Geometric descriptors for beta turns

Abstract

Beta turns, in which the protein backbone abruptly changes direction over four amino acid residues, are the most common type of protein secondary structure after alpha helices and beta sheets and play key structural and functional roles. Previous work has produced classification systems for turn geometry at multiple levels of precision, but these operate in backbone dihedral‐angle (Ramachandran) space, and the absence of a local Euclidean‐space coordinate system and structural alignment for turns, or of any systematic Euclidean‐space characterization of turn backbone shape, presents challenges for the visualization, comparison and analysis of the wide range of turn conformations and the design of turns and the structures that incorporate them. This work derives a turn‐local coordinate system that implicitly aligns turns, together with a set of geometric descriptors that characterize the bulk BB shapes of turns and describe modes of structural variation not explicitly captured by existing systems. These modes are shown to be meaningful by the demonstration of clear relationships between descriptor values and the electrostatic energy of the beta‐turn H‐bond, the overrepresentations of key side‐chain motifs, and the structural contexts of turns. Geometric turn descriptors complement Ramachandran‐space classifications, and they can be used to select turn structures for compatibility with particular side‐chain interactions or contexts. Potential applications include protein design and other tasks in which an enhanced Euclidean‐space characterization of turns may improve understanding or performance. The web‐based tools ExploreTurns, MapTurns, and ProfileTurn, available at www.betaturn.com, incorporate turn‐local coordinates and turn descriptors and demonstrate their utility.

1. INTRODUCTION

Abrupt changes in the direction of the protein backbone (BB) chain are accomplished by tight turns of types {δ, γ, β, α, π}, with lengths of {2, 3, 4, 5, 6} amino acid (AA) residues (Koch & Klebe, 2009). Of these types, the four‐residue beta turn is found most frequently; it constitutes the most common protein secondary structure after alpha helices and beta sheets. Either individually or in overlapping multiples (Isogai et al., 1980), beta turns are found in loops that link segments of repetitive BB structure, and they are also very common in the extended loops of intrinsically disordered regions. Turns exhibit a wide range of BB conformations which, taken together with the variety of structural interactions that their side‐chains (SCs) can mediate, enable them to play crucial roles in multiple contexts in proteins; for an overview, see Roles of beta turns in proteins at www.betaturn.com.

Beta turns were first described by Venkatachalam (1968), who determined the allowed geometries of four‐residue chain segments that exhibit a BB H‐bond between their first and fourth residues (4>1 H‐bond). Since this study, a substantial body of analytical work (see short reviews in D'Arminio et al., (2023); Shapovalov et al., (2019)) has evolved the beta‐turn definition and identified and classified new geometries. A current definition (Shapovalov et al., 2019), applied here, specifies a four‐residue BB segment in which the distance between the alpha carbons of the first and last residues is no greater than 7 Å and the two central residues are not part of helices or strands. In about half of the segments that meet this definition, the BB NH group of the fourth turn residue (MCN4) donates an H‐bond to the BB carbonyl oxygen of the first residue (MCO1), and the presence of this 4>1 H‐bond by itself constitutes an alternative, more restrictive definition.

Three beta‐turn classification systems developed in previous studies are employed in this work; each partition turns by BB geometry in Ramachandran (dihedral‐angle) space at a different level of precision. These systems, summarized in the ExploreTurns (Newell, 2024) online help at www.betaturn.com, are referred to here, in order of increasing precision, as the “classical” type system (Hutchinson & Thornton, 1994; Hutchinson & Thornton, ¹⁹⁹⁶; Richardson, ¹⁹⁸¹; Wilmot & Thornton, ¹⁹⁸⁸), the BB clusters (Shapovalov et al., 2019), and the Ramachandran‐type system (Shapovalov et al., 2019), with the last system based on the map of Ramachandran space due to Hollingsworth and Karplus (Hollingsworth & Karplus, 2010).

The inherent difficulty of visualizing and contrasting the wide range of beta‐turn geometries in Ramachandran space (de Brevern, ²⁰²²) motivated the construction, in this work, of a turn‐local coordinate system that could provide a common Euclidean‐space framework for structural analysis and comparison. The derivation of geometric descriptors to characterize the BB shapes of turns was suggested by the observation that while the shape of a protein helix, which can be described as a series of overlapping turns, has been characterized by parameters (including twist, rise, and pitch (Hauser et al., 2017)) which vary between helix types and also with the irregularities of helices within each type, no simple set of geometric descriptors has been developed to describe the BB shapes of beta turns, which vary between and within turn types or BB clusters. Since this project began, a parametric model for protein BB structures constrained by H‐bonds has been developed and applied to beta turns to explore their allowed conformations (Hassan & Coutsias, 2021), but this model is not designed to provide a Euclidean‐space characterization of turn shape, which is the goal of the present work.

The case for the development of geometric descriptors for beta turns was bolstered by the need for structural discriminators within types and clusters. The set of turns with both central residues in or near the right‐handed helical conformation, which constitutes close to half of all turns, is represented by a single classical type (I) or a single BB cluster (with Ramachandran‐type label AD) (Shapovalov et al., 2019), yet, as shown below, type I/cluster AD includes a range of conformations that are compatible to widely varying degrees with key SC motifs, or compatible with different SC motifs entirely, and can play distinct roles in the structural contexts that incorporate beta turns. The presence of a meaningful range of BB structures within type I/cluster AD, and also, to varying degrees, within other, smaller types/clusters, prompted a search for a set of descriptors capable of characterizing modes of geometric variation not explicitly captured by Ramachandran‐space systems.

The approach taken here to the development of turn descriptors differs from that taken to the development of helix parameters (Hassan & Coutsias, 2021; Hauser et al., ²⁰¹⁷). Unless extended to include perturbations, helix models involving a small number of parameters describe idealized structures that are fit to actual helices to estimate their parameters, or can be used to generate sets of BB dihedral angles to describe the conformations of an artificially constrained subset of natural structures. By contrast, the approach taken here is to characterize the bulk BB shape of a beta‐turn (informally, the shape of a line fit to the turn's BB's atoms) with a set of geometric descriptors, based on measurements of actual structures, which represent key dimensions of the turn's BB, including its approximate extents in the three Euclidean‐space dimensions (represented by “span,” “bulge,” and “warp”), an additional degree of freedom in the measure of span (the “half‐spans”), and a measure of the turn's asymmetry in its plane (“skew”).

