Theoretical Studies on the Origin of β-sheet Twisting - The Journal of

Right-handed twisting is a fundamental structural feature of β-pleated sheets in globular proteins which is critical for their geometry and function...
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J. Phys. Chem. B 2000, 104, 11296-11307

Theoretical Studies on the Origin of β-sheet Twisting Igor L. Shamovsky,*,† Gregory M. Ross,‡ and Richard J. Riopelle† Departments of Medicine and Physiology, Queen’s UniVersity, Kingston, Ontario, Canada K7L 2V7 ReceiVed: July 20, 2000; In Final Form: September 13, 2000

Right-handed twisting is a fundamental structural feature of β-pleated sheets in globular proteins which is critical for their geometry and function. The origin of this twisting is poorly understood and has represented a challenge for theoretical chemistry for almost 30 years. Density functional theory using the B3LYP exchangecorrelation functional and the split-valence 6-31G** basis set has been utilized to investigate the structure and conformational transitions of single and double-stranded antiparallel β-sheet models to determine the driving force for the right-handed twisting. Right-handed twisting is found to be an intrinsic property of a peptide main chain because of the difference in rotational potentials around N(sp2)-CR(sp3) and C(sp2)CR(sp3) bonds. The difference arises from a tendency of the single CR(sp3)-C(sp2) bonds to eclipse the lone pair of atoms N(sp2), which results in decreasing absolute values of dihedral angles φ but not ψ. This tendency is suppressed by hydrogen bonding between adjacent CO and NH groups within single β-strands, and released only when these bonds are disrupted by the interstrand CO‚‚‚HN hydrogen bonding. The results obtained constitute the following paradigm of the origin of β-sheet twist: although right-handed twisting of β-sheets in globular proteins is an inherent property of the peptide backbone within single β-strands, it is unleashed by the interstrand hydrogen bonding in multistranded β-sheets. The observed pleating, right-handed twisting, skewed mutual orientation of β-strands, and intrinsic conformational variability of double-stranded antiparallel β-sheet motifs in globular proteins are explained from the first principles.

Introduction Two types of regular secondary structural elements of peptides, R-helices and β-sheets, dominate in globular proteins. The spacial arrangement of these major elements represents a basis for structural classification of proteins.1-3 The R-helix is a particularly stable and rigid arrangement of the polypeptide chain; accordingly, its structural features are well described and understood.4-6 On the contrary, β-pleated sheets exhibit enormous diversity of structural forms.6-9 Their conformations in native globular proteins deviate considerably from a purely extended, classical all-trans form proposed by Pauling and Corey.10 The main types of the deviation have been classified as “twisting”, “bulging” and “bending”.11 Spectacular conformational variability of β-sheet motifs in globular proteins is thought to be important for folding, stability and function,9,12 but the nature of this variability is unknown. The β-pleated sheets in globular proteins invariably exhibit a pronounced right-handed twist, when viewed along the polypeptide strands.3,6,7 Twisted β-pleated sheets are extremely important architectural blocks of globular proteins. The β-pleated sheets consist of two or more extended polypeptide chains, β-strands, which are always twisted in the same direction in globular proteins; therefore the twist of the whole sheet and the twist of individual strands are thought to be closely related.7,9 Figure 1 illustrates geometry of an idealized β-strand. The Cβ atoms of the odd residues of the strand significantly deviate from collinearity and tend to form a spiral twisted in the righthanded direction (θ > 0). Although enormous efforts have been * Corresponding Author. Tel: (613) 549-6666, ext 4433. Fax: (613) 548-1369. E-mail: [email protected]. † Department of Medicine. ‡ Department of Physiology.

Figure 1. Characteristic geometry of β-strand made of L-amino acids. Broken line connects the β-carbon atoms of odd residues and demonstrates the right-handed twist of the strand. Nonpolar hydrogens are not illustrated for clarity. Carbon, nitrogen, oxygen and polar hydrogen atoms are shown in green, blue, red and white, respectively. Major conformational degrees of freedom (φ and ψ) within each peptide unit, and the dihedral angle θ (defined by the solid lines) determining the direction and magnitude of the twist are illustrated. The “β-strand” conformational region of L-amino acids belongs to the second quadrant of the Ramachandran plot (φ < 0, ψ > 0), the right-handed twist of the strand occurs when θ > 0.

made to reveal the origin for this right-handed twisting, a convincing explanation is lacking. Chothia was the first investigator to propose that the right-handed twist is caused by entropic factors, because he found that the majority of the allowed conformations of alanine dipeptide exhibited righthanded twist.7 Salemme suggested that the twist is a consequence of the geometric constraint imposed by interstrand H-bonding.13 Weatherford and Salemme proposed that the effect is associated with out-of-plane deformation of the peptide group.9,14 Chou and coauthors attributed the preference for the right-handed twist to the intra- and interchain nonbonded sidechain interactions.15 Specifically, they suggested that interactions of the bulky branched side chains constitute the main reason for the large right-handed twisting. Yang and Honig concluded that tertiary interactions with the rest of the protein is the

10.1021/jp002590t CCC: $19.00 © 2000 American Chemical Society Published on Web 10/26/2000

Origin of β-sheet Twistiing dominant determinant of the sheet twist.6 A completely different concept has been proposed by Maccallum and coauthors who suggested that the twist is caused by intrastrand electrostatic interactions between carbonyl groups of the successive amino acid residues.16 Wang and coauthors disagreed with all of the previously suggested concepts and attributed the driving force for the right-handed twist to interactions between strands, although the type of interaction was not specified.17 Previous concepts on the origin of the right-handed twist do not provide a simple and clear explanation why the right-handed twist is always more favorable than the left-handed twist. The essential drawback of previous theoretical studies of β-sheets is the empirical nature of the utilized force fields. These studies were based on the implicit assumption of accuracy of the empirical force field terms, which is not necessarily the case, particularly with regard to understanding the origin of a poorly understood phenomenon. Taking into account the effects of hydration and entropy6,17 does not provide a significant advantage if the basic molecular mechanical force field is incorrect. In this paper we investigated geometrical and energetic features of elementary units of single β-strands and doublestranded antiparallel β-sheets by fully optimized density functional theory calculations at the B3LYP/6-31G** level in the second quadrant of the Ramachandran plot, and determined whether the theoretically predicted trends could be identified within the X-ray conformations of double-stranded antiparallel β-sheets. The magnitude of the difference (ψ - |φ|) within peptide units was considered as a measure of the twist of the peptide chain in the right-handed direction. The results clearly demonstrate that the origin of the right-handed twist inherent in the β-sheet motifs in globular proteins lies within conformational preferences of the peptide backbone itself, and has less to do with side-chain or interchain interactions. The righthanded twist is caused by a tendency of the lone pair of the sp2-hybridized nitrogen atom of the amide group to eclipse the single CR(sp3)-C(sp2) bond, which directly causes absolute values of φ to be less than ψ. This property of the rotational potential of φ is superseded and shadowed by intrastrand H-bonds within single β-strands, and is revealed only in β-sheets where those bonds are disrupted by interstrand H-bonding. The observed right-handed twist of β-sheet motifs in globular proteins is caused by the intrinsic property of the main chains of individual β-strands to twist in the right-handed direction. Interstrand H-bonding unleashes this property, and causes extraordinary conformational flexibility and specifically skewed mutual orientation of individual β-strands within β-sheets.

