Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

An equation of Monge-Ampère type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Ampère type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tesselations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.

Direct observation by time-resolved infrared spectroscopy of the bright and the dark excited states of the [Ru(phen)2(dppz)]2+ light-switch compound in solution and when bound to DNA

The [Ru(phen)2(dppz)]2+ complex (1) is non-emissive in water but is highly luminescent in organic solvents or when bound to DNA, making it a useful probe for DNA binding. To date, a complete mechanistic explanation for this “light-switch” effect is still lacking. With this in mind we have undertaken an ultrafast time resolved infrared (TRIR) study of 1 and directly observe marker bands between 1280–1450 cm-1, which characterise both the emissive “bright” and the non-emissive “dark” excited states of the complex, in CD3CN and D2O respectively. These characteristic spectral features are present in the [Ru(dppz)3]2+ solvent light-switch complex but absent in [Ru(phen)3]2+, which is luminescent in both solvents. DFT calculations show that the vibrational modes responsible for these characteristic bands are predominantly localised on the dppz ligand. Moreover, they reveal that certain vibrational modes of the “dark” excited state couple with vibrational modes of two coordinating water molecules, and through these to the bulk solvent, thus providing a new insight into the mechanism of the light-switch effect. We also demonstrate that the marker bands for the “bright” state are observed for both L- and D enantiomers of 1 when bound to DNA and that photo-excitation of the complex induces perturbation of the guanine and cytosine carbonyl bands. This perturbation is shown to be stronger for the L enantiomer, demonstrating the different binding site properties of the two enantiomers and the ability of this technique to determine the identity and nature of the binding site of such intercalators.

Self-assembly of telechelic tyrosine end-capped PEO and poly(alanine) polymers in aqueous solution

The self-assembly in aqueous solution of three novel telechelic conjugates comprising a central hydrophilic polymer and short (trimeric or pentameric) tyrosine end-caps has been investigated. Two of the conjugates have a central poly(oxyethylene) (polyethylene oxide, PEO) central block with different molar masses. The other conjugate has a central poly(l-alanine) (PAla) sequence in a purely amino-acid based conjugate. All three conjugates self-assemble into β-sheet based fibrillar structures, although the fibrillar morphology revealed by cryogenic-TEM is distinct for the three polymers—in particular the Tyr5-PEO6k-Tyr5 forms a population of short straight fibrils in contrast to the more diffuse fibril aggregates observed for Tyr5-PEO2k-Tyr5 and Tyr3-PAla-Tyr3. Hydrogel formation was not observed for these samples (in contrast to prior work on related systems) up to quite high concentrations, showing that it is possible to prepare solutions of peptide–polymer-peptide conjugates with hydrophobic end-caps without conformational constraints associated with hydrogelation. The Tyr5-PEO6k-Tyr5 shows significant PEO crystallization upon drying in contrast to the Tyr5-PEO2k-Tyr5 conjugate. Our findings point to the remarkable ability of short hydrophobic peptide end groups to modulate the self-assembly properties of polymers in solution in model peptide-capped “associative polymers”. Retention of fluidity at high conjugate concentration may be valuable in potential future applications of these conjugates as bioresponsive or biocompatible materials, for example exploiting the enzyme-responsiveness of the tyrosine end-groups