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Backbone-dependent rotamer library

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Backbone-dependent rotamer library for serine. Each plot shows the population of the χ1 rotamers of serine as a function of the backbone dihedral angles φ and ψ

In biochemistry, the backbone-dependent rotamer library provides the frequencies, mean dihedral angles, and standard deviations of the discrete conformations (rotamers) of the amino-acid side chains in proteins as a function of the backbone dihedral angles φ and ψ of the Ramachandran map. It is used in protein structure prediction and protein design.

History

The first version of the backbone-dependent rotamer library was developed in 1993 by Roland Dunbrack and Martin Karplus 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[1]. 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 Cohen at UCSF presented a new version of the library derived from Bayesian statistics[2]. 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 from the assuming the steric and electrostatic effects of the φ and ψ dihedral angles are independent. The the φ and ψ contributions to each φ,ψ bin were calculated with a singular value decomposition on the log probabilities of all of the populated φ,ψ bins. 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[3] and protein design[4]. The library derived from Bayesian statistics was updated in 2002[5].

Backbone-dependent rotamer library for phenylalanine. Each plot shows the population of the χ1 rotamers of phenylalanine as a function of the backbone dihedral angles φ and ψ

Many modeling programs, such as Rosetta, use the backbone-dependent rotamer library as a scoring function (usually in the form E=-ln(p(φ,ψ)) and optimize the backbone conformation of proteins by minimizing the rotamer energy with derivatives of the log probabilities with respect to φ,ψ[6]. This requires smooth probability functions with well behaved derivatives. In 2011, Shapovalov and Dunbrack published a smoothed version of the backbone-dependent rotamer library derived from kernel density estimates and kernel regressions with von Mises distribution kernels on the φ,ψ variables[7]. The treatment of the non-rotameric degrees of freedom (those dihedral angles not about sp3-sp3 bonds, such Asn and Asp χ2, Phe, Tyr, His, Trp χ2, and Gln and Glu 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.

Conformational analysis of the backbone-dependence of protein side-chain rotamer populations

Steric interactions that affect the backbone-conformation-dependent rotamer preferences of amino acid side chains, shown in a Newman projection

The effect of backbone conformation on side-chain rotamer frequencies is primarily due to steric repulsions between backbone atoms whose position is dependent on φ and ψ and the side-chain γ heavy atoms of each residue type (PDB atom types CG, CG1, CG2, OG, OG1, SG). These occur in predictable combinations that depend on the dihedrals connecting the backbone atoms to the side-chain atoms[8][2]. These steric interactions occur when the connecting dihedral angles form a pair of dihedral angles with values {-60°,+60°} or {+60°,-60°}. For example, the nitrogen atom of residue i+1 is connected to the γ heavy atom of any side chain by a connected set of 5 atoms: N(i+1)-C(i)-Cα(i)-Cβ(i)-Cγ(i). The dihedral angle N(i+1)-C(i)-Cα(i)-Cβ(i) is equal to ψ+120°, and C(i)-Cα(i)-Cβ(i)-Cγ(i) is equal to χ1-120°. When ψ is -60° and χ1 is +60° (the g+ rotamer of a side chain), there is a steric interaction between N(i+1) and Cγ because the dihedral angles connecting them are N(i+1)-C(i)-Cα(i)-Cβ(i) = ψ+120° = +60°, and C(i)-Cα(i)-Cβ(i)-Cγ(i) = χ1-120° = -60°. The same interaction occurs when ψ is 0° and χ1 is 180° (the trans rotamer of a side chain). The carbonyl oxygen of residue i plays the same role when ψ=-60° for the g+ rotamer and when ψ=180° for the trans rotamer. Finally, φ-dependent interactions occur between the side-chain γ heavy atoms in g- and g+ rotamers on the one hand, and the carbonyl carbon of residue i-1 and a γ heavy atom, and between the backbone NH of residue i and its hydrogen-bonding partner on the other.

