^{1}, Adam Liwo

^{1}and Harold A. Scheraga

^{1,a)}

### Abstract

The relevance of describing complex systems by simple coarse-grained models lies in the separation of time scales between the coarse-grained and fine or secondary degrees of freedom that are averaged out when going from an all-atom to the coarse-grained description. In this study, we propose a simple toy model with the aim of studying the variations with time, in a polypeptide backbone, of the coarse-grained (the pseudodihedral angle between subsequent atoms) and the secondary degrees of freedom (torsional angles for rotation of the peptide groups about the virtual bonds). Microcanonical and Langevin dynamics simulations carried out for this model system with a full potential (which is a function of both the coarse-grained and secondary degrees of freedom) show that, although the main motions associated with the coarse-grained degrees of freedom are low-frequency motions, the motions of the secondary degrees of freedom involve both high- and low-frequency modes in which the higher-frequency mode is superposed on the lower-frequency mode that follows the motions of the coarse-grained degrees of freedom. We found that the ratio of the frequency of the high-to low-frequency modes is from about 3:1 to about 6:1. The correlation coefficients, calculated along the simulation trajectory between these two types of degrees of freedom, indeed show a strong correlation between the fast and slow motions of the secondary and coarse-grained variables, respectively. To complement the findings of the toy-model calculations, all-atom Langevin dynamics simulations with the AMBER 99 force field and generalized Born (GB) solvation were carried out on the terminally blocked polypeptide. The coupling in the motions of the secondary and coarse-grained degrees of freedom, as revealed in the toy-model calculations, is also observed for the polypeptide. However, in contrast to that of the toy-model calculations, we observed that the higher-frequency modes of the secondary degrees of freedom are spread over a wide range of frequencies in . We also observed that the correlations between the secondary and coarse-grained degrees of freedom decrease with increasing temperature. This rationalizes the use of a temperature-dependent cumulant-based potential, such as our united-residue (UNRES) energy function for polypeptide chains, as an effective potential energy. To determine the effect of the coupling in the motions of the secondary and coarse-grained degrees of freedom on the dynamics of the latter, we also carried out microcanonical and Langevin dynamics simulations for the reduced toy model with a UNRES potential or potential of mean force (PMF) (obtained by averaging the energy surface of the toy model over the secondary degrees of freedom), and compared the results to those with the full-model system (the potential of which is a function of both the coarse-grained and secondary degrees of freedom). We found that, apparently, the coupling in the motions of the secondary and coarse-grained degrees of freedom, and averaging out the secondary degrees of freedom, does not have any implications in distorting the time scale of the coarse-grained degrees of freedom. This implies that the forces that act on the coarse-grained degrees of freedom are the same, whether they arise from the full potential or from the UNRES potential (PMF), and one can still apply the naive approach of simply using the PMF in the Lagrange equations of motion for the coarse-grained degrees of freedom of a polypeptide backbone to describe their dynamics. This suggests that the coupling between the degrees of freedom of the solvent and those of a polypeptide backbone, rather than averaging out the secondary backbone degrees of freedom, is responsible for the time-scale distortion in the coarse-grained dynamics of a polypeptide backbone.

This work was supported by grants from the National Institutes of Health (GM-14312), the National Science Foundation (MCB05-41633), and the NIH Fogarty International Center (TW7193). This research was conducted by using the resources of (a) our 800-processor Beowulf cluster at Baker Laboratory of Chemistry and Chemical Biology, Cornell University, (b) the National Science Foundation Terascale Computing System at the Pittsburgh Supercomputer Center, (c) the John von Neumann Institute for Computing at the Central Institute for Applied Mathematics, Forschungszentrum Jülich, Germany, (d) our 45-processor Beowulf cluster at the Faculty of Chemistry, University of Gdańsk, (e) the Informatics Center of the Metropolitan Academic Network (IC MAN) in Gdańsk, and (f) the Interdisciplinary Center of Mathematical and Computer Modeling (ICM) at the University of Warsaw.

I. INTRODUCTION

II. METHODS

A. UNRES model of polypeptide chains and force field

B. MD using UNRES model

C. Definition of the dihedral angles

D. MD with a simple toy model

1. Description of the model

2. Derivation of the equations of motion

3. The potential-energy function

4. Calculation of the PMF

5. Simulations with the full potential

6. Simulations with the PMF

E. All-atom MD simulations of the system

F. Analysis of the motion of UNRES and secondary degrees of freedom

1. Frequency analysis

2. Correlation analysis

3. Torsional autocorrelation function analysis

III. RESULTS AND DISCUSSION

A. Toy model with the full potential

B. Comparison of the coarse-grained dynamics generated using the full potential and the PMF

C. All-atom polypeptide

IV. CONCLUSIONS

### Key Topics

- Peptides
- 57.0
- Brownian dynamics
- 29.0
- Angular correlation
- 11.0
- Lagrangian mechanics
- 11.0
- Solvents
- 11.0

## Figures

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.

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.

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

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

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.

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.

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.

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.

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.

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.

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, .

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, .

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.

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.

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, .

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, .

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 .

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 .

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°.

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°.

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.

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.

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.

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.

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.

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

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

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

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

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

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