A modified Brent’s method for finding zeros of functions

Brent’s method, also known as zeroin, has been the most popular method for finding zeros of functions since it was developed in 1972. This method usually converges very quickly to a zero; for the occasional difficult functions encountered in practice, it typically takes $O(n)$ iterations to converge, where $n$ is the number of steps required for the bisection method to find the zero to approximately the same accuracy. While it has long been known that in theory Brent’s method could require as many as $O(n^2)$ iterations to find a zero, such behavior had never been observed in practice. In this paper, we first show that Brent’s method can indeed take $O(n^2)$ iterations to converge, by explicitly constructing such worst case functions. In particular, for double precision accuracy, Brent’s method takes $2{,}914$ iterations to find the zero of our function, compared to the $77$ iterations required by bisection. Secondly, we present a modification of Brent’s method that places a stricter complexity bound of $O(n)$ on the search for a zero. In our extensive testing, this modification appears to behave very similarly to Brent’s method for all the common functions, yet it remains at worst five times slower than the bisection method for all difficult functions, in sharp contrast to Brent’s method.     

Stabilized Carbenium Ions as Latent,Z‐type Ligands

Controlling the reactivity of transition metals using secondary, σ‐accepting ligands is an active area of investigation that is impacting molecular catalysis. Herein we describe the phosphine gold complexes [(o‐Ph₂P(C₆H₄)Acr)AuCl]⁺ ([ 3 ]⁺; Acr=9‐N‐methylacridinium) and [(o‐Ph₂P(C₆H₄)Xan)AuCl]⁺ ([ 4 ]⁺; Xan=9‐xanthylium) where the electrophilic carbenium moiety is juxtaposed with the metal atom. While only weak interactions occur between the gold atom and the carbenium moiety of these complexes, the more Lewis acidic complex [ 4 ]⁺ readily reacts with chloride to afford a trivalent phosphine gold dichloride derivative ( 7 ) in which the metal atom is covalently bound to the former carbocationic center. This anion‐induced Au^I/Au^III oxidation is accompanied by a conversion of the Lewis acidic carbocationic center in [ 4 ]⁺ into an X‐type ligand in 7 . We conclude that the carbenium moiety of this complex acts as a latent Z‐type ligand poised to increase the Lewis acidity of the gold center, a notion supported by the carbophilic reactivity of these complexes.     

Stabilized Carbenium Ions as Latent,Z‐type Ligands

Controlling the reactivity of transition metals using secondary, σ‐accepting ligands is an active area of investigation that is impacting molecular catalysis. Herein we describe the phosphine gold complexes [(o‐Ph₂P(C₆H₄)Acr)AuCl]⁺ ([ 3 ]⁺; Acr=9‐N‐methylacridinium) and [(o‐Ph₂P(C₆H₄)Xan)AuCl]⁺ ([ 4 ]⁺; Xan=9‐xanthylium) where the electrophilic carbenium moiety is juxtaposed with the metal atom. While only weak interactions occur between the gold atom and the carbenium moiety of these complexes, the more Lewis acidic complex [ 4 ]⁺ readily reacts with chloride to afford a trivalent phosphine gold dichloride derivative ( 7 ) in which the metal atom is covalently bound to the former carbocationic center. This anion‐induced Au^I/Au^III oxidation is accompanied by a conversion of the Lewis acidic carbocationic center in [ 4 ]⁺ into an X‐type ligand in 7 . We conclude that the carbenium moiety of this complex acts as a latent Z‐type ligand poised to increase the Lewis acidity of the gold center, a notion supported by the carbophilic reactivity of these complexes.     

Total Synthesis of (6R,10R,13R,14R,16R,17R,19S,20R,21R,24S, 25S,28S,30S,32R,33R,34R,36S,37S,39R)‐Azaspiracid‐3 Reveals Non‐Identity with the Natural Product

A convergent and stereoselective total synthesis of the previously assigned structure of azaspiracid‐3 has been achieved by a late‐stage Nozaki–Hiyama–Kishi coupling to form the C21?C22 bond with the C20 configuration unambiguously established from l ‐(+)‐tartaric acid. Postcoupling steps involved oxidation to an ynone, modified Stryker reduction of the alkyne, global deprotection, and oxidation of the resulting C1 primary alcohol to the carboxylic acid. The synthetic product matched naturally occurring azaspiracid‐3 by mass spectrometry, but differed both chromatographically and spectroscopically.     

