Kreĭn–Saakyan and Trace Formulas for a Pair of Canonical Differential Operators

An analog of the Kreĭn–Saakyan formula is derived for any pair of relatively prime self-adjoint extensions of a minimal symmetric canonical differential operator. This allows us to deduce a trace formula in the matrix case. I am grateful to Sh. Saakyan for his interest in this work and lively discussion. Received: December 8, 2006. Accepted: December 30, 2006.     

Mechanism of palladium-catalyzed diene cyclization/hydrosilylation: direct observation of intramolecular carbometalation

The results of kinetic, deuterium-labeling, and low-temperature NMR studies have established a mechanism for the palladium-catalyzed cyclization/hydrosilylation of dimethyl diallylmalonate (1) with triethylsilane involving rapid, irreversible conversion of the palladium silyl complex [(phen)Pd(SiEt(3))(NCAr)](+) [BAr(4)](-) [Ar = 3,5-C(6)H(3)(CF(3))(2)] (4b) and 1 to the palladium 5-hexenyl chelate complex [(phen)Pd[eta(1),eta(2)-CH(CH(2)SiEt(3))CH(2)C(CO(2)Me)(2)CH(2)CH=CH(2)]](+) [BAr(4)](-) (5), followed by intramolecular carbometalation of 5 to form the palladium cyclopentylmethyl complex trans-[(phen)Pd[CH2CHCH2C(CO2Me)2CH2CHCH2SiEt3](NCAr)]+ [BAr4]- (6), and associative silylation of 6 to release 3 and regenerate 4b.     

X-ray diffraction study of the complexes of SAICAR synthase with adenosinetriphosphate

The three-dimensional structures of two enzyme-substrate complexes of SAICAR synthase from the yeast Saccharomyces cerevisiae with adenosinetriphosphate (ATP) prepared under different conditions were studied by X-ray diffraction analysis and then refined. An enzyme molecule was shown to contain two binding sites of ATP. One of these sites is located in the central cavity of the enzyme molecule and apparently binds the ATP molecule directly involved in the enzymatic reaction. In the complexes, the phosphate groups of ATP occupying this site adopt different conformations depending on the Mg²⁺ concentration. The functional role of the second binding site located at a distance of approximately 15 Å from the first site away from the central enzyme cavity has not been understood as yet. It might be that the second site perform the regulatory role in enzyme functioning.     

X-ray diffraction study of the complex of the enzyme SAICAR synthase with substrate analogues

The three-dimensional structure of the complex of the enzyme SAICAR synthase with analogues of natural substrates, namely, phosphoribosylaminoimidazolecarboxamide (AICAR) and succinic acid, was determined and refined by methods of protein crystallography. Two AICAR-binding sites were revealed in the protein molecule. One of these sites is located in the active center of the enzyme in the vicinity of the ATP-binding site, which has been found in the complex of SAICAR synthase with ATP that had been studied earlier. The second AICAR-binding site is located at the periphery of the protein molecule and coincides with the additional ATP-binding site present in the complexes studied earlier. The binding site of succinic acid was revealed in the active center of the enzyme in the vicinity of the AICAR molecule. The electron density distribution for the AICAR molecule in the active center is indicative of the possible lability of the atomic group of the adenine base.     

X-ray diffraction study of the complex of the enzyme SAICAR synthase with the reaction product

The three-dimensional structure of the complex of the enzyme SAICAR synthase with the product of the enzymatic reaction, SAICAR, was solved and refined by methods of protein crystallography. The SAICAR-binding site in the active site of the enzyme was found. The amino-acid residues providing the binding of the reaction product with the protein were revealed. These residues were compared with those involved in the substrate binding in the complex with AICAR and succinic acid studied earlier.     

