Separation of time scale and coupling in the motion governed by the coarse-grained and fine degrees of freedom in a polypeptide backbone.

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 Calpha atoms) and the secondary degrees of freedom (torsional angles for rotation of the peptide groups about the virtual Calpha...Calpha 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 Ala10 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 Ala10 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 Ala10. 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.