The maximal solution of a restricted subadditive inequality in numerical analysis

For a fixed non-negative integerp, letU _2p = {U _2p (n)},n ≥ 0, denote the sequence that is defined by the initial conditionsU _2p (0) =U _2p (1) =U _2p (2) = =U _2p (2p) = 1 and the restricted subadditive recursion $$U_{2p} (n + 2p + 1) = \mathop {\min }\limits_{0 \leqslant l \leqslant p} (U_{2p} (n + l) + U_{2p} (n + 2p - l)),n \geqslant 0$$ U _2p is of importance in the theory of sequential search for simple real zeros of real valued continuous 2p-th derivatives In this paper, several closed form expressions forU _2p (n), n > 2p, are determined, thereby providing insight into the structure ofU _2p Two of the properties thus illuminated are (a) the existence of exactlyp + 1 limit points (1 + 1/(p + 1 +i), 0 ≤i ≤p) of the associated sequence {U _2p (n + 1)/U _2p (n)},n ≥ 0, and (b) the relevance toU _2p of the classic number theoretic function ord     

Partitioning Boolean lattices into chains of subsets

We prove that the Boolean lattice of all subsets of an n-set can be partitioned into chains of size four if and only if n9.Research supported in part by N.S.F. grant DMS-8401281.Research supported in part by N.S.F. grant DMS-8406451.     

Thermodynamics of the dissociation of 2-aminopyridinium ion in synthetic seawater and a standard for pH in marine systems

Buffer solutions composed of 2-aminopyridinium chloride and 2-aminopyridine in synthetic seawater are useful as a supplement to buffers of Tris (pH 8.2) and Bis (pH 8.8) in standardizing measurements of hydrogen ion concentration (pm _H or pH(SWS)) in oceanography. The dissociation constant of 2-amino-pyridinium ion over the range of salinities (S) from 30 to 40 has now been determined from the emf of cells without liquid junction at eight temperatures (T) from 278.15 to 313.15 K. The results fit the equation pK=2498.31/T–15.3274+2.4050 lnT+S(0.012928–2.9417×10^–5T) with a standard deviation of 0.0023. Thermodynamic constants for the dissociation process and standard reference values of pm _H and pH(SWS) were derived from the data. The pm _H of the buffer consisting of 2-aminopyridinium chloride and 2-aminopyridine (each 0.04 molal) in synthetic seawater of salinity 35 varies from 7.356 at 278.15 K to 6.601 at 313.15 K.     

Über Pterinchemie. 79. Mitteilung. 13C-und 15N-NMR.-spektroskopische Untersuchungen an der (6RS)-5, 10-Methylen-5, 6, 7, 8-tetrahydro-L-folsäure; eindeutiger Beweis ihrer Struktur

¹³C- and ¹⁵N-NMR.-Spectroscopic Studies on (6RS)-5, 10-Methylene-5, 6, 7, 8-tetrahydro-L -folic Acid; Unequivocal Proof of thier Structure The existence of a 5, 10-methylene bridge in the title compound has been established with the help of ¹⁵N-NMR, and ¹³C-NMR. spectroscopy. The simultaneous coupling of ¹³C(11) with ¹⁵N(5) and ¹⁵N(10) proves that C(11) must be bound as a methylene bridge to both N-atoms. Some other aspects of the NMR. spectra of the compound are discussed. Synthesis of the title product with the label ¹⁵N is described.     

Myoglobin-CO substate structures and dynamics: multidimensional vibrational echoes and molecular dynamics simulations

Spectrally resolved infrared stimulated vibrational echo data were obtained for sperm whale carbonmonoxymyoglobin (MbCO) at 300 K. The measured dephasing dynamics of the CO ligand are in agreement with dephasing dynamics calculated with molecular dynamics (MD) simulations for MbCO with the residue histidine-64 (His64) having its imidazole epsilon nitrogen protonated (N(epsilon)-H). The two conformational substate structures B(epsilon) and R(epsilon) observed in the MD simulations are assigned to the spectroscopic A(1) and A(3) conformational substates of MbCO, respectively, based on the agreement between the experimentally measured and calculated dephasing dynamics for these substates. In the A(1) substate, the N(epsilon)-H proton and N(delta) of His64 are approximately equidistant from the CO ligand, while in the A(3) substate, the N(epsilon)-H of His64 is oriented toward the CO, and the N(delta) is on the surface of the protein. The MD simulations show that dynamics of His64 represent the major source of vibrational dephasing of the CO ligand in the A(3) state on both femtosecond and picosecond time scales. Dephasing in the A(1) state is controlled by His64 on femtosecond time scales, and by the rest of the protein and the water solvent on longer time scales.     

