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The theoretical background and practical procedures for phase determination by the symbolic addition method are discussed. Phase determining formulas are presented for centrosymmetric and noncentrosymmetric crystals. A probability formula is employed to evaluate the reliability of phase determination for centrosymmetric crystals and a formula for the variance is utilized for the same purpose in noncentrosymmetric ones. These probability measures play a key role in overcoming the main problems involved in carrying out the procedure, namely the nonuniqueness of the internal consistency criterion as applied to the phase determining formulas, and questions concerning the proper circumstances for assigning symbols. The method is generally applicable to centrosymmetric crystals and has been successful in several applications to noncentrosymmetric ones. Some auxiliary phase information is probably required to make the symbolic addition procedure a general one for noncentrosymmetric crystals.

The preparation, crystal structures, and circular dichroism (CD) spectra of two oligomers of optically active trans-2-aminocyclohexanecarboxylic acid are reported. In the solid state, both the tetramer and the hexamer of this β-amino acid display a helical conformation that involves 14-membered-ring hydrogen bonds between a carbonyl oxygen and the amide proton of the second residue toward the N-terminus. (For comparison, the familiar α-helix observed in conventional peptides is associated with a 13-membered-ring hydrogen bond between a carbonyl oxygen and the amide proton of the fourth residue toward the C-terminus.) These crystallographic data, along with CD data obtained in methanol, suggest that the 14-helix constitutes a stable secondary structure for β-amino acid oligomers (“β-peptides”). In addition, the crystal packing pattern observed for the hexamer offers a blueprint for the design of β-peptides that might adopt a helical bundle tertiary structure.

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ABSTRACT13-hairpin structures have been crystallographically characterized only in very short acyclic peptides, in contrast to helices. The structure of the designed ,3-hairpin, t-butoxycarbonyl-Leu-Val-Val-D-Pro-Gly-Leu-Val-Val-OMe in crystals is described. The two independent molecules of the octapeptide fold into almost ideal 13-hairpin conformations with the central D-Pro-Gly segment adopting a Type 1' 13-turn conformation. The definitive characterization of a 13-hairpin has implications for de novo peptide and protein design, particularly for the development of three-and four-stranded 13-sheets.De novo peptide and protein design is based on the ability to construct peptide sequences with predictable folding patterns (1-7). Helices and 13-sheets have been the focus of considerable synthetic attention in attempts to assemble mimics for helical bundles (8-11) and all 83-motifs (12-15). Helix nucleation strategies have been successful in generating both watersoluble helices, using stabilizing side chain interactions (16)(17)(18)(19), and incorporating backbone conformational constraints in hydrophobic helices (20)(21)(22). The ready crystallizability of peptide helices nucleated with a,a-dialkylated amino acids has permitted detailed stereochemical characterization by x-ray diffraction (23-26). In contrast, strategies for construction of 13-hairpins have been less widely explored (27-32). Furthermore, crystallographic examinations have been made only in two-to five-residue acyclic peptides (33)(34)(35)(36)(37)(38)(39)(40). In this report we describe the crystal structure of a designed, synthetic 13-hairpin in an acyclic octapeptide.Analysis of 13-hairpin structures in proteins reveal that such features invariably contain Type II' or I' 13-turns as the nucleating segment of the chain reversal (41-43). We therefore designed the octapeptide t-butoxycarbonyl (Boc)-LeuVal-Val-D-Pro-Gly-Leu-Val-Val-OMe (11) such that the central D-Pro-Gly segment facilitates a Type II' 13-turn conformation. The constraint of pyrrolidine ring formation in D-Pro restricts the value of 4 to +60°+ 20°. The idealized conformational angles for the i+ 1/i+2 residues of a Type II' 13-turn are: 0j+1 = +600, qAj+, = -120°, 4i+2 = -80°, and q/i+2 = 0°( 44, 45). The valine-rich arms are expected to favor extended strand conformations in view of the established propensity of 13-branched residues to occur in 13-sheets in proteins (46-48). EXPERIMENTAL METHODSThe peptide was synthesized by conventional solution phase procedures and purified by medium pressure liquid chromatography on a C18 column (40-60 microns), followed by HPLC purification on a C18 column (10 microns) using methanolwater gradients. The peptide was fully characterized by 400 MHz 'H NMR (49).Colorless crystals in the form of very thin plates were grown by slow evaporation from a CH30H solution, to which a small amount of water was added. Almost all the crystals were composites in the form of a stack of several thin wafers. The crystal selected for data collection had a slightly sp...

Aromatic-aromatic interactions between phenylalanine side chains in peptides have been probed by the structure determination in crystals of three peptides: Boc-Val-Ala-Phe-Aib-Val-Ala-Phe-Aib-OMe, I; Boc-Val-Ala-Phe-Aib-Val-Ala-Phe-Aib-Val-Ala-Phe-Aib-OMe, II; Boc-Aib-Ala-Phe-Aib-Phe-Ala-Val-Aib-OMe, III. X-ray diffraction studies reveal that all three peptides adopt helical conformations in the solid state with the Phe side chains projecting outward. Interhelix association in the crystals is promoted by Phe-Phe interactions. A total of 15 unique aromatic pairs have been characterized in the three independent crystal structures. In peptides I and II, the aromatic side chains lie on the same face of the helix at i/i + 4 positions resulting in both intrahelix and interhelix aromatic interactions. In peptide III, the Phe side chains are placed on the opposite faces of the helix, resulting in exclusive intermolecular aromatic interactions. The distances between the centroids of aromatic pair ranges from 5.11 to 6.86 A, while the distance of closest approach of ring carbon atoms ranges from 3.27 to 4.59 A. Examples of T-shaped and parallel-displaced arrangements of aromatic pairs are observed, in addition to several examples of inclined arrangements. The results support the view that the interaction potential for a pair of aromatic rings is relatively broad and rugged with several minima of similar energies, separated by small activation barriers.

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