Synthesis of optically active 1-aza-4-oxabicyclo[4.1.0]-heptan-5-ones

The reaction of the methyl ester of 1,2-dibromopropionic acid with chiral - aminoalcohols under the conditions of the Gabriel-Cromwell reaction gave epimers of methyl 1-(-hydroxyalkyl)aziridine-2-carboxylates, which were separated by liquid chromatography. NMR spectroscopy and CD were used to determine the absolute configuration of these products. The relative rate of conversion of epimeric esters of trans-1-(-hydroxyalkyl)aziridine-2-carboxylic acids to 1-aza-4- oxabicyclo-[4.1.0]heptan-5-ones is a function of the intramolecular contacts of the reacting OH and CO₂Me groups in the lactonization of the cis isomers. The most favorable conformation of the six-membered ring in 1-aza-4-oxabicyclo[4,1,0]heptan-5-ones in solution is a distorted boat.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 7, pp. 975–980, July, 1989.     

Study of synthetic hydroxyapatite by the method of high-resolution transmission electron microscopy: Morphology and growth direction

Hydroxyapatite crystals grown from aqueous solutions have been studied by the methods of high-resolution electron microscopy and transmission electron diffraction. Processing of the experimental electron micrographs with the use of the Digital Micrograph program and the study of the corresponding Fourier trans-forms showed that the submicron microcrystals grow mainly along the [0001] direction. The (0001) and the $(01\bar 10)$ planes are perpendicular and parallel to the long edge of the crystals, respectively. The good accord between the experimental electron-microscopy images and the electron microscopy images calculated by the EMS program was attained only for crystals with the thicknesses ranging from one to five lattice periods. This allows us to state that hydroxyapatite grows from aqueous solutions in the form of very thin (with the thickness of the order of several lattice parameters) platelike crystals.     

Calcium phosphate crystallization under terrestrial and microgravity conditions

Calcium phosphate crystalline powders grown under terrestrial and space (EURECA 1992-1993 flight) conditions in the Solution Growth Facility are analyzed and compared by optical and electron microscopy (scanning and transmission), electron and X-ray microdiffraction and microanalyses. On earth, only small, micrometer size scale, spherolites of hydroxyapatite (HAP) grow. In space, the HAP spherolites reach hundreds of micrometer. Also, octacalcium phosphate (OCP) spherolites up to 3 mm have been obtained. Computer modelling of diffusion in a real chamber has been performed. It suggests high spatial supersaturation gradients at zero gravity which may provide much higher local supersaturations on earth, where convection takes place. The analyses suggest that the dramatic difference between the terrestrial and space samples should come from much lower supersaturation in space.     

Modified barrier functions (theory and methods)

The nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and/or the objective function into an equivalent problem, the classical Lagrangian which has important properties on the primal and the dual spaces.The application of the nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions (MBF's) and MBF Methods (MBFM's). Being classical Lagrangians (CL's) for an equivalent problem, the MBF's combine the best properties of the CL's and classical barrier functions (CBF's) but at the same time are free of their most essential deficiencies.Due to the excellent MBF properties, new characteristics of the dual pair convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed.The MBFM have up to a superlinear rate of convergence and are to the classical barrier functions (CBF's) method as the Multipliers Method for Augmented Lagrangians is to the Classical Penalty Function Method. Based on the dual theory associated with MBF, the method for the simultaneous solution of the dual pair convex programming problems with up to quadratic rates of convergence have been developed. The application of the MBF to linear (LP) and quadratic (QP) programming leads to a new type of multipliers methods which have a much better rate of convergence under lower computational complexity at each step as compared to the CBF methods.The numerical realization of the MBFM leads to the Newton Modified Barrier Method (NMBM). The excellent MBF properties allow us to discover that for any nondegenerate constrained optimization problem, there exists a hot start, from which the NMBM has a better rate of convergence, a better complexity bound, and is more stable than the interior point methods, which are based on the classical barrier functions.     