Fragmentation of Homoallylic Alkoxides. Preparation of 1-(3′-cyclopentenyl)-2-alkanones from 2-substituted bicyclo[2.2.1]hept-5-en-2-ols

The potassium 2-substituted bicyclo[2.2.1]hept-5-en-2-alkoxides derived from alcohols 2–9 at 30° in hexamethylphosphoric triamide (HMPA) afford 1-(3′-cyclopentenyl)-2-alkanones 10–19 via heterolytic C(1), C(2)-allylic bond cleavage in the substrate alkoxide followed by intramolecular protonation of the resultant transient allylic anion.     

Synthesis of functionalized rab GTPases by a combination of solution- or solid-phase lipopeptide synthesis with expressed protein ligation

Prenylated proteins with non-native functionalities are generally very difficult to obtain by recombinant or enzymatic means. The semisynthesis of preparative amounts of prenylated Rab guanosine triphosphatases (GTPases) from recombinant proteins and synthetic prenylated peptides depends largely on the availability of functionalised prenylated peptides corresponding to the proteins' native structure or modifications thereof. Here, we describe and compare solution-phase and solid-phase strategies for the generation of peptides corresponding to the prenylated C terminus of Rab7 GTPase. The solid-phase with utilisation of a hydrazide linker emerges as the more favourable approach. It allows a fast and practical synthesis of pure peptides and gives a high degree of flexibility in their modification. To facilitate the analysis of semisynthetic proteins, the synthesised peptides were equipped with a fluorescent group. Using the described approach, we introduced fluorophores at several different positions of the Rab7 C terminus. The position of the incorporated fluorescent groups in the peptides did not influence the protein-ligation reaction, as the generated peptides could be ligated onto thioester-tagged Rab7. However, it was found that the positioning of the fluorescent group had an influence on the functionality of the Rab7 proteins; analysis of the interaction of the semisynthetic Rab7 proteins with REP (Rab escort protein) and GDI (guanosine diphosphate dissociation inhibitor) molecules revealed that modification of the peptide side chains or of the C-terminal isoprenoid did not significantly interfere with complex formation. However, functionalisation of the C terminus was found to have an adverse effect on complex formation and stability, possibly reflecting low structural flexibility of the Rab GDI/REP molecules in the vicinity of the lipid-binding site.     

Luminol Oxidation by Hydrogen Peroxide Catalyzed by Tobacco Anionic Peroxidase: Steady-State Luminescent and Transient Kinetic Studies

The properties of a newly isolated anionic tobacco peroxidase from transgenic tobacco plants overexpressing the enzyme have been studied with respect to the chemiluminescent reaction of luminol oxidation. These were compared to the properties of horseradish peroxidase in the cooxidation of luminol and p -iodophenol, the enhanced chemiluminescence reaction. The pH, luminol and hydrogen peroxide concentrations were optimized for maximum sensitivity using the tobacco enzyme. The detection limit for the latter under the optimal conditions (2.5 m M luminol, 2 m M hydrogen peroxide, 100 m M Naborate buffer, pH 9.3) was about 0.1 p M , which is at least five times lower than that for horseradish peroxidase in enhanced chemiluminescence with p -iodophenol. The rate constants for the elementary steps of the enzyme-catalyzed reaction have been determined: k ₁= 4.9 × 10⁶ M ⁻¹ s¹, k ₂= 7.3 × 10⁶ M ⁻¹ s⁻¹, k ³= 2.1 × 10⁶ M ⁻¹ s⁻¹ (pH 9.3). The similarity of these rate constants is unusual for plant peroxidases. The high catalytic activity of tobacco peroxidase in the luminescent reaction is explained by the high reactivity of its Compound II toward luminol and the high stability of the holoenzyme with respect to heme dissociation. This seems to be a unique property of this particular enzyme among other plant peroxidases.     

A novel one-pot method for the preparation of pyrazoles by 1,3-dipolar cycloadditions of diazo compounds generated in situ

A convenient one-pot procedure for the preparation of pyrazoles by 1,3-dipolar cycloaddition of diazo compounds generated in situ has been developed. Diazo compounds derived from aldehydes were reacted with terminal alkynes to furnish regioselectively 3,5-disubstituted pyrazoles. Furthermore, the reaction of N-vinylimidazole and diazo compounds derived from aldehydes gave exclusively 3-substituted pyrazoles in a one-pot process.     

The Raman active internal vibrational modes of potassium nitrate

The Raman active internal vibrational modes of single crystal orthorhombic potassium nitrate have been studied in various polarizations. The full multiplet structure predicted by factor group analysis for the v₂ and v₃ regions has been observed for the first time. The expected site group splitting of the v₄ mode was not observed and can be assumed to be less than 0.5 cm^?1.