- Introduction to the Conformational Analysis of Protein Side Chains
- Conformational Analysis of Backbone-Conformation Independent Interactions
- Graphical Views of the Backbone-Independent Rotamer Library
- Histograms of χ
_{1}Rotamer Populations - Histograms of χ
_{1},χ_{2}Rotamer Populations - Kernel density estimates of χ
_{1}distributions - Scatterplots of χ angles
- Kernel density estimates of χ
_{2}for each χ_{1}rotamer type - Conformational Analysis of Backbone-Conformation Dependent Interactions
- Graphical Views of the Backbone-Dependent Rotamer Library
- Ramachandran plots for each χ
_{1}rotamer - Box Plots (Ramachandran Maps) of χ
_{1}Rotamer Preferences - Surface plots of rotamer probabilities
- References

This page presents a detailed description of local steric interactions that influence side-chain rotamer and χ angle choice. It includes graphical representations of the backbone-independent and backbone-dependent rotamer libraries annotated to show where local steric interactions affect rotamer populations and χ angle values. Individual figures for each amino acid type are provided in most cases. The images are all clickable for an enlarged view, and can be downloaded by right-clicking on them. They are also all available in this tar file. These figures can be reused and adapted in any format as long as a reference to this page is given (CC-BY 2.0 license).

The conformational properties of side-chain dihedral angles depend on whether the central two atoms of the dihedral are sp^{3} hybridized (like tetrahedral carbons, hydroxyl oxygens, and amine nitrogens, or not. Aromatic carbon and nitrogen atoms as well as the atoms of amides and carboxylates are sp^{2} hybridized.

As with ethane and butane, the low energy conformations of an sp^{3}-sp^{3} bonds tend to be staggered (60°, 180°, 300°), rather than eclipsed (0°, 120°, 240°) (see figure below) The gauche conformations of butane, "gauche-plus" or +60° and gauche-minus or -60° (which is the same as 300°), are about 0.9 kcal per mol higher in energy than the trans conformation (180°). At room temperature, each of these conformations would represent about 15% of structures each, while the trans conformation would be 70%.

Chains of atoms longer than butane can exhibit steric hindrances that affect the populations of the possible conformations. We can write down the nine possible conformations of pentane, their relative energies (from quantum mechanical calculations), and populations at room temperature:

t,t 0.00 kcal/mol 48.9% t,g- and g-,t 0.87 kcal/mol 11.5% each t,g+ and g+,t 0.87 kcal/mol 11.5% each g-,g- and g+,g+ 1.80 kcal/mol 2.4% each g+,g- and g-,g+ 3.55 kcal/mol 0.1% each

Each gauche dihedral is worth about 0.9 kcal/mol. A conformation with two
of them of the same sign (g-,g- or g+,g+) have energies of about 1.8 kcal/mol,
so the interactions are roughly additive. But when the gauche interactions are of opposite
sign, the energy rises to 3.55 kcal/mol. These are called "syn-pentane conformations,"
which occur when two successive dihedrals in a chain of five atoms are near
+60°,-60° or-60°,+60°. In the syn-pentane conformation,
the first and fifth atoms are too close together, and there is repulsion
between the electrons in the sp^{3} molecular orbitals.

The figure below shows the four unique structures of local energy minima of pentane. The dihedrals derived from ab initio quantum mechanics calculations are given. In the syn-pentane conformation (g+,g-) the dihedrals are significantly skewed from the values of the other conformers (close to 180° and +60°) in order to lower the steric repulsion.

If we compare the energies of pentane conformations with dihedrals {+X,+X} with those with dihedrals of {+X,-X}, the energies of the opposite sign pair are significantly higher than the same-sign pair when 0° < X < 90° degrees. This is especially important in the backbone conformation dependent analysis below. When we look at {ψ,χ_{1}} and {φ,χ_{1}} pairs of dihedrals we have to look at a range around ψ or φ=+60° or -60° that includes the range 0° to 90° and -90° to 0° respectively.

