The Explicit Sato-Tate Conjecture and Densities Pertaining to Lehmer-Type Questions

Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $\theta_p\in[0,\pi]$ to be the angle for which $a(p)=2p^{( k -1)/2}\cos \theta_p$. Let $I\subset[0,\pi]$ be a closed subinterval, and let $d\mu_{ST}=\frac{2}{\pi}\sin^2\theta d\theta$ be the Sato-Tate measure of $I$. Assuming that the symmetric power $L$-functions of $f$ satisfy certain analytic properties (all of which follow from Langlands functoriality and the Generalized Riemann Hypothesis), we prove that if $x$ is sufficiently large, then $$\left|\#\{p\leq x:\theta_p\in I\} -\mu_{ST}(I)\int_2^x\frac{dt}{\log t}\right|\ll\frac{x^{3/4}\log(N k x)}{\log x}$$ with an implied constant of $3.34$. By letting $I$ be a short interval centered at $\frac{\pi}{2}$ and counting the primes using a smooth cutoff, we compute a lower bound for the density of positive integers $n$ for which $a(n)\neq0$. In particular, if $\tau$ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that $$\lim_{x\to\infty}\frac{\#\{n\leq x:\tau(n)\neq0\}}{x}>1-1.54\times10^{-13}.$$ We also discuss the connection between the density of positive integers $n$ for which $a(n)\neq0$ and the number of representations of $n$ by certain positive-definite, integer-valued quadratic forms.Comment: 29 pages. Significant revisions, including improvements in Theorems 1.2, 1.3, and 1.5 and a more detailed account of the contour integration, are included. Acknowledgements are update

Regulation of TORC2 function and localization by Rab5 GTPases in Saccharomyces cerevisiae.

The evolutionarily conserved Target of Rapamycin (TOR) complex-2 (TORC2) is an essential regulator of plasma membrane homeostasis in budding yeast (Saccharomyces cerevisiae). In this yeast, TORC2 phosphorylates and activates the effector protein kinase Ypk1 and its paralog Ypk2. These protein kinases, in turn, carry out all the crucial functions of TORC2 by phosphorylating and thereby controlling the activity of at least a dozen downstream substrates. A previously uncharacterized interplay between the Rab5 GTPases and TORC2 signaling was uncovered through analysis of a newly suspected Ypk1 target. Muk1, one of two guanine nucleotide exchange factors for the Rab5 GTPases, was found to be a physiologically relevant Ypk1 substrate; and, genetic analysis indicates that Ypk1-mediated phosphorylation activates the guanine nucleotide exchange activity of Muk1. Second, it was demonstrated both in vivo and in vitro that the GTP-bound state of the Rab5 GTPase Vps21/Ypt51 physically associates with TORC2 and acts as a direct positive effector required for full TORC2 activity. These interrelationships provide a self-reinforcing control circuit for sustained up-regulation of TORC2-Ypk1 signaling. In this overview, we summarize the experimental basis of these findings, their implications, and speculate as to the molecular basis for Rab5-mediated TORC2 activation

Septin collar formation in budding yeast requires GTP binding and direct phosphorylation by the PAK, Cla4

Assembly at the mother–bud neck of a filamentous collar containing five septins (Cdc3, Cdc10, Cdc11, Cdc12, and Shs1) is necessary for proper morphogenesis and cytokinesis. We show that Cdc10 and Cdc12 possess GTPase activity and appropriate mutations in conserved nucleotide-binding residues abrogate GTP binding and/or hydrolysis in vitro. In vivo, mutants unable to bind GTP prevent septin collar formation, whereas mutants that block GTP hydrolysis do not. GTP binding-defective Cdc10 and Cdc12 form soluble heteromeric complexes with other septins both in yeast and in bacteria; yet, unlike wild-type, mutant complexes do not bind GTP and do not assemble into filaments in vitro. Absence of a p21-activated protein kinase (Cla4) perturbs septin collar formation. This defect is greatly exacerbated when combined with GTP binding-defective septins; conversely, the septin collar assembly defect of such mutants is suppressed efficiently by CLA4 overexpression. Cla4 interacts directly with and phosphorylates certain septins in vitro and in vivo. Thus, septin collar formation may correspond to septin filament assembly, and requires both GTP binding and Cla4-mediated phosphorylation of septins

Septin Stability and Recycling during Dynamic Structural Transitions in Cell Division and Development

SummarySeptins are conserved proteins found in hetero-oligomeric complexes that are incorporated into distinct structures during cell division and differentiation; yeast septins Cdc3, Cdc10, Cdc11, and Cdc12 form hetero-octamers and polymerize into filaments, which form a “collar” at the mother-bud neck [1]. Posttranslational modifications, nucleotide binding, and protein-protein and protein-lipid interactions influence assembly and disassembly of septin structures [2], but whether individual septins are used repeatedly to build higher-order assemblies was not known. We used fluorescence-based pulse-chase methods to visualize the fate of pre-existing (old) and newly synthesized (new) molecules of two septins, Cdc10 and Cdc12. They were recycled through multiple mitotic divisions, and old and new molecules were incorporated indistinguishably into the collar. Likewise, old and new subunits intermixed within hetero-octamers, indicating that exchange occurs at this organizational level. Remarkably, in meiosis, Cdc10 made during vegetative growth was reutilized to build sporulation-specific structures and reused again during spore germination for budding and during subsequent mitotic divisions. Although Cdc12 also persisted during sporulation, it was excluded from septin structures and replaced by another subunit, Spr3; only new Cdc12 populated the collar of germinating spores. Thus, mechanisms governing septin incorporation are specific to each subunit and to the developmental state of the cell

Systems biology approaches in cell signaling research

The use of methods for global and quantitative analysis of cells is providing new systems-level insights into signal transduction processes. Recent studies reveal important information about the rates of signal transmission and propagation, help establish some general regulatory characteristics of multi-tiered signaling cascades, and illuminate the combinatorial nature of signaling specificity in cell differentiation

Function and regulation in MAPK signaling pathways: Lessons learned from the yeast Saccharomyces cerevisiae

AbstractSignaling pathways that activate different mitogen-activated protein kinases (MAPKs) elicit many of the responses that are evoked in cells by changes in certain environmental conditions and upon exposure to a variety of hormonal and other stimuli. These pathways were first elucidated in the unicellular eukaryote Saccharomyces cerevisiae (budding yeast). Studies of MAPK pathways in this organism continue to be especially informative in revealing the molecular mechanisms by which MAPK cascades operate, propagate signals, modulate cellular processes, and are controlled by regulatory factors both internal to and external to the pathways. Here we highlight recent advances and new insights about MAPK-based signaling that have been made through studies in yeast, which provide lessons directly applicable to, and that enhance our understanding of, MAPK-mediated signaling in mammalian cells