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.

4E-BP1 phosphorylation is mediated by the FRAP-p70s6k pathway and is independent of mitogen-activated protein kinase.

It has previously been argued that the repressor of protein synthesis initiation factor 4E, 4E-BP1, is a direct in vivo target of p42mapk. However, the immunosuppressant rapamycin blocks serum-induced 4E-BP1 phosphorylation and, in parallel, p70s6k activation, with no apparent effect on p42mapk activation. Consistent with this finding, the kinetics of serum-induced 4E-BP1 phosphorylation closely follow those of p70s6k activation rather than those of p42mapk. More striking, insulin, which does not induce p42mapk activation in human 293 cells or Swiss mouse 3T3 cells, induces 4E-BP1 phosphorylation and p70s6k activation in both cell types. Anisomycin, which, like insulin, does not activate p42mapk, promotes a small parallel increase in 4E-BP1 phosphorylation and p70s6k activation. The insulin effect on 4E-BP1 phosphorylation and p70s6k activation in both cell types is blocked by SQ20006, wortmannin, and rapamycin. These three inhibitors have no effect on p42mapk activation induced by phorbol 12-tetradecanoate 13-acetate, though wortmannin partially suppresses both the p70s6k response and the 4E-BP1 response. Finally, in porcine aortic endothelial cells stably transfected with either the wild-type platelet-derived growth factor receptor or a mutant receptor bearing the double point mutation 740F/751F, p42mapk activation in response to platelet-derived growth factor is unimpaired, but increased 4E-BP1 phosphorylation is ablated, as previously reported for p70s6k. The data presented here demonstrate that 4E-BP1 phosphorylation is mediated by the FRAP-p70s6k pathway and is independent of mitogen-activated protein kinase.

Cell membrane fluctuations are regulated by medium macroviscosity: Evidence for a metabolic driving force

Extracellular fluid macroviscosity (EFM), modified by macromolecular cosolvents as occurs in body fluids, has been shown to affect cell membrane protein activities but not isolated proteins. In search for the mechanism of this phenomenon, we examined the effect of EFM on mechanical fluctuations of the cell membrane of human erythrocytes. The macroviscosity of the external medium was varied by adding to it various macromolecules [dextrans (70, 500, and 2,000 kDa), polyethylene glycol (20 kDa), and carboxymethyl-cellulose (100 kDa)], which differ in size, chemical nature, and in their capacity to increase fluid viscosity. The parameters of cell membrane fluctuations (maximal amplitude and half-width of amplitude distribution) were diminished with the elevation of solvent macroviscosity, regardless of the cosolvent used to increase EFM. Because thermally driven membrane fluctuations cannot be damped by elevation of EFM, the existence of a metabolic driving force is suggested. This is supported by the finding that in ATP-depleted red blood cells elevation of EMF did not affect cell membrane fluctuations. This study demonstrates that (i) EFM is a regulator of membrane dynamics, providing a possible mechanism by which EFM affects cell membrane activities; and (ii) cell membrane fluctuations are driven by a metabolic driving force in addition to the thermal one.