^{1,a)}, Cristina Gellini

^{1}, Giangaetano Pietraperzia

^{1}, Edoardo Giovannelli

^{1}and Gianni Cardini

^{1}

### Abstract

We propose a path-breaking route to the enhancement of unidirectional nonequilibrium simulations for the calculation of free energy differences via Jarzynski's equality [C. Jarzynski, Phys. Rev. Lett.78, 2690 (Year: 1997)]10.1103/PhysRevLett.78.2690. One of the most important limitations of unidirectional nonequilibrium simulations is the amount of realizations necessary to reach suitable convergence of the work exponential average featuring the Jarzynski's relationship. In this respect, a significant improvement of the performances could be obtained by finding a way of stopping trajectories with negligible contribution to the work exponential average, before their normal end. This is achieved using path-breaking schemes which are essentially based on periodic checks of the work dissipated during the pulling trajectories. Such schemes can be based either on breaking trajectories whose dissipated work exceeds a given threshold or on breaking trajectories with a probability increasing with the dissipated work. In both cases, the computer time needed to carry out a series of nonequilibrium trajectories is reduced up to a factor ranging from 2 to more than 10, at least for the processes under consideration in the present study. The efficiency depends on several aspects, such as the type of process, the number of check-points along the pathway and the pulling rate as well. The method is illustrated through radically different processes, i.e., the helix-coil transition of deca-alanine and the pulling of the distance between two methane molecules in water solution.

This work was supported by European Union Contract RII3-CT-2003-506350 and by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca.

I. INTRODUCTION

II. PATH-BREAKING IN NONEQUILIBRIUM SIMULATIONS

A. Theoretical framework

B. An illustration of the path-breaking scheme

III. TEST CASES

A. Helix-coil transition of deca-alanine

B. Potential of mean force of two methane molecules in water solution

IV. CONCLUDING REMARKS

### Key Topics

- Free energy
- 47.0
- Methane
- 7.0
- Molecular dynamics
- 5.0
- Monte Carlo methods
- 4.0
- Solution processes
- 4.0

## Figures

(a) Schematic representation of possible work functions W λ in relation to the applicability of path-breaking schemes. The red-dashed line represents ΔF λ along the pathway, while the arrow marks the dissipation threshold, W th. Trajectories whose work functions are above the upward arrow ( ) are broken. Vice versa, trajectories whose work functions are below the upward arrow ( ) are not broken. A-type trajectories, broken at half pathway, give negligible contribution to . B-type trajectories, not broken at half pathway, give significant contribution to . C-type trajectories, not broken at half pathway, give negligible contribution to . D-type trajectories, broken at half pathway, give significant contribution to . (b) Representation of a path-breaking scheme with four check-points at Λ1, Λ2, Λ3, and Λ4. Work functions W λ associated with nonequilibrium trajectories are reported with black lines. The trajectories above the dissipation threshold limit, indicated by the upward arrows, are broken. Only the trajectories with low dissipation can reach λτ. The red-dashed line represents ΔF λ.

(a) Schematic representation of possible work functions W λ in relation to the applicability of path-breaking schemes. The red-dashed line represents ΔF λ along the pathway, while the arrow marks the dissipation threshold, W th. Trajectories whose work functions are above the upward arrow ( ) are broken. Vice versa, trajectories whose work functions are below the upward arrow ( ) are not broken. A-type trajectories, broken at half pathway, give negligible contribution to . B-type trajectories, not broken at half pathway, give significant contribution to . C-type trajectories, not broken at half pathway, give negligible contribution to . D-type trajectories, broken at half pathway, give significant contribution to . (b) Representation of a path-breaking scheme with four check-points at Λ1, Λ2, Λ3, and Λ4. Work functions W λ associated with nonequilibrium trajectories are reported with black lines. The trajectories above the dissipation threshold limit, indicated by the upward arrows, are broken. Only the trajectories with low dissipation can reach λτ. The red-dashed line represents ΔF λ.

