Balancing the n-Cube: A Census of Colorings

Weights of 1 or 0 are assigned to the vertices of the n-cube in n-dimensional Euclidean space. Such an n-cube is called balanced if its center of mass coincides precisely with its geometric center. The seldom-used n-variable form of Pólya's enumeration theorem is applied to express the number N _{n, 2k} of balanced configurations with 2k vertices of weight 1 in terms of certain partitions of 2k. A system of linear equations of Vandermonde type is obtained, from which recurrence relations are derived which are computationally efficient for fixed k. It is shown how the numbers N _{n, 2k} depend on the numbers A _{n, 2k} of specially restricted configurations. A table of values of N _{n, 2k} and A _{n, 2k} is provided for n = 3, 4, 5, and 6. The case in which arbitrary, nonnegative, integral weights are allowed is also treated. Finally, alternative derivations of the main results are developed from the perspective of superposition.     

H-1 NMR and Conformations of Halogenated 1,4-Dioxanes and 2,3-Dihydro-1,4-Dioxins

The conformations of halogenated 1,4-dioxanes have generally been studied by X-ray diffraction^1,2 in order to obtain values for the ring dihedral angles. The simple and widely used R value technique^3,4 requires two vicinal coupling constants, J-trans and J-cis, and is not useable when only one vicinal J value results, as in the halogenated dioxanes 1–5 and dihydrodioxins 7–9 listed in Tables 1 and 2.     

On the distribution of certain algebraic integers

Mathematische Zeitschrift -     

Exploring the substrate specificity of OxyB, a phenol coupling P450 enzyme involved in vancomycin biosynthesis

OxyB is a cytochrome P450 enzyme that catalyzes the first oxidative phenol coupling reaction during vancomycin biosynthesis. The preferred substrate is a linear peptide linked as a C-terminal thioester to a peptide carrier protein (PCP) domain of the glycopeptide antibiotic non-ribosomal peptide synthetase. Previous studies have shown that OxyB can efficiently oxidize a model hexapeptide-PCP conjugate (R-Leu(1)-R-Tyr(2)-S-Asn(3)-R-Hpg(4)-R-Hpg(5)-S-Tyr(6)-S-PCP) (Hpg = 4-hydroxyphenylglycine) into a macrocyclic product by phenolic coupling of the aromatic rings in residues-4 and -6. In this work, the substrate specificity of OxyB has been explored using a series of N-terminally truncated peptides related in sequence to this model hexapeptide-PCP conjugate. Deletion of one or three residues from the N-terminus afforded a penta- (Ac-Tyr-Asn-Hpg-Hpg-Tyr-S-PCP) and a tri- (Ac-Hpg-Hpg-Tyr-S-PCP) peptide that were also efficiently transformed into the corresponding macrocyclic cross-linked product by OxyB. The tripeptide, representing the core of the macrocycle in vancomycin created by OxyB, is thus sufficient, as a thioester with the PCP domain, for phenol coupling to occur. The related tetrapeptide-PCP thioester was not cyclized by OxyB, neither was a related model hexapeptide containing tryptophan in place of tyrosine-6, nor were tripeptides (related to the natural product K-13) with the sequence Ac-Tyr-Tyr-Tyr-S-PCP cross-linked by OxyB.     

Buffer Standards for the Physiological pH of Zwitterionic Compounds,MOBS and TABS from 5 to 55°C

The values of the second dissociation constant, pK₂, and related thermodynamic quantities of 4-(N-morpholino)butanesulfonic acid (MOBS) and N-tris(hydroxymethyl)-4-aminobutanesulfonic acid (TABS) have already been reported over the temperature range 5–55°C including 37{°}C. This paper reports the pH values of twelve equimolal buffer solutions at designated pH (s) with the following compositions: (a) mixtures of MOBS (0.05 mol-kg^–1) + NaMOBS (0.05 mol-kg^–1); (b) MOBS (0.08 mol-kg^–1) + NaMOBS (0.08 mol-kg^–1); (c) MOBS (0.08 mol-kg^–1) + NaMOBS (0.08 mol-kg^–1) + NaCl (0.08 mol-kg^–1); (d) TABS (0.05 mol-kg^–1) + NaTABS (0.05 mol-kg^–1); and (e) TABS (0.08 mol-kg^–1) + NaTABS (0.08 mol-kg^–1); and (f) TABS (0.08 mol-kg^–1) + NaTABS (0.08 mol-kg^–1) + NaCl (0.08 mol-kg^–1). Two buffer solutions have ionic strengths I= 0.05 mol-kg^–1, another two have I=0.08 mol-kg^–1, and the remaining two buffer solutions have I= 0.16 mol-kg^–1, which is close to that of the clinical fluids (blood serum). These buffers have been recommended as a useful pH standard for the measurements of physiological solutions. Conventional pH values of all six buffer solutions from 5–55°C, as well as those obtained from the liquid junction potential correction at 25 and 37{°}C have been calculated. The flowing-junction calomel cell has been utilized to measure E_j, the liquid junction potential.     

Controlled synthesis of (S,S)-2,7-diaminosuberic acid: a method for regioselective construction of dicarba analogues of multicystine-containing peptides

A method to facilitate regioselective formation of multiple dicarba isosteres of cystine is described. A sequence of ruthenium-catalyzed cross metathesis and rhodium-catalyzed hydrogenation of nonproteinaceous allylglycine derivatives has been developed to achieve high-yielding and unambiguous formation of diaminosuberic acid derivatives. Allylglycine derivatives readily undergo ruthenium-catalyzed metathesis and hydrogenation to yield diaminosuberic acid derivatives in near quantitative yield. Under the same experimental conditions, prenylglycine was found to be inert to both Grubbs' and Wilkinson's catalyzed metathesis and hydrogenation, respectively, but was readily activated for metathesis via cross metathesis with Z-butene. Subsequent cross metathesis of the metathesis-formed crotylglycine derivative, followed by hydrogenation, yielded the second diaminosuberic acid derivative in excellent yield.     