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Figure 2. Correlation of the magnitude of the twist (θ) of β-strand and the difference (ψ - |φ|) within the second quadrant of the Ramachandran plot. Error bars signify (3σ. The right-handed twist (θ > 0) occurs when ψ > |φ|.

Methods Ab Initio Calculations. Structure and conformational flexibility of six model molecular systems I-VI (Figure 3) mimicking β-strands were studied by fully optimized ab initio calculations using the split-valence 6-31G** basis set.18 Electron correlation was taken into account by the density functional theory (DFT) method19,20 using Becke’s three-parameter exchange functional with the Lee-Yang-Parr’s correlation functional (B3LYP).21,22 This particular hybrid GGA functional is most widely used and has been demonstrated to yield remarkable accuracy in structure, dipole moments, thermochemistry and vibrational frequencies in various molecular systems with hydrogen bonds.23-30 Conventional solid ab initio methods of correlation energy correction, namely second-order MøllerPlesset perturbation theory (MP2)31 and quadratic configuration interaction including single, double and triple excitations

Figure 3. Structure of molecular systems under study. Molecules I (L-alanine dipeptide) and II (glycine dipeptide) simulate single β-strand units. Their H-bonded dimers III-VI simulate double-stranded antiparallel β-sheet motifs. Dimers III and IV mimic small H-bonded rings; V and VI, large H-bonded rings, both rings being characteristic for antiparallel β-sheet motifs in proteins. The broken lines indicate intermolecular H-bonds.

(QCISD(T)),32 were used to support the key predictions made in the present study based on the DFT calculations. Calculations were performed with the Gaussian-98 program33 on the IBM SP2 high-performance computer available through the High Performance Computing Virtual Laboratory (HPCVL) at Queen’s University at Kingston. Molecular Systems. Models I and II represent single β-strand units of L-Ala and Gly, respectively, whereas models III-VI are double-stranded antiparallel β-sheet units built from those

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Figure 4. Basic unit of β-strand. Symbols φ and ψ designate torsional angles C-N-CR-C and N-CR-C-N within peptide backbone, respectively; d1 and d2 denote two alternative intrastrand H-bonds. The C5 conformer exhibits H-bonding d1; the C7eq conformer, d2.34,35

single β-strand models. The molecular units I (known as L-alanine dipeptide) and II (glycine dipeptide) are somewhat smaller than the conventional N-acetyl-N′-methylamide derivatives of single amino acids which have been extensively studied by theoretical methods,28,29,34-36 since the terminal methyl groups have been demonstrated to play an insignificant role in the structure and energetics of peptide units in gas phase.34,35 The C2 point group symmetry was assumed for the doublestranded models. Two basic types of interstrand H-bonding within β-pleated sheets were utilized in the double-stranded models; III and IV correspond to the so-called “small” hydrogen-bonded ring, V and VI to the “large” hydrogen-bonded ring.9 Alternation of these two basic H-bond patterns (H-bonded rings) is an inherent feature of antiparallel β-sheet motifs in proteins.9 Two possible types of intrastrand H-bonds29,34,35 within β-sheets (d1 and d2) are illustrated in Figure 4. The strategy for ab initio geometry optimization was focused on determining fully relaxed paths for conformational alterations within the model molecules in addition to defining the actual equilibrium structures. Accordingly, the length of one of the intrastrand H-bonds (d1 for I-IV or d2 for V-VI) was fixed to discrete values ranging from 2.2 to 4.5 Å, while the rest of the geometric variables were optimized. This strategy led to conformational alterations within the model molecules which require minimal energy, such that they were directed along the minimum-energy valley on the energy hypersurface. Trans configuration of the amide groups was assumed in all of the systems analyzed. Measurement of the Twist. Conformations of amino acids constituting β-pleated sheets in native globular proteins are located within the second quadrant of the Ramachandran plot (-180° < φ < 0°, 0° < ψ < 180°). Direction and magnitude of the local twist of the strand are explicitly determined by the value of the dihedral angle Cβ(i)-CR(i)-CR(i + 2)-Cβ(i + 2) (Figure 1), such that the left- and right-handed twists are associated with the negative (θ < 0) and positive (θ > 0) values of this angle, respectively. On the other hand, the local twist of the strand is obviously determined by the major conformational degrees of freedom of the strand, namely torsional angles φ(CN-CR-C) and ψ(N-CR-C-N). Figure 2 demonstrates a strong correlation between the twist (θ) and the difference (ψ - |φ|). This graph was obtained by systematic twodimensional scanning of the second quadrant of the Ramachandran plot with the 10° grid spacing using the poly-Ala strand. All residues of the strand were maintained in the identical (φ,ψ)conformations and possessed the natural (L) configuration and the planar trans-peptide groups with standard dimensions.37 Because of the strong correlation between magnitudes of θ and (ψ - |φ|) within the second quadrant of the Ramachandran plot, we will use the difference (ψ - |φ|) as a measure of the twist of β-strands in the right-handed direction. Specifically, the righthanded twist (θ > 0) is observed within the second quadrant of the Ramachandran plot only when (ψ - |φ|) > 0. Therefore,

Figure 5. Fully relaxed conformational transitions within single β-strand units in models I (b) and II (2) as predicted by DFT calculations at the B3LYP/6-31G** level. Two minima (C5 and C7eq) correspond to two alternative intrastrand H-bond patterns.

