^{1,a)}, Sara de Lorenzo

^{1,2}, Josep Maria Huguet

^{1}and Felix Ritort

^{1,2,b)}

### Abstract

A main goal of single-molecule experiments is to evaluate equilibrium free energy differences by applying fluctuation relations to repeated work measurements along irreversible processes. We quantify the error that is made in a free energy estimate by means of the Jarzynski equality when the accumulated work expended on the whole system (including the instrument) is erroneously replaced by the work transferred to the subsystem consisting of the sole molecular construct. We find that the error may be as large as 100%, depending on the number of experiments and on the bandwidth of the data acquisition apparatus. Our theoretical estimate is validated by numerical simulations and pulling experiments on DNA hairpins using optical tweezers.

The authors gratefully acknowledge financial support through Grant Nos. FIS2007-61433 and NAN2004-9348 from Spanish Research Council, Grant No. SGR05-00688 from the Catalan Government and Grant No. RGP55/2008 from Human Frontiers Science Program.

I. INTRODUCTION

II. A TOY MODEL

A. Accumulated vs transferred work

B. The reversible work

C. Jarzynski estimator

D. A numerical test

III. AN EXPERIMENTAL TEST

A. Bidirectional methods

B. Role of the data analysis technique

IV. CONCLUSION

### Key Topics

- Free energy
- 13.0
- Data analysis
- 6.0
- DNA
- 6.0
- Data acquisition
- 5.0
- Optical tweezers
- 5.0

## Figures

Schematic definition of the model under study. The pipette is at rest with respect to the thermal bath, while the trap is moving with velocity . The trap and the system are approximated by two harmonic potentials with stiffness and , respectively. The rest length of the trap spring is 0, while the rest length of the molecule spring is if the hairpin is closed and if it is open .

Schematic definition of the model under study. The pipette is at rest with respect to the thermal bath, while the trap is moving with velocity . The trap and the system are approximated by two harmonic potentials with stiffness and , respectively. The rest length of the trap spring is 0, while the rest length of the molecule spring is if the hairpin is closed and if it is open .

(a) A typical FDC obtained by numerical simulation of Eq. (9). The shaded area is equivalent to the accumulated work [see Eq. (11)]. (b) The FEC associated to the pulling experiment represented in (a). The shaded area is equivalent to the transferred work [see Eq. (12)].

(a) A typical FDC obtained by numerical simulation of Eq. (9). The shaded area is equivalent to the accumulated work [see Eq. (11)]. (b) The FEC associated to the pulling experiment represented in (a). The shaded area is equivalent to the transferred work [see Eq. (12)].

Dependence on the sample size of the mode of (i.e., the maximum of the distribution for , see Appendix B). The dimensionless variable is , where and are the mean and standard deviations of the normally distributed transferred work . The represented curve is the numerical solution to Eq. (B9).

Dependence on the sample size of the mode of (i.e., the maximum of the distribution for , see Appendix B). The dimensionless variable is , where and are the mean and standard deviations of the normally distributed transferred work . The represented curve is the numerical solution to Eq. (B9).

Numerical test of Eq. (25). The theoretical prediction is compared to the results of numerical simulations of Eq. (9). In abscissa, is the standard deviation of the transferred work values ; different values of are obtained by varying the filter applied to the data. In ordinate, we represent the error (due to the erroneous use of in the Jarzynski estimator) (in units) on the determination of the free energy of formation of the hairpin. Each point represents the result of the analysis of trajectories.

Numerical test of Eq. (25). The theoretical prediction is compared to the results of numerical simulations of Eq. (9). In abscissa, is the standard deviation of the transferred work values ; different values of are obtained by varying the filter applied to the data. In ordinate, we represent the error (due to the erroneous use of in the Jarzynski estimator) (in units) on the determination of the free energy of formation of the hairpin. Each point represents the result of the analysis of trajectories.

