Estimating Invariant Probability Densities for Dynamical Systems

Knowing a probability density (ideally, an invariant density) for the trajectories of a dynamical system allows many significant estimates to be made, from the well-known dynamical invariants such as Lyapunov exponents and mutual information to conditional probabilities which are potentially more suitable for prediction than the single number produced by most predictors. Densities on typical attractors have properties, such as singularity with respect to Lebesgue measure, which make standard density estimators less useful than one would hope. In this paper we present a new method of estimating densities which can smooth in a way that tends to preserve fractal structure down to some level, and that also maintains invariance. We demonstrate with applications to real and artificial data.     

Geometric and Combinatorial Realizations of Crystal Graphs

For irreducible integrable highest weight modules of the finite and affine Lie algebras of type A and D, we define an isomorphism between the geometric realization of the crystal graphs in terms of irreducible components of Nakajima quiver varieties and the combinatorial realizations in terms of Young tableaux and Young walls. For type A_n⁽¹⁾, we extend the Young wall construction to arbitrary level, describing a combinatorial realization of the crystals in terms of new objects which we call Young pyramids. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary 16G10, 17B37. Alistair Savage: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada and was partially conducted by the author for the Clay Mathematics Institute.     

Stereoselective reductase-catalysed deoxygenation of sulfoxides in aerobic and anaerobic bacteria

Direct and indirect evidence, of unexpected stereoselective reductase-catalysed deoxygenations of sulfoxides, was found. The deoxygenations proceeded simultaneously, with the expected dioxygenase-catalysed asymmetric sulfoxidation of sulfides, during some biotransformations with the aerobic bacterium Pseudomonas putida UV4. Stereoselective reductase-catalysed asymmetric deoxygenation of racemic alkylaryl, dialkyl and phenolic sulfoxides was observed, without evidence of the reverse sulfoxidation reaction, using anaerobic bacterial strains. A purified dimethyl sulfoxide reductase, obtained from the intact cells of the anaerobic bacterium Citrobacter braakii DMSO 11, yielded, from the corresponding racemates, enantiopure alkylaryl sulfoxide and thiosulfinate samples.     

Primary amido substituted diborane4 compounds and imidodiborate4 anions

Treatment of the diborane(4) compound B(2)(NMe(2))(4) with aniline or 2,6-dimethylaniline results in the primary amido compounds B(2)(NHR)(4)(R = Ph, 2,6-Me(2)C(6)H(3)); subsequent treatment with n-BuLi in toluene in each case affords the first examples of anionic imidodiborates namely Li(4)(thf)(6)B(2)(NPh)(4) and Li(4)(thf)(4)B(2)(N-2,6-Me(2)C(6)H(3))(4); all complexes have been characterised crystallographically.     

Selective, controllable, and reversible aggregation of polystyrene latex microspheres via DNA hybridization

The directed three-dimensional self-assembly of microstructures and nanostructures through the selective hybridization of DNA is the focus of great interest toward the fabrication of new materials. Single-stranded DNA is covalently attached to polystyrene latex microspheres. Single-stranded DNA can function as a sequence-selective Velcro by only bonding to another strand of DNA that has a complementary sequence. The attachment of the DNA increases the charge stabilization of the microspheres and allows controllable aggregation of microspheres by hybridization of complementary DNA sequences. In a mixture of microspheres derivatized with different sequences of DNA, microspheres with complementary DNA form aggregates, while microspheres with noncomplementary sequences remain suspended. The process is reversible by heating, with a characteristic "aggregate dissociation temperature" that is predictably dependent on salt concentration, and the evolution of aggregate dissociation with temperature is observed with optical microscopy.     

