Non-protein amino acids in peptide design

An overview of the use of non-protein amino acids in the design of conformationally well-defined peptides, based on work from the author’s laboratory, is discussed. The crystal structures of several designed oligopeptides illustrate the useα-aminoisobutyric acid (Aib) in the construction of helices, D-amino acids in the design of helix termination segments and^DPro-Xxx segments for nucleating ofβ-hairpin structures.β- andγ-amino acid residues have been used to expand the range of designed polypeptide structures. Dedicated to Professor C N R Rao on his 70th birthday     

Aromatic-aromatic interactions in crystal structures of helical peptide scaffolds containing projecting phenylalanine residues

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.     

Weak edge-degree domination in hypergraphs

In this paper we extend the notion of weak degree domination in graphs to hypergraphs and find relationships among the domination number, the weak edge-degree domination number, the independent domination number and the independence number of a given hypergraph.     

Speeding up Exponentiation using an Untrusted Computational Resource

We present protocols for speeding up fixed-base variable-exponent exponentiation and variable-base fixed-exponent exponentiation using an untrusted computational resource. In the fixed-base protocols, the exponent may be blinded. In the variable-base protocols, the base may be blinded. The protocols are described for exponentiation in a cyclic group. We describe how to extend them to exponentiation modulo an integer where the modulus is the product of primes with single multiplicity. The protocols provide a speedup of over the square-and-multiply algorithm, where k is the bitlength of the exponent. One application of the protocols is to speed up exponentiation-based verification in discrete log-based signature and credential schemes. The protocols also allow signature verifiers to dynamically choose, for each message, the amount of work it would like to perform to verify the signature. This results in a work-security tradeoff. We introduce a fifth protocol to perform variable-base variable- exponent exponentiation, which also has this feature. Our model allows the trusted resource to perform computations in its idle time. The protocols facilitate the offloading of work to the offline stage, such that the work the trusted resource performs when it has to do an exponentiation is smaller. Our protocols are unconditionally secure.     

Nucleation, growth, and form in crystals of peptide helices

A model for the nucleation of crystallization in peptide helices is presented. The crystal structures of four polymorphic forms of a hydrophobic helical decapeptide Boc-Leu-Aib-Phe-Phe-Leu-Aib-Ala-Ala-Leu-Aib-OMe (I) exemplify alternative packing modes in cylindrical molecules. Three crystal forms of peptide I are monoclinic P2(1), while one is orthorhombic P22121. The five different helical molecules characterized have very similar backbone conformations over much of the peptide length. A survey of 117 helical peptide structures with a length >/=8 residues reveals a preponderance of the triclinic (P1), monoclinic (P2(1)), and orthorhombic (P2(1)2(1)2(1)) crystal forms. Models for the formation of critical nuclei are based on helix association driven by solvophobic forces, resulting in the formation of raftlike structures. Raft association can be further driven by the imperative of minimizing solvent accessible surface area with the formation of blocks, which can be subsequently fitted in Lego set fashion by multiple hydrogen bond interactions in the head-to-tail region. This model provides a rationalization for observed crystal formation based on a postulated structure for an embryonic nucleus, which is determined by aggregation patterns and unconstrained by the dictates of symmetry.     

Hybrid peptides: expanding the beta turn in peptide hairpins by the insertion of beta-, gamma-, and delta-residues

The beta turn segment in designed peptide hairpins has been expanded by the insertion of beta-, gamma- and delta-amino acids at the i+2 position. The model octapeptides Boc-Leu-Phe-Val-DPro-Ac6c-Leu-Phe-Val-OMe (1), Boc-Leu-Phe-Val-DPro-beta3-Ac6c-Leu-Phe-Val-OMe (2), and Boc-Leu-Phe-Val-DPro-Gpn-Leu-Phe-Val-OMe (3) have been shown to adopt beta hairpin conformations in methanol by the observation of key diagnostic nuclear Overhauser effects. Boc-Leu-Val-Val-DPro-delta-Ava-Leu-Val-Val-OMe (4) adopts a beta hairpin conformation in crystals; this is stabilized by three cross-strand hydrogen bonds as demonstrated by X-ray diffraction. The canonical C10 turn in an alpha-alpha segment is expanded to C11, C12, and C13 turns in alpha-beta, alpha-gamma, and alpha-delta segments, respectively. The crystal structures of Piv-LPro-beta3-Ac6c-NHMe (5) and Boc-Ac6c-Gpn-Ac6c-OMe (6) reveal intramolecularly hydrogen-bonded C11 and C12 conformations, respectively. Computer modeling of octapeptide sequences that contain centrally positioned hybrid-turn segments, by using turn parameters derived from the structures of peptides 5 and 6, establishes the stereochemical acceptability of the beta hairpins in the cases of peptides 2 and 3. Accommodation of omega-amino acids into the turn segments is achieved by the adoption of gauche conformations around the backbone C--C bonds.     

Diproline templates as folding nuclei in designed peptides. Conformational analysis of synthetic peptide helices containing amino terminal Pro-Pro segments

The effect of N-terminal diproline segments in nucleating helical folding in designed peptides has been studied in two model sequences Piv-Pro-Pro-Aib-Leu-Aib-Phe-OMe (1) and Boc-Aib-Pro-Pro-Aib-Val-Ala-Phe-OMe (2). The structure of 1 in crystals, determined by X-ray diffraction, reveals a helical (alphaR) conformation for the segment residues 2 to 5, stabilized by one 4-->1 hydrogen bond and two 5-->1 interactions. The N-terminus residue, Pro(1) adopts a polyproline II (P(II)) conformation. NMR studies in three different solvent systems support a conformation similar to that observed in crystals. In the apolar solvent CDCl3, NOE data favor the population of both completely helical and partially unfolded structures. In the former, the Pro-Pro segment adopts an alphaR-alphaR conformation, whereas in the latter, a P(II)-alphaR structure is established. The conformational equilibrium shifts in favor of the P(II)-alphaR structure in solvents like methanol and DMSO. A significant population of the Pro(1)-Pro(2) cis conformer is also observed. The NMR results are consistent with the population of at least three conformational states about Pro-Pro segment: trans alphaR-alphaR, trans P(II)-alphaR and cis P(II)-alphaR. Of these, the two trans conformers are in rapid dynamic exchange on the NMR time scale, whereas the interconversion between cis and trans form is slow. Similar results are obtained with peptide 2. Analysis of 462 diproline segments in protein crystal structures reveals 25 examples of the alphaR-alphaR conformation followed by a helix. Modeling and energy minimization studies suggest that both P(II)-alphaR and alphaR-alphaR conformations have very similar energies in the model hexapeptide 1.