Ground state in the energy super-critical Gross-Pitaevskii equation with a harmonic potential

The energy super-critical Gross--Pitaevskii equation with a harmonic potential is revisited in the particular case of cubic focusing nonlinearity and dimension d > 4. In order to prove the existence of a ground state (a positive, radially symmetric solution in the energy space), we develop the shooting method and deal with a one-parameter family of classical solutions to an initial-value problem for the stationary equation. We prove that the solution curve (the graph of the eigenvalue parameter versus the supremum) is oscillatory for d = 13. Compared to the existing literature, rigorous asymptotics are derived by constructing three families of solutions to the stationary equation with functional-analytic rather than geometric methods.Comment: 42 page

Dimeric peroxiredoxins are druggable targets in human Burkitt lymphoma

Burkitt lymphoma is a fast-growing tumor derived from germinal center B cells. It is mainly treated with aggressive chemotherapy, therefore novel therapeutic approaches are needed due to treatment toxicity and developing resistance. Disturbance of red-ox homeostasis has recently emerged as an efficient antitumor strategy. Peroxiredoxins (PRDXs) are thioredoxin-family antioxidant enzymes that scavenge cellular peroxides and contribute to red-ox homeostasis. PRDXs are robustly expressed in various malignancies and critically involved in cell proliferation, differentiation and apoptosis. To elucidate potential role of PRDXs in lymphoma, we studied their expression level in B cell-derived primary lymphoma cells as well as in cell lines. We found that PRDX1 and PRDX2 are upregulated in tumor B cells as compared with normal counterparts. Concomitant knockdown of PRDX1 and PRDX2 significantly attenuated the growth rate of lymphoma cells. Furthermore, in human Burkitt lymphoma cell lines, we isolated dimeric 2-cysteine peroxiredoxins as targets for SK053, a novel thiol-specific small-molecule peptidomimetic with antitumor activity. We observed that treatment of lymphoma cells with SK053 triggers formation of covalent PRDX dimers, accumulation of intracellular reactive oxygen species, phosphorylation of ERK1/2 and AKT and leads to cell cycle arrest and apoptosis. Based on site-directed mutagenesis and modeling studies, we propose a mechanism of SK053-mediated PRDX crosslinking, involving double thioalkylation of active site cysteine residues. Altogether, our results suggest that peroxiredoxins are novel therapeutic targets in Burkitt lymphoma and provide the basis for new approaches to the treatment of this disease