Unlike the helix parameters, the descriptors proposed here are not associated with a geometric construction, so they cannot be used by themselves to generate the BB dihedral angles that fully describe the BB conformation of any turn structure, ideal or otherwise. However, since descriptors are measured from actual structures, they can characterize the bulk BB shapes of all beta turns without a need for fitting, and when they are specified in the ExploreTurns tool, they can be used to select turn structures with desired bulk BB shapes from a database with deep coverage of turn geometries, while dihedral‐angle conditions can be applied with any desired precision to choose from available BB conformations. Geometric turn descriptors provide a continuous‐valued, Euclidean‐space classification for use in combination with the Ramachandran‐space classifiers, with the descriptors specifying bulk BB shape while the Ramachandran‐space systems apply dihedral‐angle criteria which specify the conformation of the turn's BB at the desired precision.

The derivation of geometric descriptors begins with the construction of a turn‐local coordinate system, which is achieved via a set of geometric definitions, and the representation of turns in the local system implicitly aligns them. Using the definitions, the turn descriptors are then derived, their modes of variation are demonstrated with example structures, and their distributions within each classical type and BB cluster are characterized. Descriptor distributions are also heat‐mapped across the Ramachandran‐angle spaces of the two central turn residues, revealing the bond rotations associated with descriptor variations in each type or cluster.

The fractional overrepresentation of key sequence motifs in turn is shown to vary dramatically with descriptor values, demonstrating the utility of descriptors as structural discriminators that can support geometric tuning for compatibility with SC interactions. Relationships are also demonstrated between descriptor values and the electrostatic energy of the 4>1 H‐bond.

The roles and contexts of descriptor regimes are explored, as are the effects of cis‐peptide bonds on descriptor values. The dependencies between descriptors are evaluated, and the potential utility of descriptors in structure validation is investigated.

Many of the results presented here were generated with ExploreTurns (Newell, 2024), a web facility for the comprehensive exploration, analysis, geometric tuning, and retrieval of beta turns and their contexts from the dataset compiled for this study. A second tool, MapTurns (Newell, 2024), produces interactive, 3D conformational heatmaps of the BB and SC structure, H‐bonding, and contexts associated with sequence motifs in beta turns, and a third facility, ProfileTurn, profiles a turn uploaded by the user and evaluates the compatibility between its geometry and sequence motif content. Extensive online support is available within all three tools, which are available at www.betaturn.com.

2. METHODS

2.1. Turn‐local coordinate system and alignment

The derivation of turn descriptors begins with a set of geometric definitions that establish a common turn‐local Euclidean‐space coordinate system. The span line is the line between a turn's first and last alpha carbons ($C_{\alpha 1}$ and $C_{\alpha 4}$), and the span, already used in the beta‐turn definition, is its length. The turn center is the midpoint of ($C_2$→$N_3$), the middle peptide bond in the turn. The turn plane contains the span line and passes through the turn center, and the perpendicular dropped from the turn center to the span line defines the turn axis. Using these definitions, a local, orthogonal right‐handed coordinate system is established. The system's origin lies at the turn center, while the x‐axis lies in the turn plane, collinear with the turn axis, with its positive sense oriented from the center toward the span line. The y‐axis lies in the turn plane, parallel to the span, with its positive sense oriented toward the C‐terminal half of the turn. Finally, the z‐axis is perpendicular to the turn plane, with its positive sense determined by the cross‐product of the x‐ and y‐axes. The geometric definitions and the turn‐local coordinate system are illustrated in Figure 1.

FIGURE 1.

Examples of turns that show geometric descriptor values near their limits in the dataset, with the geometric definitions labeled in red and the geometric descriptors labeled in blue. While extreme descriptor values are correlated with an increased frequency of geometric criteria with outliers in PDB validation checks (Supplementary Section 1.3), none of these structures show outliers. A white bar representing the span line is added to all structures in this and subsequent figures. (a) 5HKQ_I_74 (top) and 4YZZ_A_239 (bottom) represent the ranges of both bulge and span, since the two descriptors exhibit strong negative correlation (Supplementary Section 1.2). (b) 5TEE_A_23 (top), and 3NSW_A_44 (bottom) represent strong positive (C‐terminal) and negative (N‐terminal) skew. N‐and C‐terminal half spans are also given. (c) 5BV8_A_1285 (top) and 3S9D_B_117 (bottom) represent the limits of warp in classical type I, and demonstrate the wide warp range that occurs within this type (and its associated BB cluster, AD_1). The BB atoms in the main‐chain path of the turn between C _α1 and C _α4 are labeled; all labeled atoms except C _α1 and C _α4 are used to compute warp. To view any structure in the ExploreTurns tool, load the entire database by clicking Load Turns Matching Criteria without any selection criteria, copy the structure's address into the PDB address box, and click Browse Structures.

The atomic coordinates of each turn are transformed from the global protein system in the turn's PDB file to the turn‐local system, implicitly aligning the structures. Although this alignment is not required for the computation of geometric descriptors, it is needed in the ExploreTurns and MapTurns tools to orient structures for display and comparison and for structural clustering. The alignment's accuracy was tested by a comparison between implicit pairwise turn alignments and the best alignments generated by an RMSD‐based perturbative search procedure. In this process, the average distance between the corresponding BB atoms of a randomly chosen, implicitly aligned turn pair was compared to the distance obtained for the best alignment generated by a perturbative search applied to the pair which seeks the minimum average interatomic distance by introducing small variations in the relative position and orientation of the two turns. This procedure was applied to a random sample of 100 million turn pairs to generate an estimate of the average positional error of the implicit alignments. Since this estimate (0.17 Å) is comparable to the errors expected for the positions of well‐determined atoms in well‐refined structures (0.1–0.2 Å; overall RMSDs for independently‐determined 2 Å structures have been measured as 0.5–0.8 Å) (Richardson, 1981), the accuracy of the implicitly established alignment was judged to be satisfactory.

