First, we needed to determine which subset of helix N-cap conformations would be likely to undergo backrubs and thus merited further examination. To do so, we compared amino acid preferences for α-helix N-caps and 3₁₀-helix N-caps relative to general-case protein structure, using a non-redundant, quality-filtered set of structures, the Top5200 database (see Methods). Asn/Asp/Ser/Thr were indeed found to be strongly preferred (by factors of 2.5–3) at α-helix N-caps relative to protein structure in general (Figure 2). Gly is next most common, but cannot form or be influenced by an N-cap H-bond. Pro is the outstandingly preferred residue at 3₁₀ N-caps (Figure 2), while for α-helix Pro is disfavored at the N-cap but occurs preferentially at N-cap i+1. These comparative preferences are detailed further in Text S1, with the conclusion that helix N-termini might well respond to a deletion in the preceding loop by transforming from a classic α-helix with Pro at N-cap i+1 to a tighter-wound 3₁₀-helix with the Pro as N-cap (Figure S1 in Text S1). As an additional factor in our choice of examples, a 3₁₀-helix N-cap would have an i+2 sidechain-backbone H-bond rather than the i+3 common at α-helix N-caps, introducing another complicating issue. Therefore, we decided to focus on Asn/Asp vs. Ser/Thr α-helix N-caps with i+3 sidechain-backbone H-bonds in this study.

With this specific motif in mind, we sought to obtain a preponderance of evidence for a backrub relationship in the form of numerous examples. Therefore, we performed a stringent structural motif search of the Top5200, resulting in identification of 429 Asn/Asp N-caps and a matching sample choice of 500 Ser/Thr N-caps (out of 3208; see Methods).

The backbone conformations differ consistently: the longer Asn/Asp sidechains rotate the first turn's backbone away from residue i+3, while the shorter Ser/Thr sidechains pull the first turn's backbone toward i+3 in order to form the N-cap H-bond successfully (Figure 3, Table 1). The Dataset S1 kinemage shows the full sets of Asn/Asp and Ser/Thr N-caps and animates between them. When average Asn/Asp and Ser/Thr structures are superimposed using the C_αs surrounding the N-cap in the first turn (N-cap i−1 and i+1 to i+3), all C_αs match well except the N-cap C_α itself (Table 1). The conformational difference at the N-cap position is well modeled by a backrub rotation of about 11°, similar to shifts typical of rotamer-change backrubs. Furthermore, for both N/D and S/T, the C_β deviations [23] and the C_α-C_β-C_γ bond angle distribution at N-caps are close to the general case distribution (Figure S2 in Text S1; further details in Text S1). This means the observed C_β shifts and further leveraged sidechain shifts can be attributed primarily to backbone motion rather than altered covalent sidechain geometry.

The median Asn N-cap sidechain built onto the average Ser N-cap backbone results in a steric clash in which the van der Waals radii of the N-cap sidechain O_δ1 and the i+3 backbone amide H overlap by >0.45 Å. No χ dihedral adjustments can alleviate this clash without abolishing the N-cap sidechain-mainchain H-bond or introducing clashes to other nearby atoms whose positions are fixed for this motif (e.g. the i+4 C_β). Therefore, the backrub away from the first turn by Asn N-caps is indeed forced by steric repulsion.

We also examined two control cases with similar backbone geometry but different sidechain-mainchain interactions. First, we identified 538 α-helix N-caps with any amino acid type except Asn/Asp/Ser/Thr, in which case the i+3 sidechain-backbone H-bond is absent. Second, we chose 500 examples of mid-α-helix structure flanked by at least four α-helical residues in both directions, in which case the i+3 sidechain-backbone H-bond of an N-cap is satisfied by a usual i+4 backbone-backbone α-helical H-bond. The Dataset S1 kinemage shows that the average C_α atoms for both control categories are in between the average C_α atoms for the Asn/Asp and Ser/Thr categories at the N-cap (or structurally equivalent) residue. This confirms that Asn/Asp and Ser/Thr N-caps are backrub-mediated excursions in opposite directions from equilibrium N-cap/helix structure.

We next wished to test whether a simple energy function based on molecular-mechanics terms from Amber [24] and a solvation term from EEF1 [25] would recapitulate these empirically observed changes, given the chance to access them via a backrub. This question is important for solidifying the connection between natural protein evolution and computational protein design. We therefore turned to Backrub Dead-End Elimination (BRDEE) [10], a protein design algorithm that incorporates backrubs in a provably-accurate dead-end elimination framework. BRDEE is freely available as part of the Open Source Protein Redesign for You (OSPREY) [26] suite of protein design software.

