Introduction
Basics
Torsion or Dihedral Angle For A Group of Four Atoms
A torsion angle or dihedral angle is the angle between two planes. For four consecutively bonded atoms ABCD, atoms ABC define the first plane and atoms BCD define the second plane. The angle between these two planes (ABC and BCD) is a dihedral or torsion angle. The positive rotation is the clockwise rotation of the vector CD relative to the vector AB when looking in the direction of the BC vector:
When all 4 atoms lie in the same plane and A and D atoms are on the same side relative to BC vector, the torsion angle is zero:
If they are on opposite sides then the angle is 180° or 180°:
The torsion angle, ABCD only alters the distance between atoms A and D; the other interatomic distances are constrained by approximately constant bond lengths and bond angles.
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Protein Backbone Torsion Angles, φ and ψ
The backbone torsion angles are ω, φ and ψ. The φ dihedral angle for residue i is defined by C_{i1}N_{i}Cα_{i}C_{i}; the ψ dihedral angle for residue i is defined by N_{i}Cα_{i}C_{i}N_{i+1}; the ω dihedral angle for residue i is defined by Cα_{i1}C_{i1}N_{i}Cα_{i}. ω is almost always near 180°; although there is some variation dependent on the values of ψ_{i1} and φ. For more details, click here.
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Protein SideChain Torsion Angles, χ_{1}, χ_{2}, χ_{3} and χ_{4}
The sidechain torsion angles, χ_{1}, χ_{2}, χ_{3} and χ_{4} define a sidechain conformation. For example, in the case of lysine, χ_{1} is NCαCβCγ and defines a rotation around the CαCβ bond. χ_{2} is CαCβCγCδ and defines a rotation around the CβCγ bond. χ_{3} is CβCγCδCε and defines a rotation around the CγCδ bond. χ_{4} is CγCδCεNζ and defines rotation around the CδCε.
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Traditional Rotamer Library
Rotamers, rotameric χ
Most sidechain χ torsion angles, are centered on sp3sp3 hybridized bonds, and exhibit three narrow, approximately symmetric peaks in their probability density distributions. For example, here is the backboneindependent density for methionine χ_{1}:
MET χ_{1} has gauche+ (g+), trans (t), and gauche (g) peaks at approximately 60°;, 180°, and 300° respectively. 300° for g is the same as 60°. The location and shape of these peaks vary somewhat (by at most 20°) a residue backbone conformation changes, and their standard deviations are at usually less than 10° in highresolution structures. It therefore makes sense to define a set of discrete sidechain conformations (χ_{1}, χ_{2}, χ_{3} and χ_{4}), also known as rotamers. We refer to such χ degrees of freedom as rotameric. The prevailing majority of sidechain χ angles are rotameric, comprising the following:
ARG χ1, χ2, χ3, χ4 HIS χ1 PRO χ1
ASN χ1 ILE χ1 SER χ1
ASP χ1 LEU χ1, χ2 THR χ1
CYS χ1 LYS χ1, χ2, χ3, χ4 TRP χ1
GLN χ1, χ2 MET χ1, χ2, χ3 TYR χ1
GLU χ1, χ2 PHE χ1 VAL χ1
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Nonrotameric χ.
Nevertheless, not all sidechain χ angles adhere to a concept of a rotamer. Nonrotameric degrees of freedom in protein side chains are centered on sp3sp2 bonds, and exhibit broad and often asymmetric probability density distributions. These nonrotameric degrees of freedom exist in ASN, ASP, GLN, GLU and the aromatic amino acids PHE, TYR, HIS and TRP. Each of these residue types has one or two (in the case of GLN and GLU) rotameric degrees of freedom nearer the backbone, while the terminal degree of freedom is nonrotameric. The shape and location of the distributions of these nonrotameric dihedral angles vary depending on the backbone conformation and the rotamer of the nonterminal degrees of freedom. As examples, we show the backboneindependent χ_{2} probability densities for asparagine and tryptophan for each of the three χ_{1} rotamers (<g+>, <t>, <g>) of these residue types and χ_{3} densities for all 9 χ_{1}, χ_{2} rotamers of glutamine (<g+, g+>, <g+, t>, ... and <g, g>):
Here is a gif animation illustrating how χ_{2} density varies as a function of φ and ψ for the trans χ_{1} rotamer of ASN. The Ramachandran density for the trans χ_{1} of ASN is shown in the inset in the topright corner, ρ(φ, ψ  r = <t>) showing the major secondarystructure conformational regions: αhelix, antiparallel βsheet, parallel βsheet and lefthanded helix. A brighter color corresponds to a higher density. The red box on the inset figure and the numeric values in the title indicate the current backbone conformation (φ, ψ). As the backbone conformation changes we can see several different χ_{2} populations dominating in different (φ, ψ) regions. The positions of their maximum values move with φ and ψ and sometimes they coalesce into a single wide population. You can watch movies for all nonrotameric degrees of freedom here.