2.2. Descriptor definitions

The geometric descriptors are defined here and illustrated in Figure 1.

The pre‐existing span descriptor measures the distance [Å] between $C_{\alpha 1}$ and $C_{\alpha 4}$, and a 7 Å threshold for span is part of the beta‐turn definition. Two new span‐related descriptors are defined: the N‐ and C‐terminal half‐spans measure the distances between the turn axis and $C_{\alpha 1}$ and $C_{\alpha 4}$ respectively, and quantify the separate “widths” of the N‐ and C‐terminal halves of the turn, providing an additional degree of freedom in the measurement of span. Span values in the dataset range from a minimum of 3.13 Å to the 7 Å limit set by the beta‐turn definition, while N‐terminal half‐spans range from 0 to 4.76 Å, and C‐terminal half‐spans from 0.76 to 4.82 Å.

Bulge is a measure of the magnitude of the excursion that the BB makes from the turn span as it bends to form the turn. Bulge is defined as the distance [Å] from the turn center to the span line. Bulge values in the dataset range from 1.28 to 4.7 Å.

Skew is a dimensionless measure of the asymmetry of the projection of a beta turn's BB onto the turning plane. Skew is defined as the directed displacement along the span line between the line's center and the point of intersection between the turn axis and the line, divided by half the span. Skew is negative when the axis intersects the span line in its N‐terminal half, indicating a “lean” in the N‐terminal direction, or positive when the axis intersects the line in its C‐terminal half, producing a C‐terminal lean. Skew is expressed in terms of the N‐ and C‐terminal half‐spans ($S_N$ and $S_C$) as:

$$\text{skew}=\frac{S_N-S_C}{S_N+S_C}$$ (1)

Skew is positive when $S_N$ is larger than $S_C$, negative when the reverse is true, or zero when the half‐spans are equal. A turn that skews so strongly N‐terminally that its axis intersects the span line at the position of $C_{\alpha 1}$ ($S_N$ = 0) would exhibit a skew of −1, while a turn skewed C‐terminally so that its axis intersects the line at $C_{\alpha 4}$ ($S_C$ = 0) would show a skew of +1. Skew values in the dataset range from −1 to 0.78.

Warp measures the departure from flatness of a beta turn's BB in the z‐dimension of the turn coordinate system, perpendicular to the turn plane. Warp [Å] is defined as the average excursion of the turn's BB above or below the turn plane between $C_{\alpha 1}$ and $C_{\alpha 4}$, excluding BB carbonyl oxygen atoms as well as $C_{\alpha 1}$ and $C_{\alpha 4}$ themselves (since they lie within the plane). Warp is computed as the average absolute value of the z‐components of the BB atoms ($C_1$, $N_2$, $C_{\alpha 2}$, $C_2$, $N_3$, $C_{\alpha 3}$, $C_3$, $N_4$). Warp values in the dataset range from 0.14 to 1.25 Å. The differences in BB conformation between turn types or BB clusters affect the interpretation and utility of the warp descriptor (see Supplementary Section 2.1).

2.3. Other methods

2.3.1. The beta‐turn dataset

Beta turns are defined here as four‐residue BB segments with a distance of no more than 7 Å between the alpha carbons of the first and fourth residues and central residues with DSSP (Kabsch & Sander, 1983) codes outside the set {H, G, I, E} that specifies helix or strand. Turn/tail structures were extracted from their PDB files (Berman et al., 2000) with the aid of a list of peptide chains with a maximum mutual identity of 25% obtained from PISCES (Wang & Dunbrack, 2003). Since sequence motif detection identified multiple motif “artifacts” generated by remaining local redundancy in the data, an additional step of turn‐local redundancy screening was applied: structures were clustered and filtered to reduce redundancy within five residues of turns to 40% or less; this threshold was chosen because it effectively reduced artifacts while preserving a dataset of sufficient size.

Turn structures were screened for quality by requiring that there be no missing residues within 5 residues of any turn and excluding structures with bond lengths that constituted outliers from established values. The final dataset of 102,192 turns was compiled with resolution and R‐value cutoffs of 2.0 Å and 0.25 respectively; average values in the dataset are 1.6 Å and 0.20.

2.3.2. H‐bond definition

H‐bonds are defined by both energy and geometric criteria. The electrostatic (dipole–dipole) energy of a BB H‐bond is computed as (Kabsch & Sander, 1983):

$$\begin{array}{c}E=27.888\left(\frac{1}{r_{\mathrm{ON}}}+\frac{1}{r_{\mathrm{CH}}}-\frac{1}{r_{\mathrm{OH}}}-\frac{1}{r_{\mathrm{CN}}}\right)\end{array}$$ (2)

where the r's represent the distances (Å) between the pairs of atoms indicated in the subscripts, and E is expressed in kcal/mol. The DSSP energy cutoff of −0.5 kcal/mol is applied.

Geometric H‐bond criteria include a maximum H…O distance of 2.6 Å, a minimum N–H…O (donor) angle of 100°, and a minimum H…O=C (acceptor) angle of 90°.

The measurement of H‐bond energy and geometric criteria depends on the presence of amide hydrogens in the structures, which are added to the PDB files using Reduce (Word et al., 1999). Coordinates of hydrogen atoms (and all other atomic coordinates) are extracted for processing with Biopython (Cock et al., 2009). For about 0.5% of residues, BB amide hydrogen atomic coordinates are not available, preventing the evaluation of H‐bonds.

2.3.3. Sequence motif overrepresentation

Statistical models (Bishop et al., 2007; Hu et al., ¹⁹⁹³; Newell, ²⁰¹⁵) are used to evaluate the fractional overrepresentation of single‐AA and pair sequence motifs in structure sets. Fractional overrepresentation is expressed as $\left(O-E\right)/E$, where $O$ is a motif's observed count in the set and $E$ is its expected count under a suitable null model.

