Notations and conventions in molecular spectroscopy: part 2. Symmetry notation

The field of Molecular Spectroscopy was surveyed in order to determine a set of conventions and symbols which are in common use in the spectroscopic literature. This document, which is Part 2 in a series, establishes the notations and conventions used for the description of symmetry in rigid molecules, using the Schoenflies notation. It deals firstly with the symmetry operators of the molecular point groups (also drawing attention to the difference between symmetry operators and elements). The conventions and notations of the molecular point groups are then established, followed by those of the representations of these groups as used in molecular spectroscopy. Further parts will follow, dealing inter alia with permutation and permutation-inversion symmetry notation, vibration-rotation spectroscopy and electronic spectroscopy.

Notations and conventions in molecular spectroscopy: part 3. Permutation and permutation-inversion symmetry notation

The field of Molecular Spectroscopy was surveyed in order to determine a set of conventions and symbols which are in common use in the spectroscopic literature. This document, which is Part 3 in a series, deals with symmetry notation referring to groups that involve nuclear permutations and the inversion operation. Further parts will follow, dealing inter alia with vibration-rotation spectroscopy and electronic spectroscopy.

Testing symmetry and homogeneity in the almost ideal demand system with co-integrated data using fully modified estimation and the bootstrap

Conventional seemingly unrelated estimation of the almost ideal demand system is shown to lead to small sample bias and distortions in the size of a Wald test for symmetry and homogeneity when the data are co-integrated. A fully modified estimator is developed in an attempt to remedy these problems. It is shown that this estimator reduces the small sample bias but fails to eliminate the size distortion.. Bootstrapping is shown to be ineffective as a method of removing small sample bias in both the conventional and fully modified estimators. Bootstrapping is effective, however, as a method of removing. size distortion and performs equally well in this respect with both estimators.

Amino acid pairing preferences in parallel beta-sheets in proteins

Statistical approaches have been applied to examine amino acid pairing preferences within parallel beta-sheets. The main chain hydrogen bonding pattern in parallel beta-sheets means that, for each residue pair, only one of the residues is involved in main chain hydrogen bonding with the strand containing the partner residue. We call this the hydrogen bonded (HB) residue and the partner residue the non-hydrogen bonded (nHB) residue, and differentiate between the favorability of a pair and that of its reverse pair, e.g. Asn(HB)-Thr(nHB)versus Thr(HB)-Asn(nHB). Significantly (p < or = 0.000001) favoured pairings were rationalised using stereochemical arguments. For instance, Asn(HB)-Thr(nHB) and Arg(HB)-Thr(nHB) were favoured pairs, where the residues adopted favoured chi1 rotamer positions that allowed side-chain interactions to occur. In contrast, Thr(HB)-Asn(nHB) and Thr(HB)-Arg(nHB) were not significantly favoured, and could only form side-chain interactions if the residues involved adopted less favourable chi1 conformations. The favourability of hydrophobic pairs e.g. Ile(HB)-Ile(nHB), Val(HB)-Val(nHB) and Leu(HB)-Ile(nHB) was explained by the residues adopting their most preferred chi1 and chi2 conformations, which enabled them to form nested arrangements. Cysteine-cysteine pairs are significantly favoured, although these do not form intrasheet disulphide bridges. Interactions between positively and negatively charged residues were asymmetrically preferred: those with the negatively charged residue at the HB position were more favoured. This trend was accounted for by the presence of general electrostatic interactions, which, based on analysis of distances between charged atoms, were likely to be stronger when the negatively charged residue is the HB partner. The Arg(HB)-Asp(nHB) interaction was an exception to this trend and its favorability was rationalised by the formation of specific side-chain interactions. This research provides rules that could be applied to protein structure prediction, comparative modelling and protein engineering and design. The methods used to analyse the pairing preferences are automated and detailed results are available (http://www.rubic.rdg.ac.uk/betapairprefsparallel/).

