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Path-breaking schemes for nonequilibrium free energy calculations
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10.1063/1.4808037
/content/aip/journal/jcp/138/21/10.1063/1.4808037
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/21/10.1063/1.4808037

Figures

Image of FIG. 1.
FIG. 1.

(a) Schematic representation of possible work functions in relation to the applicability of path-breaking schemes. The red-dashed line represents Δ along the pathway, while the arrow marks the dissipation threshold, . 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 Λ, Λ, Λ, and Λ. Work functions 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 λ. The red-dashed line represents Δ .

Image of FIG. 2.
FIG. 2.

Toy model. Distribution functions obtained from 10 Gaussianly distributed work samples. () is the standard work distribution. ′() is the work distribution obtained by using the PTS. The quantity ′() () is obtained by back-weighting ′() and corresponds to () in the limit of an infinite number of work samples. The free energy difference Δ = −4.58, obtained by the standard method (Eq. (4) ), is marked with an arrow.

Image of FIG. 3.
FIG. 3.

Deca-alanine. Left panels: Δ 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 = 53.33, 80, 160 m s (in right panels too). For each pulling velocity, PMFs obtained from simulations using different check-point numbers are reported (for the sake of clarity the PMFs are shifted along the -axis: from bottom to top = 1, 4, 9, 13, 19, 26). The exact PMF taken from Ref. 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 values. Actually, only one value is obtained from standard simulations; this value is repeated along the -axis for the sake of comparison with PTS data. Lines are guides for eyes.

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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

Image of FIG. 6.
FIG. 6.

Deca-alanine. Δ 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 = 160 m s, while PTS simulations (colored lines) are performed with pulling velocities and parameters reported in the legend. (b) Standard simulations are performed with = 80 m s (see the legend for PTS simulation parameters). (c) Standard simulations are performed with = 53.33 m s (see the legend for PTS simulation parameters). Error bars are calculated as explained in the text. The exact PMF taken from Ref. is also reported for comparison (dashed line). For the sake of clarity, the PMFs are shifted along the -axis.

Image of FIG. 7.
FIG. 7.

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

Image of FIG. 8.
FIG. 8.

Deca-alanine. Left panels: Δ 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 = 53.33, 80, 160 m s (in right panels too). For each pulling velocity, PMFs obtained from simulations using different check-point numbers are reported (for the sake of clarity the PMFs are shifted along the -axis: from bottom to top = 1, 4, 9, 13, 19, 26). The exact PMF taken from Ref. 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 values. Blue circles are from the standard method. Actually, only one value is obtained from standard simulations; this value is repeated along the -axis for the sake of comparison with FTS and PTS data. Lines are guides for eyes.

Image of FIG. 9.
FIG. 9.

Methane into water. (a) Δ 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 are reported (for the sake of clarity the PMFs are shifted along the y-axis: from bottom to top = 1, 3, 5, 7, 9, 11). Error bars are calculated as explained in the text. The exact PMF taken from Ref. is also reported for comparison (dashed line). (b) as a function of 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 -axis to compare better the data. (c) Computer-time efficiency ratio / as a function of .

Tables

Generic image for table
Table I.

Illustration of the PTS through a toy model. A number of work samples ( column) are generated according to a Gaussian distribution with known mean and variance. Free energy differences Δ 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 Δ , are also reported (last column). Note that the five work samples employed to evaluate the free energy guess are included in the percentages.

Generic image for table
Table II.

Simulation-parameter correlations. and 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.

Generic image for table
Table III.

Free energy estimates, , and related errors (in kJ mol) for various pulling velocities (in m s). 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, , are also reported.

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/content/aip/journal/jcp/138/21/10.1063/1.4808037
2013-06-06
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Path-breaking schemes for nonequilibrium free energy calculations
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/21/10.1063/1.4808037
10.1063/1.4808037
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