Analysis of internally restricted correlated rotations in peptides and proteins using ¹³C and ¹⁵N NMR relaxation data

The study of protein internal motions from analysis of ¹³C and ¹⁵N NMR relaxation data and auto- and cross-correlation spectral densities is being pursued in many labs. Model-free approaches and derived motional order parameters are normally used to interpret NMR relaxation data and internal mobility in proteins and peptides. Correlated motions can substantially modify the behavior of NMR auto- and cross-correlation spectral density functions and the values of derived motional order parameters. Here, a simple model is proposed to describe small amplitude (less than about 60° or 1 rad), internally restricted correlated rotations (IRCR) in peptides and proteins in order to analyze order parameters. Bond rotations are represented by vectors whose motions are correlated by a correlation coefficient, c_ij, which is the cosine of the angle between these vectors. Order parameters for NH, C_αH and CβH bond motions have been calculated from molecular dynamics simulations performed on short peptides with well-defined α-helix and β-sheet structures in order to derive values for c_ij. General equations relating dipolar auto- and cross-correlation order parameters for C_αH, C_βH, and NH bonds to c_ij have been derived. The sign of c_ij depends on the specific motional correlation within a particular molecular conformation. For glycine φ, ψ rotations, the sign of c_ij can be derived from analysis of dipolar auto- and cross-correlation order parameters. Long-range motional correlations are observed with hydrogen bonds modulating internal mobility. In general, backbone NH order parameters, S²_NH, are more sensitive to structure than are C_αH order parameters, S²_CH. S²_CH is increased and S²_NH is decreased when correlation coefficients c_ψ′φ and c_φψ are decreased and increased, respectively.