The structures and electronic configuration of compound I intermediates of Helicobacter pylori and Penicillium vitale catalases determined by X-ray crystallography and QM/MM density functional theory calculations

The structures of Helicobacter pylori (HPC) and Penicillium vitale (PVC) catalases, each with two subunits in the crystal asymmetric unit, oxidized with peroxoacetic acid are reported at 1.8 and 1.7 A resolution, respectively. Despite the similar oxidation conditions employed, the iron-oxygen coordination length is 1.72 A for PVC, close to what is expected for a Fe=O double bond, and 1.80 and 1.85 A for HPC, suggestive of a Fe-O single bond. The structure and electronic configuration of the oxoferryl heme and immediate protein environment is investigated further by QM/MM density functional theory calculations. Four different active site electronic configurations are considered, Por*+-FeIV=O, Por*+-FeIV=O...HisH+, Por*+-FeIV-OH+ and Por-FeIV-OH (a protein radical is assumed in the latter configuration). The electronic structure of the primary oxidized species, Por*+-FeIV=O, differs qualitatively between HPC and PVC with an A2u-like porphyrin radical delocalized on the porphyrin in HPC and a mixed A1u-like "fluctuating" radical partially delocalized over the essential distal histidine, the porphyrin, and, to a lesser extent, the proximal tyrosine residue. This difference is rationalized in terms of HPC containing heme b and PVC containing heme d. It is concluded that compound I of PVC contains an oxoferryl Por*+-FeIV=O species with partial protonation of the distal histidine and compound I of HPC contains a hydroxoferryl Por-FeIV-OH with the second oxidation equivalent delocalized as a protein radical. The findings support the idea that there is a relation between radical migration to the protein and protonation of the oxoferryl bond in catalase.     

Three-dimensional structure of the enzyme dimanganese catalase from Thermus Thermophilus at 1 Å resolution

The crystal structures of two forms of the enzyme dimanganese catalase from Thermus Thermophilus (native and inhibited by chloride) were studied by X-ray diffraction analysis at 1.05 and 0.98 Å resolution, respectively. The atomic models of the molecules were refined to the R factors 9.8 and 10%, respectively. The three-dimensional molecular structures are characterized in detail. The analysis of electron-density distributions in the active centers of the native and inhibited enzyme forms revealed that the most flexible side chains of the amino acid residues Lys162 and Glu36 exist in two interrelated conformations. This allowed us to obtain the structural data necessary for understanding the mechanism of enzymatic activity of the dimanganese catalase.     

X-ray diffraction study of <Emphasis Type="Italic">Penicillium Vitale</Emphasis> catalase in the complex with aminotriazole

The three-dimensional structure of the enzyme catalase from Penicillium vitale in a complex with the inhibitor aminotriazole was solved and refined by protein X-ray crystallography methods. An analysis of the three-dimensional structure of the complex showed that the inhibition of the enzyme occurs as a result of the covalent binding of aminotriazole to the amino-acid residue His64 in the active site of the enzyme. An investigation of the three-dimensional structure of the complex resulted in the amino-acid residues being more precisely identified. The binding sites of saccharide residues and calcium ions in the protein molecule were found.     

Multivariate density estimation using dimension reducing information and tail flattening transformations for truncated or censored data

This paper introduces a multivariate density estimator for truncated and censored data with special emphasis on extreme values based on survival analysis. A local constant density estimator is considered. We extend this estimator by means of tail flattening transformation, dimension reducing prior knowledge and a combination of both. The asymptotic theory is derived for the proposed estimators. It shows that the extensions might improve the performance of the density estimator when the transformation and the prior knowledge is not too far away from the true distribution. A simulation study shows that the density estimator based on tail flattening transformation and prior knowledge substantially outperforms the one without prior knowledge, and therefore confirms the asymptotic results. The proposed estimators are illustrated and compared in a data study of fire insurance claims.     

Multivariate density estimation using dimension reducing information and tail flattening transformations

We propose a nonparametric multiplicative bias corrected transformation estimator designed for heavy tailed data. The multiplicative correction is based on prior knowledge and has a dimension reducing effect at the same time as the original dimension of the estimation problem is retained. Adding a tail flattening transformation improves the estimation significantly-particularly in the tail-and provides significant graphical advantages by allowing the density estimation to be visualized in a simple way. The combined method is demonstrated on a fire insurance data set and in a data-driven simulation study.