Theory of protein molecule self-organization. III. A calculating method for the probabilities of the secondary structure formation in an unfolded polypeptide chain

A mathematical model is developed adequately describing an unfolded polypeptide chain without long-range interactions in which fluctuating hydrogen-bonded α-helices, β-bends, fragments of helices 3₁₀, and other local structures are formed. The obtained model is a modification of a one-dimensional Ising model for a heteropolymer and allows one to determine the probability of formation of different secondary structures in various parts of a polypeptide chain, provided the whole set of structural thermodynamic parameters exists.     

Cooperative effects in α-helices: An ab initio molecular-orbital study

Some properties of α-helices of polyclycine and polyalanine, up to the decapeptide, were investigated by ab initio molecular-orbital calculations. These helices were found to be unstable relative to the corresponding “fully extended chain” conformation. The electric field of helices of 8–10 residues is about 20% stronger than that of models built from noninteracting monomers, for example. This is a result of cooperativity, which is essentially governed by the intramolecular hydrogen bonds. The cooperativity is manifest in all properties of the helices: relative stability, dipole moment, proton affinity, electrical potential. The electric potential of helices of three and four residues is such that their instability can be compensated for by a single charged group acting as an “initiator.” The computed proton affinity of the (Ala)₈ α-helix is about 45 kcal/mol larger than that of formamide, which confirms that long helices may be protonated at the carboxyl end in solution.     

Novel protein structural motifs containing two-turn and longer 3(10)-helices.

The 3(10)-helix constitutes a small but significant fraction of secondary structural elements in proteins. Protein data base surveys have shown these helices to be present as alpha-helical extensions, in loops and as connectors between beta-strands. The present work focuses on two-turn and longer 3(10)-helices where we establish that two-turn and longer 3(10) helices, unlike the more abundant single-turn 3(10)-helices, frequently occur independent of any other contiguous secondary structural elements. More importantly, a large fraction of these independent two-turn and longer 3(10)-helices, along with alpha-helices and beta-strands, are found to form novel super-secondary structural motifs in several proteins with possible implications for protein folding, local conformational relaxation and biological functions.     

3(10)-helices in proteins are parahelices

The 3(10)-helix is characterized by having at least two consecutive hydrogen bonds between the main-chain carbonyl oxygen of residue i and the main-chain amide hydrogen of residue i + 3. The helical parameters--pitch, residues per turn, radius, and root mean square deviation (rmsd) from the best-fit helix--were determined by using the HELFIT program. All 3(10)-helices were classified as regular or irregular based on rmsd/(N - 1)1/2 where N is the helix length. For both there are systematic, position-specific shifts in the backbone dihedral angles. The average phi, psi shift systematically from approximately -58 degrees, approximately -32 degrees to approximately -90 degrees, approximately -4 degrees for helices 5, 6, and 7 residues long. The same general pattern is seen for helices, N = 8 and 9; however, in N = 9, the trend is repeated with residues 6, 7, and 8 approximately repeating the phi, psi of residues 2, 3, and 4. The residues per turn and radius of regular 3(10)-helices decrease with increasing length of helix, while the helix pitch and rise per residue increase. That is, regular 3(10)-helices become thinner and longer as N increases from 5 to 8. The fraction of regular 3(10)-helices decreases linearly with helix length. All longer helices, N > or = 9 are irregular. Energy minimizations show that regular helices become less stable with increasing helix length. These findings indicate that the definition of 3(10)-helices in terms of average, uniform dihedral angles is not appropriate and that it is inherently unstable for a polypeptide to form an extended, regular 3(10)-helix. The 3(10)-helices observed in proteins are better referred to parahelices.     