the question “why θ > 0 in β-strands” reduces to the question “why ψ > |φ|”. Validation from Experimental Data. Structural features of the double-stranded β-sheets obtained by DFT calculations were validated using the set of highly refined X-ray crystallographic structures of regular double-stranded antiparallel β-sheets in nonhomologous proteins deposited in Protein Data Bank and listed in ref 6. Three double-stranded β-sheets from nerve growth factor38 were added to this set. All of the native β-sheets within the set were twisted in the right-handed direction.6 The set initially contained a total of 540 amino acid residues comprising 43 double-stranded antiparallel β-sheets in 23 globular proteins. Proline (9 residues) was excluded from the pool. The first and the last amino acids from each strand were also excluded from the set to avoid boundary effects. Residues exhibiting conformations outside the second quadrant of the Ramachandran plot were also excluded. Positions of hydrogen atoms within amide groups were calculated assuming planar configuration of nitrogen atoms, and using average bond length N-H of 1.02 Å and bond angle CR-N-H of 115.1° (obtained from present DFT calculations). Results and Discussion Ab Initio Calculations. Fully optimized DFT calculations of the molecules I and II at the B3LYP/6-31G** level revealed that they each have two low-energy conformers within the second quadrant of the Ramachandran plot, C5 and C7eq, consistent with experimental studies of N-acetyl-N′-methylamides of L-alanine and glycine in nonpolar solvents by NMR and IR spectroscopy39 and previous theoretical considerations.28,29,34-36,40 Figure 5 illustrates the minimum-energy conformational transition within molecules I and II upon the increase of the H-bond distance d1. The two energy minima of the single β-strand units correspond to two H-bonded conformations. The first, all-trans conformation with the typical H-bond length of d1 ≈ 2.15 Å, formally belongs to a β-sheet conformational region41 and is referred to as conformation C5. The second conformation, known as C7eq, is more stable and corresponds to ψ and φ of approximately -80° and +70°,

Origin of β-sheet Twistiing

Figure 6. Fully relaxed conformational paths in models I (A) and II (B) from C5 to C7eq as predicted by DFT calculations at the B3LYP/ 6-31G** level. (b) denotes torsional angle φ; (O), ψ.

respectively. This conformation formally belongs to the 2.27 ribbon region and possesses an alternative H-bond (d2) of about 2.05 Å long.42 The saddle-point structure (C7eq/C5) separating conformations C7eq and C5 in the models I and II is above the deeper minimum by 10.7 and 11.8 kJ/mol, respectively. In the saddle-point structure, both H-bonds are broken: d1 ) 2.7, d2 ) 3.4 Å in I, and d1 ) 2.8, d2 ) 4.1 Å in II. Predicted geometrical features and relative energies of the equilibrium conformations C5 and C7eq, and the C7eq/C5 saddle-point structure agree with previous ab initio calculations.28,29,34,35 Specifically, present DFT computations at the B3LYP/6-31G** level predict the difference (ψ - |φ|) in conformations C5, C7eq and C7eq/C5 of molecule I to be +7.9°, -8.7° and +18.1°, respectively, and 0.0°, -13.9° and +41.8°, respectively in molecule II. Classical ab initio studies by Head-Gordon et al.35 at the HF/6-31+G* level predicted similar magnitudes of the difference, +4.6°, -7.7° and +32.7° for I, and 0.0°, -17.8° and +41.4° for II, respectively. As seen, the saddle-point structures of both I and II are predicted to exhibit spectacular magnitudes of the right-handed twist. The conformational transition within molecules I and II is associated with switching between the two alternative intrastrand H-bonding patterns (Figure 4). Intramolecular H-bonding in both C7eq and C5 conformations is weak and easily disrupted even in the gas phase at room temperature, as has been recently demonstrated by DFT molecular dynamics calculations.40 Figure 6 gives the optimized angles φ and ψ as one-dimensional functions of the H-bond length d1 in models I and II. As seen,

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Figure 7. Fully relaxed conformational profiles of the double-stranded β-sheet units along the d1 coordinate as predicted by DFT calculations at the B3LYP/6-31G** level. Models III (b) and IV (2) in panel A designating the small H-bonded ring do not exhibit the first equilibrium conformation (C5) at d1 ) 2.1 Å. Models V (O) and VI (4) in panel B designating the large H-bonded ring do not exhibit the second equilibrium conformation (C7eq) at d1 ) 3.7 Å.

|φ| is significantly lower than ψ along the minimum-energy path except for the region of very high d1 values (more than 3.3 Å) and a discrete region around d1 ) 2.43 Å for I where |φ| ≈ ψ ≈ 150°. The average difference (ψ - |φ|) obtained from the reference conformations along the d1 coordinate is 7.1° and 18.1° for models I and II, respectively. The positive magnitude of this difference leads to the right-handed twist of the peptide chain along the minimum-energy conformational path and indicates that the origin for this twisting (i) lies within a single amino acid unit and (ii) is not associated with the amino acid side chain, contrary to current opinion. We suggest that the cause for the absolute value of φ to be less than ψ is determined by the difference in corresponding rotational potentials, which directly leads to the different behavior of φ and ψ along the conformational transition path. Optimal conformational path of φ has a striking plateau around -80° at d1 > 2.8 Å, which is not characteristic for ψ. Within each of the transition paths (Figure 6), there is an area in which small changes in H-bond length d1 results in dramatic conformational alterations in the molecules. This area of notably high flexibility is located at d1 approximately equal to 2.7 and 2.8 Å for molecules I and II, respectively, and associated with the saddle-point structures. At this particular point, conformation of β-strand can significantly alter without changes in energy.

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Figure 8. Fully relaxed conformational paths in models III (A) and IV (B) as predicted by DFT calculations at the B3LYP/6-31G** level. Symbol (b) denotes torsional angle φ; (O), ψ.

Figure 9. Fully relaxed conformational paths in models V (A) and VI (B) as predicted by DFT calculations at the B3LYP/6-31G** level. Symbol (b) denotes torsional angle φ; (O), ψ.

Unlike the single β-strand models I and II, double-stranded models III-VI have the only equilibrium conformation. The fully relaxed energy profiles of these models along the d1 coordinate are presented in Figure 7. In models III and IV (Figure 7A), the equilibrium conformation (C7eq) possesses the d2 H-bond, whereas models V and VI exhibit only conformation C5 with the d1 H-bond (Figure 7B). The fully relaxed conformational paths along the d1 coordinate are illustrated in Figures 8 (for III and IV) and 9 (for V and VI). The conformational paths for dimers III and IV are very similar to the corresponding paths for monomers I and II, respectively (compare Figures 6A with 8A, and 6B with 8B). However, there are two marked effects of the interstrand H-bonding on the conformational paths: (i) dramatic conformational alterations within III and IV take place at the same magnitude of d1 ≈ 2.7 Å; (ii) these alterations are more abrupt within the double-stranded models. Thus, the area of especially high conformational flexibility exists also within the double-stranded models III and IV, even though there is no saddle-point structure in this area. Conformational alterations within the alternatively H-bonded models V and VI are more smooth (Figure 9). In all of the double-stranded models, absolute value of φ is significantly less than ψ along the minimum-energy conformational paths except for the boundary region where d1 is longer than 3.3 Å. This particular property, which unequivocally determines the right-handed twist of the double-stranded models regardless of the magnitude of d1 (for 2.0 < d1 < 3.3 Å), arises from the single-strand systems

