Packing Regularities in Biological Structures Relate to Their Dynamics

Abstract

The high packing density inside proteins leads to certain geometric regularities and also is one of the most important contributors to the high extent of cooperativity manifested by proteins in their cohesive domain motions. The orientations between neighboring non-bonded residues in proteins substantially follow the similar geometric regularities, regardless of whether the residues are on the surface or buried -a direct result of hydrophobicity forces. These orientations are relatively fixed and correspond closely to small deformations from those of the face-centered cubic lattice, which is the way in which identical spheres pack at the highest density. Packing density also is related to the extent of conservation of residues, and we show this relationship for residue packing densities by averaging over a large sample or residue packings. There are three regimes: 1) over a broad range of packing densities the relationship between sequence entropy and inverse packing density is nearly linear, 2) over a limited range of low packing densities the sequence entropy is nearly constant, and 3) at extremely low packing densities the sequence entropy is highly variable. These packing results provide important justification for the simple elastic network models that have been shown for a large number of proteins to represent protein dynamics so successfully, even when the models are extremely coarse-grained. Elastic network models for polymeric chains are simple and could be combined with these protein elastic networks to represent partially denatured parts of proteins. Finally, we show results of applications of the elastic network model to study the functional motions of the ribosome, based on its known structure. These results indicate expected correlations among its components for the step-wise processing steps in protein synthesis, and suggest ways to use these elastic network models to develop more detailed mechanisms -an important possibility, since most experiments yield only static structures.

A mechanistic stress model of protein evolution accounts for site-specific evolutionary rates and their relationship with packing density and flexibility

“…The Pearson correlation coefficient R follows the same trend. We note, that even though smaller, the correlation coefficients for rate vs. MSF are significant, which agrees with previous findings [14,32,13,15]. However, partial correlations (pR) show that once stress (MLmS) is controlled for, the rate-MSF correlation almost disappears: the sequence-flexibility correlation is indirect.…”

“…Regarding the sequence-flexibility relationship, previous empirical correlation studies had already found that the sequence-flexibility relationship is nonlinear and either dismissed the nonlinear parts or attempted an interpretation in terms of different selection regimes [14,32,13]. We found the nonlinearity follows naturally from the Stress Model: evolutionary rates depend nonlinearly on MSF because they depend (approximately) linearly on MLmS , and MSF  ≈ 1/ MLmS .…”

“…Even though previous sequence-flexibility studies used Pearson correlations, which, rigorously, make sense only for linear relationships, they already found nonlinear sequence-flexibility plots similar to those of Figure 1 (left panels) [14,32,13]. In spite of this, they either dismissed the nonlinear part [14]or interpreted it in terms of different selection regimes [13].…”

Elastic network models capture the motions apparent within ensembles of RNA structures

“…ENMs, in particular the anisotropic network model (ANM), a type of ENM that provides the directions of motions in addition to relative mobility, have proven useful for obtaining the dominant functional motions of proteins in many studies (Bahar and Rader 2005;Ma 2005;Zheng and Brooks 2005;Sen et al 2006;Tama and Brooks 2006;Jernigan and Kloczkowski 2007;Yang et al 2007Yang et al , 2008. It has been shown that the motions computed using ENMs correspond well to the principal components of molecular dynamics (MD) trajectories (Hayward and de Groot 2008), can aid in molecular structure determination (Wu and Ma 2004), refining of structural models (Delarue and Dumas 2004), and in flexible docking (May and Zacharias 2008;Gerek and Ozkan 2010).…”

Protein Conformational Disorder and Enzyme Catalysis

“…Further facilitating evolution in IDPs is their large fraction of solvent exposed residues; surface residues are generally subject to more frequent substitution than amino acids in the buried protein core [141]. Indeed, sequence variability correlates with protein packing density [142], implying that loosely packed proteins evolve more rapidly than proteins with high tertiary structural content. By comparing genetic distance in protein families, proteins having sizable disordered regions were found to In only two cases did disordered proteins evolve more slowly than their ordered counterparts.…”