Identification of WNT/β-CATENIN Signaling Pathway Components in Human Cumulus Cells

Signaling via the conserved WNT/β-CATENIN pathway controls diverse developmental processes. To explore its potential role in the ovary, we investigated the expression of WNTs, frizzled (FZD) receptors and other pathway components in human cumulus cells obtained from oocytes collected for in vitro fertilization. Proteins were detected in cultured cells using immunofluorescence microscopy. Protein–protein interactions were analyzed by means of immunoprecipitation. WNT2, FZD2, FZD3 and FZD9 were identified but WNT1, WNT4 and FZD4 were not detected. WNT2 is co-expressed with FZD2, FZD3 and FZD9. Co-immunoprecipitation using WNT2 antibody demonstrated that WNT2 interacts with both FZD3 and FZD9, but only FZD9 antibody precipitated WNT2. We also identified DVL (disheveled), AXIN, GSK-3β (glycogen synthase kinase-3β) and β-CATENIN. β-CATENIN is concentrated in the plasma membranes. DVL co-localizes with FZD9 and AXIN in the membranes, but GSK-3β has little co-localization with AXIN and β-CATENIN. Interestingly, β-CATENIN is highly co-localized with FZD9 and AXIN. CDH1 (E-cadherin) was also detected in the plasma membranes and cytoplasm, co-localized with β-CATENIN, and CDH1 antibody precipitated β-CATENIN. The results suggest that WNT2 could act through its receptor FZD9 to regulate the β-CATENIN pathway in cumulus cells, recruiting β-CATENIN into plasma membranes and promoting the formation of adherens junctions involving CDH1

Einstein boundary conditions for the 3+1 Einstein equations

In the 3+1 framework of the Einstein equations for the case of vanishing shift vector and arbitrary lapse, we calculate explicitly the four boundary equations arising from the vanishing of the projection of the Einstein tensor along the normal to the boundary surface of the initial-boundary value problem. Such conditions take the form of evolution equations along (as opposed to across) the boundary for certain components of the extrinsic curvature and for certain space-derivatives of the intrinsic metric. We argue that, in general, such boundary conditions do not follow necessarily from the evolution equations and the initial data, but need to be imposed on the boundary values of the fundamental variables. Using the Einstein-Christoffel formulation, which is strongly hyperbolic, we show how three of the boundary equations should be used to prescribe the values of some incoming characteristic fields. Additionally, we show that the fourth one imposes conditions on some outgoing fields.Comment: Revtex 4, 6 pages, text and references added, typos corrected, to appear in Phys. Rev.

Evolution systems for non-linear perturbations of background geometries

The formulation of the initial value problem for the Einstein equations is at the heart of obtaining interesting new solutions using numerical relativity and still very much under theoretical and applied scrutiny. We develop a specialised background geometry approach, for systems where there is non-trivial a priori knowledge about the spacetime under study. The background three-geometry and associated connection are used to express the ADM evolution equations in terms of physical non-linear deviations from that background. Expressing the equations in first order form leads naturally to a system closely linked to the Einstein-Christoffel system, introduced by Anderson and York, and sharing its hyperbolicity properties. We illustrate the drastic alteration of the source structure of the equations, and discuss why this is likely to be numerically advantageous.Comment: 12 pages, 3 figures, Revtex v3.0. Revised version to appear in Physical Review

Hyperboloidal evolution of test fields in three spatial dimensions

We present the numerical implementation of a clean solution to the outer boundary and radiation extraction problems within the 3+1 formalism for hyperbolic partial differential equations on a given background. Our approach is based on compactification at null infinity in hyperboloidal scri fixing coordinates. We report numerical tests for the particular example of a scalar wave equation on Minkowski and Schwarzschild backgrounds. We address issues related to the implementation of the hyperboloidal approach for the Einstein equations, such as nonlinear source functions, matching, and evaluation of formally singular terms at null infinity.Comment: 10 pages, 8 figure

Spin effects in gravitational radiation backreaction II. Finite mass effects

A convenient formalism for averaging the losses produced by gravitational radiation backreaction over one orbital period was developed in an earlier paper. In the present paper we generalize this formalism to include the case of a closed system composed from two bodies of comparable masses, one of them having the spin S. We employ the equations of motion given by Barker and O'Connell, where terms up to linear order in the spin (the spin-orbit interaction terms) are kept. To obtain the radiative losses up to terms linear in the spin, the equations of motion are taken to the same order. Then the magnitude L of the angular momentum L, the angle kappa subtended by S and L and the energy E are conserved. The analysis of the radial motion leads to a new parametrization of the orbit. From the instantaneous gravitational radiation losses computed by Kidder the leading terms and the spin-orbit terms are taken. Following Apostolatos, Cutler, Sussman and Thorne, the evolution of the vectors S and L in the momentary plane spanned by these vectors is separated from the evolution of the plane in space. The radiation-induced change in the spin is smaller than the leading-order spin terms in the momentary angular momentum loss. This enables us to compute the averaged losses in the constants of motion E, L and L_S=L cos kappa. In the latter, the radiative spin loss terms average to zero. An alternative description using the orbital elements a,e and kappa is given. The finite mass effects contribute terms, comparable in magnitude, to the basic, test-particle spin terms in the averaged losses.Comment: 12 pages, 1 figure, Phys.Rev.D15, March, 199