ASN, r = <t>
The nonrotameric degrees of freedom are:
ASN χ2 GLN χ3 PHE χ2 HIS χ2
ASP χ2 GLU χ3 TYR χ2 TRP χ2
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Traditional backbonedependent rotamer library
Under the term "traditional rotamer library", we mean a library of discrete sidechain conformations only, i.e. rotamers. The traditional rotamer library includes: rotamer frequencies, their mean torsion angles and standard deviations. These values vary significantly as a residue backbone conformation changes. Therefore, a traditional backbonedependent rotamer library contains values for rotamer frequencies, mean χ angles and standard deviations as a function of the backbone torsion angles, φ and ψ.
In this traditional rotamer model, each sidechain χ has a set of discrete conformations, i.e. rotamers. For instance, serine has only one degree of freedom, χ_{1}. Its χ_{1} has 3 rotamers with mean values about +60°, 180° and 60°. Therefore, for serine there are 3 rotamers in total: <g+>, <t> and <g>. In contrast, leucine has two sidechain degrees of freedom, i.e. χ_{1} and χ_{2}. Each of them has its own 3 rotamers, g+, t and g, producing 3 × 3 = 9 rotamers in total: <g+, g+>, <g+, t>, <g+, g>, <t, g+>, <t, t>, <t, g>, <g, g+>, <g, t> and <g, g>.
We can designate g+, t and g rotamers as 1, 2 and 3 respectively for serine χ_{1}, leucine χ_{1} and leucine χ_{2}. Thus serine has <1>, <2> and <3> rotamers while leucine has <1, 1>, <1, 2>, ..., <3, 3>. We can use the same number designations for the rotamers of the 18 standard amino acids with flexible side chains. For example, arginine and lysine have 3 × 3 × 3 × 3 = 81 rotamers in total (<1, 1, 1, 1>, <1, 1, 1, 2> ... <3, 3, 3, 3>). For the nonrotameric degrees of freedom, the angular space can be divided into some number of bins to approximate a rotamer model. For example, for ASN χ_{2} in 1997 we used three rotamers to represent the dihedral angle distribution. In 2002, we expanded this to six rotamers for χ_{2}, so ASN had 3 × 6 = 18 (<1, 1>, <1, 2>, ... <3, 6>) rotamers. For the 2010 rotamer names, definitions and total counts of 18 amino acids please refer to the table below:
Rotamer library data and Rotamer definitions
^{*} TPR (trans proline), CPR (cis proline) and CYH (nondisulfidebonded Cys), CYD (disulfidebonded Cys) percentages are calculated relative to the total number of PRO and CYS respectively. Each rotameric degree of freedom, χ has the rotamer definitions: g+ = [0°, 120°), t = [120°,240°), g = [240°,360°). ^{†} PRO, CPR, and TPR have only g+ and g rotamers.
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Rotameric Vs Nonrotameric χ angles
In the new, 2010 library, we strictly distinguish between rotameric and nonrotameric degrees of sidechain freedom based on the hybridization state of the atoms involved in a corresponding torsion angle, χ.
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2010 Library
Aims and Features
There were several aims in deriving a new backbonedependent rotamer library:
 taking advantage of the much larger dataset that is available now than at the time of the last library (2002)
 using electron density calculations to remove highly dynamic side chains (or protein segments) that have uncertain conformations or coordinates (Shapovalov and
Dunbrack, 2007)
 deriving accurate and smooth density estimates of rotamer populations and their relative frequencies, including rare rotamers, as a continuous function of backbone dihedral angles
 deriving smooth estimates of the mean values and variances of rotameric sidechain dihedral angles
 improving the treatment of nonrotameric degrees of freedom, i.e. those are not well described by the rotamer model
 employing methods producing meaningful estimates of rotamer frequencies, dihedral angles means and variances in the Ramachandran areas lacking experimental data.
By smooth estimates, we mean estimates that are mathematically smooth functions, i.e. continuously differentiable functions.
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Methods in a Nutshell
We applied adaptive kernel density estimates to compute backbonedependent probabilities of sidechain rotamers and adaptive kernel regression to estimate backbonedependent mean χ angles and their standard deviations. The central concept of these methods is to put a bellshaped curve, i.e. a kernel function on top of each experimental data point from a data set. This action converts a set of discrete data points to a set of continuous objects, i.e. smooth functions. The bellshaped curve has a bandwidth which locally adapts depending on the amount of available data. Since we are deriving for rotamer probabilities, means and standard deviations as a function of φ and ψ, we used the periodic von Mises probability distribution as the kernel choice rather than Gaussians or other nonperiodic kernels. For the nonrotameric degrees of freedom, we developed an innovative adaptive kernel regression of adaptive kernel density. Here we place a kernel on top of each experimental data point not only in φ and ψ space but also in χ space in such a way that for any φ and ψ a combination of χ kernels leads to a local estimate of χ probability density. We chose an appropriate method to adapt these kernels in the φ,ψ space and χ space as well to produce statistically sound estimates and at the same time not to lose local details of the side chain model. For a complete description of the methods, please click here.