Amino acid pairing preferences in parallel beta-sheets in proteins

Statistical approaches have been applied to examine amino acid pairing preferences within parallel beta-sheets. The main chain hydrogen bonding pattern in parallel beta-sheets means that, for each residue pair, only one of the residues is involved in main chain hydrogen bonding with the strand containing the partner residue. We call this the hydrogen bonded (HB) residue and the partner residue the non-hydrogen bonded (nHB) residue, and differentiate between the favourability of a pair and that of its reverse pair, e.g. Asn(HB)-Thr(nHB) versus Thr(HB)-Asn(nHB). Significantly (p <= 0.000001) favoured pairings were rationalised using stereochemical arguments. For instance, Asn(HB)-Thr(nHB) and Arg(HB)-Thr(nHB) were favoured pairs, where the residues adopted favoured chi(1) rotamer positions that allowed side-chain interactions to occur. In contrast, Thr(HB)-Asn(nHB) and Thr(HB)-Arg(nHB) were not significantly favoured, and could only form side-chain interactions if the residues involved adopted less favourable chi(1) conformations. The favourability of hydrophobic pairs e.g. Ile(HB)-Ile(nHB), Val(HB)-Val(nHB) and Leu(HB)-Ile(nHB) was explained by the residues adopting their most preferred chi(1) and chi(2) conformations, which enabled them to form nested arrangements. Cysteine-cysteine pairs are significantly favoured, although these do not form intrasheet disulphide bridges. Interactions between positively and negatively charged residues were asymmetrically preferred: those with the negatively charged residue at the HB position were more favoured. This trend was accounted for by the presence of general electrostatic interactions, which, based on analysis of distances between charged atoms, were likely to be stronger when the negatively charged residue is the HB partner. The Arg(HB)-Asp(nHB) interaction was an exception to this trend and its favourability was rationalised by the formation of specific side-chain interactions. This research provides rules that could be applied to protein structure prediction, comparative modelling and protein engineering and design. The methods used to analyse the pairing preferences are automated and detailed results are available (http:// www.rubic.rdg.ac.uk/betapairprefsparallel/). (c) 2005 Elsevier Ltd. All rights reserved.

Shallow water flow with cylindrical symmetry

A numerical scheme is presented tor the solution of the shallow water equations in a single radial coordinate. This can prove useful when testing codes for the two-dimensional shallow water equations. The scheme is applied with success to problems involving converging and diverging bores.

Rotational symmetry of a methane molecule and the bond angle

The rotational symmetry of a methane molecule can be used to great advantage to calculate the bond angle. The problem is worked out in this article.

From symmetry breaking to Poisson Point Process in 2D Voronoi Tessellations: the generic nature of hexagons

We bridge the properties of the regular triangular, square, and hexagonal honeycomb Voronoi tessellations of the plane to the Poisson-Voronoi case, thus analyzing in a common framework symmetry breaking processes and the approach to uniform random distributions of tessellation-generating points. We resort to ensemble simulations of tessellations generated by points whose regular positions are perturbed through a Gaussian noise, whose variance is given by the parameter α2 times the square of the inverse of the average density of points. We analyze the number of sides, the area, and the perimeter of the Voronoi cells. For all valuesα >0, hexagons constitute the most common class of cells, and 2-parameter gamma distributions provide an efficient description of the statistical properties of the analyzed geometrical characteristics. The introduction of noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α = 0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise withα <0.12. For all tessellations and for small values of α, we observe a linear dependence on α of the ensemble mean of the standard deviation of the area and perimeter of the cells. Already for a moderate amount of Gaussian noise (α >0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α >2, results converge to those of Poisson-Voronoi tessellations. The geometrical properties of n-sided cells change with α until the Poisson- Voronoi limit is reached for α > 2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established. This law allows for an easy link to the Lewis law for areas and agrees with exact asymptotic results. Finally, for α >1, the ensemble mean of the cells area and perimeter restricted to the hexagonal cells agree remarkably well with the full ensemble mean; this reinforces the idea that hexagons, beyond their ubiquitous numerical prominence, can be interpreted as typical polygons in 2D Voronoi tessellations.

Symmetry breaking, mixing, instability, and low frequency variability in a minimal Lorenz-like system

Starting from the classical Saltzman two-dimensional convection equations, we derive via a severe spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled generalized Lorenz system. The consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables, with the ensuing modifications to the structural properties of the attractor and of the spectral features. When the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time scale of O(Ec−1) is introduced in the system, as shown with standard multiscale analysis and made clear by several numerical evidences. Moreover, the system is ergodic and hyperbolic, the slow variables feature long-term memory with 1/f3/2 power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics, such as the interaction between slow and fast variables, the presence of long-term memory, and the associated extreme value statistics. This analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic. In fact, this leads to spurious invariances that affect essential dynamical properties (ergodicity, hyperbolicity) and that cause the model losing ability in describing intrinsically multiscale processes.