Past and future gauge in numerical relativity

Numerical relativity describes a discrete initial value problem for general relativity. A choice of gauge involves slicing space-time into space-like hypersurfaces. This introduces past and future gauge relative to the hypersurface of present time. Here, we propose solving the discretized Einstein equations with a choice of gauge in the future and a dynamical gauge in the past. The method is illustrated on a polarized Gowdy wave.Comment: To appear in Class Quantum Grav, Let

Gravitational radiation from compact binary systems: gravitational waveforms and energy loss to second post-Newtonian order

We derive the gravitational waveform and gravitational-wave energy flux generated by a binary star system of compact objects (neutron stars or black holes), accurate through second post-Newtonian order ($O[(v/c)^4] \sim O[(Gm/rc^2)^2]$) beyond the lowest-order quadrupole approximation. We cast the Einstein equations into the form of a flat-spacetime wave equation together with a harmonic gauge condition, and solve it formally as a retarded integral over the past null cone of the chosen field point. The part of this integral that involves the matter sources and the near-zone gravitational field is evaluated in terms of multipole moments using standard techniques; the remainder of the retarded integral, extending over the radiation zone, is evaluated in a novel way. The result is a manifestly convergent and finite procedure for calculating gravitational radiation to arbitrary orders in a post-Newtonian expansion. Through second post-Newtonian order, the radiation is also shown to propagate toward the observer along true null rays of the asymptotically Schwarzschild spacetime, despite having been derived using flat spacetime wave equations. The method cures defects that plagued previous ``brute- force'' slow-motion approaches to the generation of gravitational radiation, and yields results that agree perfectly with those recently obtained by a mixed post-Minkowskian post-Newtonian method. We display explicit formulae for the gravitational waveform and the energy flux for two-body systems, both in arbitrary orbits and in circular orbits. In an appendix, we extend the formalism to bodies with finite spatial extent, and derive the spin corrections to the waveform and energy loss.Comment: 59 pages ReVTeX; Physical Review D, in press; figures available on request to [email protected]

Connexin Expression and Gap Junctional Coupling in Human Cumulus Cells: Contribution to Embryo Quality

Gap junctional coupling among cumulus cells is important for oogenesis since its deficiency in mice leads to impaired folliculogenesis. Multiple connexins (Cx), the subunits of gap junction channels, have been found within ovarian follicles in several species but little is known about the connexins in human follicles. The aim of this study was to determine which connexins contribute to gap junctions in human cumulus cells and to explore the possible relationship between connexin expression and pregnancy outcome from in vitro fertilization (IVF). Cumulus cells were obtained from IVF patients undergoing intracytoplasmic sperm injection (ICSI). Connexin expression was examined by RT-PCR and confocal microscopy. Cx43 was quantified by immunoblotting and gap junctional coupling was measured by patch-clamp electrophysiology. All but five of 20 connexin mRNAs were detected. Of the connexin proteins detected, Cx43 forms numerous gap junction-like plaques but Cx26, Cx30, Cx30.3, Cx32, and Cx40 appeared to be restricted to the cytoplasm. The strength of gap junctional conductance varied between patients and was significantly and positively correlated with Cx43 level, but neither was correlated with patient age. Interestingly, Cx43 level and intercellular conductance were positively correlated with embryo quality as judged by cleavage rate and morphology, and were significantly higher in patients who became pregnant than in those who did not. Thus, despite the presence of multiple connexins, Cx43 is a major contributor to gap junctions in human cumulus cells and its expression level may influence pregnancy outcome after ICSI

Solving the Darwin problem in the first post-Newtonian approximation of general relativity

We analytically calculate the equilibrium sequence of the corotating binary stars of incompressible fluid in the first post-Newtonian(PN) approximation of general relativity. By calculating the total energy and total angular momentum of the system as a function of the orbital separation, we investigate the innermost stable circular orbit for corotating binary(we call it ISCCO). It is found that by the first PN effect, the orbital separation of the binary at the ISCCO becomes small with increase of the compactness of each star, and as a result, the orbital angular velocity at the ISCCO increases. These behaviors agree with previous numerical works.Comment: 33 pages, revtex, 4 figures(eps), accepted for publication in Phys. Rev.

Geometrical optics analysis of the short-time stability properties of the Einstein evolution equations

Many alternative formulations of Einstein's evolution have lately been examined, in an effort to discover one which yields slow growth of constraint-violating errors. In this paper, rather than directly search for well-behaved formulations, we instead develop analytic tools to discover which formulations are particularly ill-behaved. Specifically, we examine the growth of approximate (geometric-optics) solutions, studied only in the future domain of dependence of the initial data slice (e.g. we study transients). By evaluating the amplification of transients a given formulation will produce, we may therefore eliminate from consideration the most pathological formulations (e.g. those with numerically-unacceptable amplification). This technique has the potential to provide surprisingly tight constraints on the set of formulations one can safely apply. To illustrate the application of these techniques to practical examples, we apply our technique to the 2-parameter family of evolution equations proposed by Kidder, Scheel, and Teukolsky, focusing in particular on flat space (in Rindler coordinates) and Schwarzchild (in Painleve-Gullstrand coordinates).Comment: Submitted to Phys. Rev.