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Continuous functions Vs Subsequent Discretization on a Grid
We developed methods that enable estimation estimation of rotamer statistics or nonrotameric χ density as continuous function of φ and ψ, i.e. at any numerical values of φ and ψ. However, we evaluated the rotamer probabilities and angle statistics on a 10° grid in φ and ψ in order to provide a rotamer library that would be compatible with methods that require a fast lookup of these values, such as Rosetta or SCWRL. The text files (which can be converted to binary files) are a good tradeoff between memory overhead and experimental precision in structures. The advantages are backward compatibility, relatively low memory overhead, no CPU overhead, no restriction to a particular platform, operating system, computer language. There are several disadvantages. Users have to implement their own parsers. There is reduced accuracy of the estimated statistics at points not on the grid and decreased accuracy of derivatives when these are calculated from neighboring grid values. In the future, we will provide code that produces rotamer statistics at any value of φ and ψ, but this requires significant optimization and accurate numerical methods for derivative computation, etc.
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2010: Traditional Rotamer Model Vs. New Model of χ Densities
In 2010 we provide two different types of backbonedependent sidechain models: 1) the traditional rotamer model as in 1997 and 2002 for both rotameric and nonrotameric degrees of freedom (in 30° bins); 2) the traditional model for the rotameric degrees of freedom but also full probability density estimates for the nonrotameric degrees of freedom as a function of φ and ψ. The traditional model provides support for existing applications. Its format has remained the same as the 2002 libraries. While the traditional 2010 library demonstrates improved performance over the 2002 library, there are applications where a backbonedependent model of χ densities may provide increased accuracy. It is also possible to use both, the traditional rotamer library in the first stage of modeling and full χ densities at the second, refinement stage.
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Download Options for 2010 "Traditional Rotamer Library" Packages
For a definition and description of Traditional Rotamer Library, see above.
The traditional backbonedependent rotamer library is available as a single larger file. There are 18 standard amino acid types with (glycine and alanine do not have flexible side chains). There are three choices to make:
 Choice #1: subclasses for proline and cysteine or not
 PRO is a single category trans or cis. CYS is a single category whether or not disulfidebonded.
In total 18 residue categories.
 PRO has 3 categories: TRP (trans PRO), CPR (cis PRO) and PRO (TRP+CPR).
CYS has 3 categories: CYH (nondisulfidebonded CYS), CYD (disulfidebonded CYS) and CYS (CYH+CYD). In total 22 residue categories.
 Choice #2: rotamer bins for 8 nonrotameric χ are
 backbonedependent  vary with φ, ψ such that the mode is the center of a bin
 backboneindependent  fixed for all φ, ψ
 Choice #3: the level of smoothness of the rotamer library. 0% contains the optimized kernel width according to the maximum log likelihoods, while 2%, 5%, 10%, 20% and 25% are increasing levels of smoothness. The 5% libraries represent a good tradeoff of rotamer detail and smoothly varying probabilities and dihedral angles:
 5% (default choice)
 0%
 2%
 10%
 20%
 25%
A user decides whether additional subclasses are needed for proline and cysteine in their specific applications. While the differences between trans PRO and cis PRO or between nondisulfidebonded and disulfidebonded CYS are not as drastic as between different residue types, they are still significant and may lead to improved accuracy.
As discussed above, the density distributions for the nonrotameric χ angles demonstrate backbone dependence, i.e. the shape and location of the density peaks significantly vary as φ and ψ change. For this reason the "rotamer" definitions, i.e. their left and right limits vary as a function of φ and ψ. We provide two options. The first option has dynamic or backbonedependent "rotamer" definitions for the 8 nonrotameric χ angles. The second option has static or backboneindependent "rotamer" definitions. In applications where a user does not care about assigning a rotamer type to an experimental sidechain conformation or does not need to calculate rotamerspecific derivatives, the first option with backbonedependent "rotamer" definitions is preferable, since the mode of the distribution is centered in the maximum probability bin. In contrast, for the applications where a user wants to query a likelihood of some sidechain conformation or wants a rotamer, for example GLN <1,3,7> to remain the same, i.e. with the same left and right limits, the second choice with the backboneindependent definitions is desirable. We provide 8 small files for each nonrotameric χ with these backboneindependent definitions.