As input to the algorithm, we prepared two versions of an idealized helix N-cap motif (see Methods), one with a short sidechain (Ser) and another with a long sidechain (Asn). We then used BRDEE to compute the lowest-energy model for each template, allowing backrubs and rotamer changes at the N-cap as well as small i+3 peptide rotations (see Methods).

The lowest-energy Ser N-cap shifted “forward” whereas the lowest-energy Asn N-cap shifted “backward” in order to establish comparable hydrogen bonds (Table 1) in a manner remarkably similar to the empirically observed structures (Figure 3). In particular, the computed and observed C_α and C_β shifts and inferred backrub angles are of similar magnitude and directionality (Table 1).

The changes in flanking N-C_α-C angles (τ) for the BRDEE lowest-energy conformations recapitulate the changes for the crystal structures in terms of directionality: Δτ<0 for i−1 and Δτ>0 for i+1 from Ser/Thr to Asn/Asp (Table 1). The magnitudes are somewhat exaggerated for the computed models, but this is expected because the backrub paradigm somewhat unrealistically redirects strain to these angles as a simplification. That said, we find some evidence of actual systematic bond angle adjustment in the Δτ of almost 3° between the two crystal structure populations (Table 1). There is precedent for context-dependent τ variation with φ,ψ [27], [28]; here the variation appears to be dependent on the presence of a local motif where sidechain-mainchain H-bonding may energetically counteract the bond angle strain. At any rate, actual structures almost certainly undergo more complicated backbone changes that propagate to the i±2 residues and alleviate i±1 τ angles. However, those additional changes are quite small, and thus the backrub remains an efficient yet remarkably accurate model for conformational changes at N-caps and elsewhere.

Additional comparison and contrast of the computed models and experimental structures can be found in Text S1.

The BRDEE results recapitulate the average crystal structures, confirming the hypothesis that Ser/Thr→Asn/Asp mutations at N-caps are well modeled by a backrub relationship. More generally, this implies that the backrub may reasonably accompany mutations during natural evolution or in silico protein engineering.

A stringent structural motif search, similar to that described for N-caps above, identified 321 Phe/Tyr residues with “plus” χ₁ rotamers in antiparallel β-sheet (see Methods). Aromatics are about three-fold as common in antiparallel vs. parallel β-sheet, and are about twice as likely to adopt a plus χ₁ rotamer when they do occur in antiparallel vs. parallel β-sheet (data from Top5200), so we focused on antiparallel β-sheet in this study.

In 72 examples the amino acid on the opposite strand is a Gly, in which case the aromatic sidechain moves downward to contact the Gly C_α H. In the other 249 examples the C_β H atoms of the amino acid on the opposite strand push the aromatic ring upward (Figure 4, Table 2, Dataset S2 kinemage). Pro cannot provide both β H-bonds, but all other non-Gly residues are equivalent in this role, since their sidechains must avoid the aromatic ring and present only C_β H atoms toward it. The average C_β deviation from ideality (0.05–0.06 Å) is far less than the outlier threshold (0.25 Å) [23], and the average change in aromatic C_α-C_β-C_γ bond angle is very small (0.6°, <1σ) (Figure S3 in Text S1), resulting in a <0.05 Å shift of the C_ζ contact point. These observations argue against the possibility that this large movement of the planar aromatic group is produced just by a bond-angle “hinge” with C_α or C_β as the pivot. Rather, a dipeptide backrub rotation of about 11° (presumably 5–6° from neutral in each direction) almost perfectly interrelates the two average conformations.

Fewer examples, and more variation for β-sheet than for α-helix conformation, prevented the ideal-start calculation used for the N-cap case. Instead, low-energy conformations were computed by BRDEE for several examples judged to be appropriately representative of their respective type (across from Gly or across from other) (see Methods). In all cases, the lowest-energy conformation appears to match the average crystal structure very well, whether across from Gly or from some other amino acid with a C_β atom (Figure 4). Computed shifts in C_α, C_β, C_ζ position between the computed wildtype and mutant structures are similar to shifts between the two crystal structure populations, albeit systematically a bit smaller (Table 2, Text S1). As with N-caps, though to a slightly lesser extent here in extended β conformation, in silico backrubs place more strain into the i±1 τ angles than do real structures (Table 2); however, the discrepancies are on the same order as the standard deviations of the experimental τ distributions, which are around 3° (data not shown).

These results confirm both that BRDEE in conjunction with a simple force field reproduces natural conformations well and also that backrubs model the relationship well.

