Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Lessons from the past: Evolutionary impacts of mass extinctions

Mass extinctions have played many evolutionary roles, involving differential survivorship or selectivity of taxa and traits, the disruption or preservation of evolutionary trends and ecosystem organization, and the promotion of taxonomic and morphological diversifications—often along unexpected trajectories—after the destruction or marginalization of once-dominant clades. The fossil record suggests that survivorship during mass extinctions is not strictly random, but it often fails to coincide with factors promoting survival during times of low extinction intensity. Although of very serious concern, present-day extinctions have not yet achieved the intensities seen in the Big Five mass extinctions of the geologic past, which each removed ≥50% of the subset of relatively abundant marine invertebrate genera. The best comparisons for predictive purposes therefore will involve factors such as differential extinction intensities among regions, clades, and functional groups, rules governing postextinction biotic interchanges and evolutionary dynamics, and analyses of the factors that cause taxa and evolutionary trends to continue unabated, to suffer setbacks but resume along the same trajectory, to survive only to fall into a marginal role or disappear (“dead clade walking”), or to undergo a burst of diversification. These issues need to be addressed in a spatially explicit framework, because the fossil record suggests regional differences in postextinction diversification dynamics and biotic interchanges. Postextinction diversifications lag far behind the initial taxonomic and morphological impoverishment and homogenization; they do not simply reoccupy vacated adaptive peaks, but explore opportunities as opened and constrained by intrinsic biotic factors and the ecological and evolutionary context of the radiation.

The law of mass action governs antigen-stimulated cytolytic activity of CD8+ cytotoxic T lymphocytes.

An analysis of the initial antigen-recognition step in the destruction of target cells by CD8+ cytolytic T lymphocytes (CTLs) shows that a relationship in the form of the law of mass action can be used to describe interactions between antigen-specific receptors on T cells (TCRs) and their natural ligands on target cells (peptide-major histocompatibility protein complexes, termed pepMHC complexes), even though these reactants are confined to their respective cell membranes. For a designated level of lysis and receptor affinities below about 5 X 10(6) M-1, the product of the required number of pepMHC complexes per target cell ("epitope density") and TCR affinity for pepMHC complexes is constant; therefore, over this range TCR affinities can be predicted from epitope densities (or vice versa). At higher receptor affinities ("affinity ceiling") the epitope density required for half-maximal lysis reaches a lower limit of less than 10 complexes per target cell.

Broken symmetry and strangeness of the semiconductor impurity band metal-insulator transition

The filamentary model of the metal-insulator transition in randomly doped semiconductor impurity bands is geometrically equivalent to similar models for continuous transitions in dilute antiferromagnets and even to the λ transition in liquid He, but the critical behaviors are different. The origin of these differences lies in two factors: quantum statistics and the presence of long range Coulomb forces on both sides of the transition in the electrical case. In the latter case, in addition to the main transition, there are two satellite transitions associated with disappearance of the filamentary structure in both insulating and metallic phases. These two satellite transitions were first identified by Fritzsche in 1958, and their physical origin is explained here in geometrical and topological terms that facilitate calculation of critical exponents.

DNA binding sites for the Mlc and NagC proteins: regulation of nagE, encoding the N-acetylglucosamine-specific transporter in Escherichia coli

The NagC and Mlc proteins are homologous transcriptional regulators that control the expression of several phosphotransferase system (PTS) genes in Escherichia coli. NagC represses nagE, encoding the N-acetylglucosamine-specific transporter, while Mlc represses three PTS operons, ptsG, manXYZ and ptsHIcrr, involved in the uptake of glucose. NagC and Mlc can bind to each others operator, at least in vitro. A binding site selection procedure was used to try to distinguish NagC and Mlc sites. The major difference was that all selected NagC binding sites had a G or a C at positions +11/–11 from the centre of symmetry. This is also the case for most native NagC sites, but not the nagE operator, which thus looks like a potential Mlc target. The nagE operator does exhibit a higher affinity for Mlc than NagC, but no regulation of nagE by physiological concentrations of Mlc was detected in vivo. Regulation of wild-type nagE by NagC is achieved because of the chelation effect due to a second high affinity NagC operator covering the nagB promoter. Replacing the A/T at +11/–11 with C/G allows repression by NagC in the absence of the nagB operator.

DNA-bound transcription factor complexes analysed by mass-spectrometry: binding of novel proteins to the human c-fos SRE and related sequences

Transcription factors control eukaryotic polymerase II function by influencing the recruitment of multiprotein complexes to promoters and their subsequent integrated function. The complexity of the functional ‘transcriptosome’ has necessitated biochemical fractionation and subsequent protein sequencing on a grand scale to identify individual components. As a consequence, much is now known of the basal transcription complex. In contrast, less is known about the complexes formed at distal promoter elements. The c-fos SRE, for example, is known to bind Serum Response Factor (SRF) and ternary complex factors such as Elk-1. Their interaction with other factors at the SRE is implied but, to date, none have been identified. Here we describe the use of mass-spectrometric sequencing to identify six proteins, SRF, Elk-1 and four novel proteins, captured on SRE duplexes linked to magnetic beads. This approach is generally applicable to the characterisation of nucleic acid-bound protein complexes and the post-translational modification of their components.

Even-odd alternation in mass spectrum of thymine and uracil clusters: Evidence of intracluster photodimerization

Multiphoton ionization of thymine and uracil clusters generated by a supersonic molecular beam gave rise to a remarkable alternation of mass spectral intensities between even- and odd-numbered clusters. Such alternation was observed in clusters of up to 30 molecules. Excitation to the two lowest electronically excited states seemed to be a strong prerequisite. In view of the well known photodimerization reaction of thymine and uracil in the bulk phase, it is proposed that such alternation in the mass spectral intensity resulted from formation of photodimer units within the cluster on intense UV irradiation. Several analogues of thymine with no known propensity for photodimerization in the bulk phase did not exhibit any sign of such alternation in the cluster mass spectrum. The intrinsic UV window for photodimerization, and hence photoinduced mammalian mutagenesis, was estimated to be approximately 210–280 nm, significantly narrower than the previously reported bulk values of 150–300 nm.

Symmetry, stability, and dynamics of multidomain and multicomponent protein systems

Symmetry is commonly observed in many biological systems. Here we discuss representative examples of the role of symmetry in structural molecular biology. Point group symmetries are observed in many protein oligomers whose three-dimensional atomic structures have been elucidated by x-ray crystallography. Approximate symmetry also occurs in multidomain proteins. Symmetry often confers stability on the molecular system and results in economical usage of basic components to build the macromolecular structure. Symmetry is also associated with cooperativity. Mild perturbation from perfect symmetry may be essential in some systems for dynamic functions.

Symmetry and the energy landscapes of biomolecules

The role of symmetry in the folding of proteins is discussed using energy landscape theory. An analytical argument shows it is much easier to find sequences with funneled energy landscape capable of fast folding if the structure is symmetric. The analogy with phase transitions of small clusters with magic numbers is discussed.

Mass vaccination to control chickenpox: the influence of zoster.

The impact of transmission events from patients with shingles (zoster) on the epidemiology of varicella is examined before and after the introduction of mass immunization by using a stochastic mathematical model of transmission dynamics. Reactivation of the virus is shown to damp stochastic fluctuations and move the dynamics toward simple annual oscillations. The force of infection due to zoster cases is estimated by comparison of simulated and observed incidence time series. The presence of infectious zoster cases reduces the tendency for mass immunization to increase varicella incidence at older ages when disease severity is typically greater.