These interactions occur in any hydrocarbon chain and more generally in any chain of five heavy atoms, and can be used to find conformations of protein side chains with repulsive steric interactions with backbone atoms. These interactions can be backbone-conformation independent (the delta heavy atoms with backbone N or C form a 5 atom chain, e.g. N-CA-CB-CG-CD) or backbone-conformation dependent (e.g. C_{i-1}-N-CA-CB-CG). The backbone-conformation independent interactions depend on χ_{1} and χ_{2}. The backbone-conformation dependent interactions depend on φ or ψ and χ_{1}. In the tables below, the various combinations of dihedrals that produce local side-chain/backbone interactions are described.

We define the χ_{1} rotamers ("r1") for all side chains (except Pro) as follows

0° <= χ_{1}< 120° g+ 120° <= χ_{1}< 240° t -120° <= χ_{1}< 0° g-

For Proline,

χThe three χ_{1}>= 0° g+ (or CG-endo) χ_{1}< 0° g- (or CG-exo)

Because of close steric interactions between the delta carbons and the backbone N and C atoms of residue i, some conformations have very low frequency.

In the Newman diagram above, with the alpha carbon (not visible) behind the beta carbon, syn-pentane interactions between delta carbons and the local backbone are shown in red. In each of these cases, the dihedrals (N or C)-CA-CB-CG and CA-CB-CG-CD form approximately +60°,-60° or -60°,+60° pairs of dihedrals.

Derivation of rotamer states with syn-pentane interactions for different types of side chains is detailed just below.

We expect low percentages and large deviations from standard
rotamer values (χ=+60°,180°,-60°) for rotamers with
χ_{1},χ_{2} syn-pentane interactions with the backbone N and
C. These are marked in bold red type in the following tables

We expect 8 backbone/CD1 (or CD2) interactions for Leu ( (CD1+CD2) x (N and C) x 2 syn-pentane interactions (+60°,-60° or -60°,+60°) ). Two of these occur for the g+,g- rotamer.

Aromatic side-chain χ_{2}'s are perturbed by interaction
between XD1 and XD2 and backbone N and C. Without perturbation,
χ_{2} would be near +90° or -90°. When r1 is trans, interaction
between backbone N and XD2 and XD1 at χ_{2}=120° or -60° pushes
the average for χ_{2} to values below 90° or below -90°
respectively. Similarly, for r1 of g-, the averages are pushed to
values above χ_{2}=90° and χ_{2}=-90° by interactions of N
and XD1 and XD2 when χ_{2} is 60° or -120°. For g+ rotamers,
interactions at χ_{2}=60° and 120° exert steric conflict about
equally, so that χ_{2} averages 90°.

The backbone-independent library can be visualized graphically in a number of ways. These images are all contained in the tar file bbdep_figures.tar.gz.

Rotamer populations for χ_{1} dihedrals for all side chains are contained in the following images. For most side chains, g- rotamers are the most common (-120°<χ_{1}<0°). Exceptions include serine (g+ preferred) because of hydrogen bond interactions with C=O of residue i-1. Another is valine, where trans is the most populated rotamer because of backbone-dependent syn-pentane interactions at heavily populated regions of the φ,ψ map. Isoleucine is quite similar to valine, but since the χ_{1} is defined differently for isoleucine (CG1 at χ_{1}, CG2 at χ_{1}-120°) than in valine (CG1 at χ_{1}, and CG2 at
χ_{1}+120°), the equivalent rotamer for Ile is g-. Threonine has a χ_{1} angle defined like that of isoleucine. Thr is like Ile in its χ_{1} distribution, but in addition, like serine it has a large increase in g+ rotamers because of hydrogen bonding effects.

For all side chain types with sp^{3}-sp^{3} bonds at χ_{1} and χ_{2} (Arg, Lys, Glu, Gln, Met, Ile, Leu),
we can plot the r1,r2 rotamer populations as histograms. In all of these figures, syn-pentane rotamers are represented by solid red bars
and non-syn-pentane rotamers are represented by blue bars.

The following figures provide kernel density estimates of the χ_{1} dihedral angle for each residue type.