For a single‐AA sequence motif, the null model specifies that the probability $P$ of the motif's occurrence in the set is equal to the position‐independent probability of the occurrence of the motif's AA anywhere in proteins, which is computed as the overall abundance fraction of the AA in the complete set of protein chains used in the study: $P=C_{\mathrm{AA}}/T$, where $C_{\mathrm{AA}}$ is the count of the AA in the set of all chains and $T$ is the total number of residues in the chain set. The motif's expected count in a structure set is then $E=\mathit{NP}$, where $N$ is the size of the set.

The null model for a pair sequence motif, which represents independence between the occurrences of the component AAs in the pair, specifies that the probability $P$ of the motif's occurrence in a structure set is equal to the product of the independent probabilities of the occurrences of its component AAs in the set: $P=\left(N_1/N\right)*\left(N_2/N\right)$, where $N_1$ and $N_2$ are the counts of the component AAs in the set and $N$ is the set size. The motif's expected count is $E=\mathit{NP}=\left(N_1*N_2\right)/N$.

A larger fractional overrepresentation for a single‐AA motif signals a higher likelihood that the motif occurs more commonly in the structure set than in proteins generally, and may therefore play a structural or functional role. A larger overrepresentation for a pair motif signals a greater tendency for its two component AAs to occur together, which indicates a higher likelihood of a synergy between the AAs that reflects an interaction such as a SC/SC H‐bond. P‐values for sequence motifs are not computed here, but are available from the ExploreTurns tool.

3. RESULTS

3.1. Descriptor ranges in structures

Figure 1 shows examples of turns with descriptor values near the limits of span, bulge, skew, and warp in the dataset. The figure also illustrates the geometric definitions, turn‐local coordinate system, and geometric descriptors.

3.2. Descriptor distributions

Figure 2 plots the distributions of span, bulge, skew, and warp in the classical types; descriptor distributions for the BB clusters are plotted in Supplementary Section 1.1. See the figure captions for brief comparative descriptions of the distributions, as well as descriptions of the partitioning of the distributions in the classical types into those of the BB clusters. Mean and mode descriptor values in the BB clusters can be more extreme than those in any classical type; for example, the warp distribution for cluster AG_9 falls to the right of that of any type (Figure S1d), while cluster cisDP_18 shows mean (negative) skew more than twice as great as any type (Figure S2c).

FIGURE 2.

Distributions of the geometric descriptors in the classical turn types. Interpolated histograms of the distributions of the geometric descriptors in the structurally classified classical turn types and the structurally unclassified type IV. The vertical axis measures turn fraction within each type. (a) Span: The type I' distribution peaks sharply near 5.3 Å, reflecting the common role of turns of this type as chain‐reversers in 2:2 beta hairpins, and contrasting with the broader peak of the type I distribution, which is consistent with the wider range of type I roles. Types VIb, VIII, and IV all show substantial abundance out to the 7 Å limit of span, reflecting the scarcity of the constraining 4 >1 H‐bond in these types. (b) Bulge: Types VIa2 and VIb extend well to the right of all other types, and types II/II' show greater bulge than I/I′. All bulge distributions exhibit negative Pearson skewness (Doane & Seward, 2011), with more substantial tails on the left, indicating that the upper limit of bulge is “harder” than its lower limit: High‐bulge/low‐span (tighter) turns are more likely to be associated with conformational strain and steric clashes within the turn than low bulge/high span turns. (c) Skew: Types {I, I', VIb} and {II, II', VIa1} can be formed into two loose groups that peak near skew 0 and 0.1 respectively, while types VIII and VIa2 peak to the left and right of these groups, and the structurally unclassified type IV covers the range of all structurally classified types. (d) The warp distributions form five groups from left to right, consisting of types {II, II'}, {IV, VIII}, VIb alone, {VIa1, VIa2}, and {I, I'}. Groups {II, II'} and {IV, VIII} exhibit positive Pearson skewness, with long tails to the right, while {I, I'} shows negative skewness. The similarity between the type IV and VIII distributions in the bulge and warp plots is notable.

3.3. Ramachandran‐space descriptor heatmaps

The distributions of geometric descriptor values across the Ramachandran spaces of the BB dihedral angles of the two central turn residues characterize the bond rotations associated with the variations of each descriptor. Figure 3 presents six notable examples of heatmaps of descriptor distributions in the classical turn types or BB clusters, with descriptor values colored on relative scales within each plot according to the legend. The bond rotations shown in the plots were interpreted in Euclidean space with the aid of a molecular model of a beta‐turn; to see where the bonds described here lie within a turn, see the labeled structure in Figure 1c.

FIGURE 3.

Ramachandran‐space heatmaps characterize the bond rotations associated with the variations of geometric descriptors. Descriptor values are colored on relative scales within each plot according to the legend; actual values can be obtained from ExploreTurns. (a, b) Bulge versus the dihedral angles of the second residue in turns of classical types II and VIb. (c, d) The correlated gradients of warp and skew at the third residue of type VIII turns. (e) The falling gradient of warp across a range of ψ values as φ becomes more obtuse (more negative) at turn residue 3 in the global turn set. (f) Warp variation at residue 3 in BB cluster AD_1, which includes about half of all turns. The ideal conformation for a $3_{10}$ helix is marked on the plot; the helix can be viewed as an extension of a high‐warp beta‐turn.

Figure 3a shows that at the second residue of type II turns, rising $\psi$ reduces bulge; this occurs due to the rotation of the $C_{\alpha 2}$→$N_2$ bond “outwards and upwards” (toward –y, +z) as ${\psi }_2$ increases; this rotation brings all three bonds in the first peptide‐bond plane ($C_{\alpha 2}$→$N_2$, $N_2$→$C_1$, $C_1$→$C_{\mathrm{\alpha }1}$, oriented in the N‐terminal direction) along with it, which reduces bulge by reducing the x‐components of these bonds. In Figure 3b, the second residue of type VIb turns shows a more uniform gradient of falling bulge as $\psi$ rises, which is associated with an almost directly outwards (toward –y) rotation of $C_{\alpha 2}$→$N_2$.

