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We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems. We find that the class of systems of differential equations studied is naturally divided into two cases on the basis of the type of a function that they contain. In each case, the dimension of the Lie algebra generated by the infinitesimal symmetries is greater than 2, and for this reason we present the structures and one-dimensional optimal systems of these Lie algebras. The reduced systems corresponding to the optimal systems are also obtained. Explicit group invariant solutions are found for particular cases., Comment: 14 pages. To appear in Applied Mathematics and Computation

The wnt protein family has important members involved in cell differentiation, proliferation and plasticity expression; however, little is known about its biosynthesis processes. On the other hand, an increase in the intracerebral cyclic adenosine 3', 5'-monophosphate (cAMP) level leads to synaptic plasticity via the de novo synthesis of any protein. Here, the effect of dibutyryl cAMP (dbcAMP), a membrane permeability cAMP analog, on the wnt family was investigated in rat primary-cultured glial cells containing astrocytes and microglia. Among wnt3a, 4, 5a, 7a and 11 mRNA, only wnt4 expression was increased by longer treatment (24 h), compared with short treatment (2 h), with dbcAMP in a concentration-dependent manner, and its effect reached statistical significance at 1 mM. In cultures of isolated astrocytes or microglia, wnt4 expression was not affected by 1 mM dbcAMP for 24 h, and microglial wnt4 protein was undetectable even when cells were treated with the drug. Mixed glial cells treated for 24 h with 1 mM dbcAMP showed significantly increased wnt4 protein, as well as mRNA. Immunofluorescence manifested that cells that expressed wnt4 protein were astrocytes, but not microglia. Intraperitoneal injection of 1.25 mg/kg rolipram, a phosphodiesterase (PDE) IV inhibitor that can pass through the blood brain barrier and inhibits cAMP degradation specifically, showed a tendency to increase wnt4 expression in the adult rat brain after 24 h, and the increases in wnt4 mRNA and protein levels reached statistical significance in the hippocampus and striatum, respectively. This is the first finding to help elucidate the selective biosynthesis of central wnt4 through cAMP-stimulated microglia and astrocytes interaction.

A sensitive and specific analytical method for a potent antitumor agent, TZT-1027, in plasma has been developed using liquid chromatography–mass spectrometry (LC–MS) with [2H4]TZT-1027 as an internal standard (I.S.). A plasma sample was purified by solid-phase extraction on a C18 cartridge, followed by solvent extraction with diethyl ether. The extract was then injected into the LC–MS system. Chromatography was carried out on a C18 reversed-phase column using acetonitrile–0.05% trifluoroacetic acid (TFA) (55:45) as a mobile phase. Mass spectrometric analysis was performed in atmospheric pressure chemical ionization (APCI) mode with positive ion detection, and the protonated molecular ions ([M+H]+) of TZT-1027 and I.S. were monitored to allow quantitation. The method was applied to the determination of TZT-1027 in human, monkey, dog, rat and mouse plasma. As far as the sample preparation was concerned, good recoveries (73.5–99.1%) were obtained. The calibration curves were linear over the range of 0.25–100 ng per 1 ml of human, dog and rat plasma, per 0.5 ml of monkey plasma, and per 0.1 ml of mouse plasma. From the intra- and inter-day accuracy and precision, the present method satisfies the accepted criteria for bioanalytical method validation. TZT-1027 was stable when stored below −15°C for 6 months in human plasma and for 3 weeks in plasma from other species. TZT-1027 was also stable in plasma through at least three freeze–thaw cycles.

We give quotients V // G = Spec ( C [ V ] G ) for some prehomogeneous vector spaces ( G × H , V ).