Toy model. Distribution functions obtained from 106 Gaussianly distributed work samples. P(W) is the standard work distribution. P′(W) is the work distribution obtained by using the PTS. The quantity P′(W)f −1(W) is obtained by back-weighting P′(W) and corresponds to P(W) in the limit of an infinite number of work samples. The free energy difference ΔF λ = −4.58, obtained by the standard method (Eq. (4) ), is marked with an arrow.

Toy model. Distribution functions obtained from 106 Gaussianly distributed work samples. P(W) is the standard work distribution. P′(W) is the work distribution obtained by using the PTS. The quantity P′(W)f −1(W) is obtained by back-weighting P′(W) and corresponds to P(W) in the limit of an infinite number of work samples. The free energy difference ΔF λ = −4.58, obtained by the standard method (Eq. (4) ), is marked with an arrow.

Deca-alanine. Left panels: ΔF λ as a function of λ (end-to-end distance) calculated from PTS and standard simulations (red and black lines, respectively). From bottom to top panel: data from simulations realized with pulling velocities v = 53.33, 80, 160 m s−1 (in right panels too). For each pulling velocity, PMFs obtained from simulations using different check-point numbers M are reported (for the sake of clarity the PMFs are shifted along the y-axis: from bottom to top M = 1, 4, 9, 13, 19, 26). The exact PMF taken from Ref. 42 is also reported for comparison (dashed line). Error bars are calculated as explained in the text (in right panels too). Right panels: calculated from PTS and standard simulations (red and black circles, respectively) using all the M values. Actually, only one value is obtained from standard simulations; this value is repeated along the x-axis for the sake of comparison with PTS data. Lines are guides for eyes.

Deca-alanine. Left panels: ΔF λ as a function of λ (end-to-end distance) calculated from PTS and standard simulations (red and black lines, respectively). From bottom to top panel: data from simulations realized with pulling velocities v = 53.33, 80, 160 m s−1 (in right panels too). For each pulling velocity, PMFs obtained from simulations using different check-point numbers M are reported (for the sake of clarity the PMFs are shifted along the y-axis: from bottom to top M = 1, 4, 9, 13, 19, 26). The exact PMF taken from Ref. 42 is also reported for comparison (dashed line). Error bars are calculated as explained in the text (in right panels too). Right panels: calculated from PTS and standard simulations (red and black circles, respectively) using all the M values. Actually, only one value is obtained from standard simulations; this value is repeated along the x-axis for the sake of comparison with PTS data. Lines are guides for eyes.

Deca-alanine. (a) Fraction n of trajectories that reach the various check-points displaced along the pathway λ in PTS simulations performed at v = 53.33 m s−1 and different values of the M parameter (see the legend). The abscissa of the circles correspond to the check-point positions. For example, for the M = 4 simulation, 4 circles (check-points) plus one circle at λτ are reported. The lines are guides for eyes (in (b) too). (b) Fraction n of trajectories that reach the end coordinate λτ in PTS simulations performed at v = 53.33 m s−1 and all the M values.

Deca-alanine. (a) Fraction n of trajectories that reach the various check-points displaced along the pathway λ in PTS simulations performed at v = 53.33 m s−1 and different values of the M parameter (see the legend). The abscissa of the circles correspond to the check-point positions. For example, for the M = 4 simulation, 4 circles (check-points) plus one circle at λτ are reported. The lines are guides for eyes (in (b) too). (b) Fraction n of trajectories that reach the end coordinate λτ in PTS simulations performed at v = 53.33 m s−1 and all the M values.

Deca-alanine. Computer-time efficiency ratio T ST/T PTS for PTS simulations as a function of the M simulation parameter. Results for different pulling velocities are given (see the legend; in m s−1 units). For the sake of comparison, the maximum theoretical ratio, M + 1, is also reported (dashed line). The lines are guides for eyes.

