^{1,a)}, Felix Ritort

^{2}, Carlos Bustamante

^{3}, Martin Karplus

^{1,4}and Gavin E. Crooks

^{5,b)}

### Abstract

The Jarzynski equality and the fluctuation theorem relate equilibrium free energy differences to nonequilibrium measurements of the work. These relations extend to single-molecule experiments that have probed the finite-time thermodynamics of proteins and nucleic acids. The effects of experimental error and instrument noise have not been considered previously. Here, we present a Bayesian formalism for estimating free energy changes from nonequilibrium work measurements that compensates for instrument noise and combines data from multiple driving protocols. We reanalyze a recent set of experiments in which a single RNA hairpin is unfolded and refolded using optical tweezers at three different rates. Interestingly, the fastest and farthest-from-equilibrium measurements contain the least instrumental noise and, therefore, provide a more accurate estimate of the free energies than a few slow, more noisy, near-equilibrium measurements. The methods we propose here will extend the scope of single-molecule experiments; they can be used in the analysis of data from measurements with atomic force microscopy, optical, and magnetic tweezers.

This research was supported by the U.S. Department of Energy, under Contract No. DE-AC02-05CH11231. The research of F.R. was supported by the Spanish and Catalan Research Councils (Grant Nos. FIS2004-3454, NAN2004-09348, and SGR05-00688). The research of C.B. was supported by NIH Grant No. GM 32543 and U.S. Department of Energy Grant No. AC0376Sf00098. The research of M.K. at Harvard was supported in part by a grant from the NIH.

I. INTRODUCTION

II. POSTERIOR FREE ENERGY ESTIMATE

III. EXPERIMENTAL ERRORS

IV. APPLICATION AND DISCUSSION

## Figures

Nonequilibrium work measurements for folding and unfolding a RNA hairpin (Ref. 2). A single RNA molecule is attached between two beads via hybrid DNA/RNA linkers. One bead is captured in an optical laser trap that can measure the applied force on the bead. The other bead is attached to a piezoelectric actuator, which is used to irreversibly unfold and refold the hairpin (Refs. 2–11).

Nonequilibrium work measurements for folding and unfolding a RNA hairpin (Ref. 2). A single RNA molecule is attached between two beads via hybrid DNA/RNA linkers. One bead is captured in an optical laser trap that can measure the applied force on the bead. The other bead is attached to a piezoelectric actuator, which is used to irreversibly unfold and refold the hairpin (Refs. 2–11).

Typical force-extension curves in the unfolding (solid lines) and folding (dashed lines) of a 20 base pairs RNA hairpin. Different colors correspond to different unfolding-folding cycles. The rip in force observed around 15 pN corresponds to the cooperative unfolding/folding transition. The area below the force-extension curve is equal to the mechanical work done on the RNA hairpin. Because the transformations are irreversible, the work performed varies from one unfolding or refolding measurement to the next. Drift effects observed in force-extension curves arise from different causes, including air currents, mechanical vibrations, and temperature changes.

Typical force-extension curves in the unfolding (solid lines) and folding (dashed lines) of a 20 base pairs RNA hairpin. Different colors correspond to different unfolding-folding cycles. The rip in force observed around 15 pN corresponds to the cooperative unfolding/folding transition. The area below the force-extension curve is equal to the mechanical work done on the RNA hairpin. Because the transformations are irreversible, the work performed varies from one unfolding or refolding measurement to the next. Drift effects observed in force-extension curves arise from different causes, including air currents, mechanical vibrations, and temperature changes.

The standard logistic function, .

The standard logistic function, .

The approximation of the standard logistic distribution by the Gaussian distribution with zero mean and standard deviation .

The approximation of the standard logistic distribution by the Gaussian distribution with zero mean and standard deviation .

Posterior free energy given two work measurements, one from each of two conjugate protocols with values . The posterior variance, , is minimized when the rectified work variables coincide and increases quadratically with separation.

Posterior free energy given two work measurements, one from each of two conjugate protocols with values . The posterior variance, , is minimized when the rectified work variables coincide and increases quadratically with separation.

The approximation of the sigmoidal function [Eq. (A1)] by the logistic function , where [Eq. (A7)]. The absolute difference between the functions is always less than 0.02.

The approximation of the sigmoidal function [Eq. (A1)] by the logistic function , where [Eq. (A7)]. The absolute difference between the functions is always less than 0.02.

(a) Histograms of work measurements for folding and unfolding an RNA hairpin at three different rates. Observations are binned into integers centered at intervals. These data correspond to Fig. 2 of Collin *et al.* (Ref. 2) Note that Eq. (3) predicts that the folding and unfolding work distributions cross at the free energy change. (b) The posterior distribution of the error correction factor [Eq. (16)]. (c) Posterior free energy derived from the data in (a), both with [solid line, Eq. (16)] and without [dashed line, Eq. (11)] correction for measurement noise. Notice that the correction is substantial for the slowest experiment (1.5 pN/s) and minor for the intermediate rate, and the corrected and uncorrected posteriors are indistinguishable (at this scale) for the fastest rate. The most reliable free energy estimate is obtained by combining the three separate noise-corrected free energy posterior distributions.

(a) Histograms of work measurements for folding and unfolding an RNA hairpin at three different rates. Observations are binned into integers centered at intervals. These data correspond to Fig. 2 of Collin *et al.* (Ref. 2) Note that Eq. (3) predicts that the folding and unfolding work distributions cross at the free energy change. (b) The posterior distribution of the error correction factor [Eq. (16)]. (c) Posterior free energy derived from the data in (a), both with [solid line, Eq. (16)] and without [dashed line, Eq. (11)] correction for measurement noise. Notice that the correction is substantial for the slowest experiment (1.5 pN/s) and minor for the intermediate rate, and the corrected and uncorrected posteriors are indistinguishable (at this scale) for the fastest rate. The most reliable free energy estimate is obtained by combining the three separate noise-corrected free energy posterior distributions.

## Tables

Summary of results graphed in Fig. 7. and : number of unfolding and refolding work measurements at each pulling rate, respectively. : posterior mean free energy estimate with 95% confidence intervals, both corrected and uncorrected for measurement error. : posterior mean estimate of the noise correction factor, with 95% confidence intervals.

Summary of results graphed in Fig. 7. and : number of unfolding and refolding work measurements at each pulling rate, respectively. : posterior mean free energy estimate with 95% confidence intervals, both corrected and uncorrected for measurement error. : posterior mean estimate of the noise correction factor, with 95% confidence intervals.

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