Combinatorics of Matrix Factorizations and Integrable Systems

We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a combinatorial–geometric way using a diagram of a cube. In this representation, vertices of the cube represent eigenvectors, edges are labeled by differences of locations of zeroes and poles of the determinant of our matrix function, and each face corresponds to a particular choice of a coordinate system on the space of such functions. Moreover, for each face this labeling encodes, in a neat and efficient way, a generating function for the expressions of the remaining four eigenvectors that label the opposing face of the cube in terms of the coordinates represented by the chosen face. The main motivation behind this work is that when our matrix is a Lax matrix of a discrete integrable system, such generating functions can be interpreted as Lagrangians of the system, and a choice of a particular face corresponds to a choice of the direction of the motion.     

A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, non-positively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour.     

Symbols of truncated Toeplitz operators

We consider three topics connected with coinvariant subspaces of the backward shift operator in Hardy spaces Hp:

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properties of truncated Toeplitz operators;

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Carleson-type embedding theorems for the coinvariant subspaces;

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factorizations of pseudocontinuable functions from H¹.

These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by D. Sarason.     

Postprocessing of docked protein-ligand complexes using implicit solvation models

Molecular docking plays an important role in drug discovery as a tool for the structure-based design of small organic ligands for macromolecules. Possible applications of docking are identification of the bioactive conformation of a protein-ligand complex and the ranking of different ligands with respect to their strength of binding to a particular target. We have investigated the effect of implicit water on the postprocessing of binding poses generated by molecular docking using MM-PB/GB-SA (molecular mechanics Poisson-Boltzmann and generalized Born surface area) methodology. The investigation was divided into three parts: geometry optimization, pose selection, and estimation of the relative binding energies of docked protein-ligand complexes. Appropriate geometry optimization afforded more accurate binding poses for 20% of the complexes investigated. The time required for this step was greatly reduced by minimizing the energy of the binding site using GB solvation models rather than minimizing the entire complex using the PB model. By optimizing the geometries of docking poses using the GB(HCT+SA) model then calculating their free energies of binding using the PB implicit solvent model, binding poses similar to those observed in crystal structures were obtained. Rescoring of these poses according to their calculated binding energies resulted in improved correlations with experimental binding data. These correlations could be further improved by applying the postprocessing to several of the most highly ranked poses rather than focusing exclusively on the top-scored pose. The postprocessing protocol was successfully applied to the analysis of a set of Factor Xa inhibitors and a set of glycopeptide ligands for the class II major histocompatibility complex (MHC) A(q) protein. These results indicate that the protocol for the postprocessing of docked protein-ligand complexes developed in this paper may be generally useful for structure-based design in drug discovery.     

Poly(vinyl alcohol) physical hydrogels: new vista on a long serving biomaterial

Poly(vinyl alcohol), PVA, and physical hydrogels derived thereof have an excellent safety profile and a successful history of biomedical applications. However, these materials are hardly in the focus of biomedical research, largely due to poor opportunities in nano- and micro-scale design associated with PVA hydrogels in their current form. In this review we aim to demonstrate that with PVA, a (sub)molecular control over polymer chemistry translates into fine-tuned supramolecular association of chains and this, in turn, defines macroscopic properties of the material. This nano- to micro- to macro- translation of control is unique for PVA and can now be accomplished using modern tools of macromolecular design. We believe that this strategy affords functionalized PVA physical hydrogels which meet the demands of modern nanobiotechnology and have a potential to become an indispensable tool in the design of biomaterials.     

Electrical conductivity of PUR/MWCNT nanocomposites in the molten state, during crystallization and in the solid state

The electrical conductivity of a nanocomposite constituted of multiwalled carbon nanotubes (MWCNT) dispersed in a semicrystalline polyurethane matrix, is investigated during cooling from the melt to the solid state. The same percolation threshold, ?_c = 0.85 wt.%, is obtained in the molten and in the solid state, although the exponent t of the percolation equation is significantly higher in the solid state. A remarkable increase of the conductivity during crystallization is observed for nanocomposites of MWCNT content above ?_c, but for contents below ?_c the conductivity decreases. Combined conductivity and PVT results, lead us to discard the hypothesis of an increase of the density of the conductive network (associated with volume shrinkage) as being the cause of the conductivity enhancement during crystallization. Instead, the analysis of the parameters of the percolation equation for the molten and the solid state, suggests a transition from a less effective conductive network to a more performing three dimensional network.     

NCN-chelated organoantimony(III) and organobismuth(III) phosphates: synthesis and solid-state and solution structures

.Organoantimony(III) and organobismuth(III) phosphates (LM)(3)(PO(4))(2) [M = Sb (3) and Bi (4)], containing the NCN-chelating ligand L [L = 2,6-(CH(2)NMe(2))(2)C(6)H(3)], were prepared by the simple treatment of parent oxides 1 and 2 with H(3)PO(4). Both compounds were characterized by elemental analysis, electrospray ionization mass spectrometry, and IR and NMR spectroscopy and in the case of 3 by X-ray diffraction techniques. Compound 3 has an interesting behavior in solution, i.e., the formation of two possible conformational isomers, which was studied by (1)H, (13)C, and (31)P NMR spectroscopy.     

Phosphorescence vs fluorescence in cyclometalated platinum(II) and iridium(III) complexes of (oligo)thienylpyridines

Two newly prepared oligothienylpyridines, 5-(2-pyridyl)-5'-dodecyl-2,2'-bithiophene, HL(2), and 5-(2-pyridyl)-5'-dodecyl-2,2':5',2'-ter-thiophene, HL(3), bind to platinum(II) and iridium(III) as N∧C-coordinating ligands, cyclometallating at position C(4) in the thiophene ring adjacent to the pyridine, leaving a chain of either one or two pendent thiophenes. The synthesis of complexes of the form [PtL(n)(acac)] and [Ir(L(n))(2)(acac)] (n = 2 or 3) is described. The absorption and luminescence properties of these four new complexes are compared with the behavior of the known complexes [PtL(1)(acac)] and [Ir(L(1))(2)(acac)] {HL(1) = 2-(2-thienyl)pyridine}, and the profound differences in behavior are interpreted with the aid of time-dependent density functional theory (TD-DFT) calculations. Whereas [PtL(1)(acac)] displays solely intense phosphorescence from a triplet state of mixed ππ*/MLCT character, the phosphorescence of [PtL(2)(acac)] and [PtL(3)(acac)] is weak, strongly red shifted, and accompanied by higher-energy fluorescence. TD-DFT reveals that this difference is probably due to the metal character in the lowest-energy excited states being strongly attenuated upon introduction of the additional thienyl rings, such that the spin-orbit coupling effect of the metal in promoting intersystem crossing is reduced. A similar pattern of behavior is observed for the iridium complexes, except that the changeover to dual emission is delayed to the terthiophene complex [Ir(L(3))(2)(acac)], reflecting the higher degree of metal character in the frontier orbitals of the iridium complexes than their platinum counterparts.