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Reaction coordinates and transition pathways of rare events via forward flux sampling
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10.1063/1.2776270
/content/aip/journal/jcp/127/16/10.1063/1.2776270
http://aip.metastore.ingenta.com/content/aip/journal/jcp/127/16/10.1063/1.2776270

Figures

Image of FIG. 1.
FIG. 1.

(a) A schematic view of the generation of branched paths (thick lines) using the branched growth (BG) sampling method. The first stage involves the simulation run in the basin shown by the dotted line. Starting points for the subsequent generation of branched paths are marked with the black circle at . The second stage corresponds to the trial runs fired from ; those that reached the next interface are shown by the thick line and those which failed to reach are shown by the dotted line. In this example, , , and . for all points collected at . for the points at are estimated from Eq. (5) as follows: , , and . The value for point 1 at is then obtained from . (b) A sketch of the committor function in one dimension. for points inside and near basin because trajectories started thereof end in ; for points in or near basin because trajectories started thereof end in . The statistical modeling of is restricted to the intermediate, transitional region of the dashed line illustrates a sample linear fit. The configurations for which are called transitions states.

Image of FIG. 2.
FIG. 2.

(Color) Contour graph of . The color scheme changes from highest (black) to lowest (white) elevations. The initial and final regions are shown by the circles labeled and , respectively. The initial guess vector for the FFS-type simulation is shown by the dotted lines (blue). The thick line (red) shows the TS dividing surface. Two branched paths are also shown: (●) black and red. These state points were used to determine the history for the reaction coordinate estimation.

Image of FIG. 3.
FIG. 3.

(Color) Estimated reaction coordinate isolines for model system . The thick lines (red) are the predicted committors from LSE. The committor values appear as labels over the lines. The contour of the free-energy surface is shown only for reference. The color scheme changes from highest (black) to lowest (white) elevations.

Image of FIG. 4.
FIG. 4.

(Color) Contour graph of . The color scheme changes from highest (black) to lowest (white) elevations. The initial and final regions are labeled as and , respectively. The initial guess vector for the FFS-type simulation is shown by the dotted lines (blue). The thick line (red) shows the TS dividing surface. Two branched paths are also shown: (●) black and (엯) red. These state points were used to determine the history for the reaction coordinate estimation.

Image of FIG. 5.
FIG. 5.

(Color) Estimated reaction coordinate isolines for model system . The thick lines (red) are the predicted committors from LSE, whose values appear as labels. Contour of the free-energy surface is shown only for reference. The color scheme changes from highest (black) to lowest (white) elevations.

Image of FIG. 6.
FIG. 6.

(Color) Estimated reaction coordinate isolines for the genetic switch model. The thick lines (red) are the predicted committors from LSE, whose values appear as labels. Contour of the free-energy landscape ( vs ) is shown only for visual reference. The color scheme changes from highest (gray/red) to lowest (black/blue) elevations. The TS dividing surface passes through a line at an angle, with origin at .

Image of FIG. 7.
FIG. 7.

(Color) Estimated reaction coordinate isolines for the folding of the 48-mer lattice protein model at . The thin dotted lines (red) are the predicted committors from FFS-LSE with as initial guess for the order parameter. The thick lines (blue) are the predicted committors from FFS-LSE with as initial guess for the order parameter. The committor values are shown as labels. Contour plot of the free-energy landscape [energy vs nativeness (NC)] is shown only for visual reference. The color scheme changes from highest (red) to lowest (blue) elevations. The initial region is enclosed by the circle labeled . The most probable visited transition state region is enclosed by the ellipse with center at (,).

Image of FIG. 8.
FIG. 8.

(Color online) (a) Nativeness (NC) and (b) conformational energy histograms for surface corresponding to the 48-mer lattice protein model at . The histograms were accumulated from the states collected in the [0.4, 0.6] surface region during FFS-type simulation (DFFS scheme). The superimposed normal distribution (dotted/red line) that would result from sampling on the exact surface is also shown. The exact Gaussian distribution obtained from committor analysis is shown by the solid line (blue).

Image of FIG. 9.
FIG. 9.

(Color online) histogram for the states collected at the TSE for the 48-mer lattice protein folding at from Ref. 3. values were estimated from the reaction coordinate model: . The bin width is 0.01. The thick line (red) shows the normal distribution ( and ) obtained from the histogram. The exact normal distribution (dotted/blue line) (, ) that would result from sampling on the exact surface is also shown. The standard error in the computed reaction coordinate model [Eq. (20)] is .

Image of FIG. 10.
FIG. 10.

Measured efficiency for the 48-mer lattice protein folding at : (—◇) and (—●) . Simulation results were obtained with 100 DFFS sampling blocks with starting points per block. Interfaces were unequally spaced for . The efficiency plateau is marked by the solid horizontal line.

Image of FIG. 11.
FIG. 11.

(Color) Estimated reaction coordinate isolines for the 64-mer lattice protein model folding at . Thin dotted (red) lines are the predicted committors from FFS-LSE, whose values appear as labels. Contour plot of the free-energy landscape [energy vs nativeness (NC)] is shown only for visual reference. The color scheme changes from highest (red) to lowest (blue) elevations. The initial region is delimited by the dotted line labeled . The most probable visited transition state region is bounded by the ellipse with center at (, ).

Image of FIG. 12.
FIG. 12.

(Color) Representative snapshots for the folding event of the 64-mer lattice protein at . The simulation trajectory in the unfolded basin is shown by the thick dotted line. Trial runs fired from that successfully reached the next interface are shown by the thick line and those which failed to reach are shown by the dotted line. Amino acid contacts are coded by the color of the residue: gray/blue indicates native contacts, whereas black/red designates the native contacts listed in Table VII, which have higher probability to belong to the transition state. The final frame shows the native state conformation.

Tables

Generic image for table
Table I.

LSE parameters and analysis of variance for the reaction coordinate model of the two-dimensional energy surface .

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Table II.

LSE parameters and analysis of variance for the reaction coordinate model of the two-dimensional energy surface .

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Table III.

Reactions involved for the genetic switch. Forward and backward rate constants and are also given (Refs. 2 and 18).

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Table IV.

LSE parameters and analysis of variance for the reaction coordinate model of the genetic switch.

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Table V.

LSE parameters and analysis of variance for the reaction coordinate model of the lattice protein folding with as initial guess of the order parameter for the FFS-LSE method.

Generic image for table
Table VI.

LSE parameters and analysis of variance for the reaction coordinate model of the lattice protein folding with as initial guess of the order parameter for the FFS-LSE method.

Generic image for table
Table VII.

Most probable native contacts found in the transition state ensemble for 64-mer lattice protein at . They are listed in order of decreasing probability.

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/content/aip/journal/jcp/127/16/10.1063/1.2776270
2007-10-22
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Reaction coordinates and transition pathways of rare events via forward flux sampling
http://aip.metastore.ingenta.com/content/aip/journal/jcp/127/16/10.1063/1.2776270
10.1063/1.2776270
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