Stereochemistry and reactivity of the HPA-imine Mannich intermediate

Homophthalic anhydride (HPA) typically reacts rapidly with benzalimines to afford the formal [4+2] adduct, a 1,2,3,4-tetrahydroisoquinolin-1-one-4-carboxylic acid. The stereochemical outcome of this reaction is consistent with an open transition state comprising an iminium species and enolized HPA, leading to a short-lived amino-anhydride intermediate. In the case of N-tert-butylbenzalimine, this Mannich-type intermediate, which would normally cyclize at low temperature to a single isomer of the delta-lactam, is intercepted by base treatment to afford beta-lactam products. A pathway featuring ketene formation followed by ring closure is implicated.     

Increasing the attraction area of the global minimum in the binary optimization problem

The problem of binary minimization of a quadratic functional in the configuration space is discussed. In order to increase the efficiency of the random-search algorithm it is proposed to change the energy functional by raising to a power the matrix it is based on. We demonstrate that this brings about changes of the energy surface: deep minima displace slightly in the space and become still deeper and their attraction areas grow significantly. Experiments show that this approach results in a considerable displacement of the spectrum of the sought-for minima to the area of greater depth, and the probability of finding the global minimum increases abruptly (by a factor of 10³ in the case of the 10 × 10 Edwards–Anderson spin glass).     

1.5-Q-superlinear convergence of an exterior-point method for constrained optimization

We introduce and analyze an exterior-point method (EPM) for constrained optimization problems with both inequality constraints and equations. We show that under the standard second-order optimality conditions the EPM converges to the primal–dual solution with 1.5-Q-superlinear rate. Dedicated to Professor Gil Strang on the occasion on his 70th birthday.     

Quantum mechanical/molecular mechanical study on the mechanism of the enzymatic Baeyer-Villiger reaction

We report a combined quantum mechanical/molecular mechanical (QM/MM) study on the mechanism of the enzymatic Baeyer-Villiger reaction catalyzed by cyclohexanone monooxygenase (CHMO). In QM/MM geometry optimizations and reaction path calculations, density functional theory (B3LYP/TZVP) is used to describe the QM region consisting of the substrate (cyclohexanone), the isoalloxazine ring of C4a-peroxyflavin, the side chain of Arg-329, and the nicotinamide ring and the adjacent ribose of NADP(+), while the remainder of the enzyme is represented by the CHARMM force field. QM/MM molecular dynamics simulations and free energy calculations at the semiempirical OM3/CHARMM level employ the same QM/MM partitioning. According to the QM/MM calculations, the enzyme-reactant complex contains an anionic deprotonated C4a-peroxyflavin that is stabilized by strong hydrogen bonds with the Arg-329 residue and the NADP(+) cofactor. The CHMO-catalyzed reaction proceeds via a Criegee intermediate having pronounced anionic character. The initial addition reaction has to overcome an energy barrier of about 9 kcal/mol. The formed Criegee intermediate occupies a shallow minimum on the QM/MM potential energy surface and can undergo fragmentation to the lactone product by surmounting a second energy barrier of about 7 kcal/mol. The transition state for the latter migration step is the highest point on the QM/MM energy profile. Gas-phase reoptimizations of the QM region lead to higher barriers and confirm the crucial role of the Arg-329 residue and the NADP(+) cofactor for the catalytic efficiency of CHMO. QM/MM calculations for the CHMO-catalyzed oxidation of 4-methylcyclohexanone reproduce and rationalize the experimentally observed (S)-enantioselectivity for this substrate, which is governed by the conformational preferences of the corresponding Criegee intermediate and the subsequent transition state for the migration step.     

Stability of linear difference equations with unmodelled higher order terms

Dedicated to Gerry Ladas on his sixtieth birthday. We consder the case when the tru linear difference equation may difer from the nominal one. Neither higher order coefficients nor the order of the true equation are konwn: the only information available is that these higher order coefficients ae bounded in l ₁norm. We obtain necessary and sufficient conditionsfor stability of the entire family of such equations.