Supramolecular allosteric cofacial porphyrin complexes

Nature routinely uses cooperative interactions to regulate cellular activity. For years, chemists have designed synthetic systems that aim toward harnessing the reactivity common to natural biological systems. By learning how to control these interactions in situ, one begins to allow for the preparation of man-made biomimetic systems that can efficiently mimic the interactions found in Nature. To this end, we have designed a synthetic protocol for the preparation of flexible metal-directed supramolecular cofacial porphyrin complexes which are readily obtained in greater than 90% yield through the use of new hemilabile porphyrin ligands with bifunctional ether-phosphine or thioether-phosphine substituents at the 5 and 15 positions on the porphyrin ring. The resulting architectures contain two hemilabile ligand-metal domains (RhI or CuI sites) and two cofacially aligned porphyrins (ZnII sites), offering orthogonal functionalities and allowing these multimetallic complexes to exist in two states, "condensed" or "open". Combining the ether-phosphine ligand with the appropriate RhI or CuI transition-metal precursors results in "open" macrocyclic products. In contrast, reacting the thioether-phosphine ligand with RhI or CuI precursors yields condensed structures that can be converted into their "open" macrocyclic forms via introduction of additional ancillary ligands. The change in cavity size that occurs allows these structures to function as allosteric catalysts for the acyl transfer reaction between X-pyridylcarbinol (where X = 2, 3, or 4) and 1-acetylimidazole. For 3- and 4-pyridylcarbinol, the "open" macrocycle accelerates the acyl transfer reaction more than the condensed analogue and significantly more than the porphyrin monomer. In contrast, an allosteric effect was not observed for 2-pyridylcarbinol, which is expected to be a weaker binder and is unfavorably constrained inside the macrocyclic cavity.     

Exchange coupling mediated through-bonds and through-space in conformationally constrained polyradical scaffolds: calix[4]arene nitroxide tetraradicals and diradical

Calix[4]arenes constrained to the 1,3-alternate conformation and functionalized at the upper rim with four and two tert-butylnitroxides have been synthesized and characterized by X-ray crystallography, magnetic resonance (EPR and (1)H NMR) spectroscopy, and magnetic studies. The 1,3-alternate nitroxide tetraradical and diradical provide unique polyradical scaffolds for dissection of the through-bond and through-space intramolecular exchange couplings. In addition, detailed magnetic studies of the previously reported calix[4]arene nitroxide tetraradical, which possesses cone conformation in solution, reveal conformational dependence of exchange coupling. Through-bond coupling between the adjacent nitroxide radicals is mediated by the nitroxide-m-phenylene-CH(2)-m-phenylene-nitroxide coupling pathway, and through-space coupling is found between the diagonal nitroxide radicals at the conformationally constrained N...N distance of 5-6 A. Magnetic studies of the calix[4]arene polyradical scaffolds in frozen solutions show that the through-bond exchange coupling in the 1,3-alternate calix[4]arene tetraradical is antiferromagnetic, while that in cone calix[4]arene tetraradical is ferromagnetic. The through-space exchange couplings are antiferromagnetic in both cone and 1,3-alternate calix[4]arene tetraradical, as well as in the 1,3-alternate calix[4]arene diradical. The exchange coupling constants (|J/k|) are of the order of 1 K.     

A Notion of Stopping Line for Set-Indexed Processes

In this paper, we introduce the class of -stopping lines which generalize the planar stopping lines in Merzbach [(1980), Stochastic Process. Appl. 10, 49–63] by replacing the positive quadrant of the plane by a collection of compact subsets of a fixed topological space. Our notion of stopping line also compliments and generalizes the stopping sets defined in Ivanoff and Merzbach [(1995), Stochastic Process. Appl. 57, 83–98].      

Polymers designed to control nucleation and growth of inorganic crystals from aqueous media

Water soluble graft polymers prepared by copolymerization of either methacrylic acid (MAA) or vinylsulfonic acid (VS) with α‐methoxy‐ω‐methacroyl‐oligo(oxyethylene)s (PEO_n‐MA) serve to control nucleation and crystal growth during precipitation of inorganic crystals from aqueous media. Precipitation of zinc oxide crystals (‘zincite’) is used as example for such mineralization processes. Homogeneous and narrow crystal size distributions are obtained in presence of ppm‐amounts of graft copolymers. Copolymer is incorporated into the crystals demonstrated by using latex particles with ‐CO₂H‐group rich surfaces as controlling additives. Incorporation of these particles leads to single crystals with pores of the size of the particles (‘Swiss cheese’ morphologies).     

Reconfiguration of connected graph partitions

Motivated by recent computational models for redistricting and detection of gerrymandering, we study the following problem on graph partitions. Given a graph $G$ and an integer $k\ge 1$, a $k$-district map of $G$ is a partition of $V\left(G\right)$ into $k$ nonempty subsets, called districts, each of which induces a connected subgraph of $G$. A switch is an operation that modifies a $k$-district map by reassigning a subset of vertices from one district to an adjacent district; a 1-switch is a switch that moves a single vertex. We study the connectivity of the configuration space of all $k$-district maps of a graph $G$ under 1-switch operations. We give a combinatorial characterization for the connectedness of this space that can be tested efficiently. We prove that it is PSPACE-complete to decide whether there exists a sequence of 1-switches that takes a given $k$-district map into another; and NP-hard to find the shortest such sequence (even if a sequence of polynomial lengths is known to exist). We also present efficient algorithms for computing a sequence of 1-switches that take a given $k$-district map into another when the space is connected, and show that these algorithms perform a worst-case optimal number of switches up to constant factors.