A result about the density of iterated line intersections in the plane

Let S be a finite set of points in the plane and let be the set of intersection points between pairs of lines passing through any two points in S. We characterize all configurations of points S such that iteration of the above operation produces a dense set. We also discuss partial results on the characterization of those finite point-sets with rational coordinates that generate all of through iteration of .     

Expeditious and Divergent Total Syntheses of Aspidosperma Alkaloids Exploiting Iridium(I)‐Catalyzed Generation of Reactive Enamine Intermediates

A new approach for the divergent total syntheses of (±)‐vincaminorine, (±)‐N‐methylquebrachamine, (±)‐quebrachamine, (±)‐minovine and (±)‐vincadifformine, each in less than 10 linear steps starting from a single δ‐lactam building block, is reported. Key to our route design is the late‐stage generation of reactive enamine functionality from stable indole‐linked δ‐lactams via a highly chemoselective iridium(I)‐catalyzed reduction. The efficiently formed secodine intermediates subsequently undergo either a formal Diels–Alder cycloaddition or a competitive Michael addition/reduction to access aspidosperma‐type alkaloids in excellent diastereoselectivities. Product selectivity could be controlled by changing the indole N‐protecting group in the reductive cyclization precursors. An asymmetric variant of this synthetic strategy for the synthesis of (+)‐20‐epi‐ibophyllidine is also described.     

A Synthetic Supramolecular Receptor for the Hydrosulfide Anion

Hydrogen sulfide (H₂S) has emerged as a crucial biomolecule in physiology and cellular signaling. Key challenges associated with developing new chemical tools for understanding the biological roles of H₂S include developing platforms that enable reversible binding of this important biomolecule. The first synthetic small molecule receptor for the hydrosulfide anion, HS^?, using only reversible, hydrogen‐bonding interactions in a series of bis(ethynylaniline) derivatives, is reported. Binding constants of up to 90 300±8700 m ^?1 were obtained in MeCN. The fundamental science of reversible sulfide binding, in this case featuring a key CH???S hydrogen bond, will expand the possibility for discovery of sulfide protein targets and molecular recognition agents.     

A new cytotoxic diterpenoid glycoside from the leaves of Blumea lacera and its effects on apoptosis and cell cycle

A new diterpenoid glycoside, 6E,10E,14Z-(3S)-17-hydroxygeranyllinalool-17-O-β-d-glucopyranosyl-(1?→?2)-[α-l-rhamnopyranosyl-(1?→?6)]-β-d-glucopyranoside (1) together with the known diterpenoid glycoside (2) and two known flavonoid glycosides (3, 4) were isolated from the methanol extract of Blumea lacera leaves. The structures were determined by the interpretation of their spectroscopic data and comparison with the literature. All compounds were isolated for the first time from B. lacera and evaluated for their cytotoxic activity. Only the new compound (1) showed strong cytotoxic activity with the lowest IC₅₀ value (8.3 μM) being displayed against MCF-7 breast cancer cells. In apoptosis and cell cycle analysis, 1 revealed strong apoptotic activity against MCF-7 cells (45.5% AV⁺/PI^?) after 24 h, but showed no arresting of any of the cell cycle phases in MCF-7.     

Selective and Potent Proteomimetic Inhibitors of Intracellular Protein–Protein Interactions

Inhibition of protein–protein interactions (PPIs) represents a major challenge in chemical biology and drug discovery. α‐Helix mediated PPIs may be amenable to modulation using generic chemotypes, termed “proteomimetics”, which can be assembled in a modular manner to reproduce the vectoral presentation of key side chains found on a helical motif from one partner within the PPI. In this work, it is demonstrated that by using a library of N‐alkylated aromatic oligoamide helix mimetics, potent helix mimetics which reproduce their biophysical binding selectivity in a cellular context can be identified.     

Electrochemical Synthesis of Isoxazolines: Method and Mechanism

An electrochemical method for the green and practical synthesis of a broad range of substituted isoxazoline cores is presented. Both aryl and more challenging alkyl aldoximes are converted to the desired isoxazoline in an electrochemically enabled regio- and diastereoselective reaction with electron-deficient alkenes. Additionally, in-situ reaction monitoring methods compatible with electrochemistry equipment have been developed in order to probe the reaction pathway. Supporting analyses from kinetic (time-course) modelling and density functional theory support a stepwise, radical-mediated mechanism, and discounts hypothesised involvement of closed shell [3+2] cycloaddition pathways.     

Plateau's Problem as a Singular Limit of Capillarity Problems

Soap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a homotopic spanning condition. This point of view introduces a length scale in the classical Plateau's problem, which is in turn recovered in the vanishing volume limit. This approximation of area minimizing hypersurfaces leads to an energy based selection principle for Plateau's problem, points at physical features of soap films that are unaccessible by simply looking at minimal surfaces, and opens several challenging questions. © 2020 Wiley Periodicals LLC.     

On a Model of Nonlocal Continuum Mechanics Part II: Structure, Asymptotics, and Computations

We consider the asymptotic behavior and local structure of solutions to the nonlocal variational problem developed in the companion article to this work, On a Model of Nonlocal Continuum Mechanics Part I: Existence and Regularity. After a brief review of the basic setup and results of Part I, we conduct a thorough analysis of the phase plane related to an integro-differential Euler--Lagrange equation and classify all admissible structures that arise as energy minimizing strain states. We find that for highly elastic materials with relatively weak particle-particle interactions, the maximum number of internal phase boundaries is two. Moreover, we also develop explicit bounds for the number of internal phase boundaries supported by any material and show that this bound is essentially inversely related to the particle size. To understand the question of asymptotics, we utilize the Young measure and show that in the sense of energetics and averages, minimizers of the full nonlocal problem converge to minimizers of two limiting problems corresponding to both the large and small particle limits. In fact, in the small particle limit, we find that the minimizing fields converge, up to a subsequence in strong-L^p, for 1 ≤ p < ∞, to fields that support either a single internal phase boundary, or two internal phase boundaries that are distributed symmetrically about the body midpoint. We close this work with some computations that illustrate these asymptotic limits and provide insight into the notion of nonlocal metastability. This revised version was published online in August 2006 with corrections to the Cover Date.     

Solution structural characterization of an array of nanoscale aqueous inorganic Ga13–xInx (0 ≤ x ≤ 6) clusters by 1H-NMR and QM computations

NMR spectroscopy is the go-to technique for determining the solution structures of organic, organometallic, and even macromolecular species. However, structure determination of nanoscale aqueous inorganic clusters by NMR spectroscopy remains an unexplored territory. The few hydroxo-bridged inorganic species well characterized by ¹H Nuclear Magnetic Resonance spectroscopy (¹H-NMR) do not provide enough information for signal assignment and prediction of new samples. ¹H-NMR and quantum mechanical (QM) computations were used to characterize the NMR spectra of the entire array of inorganic flat-Ga_13–xIn_x (0 ≤ x ≤ 6) nanoscale clusters in solution. A brief review of the known signals for μ₂-OH and μ₃-OH bridges gives expected ranges for certain types of protons, but does not give enough information for exact peak assignment. Integration values and NOESY data were used to assign the peaks of several cluster species with simple ¹H-NMR spectra. Computations agree with these hydroxide signal assignments and allow for assignment of the complex spectra arising from the remaining cluster species. This work shows that ¹H-NMR spectroscopy provides a variety of information about the solution behavior of inorganic species previously thought to be inaccessible by NMR due to fast ligand and/or proton exchange in wet solvents.