Protein allostery requires dynamical structural correlations. Physical originof which, however, remain elusive despite intensive studies during last twodecades. Based on analysis of molecular dynamics (MD) simulation trajectoriesfor ten proteins with different sizes and folds, we found that nonlinearbackbone torsional pair (BTP) correlations, which are spatially morelong-ranged and are mainly executed by loop residues, exist extensively in mostanalyzed proteins. Examination of torsional motion for correlated BTPssuggested that aharmonic torsional state transitions are essential for suchnon-linear correlations, which correspondingly occur on widely different andrelatively longer time scales. In contrast, BTP correlations between backbonetorsions in stable $\alpha$ helices and $\beta$ strands are mainly linear andspatially more short-ranged, and are more likely to associate with intra-welltorsional dynamics. Further analysis revealed that the direct cause ofnon-linear contributions are heterogeneous, and in extreme cases canceling,linear correlations associated with different torsional states of participatingtorsions. Therefore, torsional state transitions of participating torsions fora correlated BTP are only necessary but not sufficient condition forsignificant non-linear contributions. These findings implicate a general searchstrategy for novel allosteric modulation of protein activities. Meanwhile, itwas suggested that ensemble averaged correlation calculation and static contactnetwork analysis, while insightful, are not sufficient to elucidate mechanismsunderlying allosteric signal transmission in general, dynamical and time scaleresolved analysis are essential.
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