The warp and skew heatmaps for the third residue in type VIII turns, displayed in Figure 3c,d, show broad, highly correlated gradients of falling warp and increasingly negative (N‐terminal) skew from lower‐right to upper‐left in the plots, as the residue opens into a more extended conformation, which both flattens the turn and generates N‐terminal skew by increasing the C‐terminal half‐span.

The most characteristic feature of high warp in turns is the orientation of the turn's middle peptide bond ($C_2$→$N_3$, oriented in the C‐terminal direction) approximately parallel to the z‐axis, and the global warp distribution at residue 3, displayed in Figure 3e, shows a falling gradient over a wide range of $\psi$ as $\phi$ becomes increasingly obtuse below −60°, flattening the BB as $C_2$→$N_3$ rotates down toward the turn plane.

The type I‐associated BB cluster AD_1 includes about half of all turns, and Figure 3f shows that at the third residue of turns in this cluster warp rises as $\phi$ becomes increasingly acute (less negative), supporting the sharper BB reversals required at $C_{\alpha 3}$ to direct the BB back down toward the turn plane after the large z‐dimensional excursions associated with high warp. In this plot, warp is maximal near the line along which $\left(\phi ,\psi \right)$ sums to −75°, ideal for a $3_{10}$ helix, demonstrating the relationship between warp and helical character; a $3_{10}$ helix can be viewed as an extension of a high‐warp beta turn (Pal et al., 2002).

3.4. H‐bond energy versus descriptor value

Figure 4 plots the electrostatic (dipole–dipole) energy of the 4>1 BB H‐bond versus geometric descriptor values for classical types I and II, the largest types in which the H‐bond is common. In type I, energy drops steadily with increasing warp, from 0.3 kcal/mol for the flattest turns to −1.8 kcal/mol at high warp (Figure 4a), while in type II energy shows a gradual increase as warp rises (Figure 4b). The negative correlation ($r$ = −0.61) between H‐bond energy and warp in type I can be rationalized by the observation that the 4>1 H‐bond donor and acceptor typically show a much poorer alignment, with a greater H…0 distance, in flatter versus higher‐warp turns due to rotations of the first and third peptide‐bond planes as warp falls; the primary factor is likely the rotation of the $C_1$→$O_1$ H‐bond acceptor toward −z, but the $N_4$→$H$ donor can also exhibit a limited rotation toward +z. The incompatibility between low warp and the 4>1 H‐bond in type I turns is a major contributor to the much lower frequency of this H‐bond in type I (71%) than in the other three largest classical types that commonly exhibit it: I' (95%), II (86%), and II' (92%); H‐bond frequency in the 47% of type I turns with warp greater than 0.75 is 90%.

FIGURE 4.

Electrostatic (dipole–dipole) energy of the 4>1 beta‐turn H‐bond versus descriptor value for warp, span, skew, and bulge in classical types I (top) and II (bottom). (a, b) H‐bond energy versus warp. (c, d) H‐bond energy versus span. (e, f) H‐bond energy versus skew (g, h) H‐bond energy versus bulge.

In type I (Figure 4c), energy falls as span increases until a minimum is reached near span 5.4 Å, but then reverses its trend and rises steadily, while in type II energy remains at its minimum as span increases until about 5.4 Å, then rises steadily (Figure 4d). The rise in H‐bond energy above 5.4 Å in both types can be rationalized by the expected increase in H‐bond length with increasing span, but the initial energy drop with rising span in type I defies this; it is likely an indirect effect associated with the correlation between span and warp in this type (Supplementary Section 1.2): lower span is linked with lower warp, which is associated with weak H‐bonds.

H‐bond energy is only weakly sensitive to skew in type II (Figure 4f), with low energy across the skew range, but in type I (Figure 4e) energy has its minimum at skew zero and rises dramatically with absolute skew.

The steady drop in H‐bond energy in type II as bulge increases (Figure 4h) can be rationalized as an indirect effect, associated with the negative correlation between bulge and span, that is consistent with the behavior of a simple wire‐frame model for the turn: in this model, an increase in bulge is associated with a drop in span due to conservation of BB length, and reduced span shortens and strengthens the H‐bond. However, this picture does not hold in type I (Figure 4g), where energy falls to a minimum near 3.2 Å, then begins a steady rise from 3.3 Å upwards. This behavior can also be rationalized as an indirect effect, this time associated with the strong negative correlation between bulge and warp in this type (r = −0.77): higher bulge is linked with lower warp, which is incompatible with a strong H‐bond.

3.5. Sequence preference versus descriptor value

The categories of the classical turn types and BB clusters specify ranges of turn geometries which are commonly separated by energy barriers established by BB sterics within the turn, and factors such as a turn's AA content and the influence of the structures in a protein external to the turn affect a turn's choice of conformation within these ranges. Geometric descriptors can discriminate between the conformations within types and clusters, and the overrepresentations of multiple key sequence motifs in turns and their BB neighborhoods show clear relationships to descriptor values. Figures 5, 6, 7 present plots of fractional overrepresentation vs. span, bulge, skew and warp for sequence motifs which commonly form SC/BB H‐bonds (Figures 5, 6), SC/SC H‐bonds (Figure 7a,b), or pi‐stacking interactions (Figure 7c); see the figure captions for details. As the plots show, geometric descriptors can serve as structural discriminators within classical types or BB clusters that dramatically increase the specificity of measurements of sequence preferences in turns and their BB neighborhoods. For example, while the sequence motif that specifies Thr at turn position 3 (T3) is overrepresented by 48% in the set of all type I turns, Figure 6e shows that its overrepresentation peaks near 320% at low warp, reflecting roles that include SC/BB H‐bonds linking the flat (for type I) “chair back” turn in the type 2 beta‐bulge loop (BBL2) (Milner‐White, 1987) to the motif's bulge. Similarly, the pair motif that specifies Asp at turn position 1 and Thr at position 3 (D1T3), associated with SC/SC H‐bonding above the turn plane, is overrepresented by just 9% in the set of all type I turns, but Figure 7b shows that it peaks near 100% at high warp, which is likely favorable because it raises Thr's SC into position for H‐bonding with Asp.