because of obvious similarity between the minimum-energy paths of double-stranded and single-strand models. Figure 10 illustrates the equilibrium conformations of the double-stranded model systems III-VI. As seen, the d1 H-bond is broken in III and IV, while d2 is broken in V and VI; and instead of these intrastrand H-bonds, corresponding groups are involved in interstrand H-bonding. Consequently, intermolecular H-bonding between β-strands, which includes two close CO‚‚‚ HN H-bonds, destroys intramolecular H-bonding between participating functional groups. Similar disruption of intramolecular H-bonding in alanine dipeptide by H-bonding to water molecules has been demonstrated by various theoretical approaches29,36 and is consistent with NMR studies in water.36,43 The reason for this disruption is the electrostatic repulsion O‚‚‚O and H‚‚‚H within H-bonded rings. Interaction between the H-bonded hydrogen atoms within small H-bonded rings of double-stranded antiparallel β-sheet motifs has been recently detected by H1 NMR studies.44 The repulsion of close intermolecular H-bonds not only destroys the intramolecular Hbonding but also leads to a distinct skewed geometry of the small H-bonded rings. Since the O‚‚‚O repulsion is stronger than H‚‚‚H, ratio between optimal distances O‚‚‚O and H‚‚‚H within small H-bonded rings is significantly higher than 1.0 (Figure 10A,B). This ratio slightly varies along the path between 1.20 and 1.41 for III, and between 1.18 and 1.49 for IV. The area of high conformational flexibility at d1 ) 2.7 Å within III and IV corresponds to the ratios of 1.24 and 1.21, respectively. Similar skewed geometry of dimers possessing two adjacent

Origin of β-sheet Twistiing

Figure 10. Structures of the double-stranded antiparallel β-sheet models III (A), IV (B), V (C) and VI (D) obtained by fully optimized DFT calculations at the B3LYP/6-31G** level. Carbon, nitrogen, oxygen and hydrogen atoms are shown in green, blue, red and white, respectively. The broken lines indicate H-bonds.

H-bonds, in which the O‚‚‚O distance is significantly longer than H‚‚‚H, has been found in other systems.26 The magnitude of the ratio O‚‚‚O/H‚‚‚H is lower than 1.0 in the large H-bonded ring models V and VI and varies from 0.74 to 0.91 along the minimum-energy path for not too small values of d2. At d1 ) 2.7 Å, the ratio is approximately 0.75. Distances O‚‚‚O and H‚‚‚H are normally longer in large rather than in small H-bonded rings, therefore the influence of the electrostatic repulsion between these two pairs of atoms on the preferred orientation of the strands in the large H-bonded ring models is

J. Phys. Chem. B, Vol. 104, No. 47, 2000 11301 not critical. The driving force for the skewed geometry of the large H-bonded rings is a weak intermolecular H-bonding CO‚‚‚HCR (Figure 10C,D), which tends to make the O‚‚‚O/ H‚‚‚H ratio below 1.0. This particular H-bonding is ubiquitous in β-pleated sheets.45 An alternative orientation between strands within large H-bonded ring models with the ratios higher than 1.0 would result in an electrostatically unfavorable contact NH‚‚‚HCR instead of favorable CO‚‚‚HCR. Factors Determining Structure of β-Sheets. The obtained ab initio results allow us to suggest the following model describing structural determinants of β-sheets in globular proteins (Figure 11). The structure of a hypothetical single β-strand is primarily determined by weak intrastrand H-bonding d1 within each amino acid unit (Figure 11A). The resulting equilibrium structure is similar to the classical all-trans conformation of the peptide chain described by Pauling and Corey,10 with little right-handed twisting and very minor pleat (according to the present and published35 results, a single poly-Gly β-strand is not expected to exhibit any twist or pleat, since φ ) ψ ) 180°). However, the B3LYP/6-31G** transition energy curves (Figure 5) indicate that this structure is unstable and readily undergoes a conformational transition at room temperature to the left-handed 2.27 ribbon.40 The latter conformation is also unstable;42 accordingly, its existence has never been confirmed. Thus, the fully extended conformation of a single β-strand would be even less stable than the 2.27 ribbon. In the double-stranded or multistranded β-sheets, strong interstrand H-bonding disrupts the intrastrand H-bonds within small and large H-bonded rings because of electrostatic repulsion between adjacent H-bonded CO‚‚‚HN pairs (Figure 11A,B). This particular repulsion tends to increase the d1 distance within small H-bonded rings (Figure 11A) and form the alternative H-bond d2. Likewise, intrastrand H-bonds d2 within large H-bonded rings are disrupted by interstrand H-bonding (Figure 11B), which in turn leads to H-bond d1. As a result of these opposite tendencies, both d1 and d2 are to deviate from their optimal length of 2 Å,

Figure 11. The interplay between intra- and interstrand H-bonding within antiparallel β-sheets releases pleating and right-handed twisting hidden in individual β-strands, and causes the skewed orientation of adjacent β-strands and extraordinary conformational variability. (A) the first type of intrastrand H-bonding (d1) is disrupted by the electrostatic repulsions O‚‚‚O and H‚‚‚H between adjacent H-bonded functional groups within small H-bonded rings, which tend to form H-bonds d2. (B) the second type of intrastrand H-bonding (d2) is disrupted by similar electrostatic repulsions within large H-bonded rings which tend to form H-bonds d1. Putative H-bonds CO‚‚‚HCR are not shown for simplicity. (C) the result of reciprocal cancellation of the two opposite tendencies (A and B): neither alternative intrastrand H-bonding exists, and individual β-strands are shifted toward amino ends. The broken lines indicate H-bonds.

