In the eye of the beholder: Inhomogeneous distribution of high-resolution shapes within the random-walk ensemble

Expected distributions of root-mean-square positional deviations in proteins

The atom positional root-mean-square deviation (RMSD) is a standard tool for comparing the similarity of two molecular structures. It is used to characterize the quality of biomolecular simulations, to cluster conformations, and as a reaction coordinate for conformational changes. This work presents an approximate analytic form for the expected distribution of RMSD values for a protein or polymer fluctuating about a stable native structure. The mean and maximum of the expected distribution are independent of chain length for long chains and linearly proportional to the average atom positional root-mean-square fluctuations (RMSF). To approximate the RMSD distribution for random-coil or unfolded ensembles, numerical distributions of RMSD were generated for ensembles of self-avoiding and non-self-avoiding random walks. In both cases, for all reference structures tested for chains more than three monomers long, the distributions have a maximum distant from the origin with a power-law dependence on chain length. The purely entropic nature of this result implies that care must be taken when interpreting stable high-RMSD regions of the free-energy landscape as "intermediates" or well-defined stable states.

Determination of ensemble-average pairwise root mean-square deviation from experimental B-factors

Root mean-square deviation (RMSD) after roto-translational least-squares fitting is a measure of global structural similarity of macromolecules used commonly. On the other hand, experimental x-ray B-factors are used frequently to study local structural heterogeneity and dynamics in macromolecules by providing direct information about root mean-square fluctuations (RMSF) that can also be calculated from molecular dynamics simulations. We provide a mathematical derivation showing that, given a set of conservative assumptions, a root mean-square ensemble-average of an all-against-all distribution of pairwise RMSD for a single molecular species, <RMSD(2)>(1/2), is directly related to average B-factors (<B>) and <RMSF(2)>(1/2). We show this relationship and explore its limits of validity on a heterogeneous ensemble of structures taken from molecular dynamics simulations of villin headpiece generated using distributed-computing techniques and the Folding@Home cluster. Our results provide a basis for quantifying global structural diversity of macromolecules in crystals directly from x-ray experiments, and we show this on a large set of structures taken from the Protein Data Bank. In particular, we show that the ensemble-average pairwise backbone RMSD for a microscopic ensemble underlying a typical protein x-ray structure is approximately 1.1 A, under the assumption that the principal contribution to experimental B-factors is conformational variability.