$${\mathcal{M}}$$ -decomposability and symmetric unimodal densities in one dimension

In this paper, we introduce the notion of -decomposability of probability density functions in one dimension. Using -decomposability, we derive an inequality that applies to all symmetric unimodal densities. Our inequality involves only the standard deviation of the densities concerned. The concept of -decomposability can be used as a non-parametric criterion for mode-finding and cluster analysis.     

A continuous family of partition statistics equidistributed with length

This article investigates a remarkable generalization of the generating function that enumerates partitions by area and number of parts. This generating function is given by the infinite product _∏i?11/(1−tqi). We give uncountably many new combinatorial interpretations of this infinite product involving partition statistics that arose originally in the context of Hilbert schemes. We construct explicit bijections proving that all of these statistics are equidistributed with the length statistic on partitions of n. Our bijections employ various combinatorial constructions involving cylindrical lattice paths, Eulerian tours on directed multigraphs, and oriented trees.     

On the hardness of sampling independent sets beyond the tree threshold

We consider local Markov chain Monte–Carlo algorithms for sampling from the weighted distribution of independent sets with activity λ, where the weight of an independent set I is λ^|I|. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and λ < λ _c (d), where λ _c (d) is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the d-regular infinite tree. We show that for d ≥ 3, λ just above λ _c (d) with high probability over d-regular bipartite graphs, any local Markov chain Monte–Carlo algorithm takes exponential time before getting close to the stationary distribution. Our results provide a rigorous justification for “replica” method heuristics. These heuristics were invented in theoretical physics and are used in order to derive predictions on Gibbs measures on random graphs in terms of Gibbs measures on trees. A major theoretical challenge in recent years is to provide rigorous proofs for the correctness of such predictions. Our results establish such rigorous proofs for the case of hard-core model on bipartite graphs. We conjecture that λ _c is in fact the exact threshold for this computational problem, i.e., that for λ > λ _c it is NP-hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph.     

Rook-by-rook rook theory: Bijective proofs of rook and hit equivalences

Suppose μ and ν are integer partitions of n, and N>n. It is well known that the Ferrers boards associated to μ and ν are rook-equivalent iff the multisets [μ_i+i:1iN] and [ν_i+i:1iN] are equal. We use the Garsia–Milne involution principle to produce a bijective proof of this theorem in which non-attacking rook placements for μ are explicitly matched with corresponding placements for ν. One byproduct is a direct combinatorial proof that the matrix of Stirling numbers of the first kind is the inverse of the matrix of Stirling numbers of the second kind. We also prove q-analogues and p,q-analogues of these results. We also use the Garsia–Milne involution principle to show that for any two rook boards B and B^′, if B and B^′ are bijectively rook-equivalent, then B and B^′ are bijectively hit-equivalent.     

Biosynthesis of 2-deoxystreptamine-containing aminoglycoside antibiotics

The 2-deoxystreptamine-containing aminoglycosides are an important class of clinically valuable antibiotics. A deep understanding of the biosynthesis of these natural products is required to enable efforts to rationally manipulate and engineer the biological production of novel aminoglycosides. This review discusses the development of our biosynthetic knowledge over the past half-century, with emphasis on the relatively recent contributions of molecular biology to the elucidation of these biosynthetic pathways.     

Potential-, pH-, and isotope-dependence of proton-coupled electron transfer of an osmium aquo complex attached to an electrode