FIGURE 5.

Examples of relationships between sequence preferences and span, bulge and skew. (a, b) Plots of fractional overrepresentation versus span for Asp and Thr at turn position 4 (D4 and T4) in type II turns show prominent maxima that likely correspond to the optimal spans for SC/BB H‐bonding with the main‐chain NH group of the first turn residue (MCN1, for D4) or the main‐chain carbonyl group of that residue (MCO1, for T4). (c) The overrepresentation of Ser at position 1 (S1) drops as bulge rises in type VIII, likely reflecting the lengthening and weakening of the H‐bond between Ser's SC and MCN3 (adjacent to the turn center) in this ST turn/motif (Wan & Milner‐White, ^1999a). (d) S1's overrepresentation also falls as bulge increases in the type VIII‐associated BB cluster AB1_5. (e, f) While Asp at position 1 (D1) in type I turns shows peak overrepresentation near zero skew, Ser at position 1 (S1) shows a negative‐skew bias; this contrast may reflect the suitability of a short N‐terminal half‐span (which promotes negative skew) for optimal H‐bonding between Ser's shorter SC and the BB NH group at the turn's C‐terminus (MCN4).

FIGURE 6.

Examples of relationships between sequence preferences and warp. (a, b) Asp and Asn at turn position 1 (D1 and N1) in type I turns show peak fractional overrepresentation at high warp, reflecting the compatibility of this geometry with SC/BB H‐bonding with the main‐chain NH groups of the third and fourth residues (MCN3/MCN4) in Asx turns/motifs (Wan & Milner‐White, ^1999b) in multiple contexts, including open loops and the type 1 beta‐bulge loop (Milner‐White, 1987) (BBL1) (see Figure 9g). The favorability of high warp for these motifs likely reflects an indirect effect associated with the negative correlation between warp and bulge (Supplementary Section 1.2); structural influences that produce high warp also yield low bulge, and this results in a shorter distance between these SCs and MCN3 adjacent to the turn center. (c) In contrast to D1/N1, Asp just before the turn (D‐1) peaks at low warp, reflecting compatibility with structures, such as the type 2 beta‐bulge loop (BBL2, Figure 9h) and the alpha‐beta loop (Leader & Milner‐White, 2009), in which flatter turns orient at approximately 90° to the incoming BB; D‐1's SC supports these “upright turns” by H‐bonding with MCN2/MCN3. (d) The plot for Asp at turn position 3 (D3) reflects multiple roles: At minimum warp, the motif links the upright turns in BBL2s to the main‐chain NH group just after the turn (MCN+1) in the loop's bulge, at mid/high warp D3 H‐bonds with its own BB NH group and/or MCN+1 in BBL1s and other structures, while at maximum warp the motif H‐bonds with MCN+1 and the SC immediately after the turn in the BBL1s of beta propellers (Pons et al., 2003). (e) At low warp, Thr at position 3 (T3) forms SC/BB and SC/SC H‐bonds in BBL2s (see Figure 9h); this motif is four times as common as D3 in this context. (f) Ser at turn position 3 (S3) peaks at low warp, reflecting a role like that of D3/T3 in BBL2s, and at high warp, reflecting roles including SC/SC H‐bonding with turn position 1 in multiple contexts and SC/BB H‐bonding with MCN+1 in structures such as BBL1s.

FIGURE 7.

Examples of relationships between pair sequence preferences and warp. (a, b) Arg1 + Asp3 (R1D3) and Asp1 + Thr3 (D1T3) show strong preferences for high warp, which likely optimizes SC positioning for H‐bonding (4A56_A_55 and 3AL2_A_1301). (c) Phe2 + Phe3 (F2F3) favors lower warp, which brings the SCs closer for perpendicular pi‐stacking (3PYC_A_29). See the Figure 1 caption for ExploreTurns structure viewing directions.

With the guidance of built‐in motif overrepresentation plots like those shown here, geometric descriptors can be used in ExploreTurns to tune structures for compatibility with particular sequence motifs, and the facility's motif detection tools can be used to identify the motifs important in particular turn types, clusters or BB motifs. Sequence motifs can be comprehensively explored in the interactive, 3D conformational heatmaps generated by the MapTurns tool; to open a motif's map, enter its label into the Sequence motif box in ExploreTurns and click Map Motif.

3.6. Effect of cis‐peptide bonds on bulge and skew

Figure 8 demonstrates strong relationships between the occurrence of cis‐peptide bonds in a beta‐turn and the turn's skew and bulge, and these effects show a degree of superposition when cis‐peptide bonds are combined (see the figure caption for details).

FIGURE 8.

cis ‐peptide bonds influence bulge and skew and show a degree of superposition. (a) Turns with a cis conformation at the first peptide bond (cis ₁) exhibit low average bulge (2.89 Å vs. the 3.45 Å global avg.) and a negative‐skew bias (3O4P_A_140). (b) cis ₂ turns show high bulge with a minor positive‐skew bias (1SG6_A_99). (c) cis ₃ turns show low bulge and a positive skew bias (3UCP_A_685). (d, f) In cis ₁ cis ₂ and cis ₂ cis ₃ turns, the low‐bulge biases of the component cis ₁ and cis ₃ peptide bonds effectively cancel the high‐bulge bias of cis ₂, yielding bulges close to the global mean, while skew biases of the same signs as the cis ₁ and cis ₃ components persist (enhanced in cis ₁ cis ₂) when combined with cis ₂'s near‐neutral skew (3LCC_A_180, 4OW5_A_102). (e) In cis ₁ cis ₃ turns, the low‐bulge biases of the cis components show an additive effect, resulting in extreme low bulge, while cis ₁'s larger negative‐skew bias is reduced by combination with cis ₃'s smaller positive bias (1ZR6_A_397). Note that sample sizes are small for the combination cis turns.

