Backbone-dependent rotamer library

History

The first backbone-dependent rotamer library was developed in 1993 by Roland Dunbrack to assist the prediction of the Cartesian coordinates of a protein's side chains given the experimentally determined or predicted Cartesian coordinates of its main chain. The library was derived from the structures of 132 proteins from the Protein Data Bank with resolution of 2.0 Å or better. The library provided the counts and frequencies of χ₁ or χ₁+χ₂ rotamers of 18 amino acids (excluding glycine and alanine residue types, since they do not have a χ₁ dihedral) for each 20° x 20° bin of the Ramachandran map (φ,ψ = -180° to -160°, -160° to -140° etc.).

In 1997, Dunbrack and Fred E. Cohen at the University of California, San Francisco presented a backbone-dependent rotamer library derived from Bayesian statistics. The Bayesian approach provided the opportunity for the definition of a Bayesian prior for the frequencies of rotamers in each 10° x 10° bin derived by assuming that the steric and electrostatic effects of the φ and ψ dihedral angles are independent. In addition, a periodic kernel with 180° periodicity was used to count side chains 180° away in each direction from the bin of interest. As an exponent of a sin² function, it behaved much like a von Mises distribution commonly used in directional statistics. The 1997 library was made publicly available via the World Wide Web in 1997, and found early use in protein structure prediction and protein design. The library derived from Bayesian statistics was updated in 2002

Many modeling programs, such as Rosetta, use a backbone-dependent rotamer library as a scoring function (usually in the form E=-ln(p(rotamer(i) | φ,ψ)) for the ith rotamer, and optimize the backbone conformation of proteins by minimizing the rotamer energy with derivatives of the log probabilities with respect to φ,ψ. This requires smooth probability functions with smooth derivatives, because most mathematical optimization algorithms use first and sometimes second derivatives and will get stuck in local minima on rough surfaces. In 2011, Shapovalov and Dunbrack published a smoothed backbone-dependent rotamer library derived from kernel density estimates and kernel regressions with von Mises distribution kernels on the φ,ψ variables. The treatment of the non-rotameric degrees of freedom (those dihedral angles not about sp³-sp³ bonds, such as asparagine and aspartate χ₂, phenylalanine, tyrosine, histidine, tryptophan χ₂, and glutamine and glutamate χ₃) was improved by modeling the dihedral angle probability density of each of these dihedral angles as a function of χ₁ rotamer (or χ₁ and χ₂ for Gln and Glu) and φ,ψ. The functions are essentially regressions of a periodic probability density on a torus.

In addition to statistical analysis of structures in the Protein Data Bank, backbone-dependent rotamer libraries can also be derived from molecular dynamics simulations of proteins, as demonstrated by the Dynameomics Library from Valerie Daggett's research group. Because these libraries are based on sampling from simulations, they can generate far larger numbers of data points across regions of the Ramachandran map that are sparsely populated in experimental structures, leading to higher statistical significance in these regions. Rotamer libraries derived from simulations are dependent on the force field used in the simulations. The Dynameomics Library is built on simulations using the ENCAD force field of Levitt et al. from 1995.