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.

T1/ST2 is preferentially expressed on murine Th2 cells, independent of interleukin 4, interleukin 5, and interleukin 10, and important for Th2 effector function

T helper (Th) cells can be categorized according to their cytokine expression. The differential induction of Th cells expressing Th1 and/or Th2 cytokines is key to the regulation of both protective and pathological immune responses. Cytokines are expressed transiently and there is a lack of stably expressed surface molecules, significant for functionally different types of Th cells. Such molecules are of utmost importance for the analysis and selective functional modulation of Th subsets and will provide new therapeutic strategies for the treatment of allergic or autoimmune diseases. To this end, we have identified potential target genes preferentially expressed in Th2 cells, expressing interleukin (IL)-4, IL-5, and/or IL-10, but not interferon-γ. One such gene, T1/ST2, is expressed stably on both Th2 clones and Th2-polarized cells activated in vivo or in vitro. T1/ST2 expression is independent of induction by IL-4, IL-5, or IL-10. T1/ST2 plays a critical role in Th2 effector function. Administration of either a mAb against T1/ST2 or recombinant T1/ST2 fusion protein attenuates eosinophilic inflammation of the airways and suppresses IL-4 and IL-5 production in vivo following adoptive transfer of Th2 cells.

Differential binding to HLA-C of p50-activating and p58-inhibitory natural killer cell receptors

Natural killer (NK) cell cytotoxicity is regulated in large part by the expression of NK cell receptors able to bind class I major histocompatibility complex glycoproteins. The receptors associated with recognition of HLA-C allospecificities are the two-domain Ig-like molecules, p50 and p58 proteins, with highly homologous extracellular domains but differing in that they have either an activating or inhibitory function, respectively, depending on the transmembrane domain and cytoplasmic tails that they possess. We have compared the binding to HLA-Cw7 of an inhibitory p58 molecule, NKAT2, the highly homologous activating p50 molecule, clone 49, and a second activating p50 molecule, clone 39, which has homologies to both NKAT1 and NKAT2. NKAT2 binds to HLA-Cw7 with very rapid association and dissociation rates. However, the p50 receptors bind only very weakly, if at all, to HLA-C. The molecular basis of this difference is analyzed, and the functional significance of these observations is discussed.

Protein binding and signaling properties of RIN1 suggest a unique effector function

Human RIN1 was first characterized as a RAS binding protein based on the properties of its carboxyl-terminal domain. We now show that full-length RIN1 interacts with activated RAS in mammalian cells and defines a minimum region of 434 aa required for efficient RAS binding. RIN1 interacts with the “effector domain” of RAS and employs some RAS determinants that are common to, and others that are distinct from, those required for the binding of RAF1, a known RAS effector. The same domain of RIN1 that binds RAS also interacts with 14-3-3 proteins, extending the similarity between RIN1 and other RAS effectors. When expressed in mammalian cells, the RAS binding domain of RIN1 can act as a dominant negative signal transduction blocker. The amino-terminal domain of RIN1 contains a proline-rich sequence similar to consensus Src homology 3 (SH3) binding regions. This RIN1 sequence shows preferential binding to the ABL–SH3 domain in vitro. Moreover, the amino-terminal domain of RIN1 directly associates with, and is tyrosine phosphorylated by, c-ABL. In addition, RIN1 encodes a functional SH2 domain that has the potential to activate downstream signals. These data suggest that RIN1 is able to mediate multiple signals. A differential pattern of expression and alternate splicing indicate several levels of RIN1 regulation.

Reciprocal subtraction differential RNA display: An efficient and rapid procedure for isolating differentially expressed gene sequences

A reciprocal subtraction differential RNA display (RSDD) approach has been developed that permits the rapid and efficient identification and cloning of both abundant and rare differentially expressed genes. RSDD comprises reciprocal subtraction of cDNA libraries followed by differential RNA display. The RSDD strategy was applied to analyze the gene expression alterations resulting during cancer progression as adenovirus-transformed rodent cells developed an aggressive transformed state, as documented by elevated anchorage-independence and enhanced in vivo oncogenesis in nude mice. This approach resulted in the identification and cloning of both known and a high proportion (>65%) of unknown sequences, including cDNAs displaying elevated expression as a function of progression (progression-elevated gene) and cDNAs displaying suppressed expression as a function of progression (progression-suppressed gene). Sixteen differentially expressed genes, including five unknown progression-elevated genes and six unknown progression-suppressed genes, have been characterized. The RSDD scheme should find wide application for the effective detection and isolation of differentially expressed genes.