Experimental test of Eq. (25). In abscissa, is the standard deviation of the transferred work values ; different values of are obtained by varying the stiffness of the trap and the bandwidth. In ordinate, we represent the error (due to the erroneous use of in the Jarzynski estimator) on the determination of the hairpin energy levels. See Table I for further details about the data.

Experimental test of Eq. (25). In abscissa, is the standard deviation of the transferred work values ; different values of are obtained by varying the stiffness of the trap and the bandwidth. In ordinate, we represent the error (due to the erroneous use of in the Jarzynski estimator) on the determination of the hairpin energy levels. See Table I for further details about the data.

(a) An experimental FDC observed with a high-frequency (20 kHz) and a low-frequency (1 kHz) data acquisition system. The area under the curve, which is a measure of the accumulated work , practically does not change. (b) The FEC associated to the pulling experiment represented in (a). The area under the curve, which represents the transferred work , depends on the frequency of the data acquisition system because of the large fluctuations of the integration extrema. Insets: magnified views of the region around the maximum of the force.

(a) An experimental FDC observed with a high-frequency (20 kHz) and a low-frequency (1 kHz) data acquisition system. The area under the curve, which is a measure of the accumulated work , practically does not change. (b) The FEC associated to the pulling experiment represented in (a). The area under the curve, which represents the transferred work , depends on the frequency of the data acquisition system because of the large fluctuations of the integration extrema. Insets: magnified views of the region around the maximum of the force.

Graph of using high- and low-frequency data, accumulated and transferred work. Data have been shifted along the horizontal axis to be easily compared. Data for the accumulated work (circles and squares) fall into a (bandwidth-independent) straight line of slope 1.00(8) in quantitative agreement with the prediction by the fluctuation relation Eq. (26). However data for the transferred work (triangles and rhombs) exhibit bandwidth-dependent very small slopes (around 0.03) that exclude the validity of an equivalent relation to Eq. (26) for the transferred work.

Graph of using high- and low-frequency data, accumulated and transferred work. Data have been shifted along the horizontal axis to be easily compared. Data for the accumulated work (circles and squares) fall into a (bandwidth-independent) straight line of slope 1.00(8) in quantitative agreement with the prediction by the fluctuation relation Eq. (26). However data for the transferred work (triangles and rhombs) exhibit bandwidth-dependent very small slopes (around 0.03) that exclude the validity of an equivalent relation to Eq. (26) for the transferred work.

Comparison between the histogram of the transferred work in one of the experiments reported in Table I and the normal distribution that better approximates it.

Comparison between the histogram of the transferred work in one of the experiments reported in Table I and the normal distribution that better approximates it.

Comparison between the histogram of the accumulated work in one of the experiments reported in Table I and the Gumbel distribution that better approximates it.

Comparison between the histogram of the accumulated work in one of the experiments reported in Table I and the Gumbel distribution that better approximates it.

## Tables

Experimental results: Comparison between the experimental (also shown in Fig. 5) and the theoretical [based on Eq. (25)] values of . The data sets labeled “1 kHz” and “20 kHz” refer to the same experiment, with the standard (low-frequency) and the new (high-frequency) data acquisition system. The stiffness of the trap is measured in , while is the number of trajectories.

Experimental results: Comparison between the experimental (also shown in Fig. 5) and the theoretical [based on Eq. (25)] values of . The data sets labeled “1 kHz” and “20 kHz” refer to the same experiment, with the standard (low-frequency) and the new (high-frequency) data acquisition system. The stiffness of the trap is measured in , while is the number of trajectories.

Experimental results. The data sets labeled “1 kHz” and “20 kHz” refer to the same experiment, with the standard (low-frequency) and the new (high-frequency) data acquisition system. The data sets labeled “ave ” are obtained from 20 kHz data by averaging over points.

Experimental results. The data sets labeled “1 kHz” and “20 kHz” refer to the same experiment, with the standard (low-frequency) and the new (high-frequency) data acquisition system. The data sets labeled “ave ” are obtained from 20 kHz data by averaging over points.

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