Kinetic steps for α-helix formation

The kinetics of α-helix formation in polyalanine and polyglycine eicosamers (20-mers) were examined using torsional-coordinate molecular dynamics (MD). Of one hundred fifty-five MD experiments on extended (Ala)₂₀ carried out for 0.5 ns each, 129 (83%) formed a persistent α-helix. In contrast, the extended state of (Gly)₂₀ only formed a right-handed α-helix in two of the 20 MD experiments (10%), and these helices were not as long or as persistent as those of polyalanine. These simulations show helix formation to be a competition between the rates of (a) forming local hydrogen bonds (i.e. hydrogen bonds between any residue i and its i + 2, i + 3, i + 4, or i + 5th neighbor) and (b) forming nonlocal hydrogen bonds (HBs) between residues widely separated in sequence. Local HBs grow rapidly into an α-helix; but nonlocal HBs usually retard helix formation by “trapping” the polymer in irregular, “balled-up” structures. Most trajectories formed some nonlocal HBs, sometimes as many as eight. But, for (Ala)₂₀, most of these eventually rearranged to form local HBs that lead to α-helices. A simple kinetic model describes the rate of converting nonlocal HBs into α-helices. Torsional-coordinate MD speeds folding by eliminating bond and angle degrees of freedom and reducing dynamical friction. Thus, the observed 210 ps half-life for helix formation is likely to be a lower bound on the real rate. However, we believe the sequential steps observed here mirror those of real systems. Proteins 33:343–357, 1998. © 1998 Wiley-Liss, Inc.     

Theory of protein molecule self-organization. II. A comparison of calculated thermodynamic parameters of local secondary structures with experiments

Constants of the helix–coil transition for all natural amino acid residues are evaluated on the basis of thermodynamic parameters obtained in paper I of this series. The specific effects at the termini of the helices are also considered as well as the parameters controlling the formation of β-bends in the unfolded protein chain. Evaluated s constants of the helix–coil transition agree with the experimental data on helix–coil transitions of synthetic polypeptides in water. Only a very qualitative correlation exists between s constants (both experimental and theoretical) and the occurrence of corresponding residues in internal turns of α-helices in globular proteins: residues with s > 1 occur in helices as a rule more often than residues with s < 1. At the same time a direct correlation is demonstrated between theoretical parameters of residue incorporation into α-helical termini and β-bends in an unfolded polypeptide chain and the occurrence of residues in corresponding positions of the globular protein secondary structures.     

Variants of 3(10)-helices in proteins

An analysis of the shortest 3(10)-helices, containing three helical residues and two flanking capping residues that participate in two consecutive i + 3 --> i hydrogen bonds, shows that not all helices belong to the classic 3(10)-helix, where the three central residues adopt the right-handed helical conformation (alpha(R)). Three variants identified are: 3L10-helix with all residues in the left-handed helical region (alpha(L)), 3EL10-helix where the first residue is in the extended region followed by two residues in the alpha(L) conformation, and its mirror-image, the 3E'R10-helix. In the context of these helices, as well as the equivalent variants of alpha-helices, the length dependence of the handedness of secondary structures in protein structure is discussed. There are considerable differences in the amino acid preferences at different positions in the various types of 3(10)-helices. Each type of 3(10)-helix can be thought to be made up of an extension of a particular type of beta-turn (made up of residues i to i + 3) such that the (i + 3)th residue assumes the same conformation as the preceding residue. Distinct residue preferences at i and i + 3 positions seem to decide whether a particular stretch of four residues will be a beta-turn or a 3(10)-helix in the folded structure.     

Temperature coefficients of amide proton NMR resonance frequencies in trifluoroethanol: A monitor of intramolecular hydrogen bonds in helical peptides?

Summary 2D ¹H NMR spectroscopy of two -helical peptides which differ in their amphipathicity has been used to investigate the relationships between amide-proton chemical shifts, amide-proton exchange rates, temperature, and trifluoroethanol (TFE) concentration. In 50% TFE, in which the peptides are maximally helical, the amide-proton chemical shift and temperature coefficient patterns are very similar to each other in each peptide. Temperature coefficients from –10 to –6 ppb/K, usually indicative of the lack of intramolecular hydrogen bonds, were observed even for hydrophobic amino acids in the center of the -helices. However, slow hydrogen isotope exchange for residues from 4 to 16 in both 18-mer helices indicates intact intramolecular hydrogen bonds over most of the length of these peptides. Based on these anomalous observations, we suggest that the pattern of amide-proton shifts in -helices in H₂O/TFE solvents is dominated by bifurcated intermolecular hydrogen-bond formation between the backbone carbonyl groups and TFE. The amide-proton chemical shift changes with increasing temperature may be interpreted by a disruption of intermolecular hydrogen bonds between carbonyl groups and the TFE in TFE/water rather than by the length of intramolecular hydrogen bonds in -helices. Supplementary Material is available upon request, comprising seven pages with listings of experimental details and the NMR shift data for the two peptides.     

