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Bayesian estimates of free energies from nonequilibrium work data in the presence of instrument noise
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Image of FIG. 1.
FIG. 1.

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).

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

The standard logistic function, .

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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.

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

(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.


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Table I.

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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Bayesian estimates of free energies from nonequilibrium work data in the presence of instrument noise