Revisiting the Ramachandran plot: hard-sphere repulsion, electrostatics, and H-bonding in the alpha-helix

Abstract

What determines the shape of the allowed regions in the Ramachandran plot? Although Ramachandran explained these regions in terms of 1-4 hard-sphere repulsions, there are discrepancies with the data where, in particular, the alphaR, alphaL, and beta-strand regions are diagonal. The alphaR-region also varies along the alpha-helix where it is constrained at the center and the amino terminus but diffuse at the carboxyl terminus. By analyzing a high-resolution database of protein structures, we find that certain 1-4 hard-sphere repulsions in the standard steric map of Ramachandran do not affect the statistical distributions. By ignoring these steric clashes (NH(i+1) and O(i-1)C), we identify a revised set of steric clashes (CbetaO, O(i-1)N(i+1), CbetaN(i+1), O(i-1)Cbeta, and O(i-1)O) that produce a better match with the data. We also find that the strictly forbidden region in the Ramachandran plot is excluded by multiple steric clashes, whereas the outlier region is excluded by only one significant steric clash. However, steric clashes alone do not account for the diagonal regions. Using electrostatics to analyze the conformational dependence of specific interatomic interactions, we find that the diagonal shape of the alphaR and alphaL-regions also depends on the optimization of the NH(i+1) and O(i-1)C interactions, and the diagonal beta-strand region is due to the alignment of the CO and NH dipoles. Finally, we reproduce the variation of the Ramachandran plot along the alpha-helix in a simple model that uses only H-bonding constraints. This allows us to rationalize the difference between the amino terminus and the carboxyl terminus of the alpha-helix in terms of backbone entropy.

Figure 1.

Schematic of the ϕ–ψ angles. (A) The schematic of the alanine dipeptide that represents the protein backbone parameterized by the ϕ = C-N-C-C^α-C and ψ = N-C^α-C-N dihedral angles. (B) The original Ramachandran steric map (Ramachandran et al. 1963) where the specific hard-sphere repulsions (dashed line) identified by Mandel et al. (1977) define the allowed regions (gray): α_L, α_R, and β regions.

Figure 2.

Ramachandran plots. (A) All residues excluding Pro, Gly, and pre-Pro; (B) residues in the center of the α-helix, which are more constrained than for all residues; (C) the Ncap residue; and (D) the Ccap residue in the α-helix, which are scattered throughout the entire allowed region.

Figure 3.

Distributions of interatomic distances [Å] parameterized by ϕ [°]. The ideal curves (gray) are calculated using Engh and Huber (1991) geometry. The vdW diameters (dashed line) are taken from Word et al. (1999). (A) O_i−1···H^α versus ϕ; (B) O_i−1···C versus ϕ; (C) C_i−1···C^β versus ϕ; and (D) O_i−1···C^β versus ϕ. The ϕ frequency distribution is shown at the bottom of D.

Figure 4.

Distributions of interatomic distances [Å] parameterized by ψ [°]. The ideal curves (gray) are calculated using Engh and Huber (1991) geometry. The vdW diameter (dashed line) are taken from Word et al. (1999). (A) C^β···H_i+1 versus ξ; (B) C^β···N_i+1 versus ξ; (C) N···H_i+1 versus ξ; and (D) C^β···O versus ξ. The ξ frequency distribution is shown at the bottom of D.

Figure 5.

Revised steric map. (A) The steric clashes (dashed blue lines) that best match the data. d(O_i−1···O) = 2.7Å, d(O_i−1···N_i+1) = 2.7 Å, and d(H···H_i+1) = 1.6 Å. (B) Schematic of the revised steric map showing steric restrictions (dashed blue lines) and sterically allowed regions (dark blue). The revised steric map gives diagonal boundaries for the α_R, α_L, and β regions and defines a more realistic upper boundary for the α_R-region. Diagonal α_R and α_L regions (red region) from the dipole–dipole analysis (Fig. 7G ▶) are defined mainly by the attractive O_i−1···C and N···H_i+1 interactions (red lines). The diagonal β-strand region (yellow) is induced by aligning the CO···HN dipole–dipole interaction. Regions that are only excluded by a single steric clash (light blue) accounts for the outlier region in Lovell et al. (2003).

Figure 6.

Contour plots of the ϕ–ξ codependent interactions. The contours of constant distance [Å] are shown as functions of the ϕ–ξ angles [°]. These interactions can be grouped in terms of dipole–dipole interactions in the alanine dipeptide (see Fig. 1A ▶) where the contour plots within each group are geometrically similar. In terms of the CO_i−1···CO dipole–dipole interaction, they are (A) O_i−1···O, (B) C_i−1···O. In terms of the NH···NH_i+1 interaction, they are (C) H···H_i+1 and (D) H···N_i+1. In terms of the CO_i−1···NH_i+1 interaction, they are (E) O_i−1···N_i+1, (F) O_i−1···H_i+1, (G) C_i−1···H_i+1, and (H) C_i−1···N_i+1. In terms of the CO···HN interaction, the only ϕ–ξ codependent distance is (I) O···H. All contour plots possess a twofold inversion symmetry through the point ϕ = ξ = 0°. The sterically excluded regions are defined as the regions where the interatomic distance is smaller than the corresponding vdW diameter (see Table 1).

Figure 7.

Contour plots of the dipole–dipole interactions [kcal/mole] as a function of the ϕ–ξ angles [°]. Energy plots of (A) the Lennard-Jones 12–6 potentials of the revised set of steric clashes; (B) all electrostatic interactions; the individual dipole–dipole interactions of (C) CO_i−1···CO; (D) NH···NH_i+1; (E) CO···NH; and (F) CO_i−1···NH_i+1. (G) The combination of the CO_i−1···CO, NH···NH_i+1 and CO···NH dipole–dipole interactions produces clear diagonal minima in the α_R, α_L, and β regions.

Figure 8.

H-bonding in the amino terminus and carboxyl terminus of the α-helix. (A) Carboxyl terminus showing the carboxy-terminal residues (red) and the H-bonds (red) used in the model. (B) The amino terminus showing the amino-terminal residues (red). (C) The schematic of the allowed region in C1 residue (solid red), which is due to steric constraints (black), electrostatics (red outline), and formation of H-bonds that bring the two H atoms together (blue; see also A). (D) The schematic of the H-bonding constraints on the N1 residue (see also B).

Figure 9.

The Ramachandran plot of the amino-terminal residues. The left column gives the observed distribution. The right column gives the energy map of the H-bonding constraints and Lennard-Jones potential. The Ramachandran plot has been truncated for clarity.

Figure 10.

The Ramachandran plot of the carboxyl-terminal residues. The left column gives the observed distribution. The right column gives the energy map of the H-bonding constraints and Lennard-Jones potential. The Ramachandran plot has been truncated for clarity.