Density lower bound estimate for local minimizer of free interface problem with volume constraint

We prove a density lower bound for some functionals involving bulk and interfacial energies. The bulk energies are convex functions with p-power growth not subjected to any further structure conditions. The interface \(\partial E\) is the boundary of a set \(E\subset \Omega \) such that \(|E|=d\) is prescribed. Then we get \(\mathcal {H}^{n-1}((\partial E{\setminus }\partial E^*)\cup \Omega )=0\).

     

The Impact of pH on Catalytically Critical Protein Conformational Changes: The Case of the Urease,a Nickel Enzyme

Urease uses a cluster of two Ni^II ions to activate a water molecule for urea hydrolysis. The key to this unsurpassed enzyme is a change in the conformation of a flexible structural motif, the mobile flap, which must be able to move from an open to a closed conformation to stabilize the chelating interaction of urea with the Ni^II cluster. This conformational change brings the imidazole side chain functionality of a critical histidine residue, αHis323, in close proximity to the site that holds the transition state structure of the reaction, facilitating its evolution to the products. Herein, we describe the influence of the solution pH in modulating the conformation of the mobile flap. High-resolution crystal structures of urease inhibited in the presence of N-(n-butyl)phosphoric triamide (NBPTO) at pH 6.5 and pH 7.5 are described and compared to the analogous structure obtained at pH 7.0. The kinetics of urease in the absence and presence of NBPTO are investigated by a calorimetric assay in the pH 6.0–8.0 range. The results indicate that pH modulates the protonation state of αHis323, which was revealed to have pK_a=6.6, and consequently the conformation of the mobile flap. Two additional residues (αAsp224 and αArg339) are shown to be key factors for the conformational change. The role of pH in modulating the catalysis of urea hydrolysis is clarified through the molecular and structural details of the interplay between protein conformation and solution acidity in the paradigmatic case of a metalloenzyme.     

Repdigits as sums of three Pell numbers

In this paper, all base 10 repdigits expressible as sums of three Pell numbers are found.     

Squares and factorials in products of factorials

We study a question of Erd?s and Graham on products of factorials being a square or again a factorial.     

Synthesis and Biological Evaluation of RGD Peptidomimetic–Paclitaxel Conjugates Bearing Lysosomally Cleavable Linkers

Two small‐molecule–drug conjugates (SMDCs, 6 and 7 ) featuring lysosomally cleavable linkers (namely the Val–Ala and Phe–Lys peptide sequences) were synthesized by conjugation of the α_vβ₃‐integrin ligand cyclo[DKP–RGD]‐CH₂NH₂ ( 2 ) to the anticancer drug paclitaxel (PTX). A third cyclo[DKP–RGD]–PTX conjugate with a nonpeptide “uncleavable” linker ( 8 ) was also synthesized to be tested as a negative control. These three SMDCs were able to inhibit biotinylated vitronectin binding to the purified α_Vβ₃‐integrin receptor at nanomolar concentrations and showed good stability at pH 7.4 and pH 5.5. Cleavage of the two peptide linkers was observed in the presence of lysosomal enzymes, whereas conjugate 8 , which possesses a nonpeptide “uncleavable” linker, remained intact under these conditions. The antiproliferative activities of the conjugates were evaluated against two isogenic cell lines expressing the integrin receptor at different levels: the acute lymphoblastic leukemia cell line CCRF‐CEM (α_Vβ₃?) and its subclone CCRF‐CEM α_Vβ₃ (α_Vβ₃+). Fairly effective integrin targeting was displayed by the cyclo[DKP–RGD]–Val–Ala–PTX conjugate ( 6 ), which was found to differentially inhibit proliferation in antigen‐positive CCRF‐CEM α_Vβ₃ versus antigen‐negative isogenic CCRF‐CEM cells. The total lack of activity displayed by the “uncleavable” cyclo[DKP–RGD]–PTX conjugate ( 8 ) clearly demonstrates the importance of the peptide linker for achieving the selective release of the cytotoxic payload.     

KAM for Reversible Derivative Wave Equations

We prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations.     

New relationship of displacement signal at quadrant photodiode: Control signal analysis and simulation of a laser tracker

New relationship of displacement signal using opposite sectors on a quadrant photodiode is derived. Standard and new displacement signals are analyzed in details. Through MATLAB^® laser tracking simulation models, based on common and suggested approaches, detailed analysis is performed, and it is shown that better results for the new relationship signal processing are obtained. Within new relationship of displacement signal, the sensitivity of the system to the displacement of the spot increases and, hence, provides better accuracy in positioning up to 30%.