11302 J. Phys. Chem. B, Vol. 104, No. 47, 2000 and therefore, H-bond length d1 in β-sheets is expected to exhibit intermediate magnitudes around 2.7 Å (Figure 11C). The latter conformational area is characterized by the following three features (Figures 6, 8, and 9). First, β-strands are significantly twisted in right-handed direction because optimal |φ| is much lower than ψ (Figure 8).35 Second, β-strands are remarkably pleated, since optimal dihedrals φ and ψ are significantly deviated from 180° (Figure 8).35 Third, β-strands are very much conformationally variable as this particular area is characterized by enormous conformational flexibility. Therefore, conformational variability of β-pleated sheets arises from reciprocal cancellation of intrastrand H-bonding effects, such that conformational preferences within β-strands are governed by the remaining weaker intrastrand interactions. In addition to these conformational features, we predict that strands within β-pleated sheets are to be markedly shifted from each other toward amino ends (Figure 11C). This shift arises from the skewed equilibrium geometries of the small and large H-bonded rings found within the double-stranded models under study. Indeed, this particular shift should cause alternations of the O‚‚‚O/H‚‚‚H ratio along the stranded peptide chains (above 1.0 and below 1.0 for small and large H-bonded rings, respectively), consistent with the optimal geometries of the H-bonded rings, which are, therefore, preorganized for their incorporation into a β-sheet motif. Validation from Experimental Data. Distribution of distance d1 in the set of experimentally observed structures of double-stranded antiparallel β-sheets is presented in Figure 12A. The mean value of d1 is 2.626 ( 0.017 Å. As seen, the intermediate area where both intrastrand H-bonds d1 and d2 are disrupted is most populated in the observed conformational pool. An asymmetrical shape of this distribution suggests that it is likely to represent a mixture of two Gaussian-like populations. Figure 12B illustrates the best fit of the observed distribution by two Gauss functions centered at d1 ≈ 2.44 and 2.84 Å. The hypothetical energy profile of β-sheets which would produce this particular distribution at room-temperature assuming Boltzmann’s probability functional is presented in Figure 12B. The selected pool of experimentally observed conformations (Figure 12A) supports the present concept for the structural determinants of β-sheets which arises solely from ab initio calculations (Figure 11). The effect of interstrand H-bonding on the conformational equilibrium within β-strands is to move the two energy minima inherent in single β-strands (Figure 5) nearer to each other along the d1 coordinate and locate them close to the intermediate point of high conformational flexibility with d1 ) 2.7 Å. The consequence of this effect is an asymmetrical shape of the observed distribution of distances d1, which is still affected by the two-minima potential energy function. This indicates that traces of both intrastrand H-bonding still exist within double-stranded β-sheets (Figure 12B), which is consistent with gradual declining of electrostatic contributions to H-bonds.46 This being the case, the following two factors are responsible for conformational flexibility of β-sheet motifs. First, the existence of two close energy minima along the d1 coordinate results in a shallow energy surface. Second, the location of these minima close to the point of high intrinsic flexibility of individual β-strands makes β-sheets particularly variable. Remarkably, traces of the energy barrier at d1 ) 2.7 Å characteristic for the single β-strands are seen in the predicted conformational energy profile for β-sheets (Figure 12B). From this particular point, both decrease and increase of d1 are somewhat supported by intrastrand H-bonding. Consequently, this is the intimate balance between intra- and interstrand

Shamovsky et. al.

Figure 12. Distribution of H-bond length d1 in the preselected set of X-ray crystallographic structures of regular double-stranded antiparallel β-sheets extracted from Protein Data Bank. (A) the histogram illustrates relative population of the indicated interval of the H-bond length d1 within the set. (B) our interpretation of the observed distribution. The total population is approximated by two Gauss functions (dotted lines). Conformational energy of β-sheets in proteins (bolded line) is estimated from the sum of the two best-fitted Gauss-like populations (thin line) using Boltzmann’s conformational probability distribution at room temperature.

H-bonding that causes striking conformational variability of β-sheets in proteins. Figure 13 illustrates the observed dependence of average dihedral angles φ and ψ on the distance d1 derived from the selected conformational pool. As seen, both angles decrease with the increase of d1 in the interval from 2.0 to 3.0 Å, in agreement with ab initio predictions. Interval of d1 between 3.0 and 3.2 Å where angle ψ increases is marginally populated (Figure 12A). The average magnitude of |φ| is smaller than that of ψ; the mean average difference (ψ - |φ|) for the whole significantly populated interval 2.0 < d1 < 3.0 Å is 12.9 ( 1.1°. The average differences (ψ - |φ|) obtained by ab initio calculations for the reference minimum-energy conformations of the model systems corresponding to the same interval of d1 are 10.6°, 10.9° and 10.8° for alanine-based models I, III and V, respectively; and 26.1°,18.1° and 17.7° for glycine-based models II, IV and VI, respectively. Since glycine residues are only slightly populated in the selected conformational pool (3 residues), the theoretically obtained average difference from the alanine-based models (11°) represents a more appropriate comparison with the experimental data. The striking match of the theoretically predicted and experimentally observed average magnitudes of the right-handed

Origin of β-sheet Twistiing

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Figure 14. Structure of nonpeptidic molecules mimicking specific torsional potentials of models I (molecules VII (φ) and VIII (ψ)) and II (molecules IX (φ) and X (ψ)).

Figure 13. Experimentally observed trends of mean magnitudes of φ and ψ in double-stranded antiparallel β-sheets estimated within corresponding intervals of d1 of 0.2 Å wide using the preselected set of β-sheet conformations. Error bars signify standard errors of mean values.

twisting indicates that important factors determining the twisting of double-stranded antiparallel β-sheets in proteins are all taken into account within the model systems under study. A more detailed similarity between the observed and predicted twisting along the d1 coordinate was not expected, as the observed dependence arises from a mixture of two different trends, each being characteristic either for small or large H-bonded ring (compare Figures 8A and 9A). Moreover, the observed conformations of β-sheets are affected by local constraints arising from strong side-chain interactions. Indeed, close examination of the β-sheet residues falling into the interval 3.0 Å < d1 < 3.2 Å within the Protein Data Bank indicates that they are all involved in strong side-chain interactions within proteins (ionic, hydrophobic or disulfide bonding; data not shown). According to the optimized geometries of model III, side chains of the strands move nearer to each other at large values of d1 (Cβ‚‚‚Cβ′ distance becomes shorter than 4.5 Å at d1 > 3.0 Å), which makes this area ideal for attractive sidechain interactions. Therefore, the observed increase of the average value of ψ in this interval (Figure 13) is likely to be an artifact; this interval is only slightly populated by authentic β-sheet conformations but contaminated with conformations distorted by local side-chain interactions. The predicted abrupt conformational alterations at d1 ≈ 2.7 Å cannot be seen in the conformational pool (Figure 13) because conformations at both sides of the transition are averaged in the pool, which leads to intermediate values of φ and ψ in the interval 2.6 Å < d1 < 2.8 Å. The predicted shift between individual β-strands in doublestranded β-sheets is in quantitative agreement with experimental data as well. The ratio O‚‚‚O/H‚‚‚H actually alternates along the peptide chains within antiparallel β-sheets motifs exactly as expected (Figure 11C), and the mean values of the two alternating ratios are 1.245 ( 0.015 for small H-bonded rings and 0.794 ( 0.009 for large H-bonded rings (compare with energy-optimized values of 1.24 and 0.75 at d1 ) 2.7 Å for double-stranded model units III and V, respectively). Origin of β-Sheet Twisting. The demonstrated match of theoretically predicted and experimentally observed structural