An osmium complex, [OsII(bpy)2(4-aminomethylpyridine)(H2O)]2+, is attached to a mixed self-assembled monolayer on a gold electrode. The complex exhibits 1-electron, 1-proton redox chemistry (OsIII(OH)/OsII(H2O)) at pHs and potentials that are experimentally accessible with gold electrodes in aqueous electrolytes. The thermodynamic behavior and kinetic behavior of the system are investigated as a function of pH in both H2O and D2O. The two formal potentials and two pKa values are relatively constant for two chain lengths in H2O and in D2O. The standard rate constants at all pHs are strongly and uniformly affected by chain length, indicating that electronic coupling is the dominant factor controlling the rate of electron transfer. In both H2O and D2O, the standard rate constant is weakly dependent on the pH, exhibiting a minimum value midway between the pKa values. The kinetic isotope effect is small; standard rate constants decrease by roughly a factor of 2 in D2O over a wide range of pHs, but not at the more acidic pHs. The Tafel plots and plots of the transfer coefficient vs overpotential are asymmetrical at all pHs. These results are interpreted in terms of a larger reorganization energy for the OsII species and a smaller reorganization energy for the OsIII species. The OsIII reorganization energy is constant at all pHs in both H2O and D2O. The pH dependence of the OsII reorganization energy accounts for some or all of the pH dependence of the standard rate constant in H2O and D2O. The data deviate substantially from predictions of the stepwise proton-coupled electron-transfer mechanism. The observation of a kinetic isotope effect supports the concerted mechanism.     

Targeting retroviral Zn finger-DNA interactions: a small-molecule approach using the electrophilic nature of trans-platinum-nucleobase compounds

Noncovalent interactions are ubiquitous in ternary systems involving metal ions, DNA/RNA, and proteins and represent a structural motif for design of selective inhibitors of biological function. This contribution shows that small molecules containing platinated purine nucleobases mimic the natural DNA(RNA)-tryptophan recognition interaction of zinc finger peptides, specifically the C-terminal finger of HIV NCp7 protein. Interaction with platinum results in Zn ejection from the peptide accompanied by loss of tertiary structure. Targeting the NCp7-DNA interaction for drug design represents a conceptual advance over electrophiles designed for chemical attack on the zinc finger alone. These results demonstrate examples of a new platinum structural class targeting specific biological processes, distinct from the bifunctional DNA-DNA binding of cytotoxic agents like cisplatin. The results confirm the validity of a chemical biological approach for metallodrug design for selective ternary DNA(RNA)-protein interactions.     

Freezing halide ion solutions and the release of interhalogens to the atmosphere

The effect of freezing on a variety of acidified and neutral, nitrite ion and halide-containing mixtures has been investigated using UV/vis spectroscopy. Several trihalide ions were formed and monitored, including I(2)Cl(-), I(2)Br(-), ICl(2)(-) and IBr(2)(-). A mechanism to explain the observations is given in terms of steps involving INO and the nitroacidium ion, [H(2)ONO](+). The transformation of sea salt components to specific trihalide ions by freezing represents a potentially important process in a polar atmospheric context. This is because the dichloro- and dibromo-trihalide ions can release chlorine- and bromine-containing gases, which are key intermediates in ozone destruction.     

Utilizing relativistic effective core potentials for accurate calculations of molecular polarizabilities on transition metal compounds

The requirements necessary to extend an ECP basis set for the calculation of electric and linear optical properties to the transition metals are studied. Previously an augmentation of the SBK basis set for 39 elements with s and p electron only valences (H-Rn, excluding Ga, In, and Tl) [J. Comput. Chem., 2005, 26, 1464-1471] was presented. In this work, electric dipole moments, polarizabilities, and anisotropies of selected metal hydrides, sulfides, and bromides, cisplatin, and the Fe, Ru, and Os metallocene derivatives along with many other systems are calculated and discussed. ECP calculations of molecules containing 3d and 4d metal centers among main group atoms have good agreement, often within 1-2% of the all-electron result at the time-dependent Hartree-Fock (TDHF)/Sadlej level of theory. Molecules with a 5d metal center have a large difference from and are more accurate than the all-electron results due to the inclusion of relativistic effects in the ECPs. The polarizability as calculated with and without ECPs of metallic clusters and surfaces is increasingly different as atomic number increases, again due to a lack of relativistic effects in the all-electron calculations. The augmented ECP calculations are consistent with relativistic all-electron results, while the Sadlej calculations are consistent with other nonrelativistic results. Both relativistic and basis set effects are less noticeable when the metal is in a formally positive state.