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.

Differential Roles of Syntaxin 7 and Syntaxin 8 in Endosomal Trafficking

To understand molecular mechanisms that regulate the intricate and dynamic organization of the endosomal compartment, it is important to establish the morphology, molecular composition, and functions of the different organelles involved in endosomal trafficking. Syntaxins and vesicle-associated membrane protein (VAMP) families, also known as soluble N-ethylmaleimide-sensitive factor (NSF) attachment protein receptors (SNAREs), have been implicated in mediating membrane fusion and may play a role in determining the specificity of vesicular trafficking. Although several SNAREs, including VAMP3/cellubrevin, VAMP8/endobrevin, syntaxin 13, and syntaxin 7, have been localized to the endosomal membranes, their precise localization, biochemical interactions, and function remain unclear. Furthermore, little is known about SNAREs involved in lysosomal trafficking. So far, only one SNARE, VAMP7, has been localized to late endosomes (LEs), where it is proposed to mediate trafficking of epidermal growth factor receptor to LEs and lysosomes. Here we characterize the localization and function of two additional endosomal syntaxins, syntaxins 7 and 8, and propose that they mediate distinct steps of endosomal protein trafficking. Both syntaxins are found in SNARE complexes that are dissociated by α-soluble NSF attachment protein and NSF. Syntaxin 7 is mainly localized to vacuolar early endosomes (EEs) and may be involved in protein trafficking from the plasma membrane to the EE as well as in homotypic fusion of endocytic organelles. In contrast, syntaxin 8 is likely to function in clathrin-independent vesicular transport and membrane fusion events necessary for protein transport from EEs to LEs.

Differential In Vivo Regulation of Steroid Hormone Receptor Activation by Cdc37p

The CDC37 gene is essential for the activity of p60v-src when expressed in yeast cells. Since the activation pathway for p60v-src and steroid hormone receptors is similar, the present study analyzed the hormone-dependent transactivation by androgen receptors and glucocorticoid receptors in yeast cells expressing a mutant version of the CDC37 gene. In this mutant, hormone-dependent transactivation by androgen receptors was defective at both permissive and restrictive temperatures, although transactivation by glucocorticoid receptors was mildly defective only at the restrictive temperature. Cdc37p appears to function via the androgen receptor ligand-binding domain, although it does not influence receptor hormone-binding affinity. Models for Cdc37p regulation of steroid hormone receptors are discussed.

A T42A Ran Mutation: Differential Interactions with Effectors and Regulators, and Defect in Nuclear Protein Import

Ran, the small, predominantly nuclear GTPase, has been implicated in the regulation of a variety of cellular processes including cell cycle progression, nuclear-cytoplasmic trafficking of RNA and protein, nuclear structure, and DNA synthesis. It is not known whether Ran functions directly in each process or whether many of its roles may be secondary to a direct role in only one, for example, nuclear protein import. To identify biochemical links between Ran and its functional target(s), we have generated and examined the properties of a putative Ran effector mutation, T42A-Ran. T42A-Ran binds guanine nucleotides as well as wild-type Ran and responds as well as wild-type Ran to GTP or GDP exchange stimulated by the Ran-specific guanine nucleotide exchange factor, RCC1. T42A-Ran·GDP also retains the ability to bind p10/NTF2, a component of the nuclear import pathway. In contrast to wild-type Ran, T42A-Ran·GTP binds very weakly or not detectably to three proposed Ran effectors, Ran-binding protein 1 (RanBP1), Ran-binding protein 2 (RanBP2, a nucleoporin), and karyopherin β (a component of the nuclear protein import pathway), and is not stimulated to hydrolyze bound GTP by Ran GTPase-activating protein, RanGAP1. Also in contrast to wild-type Ran, T42A-Ran does not stimulate nuclear protein import in a digitonin permeabilized cell assay and also inhibits wild-type Ran function in this system. However, the T42A mutation does not block the docking of karyophilic substrates at the nuclear pore. These properties of T42A-Ran are consistent with its classification as an effector mutant and define the exposed region of Ran containing the mutation as a probable effector loop.