features of double-stranded antiparallel β-sheets (such as pleating, right-handed twisting, shift of individual β-strands toward amino ends and spectacular conformational variability) indicates that the size of the models and theoretical level of approximation used are sufficient for an adequate description of the observed effects but does not imply that the origin of the right-handed twisting is understood. According to general considerations (Figure 2) and the obtained ab initio results (Figures 6, 8, and 9), understanding of the origin of the right-handed twisting in β-sheets reduces to identifying of the precise reason for the absolute value of φ to be below ψ for intermediate values of d1 in dipeptide models I and II. Since this relation exists regardless of intra- or intermolecular H-bonding, one would hypothesize that the difference in rotational potentials associated with φ and ψ is directly responsible for |φ| to be below ψ. Rotational potentials are usually much softer than H-bonding interactions and, obviously, are suppressed by the latter. Therefore, the physical reason for this difference was further investigated in similar molecular systems without complicating H-bonding effects by DFT and conventional ab initio calculations. Molecules VII-X (Figure 14) have torsional interactions similar to peptidic units I and II but have no H-bonding. Molecules VII and VIII simulate specific torsional potentials of model I; molecules IX and X, those of model II. Geometries of molecules VII-X were fully optimized at the B3LYP/631G** level of theory, and then one-dimensional conformational profiles were obtained by single-point calculations at the same DFT level by incremental alterations of the specific dihedral angle, φ or ψ, from -180° to 180° (Figure 14). The conformational profiles of φ and ψ are combined in Figure 15 such that angles φ and ψ are of the opposite signs to illustrate the situation in β-strands. Profiles 15A and 15B simulate torsional interactions in the alanine- and glycine-based molecules, respectively. They span the second and the fourth quadrants of the Ramachandran plot. The alanine-based molecules VII and VIII reveal conformational preference toward the second quadrant (Figure 15A), whereas glycine-based molecules IX and X do not (Figure 15B). Basically, this result is consistent with the well-established fact that the fourth quadrant is prohibited for chiral L-amino acids because of repulsion of the side chains from CO and NH groups.35,41 In the second quadrant, the tendency of φ to be around -90° obviously exists in VII and IX, whereas dihedral ψ tends to be equal either to 120° or 180° in VIII and X, respectively. The asymmetry of torsional potentials of φ and ψ clearly causes magnitudes of |φ| to be below |ψ| in peptide units within β-strands. The difference in optimal values of |ψ| and |φ| is very impressive for glycine-based models (90°; Figure 15B), which explains the overall higher right-handed twisting

11304 J. Phys. Chem. B, Vol. 104, No. 47, 2000

Figure 15. Torsional profiles of molecules VII-X obtained by singlepoint DFT calculations at the B3LYP/6-31G** level using incremental (10°) alterations of the specific torsional angle (φ, bolded line; ψ, thin line). (A) rotational potentials in L-alanine-based molecules VII and VIII (full and empty circles, respectively), (B) rotational potentials in glycine-based molecules IX and X (full and empty triangles, respectively). Roman numerals II and IV indicate the corresponding quadrant of the Ramachandran plot. High conformational energy of the alaninebased molecules (A) within the fourth quadrant illustrates the wellknown fact that this region is prohibited for L-amino acids.

in glycine-based than in alanine-based models predicted by conventional ab initio35 and DFT calculations (compare Figures 6B, 8B, and 9B with 6A, 8A, and 9A). The rotational potential of φ in VII has a remarkably flat minimum between -90° and -150° (Figure 15A). The conformations of VII flanking this region are equivalent, since they are mirror images of each other. This represents a limitation of the model VII as these conformations are not equivalent for the alanine-based peptide models I, III and V, where one of the two methyl groups is replaced by the amide group. Accordingly, the DFT studies of the simplified model VII only suggest that there should be a tendency of the torsional potential of φ to be flat between -90° and -150°, but the actual alanine-based peptide units manifest higher stability of the flanking conformation at φ ) -90° (Figures 6A, 8A, and 9A; see below). Intrastrand H-bonding d1 overrides the asymmetry in torsional potentials in single β-strands as it maintains both φ and ψ close to 180°. When intrastrand H-bonds are broken within β-sheets, the intrinsic asymmetry of torsional potentials is unleashed and leads to profound right-handed twisting of peptide backbones. The particular symmetry of the torsional potential defined by ψ is well described and understood. Rotation around the C(sp3)sC(sp2) bond is determined by a general tendency of a

Shamovsky et. al.

Figure 16. Newman projections illustrating the φ and ψ values for the key conformations of the peptide backbone within β-sheet motifs. A and B denote conformations corresponding to dihedral angle ψ; C, D, and E, φ. π-electron density within peptide bond is double-hatched, doubly occupied pz-orbital is single-hatched, σ-electron density of single bonds eclipsing a partially double bond is black. Torsional potentials ψ of L-alanine-based models I and VIII exhibit a minimum at conformation A, whereas those of glycine-based models II and X exhibit a minimum at conformation B. Torsional potentials φ of L-alanine-based models I and VII represent a mixture of three stabilizing interactions (designated in C, D and E), whereas those of glycine-based models II and IX represent a mixture of two interactions (C and D). R designates amino acid side chain.

single bond to eclipse a double bond,47-50 such that in the eclipsed conformation, the Coulomb repulsion between torsional σ- and π-MO is minimized. Since two bonds, CdO and CsN, from both sides of the C(sp2) atom have a partial double-bond character in amide group,10 the existence of either a 120° or 180° minimum is expected (Figure 16A,B, respectively).50 On the other hand, the origin of the energy minimum at -90° of the torsional potential around the C(sp3)-N(sp2) bond (Figure 16C) is not understood, although this tendency has been noted before, e.g. in recent ab initio studies on fragments of 20-(S)camptothecin at the HF/4-31G** level.51 To understand the physical reason causing energy minimum on φ to be located at -90° in peptide units, rotational potentials in 8 simple test molecules H2M-CH2R (M ) N(sp2) or C(sp2); R ) H or CH3; with singly- or doubly occupied pz-orbital on atom M) were studied by fully optimized ab initio calculations at the B3LYP/6-31G**, MP2/6-31G** and MP2/6-311G(2d,p) levels of theory, as well as at the single-point QCISD(T)/6311G(2d,p)//MP2/6-311G(2d,p) level. The sp2-hybridization of the center M was ensured by keeping its planar geometry when performing geometry optimization of the molecules. The latter constraint made torsional profiles of the molecules relevant to rotational potentials in peptides. The particular structure of the test molecules was chosen to avoid π-bonds which would complicate the rotational potentials. The methyl group (when R ) H) is found to exhibit no significant barrier hindering torsional rotation around the sp2hybridized center M in molecules H2N-CH3, [H2N-CH3]+,

Origin of β-sheet Twistiing

J. Phys. Chem. B, Vol. 104, No. 47, 2000 11305 TABLE 1: Ab Initio Relative Energies (kJ/mol) of Two Conformations (E and P) of the Molecules H2M-CH2CH3 and (CH3)2M-CH2CH3 (M ) N or C) Obtained at Different Levels of Theorya Vv N E B3LYP/6-31G(d,p) MP2/6-31G(d,p)b MP2/6-311G(2d,p)b QCISD(T)/6-311G(2d,p)// MP2/6-311G(2d,p) B3LYP/6-31G(d,p) MP2/6-31G(d,p)b QCISD(T)/6-31G(d,p)// MP2/6-31G(d,p)

Figure 17. Unraveling the origin of β-sheet twisting: torsional profiles of molecules H2M-CH2CH3 obtained by fully relaxed DFT calculations maintaining planarity of the trigonal center M at the B3LYP/6-31G** level: (b) H2N-CH2CH3, (9) [H2N-CH2CH3]+, (O) [H2C-CH2CH3]-, (0) H2C-CH2CH3. Molecules with doubly occupied pz-orbital on M(sp2) demonstrate preferred conformations at 90° and -90° where this MO eclipses the single C-C bond. Molecules with the singly occupied pz-orbital on M do not exhibit a noticeable conformational preference.