Synthesis of a Ferrolite: A Zeolitic All‐Iron Framework

Crystals of the first sodalite‐type zeolite containing an all‐iron framework, a ferrolite, Ba₈(Fe₁₂O₂₄)Na_y(OH)₆?x H₂O, were synthesized using the hydroflux method in nearly quantitative yield. Ba₈(Fe₁₂O₂₄)Na_y(OH)₆?x H₂O crystallizes in the cubic space group with a=10.0476(1) Å. Slightly distorted FeO₄ tetrahedra are linked to form Fe₄O₄ and Fe₆O₆ rings, which in turn yield channels and internal cavities that are characteristic of the sodalite structure. Barium, sodium, and hydroxide ions and water molecules are found in the channels and provide charge balance. Magnetic measurements indicate that the ferrolite exhibits magnetic order up to at least 700 K, with the field‐cooled and zero‐field‐cooled curves diverging. Analysis of the ⁵⁷Fe Mössbauer spectra revealed two spectral components that have equal spectral areas, indicating the presence of two subsets of iron centers in the structure. Dehydrated versions of the ferrolite were also prepared by heating the sample.     

The role of liquid chromatography in proteomics

Proteomics represents a significant challenge to separation scientists because of the diversity and complexity of proteins and peptides present in biological systems. Mass spectrometry as the central enabling technology in proteomics allows detection and identification of thousands of proteins and peptides in a single experiment. Liquid chromatography is recognized as an indispensable tool in proteomics research since it provides high-speed, high-resolution and high-sensitivity separation of macromolecules. In addition, the unique features of chromatography enable the detection of low-abundance species such as post-translationally modified proteins. Components such as phosphorylated proteins are often present in complex mixtures at vanishingly small concentrations. New chromatographic methods are needed to solve these analytical challenges, which are clearly formidable, but not insurmountable. This review covers recent advances in liquid chromatography, as it has impacted the area of proteomics. The future prospects for emerging chromatographic technologies such as monolithic capillary columns, high temperature chromatography and capillary electrochromatography are discussed.     

Enhanced Methylarginine Characterization by Post-Translational Modification-Specific Targeted Data Acquisition and Electron-Transfer Dissociation Mass Spectrometry

When localizing protein post-translational modifications (PTMs) using liquid-chromatography (LC)-tandem mass spectrometry (MS/MS), existing implementations are limited by inefficient selection of PTM-carrying peptides for MS/MS, particularly when PTM site occupancy is sub-stoichiometric. The present contribution describes a method by which peptides carrying specific PTMs of interest-in this study, methylarginines-may be selectively targeted for MS/MS: peptide features are extracted from high mass accuracy single-stage MS data, searched against theoretical PTM-carrying peptide masses, and matching features are subjected to targeted data acquisition LC-MS/MS. Using trypsin digested Saccharomyces cerevisiae Npl3, in which evidence is presented for 18 methylarginine sites-17 of which fall within a glycine-arginine-rich (GAR) domain spanning <120 amino acids-it is shown that this approach outperforms conventional data dependent acquisition (DDA): when applied to a complex protein mixture featuring in vivo methylated Npl3, 95% more (P=0.030) methylarginine-carrying peptides are selected for MS/MS than DDA, leading to an 86% increase (P=0.044) in the number of methylated peptides producing Mascot ion scores ≥20 following electron-transfer dissociation (ETD). Notably, significantly more low abundance arginine methylated peptides (maximum ion intensities <6×10(4) cps) are selected for MS/MS using this approach relative to DDA (50% more in a digest of purified in vitro methylated Npl3). It is also demonstrated that relative to collision-induced dissociation (CID), ETD facilitates a 586% increase (P=0.016) in average Mascot ion scores of methylarginine-carrying peptides. The present PTM-specific targeted data acquisition approach, though described using methylarginine, is applicable to any ionizable PTM of known mass.     

Pulsed-valve chemical ionization for gas chromatography/fourier-transform mass spectrometry

The pressure requirements for chemical ionization g.c./F.t.m.s. which restrict mass resolution and accuracy are overcome through use of a pulsed valve that provides momentary reagent gas pressures. For alternate electron impact (EI)/chemical ionization (c.i.) g.c./F.t.m.s., similar resolution for both e.i. and c.i. data is demonstrated. The efficiency of chemical ionization with the pulsed valve is similar to static high pressure c.i. measurements of several model compounds. Results from the analysis of peppermint oil and a fuel additive illustrate the potential information available from a single g.c./F.t.m.s. experiment.