Side-chain/main-chain steric interactions that affect the Ramachandran plot distributions of amino acids. The data are for the amino acid lysine

The φ,ψ-dependent interactions of backbone atoms and side-chain Cγ atoms can be observed in the distribution of observations in the Ramachandran plot of each each χ1 rotamer (marked in the figure). At these positions, the Ramachandran populations of the rotamers are significantly reduced. They can be summarized as follows:

φ,ψ-dependence of backbone/side-chain interactions
Rotamer N(i+1) O(i)
g+ ψ = -60° ψ = +120°
trans ψ = 180° ψ = 0°
Rotamer C(i-1) HBond to NH(i)
g+ φ = +60° φ = -120°
g- φ = -180° φ = 0°
Backbone-dependent rotamer library for valine. Each plot shows the population of the χ1 rotamers of valine as a function of the backbone dihedral angles φ and ψ

Side-chain types with two heavy atoms (Val, Ile, Thr) have backbone-dependent interactions with both heavy atoms. Val has CG1 at χ1 and CG2 at χ1+120°. Because Val g+ and g- conformations have steric interactions with the backbone near ψ=120° and -60° (the most populated ψ ranges), Val is the only amino acid where the t rotamer (χ1~180°) is the most common. At most values of φ and ψ, only one rotamer of Val is allowed (shown in figure). Ile has CG1 at χ1 and CG2 at χ1-120°. Thr has OG1 at χ1 and CG2 at χ1-120°.


Uses

The backbone-dependent rotamer library is used in a number of programs for protein structure prediction and computational design, including:

References

  1. ^ Dunbrack, RL, Jr.; Karplus, M (1993). "Backbone-dependent rotamer library for proteins. Application to side-chain prediction". Journal of molecular biology. 230: 543–74. doi:10.1006/jmbi.1993.1170. PMID 8464064.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  2. ^ a b Dunbrack, RL, Jr.; Cohen, FE (1997). "Bayesian statistical analysis of protein side-chain rotamer preferences". Protein science. 6 (8): 1661–81. doi:10.1002/pro.5560060807. PMID 9260279.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. ^ Bower, MJ; Cohen, FE; Dunbrack, RL, Jr (1997). "Prediction of protein side-chain rotamers from a backbone-dependent rotamer library: a new homology modeling tool". Journal of molecular biology. 267: 1268–82. doi:10.1006/jmbi.1997.0926. PMID 9150411.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  4. ^ Kuhlman, B; Baker, D (2000). "Native protein sequences are close to optimal for their structures". Proceedings of the National Academy of Sciences of the United States of America. 97: 10383–8. doi:10.1073/pnas.97.19.10383. PMID 10984534.
  5. ^ Dunbrack, RL, Jr (2002). "Rotamer libraries in the 21st century". Current opinion in structural biology. 12 (4): 431–40. doi:10.1016/s0959-440x(02)00344-5. PMID 12163064.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  6. ^ Alford, RF; Leaver-Fay, A; Jeliazkov, JR; O'Meara, MJ; DiMaio, FP; Park, H; Shapovalov, MV; Renfrew, PD; Mulligan, VK; Kappel, K; Labonte, JW; Pacella, MS; Bonneau, R; Bradley, P; Dunbrack, RL; Das, R; Baker, D; Kuhlman, B; Kortemme, T; Gray, JJ (13 June 2017). "The Rosetta All-Atom Energy Function for Macromolecular Modeling and Design". Journal of chemical theory and computation. 13 (6): 3031–3048. doi:10.1021/acs.jctc.7b00125. PMID 28430426.
  7. ^ Shapovalov, MV; Dunbrack, RL, Jr (2011). "A smoothed backbone-dependent rotamer library for proteins derived from adaptive kernel density estimates and regressions". Structure (Cell Press). 19: 844–58. doi:10.1016/j.str.2011.03.019. PMID 21645855.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  8. ^ Dunbrack, RL, Jr.; Karplus, M (1994). "Conformational analysis of the backbone-dependent rotamer preferences of protein sidechains". Nature Structural Biology. 1 (5): 334–340. doi:10.1038/nsb0594-334. ISSN 1545-9985. PMID 7664040.{{cite journal}}: CS1 maint: multiple names: authors list (link)
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