Comput Chem 1997, 18, 1834, a FORTRAN suite to simulate complex biosystems at the atomistic level. The previous release of the ORAC code included multiple time steps integration, smooth particle mesh Ewald method, constant pressure and constant temperature simulations. The present release has been supplemented with the most advanced techniques for enhanced sampling in atomistic systems including replica exchange with solute tempering, metadynamics and steered molecular dynamics. All these computational technologies have been implemented for parallel architectures using the standard MPI communication protocol. ORAC is an open-source program distributed free of charge under the GNU general public license (GPL) at

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“…The MLE is an iterative method that weights the measurements to remove statistical bias and minimize the subsequent variance [28][29][30][31] …”

Section: Improving On Exponential Samplingmentioning

confidence: 99%

Different approaches for evaluating exponentially weighted nonequilibrium relations

Reid

Cunning

Searles

2010

The Journal of Chemical Physics

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The Kawasaki identity ͑KI͒ and the Jarzynski equality ͑JE͒ are important nonequilibrium relations. Both of these relations take the form of an ensemble average of an exponential function and can exhibit convergence problems when the average of the exponent differs greatly from the log of the average of the exponential function. In this work, we re-express these relations so that only selected regions need to be evaluated in an attempt to avoid these convergence issues. In the context of measuring free energies, we compare our method to the JE and the literature standard approach, the maximum likelihood estimator ͑MLE͒, and show that in a system with asymmetric work distributions it can perform as well as the MLE. For the KI, we derive an analog to the MLE to compare with our relation and show that these two new relations improve on the KI and are complimentary to each other.

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Calculation of the potential of mean force from nonequilibrium measurements via maximum likelihood estimators

Cited by 31 publications

References 21 publications

Optimized Free Energies from Bidirectional Single-Molecule Force Spectroscopy

Optimized Free Energies from Bidirectional Single-Molecule Force Spectroscopy

ORAC: A molecular dynamics simulation program to explore free energy surfaces in biomolecular systems at the atomistic level

Different approaches for evaluating exponentially weighted nonequilibrium relations

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