Some new results on the well-posedness of hyperbolic evolution equations

We summarize some recently obtained results on the well-posedness of hyperbolic linear evolution equations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

PROCOS: computational analysis of protein-protein complexes

One of the main challenges in protein-protein docking is a meaningful evaluation of the many putative solutions. Here we present a program (PROCOS) that calculates a probability-like measure to be native for a given complex. In contrast to scores often used for analyzing complex structures, the calculated probabilities offer the advantage of providing a fixed range of expected values. This will allow, in principle, the comparison of models corresponding to different targets that were solved with the same algorithm. Judgments are based on distributions of properties derived from a large database of native and false complexes. For complex analysis PROCOS uses these property distributions of native and false complexes together with a support vector machine (SVM). PROCOS was compared to the established scoring schemes of ZRANK and DFIRE. Employing a set of experimentally solved native complexes, high probability values above 50% were obtained for 90% of these structures. Next, the performance of PROCOS was tested on the 40 binary targets of the Dockground decoy set, on 14 targets of the RosettaDock decoy set and on 9 targets that participated in the CAPRI scoring evaluation. Again the advantage of using a probability-based scoring system becomes apparent and a reasonable number of near native complexes was found within the top ranked complexes. In conclusion, a novel fully automated method is presented that allows the reliable evaluation of protein-protein complexes.     

From hetero- to homochiral bis(metallahelicene)s based on a Pt(III)-Pt(III) bonded scaffold: isomerization, structure, and chiroptical properties

Hetero- and homochiral diastereomeric bis(metallahelicene)s have been synthesized. They possess a rare Pt(III)-Pt(III) scaffold bridged by benzoato ligands. It is shown that heterochiral (P,M)-bis(Pt(III)-[6]helicene) can isomerize into the homochiral (P,P)- and (M,M)-bis(Pt(III)-[6]helicene). A theoretical study shows a unique σ-π conjugation between the two π-helices and the σ-Pt(III)-Pt(III) scaffold that impacts the strong chiroptical properties.     

Impact of sugar stereochemistry on natural killer T cell stimulation by bacterial glycolipids

Natural killer T (NKT) cells recognize glycolipids produced by Sphingomonas bacteria, and these glycolipids contain C6-oxidized sugars, either glucuronic acid or galacturonic acid, linked to ceramides. Glycolipids with gluco stereochemistry are the most prevalent. Multiple studies have demonstrated that galactosylceramides are more potent stimulators of NKT cells than their glucose isomers. To determine if this stereoselectivity is retained in the context of the C6-oxidized sugars found in bacterial glycolipids, we prepared two sets of gluco and galacto-glycolipids oxidized at their C6 positions and compared their NKT stimulatory properties. In the context of carboxylic acid groups at C6, gluco stereochemistry gave the more potent responses. We also prepared bacterial glycolipids containing more complex ceramide groups to determine if these chains impact NKT cell responses.     

Tame linear extension operators for smooth Whitney functions

For a compact set K⊆Rd we present a rather easy construction of a linear extension operator E:E(K)→C^∞(Rd) for the space of Whitney jets E(K) which satisfies linear tame continuity estimates , where ‖⋅s_‖ denotes the s-th Whitney norm. The construction turns out to be possible if and only if the local Markov inequality LMI(s) introduced by Bos and Milman holds for every s>r on K. In particular, E(K) admits a tame linear extension operator if and only if the local Markov inequality LMI(s) holds on K for some s?1.     

A partition-based relaxation for Steiner trees

The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G = (V, E), a set of terminals \({R\subseteq V}\) , and non-negative costs c _e for all edges \({e \in E}\) . Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The vertices \({V \backslash R}\) are called Steiner vertices. The best approximation algorithm known for the Steiner tree problem is a greedy algorithm due to Robins and Zelikovsky (SIAM J Discrete Math 19(1):122–134, 2005); it achieves a performance guarantee of \({1+\frac{\ln 3}{2}\approx 1.55}\) . The best known linear programming (LP)-based algorithm, on the other hand, is due to Goemans and Bertsimas (Math Program 60:145–166, 1993) and achieves an approximation ratio of 2?2/|R|. In this paper we establish a link between greedy and LP-based approaches by showing that Robins and Zelikovsky’s algorithm can be viewed as an iterated primal-dual algorithm with respect to a novel LP relaxation. The LP used in the first iteration is stronger than the well-known bidirected cut relaxation. An instance is b-quasi-bipartite if each connected component of \({G \backslash R}\) has at most b vertices. We show that Robins’ and Zelikovsky’s algorithm has an approximation ratio better than \({1+\frac{\ln 3}{2}}\) for such instances, and we prove that the integrality gap of our LP is between \({\frac{8}{7}}\) and \({\frac{2b+1}{b+1}}\) .     