The skew biases of cis‐peptide bonds can be compounded by combination with biased Ramachandran types. For example, the combination of a cis conformation at the first peptide bond (cis ₁ , avg. skew −0.25) with the negatively‐biased type AB (−0.18), yields cisAB (−0.41), and the combination of cis ₃ (0.16) with Ba (0.20) yields Bacis (0.36), in which the biases add almost exactly. The Ramachandran type that specifies cis‐polyPro conformations at both central turn positions (cisPcisP) shows extreme skew bias (−0.46). To view a Ramachandran‐region map in ExploreTurns, click Ramachandran Types; turns of a particular Ramachandran type can be browsed and profiled by entering the type name in the Type or BB cluster box and clicking Load Turns Matching Criteria, and turns with particular cis‐peptide bonds be viewed by selecting the bonds in the cis _n (y/n) boxes.

3.7. Descriptor correlations

The structural constraints imposed by bond lengths and angles in the beta‐turn BB impose relationships between the descriptors that reflect geometric tradeoffs in turn design. Supplementary Section 1.2 presents a table of descriptor cross‐correlations and rationalizes several key examples. Notable descriptor relationships include negative correlations between span and bulge in all classical types and bulge and warp in all types except VIa2 and VIb. Both of these relationships can be rationalized by the conservation of BB length in a simple wire‐arch model for turns, but a third relationship, the positive correlation between warp and span in types (I, I', II, II', IV and VIa1), cannot be explained by a wire‐arch model and likely reflects the effects of bond rotations which can widen the turn as warp increases.

3.8. Roles and contexts of descriptor variations

Turn descriptors reflect geometric variations which effect the suitability of turns for particular roles and contexts. The bulge and span descriptors correlate strongly with a turn's curvature, and turns of high bulge/low span are found where the BB must reverse within spaces that are narrow in the y‐dimension of turn coordinates. Bulge provides conformational range in the x dimension; turns with a higher bulge can project the BB and SC groups of their central residues further towards −x for structural interactions, participation in ligand binding or active sites, or to increase solvent exposure. On the opposite end of the bulge/span scale, the most common context for turns with the largest spans/smallest bulges is extended loops.

Skew is a measure of a turn's asymmetry across the y dimension, while the half‐spans that determine skew reflect some flexibility in the positioning of BB and SC groups on each side of the turn axis, which can be used to satisfy packing constraints or support interactions. Ramachandran types which specify a more extended conformation at one central turn residue are associated with skew biases, since they lengthen one half‐span. For example, Ramachandran types AB, AZ and AP show negative‐skew biases due to their {B, Z, P} conformations at the third turn residue; types AB and AZ are largely responsible for the negative‐skew bias of classical type VIII. Extended conformations at the second turn residue have the opposite effect on skew: types {Ba, Za, Pa, Pd} show positive‐skew biases, and Pa and Pd are responsible for most of classical type II's positive bias.

Warp measures the magnitude of the excursions a turn's BB makes above and below the turn plane; low warp can accommodate tight packing in the z‐dimension, while high warp enables a turn's BB and SC groups to interact with structures farther above and/or below the plane.

Just as the values of all four descriptors show particular regimes that are compatible with key SC motifs, they also exhibit compatibilities with the structures in a protein that incorporate turns, including beta hairpins, local H‐bonded motifs (Leader & Milner‐White, 2009) and ligand‐binding sites. Turn geometry and context are interrelated; a turn's AA composition can bias its geometry in favor of a particular context, while the steric contacts, BB stress, H‐bonding, and other interactions associated with a context affect a turn's conformation, influencing the values of its geometric descriptors and, when the effect is strong enough, its choice of classical type or BB cluster.

Figure 9 displays examples of structural contexts associated with higher or lower descriptor values. The 2:4 beta‐hairpin (Sibanda & Thornton, 1985) (Figure 9f) demonstrates a notable relationship between warp and beta‐turn context. Most chain‐reversing turns in 2:2 hairpins (the most common hairpin type) are of types I' or II'; turns of type I are much less common due to a geometric clash between the right‐handed inter‐strand twist of the hairpin's beta ladder and the left‐handed twist between the turn's incoming and outgoing BB segments (Richardson, 1981). This clash is reduced in low‐warp turns because the turn's twist is reduced, rendering its structure more compatible with the ladder, but the conformational changes associated with warp reduction commonly disrupt the turn's 4>1 H‐bond (Figure 4a), yielding a 2:4 hairpin in place of the more common 2:2 type. As a result, more than half of 2:4 hairpins in the beta‐turn dataset in effect owe their existence to the compatibility between low‐warp and the ladder's twist and the inverse relationship between warp and H‐bond energy.

FIGURE 9.

Roles and contexts of descriptor variations. See the Figure 1 caption for ExploreTurns structure viewing directions. (a) Chain‐reversing loops in 4:4 beta hairpins are commonly formed of high‐bulge type VIb turns (3AKS_A_81). (b) The low bulge of a type IV (Ramachandran type dA) turn in a helix‐entrance loop positions the Asp SC at turn position 3 (D3) to mediate a network of H‐bonds binding the loop (5UJ6_A_542). (c) The positive‐skew geometry of a type IV (Ba) turn, with its shorter C‐terminal half‐span, helps position turn and helix groups for DTP/DNA binding in DNA polymerase (5KFZ_A_14). (d) The shorter N‐terminal half‐span of a negatively skewed turn helps position the helix with respect to the turn for C‐capping and Cd binding (3B40_A_50). (e) The short C‐terminal half‐span of a positively‐skewed type VIa2 (BcisA) turn (avg. skew 0.25 for the type) is compatible with a shorter BB H‐bond that secures the geometry of a helix‐terminal loop (3V5C_A_287). (f) The low warp of a type I turn is compatible with the ladder component of a beta‐hairpin but incompatible with the turn's 4>1 H‐bond, yielding a 2:4 hairpin in place of the more common 2:2 type (4ZO2_A_102). (g) A high‐warp type I turn is compatible with a BBL1's interior 4>1 BB H‐bond as the turn forms the back of the loop's “reclined chair”; the motif is shown here with Asp at position 1 (D1, see overrepresentation plot in Figure 6a) (4K9Z_A_34). (h) A low‐warp type I turn is compatible with the “upright chair” conformation of a BBL2 and the motif's characteristic 4>−1 H‐bond. The loop is shown with its chair‐back turn supported by SC/BB H‐bonds mediated by Asp just before the turn and Thr at turn position 3 (D‐1 and T3, see Figure 6c, e) (3SCY_A_360).