We first noticed that backrubs may explain backbone adjustments upon mutation between short and long N-caps sidechains while examining N-caps in T4 lysozyme. Visual analysis using the BACKRUB tool in KiNG [39] revealed that a modest backrub (about 7°) nicely models the relationship between the Thr59 N-cap in the wildtype structure (2lzm) and the Thr59→Asn N-cap in the mutant structure (1lyg) [40]. Intriguingly, as well as being one of the most common N-caps, Ser is the most common amino-acid type for backrubs between alternate conformations in crystal structures [9], perhaps because it has many distinct possibilities for sidechain-backbone H-bonding.

N-caps were identified using a helix recognition algorithm based loosely on DSSP [41] that combined consecutive α-like (i+4 mainchain-mainchain H-bond) and/or 3₁₀-like (i+3 mainchain-mainchain H-bond) helical turns. To approximate the visually determined definition of an N-cap as the first residue within the C_α cylinder of the helix [17] but extend it to a much larger data set, residues were added to the N-terminus as putative N-caps if the distance from their C_α to the i+3 C_α was less than 5.9 Å. We chose this distance cutoff in order to include true N-caps at helix ends that were somewhat but not excessively stretched, i.e. with relatively weak mainchain-mainchain H-bonds but within the helix C_α cylinder and close enough to contribute with a sidechain-mainchain capping interaction. No amendments to the DSSP algorithm were made for C-caps, which we did not specifically examine in this study. Helices were required to include at least five residues including N- and C-caps.

We also took precautions to help offset any leniency in helix extension introduced in the steps described above. To ensure that a tightly defined subset of N-caps was being examined, H-bond pseudo-energies were computed as in DSSP with the standard −0.5 kcal/mol cutoff. N-caps with an i+4 but not an i+3 mainchain-mainchain H-bond were labeled α, N-caps with an i+3 but not an i+4 mainchain-mainchain H-bond were labeled 3₁₀, and N-caps with both i+3 and i+4 mainchain-mainchain H-bonds were labeled bifurcated α/3₁₀ and were ignored in this study.

For N-cap sequence propensity analysis, all α and 3₁₀ N-caps of each amino acid type were compared to the general case of all residues in the Top5200 regardless of backbone conformation. For N-cap backrub analysis, on the other hand, only α N-caps with i+3 but not i+2 sidechain-backbone H-bonds were used. These N-caps are found in the “extended” region of Ramachandran space, with two peaks of occurrence near (−150°,170°) and (−80°,170°) [42]. We required φ,ψ values near the “right-most” of these peaks, i.e. between (−100°,155°) and (−60°,180°), because it is the more populated of the two peaks for both the Asn/Asp and Ser/Thr groups and because the restriction provides a consistent depiction of the subset of N-caps that could undergo backrubs to relate to each other.

As described in Results, we also created two control categories. For the “other N-caps” category, the criteria were the same as for Asn/Asp or Ser/Thr N-caps, except we required some other amino acid identity at the N-cap position. For the mid-helix category, we required strictly α-helical φ,ψ values between (−65°,−45°) and (−55°,−35°) for the central residue of interest, at least four residues labeled “H” for α-helix by DSSP [41] in each direction, and at least 13 such residues total in the helix. For the Ser/Thr, Asn/Asp, and other N-cap categories, the N-cap residue was required to have maximum mainchain or sidechain atom B-factor <40 [23]. For the mid-helix category, the maximum B-factor was 20. To attain roughly similar sample sizes for all four categories, we randomly selected 500 examples for Ser/Thr (out of 3,208) and for mid-helix (out of 17,047), to match the 429 Asn/Asp and 538 “other” N-caps.

Examples of the N-cap motif were superposed onto one another using the N-cap i−1 and i+1 to i+3 C_αs; the N-cap i C_α itself was not used for superposition to avoid biasing its final average position. For the mid-helix control category, only the i+1 to i+3 C_αs were used, because there is no structural equivalent to the N-cap i−1 residue. Root mean square deviation (RMSD)≤1 Å relative to an ideal reference N-cap (see BRDEE input below) for Asn/Asp, Ser/Thr, and other N-caps, or relative to an ideal poly-Ala helix (similar to BRDEE input) for mid-helix, was required to keep each example.

Average motifs were then generated by taking the mean position of all backbone atoms (including hydrogens) and C_β atoms from N-cap i−1 to i+3. A Ser rotamer with χ₁ = +65° and an Asn rotamer with χ₁ = +66° and χ₂ = +30° were added to reflect median dihedral angles from the data set, while maintaining the average C_β positions. Finally the average structures were superposed onto one another, again using the N-cap i−1 and i+1 to i+3 C_αs (i+1 to i+3 C_αs for mid-helix).

