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.

On the controllability of distributed systems

To “control” a system is to make it behave (hopefully) according to our “wishes,” in a way compatible with safety and ethics, at the least possible cost. The systems considered here are distributed—i.e., governed (modeled) by partial differential equations (PDEs) of evolution. Our “wish” is to drive the system in a given time, by an adequate choice of the controls, from a given initial state to a final given state, which is the target. If this can be achieved (respectively, if we can reach any “neighborhood” of the target) the system, with the controls at our disposal, is exactly (respectively, approximately) controllable. A very general (and fuzzy) idea is that the more a system is “unstable” (chaotic, turbulent) the “simplest,” or the “cheapest,” it is to achieve exact or approximate controllability. When the PDEs are the Navier–Stokes equations, it leads to conjectures, which are presented and explained. Recent results, reported in this expository paper, essentially prove the conjectures in two space dimensions. In three space dimensions, a large number of new questions arise, some new results support (without proving) the conjectures, such as generic controllability and cases of decrease of cost of control when the instability increases. Short comments are made on models arising in climatology, thermoelasticity, non-Newtonian fluids, and molecular chemistry. The Introduction of the paper and the first part of all sections are not technical. Many open questions are mentioned in the text.

Tubulin Polyglycylation: Differential Posttranslational Modification of Dynamic Cytoplasmic and Stable Axonemal Microtubules in Paramecium

Polyglycylation, a posttranslational modification of tubulin, was discovered in the highly stable axonemal microtubules of Paramecium cilia where it involves the lateral linkage of up to 34 glycine units per tubulin subunit. The observation of this type of posttranslational modification mainly in axonemes raises the question as to its relationship with axonemal organization and with microtubule stability. This led us to investigate the glycylation status of cytoplasmic microtubules that correspond to the dynamic microtubules in Paramecium. Two anti-glycylated tubulin monoclonal antibodies (mAbs), TAP 952 and AXO 49, are shown here to exhibit different affinities toward mono- and polyglycylated synthetic tubulin peptides. Using immunoblotting and mass spectrometry, we show that cytoplasmic tubulin is glycylated. In contrast to the highly glycylated axonemal tubulin, which is recognized by the two mAbs, cytoplasmic tubulin reacts exclusively with TAP 952, and the α- and β- tubulin subunits are modified by only 1–5 and 2–9 glycine units, respectively. Our analyses suggest that most of the cytoplasmic tubulin contains side chain lengths of 1 or 2 glycine units distributed on several glycylation sites. The subcellular partition of distinct polyglycylated tubulin isoforms between cytoplasmic and axonemal compartments implies the existence of regulatory mechanisms for glycylation. By following axonemal tubulin immunoreactivity with anti-glycylated tubulin mAbs upon incubation with a Paramecium cellular extract, the presence of a deglycylation enzyme is revealed in the cytoplasm of this organism. These observations establish that polyglycylation is reversible and indicate that, in vivo, an equilibrium between glycylating and deglycylating enzymes might be responsible for the length of the oligoglycine side chains of tubulin.

Transition-state structure as a unifying basis in protein-folding mechanisms: Contact order, chain topology, stability, and the extended nucleus mechanism

I attempt to reconcile apparently conflicting factors and mechanisms that have been proposed to determine the rate constant for two-state folding of small proteins, on the basis of general features of the structures of transition states. Φ-Value analysis implies a transition state for folding that resembles an expanded and distorted native structure, which is built around an extended nucleus. The nucleus is composed predominantly of elements of partly or well-formed native secondary structure that are stabilized by local and long-range tertiary interactions. These long-range interactions give rise to connecting loops, frequently containing the native loops that are poorly structured. I derive an equation that relates differences in the contact order of a protein to changes in the length of linking loops, which, in turn, is directly related to the unfavorable free energy of the loops in the transition state. Kinetic data on loop extension mutants of CI2 and α-spectrin SH3 domain fit the equation qualitatively. The rate of folding depends primarily on the interactions that directly stabilize the nucleus, especially those in native-like secondary structure and those resulting from the entropy loss from the connecting loops, which vary with contact order. This partitioning of energy accounts for the success of some algorithms that predict folding rates, because they use these principles either explicitly or implicitly. The extended nucleus model thus unifies the observations of rate depending on both stability and topology.

Differential Expression of Genes for Cyclin-Dependent Protein Kinases in Rice Plants1

Cyclin-dependent protein kinases (CDKs) play key roles in regulating the eukaryotic cell cycle. We have analyzed the expression of four rice (Oryza sativa) CDK genes, cdc2Os1, cdc2Os2, cdc2Os3, and R2, by in situ hybridization of sections of root apices. Transcripts of cdc2Os1, cdc2Os2, and R2 were detected uniformly in the dividing region of the root apex. cdc2Os1 and cdc2Os2 were also expressed in differentiated cells such as those in the sclerenchyma, pericycle, and parenchyma of the central cylinder. By contrast, signals corresponding to transcripts of cdc2Os3 were distributed only in patches in the dividing region. Counterstaining of sections with 4′,6-diamidino-2-phenylindole and double-target in situ hybridization with a probe for histone H4 transcripts revealed that cdc2Os3 transcripts were abundant from the G2 to the M phase, but were less abundant or absent during the S phase. The levels of the Cdc2Os3 protein and its associated histone H1-kinase activity were reduced by treatment of cultured cells with hydroxyurea, which blocks cycling cells at the onset of the S phase. Our results suggest that domains other than the conserved amino acid sequence (the PSTAIRE motif) have important roles in the function of non-PSTAIRE CDKs in distinct cell-cycle phases.

Tomato Fructokinases Exhibit Differential Expression and Substrate Regulation1

Two divergent genes encoding fructokinase, Frk1 and Frk2, have been previously shown to be expressed in tomato (Lycopersicon esculentum L.) and have now been further characterized with regard to their spatial expression and the enzymic properties of the encoded proteins. Frk1 and Frk2 mRNA levels were coordinately induced by exogenous sugar, indicating that both belong to the growing class of sugar-regulated genes. However, in situ hybridization indicated that Frk1 and Frk2 were expressed in a spatially distinct manner, with Frk2 mRNA primarily localized in cells of the fruit pericarp, which store starch, and Frk1 mRNA distributed ubiquitously in pericarp tissue. To evaluate the biochemical characteristics of the products of the Frk1 and Frk2 genes, each cDNA was expressed in a mutant yeast (Saccharomyces cerevisiae) line defective in hexose phosphorylation and unable to grow on glucose or fructose (Fru). Both Frk1 and Frk2 proteins expressed in yeast conferred the ability to grow on Fru and exhibited fructokinase activity in vitro. Although both Frk1 and Frk2 both utilized Fru as a substrate, only Frk2 activity was inhibited at high Fru concentrations. These results indicate that Frk2 can be distinguished from Frk1 by its sensitivity to substrate inhibition and by its temporal and spatial pattern of expression, which suggests that it plays a primary role in plant cells specialized for starch storage.