Structure of CO2 adsorbed on the KCl(100) surface

The structure and dynamics of the adsorbate CO(2)/KCl(100) from a diluted phase to a saturated monolayer have been investigated with He atom scattering (HAS), low-energy electron diffraction (LEED), and polarization dependent infrared spectroscopy (PIRS). Two adsorbate phases with different CO(2) coverage have been found. The low-coverage phase is disordered at temperatures near 80 K and becomes at least partially ordered at lower temperatures, characterized by a (2√2×√2)R45° diffraction pattern. The saturated 2D phase has a high long-range order and exhibits (6√2×√2)R45° symmetry. Its isosteric heat of adsorption is 26 ± 4 kJ mol(-1). According to PIRS, the molecules are oriented nearly parallel to the surface, the average tilt angle in the saturated monolayer phase is 10° with respect to the surface plane. For both phases, structure models are proposed by means of potential calculations. For the saturated monolayer phase, a striped herringbone structure with 12 inequivalent molecules is deduced. The simulation of infrared spectra based on the proposed structures and the vibrational exciton approach gives reasonable agreement between experimental and simulated infrared spectra.     

Calculation of molecular g-tensors using the zeroth-order regular approximation and density functional theory: expectation value versus linear response approaches

Density functional theory (DFT) calculations of molecular g-tensors were implemented as a second derivative property within the two-component relativistic zeroth-order regular approximation (ZORA). g-tensors were computed for systems ranging from light atomic radicals to molecules with heavy d and f block elements. For comparison, computations were also performed with a ZORA first-order derivative approach and with a second derivative method based on the Pauli Hamiltonian. In each set of computations, Slater-type basis sets have been used. The new ZORA implementation allows for non-hybrid and hybrid DFT calculations. A comparison of the PBE non-hybrid and the PBE0 hybrid functional yielded mixed results for our test set. For the lanthanide complex $[\hbox{Ce}(\hbox{DPA})_3]^{3-}$ (DPA = pyridine-2,6-dicarboxylate), calculations of the g-tensor were used to estimate paramagnetic NMR pseudocontact shifts for protons and carbon atoms in the ligands. The results are in reasonable agreement with experimental data.     

Tuning of switching properties and excited-state dynamics of fulgides by structural modifications

The ultrafast photo-induced dynamics of the E-isomers of four selected photochromic fulgides with distinct structural motifs have been elucidated by femtosecond broadband transient absorption spectroscopy in n-hexane as solvent. E→C and E→Z isomerisations, respectively, with time constants of ～0.12 ± 0.02 ps and ～0.34 ± 0.03 ps taking place in parallel were found for derivatives with a methyl substituent at the central hexatriene (HT) unit. In contrast, fulgides with increased steric constraints by an iso-propyl substituent or by intramolecular bridging displayed virtually zero E→Z isomerisation, but instead a desired accelerated and more efficient ring closure in a reaction time of only ～50 ± 10 fs. Both photoisomerisations appear to follow excited-state pathways with distinctive conical intersections. For the ring closure, direct barrierless pathways with steep downhill gradients are likely. Furthermore, the results indicate conformer-specific reactions, with ring closure exclusively by the E(α) conformer and E→Z isomerisation predominantly by the E(β) conformer, because the E(α)→Z channel is unfavoured by the faster and kinetically more competitive E(α)→C reaction. DFT calculations of the equilibrium structures showed that the sterically demanding groups at the HT unit shift the conformer equilibria towards the E(α) conformers. At the same time, they appear to cause a favourable pre-orientation of the furyl unit that accelerates the conrotatory ring closure in the E(α)→C reaction. Benzo-annulation of the furyl unit has little effect on the observed dynamics. Overall, the results demonstrate how the excited-state dynamics and thereby the photoswitching properties of fulgides can be successfully tuned and improved by structural modifications at the chromophores.     

Electronic and steric effects on the photo-induced C→E ring-opening of structurally modified furylfulgides

The ultrafast C→E ring-opening reactions of four selectively modified furylfulgides have been studied by means of ultrafast broadband transient absorption spectroscopy after femtosecond laser excitation at λ = 500 nm. A large difference in the dynamics was found in the case of benzannulation at the furyl moiety as an example for an electronic effect by extension of the conjugated π-electron system compared to furylfulgides carrying sterically different alkyl substituents at the central cyclohexadiene (CHD) ring. The measured very similar spectro-temporal absorption maps for the furylfulgides with a methyl or isopropyl group at the CHD ring or an intramolecular alkyl bridge from the CHD to the furyl moiety showed two distinctive excited-state absorptions with slightly different decay times. The first time constant (τ(1) = 0.39-0.57 ps) was assigned to the rapid departure of the excited wavepacket from the Franck-Condon region. The slightly longer second decay time of τ(2) = 0.66-0.92 ps, depending on the compound, was attributed to the electronic deactivation and ring-opening through a conical intersection to the S(0) state. In contrast, the benzannulation at the furyl moiety was found to lead to a bi-phasic excited-state decay with τ(2) = 4.7 ps and a much slower additional contribution of τ(3) = 17.4 ps, ≈25 times longer compared to the normal furylfulgides. The drastic change is attributed to a trapping of excited molecules in a local potential energy minimum en route to the conical intersection.