While χ_{1} is always rotameric, the χ_{2} distributions of Asn, Asp, Phe, Tyr, His, and Trp and the
χ3 distributions of Glu and Gln are non-rotameric. The difference can be observed in the χ_{1},χ_{2} scatterplots
of Arg and Asn. The χ_{1},χ_{2} values for Arg occur in nine bins corresponding to the g+,t,g- rotamers of χ_{1} and χ_{2} of Arg.
The distributions of χ_{2} for each χ_{1} rotamer of Asn vary from one another, but all three
are broadly distributed (right panel). The plots that follow are the χ_{2} kernel density
estimate for each residue type. For the rotameric χ_{2}, you can see the deviation of χ_{2} from the
common rotameric value (-60°, 180°, 300°) when there are syn-pentane interactions (e.g., for {g+,g+},
{g+,g-}, {t,g-}, {g-,g+} conformations). For example, for Arg, in the left panel, the first peak for {g+,g+} is at
χ_{2} of around 90°.

**Scatterplots of χ _{1},χ_{2} angle pairs for rotameric degrees of freedom**

For χ_{1},χ_{2} rotamers, we expect the {g+,g+}, {g+,g-}, {t,g-}, and {g-,g+} rotamers to be low in population.

For χ_{2},χ_{3} rotamers, we expect the {g+,g-} and {g-,g+} rotamers to be low in population

For χ_{3},χ_{4} rotamers, we expect the {g+,g-} and {g-,g+} rotamers to be low in population

Note the distortion distorted position of the {g+,g-} and {g-,g+} χ_{3},χ_{4} points for the trans χ_{2} rotamers of ARG

**Scatterplots of χ _{1},χ_{2} that consist of one rotameric degree of freedom and non-rotameric degree of freedom**

Backbone-conformation dependent interactions occur between gamma heavy atoms (CG,OG,OG1,CG1,CG2,SG)
and backbone C_{i-1} (carbonyl carbon of the previous amino acid), backbone N_{i+1}
(backbone nitrogen of the next amino acid), backbone O_{i} (backbone oxygen of the same amino acid),
and the hydrogen bond
donor to backbone N_{i} of the same amino acid ("HB" below,
assuming linear hydrogen bond between oxygen and H-N bond). These
interactions are expected to be strongly repulsive when dihedrals
connecting these heavy atoms to gamma heavy atoms occur in +60°,-60° or
-60°,+60° pairs. They will occur in a range about the φ and
ψ dihedrals that cause the connecting dihedrals to the
backbone to be near +60° or -60° (when connecting dihedral to XG is -60°
or +60° respectively). In each case, the dihedrals needed are of the
form: X1-X2-X3-X4 and X2-X3-X4-X5. So for instance for ψ
dependent interactions, the dihedrals are N_{i+1}-C-CA-CB and
C-CA-CB-XG or ψ+120° and χ_{1}-120° respectively. These
are tabulated below for instances where syn-pentane interactions are
expected. Val, Ile, and Thr are tabulated separately, since these
amino acids have 2 gamma heavy atoms. Interactions with "HB"
are expected to be weak, but the backbone-dependent rotamer library
does exhibit effects due to this interaction.

Note: we need to observe ranges of φ and ψ of 0°
to 90° and -90° to 0°, for g- and g+ dihedrals of the side chain. Since
χ_{1} is always an sp^{3}-sp^{3} dihedral with local minima around
180°, +60°, and -60°, χ_{1} (or χ_{1}-120°)
will always fall in the -90° to 0° or 0° to 90° range. φ and
ψ are both sp^{2}-sp^{3} dihedrals, which do not have barriers as
high as sp^{3}-sp^{3} dihedrals. They can take on a somewhat broader range
of dihedrals than sp^{3}-sp^{3} bonds. Hence, we locate 90° degree intervals
of φ and ψ that will conflict with certain
χ_{1} rotamers to search for likely steric conflicts.

**Val**

Val has CG1 at χ_{1} and CG2 at χ_{1}+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 (χ_{1}~180°) is the most common.