Sequence and structure patterns in proteins from an analysis of the shortest helices: implications for helix nucleation

The shortest helices (three-length 3(10) and four-length alpha), most abundant among helices of different lengths, have been analyzed from a database of protein structures. A characteristic feature of three-length 3(10)-helices is the shifted backbone conformation for the C-terminal residue (phi,psi angles: -95 degrees,0 degrees ), compared to the rest of the helix (-62 degrees,-24 degrees ). The deviation can be attributed to the release of electrostatic repulsion between the carbonyl oxygen atoms at the two C-terminal residues and further stabilization (due to a more linear geometry) of an intrahelical hydrogen bond. A consequence of this non-canonical C-terminal backbone conformation can be a potential origin of helix kinks when a 3(10)-helix is sequence-contiguous at the alpha-helix N-terminal. An analysis of hydrogen bonding, as well as hydrophobic interactions in the shortest helices shows that capping interactions, some of them not observed for longer helices, dominate at the N termini. Further, consideration of the distribution of amino acid residues indicates that the shortest helices resemble the N-terminal end of alpha-helices rather than the C terminus, implying that the folding of helices may be initiated at the N-terminal end, which does not get propagated in the case of the shortest helices. Finally, pairwise comparison of beta-turns and the shortest helices, based on correlation matrices of site-specific amino acid composition, and the relative abundance of these short secondary structural elements, leads to a helix nucleation scheme that considers the formation of an isolated beta-turn (and not an alpha-turn) as the helix nucleation step, with shortest 3(10)-helices as intermediates between the shortest alpha-helix and the beta-turn. Our results ascribe an important role played by shortest 3(10)-helices in proteins with important structural and folding implications.     

Converting a marginally hydrophobic soluble protein into a membrane protein

δ-Helices are marginally hydrophobic α-helical segments in soluble proteins that exhibit certain sequence characteristics of transmembrane (TM) helices [Cunningham, F., Rath, A., Johnson, R. M. & Deber, C. M. (2009). Distinctions between hydrophobic helices in globular proteins and TM segments as factors in protein sorting. J. Biol. Chem., 284, 5395-402]. In order to better understand the difference between δ-helices and TM helices, we have studied the insertion of five TM-like δ-helices into dog pancreas microsomal membranes. Using model constructs in which an isolated δ-helix is engineered into a bona fide membrane protein, we find that, for two δ-helices originating from secreted proteins, at least three single-nucleotide mutations are necessary to obtain efficient membrane insertion, whereas one mutation is sufficient in a δ-helix from the cytosolic protein P450BM-3. We further find that only when the entire upstream region of the mutated δ-helix in the intact cytochrome P450BM-3 is deleted does a small fraction of the truncated protein insert into microsomes. Our results suggest that upstream portions of the polypeptide, as well as embedded charged residues, protect δ-helices in globular proteins from being recognized by the signal recognition particle-Sec61 endoplasmic-reticulum-targeting machinery and that δ-helices in secreted proteins are mutationally more distant from TM helices than δ-helices in cytosolic proteins.     

The structure of the ends of α-helices in globular proteins: effect of additional hydrogen bonds and implications for helix formation

We prepared a set of about 2000 α-helices from a relational database of high-resolution three-dimensional structures of globular proteins, and identified additional main chain i ← i+3 hydrogen bonds at the ends of the helices (i.e., where the hydrogen bonding potential is not fulfilled by canonical i ← i+4 hydrogen bonds). About one-third of α-helices have such additional hydrogen bonds at the N-terminus, and more than half do so at the C-terminus. Although many of these additional hydrogen bonds at the C-terminus are associated with Schellman loops, the majority are not. We compared the dihedral angles at the termini of α-helices having or lacking the additional hydrogen bonds. Significant differences were found, especially at the C-terminus, where the dihedral angles at positions C2 and C1 in the absence of additional hydrogen bonds deviate substantially from those occurring within the α-helix. Using a novel approach we show how the structure of the C-terminus of the α-helix can emerge from that of constituent overlapping α-turns and β-turns, which individually show a variation in dihedral angles at different positions. We have also considered the direction of propagation of the α-helix using this approach. If one assumes that helices start as a single α-turn and grow by successive addition of further α-turns, the paths for growth in the N → C and C → N directions differ in a way that suggests that extension in the C → N direction is favored.     

