Tjon Lines and Scaling Limit in Four-Body Systems

The fixed-slope correlation between tetramer and trimer binding energies, observed by Tjon in the context of nuclear physics, is mainly a manifestation of the dominance of the two-nucleon force in the nuclear potential, which makes the four-body scale on the order of the three-body one. In a more general four-boson case, the correlation between tetramer and trimer binding energies has a non-fixed slope, which expresses the dependence on the new scale. The associated scaling function generates a family of Tjon lines. This conclusion relies on a recent study with weakly-bound four identical bosons, within a renormalized zero-range Faddeev-Yakubovsky formalism.     

GAP-43 mRNA detection by in situ hybridization, direct and indirect in situ RT-PCR in hippocampal and cerebellar tissue sections of adult rat brain

The growth-associated protein GAP-43 is a presynaptic membrane phosphoprotein that is expressed at high levels during development and axonal growth. To evaluate the cellular distribution of GAP-43 mRNA in the hippocampus and cerebellum of adult rats we applied in situ hybridization (ISH) as well as direct and indirect in situ RT-PCR using biotin as a reporter molecule. ISH resulted in a positive signal in most cerebellar granular cells and in 30% of hippocampal CA3 neurons. Direct in situ RT-PCR yielded cells with strong signals in every region investigated, with elevated background levels most likely related to incorporation of labeled nucleotides into non-specific amplicons through internal priming and DNA repair activity. Indirect in situ RT-PCR turned out to be the best approach for detecting GAP-43 mRNA positive cells. Cerebellar cells exhibiting a positive signal for GAP-43 mRNA were of the granular cell type (98%). Hippocampal neurons with a positive reaction for GAP-43 mRNA included all the neuron groups analyzed, namely CA1 (99%) and CA3 pyramidal cells (94%) and dentate gyrus granule cells (92%). Dentate gyrus granule cells have not tested positive for GAP-43 mRNA detection by molecular morphology analysis. These data show that in normal rats GAP-43 mRNA is present in different cell populations of hippocampal formation, supporting the role of this protein in the ongoing processes of synaptic plasticity.     

Range Corrections to Universal Tetramer Properties

The universal properties of weakly-bound tetramers close to the scaling limit are investigated by solving a subtracted set of Faddeev–Yakubovsky (FY) equations for identical bosons with a zero-range interaction. The solution demands a four-body scale independent of the trimer properties. Furthermore, the effect of a finite effective range is introduced in the FY equations, which we show produces results that are distinct from the scale variation. In particular range effects to two universal scaling functions for the tetramers are investigated. The correlation between successive tetramer energies corresponding to states within two Efimov trimer energies, proposed before and studied close to the unitary limit; and the correlation between the position of the four-atom recombination peaks. In this case, we found a shift in the scaling function due to the range, which can be associated to the shift of the data found for caesium atoms, with respect to zero-range calculations, due to a nonvanishing range in the actual experimental setups.