Determination of total arsenic in serum and packed cellsof patients with renal insufficiency

In order to investigate the arsenic level in serum and packed cells of patients with renal insufficiency, total arsenic (As) concentrations were determined with hydride generation atomic absorption spectrometry (HGAAS) in serum (S) and packed cells (PC) of 31 non-dialyzed patients. The accuracy of the method was tested by the analysis of arsenic in 3 certified reference materials. Patients showed a three-fold increase of arsenic concentrations in serum and a two-fold increase of arsenic in packed cells compared with controls. Patients (n=10) with higher serum creatinine (>2.0 mg/dL), urea (>0.70 g/L) and urinary protein (mean+/-SD: 1.12+/-0.82 g/L) showed higher arsenic concentrations (5.8+/-3.3 microg/L in serum and 18.0+/-16.7 microg/kg in packed cells) compared with those with lower creatinine (<1.6 mg/dL), urea (<0.6 g/L) and urinary protein (mean+/-SD: 0.27+/-0.82 g/L) (n=16, serum arsenic 1.2+/-1.2 microg/L, packed cells arsenic 2.6+/-1.9 microg/kg). The significant differences (both p < 0.001) in S and PC-arsenic levels of patients in group I and II implies a relationship between the arsenic level and the degree of chronic renal insufficiency.     

Mixed-metal cluster chemistry. 26 . Proclivity for “all-terminal” or “plane-of-bridging-carbonyls” ligand disposition in tungsten-triiridium clusters

Reaction of WH(CO)₃(η-C₅Me₅) with IrCl(CO)₂(4-H₂NC₆H₄Me) affords WIr₃(μ-CO)₃(CO)₈(η-C₅Me₅) in low yield. A structural study reveals a WIr₂-centred plane of bridging carbonyls, in contrast to the crystal structure of WIr₃(CO)₁₁(η-C₅H₅) (all-terminal carbonyl distribution). DFT calculations reveal an increasing proclivity to adopt an all-terminal CO disposition for clusters MIr₃(CO)₁₁(η-C₅H₅) in the gas phase on proceeding from M=Cr to Mo and then W, consistent with structural studies in the solid state for which the tungsten-containing cluster is the only all-terminal example. Increasing electron donation from the ligands in the tungsten system (either from phosphine substitution or cyclopentadienyl permethylation) suffices to impose a plane of bridging carbonyls in the ground state structure. ¹³C NMR fluxionality studies reveal that CO exchange mechanism(s) for WIr₃(CO)₁₁(η-C₅H₅) and the related tetrahedral cluster W₂Ir₂(CO)₁₀(η-C₅H₅)₂ are very fast and involve all carbonyls on the clusters. DFT calculations on MIr₃(CO)₁₁(η-C₅H₅) (M=Cr, Mo) substantiate a ‘merry-go-round’ mechanism for carbonyl scrambling in these systems, a result which is consistent with the scrambling behaviour seen in the NMR fluxionality studies on the W-containing congener.     

Synthesis, structure, and kinetics and stereochemistry of base-catalyzed hydrolysis of meso- and rac-[Co2(tmpdtne)Cl2]4+, bis(pentaamine) complexes devoid of deprotonatable NH centers

The bis(pentadentate) ligand tmpdtne binds two Co(II) centers, and the entity is readily oxidized to the dicobalt(III) derivative [Co(2)(tmpdtne)Cl(2)](4+) which has been separated into two isomeric forms. NMR studies establish these as meso and rac isomers arising from the different or same absolute configurations for the asym configuration about each Co(III) center. Each dinuclear ion base hydrolyses to the dihydroxo derivative [Co(2)(tmpdtne)(OH)(2)](4+) with retained asym configurations about each metal ion and also retained rac or meso configurations. The kinetics for the stepwise loss of the two Cl(-) ligands is uniphasic, and data are presented to show that the loss of the first chloride is rate determining and is followed by very rapid intramolecular and loss of the second Cl(-) via a hydroxo-bridged species to yield the observed dihydroxo derivative. Meso and rac forms of the latter have been crystallized. The X-ray crystal structure of the rac-dihydroxo complex is reported, and it establishes the configurations of all the complexes reported. The (1)H NMR spectra for the hydroxo ions show very high field Co-OH resonances (ca. delta-0.5 ppm) not observed previously for such ions, and this result is discussed in the context of published (1)H NMR data for bridged Co-OH-Co species. The base hydrolysis kinetics for the dichloro ions are first order in [OH(-)], and deprotonation at an alpha-CH(2) center (alpha to a pyridyl) is identified as the source of the catalysis, since there is no NH center available for deprotonation on the ligand. These data further support the new pseudoaminate base hydrolysis mechanism first reported in 2003. The values of k(OH) for the second-order base-catalyzed reaction are ca. 4.0 M(-1) s(-1) for both the rac and meso isomers, and these results are discussed in terms of the increased acidities of these 4+ cations compared to their 2+ ion counterparts.     