As shown above, the statistical methods used in the 2010 library are completely different from the ones used in 1997 or 2002. The 1997 and 2002 libraries relied on Bayesian formalism while the 2010 library takes advantage of adaptive kernel density estimates, kernel regressions and adaptive kernel regressions of densities. These kernel methods were chosen to produce smoother and more accurate libraries than in 1997/2002. The level of smoothness was separately optimized for each residue type, each degree of freedom and each library component. However, the library consists of many different components and the final performance is an interplay of separate components. Our benchmark tests in SCWRL4 and Rosetta demonstrated the additional 5% smoothness lead to improved accuracy in these applications. The 5% library is our suggested, default choice. Nevertheless, there are some applications where a user can benefit from increased or decreased smoothness. Additional testing is required for such applications. For example, there is a scenario when a user may introduce the smoothest, 25% library in initial modeling and at the final stage of modeling during the refinement switch to 0% library with the greatest amount of details.
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New Model of χ Densities
For the 8 nonrotameric χ angles, we developed a new model describing their internal properties in a statistically more appropriate way. Each nonrotameric χ is modeled with onedimensional probability density distribution varying as a function of the backbone φ and ψ. The nonrotameric χ is always the last χ in these sidechain types, i.e. χ_{n}; the preceding χ angles are rotameric, i.e. χ_{1}, χ_{2}, ... χ_{n1}. There are therefore N_{tot} = N_{1} × N_{2} × ... × N_{n1} rotamers for each side chain with a nonrotameric degree of freedom. Separately for each of the N_{tot} rotamers, we estimated the χ_{n} probability density as a function of φ and ψ, i.e. ρ(χ_{n}  r, φ, ψ).
The remaining residue types with all rotameric χ angles are modeled as a Traditional Rotamer Library. Their rotamer model allows for quick traversal through discrete sidechain conformation space which is a good statistical model for sp3sp3 hybridized bonds.
In future we may consider modeling all χ angles (whether rotameric or nonrotameric) for all residue types as backbonedependent χ densities along with the traditional rotamer model which obviously has advantages in exhaustive searches of conformation space.
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Download Options for 2010 "Model of χ Densities" Packages
The new model of χ densities has separate file(s) for each residue type. For each rotameric residue type there is always a single file with backbonedependent rotamer probabilities (Traditional Rotamer Model). For each nonrotameric residue type we provide three files. The first file contains the χ Densities. The second and third files purely belong to Traditional Rotamer Model. They are provided for convenience, so that the user can easily switch between two models for the nonrotameric residue types if needed.
 A text file with χ_{n} densities as a function of φ, ψ and rotamer, r_{n} = 1, 2 ... N_{1} × N_{2} × ... × N_{n1} along with backbonedependent rotamer probability for each such rotamer, r_{n}.
 A text file with backbonedependent rotamer probabilities for all "rotamers", r = 1, 2 ... N_{1} × N_{2} × ... × N_{n1} × N_{n}.
 A small text file with backboneindependent χ_{n} "rotamer" definitions used in 2.
We provide separate file(s) for 22 residue types:
14:
ARG, ILE, LEU, LYS, MET, SER, THR, VAL
CYS, CYH, CYD
PRO, TPR, CPR
8:
ASN, ASP, GLN, GLU, HIS, PHE, TRP, TYR
A user can choose additional smoothness applied to all components of a rotamer library at the level of
 5% (default choice)
 0%
 2%
 10%
 20%
 25%
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Format
General
Depending on what library model a user chooses from a list, several files in different text formats are available from a distributive package. The names and formats of the files are selfexplanatory. A header of each file describes what is included in a file and some important options and their values used in a generated library. When parsing a file, a user can always skip any commentary information by ignoring lines starting with "#":