Ile has CG1 at χ_{1} and CG2 at χ_{1}-120°. Thr has OG1 at χ_{1} and CG2 at χ_{1}-120°.

The backbone-dependent rotamer library can be visualized graphically in a number of ways.

Ramachandran plots for each of the three χ_{1} rotamers demonstrate
how different their backbone conformational spaces are. It is easy to observe the strongest
backbone-dependent effects that lower the population of some rotamers:

g+ rotamer at ψ= -60 and ψ= +120 t rotamer at ψ= 0 and ψ= +180 g+ rotamer at φ= -120 and φ= +60 g- rotamer at φ= -180 and φ= 0The residues with two gamma heavy atoms experience these steric hindrances for both the G1 and G2 heavy atoms.

VAL g+ rotamer = CG1 at g+, CG2 at t --> ψ= -60 120 0 180 φ= -120 +60 VAL t rotamer = CG1 at t, CG2 at g- --> ψ= 0 180 φ= -180 0 VAL g- rotamer = CG1 at g-, CG2 at g+ --> ψ= -60 120 φ= -120 +60 -180 0 THR g+ rotamer = OG1 at g+, CG2 at g- --> ψ= -60 120 φ= -120 +60 -180 0 THR t rotamer = OG1 at t, CG2 at g+ --> ψ= 0 180 -60 120 φ= -120 +60 THR g- rotamer = OG1 at g-, CG2 at t --> ψ= 0 180 φ= -180 0 ILE g+ rotamer = OG1 at g+, CG2 at g- --> ψ= -60 120 φ= -120 +60 -180 0 ILE t rotamer = OG1 at t, CG2 at g+ --> ψ= 0 180 -60 120 φ= -120 +60 ILE g- rotamer = OG1 at g-, CG2 at t --> ψ= 0 180 φ= -180 0

**Example: Ramachandran populations are lower where expected as marked with red arrows. The atom interacting with the XG atom is shown in blue type.**

The φ and ψ dependence can be illustrated simultaneously in box plots. The width of the boxes located at each φ,ψ point in a Ramachandran map are proportional to the percentage of rotamers for that φ,ψ that are a particular rotamer (one graph for each of the g+, t, and g- rotamers, labeled "+", "t", and "-" in the plots). The summed distribution (Arg, Lys, Glu, Gln, Met) is clearest (labeled "CGG"). The plot labeled "ARO" includes all four aromatic residue types (PHE, TYR, HIS, TRP). These plots are based on the 1997 data. The interactions between the XG atom and various backbone atoms is illustrated in the first plot. All of the plots can be downloaded here.

- R. L. Dunbrack, Jr. and M. Karplus. Backbone-dependent
Rotamer Library for Proteins: Application to Side-chain
prediction.
*J. Mol. Biol.,***230**, 543-574 (1993). Download PDF - R. L. Dunbrack, Jr.
*Conformational analysis of protein side chains: Empirical energy parameters for proline and development of a backbone-dependent rotamer library*. Ph. D. dissertation, Harvard University (1993). Download PDF - R. L. Dunbrack, Jr. and M. Karplus. Conformational analysis
of the backbone-dependent rotamer preferences of protein
side chains.
*Nature Structural Biology*,**1**, 334-340 (1994). Download PDF - R. L. Dunbrack, Jr. and F. E. Cohen. Bayesian statistical
analysis of protein side-chain rotamer preferences.
*Protein Science*,**6**, 1661-1681 (1997). Download PDF - R. L. Dunbrack, Jr. Rotamer libraries in the 21st century.
*Curr Opin Struct Biol***12**, 431-440 (2002). Download PDF - M. V. Shapovalov and R. L. Dunbrack, Jr. Statistical and conformational analysis of the electron density of protein side chains.
*Proteins***66**, 279-303 (2007). Download PDF - M. S. Shapovalov and R. L. Dunbrack, Jr. A smoothed backbone-dependent rotamer library for proteins derived from adaptive kernel density estimates and regressions.
*Structure*,**19**, 844-858 (2011). Download PDF