Location of Poles for the Hastings-McLeod Solution to the Second Painlev\'{e} Equation

We show that the well-known Hastings-McLeod solution to the second Painlev\'{e} equation is pole-free in the region $\arg x \in [-\frac{\pi}{3},\frac{\pi}{3}]\cup [\frac{2\pi}{3},\frac{ 4 \pi}{3}]$, which proves an important special case of a general conjecture concerning pole distributions of Painlev\'{e} transcedents proposed by Novokshenov. Our strategy is to construct explicit quasi-solutions approximating the Hastings-McLeod solution in different regions of the complex plane, and estimate the errors rigorously. The main idea is very similar to the one used to prove Dubrovin's conjecture for the first Painlev\'{e} equation, but there are various technical improvements.Comment: 31 pages, 2 figures. Minor revision, to appear in Constructive Approximatio

Vacuolar ATPase depletion contributes to dysregulation of endocytosis in bloodstream forms of Trypanosoma brucei

BACKGROUND Vacuolar H-ATPase (V-ATPase) is a highly conserved protein complex which hydrolyzes ATP and pumps protons to acidify vacuolar vesicles. Beyond its role in pH maintenance, the involvement of V-ATPase in endocytosis is well documented in mammals and plants but is less clear in Trypanosoma brucei. METHODS In this study, the subcellular localization of V-ATPase subunit B (TbVAB) of T. brucei was assessed via in situ N-terminal YFP-tagging and immunofluorescence assays. Transgenic bloodstream forms (BSF) of T. brucei were generated which comprised either a V-ATPase subunit B (TbVAB) conditional knockout or a V-ATPase subunit A (TbVAA) knockdown. Acridine orange and BCECF-AM were employed to assess the roles of V-ATPase in the pH regulation of BSF T. brucei. The endocytic activities of three markers were also characterized by flow cytometry analyses. Furthermore, trypanosomes were counted from trypanolysis treatment groups (either containing 1% or 5% NHS) and endocytosed trypanosome lytic factor (TLF) was also analyzed by an immunoblotting assay. RESULTS TbVAB was found to localize to acidocalcisomes, lysosomes and probably also to endosomes of BSF of T. brucei and was demonstrated to be essential for cell growth. TbVAB depletion neutralized acidic organelles at 24 hours post-tetracycline depletion (hpd), meanwhile the steady state intracellular pH increased from 7.016 ± 0.013 to 7.422 ± 0.058. Trypanosomes with TbVAB depletion at 24 hpd were found to take up more transferrin (2.068 ± 0.277 fold) but less tomato lectin (49.31 ± 22.57%) by endocytosis, while no significant change was detected in dextran uptake. Similar endocytic dysregulated phenotypes were also observed in TbVAA knockdown cells. In addition, TbVAB depleted trypanosomes showed a low uptake of TLF and exhibited less sensitive to lysis in both 1% and 5% NHS treatments. CONCLUSIONS TbVAB is a key component of V-ATPase and was found to play a key function in endocytosis as well as exhibiting different effects in a receptor/cargo dependent manner in BSF of T. brucei. Besides vacuolar alkalinization, the dysregulation of endocytosis in TbVAB depleted T. brucei is considered to contribute to the reduced sensitivity to lysis by normal human serum

Transient asymptotics of the modified Camassa-Holm equation

We investigate long time asymptotics of the modified Camassa-Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlev\'e-type asymptotic formulas in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a $\bar{\partial}$ nonlinear steepest descent analysis to the associated Riemann-Hilbert problem.Comment: 58 pages, 16 figures. Comments are welcom