The chain‐reversing turns in type 1 and 2 BBLs, which are both commonly of type I, demonstrate further relationships between warp and context. In the BBL1 (Figure 9g), a high‐warp turn supports the loop's 4>1 H‐bond, commonly forming a “reclined chair”, while in the BBL2 (Figure 9h), the low‐warp geometry of the turn that forms the back of the motif's “upright chair” suppresses the 4>1 H‐bond in favor of the 4>−1 alternative, which is characteristic of the loop.

3.9. Descriptors and structure quality

Since geometric descriptors enhance the Euclidean‐space picture of turn geometry, they may aid the detection and characterization of structurally suspicious conformations. Supplementary Section 1.3 investigates the relationship between descriptor values and structure quality, and finds associations between extreme descriptor values and an increased frequency of geometric quality criteria with outliers as measured by PDB validation tools (Gore et al., 2017). Outlier frequency is found to be higher when descriptor values are extreme not only for the global set of turns but also for the particular neighborhoods of the recurrent BB geometries, as represented by the BB cluster medoids.

4. CONCLUSIONS

The turn‐local coordinate system and implicit global alignment derived here resolve the difficulties with visualization inherent in the Ramachandran‐space turn representation, providing a natural framework for the comparison and analysis of the wide variety of beta‐turn structures and the contexts in which they occur, including H‐bonded motifs, ligand‐binding/active sites and supersecondary structures. The utility of this framework has been demonstrated (Newell, 2024) by the application of the ExploreTurns tool to the detection and characterization of multiple new short H‐bonded loops, the mapping of Asx N‐cap sequence preference vs. cap geometry and other tasks, and the framework also underpins the comprehensive maps of the BB and SC structure, H‐bonding and contexts associated with sequence motifs in beta turns that are generated by the MapTurns tool (Newell, 2024).

Geometric turn descriptors provide a Euclidean‐space picture of the bulk BB shapes of turns that reflects meaningful modes of geometric variation not explicitly captured by Ramachandran‐space representations. Descriptors complement existing turn classification systems, enabling a fine‐tuning of BB geometry within classical types or BB clusters that supports the selection of structures for compatibility with particular SC interactions or turn contexts, and produces major improvements in the specificity of measurements of AA sequence preferences in turns. Since close to two‐thirds of residues in protein loops are found in beta turns (Shapovalov et al., 2019), descriptors bring needed additional order to the characterization of the irregular geometries of loops, which represent about half of all protein structures (Regad et al., 2010).

Each particular set of continuous descriptor values does not specify a conformation that is constrained by its own energy well, as do some (but perhaps not all) of the discrete categories of the Ramachandran‐space classification systems. But turn descriptors can be used in ExploreTurns, in combination with the Ramachandran‐space classification systems, to select from particular ranges of BB shapes that are biased and constrained by the energy barriers associated with the categories of those systems, since the descriptor distributions in the types and BB clusters overlap but do not coincide (Figures 2, S1, and S2). For example, Figure 2 shows that if maximum bulge is the priority (and the BB conformation is otherwise acceptable), then a type VIb turn should be chosen, while Figure S1 indicates that a turn from cluster AB1_5 is appropriate if negative skew is needed, while a member of cluster Pd_2 is a good choice if flatness is a priority.

Turn‐local coordinates and geometric descriptors should prove useful in any application where an enhanced Euclidean‐space picture of turns and structural tuning can improve understanding or performance. One potential area of application is protein design, and in particular the design of loops in binding and active sites, where turns are common and even small changes in geometry may have major impacts on function. Model beta‐turn structures are employed in loop design (Jiang et al., 2024), and descriptors provide a systematic, Euclidean‐space picture of the BB shapes of turns which may inform design decisions and identify model structures that support those decisions. For example, if it were recognized that a high‐bulge turn could best position a SC in an active site, then ExploreTurns, with its descriptor‐, sequence motif‐ and context‐selection features, could be used to identify sets of candidate turns with high bulge that contain the desired SC and are embedded in loop contexts similar to the design target. Descriptors may also aid in the design of stable loops by more precisely identifying the SC interactions most likely to stabilize a particular desired geometry, or by supporting the selection of sets of model turns with geometries tuned for compatibility with particular SC interactions known to be stabilizing for that geometry.

Geometric descriptors may also prove valuable in structure validation, by improving the characterization of suspicious conformations and identifying extreme geometries that may not always be highlighted as outliers by existing validation tools.

AUTHOR CONTRIBUTIONS

Nicholas E. Newell: Conceptualization; investigation; writing – original draft; methodology; validation; visualization; writing – review and editing; software; formal analysis.

FUNDING INFORMATION

This work was funded entirely by the author.

CONFLICT OF INTEREST STATEMENT

The author declares that he has no competing interests.

Supporting information

Supplementary file 1 Newell Geometric Descriptors.

Newell NE. Geometric descriptors for beta turns. Protein Science. 2024;33(9):e5159. 10.1002/pro.5159

DATA AVAILABILITY STATEMENT

All data supporting the conclusions of this article are available from the Protein Data Bank at www.rcsb.org. The ExploreTurns, MapTurns, and ProfileTurn web tools are available at www.betaturn.com.