Synthesis and structures of (13-hydroxylamino-5,7,7,13-tetramethyl-1,4,8,11-tetraazacyclotetradec-4-ene-κ)nickel(II) tetrachlorozincate(II) and (13-azonium-5,5,7,13-tetramethyl-1,4,8,11-tetraazacyclotetradecane-κ)nickel(II) tetrachlorozincate(II) chloride 2.25 water

Zinc/acid reduction of the nitro function of (5,7,7,13-tetramethyl-13-nitro-1,4,8,11-tetraazacyclotetradec-4-ene)nickel(II) yields (13-hydroxylamino-5,7,7,13-tetramethyl-1,4,8,11-tetraazacyclotetradec-4-ene)nickel(II), isolated as the tetrachlorozincate salt, for which the structure is reported. This has singlet ground state nickel(II) in square-planar coordination by the three amine and the imine nitrogen atoms of the macrocycle. There is no interaction between the nickel(II) ion and the axially oriented hydroxylamino group.Zinc/acid reduction of the nitro function of (5,5,7,13-tetramethyl-13-nitro-1,4,8,11-tetraazacyclotetradecane)nickel(II) yields (13-amino-5,5,7,13-tetramethyl-1,4,8,11-tetraazacyclotetradecane)nickel(II), isolated from acid solution as (13-azonium-5,5,7,13-tetramethyl-1,4,8,11-tetraazacyclotetradecane-κ⁴)nickel(II) tetrachlorozincate(II) chloride 2.25 water, for which the structure is reported. The two independent cations have singlet ground state nickel(II) in square-planar coordination by the four secondary amine nitrogen atoms of the macrocycle, with the protonated amine substituent axially oriented.The cations of both compounds show disorder in the location of the axial methyl component of the gem-dimethyl group, manifest as reversal of the trimethyl substituted 1,3-amine-imine or 1,3-diamine chelate rings, arising from co-crystallisation of enantiomers.     

Fourier transform infrared matrix-isolation analysis of acetaldehyde fragmentation products after charge exchange with Ar•+ under varied ionization density conditions

The products of the Ar^?+ charge exchange ionization of acetaldehyde have been isolated and compared with related photoionization results and computational work. Acetaldehyde has been used to assess the effect of varied ion density in the ionization region of the electron bombardment matrix isolation apparatus. The amount of acetaldehyde destruction has been measured for constant gas‐sample composition and constant ionization current for two anode geometries: a pin anode and a plate anode. For the same ionization current, a pin‐shaped anode demonstrates higher precursor molecule destruction efficiency (85%) than the plate‐shaped anode (30%), resulting in substantial effect on the yield and quantity of isolated products. When the plate anode is used, the observed infrared products correspond to matrix‐isolated carbon monoxide (CO), methane (CH₄), ketene (CH₂CO), ethynyloxy radical (HCCO), formyl radical (HCO^?), acetyl radical (CH₃CO^?), vinyl alcohol (H₂C = CH‐OH), and cationic proton‐bound dimer, Ar₂H⁺. When the pin anode is used, the same products are observed with different relative proportions and new absorption features corresponding to dicarbon monoxide (CCO) and methyl radical (CH₃^?) are observed. The surprising observation of infrared absorptions corresponding to vinyl alcohol along with low yield of products anticipated through the analysis of photoelectron–photoionization coincidence measurements suggests that the initially formed fragmentation products are able to further react within the matrix‐isolation environment to influence observed product yields. Related experiments, using the isotopomer CD₃CHO, suggest that the observed products are formed via radical–radical reactions that occur under the high pressure conditions of the matrix isolation environment. Copyright © 2011 John Wiley & Sons, Ltd.     

The problem of dating quartz 2: Synchrotron generated X-ray excited optical luminescence (XEOL) from quartz

The luminescence emission of quartz is used in optically stimulated luminescence dating (OSL), however the precise origins of the emission are unclear. A suite of quartz samples were analysed using X-ray excited optical luminescence (XEOL). Radiation dose effects were observed whereby the UV emissions (3.8 and 3.4 eV) were depleted to the benefit of the red emission (1.9–2.0 eV). Samples were excited at ∼7 keV. Understanding why some quartz emit light more brightly than others will increase the efficiency and precision of OSL analyses.     

Positivity and monotonicity properties ofC 0-semigroups. I

If exp {?tH}, exp {?tK}, are self-adjoint, positivity preserving, contraction semigroups on a Hilbert space ?=L ²(X;dμ) we write (*) $$e^{ - tH} \succcurlyeq e^{ - tK} \succcurlyeq 0$$ whenever exp {?tH}-exp {?tK} is positivity preserving for allt≧0 and then we characterize the class of positive functions for which (*) always implies $$e^{ - tf(H)} \succcurlyeq e^{ - tf(K)} \succcurlyeq 0.$$ This class consists of thef∈C ^∞(0, ∞) with $$( - 1)^n f^{(n + 1)} (x) \geqq 0,x \in (0,\infty ),n = 0,1,2, \ldots .$$ In particular it contains the class of monotone operator functions. Furthermore if exp {?tH} isL ^p (X;dμ) contractive for allp∈[1, ∞] and allt>0 (or, equivalently, forp=∞ andt>0) then exp {?tf(H)} has the same property. Various applications to monotonicity properties of Green's functions are given.