Deca-alanine. Computer-time efficiency ratio T ST/T PTS for PTS simulations as a function of the M simulation parameter. Results for different pulling velocities are given (see the legend; in m s−1 units). For the sake of comparison, the maximum theoretical ratio, M + 1, is also reported (dashed line). The lines are guides for eyes.

Deca-alanine. ΔF λ as a function of λ (end-to-end distance) calculated from PTS and standard simulations whose efficiency ratio is close to 1 (see Table II ). (a) Standard simulations (black lines) are performed with v = 160 m s−1, while PTS simulations (colored lines) are performed with pulling velocities and M parameters reported in the legend. (b) Standard simulations are performed with v = 80 m s−1 (see the legend for PTS simulation parameters). (c) Standard simulations are performed with v = 53.33 m s−1 (see the legend for PTS simulation parameters). Error bars are calculated as explained in the text. The exact PMF taken from Ref. 42 is also reported for comparison (dashed line). For the sake of clarity, the PMFs are shifted along the y-axis.

Deca-alanine. ΔF λ as a function of λ (end-to-end distance) calculated from PTS and standard simulations whose efficiency ratio is close to 1 (see Table II ). (a) Standard simulations (black lines) are performed with v = 160 m s−1, while PTS simulations (colored lines) are performed with pulling velocities and M parameters reported in the legend. (b) Standard simulations are performed with v = 80 m s−1 (see the legend for PTS simulation parameters). (c) Standard simulations are performed with v = 53.33 m s−1 (see the legend for PTS simulation parameters). Error bars are calculated as explained in the text. The exact PMF taken from Ref. 42 is also reported for comparison (dashed line). For the sake of clarity, the PMFs are shifted along the y-axis.

Deca-alanine. Computer-time efficiency ratios T ST/T PTS and T ST/T FTS for PTS and FTS simulations performed at pulling velocities of 53.33, 80, and 160 m s−1 as a function of the simulation parameter M. For FTS simulations, the threshold W th is chosen to get an efficiency ratio comparable to that obtained from PTS (see text).

Deca-alanine. Computer-time efficiency ratios T ST/T PTS and T ST/T FTS for PTS and FTS simulations performed at pulling velocities of 53.33, 80, and 160 m s−1 as a function of the simulation parameter M. For FTS simulations, the threshold W th is chosen to get an efficiency ratio comparable to that obtained from PTS (see text).

Deca-alanine. Left panels: ΔF λ as a function of λ (end-to-end distance) calculated from FTS and PTS simulations (red and black lines, respectively). From bottom to top panel: data from simulations realized with pulling velocities v = 53.33, 80, 160 m s−1 (in right panels too). For each pulling velocity, PMFs obtained from simulations using different check-point numbers M are reported (for the sake of clarity the PMFs are shifted along the y-axis: from bottom to top M = 1, 4, 9, 13, 19, 26). The exact PMF taken from Ref. 42 is also reported for comparison (dashed line). Error bars are calculated as explained in the text (in right panels too). Right panels: calculated from FTS and PTS simulations (red and black circles, respectively) using all the M values. Blue circles are from the standard method. Actually, only one value is obtained from standard simulations; this value is repeated along the x-axis for the sake of comparison with FTS and PTS data. Lines are guides for eyes.

Deca-alanine. Left panels: ΔF λ as a function of λ (end-to-end distance) calculated from FTS and PTS simulations (red and black lines, respectively). From bottom to top panel: data from simulations realized with pulling velocities v = 53.33, 80, 160 m s−1 (in right panels too). For each pulling velocity, PMFs obtained from simulations using different check-point numbers M are reported (for the sake of clarity the PMFs are shifted along the y-axis: from bottom to top M = 1, 4, 9, 13, 19, 26). The exact PMF taken from Ref. 42 is also reported for comparison (dashed line). Error bars are calculated as explained in the text (in right panels too). Right panels: calculated from FTS and PTS simulations (red and black circles, respectively) using all the M values. Blue circles are from the standard method. Actually, only one value is obtained from standard simulations; this value is repeated along the x-axis for the sake of comparison with FTS and PTS data. Lines are guides for eyes.

