Formal Solutions of an Evolution Equation of Riemann Type

A general implicit solution to a multivariable quasilinear first-order partial differential equation of evolution type for a scalar function is demonstrated. A formal power series solution to the Cauchy problem for this equation is shown to possess succinct Taylor coefficients.     

Modulation of the output of an XeCl* laser by the generation of electrical solitons

A new technique is reported for exciting an XeCl^* excimer laser to produce a sequence of high recurrence frequency output pulses. An electrical soliton array is generated within the pulsed power driving circuit using a discrete-component transmission line containing nonlinear ferroelectric capacitors. The method offers several advantages over alternative electrooptical means of laser output modulation.     

Cycloadditions of isobenzofuran to a constrained template bearing neighboring dienophiles

A high yielding synthesis of the pentacyclic diene-dione 1 has enabled investigation of its reactivity as a double dienophile in Diels-Alder [4+2] cycloadditions with isobenzofuran, leading to novel and highly symmetrical three-sided cavitands 3 and 4.     

Synthesis and structural properties of patellamide A derivatives and their copper(II) compounds

The synthesis, characterization and copper(II) coordination chemistry of three new cyclic peptide ligands, PatJ(1) (cyclo-(Ile-Thr-(Gly)Thz-Ile-Thr-(Gly)Thz)), PatJ(2) (cyclo-(Ile-Thr-(Gly)Thz-(D)-Ile-Thr-(Gly)Thz)), and PatL (cyclo-(Ile-Ser-(Gly)Thz-Ile-Ser-(Gly)Thz)) are reported. All of these cyclic peptides and PatN (cyclo-(Ile-Ser-(Gly)Thz-Ile-Thr-(Gly)Thz)) are derivatives of patellamide A and have a [24]azacrown-8 macrocyclic structure. All four synthetic cyclic peptides have two thiazole rings but, in contrast to patellamide A, no oxazoline rings. The molecular structure of PatJ(1), determined by X-ray crystallography, has a saddle conformation with two close-to-coparallel thiazole rings, very similar to the geometry of patellamide D. The two coordination sites of PatJ(1) with thiazole-N and amide-N donors are each well preorganized for transition metal ion binding. The coordination of copper(II) was monitored by UV/Vis spectroscopy, and this reveals various (meta)stable mono- and dinuclear copper(II) complexes whose stoichiometry was confirmed by mass spectra. Two types of dinuclear copper(II) complexes, [Cu(2)(H(4)L)(OH(2))(n)](2+) (n=6, 8) and [Cu(2)(H(2)L)(OH(2))(n)] (n=4, 6; L=PatN, PatL, PatJ(1), PatJ(2)) have been identified and analyzed structurally by EPR spectroscopy and a combination of spectra simulations and molecular mechanics calculations (MM-EPR). The four structures are similar to each other and have a saddle conformation, that is, derived from the crystal structure of PatJ(1) by a twist of the two thiozole rings. The small but significant structural differences are characterized by the EPR simulations.     

Conformationally constrained macrocycles that mimic tripeptide beta-strands in water and aprotic solvents

The beta-strand conformation is unknown for short peptides in aqueous solution, yet it is a fundamental building block in proteins and the crucial recognition motif for proteolytic enzymes that enable formation and turnover of all proteins. To create a generalized scaffold as a peptidomimetic that is pre-organized in a beta-strand, we individually synthesized a series of 15-22-membered macrocyclic analogues of tripeptides and analyzed their structures. Each cycle is highly constrained by two trans amide bonds and a planar aromatic ring with a short nonpeptidic linker between them. A measure of this ring strain is the restricted rotation of the component tyrosinyl aromatic ring (DeltaG(rot) 76.7 kJ mol(-1) (16-membered ring), 46.1 kJ mol(-1) (17-membered ring)) evidenced by variable temperature proton NMR spectra (DMF-d(7), 200-400 K). Unusually large amide coupling constants ((3)J(NH-CHalpha) 9-10 Hz) corresponding to large dihedral angles were detected in both protic and aprotic solvents for these macrocycles, consistent with a high degree of structure in solution. The temperature dependence of all amide NH chemical shifts (Deltadelta/T 7-12 ppb/deg) precluded the presence of transannular hydrogen bonds that define alternative turn structures. Whereas similar sized conventional cyclic peptides usually exist in solution as an equilibrium mixture of multiple conformers, these macrocycles adopt a well-defined beta-strand structure even in water as revealed by 2-D NMR spectral data and by a structure calculation for the smallest (15-membered) and most constrained macrocycle. Macrocycles that are sufficiently constrained to exclusively adopt a beta-strand-mimicking structure in water may be useful pre-organized and generic templates for the design of compounds that interfere with beta-strand recognition in biology.     

Helix Nucleation by the Smallest Known α‐Helix in Water

Cyclic pentapeptides (e.g. Ac‐(cyclo‐1,5)‐[KAXAD]‐NH₂; X=Ala, 1 ; Arg, 2 ) in water adopt one α‐helical turn defined by three hydrogen bonds. NMR structure analysis reveals a slight distortion from α‐helicity at the C‐terminal aspartate caused by torsional restraints imposed by the K(i)–D(i+4) lactam bridge. To investigate this effect on helix nucleation, the more water‐soluble 2 was appended to N‐, C‐, or both termini of a palindromic peptide ARAARAARA (≤5 % helicity), resulting in 67, 92, or 100 % relative α‐helicity, as calculated from CD spectra. From the C‐terminus of peptides, 2 can nucleate at least six α‐helical turns. From the N‐terminus, imperfect alignment of the Asp5 backbone amide in 2 reduces helix nucleation, but is corrected by a second unit of 2 separated by 0–9 residues from the first. These cyclic peptides are extremely versatile helix nucleators that can be placed anywhere in 5–25 residue peptides, which correspond to most helix lengths in protein–protein interactions.     

The Multi-field Complex Bateman Equation

The multi-field generalisation of the Bateman equation arises from considerations of the continuation of String and Brane equations to the case where the base space is of higher dimension than the target space. The complex extension of this equation possesses a remarkably large invariance group, and admits a very simple implicit form for its general solution, in addition to the special case of holomorphic and anti-holomorphic explicit solutions. A class of inequivalent Lagrangians for this equation is discovered.     

Visualisation of a supersonic free-jet expansion using laser-induced fluorescence spectroscopy: Application to the measurement of rate constants at ultralow temperatures

2 Σ⁺, v^′=0) by air at 26±4 K. A value of 2.56±0.40×10^-10 molecule^-1 cm³ s^-1 was obtained, a factor of more than four higher than at room temperature, and consistent with attractive forces dominating the quenching of OH(A²Σ⁺). A d etailed computational fluid dynamics (CFD) simulation of the supersonic free-jet expansion was performed, providing a two-dimensional visualisation of temperature and density variations throughout the expansion. The CFD calculations reproduced the salient features of the experimental temperature and density profiles along the centreline. Comparison between experiment and computation has allowed validation of CFD codes. Received: 31 January 1997/Revised version: 24 March 1997     

The construction of self-dual solutions to SU(2) gauge theory

Ignoring the problem of sources and singularities, explicit expressions are constructed for the ansätze of Atiyah and Ward. These take an especially simple form in theR gauge of Yang. Some non-linear transformation properties of the self-duality equations in this gauge provide an inductive proof of the ansätze. There is a six-parameter family of these Bäcklund transformations. They take real SU(2) gauge fields into real SU(1, 1) gauge fields and vice versa.On leave of absence from Department of Mathematics, University of Durham, U.K.