Biocatalytic Strategies towards [4+2] Cycloadditions

Long sought after [4+2] cyclases have sprouted up in numerous biosynthetic pathways in recent years, raising hopes for biocatalytic solutions to cycloaddition catalysis, an important problem in chemical synthesis. In a few cases, detailed pictures of the inner workings of these catalysts have emerged, but intense efforts to gain deeper understanding are underway by means of crystallography and computational modelling. This Minireview aims to shed light on the catalytic strategies that this highly diverse family of enzymes employs to accelerate and direct the course of [4+2] cycloadditions with reference to small-molecule catalysts and designer enzymes. These catalytic strategies include oxidative or reductive triggers and lid-like movements of enzyme domains. A precise understanding of natural cycloaddition catalysts will be instrumental for customizing them for various synthetic applications.     

Contractible subgraphs, Thomassen’s conjecture and the dominating cycle conjecture for snarks

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is Hamiltonian), by Thomassen (every 4-connected line graph is Hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edge-coloring or a dominating cycle), which are known to be equivalent, are equivalent to the statement that every snark (i.e. a cyclically 4-edge-connected cubic graph of girth at least five that is not 3-edge-colorable) has a dominating cycle.We use a refinement of the contractibility technique which was introduced by Ryjá?ek and Schelp in 2003 as a common generalization and strengthening of the reduction techniques by Catlin and Veldman and of the closure concept introduced by Ryjá?ek in 1997.     

On Toughness and Hamiltonicity of 2K2‐Free Graphs

The toughness of a (noncomplete) graph G is the minimum value of t for which there is a vertex cut A whose removal yields components. Determining toughness is an NP‐hard problem for general input graphs. The toughness conjecture of Chvátal, which states that there exists a constant t such that every graph on at least three vertices with toughness at least t is hamiltonian, is still open for general graphs. We extend some known toughness results for split graphs to the more general class of 2K₂‐free graphs, that is, graphs that do not contain two vertex‐disjoint edges as an induced subgraph. We prove that the problem of determining toughness is polynomially solvable and that Chvátal's toughness conjecture is true for 2K₂‐free graphs.     

Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs

Given a function f : ℕ→ℝ, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k≤(n−f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f(k)≥2k+1, and contains a Hamilton cycle if f(k)≥2(k+1)². We conjecture that linear growth of f suffices to imply hamiltonicity.     

On the averaged quantum dynamics by white-noise hamiltonians with and without dissipation

Exact results are derived on the averaged dynamics of a class of random quantum-dynamical systems in continuous space. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time-independent and quadratic, the Weyl-Wigner symbol of the other part is a homogeneous Gaussian random field which is delta correlated in time, but smoothly correlated in position and momentum. The averaged dynamics of the resulting white-noise system is shown to be a monotone mixing increasing quantum-dynamical semigroup. Its generator is computed explicitly. Typically, in the course of time the mean energy of such a system grows linearly to infinity. In the second part of the paper an extended model is studied, which, in addition, accounts for dissipation by coupling the white-noise system linearly to a quantum-mechanical harmonic heat bath. It is demonstrated that, under suitable assumptions on the spectral density of the heat bath, the mean energy then saturates for long times.     

A note on minimum degree conditions for supereulerian graphs

A graph is called supereulerian if it has a spanning closed trail. Let G be a 2-edge-connected graph of order n such that each minimal edge cut SE(G) with |S|3 satisfies the property that each component of G−S has order at least (n−2)/5. We prove that either G is supereulerian or G belongs to one of two classes of exceptional graphs. Our results slightly improve earlier results of Catlin and Li. Furthermore, our main result implies the following strengthening of a theorem of Lai within the class of graphs with minimum degree δ4: If G is a 2-edge-connected graph of order n with δ(G)4 such that for every edge xyE(G) , we have max{d(x),d(y)}(n−2)/5−1, then either G is supereulerian or G belongs to one of two classes of exceptional graphs. We show that the condition δ(G)4 cannot be relaxed.     

Eliminating graphs by means of parallel knock-out schemes

In 1997 Lampert and Slater introduced parallel knock-out schemes, an iterative process on graphs that goes through several rounds. In each round of this process, every vertex eliminates exactly one of its neighbors. The parallel knock-out number of a graph is the minimum number of rounds after which all vertices have been eliminated (if possible). The parallel knock-out number is related to well-known concepts like perfect matchings, hamiltonian cycles, and 2-factors.We derive a number of combinatorial and algorithmic results on parallel knock-out numbers: for families of sparse graphs (like planar graphs or graphs of bounded tree-width), the parallel knock-out number grows at most logarithmically with the number n of vertices; this bound is basically tight for trees. Furthermore, there is a family of bipartite graphs for which the parallel knock-out number grows proportionally to the square root of n. We characterize trees with parallel knock-out number at most 2, and we show that the parallel knock-out number for trees can be computed in polynomial time via a dynamic programming approach (whereas in general graphs this problem is known to be NP-hard). Finally, we prove that the parallel knock-out number of a claw-free graph is either infinite or less than or equal to 2.     

