Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion

For a specific choice of the diffusion, the parabolic-elliptic Patlak-Keller-Segel system with non-linear diffusion (also referred to as the quasi-linear Smoluchowski-Poisson equation) exhibits an interesting threshold phenomenon: there is a critical mass $M_c>0$ such that all the solutions with initial data of mass smaller or equal to $M_c$ exist globally while the solution blows up in finite time for a large class of initial data with mass greater than $M_c$. Unlike in space dimension 2, finite mass self-similar blowing-up solutions are shown to exist in space dimension $d?3$

Symbolic dynamics for the NN-centre problem at negative energies

We consider the planar $N$-centre problem, with homogeneous potentials of degree -\a<0, \a \in [1,2). We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets

Establishment of a Transgenic Zebrafish Line for Superficial Skin Ablation and Functional Validation of Apoptosis Modulators In Vivo

BACKGROUND: Zebrafish skin is composed of enveloping and basal layers which form a first-line defense system against pathogens. Zebrafish epidermis contains ionocytes and mucous cells that aid secretion of acid/ions or mucous through skin. Previous studies demonstrated that fish skin is extremely sensitive to external stimuli. However, little is known about the molecular mechanisms that modulate skin cell apoptosis in zebrafish. METHODOLOGY/PRINCIPAL FINDINGS: This study aimed to create a platform to conduct conditional skin ablation and determine if it is possible to attenuate apoptotic stimuli by overexpressing potential apoptosis modulating genes in the skin of live animals. A transgenic zebrafish line of Tg(krt4:NTR-hKikGR)(cy17) (killer line), which can conditionally trigger apoptosis in superficial skin cells, was first established. When the killer line was incubated with the prodrug metrodinazole, the superficial skin displayed extensive apoptosis as judged by detection of massive TUNEL- and active caspase 3-positive signals. Great reductions in NTR-hKikGR(+) fluorescent signals accompanied epidermal cell apoptosis. This indicated that NTR-hKikGR(+) signal fluorescence can be utilized to evaluate apoptotic events in vivo. After removal of metrodinazole, the skin integrity progressively recovered and NTR-hKikGR(+) fluorescent signals gradually restored. In contrast, either crossing the killer line with testing lines or transiently injecting the killer line with testing vectors that expressed human constitutive active Akt1, mouse constitutive active Stat3, or HPV16 E6 element displayed apoptosis-resistant phenotypes to cytotoxic metrodinazole as judged by the loss of reduction in NTR-hKikGR(+) fluorescent signaling. CONCLUSION/SIGNIFICANCE: The killer/testing line binary system established in the current study demonstrates a nitroreductase/metrodinazole system that can be utilized to conditionally perform skin ablation in a real-time manner, and provides a valuable tool to visualize and quantify the anti-apoptotic potential of interesting target genes in vivo. The current work identifies a potential use for transgenic zebrafish as a high-throughput platform to validate potential apoptosis modulators in vivo