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.

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.

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.

Differential recognition of the type I interferon receptor by interferons tau and alpha is responsible for their disparate cytotoxicities.

Interferon tau (IFN tau), originally identified as a pregnancy recognition hormone, is a type I interferon that is related to the various IFN alpha species (IFN alpha s). Ovine IFN tau has antiviral activity similar to that of human IFN alpha A on the Madin-Darby bovine kidney (MDBK) cell line and is equally effective in inhibiting cell proliferation. In this study, IFN tau was found to differ from IFN alpha A in that is was > 30-fold less toxic to MDBK cells at high concentrations. Excess IFN tau did not block the cytotoxicity of IFN alpha A on MDBK cells, suggesting that these two type I IFNs recognize the type I IFN receptor differently on these cells. In direct binding studies, 125I-IFN tau had a Kd of 3.90 x 10(-10) M for receptor on MDBK cells, whereas that of 125I-IFN alpha A was 4.45 x 10(-11) M. Consistent with the higher binding affinity, IFN alpha A was severalfold more effective than IFN tau in competitive binding against 125I-IFN tau to receptor on MDBK cells. Paradoxically, the two IFNs had similar specific antiviral activities on MDBK cells. However, maximal IFN antiviral activity required only fractional occupancy of receptors, whereas toxicity was associated with maximal receptor occupancy. Hence, IFN alpha A, with the higher binding affinity, was more toxic than IFN tau. The IFNs were similar in inducing the specific phosphorylation of the type I receptor-associated tyrosine kinase Tyk2, and the transcription factors Stat1 alpha and Stat2, suggesting that phosphorylation of these signal transduction proteins is not involved in the cellular toxicity associated with type I IFNs. Experiments using synthetic peptides suggest that differences in the interaction at the N terminal of IFN tau and IFN alpha with the type I receptor complex contribute significantly to differences in high-affinity equilibrium binding of these molecules. It is postulated that such a differential recognition of the receptor is responsible for the similar antiviral but different cytotoxic effects of these IFNs. Moreover, these data imply that receptors are "spare'' with respect to certain biological properties, and we speculate that IFNs may induce a concentration-dependent selective association of receptor subunits.