# Backbonedependent rotamer library with regular rotamers
#
# phi interval, deg [180.0, 180.0]
# phi step, deg 10.0
# psi interval, deg [180.0, 180.0]
# psi step, deg 10.0
#
# Residue type MET
#
# Rotamer probability precision 0.000001
#
# Number of chi angles (degrees of freedom) 3
# Number of chi angles treated as discrete 3
# Number of bins for each discrete chi angle [3, 3, 3]
# Number of rotamers for discrete chi angles 27
# Number of chi angles treated as continuous 0
#
# TotalDatapointsNum 12240
The actual data is included on the lines not starting with "#". The library data are presented in a table form either space or tab delimited. The beginning of the table is preceded with a "#", commentary line showing selfdescriptive titles for each of its columns:
ser.bbdep.rotamers.lib
# T Phi Psi Count r1 r2 r3 r4 Probabil chi1Val chi2Val chi3Val chi4Val chi1Sig chi2Sig chi3Sig chi4Sig
#
SER 180 180 19 1 0 0 0 0.802596 68.0 0.0 0.0 0.0 8.1 0.0 0.0 0.0
SER 180 180 19 2 0 0 0 0.197211 175.4 0.0 0.0 0.0 10.1 0.0 0.0 0.0
SER 180 180 19 3 0 0 0 0.000193 62.9 0.0 0.0 0.0 8.6 0.0 0.0 0.0
asn.bbind.chi2.Definitions.lib
# r1 r2 r3 r4 P logP left chi2 right
1 1 0 0 0.236 1.443 7.000 7.526 23.000
1 2 0 0 0.195 1.634 23.000 36.529 53.000
asp.bbdep.densities.lib
# T Phi Psi Count r1 Probabil chi1Val chi1Sig 90 85 80 ... 85
#
ASP 80 10 404 3 0.630735 68.8 7.7 4.505e003 6.261e003 8.664e003 ... 3.256e003
ASP 80 10 404 1 0.342077 62.7 7.2 8.006e003 1.022e002 1.289e002 ... 6.260e003
ASP 80 10 404 2 0.027188 166.9 11.9 2.047e002 1.350e002 8.825e003 ... 3.024e002
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"Traditional Rotamer Model" Package
Main File Format
with rotamer probabilities, mean and sigma of rotameric χ angles
When the 2010 library is used according to a traditional model of discrete conformations, i.e. rotamer model, there is a large text file with merged data for either 18 or 22 residue types (see above). We preserved the original format of the 1997 or 2002 text libraries in order to support older, existing applications:
ALL.bbdep.rotamers.lib
# T Phi Psi Count r1 r2 r3 r4 Probabil chi1Val chi2Val chi3Val chi4Val chi1Sig chi2Sig chi3Sig chi4Sig
#
LEU 70 40 5884 3 2 0 0 0.668530 68.3 173.0 0.0 0.0 6.5 8.1 0.0 0.0
LEU 70 40 5884 2 1 0 0 0.296974 177.4 58.8 0.0 0.0 7.2 6.1 0.0 0.0
LEU 70 40 5884 3 1 0 0 0.014762 88.5 60.0 0.0 0.0 7.7 10.2 0.0 0.0
LEU 70 40 5884 2 2 0 0 0.009079 175.1 153.0 0.0 0.0 9.3 10.0 0.0 0.0
LEU 70 40 5884 3 3 0 0 0.007372 89.0 62.2 0.0 0.0 9.2 16.8 0.0 0.0
LEU 70 40 5884 2 3 0 0 0.001641 174.6 78.6 0.0 0.0 9.5 11.9 0.0 0.0
LEU 70 40 5884 1 1 0 0 0.001614 72.2 85.1 0.0 0.0 6.8 7.0 0.0 0.0
LEU 70 40 5884 1 3 0 0 0.000020 70.3 63.0 0.0 0.0 9.2 21.3 0.0 0.0
LEU 70 40 5884 1 2 0 0 0.000009 72.1 165.9 0.0 0.0 10.1 13.3 0.0 0.0
T  threeletter designation of an amino acid type.
Phi  torsion angle value for backbone φ, C_{i1}N_{i}Cα_{i}C_{i} in a [180, 180]° range.
Psi  torsion angle value for backbone ψ, N_{i}Cα_{i}C_{i}N_{i+1} in a [180, 180]° range.
As in the previous versions of a rotamer library there is redundancy for reported values of backbone φ and ψ since both of them cycle from 180 up to 180 included. A user can ignore either 180 or 180 for φ and ψ or use the redundant values as checkpoints to catch possible parsing errors. All statistics are provided exactly at the reported φ, ψ. For any φ, ψ off the grid points the neareast grid point can be used or a bilinear interpolation of the four nearest grid points.
Count  only to support the older text format: the number of (φ_{i}, ψ_{i}) experimental points from a data set within a 10° × 10° bin centered on the reported (φ, ψ). The data set is the set of experimental data used to generate the rotamer library.
r1  a numerical designation of a χ_{1} rotamer, i.e. 1, 2 .. N_{1}. When a residue type doesn't have χ_{1}, 0 is reported.
r2  a numerical designation of a χ_{2} rotamer, i.e. 1, 2 .. N_{2}. When a residue type doesn't have χ_{2}, 0 is reported.
r3  a numerical designation of a χ_{3} rotamer, i.e. 1, 2 .. N_{3}. When a residue type doesn't have χ_{3}, 0 is reported.
r4  a numerical designation of a χ_{4} rotamer, i.e. 1, 2 .. N_{4}. When a residue type doesn't have χ_{4}, 0 is reported.
The rotamer types are sorted according to their backbonedependent probability, Probabil, see below.
Probabil  a probability of a rotamer, r = <r_{1}, r_{2}, r_{3}, r_{4}> given a backbone conformation (φ, ψ), i.e. P(r  φ,ψ). The sum of probabilities, P(r_{j}  φ,ψ) j = 1 .. N_{1} × N_{2} × N_{3} × N_{4}, is always equal to 1 for any (φ, ψ).