Mass spectra of some naturally occurring stictane triterpenoids and their trimethylsilyl derivatives

The mass spectra of ten di- and tri-oxygenated stictane triterpenoids and their trimethylsilyl derivatives have been studied in detail. Gas chromatography mass spectrometry of the TMS derivatives on OV-17 and OV-101 columns provided useful separations and identifications of mixtures of stictanes triterpenoids in lichen extracts. The major fragmentations involve ring C, but specific cleavages in rings A, B and E are also observed which allow the substitution pattern of the skeleton to be readily determined. The formation of an intense [M – C₅H₁₁?]⁺ ion for 22α-OTMS stictanes appears specific to their ring E configuration.     

Comparison of linear-scaling semiempirical methods and combined quantum mechanical/molecular mechanical methods for enzymic reactions. II. An energy decomposition analysis

QM/MM methods have been developed as a computationally feasible solution to QM simulation of chemical processes, such as enzyme-catalyzed reactions, within a more approximate MM representation of the condensed-phase environment. However, there has been no independent method for checking the quality of this representation, especially for highly nonisotropic protein environments such as those surrounding enzyme active sites. Hence, the validity of QM/MM methods is largely untested. Here we use the possibility of performing all-QM calculations at the semiempirical PM3 level with a linear-scaling method (MOZYME) to assess the performance of a QM/MM method (PM3/AMBER94 force field). Using two model pathways for the hydride-ion transfer reaction of the enzyme dihydrofolate reductase studied previously (Titmuss et al., Chem Phys Lett 2000, 320, 169-176), we have analyzed the reaction energy contributions (QM, QM/MM, and MM) from the QM/MM results and compared them with analogous-region components calculated via an energy partitioning scheme implemented into MOZYME. This analysis further divided the MOZYME components into Coulomb, resonance and exchange energy terms. For the model in which the MM coordinates are kept fixed during the reaction, we find that the MOZYME and QM/MM total energy profiles agree very well, but that there are significant differences in the energy components. Most significantly there is a large change (approximately 16 kcal/mol) in the MOZYME MM component due to polarization of the MM region surrounding the active site, and which arises mostly from MM atoms close to (<10 A) the active-site QM region, which is not modelled explicitly by our QM/MM method. However, for the model where the MM coordinates are allowed to vary during the reaction, we find large differences in the MOZYME and QM/MM total energy profiles, with a discrepancy of 52 kcal/mol between the relative reaction (product-reactant) energies. This is largely due to a difference in the MM energies of 58 kcal/mol, of which we can attribute approximately 40 kcal/mol to geometry effects in the MM region and the remainder, as before, to MM region polarization. Contrary to the fixed-geometry model, there is no correlation of the MM energy changes with distance from the QM region, nor are they contributed by only a few residues. Overall, the results suggest that merely extending the size of the QM region in the QM/MM calculation is not a universal solution to the MOZYME- and QM/MM-method differences. They also suggest that attaching physical significance to MOZYME Coulomb, resonance and exchange components is problematic. Although we conclude that it would be possible to reparameterize the QM/MM force field to reproduce MOZYME energies, a better way to account for both the effects of the protein environment and known deficiencies in semiempirical methods would be to parameterize the force field based on data from DFT or ab initio QM linear-scaling calculations. Such a force field could be used efficiently in MD simulations to calculate free energies.