Using the same protocol, we also analyzed 388 Asn/Asp and 976 Ser/Thr examples from the second Ramachandran peak, near (−150°,170°) instead of (−80°,170°) as mentioned above. The i−1 C_α in the secondary cluster is displaced by about 1 Å relative to the primary cluster due to the differing φ values (−150° vs. −80°) before entering the helix; thus, given the magnitude of the backrub motion we hoped to capture, it was unreasonable to compute an average structure for the combined set. However, separate analysis of the secondary cluster revealed a very similar backrub relationship as seen in the primary cluster (Dataset S1 kinemage, PDB files in Dataset S3).

For β aromatics, we identified Phe/Tyr residues with “plus” χ₁ (0–120°) in antiparallel β-sheet. The “opposite” residue is in the direction of the aromatic sidechain, between the narrow pair of mainchain H-bonds. Both residues were required to have maximum mainchain or sidechain atom B-factor less than 40, and to have at least one additional β residue (according to the custom DSSP-based secondary structure identification algorithm described above) in each direction along their respective strands.

To avoid irregularities from strand ends, we defined the “fray” parameter f:

where t_N (N-ward twist) is the dihedral between C_α i−1 and i on the aromatic strand and C_α i and i+1 on the opposite strand; t_C (C-ward twist) is the dihedral between C_α i and i+1 on the aromatic strand and C_α i and i−1 on the opposite strand; p_a (aromatic pleat) is the angle between C_α i−1, i, and i+1 at the aromatic residue; and p_o (opposite pleat) is the angle between C_α i−1, i, and i+1 at the opposite residue. A difference in twist indicates that the two strands locally “pull apart” from each other; any difference in amount of pleating between the two strands is subtracted as a correction term. Fray was constrained to less than 10° for our motif search.

Examples of the β aromatic motif were superposed onto one another using the aromatic i−2, i−1, i+1, i+2 C_αs and the C_α opposite i as an “anchor” point; the aromatic i C_α itself was not used for superposition to avoid biasing its final average position. RMSD≤1 Å relative to a reference example was required to keep each example. For each category (across from Gly vs. other), 10 arbitrarily chosen reference examples were tried; the one that resulted in the fewest examples being pruned by the RMSD≤1 Å filter was deemed representative of its category (1gyhA Tyr109/Gly122 pruned only 3.8% of across-from-Gly examples, and 1avbA Phe6/Phe221 pruned only 4.2% of across-from-other examples). The observed backrub relationship was similar when manually selected reference examples judged to representative of each category or completely randomly selected reference examples were used instead (data not shown), and the RMSD between the representative reference examples described above (0.11 Å) was significantly less than the 1.0 Å RMSD filter, so it is unlikely the choice of particular reference examples pre-determined the result of an observed backrub between the sets of examples superposed onto the reference examples.

Average motifs were then generated by taking the mean position of all backbone atoms (including hydrogens) and C_β atoms from residues i±2 on both strands. Average Phe rotamers with χ₁ = +65° and χ₂ = +80° for the across-from-Gly category and with χ₁ = +66° and χ₂ = +84° for the across-from-other category were added to reflect median dihedral angles from the data set, while maintaining the average C_β positions. Finally the average structures were superposed onto one another, again using the aromatic i−2, i−1, i+1, i+2 C_αs and the C_α opposite i.

For N-cap calculations, we started from an ideal helix (φ,ψ −60°,−40°) with poly-Pro conformation (φ,ψ −80°,170°) for the N-cap and its preceding residue. All residues were Ala except for the N-cap, which was either Asn or Ser. At the N-cap position, primary backrub rotation angles from −15° to +15° in increments of 0.5° were allowed, and the default ε factor of 0.7 was used for back-rotation of the single peptides. At the N3 position, backrubs were not allowed as part of the main search. However, using KiNG [39] we manually determined that single-peptide rotation angles from −5° to +15° of the N3 peptide could help optimize the H-bonding angle of its backbone amide to the N-cap sidechain H-bonding angle and were viable in terms of sterics, Ramachandran dihedrals [23], and τ angles; therefore we included such a rotation in increments of 1°.

For β aromatic calculations, the strand twist and curl vary too much to construct averaged cases. Instead, we repeated the computational experiment several times with individual examples of the motif from different experimental structures as input coordinates. Representative examples such as 1khbA Phe144-Gly157 and 1z84A Phe171-Gln188 were chosen because they were visually relatively “middle of the pack” and were not obvious outliers in terms of geometric parameters like fray. Primary backrub rotation angles from −15° to +15° in increments of 0.5° were allowed at both residue positions, with the default ε factor of 0.7. (Backrubs at the opposite position were of less interest for this study, and indeed turned out to be much smaller than at the aromatic position.) If the original opposite residue was Gly, BRDEE was run for both the original coordinates and a Gly→Ala mutant. Likewise, if the original opposite residue was anything but Gly, BRDEE was run for both the original coordinates and an Xaa→Gly mutant.