Caution #1
Not all possible rotamer types may present in the table. If we did not have such a rotamer type in our experimental data set than its probability is absolute zero across all φ and ψ values. For such rotamer types there are no lines. Some rotamer types are so rare (a few data points in our data set) that they have 0.000000 probability reported for some (φ, ψ). Be cautious when using such rare rotamers, log(0.000000) may lead to NaN, Inf and error exception. As a solution, a user may load all rotamers up to some threshold, for example up to 99.9%, 99%, 98% or 95%. It will help to resolve the log(0.000000) error and also ignore extremely rare and physically unrealistic rotamers and speed up your calculations.
Caution #2
The rotamer probabilities are simply frequencies of a set of possible sidechain conformations which add up to 1. These frequencies are backbonedependent, i.e. vary as a function of a backbone conformation, (φ, ψ). A user can investigate any backbone conformation and have an estimate of the said sidechain conformation frequencies. However, a complete understanding has to be: most of the backbone conformations from Ramachandran map are merely physically impossible. They have large steric clashes with its own backbone and/or side chain. For such conformations a distribution of rotamer frequencies do not play a role, such backbone conformations cannot exist. For this reason, a rotamer library, i.e. a library of sidechain conformations cannot be used alone in applications involving backbone perturbation/modeling. When a protein backbone is not known, a Ramachandran probability has to be used in addition, i.e. a probability of a backbone conformation (φ, ψ) for some amino acid type, P(φ, ψ). P(φ, ψ) integrates to 1 over full ranges of φ and ψ. For downloading and learning more on neighborindependent and neighbordependent Ramachandran probabilities, a user may refer to our study here.
Caution #3
The rotamer library should not be used in a way where all rotamers are treated equally, i.e. disregarding of their probabilities or energies. Either the probabilities should be used (as log(Probabil)) or some other energy function should be used to distinguish the rotamers.
chi1Val  a mean value of a sidechain torsion angle, χ_{1}, reported for a given backbone conformation (φ, ψ) and rotamer, r = <r_{1}, r_{2}, r_{3}, r_{4}>. Most amino acid types have mean χ angles around the canonical 60°, 180° and 60° values. Due to an interaction of a side chain with its own backbone the mean χ values can deviate from the canonical ones. The reported χ angles demonstrate this. The mean χ_{1} range is [180, 180]°.
chi2Val  a mean value of a sidechain torsion angle, χ_{2} with a range of [180, 180]°.
chi3Val  a mean value of a sidechain torsion angle, χ_{3} with a range of [180, 180]°.
chi4Val  a mean value of a sidechain torsion angle, χ_{4} with a range of [180, 180]°.
Amino acid types not having χ_{2} and/or χ_{3} and/or χ_{4} have 0.0 reported across the whole column.
chi1Sig  a standard deviation, i.e. sigma = sqrt(variance) of χ_{1} for a given (φ, ψ) and rotamer, r. This standard deviation characterizes the width of χ spread around its mean χ value, chiVal. The χ spread width also varies as a function of a backbone, conformation (φ, ψ) owing to side chainbackbone interaction.
chi2Sig  a standard deviation of χ_{2}.
chi3Sig  a standard deviation of χ_{3}.
chi4Sig  a standard deviation of χ_{4}.
Amino acid types not having χ_{2} and/or χ_{3} and/or χ_{4} have 0.0 reported across the whole column.
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Optional File Formatwith backboneindependent "rotamer" definitions for nonrotameric χ angles
For each of the 8 nonrotameric χ_{n} we provide backboneindependent "rotamer" definitions, meaning the definitions are static and do not change as φ and ψ vary. Here is the format description based on glutamic acid:
glu.bbind.chi2.Definitions.lib
# r1 r2 r3 r4 P logP left chi3 right
1 1 1 0 0.406 0.900 5.500 20.276 35.500
1 1 2 0 0.218 1.525 35.500 47.346 65.500
1 1 3 0 0.072 2.631 65.500 78.259 95.500
1 1 4 0 0.041 3.187 84.500 70.468 54.500
1 1 5 0 0.056 2.888 54.500 37.366 24.500
1 1 6 0 0.207 1.575 24.500 6.564 5.500
2 1 1 0 0.423 0.860 6.500 21.331 36.500
2 1 2 0 0.237 1.440 36.500 48.692 66.500
2 1 3 0 0.081 2.512 66.500 78.719 96.500
2 1 4 0 0.034 3.377 83.500 70.282 53.500
2 1 5 0 0.038 3.266 53.500 36.204 23.500
2 1 6 0 0.186 1.680 23.500 4.851 6.500
3 1 1 0 0.350 1.049 13.500 1.292 16.500
3 1 2 0 0.240 1.427 16.500 29.564 46.500
3 1 3 0 0.102 2.280 46.500 58.574 76.500
3 1 4 0 0.038 3.273 76.500 89.996 106.500
3 1 5 0 0.063 2.764 73.500 55.812 43.500
3 1 6 0 0.206 1.578 43.500 26.206 13.500
r1  a numerical designation of a χ_{1} rotamer, i.e. 1, 2 .. N_{1}. When a residue type doesn't have χ_{1}, 0 is reported.
