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

Functional expression of the yeast alpha-factor receptor in Xenopus oocytes

The STE2 gene of the yeast Saccharomyces cerevisiae encodes a 431- residue polypeptide that has been shown by chemical cross-linking and genetic studies to be a component of the receptor for the peptide mating pheromone, alpha-factor. To demonstrate directly that the ligand binding site of the alpha-factor receptor is comprised solely of the STE2 gene product, the STE2 protein was expressed in Xenopus oocytes. Oocytes microinjected with synthetic STE2 mRNA displayed specific surface binding for 35S-labeled alpha-factor (up to 40 sites/micron2/ng RNA). Oocytes injected with either STE2 antisense RNA or heterologous receptor mRNA (nicotinic acetylcholine receptor alpha, beta, gamma, and delta subunit mRNAs) showed no binding activity (indistinguishable from uninjected control oocytes). The apparent KD (7 nM) of the alpha-factor binding sites expressed on the oocyte surface, determined by competition binding studies, agreed with the values reported for intact yeast cells and yeast plasma membrane fractions. These findings demonstrate that the STE2 gene product is the only yeast polypeptide required for biogenesis of a functional alpha-factor receptor. Electrophysiological measurements indicated that the membrane conductance of oocytes injected with STE2 mRNA, or with both STE2 and GPA1 (encoding a yeast G protein alpha-subunit) mRNAs, did not change and was not affected by pheromone binding. Thus, the alpha-factor receptor, like mammalian G protein-coupled receptors, apparently lacks activity as an intrinsic or ligand-gated ion channel. This report is the first instance in which a membrane-bound receptor from a unicellular eukaryote has been expressed in a vertebrate cell

Human fur gene encodes a yeast KEX2-like endoprotease that cleaves pro-beta-NGF in vivo.

Extracts from BSC-40 cells infected with vaccinia recombinants expressing either the yeast KEX2 prohormone endoprotease or a human structural homologue (fur gene product) contained an elevated level of a membrane-associated endoproteolytic activity that could cleave at pairs of basic amino acids (-LysArg- and -ArgArg-). The fur-directed activity (furin) shared many properties with Kex2p including activity at pH 7.3 and a requirement for calcium. By using antifurin antibodies, immunoblot analysis detected two furin translation products (90 and 96 kD), while immunofluorescence indicated localization to the Golgi apparatus. Coexpression of either Kex2p or furin with the mouse beta-nerve growth factor precursor (pro-beta-NGF) resulted in greatly enhanced conversion of the precursor to mature nerve growth factor. Thus, the sequence homology shared by furin and the yeast KEX2 prohormone processing enzyme is reflected by significant functional homology both in vitro and in vivo

Phosphorylation by the stress-activated MAPK Slt2 down-regulates the yeast TOR complex 2

Saccharomyces cerevisiae target of rapamycin (TOR) complex 2 (TORC2) is an essential regulator of plasma membrane lipid and protein homeostasis. How TORC2 activity is modulated in response to changes in the status of the cell envelope is unclear. Here we document that TORC2 subunit Avo2 is a direct target of Slt2, the mitogen-activated protein kinase (MAPK) of the cell wall integrity pathway. Activation of Slt2 by overexpression of a constitutively active allele of an upstream Slt2 activator (Pkc1) or by auxin-induced degradation of a negative Slt2 regulator (Sln1) caused hyperphosphorylation of Avo2 at its MAPK phosphoacceptor sites in a Slt2-dependent manner and diminished TORC2-mediated phosphorylation of its major downstream effector, protein kinase Ypk1. Deletion of Avo2 or expression of a phosphomimetic Avo2 allele rendered cells sensitive to two stresses (myriocin treatment and elevated exogenous acetic acid) that the cell requires Ypk1 activation by TORC2 to survive. Thus, Avo2 is necessary for optimal TORC2 activity, and Slt2-mediated phosphorylation of Avo2 down-regulates TORC2 signaling. Compared with wild-type Avo2, phosphomimetic Avo2 shows significant displacement from the plasma membrane, suggesting that Slt2 inhibits TORC2 by promoting Avo2 dissociation. Our findings are the first demonstration that TORC2 function is regulated by MAPK-mediated phosphorylation.Comment: This work was supported by National Institutes of Health (NIH) Predoctoral Traineeship GM07232 and a University of California at Berkeley MacArthur and Lakhan-Pal Graduate Fellowship to K.L.L., Erwin Schroedinger Fellowship J3787-B21 from the Austrian Science Fund to AE-A, Marie Sklodowska-Curie Action H2020-MSCA-IF-2016 InsiliCardio, GA 75083 to CMA, and NIH R01 research grant GM21841 to J

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