Digitalizing the backbone torsional dynamics of a folding protein

Abstract

This work deals with the elucidation of the dominant modes that entrain or subordinate the torsional dynamics of a peptide chain in the long-time limit relevant to the folding process. A timescale separation enables us to view local potential energy contributions as determining coarse constraints to which soft modes are subject, while long-range contributions are responsible for the deviations from the locally-optimized geometries within such constraints. These local steric constraints to backbone torsions are side-chain-dependent and incorporated by taking into account that torsional dynamics are enslaved by a discretized version in which local torsional states are viewed modulo basins of attraction in the Ramachandran (R-) maps. Thus, torsional dynamics are subordinated by the evolution of the so-called LTM (local topological constraints matrix). This basic assumption begets a ‘‘gears caricature’’ of each residue, with as many ‘‘arrested positions ’’ as there are basins in its Ramachandran map. This picture is consistent with the fact that equilibration within R-basins is incommensurably faster than the occurrence of folding events requiring interbasin transitions. Long-range interactions, encoded as a contact matrix (CM), may be determined by factorizing the LTM→CM map through the possible 3-D realizations of each LTM. Each such realization takes into account the dominant detailed torsional isomers within each R-basin. The CM time evolution eventually entrains the LTM dynamics through a feedback mechanism whereby R-basin hopping is slowed down according to the structural hierarchy to which the particular residue belongs, as inferred from the CM. This model incorporates the side-chain dependence of the local torsional entropy of the backbone, shows how the fine (geometric) structure of each R-basin is delineated by the long-range potential contributions, and ultimately reveals how the local steric constraints bias the 3-D organization. In order to test the predictive power of our treatment, the feedback LTM evolution is computed for a relatively large protein endowed with conformational plasticity, the β-lactoglobulin (N=162). The computations reveal distortions of locally-optimized geometries for α-helix patterns as long-range energetic interactions entrain the LTM dynamics in the long-time limit, eventually leading to a predominantly β-sheet motif. In accord with experimental evidence, no molten globule intermediate or hierarchical folding scenario is observed. Instead, as the CM dynamics subordinates the evolution of the LTM, locally-optimized torsional states of the predominantly α-helical intermediates are shown to be distorted while belonging to the same topological class. The dominant structural features of the native folding pattern are identifiable in the destination CM obtained in our simulations.