r2  a numerical designation of a χ_{2} rotamer, i.e. 1, 2 .. N_{2}. When a residue type doesn't have χ_{2}, 0 is reported.
r3  a numerical designation of a χ_{3} rotamer, i.e. 1, 2 .. N_{3}. When a residue type doesn't have χ_{3}, 0 is reported.
r4  a numerical designation of a χ_{4} rotamer, i.e. 1, 2 .. N_{4}. When a residue type doesn't have χ_{4}, 0 is reported.
The rotamer types are sorted according to their backboneindependent probability, P, see below.
P  a backboneindependent probability of a rotamer, r = <r_{1}, r_{2}, r_{3}, r_{4}>, i.e. P(r). The sum of probabilities, P(r_{j}), j = 1 .. N_{1} × N_{2} × N_{3} × N_{4}, is equal to 1.
logP  minus of log_{10}(P)
left  a left limit of a definition for a rotamer, r =<r_{1}, r_{2}, r_{3}, r_{4}>.
chi3  a backboneindependent mean value for χ_{n}, i.e. χ_{3} in the case of GLU, for the rotamer, r = <r_{1}, r_{2}, r_{3}, r_{4}>.
right  a right limit of a definition for the rotamer, r =<r_{1}, r_{2}, r_{3}, r_{4}>.
A χ_{n} rotamer definition is specified with an interval [left, right); any experimental χ_{n} point lying within this interval is said to be in such χ_{n} rotamer conformation.
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"New Model of χ Densities" Package
Rotameric Residue Types
In the package, "New Model of χ Densities" available for downloading, the rotameric residue types are always modeled with Traditional Rotamer Model. The only difference here is that they are provided as separate files for each type instead of one large merged file. A user can decide on which subclasses of proline or cysteine to use.
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NonRotameric Residue Types
For each nonrotameric residue types, we provide three files:
Only one file comes from the New Model of χ Densities; it contains nonrotameric χ_{n} densities as a function of φ, ψ and rotamer r, r = 1, 2 ... N_{1} × N_{2} × ... N_{n1} along with backbonedependent rotamer probability for each such rotamer, r. Its format is different but somewhat similar and described below.
The other two files belong to the Traditional Rotamer Model and contain backbonedependent rotamer probabilities and backboneindependent nonrotameric χ_{n} definitions respectively. They are provided for convenience so that a user can switch between the Traditional Rotamer Model and the New Model of χ Densities for them. Their formats are the same and described above in Traditional Rotamer Model section of Format.
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"χ_{n} Densities" File Formatwith densities for nonrotameric χ_{n}, rotamer probabilities and mean and sigma of rotameric χ angles
tyr.bbdep.densities.lib
# T Phi Psi Count r1 Probabil chi1Val chi1Sig 30 25 20 ... 145
#
TYR 60 40 1769 2 0.558459 178.1 10.3 6.533e004 6.394e004 6.673e004 ... 7.316e004
TYR 60 40 1769 3 0.418968 72.7 11.3 3.303e002 3.011e002 2.714e002 ... 3.616e002
TYR 60 40 1769 1 0.022572 75.1 12.6 6.880e004 5.504e004 4.554e004 ... 9.006e004
T  threeletter designation of an amino acid type.
Phi  torsion angle value for backbone φ, C_{i1}N_{i}Cα_{i}C_{i} in a [180, 180]° range.
Psi  torsion angle value for backbone ψ, N_{i}Cα_{i}C_{i}N_{i+1} in a [180, 180]° range.
As in the previous versions of a rotamer library there is redundancy for reported values of backbone φ and ψ since both of them cycle from 180 up to 180 included. A user can ignore either 180 or 180 for φ and ψ or use the redundant values as checkpoints to catch possible parsing errors. All statistics are provided exactly at the reported φ, ψ. Using the new methods, there is no concept of a bin, so that the statistics are not estimated in a middle of bins.
Count  only as a reference to the older text format: the number of (φ_{i}, ψ_{i}) experimental points from a data set within a 10° × 10° bin centered on the reported (φ, ψ). The data set is the set of experimental data used to generate the rotamer library. The count also provides a practical understanding how many experimental data points were available within proximity of the reported (φ, ψ) at the time the library was compiled.
r1  a numerical designation of a χ_{1} rotamer, i.e. 1, 2 .. N_{1}.
r2  a numerical designation of a χ_{2} rotamer, i.e. 1, 2 .. N_{2}.