Abundance and arrangement of single,double and bifurcated hydrogen bonds in protein α-helices: Statistical analysis of PDB-select

Statistics are collected and analyzed for the possibility of hydrogen bonding in the secondary structures of globular proteins, based on geometric criteria. Double and bifurcated bonds are considered as pairs of admissible H-bonds with two proton donors or two proton acceptors, respectively. Most of such bonds belong to peptide groups in α-helices, with O _i …N _{i + 3} nearly as frequent as O _i …N _{i + 4}; in contrast, most of the 3/10-helical segments are too short to have any. Alternating double and bifurcated bonds in α-helices form an apparently cooperative network structure. A typical α-helical segment perhaps carries two stretches of the H-bond network broken in the middle. The constituent H-bonds are nonlinear: the hydrogen atom is off the straight line connecting the proton donor and proton acceptor atoms. This deflection is larger for H _{i + 3} vs. bond line O _i −N _{i + 3} than for H _{i + 4} vs. O _i −N _{i + 4}, and though the two kinds of bond have about the same length (exceeding those typical of low-molecular compounds), O _i …N _{i + 4} must be stronger than O _i …N _{i + 3}. Double/bifurcated bonds are also not coplanar, i.e., hydrogen atoms are beyond the N…O…N (or O…N…O) plane. The text was submitted by the authors in English.     

Packing scheme of α-helices in the histone core of the nucleosome

A nucleosome histone core model is presented which is compatible with experimental data. The model consists of 28 α-helices located as predicted by others^1–4. The histone wheel resembles the one proposed by Klug et al.⁵ Most of the helices are packed nearly parallel to the DNA superhelical axis, forming a bandoleer-like structure. About 10 to 20% of the nucleosomal phosphates may be neutralized by positively charged residues in the α-helices. Disregarding the charge of the NH₂-terminals, this is sifficient for the thermodynamic stability of the nucleosome under physiological conditions. The electrostatic charge on the protein surface is assumed to be relatively fixed due to the participation of the corresponding side chains to the hydrophobically packed helices. Thus, DNA wrapping appears to be determined by the core histones not by the histone NH₂-terminals.     

Patterns in ionizable side chain interactions in protein structures

In a selected set of 44 high-resolution, non-homologous protein structures, the intramolecular hydrogen bonds or salt bridges formed by ionizable amino acid side chains were identified and analyzed. The analysis was based on the investigation of several properties of the involved residues such as their solvent exposure, their belonging to a certain secondary structural element, and their position relative to the N- and C-termini of their respective structural element. It was observed that two-thirds of the interactions made by basic or acidic side chains are hydrogen bonds to polar uncharged groups. In particular, the majority (78%) of the hydrogen bonds between ionizable side chains and main chain polar groups (sch:mch bonds) involved at least one buried atom, and in 42% of the cases both interacting atoms were buried. In α-helices, the sch:mch bonds observed in the proximity of the C- and N-termini show a clear preference for acidic and basic side chains, respectively. This appears to be due to the partial charges of peptide group atoms at the termini of α-helices, which establish energetically favorable electrostatic interactions with side chain carrying opposite charge, at distances even greater than 4.5 Å. The sch:mch interactions involving ionizable side chains that belong either to β-strands or to the central part of α-helices are based almost exclusively on basic residues. This results from the presence of main chain carbonyl oxygen atoms in the protein core which have unsatisfied hydrogen bonding capabilities.     

Differences in the amino acid distributions of 3(10)-helices and alpha-helices.

Local determinants of 3(10)-helix stabilization have been ascertained from the analysis of the crystal structure data base. We have clustered all 5-length substructures from 51 nonhomologous proteins into classes based on the conformational similarity of their backbone dihedral angles. Several clusters, derived from 3(10)-helices and multiple-turn conformations, had strong amino acid sequence patterns not evident among alpha-helices. Aspartate occurred over twice as frequently in the N-cap position of 3(10)-helices as in the N-cap position of alpha-helices. Unlike alpha-helices, 3(10)-helices had few C-termini ending in a left-handed alpha conformation; most 3(10) C-caps adopted an extended conformation. Differences in the distribution of hydrophobic residues among 3(10)- and alpha-helices were also apparent, producing amphipathic 3(10)-helices. Local interactions that stabilize 3(10)-helices can be inferred both from the strong amino acid preferences found for these short helices, as well as from the existence of substructures in which tertiary interactions replace consensus local interactions. Because the folding and unfolding of alpha-helices have been postulated to proceed through reverse-turn and 3(10)-helix intermediates, sequence differences between 3(10)- and alpha-helices can also lend insight into factors influencing alpha-helix initiation and propagation.     