Simultaneous DTA and DTG measurements on aluminium oxide monohydroxides

Simultaneous DTA and DTG curves (Mettler TA-2) have been measured for one diaspore and four different boehmites at temperatures up to 750° in a flow of dried argon. The crystallographic structures of the materials were assessed by X-ray diffraction and the degree of clustering of the elementary particles by scanning electron microscopy. To get straight base-lines a calibration curve was used to correct the DTA curves. The usefulness of the correction was established by comparison with DTA curves obtained with a Mettler TA-2000.The diaspore contained occluded water. Expulsion of the water at 300–350° brought about fragmentation of the crystals. The DTA and DTG peaks that correspond to the dehydration to alumina both lie at 560°.There is a noticeable spread in the DTA and DTG peaks for the different boehmites in the temperature range 455–530°. All the DTA peaks lie close to the corresponding DTG peaks. One of the boehmites displayed a double peak (470° and 502°). To trace the origin of the variation in peak temperature, the porous structure of the boehmite having its peak at 530° was varied by ball-milling and by dispersion into water, neither of which markedly affected the crystallographic structure and the crystallite size. Whereas ball-milling did not change the peak temperature much, dispersion into water brought about a transition from a single peak at 530° to a double peak at 450° and 490°. Prolonged storage in air led to a shift of the peaks to 480° and 520°. It is concluded that the porous structure of boehmites can profoundly affect the appearance and the temperature of their dehydration peaks.

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Zusammenfassung Simultane DTA- und DTG-Kurven (Mettler TA-2) wurden für ein Diaspor und vier verschiedene Boehmite bei Temperaturen bis zu 750° in einem Strom von getrocknetem Argon aufgenommen. Die kristallographische Struktur der Substanzen wurde mittels Röntgendiffraktion und der Anhäufungsgrad der Elementarpartikel mittels Raster-Elektronenmikroskopie bestimmt. Um gerade Grundlinien zu erhalten wurde eine Eichkurve zur Korrektur der DTA-Kurven verwendet. Die Nützlichkeit der Korrektur wurde durch Vergleiche der in einem Mettler TA-2000 Gerät erhaltenen DTA-Kurven festgestellt.Der Diaspor enthielt eingeschlossenes Wasser. Das bei 300 bis 350° erfolgte Abspalten des Wassers brachte eine Fragmentierung der Kristalle mit sich. Die DTA- und DTG-Peaks, welche der Dehydratisierung zu Aluminiumoxid entsprechen, liegen beide bei 560°.Im Temperaturbereich von 455 bis 530° läßt sich ein Verbreitern der DTA- und DTG-Peaks der verschiedenen Boehmite wahrnehmen. Einer der Boehmite zeigte einen Doppel-peak (470° und 502°). Um dem Ursprung der Änderungen der Peak-Temperaturen festzustellen wurde die poröse Struktur des Boehmits mit einem Peak bei 530° mit Hilfe einer Kugelmühle und durch Dispersion in Wasser verändert. Keine dieser Behandlungen änderte merklich die kristallographische Struktur und die Kristallgröße. Während das Vermählen in der Kugelmühle die Peak-Temperatur nicht wesentlich änderte, erbrachte die Dispergierung in Wasser einen Übergang von einem einzigen Peak bei 530° zu einem Doppelpeak bei 450 und 490°. Längeres Lagern an der Luft führte zu einer Verschiebung der Peaks nach 480° und 520°. Es wird gefolgert, daß die Porösstruktur von Boehmiten das Erscheinen und die Temperatur ihrer Dehydratisierungspeaks stark beeinflussen kann.

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Résumé Les courbes ATD et TGD d'une diaspore et de quatre boehmites différentes ont été enregistrées simultanément (Mettler TA-2) jusqu'à 750°, sous circulation d'argon desséché. La structure cristallographique de ces composés a été examinée par diffraction des rayons X et le degré de compacité a été évalué par microscopie électronique à balayage. Une courbe d'étalonnage a été utilisée pour corriger les courbes ATD et obtenir des lignes de base droites. L'utilité de cette correction est démontrée en comparant avec les courbes ATD obtenues à l'aide d'un appareil Mettler-TA-2000.La diaspore contenait de l'eau d'inclusion. L'élimination de l'eau à 300–350° a entraîné la fragmentation des cristaux. Les pics ATD et TGD qui correspondent à la déshydratation en oxyde d'aluminium se trouvent tous deux à 560°.Un étalement des pics ATD et TGD est perceptible pour les différentes boehmites dans l'intervalle de température 455–530°. Tous les pics ATD sont proches des pics TGD correspondants. L'une des boehmites a donné un pic double (470 et 502°). Afin de trouver l'origine de la différence de température entre les deux pics, la structure poreuse de la boehmite dont le pic se situait à 530° a été modifiée par traitement dans un broyeur à boulets et par dispersion dans de l'eau; ni l'un ni l'autre de ces deux traitements n'influence de façon apparente la structure cristallographique et la taille des cristaux. Alors que le traitement au broyeur à boulets ne change pas beaucoup la température du pic, la dispersion dans de l'eau fait apparaitre le passage d'un pic unique à 530° à un pic double à 450 et 490°. Le stockage prolongé dans l'air provoque un déplacement des pics à 480 et 520°. On en conclut que la structure poreuse des boehmites peut influer profondément sur l'apparition et la température de leurs pics de déshydratation.

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— ( TA-2) 750° . - , — . . , TA-2000. . 300–350°C, . — , , 560°. - 455–530°. . (470° 502°). , 530°C . . , — 530° — 450° 490°. 480° 520°. , , .

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The authors are indebted to Dr. P. F. Elbers for placing the electron microscopic facilities of the University of Utrecht at their disposal. We especially wish to acknowledge the time and energy spent by Mr. J. Pieters in the preparation of samples and the investigation with the scanning electron microscope. The assistance given by Dr. A. Duisenberg in the measurement of the line-broadening is also gratefully acknowledged.