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Separation of time scale and coupling in the motion governed by the coarse-grained and fine degrees of freedom in a polypeptide backbone
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10.1063/1.2784200
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Affiliations:
1 Baker Laboratory of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853-1301, USA
a) Author to whom correspondence should be addressed. Telephone: (607) 255-4034. FAX: (607) 254-4700, Electronic mail: has5@cornell.edu
J. Chem. Phys. 127, 155103 (2007)
/content/aip/journal/jcp/127/15/10.1063/1.2784200
http://aip.metastore.ingenta.com/content/aip/journal/jcp/127/15/10.1063/1.2784200

## Figures

FIG. 1.

UNRES representation of a polypeptide chain. Filled circles represent the united peptide groups , and open circles represent the atoms, which serve as geometric points. Ellipsoids with their centers of mass at the SC positions represent UNRES side chains. The ’s are located halfway between two consecutive atoms at positions . The conformation of the polypeptide chain can be described fully by either the coordinates of all the and vectors or by the virtual-bond angles , the virtual-bond dihedral angles , and the angles and defining the orientation of the side chain with respect to the backbone.

FIG. 2.

Definition of the dihedral angles and for rotation of the peptide groups about the virtual bonds (dashed) of a peptide unit.

FIG. 3.

Simple toy model of a terminally blocked dipeptide. Filled circles mark the positions of the centers of the peptide groups , and open circles represent the atoms. Each peptide group is represented by a plane defined by the virtual-bond vector and the vector perpendicular to . The mass of the peptide group is distributed uniformly over the rectangle spanned by the corresponding vectors and ; additionally, two massless spheres (shown as dotted circles), each one with a radius equal to , are positioned at the distance of from the center of each peptide group along the directions of and opposite to , respectively, in order to enable the introduction of friction and random forces acting on the peptide groups. A point mass equal to the mass of a methyl group is located at each atom, and each atom is surrounded by a sphere with a radius corresponding to the UNRES radius of a methyl group (Ref. 27). The virtual-bond angles , the virtual-bond dihedral angle , and the torsional angles that define the rotations of the peptide groups about the virtual bonds are also indicated.

FIG. 4.

Contour plot of the ECEPP/3 conformational energy (Ref. 38) of a terminally blocked alanine residue (a) and its third-order Fourier expansion [Eq. (36)] (b). Energies are expressed relative to the global minimum.

FIG. 5.

Potentials of mean force for rotation about the virtual-bond axes [, Eq. (40)] evaluated at (a) and (b). The values calculated at points of the grid are shown as empty circles and the curves fitted with Fourier series [Eq. (42) with ] are shown as a solid line. The numerical values of the Fourier coefficients obtained from the fit are summarized in Table II.

FIG. 6.

Variation of the angle (solid lines, all panels), [dashed lines, slightly above ; (a) and (c)] [dotted lines, slightly above ; (a) and (c)] and [dot-dashed lines, slightly below 100°; (a) and (c)] along a simulation trajectory for the simple toy model or the corresponding coarse-grained model obtained by computing the PMF of the toy model. (a) Full potential, simulation with average kinetic energy corresponding to . (b) PMF, simulation with average kinetic energy corresponding to . (c) Full potential, Langevin simulation, . (d) PMF, Langevin simulation, . (e) Full potential, Langevin simulation, . (f) PMF, Langevin simulation, .

FIG. 7.

The correlation coefficients [Eq. (45)] calculated along the simulation between the UNRES and secondary variables (circles), (squares), and (diamonds) for the simple toy model for (a) simulations with average kinetic energy corresponding to , (b) Langevin simulations at , and (c) Langevin simulations at . See the discussion in Sec. ??? for the calculation of . For each time window , the correlation coefficients are averaged over eight independent runs. For further details, see the text.

FIG. 8.

The frequency spectra of the time series of the angle (solid lines, all panels), [dashed lines; (a) and (c)] [dotted lines; (a) and (c)] and [dot-dashed lines; (a) and (c)]. The spectra were calculated numerically using the standard fast Fourier transform algorithm. It should be noted that each spectrum corresponds to average frequency histograms over eight independent runs. For further details, see the text. (a) Full potential, simulations with average kinetic energy corresponding to . (b) PMF, simulations with average kinetic energy corresponding to . (c) Full potential, Langevin simulations, . (d) PMF, Langevin simulations, . (e) Full potential, Langevin simulations, . (f) PMF, Langevin simulations, .

FIG. 9.

Comparison of the time evolution of the torsional autocorrelation functions [Eq. (49)] in the full potential [Eq. (35)] (solid lines) and in the PMF [Eq. (41)] (dashed lines) for the toy model for (a) simulations with average kinetic energy corresponding to , (b) Langevin simulations at , and (c) Langevin simulations at .

FIG. 10.

Variation of the angles , , , and along a small section of a simulation trajectory for the all-atom terminally blocked polypeptide at . It should be noted that in order to trace the real variation of an angle, not contaminated by periodicity, we have eliminated the artificial flips between near and 180°.

FIG. 11.

The correlation coefficients [Eq. (45)] calculated along the simulation between the UNRES and secondary variables (, , and ) for the all-atom terminally blocked polypeptide at . For each time window , the correlation coefficients are averaged over five independent runs. For further details, see the text.

FIG. 12.

The frequency spectrum of the time series of the angles , , , and calculated for the all-atom terminally blocked polypeptide at . It should be noted that each spectrum corresponds to average frequency histograms over five independent runs. For further details, see the text.

FIG. 13.

The correlation coefficients between and calculated at three different temperatures, , , and for the all-atom terminally blocked polypeptide. For each time window , the correlation coefficients are averaged over five independent runs. It should be noted that the correlation decreases with increase in temperature. For further details, see the text.

## Tables

Table I.

Coefficients of the Fourier expansion of the ECEPP/3 energy surface of the terminally blocked alanine residue in and [Eq. (36)].

Table II.

Coefficients of the Fourier expansion of the PMF corresponding to the toy model [ of Eq. (42)] for and .

/content/aip/journal/jcp/127/15/10.1063/1.2784200
2007-10-16
2014-04-21

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