Cooperative effects in hydrogen-bonding of protein secondary structure elements: a systematic analysis of crystal data using Secbase

A systematic analysis of the hydrogen-bonding geometry in helices and beta sheets has been performed. The distances and angles between the backbone carbonyl O and amide N atoms were correlated considering more than 1500 protein chains in crystal structures determined to a resolution better than 1.5 A. They reveal statistically significant trends in the H-bond geometry across the different secondary structural elements. The analysis has been performed using Secbase, a modular extension of Relibase (Receptor Ligand Database) which integrates information about secondary structural elements assigned to individual protein structures with the various search facilities implemented into Relibase. A comparison of the mean hydrogen-bond distances in alpha helices and 3(10) helices of increasing length shows opposing trends. Whereas in alpha helices the mean H-bond distance shrinks with increasing helix length and turn number, the corresponding mean dimension in 3(10) helices expands in a comparable series. Comparing similarly the hydrogen-bond lengths in beta sheets there is no difference to be found between the mean H-bond length in antiparallel and parallel beta sheets along the strand direction. In contrast, an interesting systematic trend appears to be given for the hydrogen bonds perpendicular to the strands bridging across an extended sheet. With increasing number of accumulated strands, which results in a growing number of back-to-back piling hydrogen bonds across the strands, a slight decrease of the mean H-bond distance is apparent in parallel beta sheets whereas such trends are obviously not given in antiparallel beta sheets. This observation suggests that cooperative effects mutually polarizing spatially well-aligned hydrogen bonds are present either in alpha helices and parallel beta sheets whereas such influences seem to be lacking in 3(10) helices and antiparallel beta sheets.     

Helix packing of leucine-rich peptides: a parallel leucine ladder in the structure of Boc-Aib-Leu-Aib-Aib-Leu-Leu-Leu-Aib-Leu-Aib-OMe.

The packing of peptide helices in crystals of the leucine-rich decapeptide Boc-Aib-Leu-Aib-Aib-Leu-Leu-Leu-Aib-Leu-Aib-OMe provides an example of ladder-like leucylleucyl interactions between neighboring molecules. The peptide molecule forms a helix with five 5----1 hydrogen bonds and two 4----1 hydrogen bonds near the C terminus. Three head-to-tail NH ... O = C hydrogen bonds between helices form continuous columns of helices in the crystal. The helicial columns associate in an antiparallel fashion, except for the association of Leu ... Leu side chains, which occurs along the diagonal of the cell where the peptide helices are parallel. The peptide, with formula C56H102N10O13, crystallizes in space group P2(1)2(1)2(1) with Z = 4 and cell parameters a = 16.774(3) A, b = 20.032(3) A and c = 20.117(3) A; overall agreement factor R = 10.7% for 2014 data with magnitude of F(obs) greater than 3 sigma (F); resolution 1.0 A.     

Intramolecular steric effects and hydrogen bonding in regular conformations of polyamino acids

A survey has been made, by using computer methods, of the types of helices which polypeptide chains can form, taking into account steric requirements and intramolecular hydrogen-bonding interactions. The influence on these two requirements, of small variations in the bond angles of the peptide residues, or of small changes in the overall dimensions of the helix (pitch and residues per turn), have been assessed for the special case of the α-helix. Criteria for the formation of acceptable hydrogen bonds have also been applied to helices of other types, viz., the 3, γ^?, ω^?, and π-helices. It was shown that the N? H … O and H … O? C angles in hydrogen bonds are sensitive to changes in either the NC^αC′ bond angle or in the rotational angles about the N? C^α and C^α? C′ bonds. However, the variants of the α-helix observed experimentally in myoglobin can all be constructed without distortion of the hydrogen bonds. For α-helices, the steric and hydrogen bonding requirements are more easily fulfilled with an NC^αC′ bond angle of 111°, rather than 109.5°. The decreased stability observed for the left-handed α-helix relative to the right-handed one for L -amino acids is due essentially only to interactions of the C^β atom of the side chains with atoms in adjacent peptide units in the backbone, and interactions with atoms in adjacent turns of the helical backbone are not significantly different in the two helices. Restrictions in the freedom of rotation of bulky side chains may have significant kinetic effects during the formation of the α-helix from the “random coil” state.     

