On the ratio of values of a polynomial

For given positive integersa andb, the equationa(x + 1)...(x + k) =b(y+1)...(y + k) in positive integers is considered. More general equations are also considered

On the equation x(x +d<SUB>1</SUB>)...(x + (k − 1)d<SUB>1</SUB>) =y(y +d<SUB>2</SUB>)...(y + (mk − 1)d<SUB>2</SUB>)

For given positive integers m ≥ 2,d1 and d2, we consider the equation of the title in positive integers x, y and k ≥ 2. We show that the equation implies that k is bounded. For a fixed k, we give conditions under which the equation implies that max(x, y) is bounded

Patronin-mediated minus end growth is required for dendritic microtubule polarity.

Microtubule minus ends are thought to be stable in cells. Surprisingly, in Drosophila and zebrafish neurons, we observed persistent minus end growth, with runs lasting over 10 min. In Drosophila, extended minus end growth depended on Patronin, and Patronin reduction disrupted dendritic minus-end-out polarity. In fly dendrites, microtubule nucleation sites localize at dendrite branch points. Therefore, we hypothesized minus end growth might be particularly important beyond branch points. Distal dendrites have mixed polarity, and reduction of Patronin lowered the number of minus-end-out microtubules. More strikingly, extra Patronin made terminal dendrites almost completely minus-end-out, indicating low Patronin normally limits minus-end-out microtubules. To determine whether minus end growth populated new dendrites with microtubules, we analyzed dendrite development and regeneration. Minus ends extended into growing dendrites in the presence of Patronin. In sum, our data suggest that Patronin facilitates sustained microtubule minus end growth, which is critical for populating dendrites with minus-end-out microtubules

Extension of Laguerre polynomials with negative arguments II

For $n \geq 3$ and $s \leq 92$, it is proved in \cite{ShSi} that, except for finitely many pairs $(n, s), G_1(x) = G_1(x, n, s)$ is either irreducible or linear factor times an irreducible polynomial. If $s \leq 30$, we determine here explicitely the set of pairs $(n, s)$ in the above assertion. This implies a new proof of the result of Nair and Shorey \cite{NaSh1} that $G_1(x)$ is irreducible for $s \leq 22$

Nicotinamide is an Endogenous Agonist for a C. elegans TRPV OSM-9 and OCR-4 Channel

TRPV ion channels are directly activated by sensory stimuli and participate in thermo-, mechano- and chemo-sensation. They are also hypothesized to respond to endogenous agonists that would modulate sensory responses. Here, we show that the nicotinamide (NAM) form of vitamin B3 is an agonist of a Caenorhabditis elegans TRPV channel. Using heterologous expression in Xenopus oocytes, we demonstrate that NAM is a soluble agonist for a channel consisting of the well-studied OSM-9 TRPV subunit and relatively uncharacterized OCR-4 TRPV subunit as well as the orthologous Drosophila Nan-Iav TRPV channel, and we examine stoichiometry of subunit assembly. Finally, we show that behaviours mediated by these C. elegans and Drosophila channels are responsive to NAM, suggesting conservation of activity of this soluble endogenous metabolite on TRPV activity. Our results in combination with the role of NAM in NAD+ metabolism suggest an intriguing link between metabolic regulation and TRPV channel activity

Pathogenic role of delta 2 tubulin in bortezomib-induced peripheral neuropathy

The pathogenesis of chemotherapy-induced peripheral neuropathy (CIPN) is poorly understood. Here, we report that the CIPN-causing drug bortezomib (Bort) promotes delta 2 tubulin (D2) accumulation while affecting microtubule stability and dynamics in sensory neurons in vitro and in vivo and that the accumulation of D2 is predominant in unmyelinated fibers and a hallmark of bortezomib-induced peripheral neuropathy (BIPN) in humans. Furthermore, while D2 overexpression was sufficient to cause axonopathy and inhibit mitochondria motility, reduction of D2 levels alleviated both axonal degeneration and the loss of mitochondria motility induced by Bort. Together, our data demonstrate that Bort, a compound structurally unrelated to tubulin poisons, affects the tubulin cytoskeleton in sensory neurons in vitro, in vivo, and in human tissue, indicating that the pathogenic mechanisms of seemingly unrelated CIPN drugs may converge on tubulin damage. The results reveal a previously unrecognized pathogenic role for D2 in BIPN that may occur through altered regulation of mitochondria motility