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.

Assembly of a Novel Cartilage Matrix Protein Filamentous Network: Molecular Basis of Differential Requirement of von Willebrand Factor A Domains

Cartilage matrix protein (CMP) is the prototype of the newly discovered matrilin family, all of which contain von Willebrand factor A domains. Although the function of matrilins remain unclear, we have shown that, in primary chondrocyte cultures, CMP (matrilin-1) forms a filamentous network, which is made up of two types of filaments, a collagen-dependent one and a collagen-independent one. In this study, we demonstrate that the collagen-independent CMP filaments are enriched in pericellular compartments, extending directly from chondrocyte membranes. Their morphology can be distinguished from that of collagen filaments by immunogold electron microscopy, and mimicked by that of self-assembled purified CMP. The assembly of CMP filaments can occur from transfection of a wild-type CMP transgene alone in skin fibroblasts, which do not produce endogenous CMP. Conversely, assembly of endogenous CMP filaments by chondrocytes can be inhibited specifically by dominant negative CMP transgenes. The two A domains within CMP serve essential but different functions during network formation. Deletion of the A2 domain converts the trimeric CMP into a mixture of monomers, dimers, and trimers, whereas deletion of the A1 domain does not affect the trimeric configuration. This suggests that the A2 domain modulates multimerization of CMP. Absence of either A domain from CMP abolishes its ability to form collagen-independent filaments. In particular, Asp22 in A1 and Asp255 in A2 are essential; double point mutation of these residues disrupts CMP network formation. These residues are part of the metal ion–dependent adhesion sites, thus a metal ion–dependent adhesion site–mediated adhesion mechanism may be applicable to matrilin assembly. Taken together, our data suggest that CMP is a bridging molecule that connects matrix components in cartilage to form an integrated matrix network.