Methane into water. (a) ΔF λ as a function of λ (methane-methane distance) calculated from PTS and standard configurational freezing simulations (red and black lines, respectively). Data obtained from simulations realized with different parameter M are reported (for the sake of clarity the PMFs are shifted along the y-axis: from bottom to top M = 1, 3, 5, 7, 9, 11). Error bars are calculated as explained in the text. The exact PMF taken from Ref. 33 is also reported for comparison (dashed line). (b) as a function of M calculated from PTS simulations (red circles; lines are guides for eyes). Black circles are from the standard configurational freezing method. Actually, only one value is obtained from standard simulations; this value is repeated along the x-axis to compare better the data. (c) Computer-time efficiency ratio T ST/T PTS as a function of M.

Methane into water. (a) ΔF λ as a function of λ (methane-methane distance) calculated from PTS and standard configurational freezing simulations (red and black lines, respectively). Data obtained from simulations realized with different parameter M are reported (for the sake of clarity the PMFs are shifted along the y-axis: from bottom to top M = 1, 3, 5, 7, 9, 11). Error bars are calculated as explained in the text. The exact PMF taken from Ref. 33 is also reported for comparison (dashed line). (b) as a function of M calculated from PTS simulations (red circles; lines are guides for eyes). Black circles are from the standard configurational freezing method. Actually, only one value is obtained from standard simulations; this value is repeated along the x-axis to compare better the data. (c) Computer-time efficiency ratio T ST/T PTS as a function of M.

## Tables

Illustration of the PTS through a toy model. A number N of work samples (N column) are generated according to a Gaussian distribution with known mean and variance. Free energy differences ΔF λ are calculated by using both the standard approach represented by Eq. (4) (ST column; arbitrary units) and the PTS (PTS column; arbitrary units). The percentages of work samples that contribute to ΔF λ, are also reported (last column). Note that the five work samples employed to evaluate the free energy guess are included in the percentages.

Illustration of the PTS through a toy model. A number N of work samples (N column) are generated according to a Gaussian distribution with known mean and variance. Free energy differences ΔF λ are calculated by using both the standard approach represented by Eq. (4) (ST column; arbitrary units) and the PTS (PTS column; arbitrary units). The percentages of work samples that contribute to ΔF λ, are also reported (last column). Note that the five work samples employed to evaluate the free energy guess are included in the percentages.

Simulation-parameter correlations. v and M parameters to be used in PTS simulations (second and third columns) necessary to get comparable efficiency (fourth column) between PTS and standard simulations as the latter simulations are performed with the pulling velocities reported in the first column.

Simulation-parameter correlations. v and M parameters to be used in PTS simulations (second and third columns) necessary to get comparable efficiency (fourth column) between PTS and standard simulations as the latter simulations are performed with the pulling velocities reported in the first column.

Free energy estimates, , and related errors (in kJ mol−1) for various pulling velocities v (in m s−1). The calculations have been done by using full sets of trajectories (2nd column) and partial sets of trajectories resulting from the exclusion of D-type trajectories (3rd, 4th, and 5th columns). Three criteria for classifying the D-type trajectories are employed (see text for details). The mean percentages of D-type trajectories, N D-traj, are also reported.

Free energy estimates, , and related errors (in kJ mol−1) for various pulling velocities v (in m s−1). The calculations have been done by using full sets of trajectories (2nd column) and partial sets of trajectories resulting from the exclusion of D-type trajectories (3rd, 4th, and 5th columns). Three criteria for classifying the D-type trajectories are employed (see text for details). The mean percentages of D-type trajectories, N D-traj, are also reported.

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