H2C-CH3 and [H2C-CH3]-, consistent with studies of a similar symmetrical system, toluene.52 Figure 17 presents the fully relaxed B3LYP/6-31G** energy profiles for torsional rotation around the M(sp2)-C(sp3) bonds in the other four molecules. They split into two sets by the type of the rotational potential. In the first set, which includes H2C-CH2CH3 and [H2N-CH2CH3]+ with singly occupied pz-orbital on the trigonal atom, there is no significant conformational preference. In the second set, which consists of H2N-CH2CH3 and [H2C-CH2CH3]-, the energy minimum corresponds to the conformation (E) in which the C-C bond eclipses the pz-orbital of the sp2-hybridized center (φ ) 90° or -90°), whereas the energy maximum corresponds to the conformation (P) in which plane M-C-C coincides with plane XY (φ ) 0° or 180°). Table 1 presents relative energies of these conformations at different levels of ab initio theory for the test molecules. The results indicate that in the molecules H2N-CH2CH3 and [H2C-CH2CH3]-, the doubly occupied pzorbital of trigonal atom M exhibits an intrinsic tendency to eclipse the single C-C bond, while the singly occupied orbital does not. The tendency of a lone pair of an sp2-hybridized heteroatom to eclipse a single bond is of general nature. This tendency definitely exists within molecules (CH3)2M-CH2CH3 (M ) N(sp2) or C(sp2) with doubly occupied pz-orbital on atom M; Table 1). Predicted features of rotational potentials around both N(sp2)-C(sp3) and O(sp2)-C(sp3) bonds in 20-(S)camptothecin are explained by this general tendency.51 The revealed conformational preference in the test molecules is likely to be caused by the Coulomb repulsion between electron pairs located at the p-orbitals of atoms M and R, which is minimized in the eclipsed conformation due to the characteristic shapes of those orbitals (Figure 18). Molecular orbitals obtained for this conformation at the HF/6-31G** level indicate that the pz-AO contribution of atom M (pz(M)) dominates in HOMO, whereas the previous orbital, (HOMO-1), is mainly built from the px-AO’s of atoms M, C and R. The single occupancy of HOMO dramatically decreases electron density of pz(M) and,

Vv C P

E

H2M-CH2CH3 0.0 3.1 0.0 0.0 2.5 0.0 0.0 1.4 0.0 0.0 2.5 0.0

v N

v C

P

E

P

E

P

13.4 10.6 N/C 17.5c

0.0 0.0 0.0 0.0

1.1 0.4 0.1 0.8

1.1 0.9 3.0 0.0

0.0 0.0 0.0 0.0

(CH3)2M-CH2CH3 0.0 12.8 0.0 22.4 0.0 13.1 0.0 22.1 0.0 12.9 0.0 21.4

0.0 2.9 0.0 4.8 0.0 4.5 0.0 4.8 0.0 4.6 0.0 4.7

a Symbol E denotes conformation where the C-C bond eclipses the pz-orbital of the sp2-hybridized center M; P denotes conformation where the M-C-C plane is perpendicular to the pz-orbital. Both conformations for each molecule were fully optimized at the B3LYP/6-31G(d,p), MP2/6-31G(d,p) and MP2/6-311G(2d,p) levels, while maintaining planarity of the center M. b The “projected” UMP2 (PUMP2) relative energies are given for the open-shell systems. N/C - geometry optimization was not converged. c This magnitude was obtained at the QCISD(T)/6-311G(2d,p)//MP2/6-31G(d,p) level. Conformation corresponding to the energy minimum is indicated in bold.

Figure 18. Newman projection demonstrating a tendency of the doubly occupied pz-orbital of the sp2-hybridized atom N within peptide backbone to eclipse single C-C bonds. A putative explanation is that the Coulomb repulsion between electron pairs located at the px(C) (black) and pz(N) (double-hatched) molecular orbitals is minimized in the illustrated conformation (E; φ ) (90°).

thereby its Coulomb repulsion from px(R) also decreases, which makes corresponding rotational potentials very shallow. Accordingly, the hydrogen atom, which does not have p-orbitals, does not demonstrate any tendency to eclipse the lone pair. These calculations confirmed that the torsional potentials within single amino acids being in the conformational region of β-pleated sheets actually tend to maintain |φ| below |ψ|. There is no surprise that the predicted conformational preference within molecules H2N-CH2CH3 and (CH3)2N-CH2CH3 toward the eclipsed conformation is relatively minor (Figure 17, Table 1), since it represents an estimate for only one term of the rotational potential φ in peptides. The additional stabilization of the eclipsed conformation (E) of the C(sp2)-N(sp2)-CR(sp3)C(sp2) motif within peptide strands comes from the loss of planarity of the nitrogen atom because of partial sp3-hybridization. The corresponding out-of-plane deformations of peptide groups within β-pleated sheets have been detected.9,14 The marked intrinsic tendency of the lone pair of the sp2hybridized atom N of the amide groups to eclipse single C-C bonds is the origin of the right-handed twisting of β-sheets in globular proteins. This tendency results in decreasing of the optimal magnitudes of |φ| in β-strands toward 90° (Figures 15 and 16C). On the other hand, since one pz-electron sitting on the trigonal atom C of the amide group does not tend to eclipse single C-C bonds (Figure 17 and Table 1), values of ψ do not