...
r(n1)  a numerical designation of a χ_{n1} rotamer, i.e. 1, 2 .. N_{n1}.
The rotamer types, r = 1, 2, ..., N_{1} × N_{2} × ... × N_{n1} are sorted according to their backbonedependent probability, Probabil, see below.
Probabil  a probability of a rotamer, r = <r_{1}, r_{2}, ... r_{n1}> given a backbone conformation (φ, ψ), i.e. P(r  φ,ψ). The sum of probabilities, P(r_{j}  φ,ψ) j = 1 .. N_{1} × N_{2} × ... × N_{n1}, is always equal to 1 for any (φ, ψ).
chi1Val  a mean value of a sidechain torsion angle, χ_{1}, reported for a given backbone conformation (φ, ψ) and rotamer, r = <r_{1}, r_{2}, ... r_{n1}>. The mean χ_{1} range is [180, 180]°.
chi2Val  a mean value of a sidechain torsion angle, χ_{2} with a range of [180, 180]°.
...
chi(n1)Val  a mean value of a sidechain torsion angle, χ_{n1} with a range of [180, 180]°.
chi1Sig  a standard deviation, i.e. sigma = sqrt(variance) of χ_{1} for a given (φ, ψ) and rotamer, r.
chi2Sig  a standard deviation of χ_{2}.
...
chi(n1)Sig  a standard deviation of χ_{n1}.
min(χ_{n})
 I.e. for all 8 nonrotameric χ_{n},

min(χ_{n}) + step(χ_{n})
 ASN χ_{2}: 180, 170, ..., 160, 170

...
 GLN χ_{3}: 180, 170, ..., 160, 170

max(χ_{n})  2 × step(χ_{n})
 TRP χ_{2}: 180, 170, ..., 160, 170

max(χ_{n})  step(χ_{n})
 HIS χ_{2}: 180, 170, ..., 160, 170

 ASP χ_{2}: 90, 85, ..., 80, 85

 GLU χ_{3}: 90, 85, ..., 80, 85

 PHE χ_{2}: 30, 25, ..., 140, 145

 TYR χ_{2}: 30, 25, ..., 140, 145

 Integrated χ_{n} density is provided for each of these χ_{n} values covering the whole χ_{n} period. For a rotamer r, r = <r_{1}, r_{2}, ... r_{n1}>, ρ(χ_{n}  r, φ, ψ) is integrated over a χ_{n} interval centered at the reported χ_{n} value. Basically, during this χ_{n} discretization, we report χ_{n} probabilities, P[ χ_{n} ± ½ × step(χ_{n}) ] at these χ_{n} values. For any nonrotameric χ_{n} we provide probabilties at exactly 36 χ_{n} values. All 36 probabilities add up to a probability of 1.
ASN χ_{2}, GLN χ_{3}, TRP χ_{2} and HIS χ_{2} have a nonsymmetrical chemical group at their torsion rotation. That is why their period is full 360° and a step of 10°. All the rest nonrotameric χ_{n} have twofold symmetry and demonstrate a period of 180° and step of 5°.

A number of columns for r1, r2 ... r(n1) or chi1Val, chi2Val ... chi(n1)Val or chi1Sig, chi2Sig ... chi(n1)Sig is the same as the number of the rotameric χ angles for a residue type. For easier parsing a user can read this number beforehand from the commentary section lines which are in the following fixed format:
glu.bbdep.densities.lib
# Number of chi angles (degrees of freedom) 3
# Number of chi angles treated as discrete 2
# Number of bins for each discrete chi angle [3, 3]
# Number of rotamers for discrete chi angles 9
# Number of chi angles treated as continuous 1
The number of the nonrotameric χ_{n} integrated probabilities, P[ χ_{n} ± ½ × step(χ_{n}) ] at the χ_{n} values is always 36. The stepsize, starting point and ending point of the χ_{n} period interval can be parsed beforehand from either of two places with a fixed format in the commentary section.
glu.bbdep.densities.lib
# chi3 interval, deg [90.0, 90.0]
# chi3 step, deg 5.0
glu.bbdep.densities.lib
# T Phi Psi Count r1 Probabil chi1Val chi1Sig 90 85 80 ... 85
We chose such starting and ending χ_{n} because in the 1997 / 2002 libraries some "rotamers" had such staring or ending positions and in addition the χ_{n} distributions are better viewed with such limits.
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Please cite this paper when publishing results based on our library:
Shapovalov, M.S., and Dunbrack, R.L., Jr. (2011). A smoothed backbonedependent rotamer library for proteins derived from adaptive kernel density estimates and regressions. Structure, 19, 844858.
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