Helix stability of oligoglycine,oligoalanine, and oligo‐β‐alanine dodecamers reflected by hydrogen‐bond persistence

Helices are important structural/recognition elements in proteins and peptides. Stability and conformational differences between helices composed of α‐ and β‐amino acids as scaffolds for mimicry of helix recognition has become a theme in medicinal chemistry. Furthermore, helices formed by β‐amino acids are experimentally more stable than those formed by α‐amino acids. This is paradoxical because the larger sizes of the hydrogen‐bonding rings required by the extra methylene groups should lead to entropic destabilization. In this study, molecular dynamics simulations using the second‐generation force field, AMOEBA (Ponder, J.W., et al., Current status of the AMOEBA polarizable force field. J Phys Chem B, 2010. 114 (8): p. 2549–64.) explored the stability and hydrogen‐bonding patterns of capped oligo‐β‐alanine, oligoalanine, and oligoglycine dodecamers in water. The MD simulations showed that oligo‐β‐alanine has strong acceptor+2 hydrogen bonds, but surprisingly did not contain a large content of 3₁₂‐helical structures, possibly due to the sparse distribution of the 3₁₂‐helical structure and other structures with acceptor+2 hydrogen bonds. On the other hand, despite its backbone flexibility, the β‐alanine dodecamer had more stable and persistent <3.0 Å hydrogen bonds. Its structure was dominated more by multicentered hydrogen bonds than either oligoglycine or oligoalanine helices. The 3₁ (PII) helical structure, prevalent in oligoglycine and oligoalanine, does not appear to be stable in oligo‐β‐alanine indicating its competition with other structures (stacking structure as indicated by MD analyses). These differences are among the factors that shape helical structural preferences and the relative stabilities of these three oligopeptides. Proteins 2014; 82:3043–3061. © 2014 Wiley Periodicals, Inc.     

High-Resolution Crystal Structures of Protein Helices Reconciled with Three-Centered Hydrogen Bonds and Multipole Electrostatics

Theoretical and experimental evidence for non-linear hydrogen bonds in protein helices is ubiquitous. In particular, amide three-centered hydrogen bonds are common features of helices in high-resolution crystal structures of proteins. These high-resolution structures (1.0 to 1.5 Å nominal crystallographic resolution) position backbone atoms without significant bias from modeling constraints and identify Φ = -62°, ψ = -43 as the consensus backbone torsional angles of protein helices. These torsional angles preserve the atomic positions of α-β carbons of the classic Pauling α-helix while allowing the amide carbonyls to form bifurcated hydrogen bonds as first suggested by Némethy et al. in 1967. Molecular dynamics simulations of a capped 12-residue oligoalanine in water with AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications), a second-generation force field that includes multipole electrostatics and polarizability, reproduces the experimentally observed high-resolution helical conformation and correctly reorients the amide-bond carbonyls into bifurcated hydrogen bonds. This simple modification of backbone torsional angles reconciles experimental and theoretical views to provide a unified view of amide three-centered hydrogen bonds as crucial components of protein helices. The reason why they have been overlooked by structural biologists depends on the small crankshaft-like changes in orientation of the amide bond that allows maintenance of the overall helical parameters (helix pitch (p) and residues per turn (n)). The Pauling 3.6₁₃ α-helix fits the high-resolution experimental data with the minor exception of the amide-carbonyl electron density, but the previously associated backbone torsional angles (Φ, Ψ) needed slight modification to be reconciled with three-atom centered H-bonds and multipole electrostatics. Thus, a new standard helix, the 3.6_13/10-, Némethy- or N-helix, is proposed. Due to the use of constraints from monopole force fields and assumed secondary structures used in low-resolution refinement of electron density of proteins, such structures in the PDB often show linear hydrogen bonding.