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.

Advances in random matrix theory, zeta functions, and sphere packing

Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues (“the energy levels”) follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These topics are distinct, so we present them separately with their own introductory remarks.

A fast marching level set method for monotonically advancing fronts.

A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial differential equation for a propagating level set function and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. This paper describes a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations, and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithographic development calculations in microchip manufacturing, and arrival time problems in control theory.

A model-based approach for assessing in vivo combination therapy interactions

We present an approach for evaluating the efficacy of combination antitumor agent schedules that accounts for order and timing of drug administration. Our model-based approach compares in vivo tumor volume data over a time course and offers a quantitative definition for additivity of drug effects, relative to which synergism and antagonism are interpreted. We begin by fitting data from individual mice receiving at most one drug to a differential equation tumor growth/drug effect model and combine individual parameter estimates to obtain population statistics. Using two null hypotheses: (i) combination therapy is consistent with additivity or (ii) combination therapy is equivalent to treating with the more effective single agent alone, we compute predicted tumor growth trajectories and their distribution for combination treated animals. We illustrate this approach by comparing entire observed and expected tumor volume trajectories for a data set in which HER-2/neu-overexpressing MCF-7 human breast cancer xenografts are treated with a humanized, anti-HER-2 monoclonal antibody (rhuMAb HER-2), doxorubicin, or one of five proposed combination therapy schedules.

P-glycoprotein function involves conformational transitions detectable by differential immunoreactivity

The MDR1 P-glycoprotein (Pgp), a member of the ATP-binding cassette family of transporters, is a transmembrane ATPase efflux pump for various lipophilic compounds, including many anti-cancer drugs. mAb UIC2, reactive with the extracellular moiety of Pgp, inhibits Pgp-mediated efflux. UIC2 reactivity with Pgp was increased by the addition of several Pgp-transported compounds or ATP-depleting agents, and by mutational inactivation of both nucleotide-binding domains (NBDs) of Pgp. UIC2 binding to Pgp mutated in both NBDs was unaffected in the presence of Pgp transport substrates or in ATP-depleted cells, whereas the reactivities of the wild-type Pgp and Pgps mutated in a single NBD were increased by these treatments to the level of the double mutant. These results indicate the existence of different Pgp conformations associated with different stages of transport-associated ATP hydrolysis and suggest trapping in a transient conformation as a mechanism for antibody-mediated inhibition of Pgp.

Differential Epitope Tagging of Actin in Transformed Drosophila Produces Distinct Effects on Myofibril Assembly and Function of the Indirect Flight Muscle

We have tested the impact of tags on the structure and function of indirect flight muscle (IFM)-specific Act88F actin by transforming mutant Drosophila melanogaster, which do not express endogenous actin in their IFMs, with tagged Act88F constructs. Epitope tagging is often the method of choice to monitor the fate of a protein when a specific antibody is not available. Studies addressing the functional significance of the closely related actin isoforms rely almost exclusively on tagged exogenous actin, because only few antibodies exist that can discriminate between isoforms. Thereby it is widely presumed that the tag does not significantly interfere with protein function. However, in most studies the tagged actin is expressed in a background of endogenous actin and, as a rule, represents only a minor fraction of the total actin. The Act88F gene encodes the only Drosophila actin isoform exclusively expressed in the highly ordered IFM. Null mutations in this gene do not affect viability, but phenotypic effects in transformants can be directly attributed to the transgene. Transgenic flies that express Act88F with either a 6x histidine tag or an 11-residue peptide derived from vesicular stomatitis virus G protein at the C terminus were flightless. Overall, the ultrastructure of the IFM resembled that of the Act88F null mutant, and only low amounts of C-terminally tagged actins were found. In contrast, expression of N-terminally tagged Act88F at amounts comparable with that of wild-type flies yielded fairly normal-looking myofibrils and partially reconstituted flight ability in the transformants. Our findings suggest that the N terminus of actin is less sensitive to modifications than the C terminus, because it can be tagged and still polymerize into functional thin filaments.

Differential requirement of the cytoplasmic subregions of γc chain in T cell development and function

The common cytokine receptor γ chain (γc), a shared component of the receptors for IL-2, IL-4, IL-7, IL-9, and IL-15, is critical for the development and function of lymphocytes. The cytoplasmic domain of γc consists of 85 aa, in which the carboxyl-terminal 48 aa are essential for its interaction with and activation of the Janus kinase, Jak3. Evidence has been provided that Jak3-independent signals might be transmitted via the residual membrane-proximal region; however, its role in vivo remains totally unknown. In the present study, we expressed mutant forms of γc, which lack either most of the cytoplasmic domain or only the membrane-distal Jak3-binding region, on a γc null background. We demonstrate that, unlike γc or Jak3 null mice, expression of the latter, but not the former mutant, restores T lymphopoiesis in vivo, accompanied by strong expression of Bcl-2. On the other hand, the in vitro functions of the restored T cells still remained impaired. These results not only reveal the hitherto unknown role of the γc membrane-proximal region, but also suggest the differential requirement of the cytoplasmic subregions of γc in T cell development and function.