11306 J. Phys. Chem. B, Vol. 104, No. 47, 2000 have a particular preference to be around 90° (Figures 15 and 16A,B). The revealed property of the lone pair of nitrogen is also the reason for a remarkable flat shape of the rotational potential described by angle φ in alanine-based models I, III, V, and VII (Figure 15A). In these molecules, energy of torsional rotation around φ is built from three overlapping tendencies, namely the tendency of the lone pair to eclipse bond CR(sp3)C(sp2) (when φ ) -90°; Figure 16C), the tendency of CR(sp3)-H bond to eclipse a partially double bond N(sp2)C(sp2) (φ ) -120°; Figure 16D), and the tendency of the lone pair to eclipse bond CR(sp3)-Cβ(sp3) (φ ) -150°; Figure 16E). Overlapping of the three close minima causes the appearing flatness of the torsional potential in the alanine-based models. However, the first term, which minimizes torsional energy at φ ) -90°, dominates in alanine-based peptide models over the third term, because at φ ) -90° the nitrogen lone pair eclipses the more positively charged atom C(sp2) (Figure 16C) than at φ ) -150° (Cβ(sp3); Figure 16E). Since the pz electron pair does not tend to eclipse atom HR, the torsional potential φ in the glycine-based models II, IV, VI and VIII loses the flatness in the area of φ ) -150° (compare Figures 16E and 15A,B), which is the origin of the increased right-handed twisting characteristic for the glycine-based β-sheets. Does the tendency of β-strands to twist in the right-handed direction break the laws of symmetry? Obviously not. The same amount of the allowed conformational space is attributed to the left-handed twist. According to the major principle of the conformational analysis, all results remain valid if the signs of all torsional angles in molecular systems are simultaneously changed. This will result in mirror images of the molecules. Consequently, the right-handed twist will become left-handed and L-amino acids become D-amino acids. Accordingly, the conformational location of β-strands will transfer from the second to the fourth quadrant of the Ramachandran plot. This transformation does not change the configuration of Gly; therefore, the left-handed twist of poly-Gly β-strands is equally possible within the fourth quadrant as the right-handed twist within the second quadrant. The fourth quadrant of the Ramachandran plot is improbable for amino acids with natural L-configuration because of repulsion of the side chain from CO and NH groups within individual residues (Figure 15A).35,41 Consequently, there is no conformational region which would induce the left-handed twist of β-strands consisting of natural amino acids other than Gly. Since β-sheets in globular proteins never consist exclusively of Gly, the left-handed twist has never been observed. Unique structural features of β-pleated sheets in proteins are the direct consequence of the effect of the intermolecular H-bonded network on the asymmetrical two-minima conformational potential of each amino acid unit (Figure 5). It has been previously demonstrated using a simple model of the planar network of rotators that the increase of thermodynamical probability of the transition state, which separates two nonequivalent energy minima inherent in each unit cell, results in a glasslike state of the whole system.53 The results of the present study indicate that this is exactly the case in β-pleated sheets (compare Figures 5 and 12B), therefore they may represent natural two-dimensional glasses. This theoretical concept explains the extraordinary conformational flexibility and variability of β-pleated sheet domains, and sheds light on their functional role in proteins. Conclusions The origin of the right-handed twisting of β-pleated sheets in proteins is described. The key determinant is the lone pair

Shamovsky et. al. of nitrogen atoms within peptide main chains. This lone pair tends to eclipse the CR(sp3)-C(sp2) bond, which directly decreases absolute values of φ but not ψ, and, thereby, results in right-handed twisting of β-sheets. The split of values of |φ| and ψ in β-strands is very significant when intrastrand Hbonding is broken by interstrand H-bonds within β-sheets. Thus, the right-handed twist of β-pleated sheets in globular proteins is an intrinsic property of the main chains of individual β-strands, which is unlocked by interstrand H-bonds. A particular environment in globular proteins may induce local irregular deformations to geometry of β-pleated sheets, thereby influencing the local magnitude of the twisting, but it is not responsible for the right-handed twist itself. Configuration of the amino acid side chains comprising β-pleated sheets determines the location of their low-energy conformational spaces. Because of intrinsic conformational restrictions of individual chiral amino acids, the second and the fourth quadrants of the Ramachandran plot are the only allowed locations for β-sheets made of L- and D-amino acids, respectively. The split of values |φ| and |ψ| within each amino acid, such that |φ| < |ψ|, unequivocally determines the direction of the twist within those conformational regions. The second quadrant is the place for the right-handed β-strands, whereas the fourth quadrant is for the left-handed β-strands. Therefore, β-strands made of natural L-amino acids invariably exhibit the right-handed twist in native globular proteins. Since both quadrants are equally acceptable for Gly, the most stable form of hypothetical pure poly-Gly β-pleated sheets is equally expected to be either right- or left-handed depending on the occupied conformational region (second versus fourth quadrant of the Ramachandran plot). When Gly is incorporated into real β-strands made of asymmetrical L-amino acids within globular prioteins, it is forced to exhibit only one of the two possible stable forms, i.e., the form which is located in the second quadrant of the Ramachandran plot and, therefore, twisted in the right-handed direction. Acknowledgment. This work was supported by a research grant from Neuroceptor Inc. (Ontario, Canada) and an operating grant to R.J.R. from the Canadian Institutes of Health Research (MT7757). G.M.R. is a Queen’s National Scholar. References and Notes (1) Levitt, M.; Chothia, C. Nature 1976, 261, 552-558. (2) Richardson, J. S. Nature 1977, 268, 495-500. (3) Richardson, J. S. AdV. Prot. Chem. 1981, 34, 167-339. (4) Pauling, L.; Corey, R. B. Proc. Natl. Acad. Sci. U.S.A. 1951, 37, 205-211. (5) Yang, A.-S.; Honig, B. J. Mol. Biol. 1995, 252, 351-365. (6) Yang, A.-S.; Honig, B. J. Mol. Biol. 1995, 252, 366-376. (7) Chothia, C. J. Mol. Biol. 1973, 75, 295-302. (8) Chothia, C. J. Mol. Biol. 1983, 163, 107-117. (9) Salemme, F. R. Prog. Biophys. Mol. Biol. 1983, 42, 95-133. (10) Pauling, L.; Corey, R. B. Proc. Natl. Acad. Sci. U.S.A. 1951, 37, 251-256. (11) Daffner, C.; Chelvanayagam, G.; Argos, P. Protein Sci. 1994, 3, 876-882. (12) Carter, C. W., Jr.; Kraut, A. J. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 283-287. (13) Salemme, F. R. J. Mol. Biol. 1981, 146, 143-156. (14) Weatherford, D. W.; Salemme, F. R. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 19-23. (15) (a) Chou, K. C.; Pottle, M.; Ne´methy, G.; Ueda, Y.; Scheraga, H. A. J. Mol. Biol. 1982, 162, 89-112. (b) Chou, K. C.; Scheraga, H. A. Proc. Natl. Acad. Sci. U.S.A. 1982, 79, 7047-7051. (c) Chou, K. C.; Ne´methy, G.; Scheraga, H. A. J. Mol. Biol. 1983, 168, 389-407. (d) Chou, K. C.; Ne´methy, G.; Scheraga, H. A. Biochemistry 1983, 22, 6213-6221. (16) Maccallum, P. H.; Poet, R.; Milner-White, E. J. J. Mol. Biol. 1995, 248, 374-384.

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