Advances in random matrix theory, zeta functions, and sphere packing

Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues (“the energy levels”) follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These topics are distinct, so we present them separately with their own introductory remarks.

Zeta functions and Eisenstein series on classical groups

We construct an Euler product from the Hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a CM field and the eigenform is holomorphic. We also prove analytic continuation of an Eisenstein series on another unitary group, containing the group just mentioned defined with such an eigenform. As an application of our methods, we prove an explicit class number formula for a totally definite hermitian form over a CM field.

Computation of Surface Electrical Potentials of Plant Cell Membranes : Correspondence to Published Zeta Potentials from Diverse Plant Sources

A Gouy-Chapman-Stern model has been developed for the computation of surface electrical potential (ψ0) of plant cell membranes in response to ionic solutes. The present model is a modification of an earlier version developed to compute the sorption of ions by wheat (Triticum aestivum L. cv Scout 66) root plasma membranes. A single set of model parameters generates values for ψ0 that correlate highly with published ζ potentials of protoplasts and plasma membrane vesicles from diverse plant sources. The model assumes ion binding to a negatively charged site (R− = 0.3074 μmol m−2) and to a neutral site (P0 = 2.4 μmol m−2) according to the reactions R− + IΖ ⇌ RIΖ−1 and P0 + IΖ ⇌ PIΖ, where IΖ represents an ion of charge Ζ. Binding constants for the negative site are 21,500 m−1 for H+, 20,000 m−1 for Al3+, 2,200 m−1 for La3+, 30 m−1 for Ca2+ and Mg2+, and 1 m−1 for Na+ and K+. Binding constants for the neutral site are 1/180 the value for binding to the negative site. Ion activities at the membrane surface, computed on the basis of ψ0, appear to determine many aspects of plant-mineral interactions, including mineral nutrition and the induction and alleviation of mineral toxicities, according to previous and ongoing studies. A computer program with instructions for the computation of ψ0, ion binding, ion concentrations, and ion activities at membrane surfaces may be requested from the authors.

Cell-surface-expressed T-cell antigen-receptor zeta chain is associated with the cytoskeleton.

The T-cell antigen receptor zeta chain plays an important role in coupling antigen recognition to several intracellular signal-transduction pathways. zeta chain can associate with certain protein tyrosine kinases and retains the capacity to transduce signals independently of the other receptor subunits. Thus, zeta chain could couple cell-surface-expressed T-cell antigen receptors to the intracellular signal-transduction apparatus by its association with various intracellular molecules in addition to tyrosine kinases. In the process of searching for zeta chain-associated molecules we observed that after lysis of resting T cells with Triton X-100, zeta chain is localized in the detergent-insoluble fraction, in addition to its presence in the detergent-soluble fraction. Treatment of T cells with cytochalasin B, an actin-depolymerizing agent, leads to the complete dissociation of zeta chain from the Triton-insoluble fraction, suggesting a linkage between zeta chain and the cytoskeletal matrix. We have also determined that cytoskeletal-associated zeta chain is expressed on the cell surface. Furthermore, a tyrosine-phosphorylated 16-kDa zeta chain was detected only in the Triton-insoluble cytoskeletal fraction of resting T cells. zeta chain also maintains its association with the cytoskeleton when expressed in COS cells, inferring that the cytoskeletal elements involved in this linkage may be ubiquitous. Finally, we have localized a 42-amino acid region in the intracytoplasmic domain of zeta chain, which is crucial for maximal interaction between zeta chain and the cytoskeleton. Anchorage of cell-surface-expressed zeta chain to the cytoskeleton in resting T cells may facilitate recycling of receptor complexes and/or allow the transduction of external stimuli into the cell.

Cellular protein kinase C isoform zeta regulates human parainfluenza virus type 3 replication.

Phosphorylation of the P proteins of nonsegmented negative-strand RNA viruses is critical for their function as transactivators of the viral RNA polymerases. Using unphosphorylated P protein of human parainfluenza virus type 3 (HPIV3) expressed in Escherichia coli, we have shown that the cellular protein kinase that phosphorylates P in vitro is biochemically and immunologically indistinguishable from cellular protein kinase C isoform zeta (PKC-zeta). Further, PKC-zeta is specifically packaged within the progeny HPIV3 virions and remains tightly associated with the ribonucleoprotein complex. The P protein seems also to be phosphorylated intracellularly by PKC-zeta, as shown by the similar protease digestion pattern of the in vitro and in vivo phosphorylated P proteins. The growth of HPIV3 in CV-1 cells is completely abrogated when a PKC-zeta-specific inhibitor pseudosubstrate peptide was delivered into cells. These data indicate that PKC-zeta plays an important role in HPIV3 gene expression by phosphorylating P protein, thus providing an opportunity to develop antiviral agents against an important human pathogen.

Pax-6 is essential for lens-specific expression of zeta-crystallin.

Pax-6 is essential for normal eye development and has been implicated as a "master gene" for lens formation in embryogenesis. Guinea pig zeta-crystallin, a taxon-specific enzyme crystallin, achieves high expression specifically in lens through use of an alternative promoter. Here we show that Pax-6 binds a site in this promoter, which is essential for lens-specific expression. Lens and lens-derived cells exhibit a tissue-specific pattern of alternative splicing of Pax-6 transcripts and Pax-6 is expressed in adult lenses and cells that support zeta-crystallin expression. These results suggest that zeta-crystallin is a natural target gene for Pax-6 and that this Pax family member has a direct role in the continuing expression of tissue-specific genes.

Mechanism of activation for Zap-70 catalytic activity.

There is a growing body of evidence, including data from human genetic and T-cell receptor function studies, which implicate a zeta-associated protein of M(r) 70,000 (Zap-70) as a critical protein tyrosine kinase in T-cell activation and development. During T-cell activation, Zap-70 becomes associated via its src homology type 2 (SH2) domains with tyrosine-phosphorylated immune-receptor tyrosine activating motif (ITAM) sequences in the cytoplasmic zeta chain of the T-cell receptor. An intriguing conundrum is how Zap-70 is catalytically activated for downstream phosphorylation events. To address this question, we have used purified Zap-70, tyrosine phosphorylated glutathione S-transferase (GST)-Zeta, and GST-Zeta-1 cytoplasmic domains, and various forms of ITAM-containing peptides to see what effect binding of zeta had upon Zap-70 tyrosine kinase activity. The catalytic activity of Zap-70 with respect to autophosphorylation increased approximately 5-fold in the presence of 125 nM phosphorylated GST-Zeta or GST-Zeta-1 cytoplasmic domain. A 20-fold activity increase was observed for phosphorylation of an exogenous substrate. Both activity increases showed a GST-Zeta concentration dependence. The increase in activity was not produced with nonphosphorylated GST-Zeta, phosphorylated zeta, or phosphorylated ITAM-containing peptides. The increase in Zap-70 activity was SH2 mediated and was inhibited by phenylphosphate, Zap-70 SH2, and an antibody specific for Zap-70 SH2 domains. Since GST-Zeta and GST-Zeta-1 exist as dimers, the data suggest Zap-70 is activated upon binding a dimeric form of phosphorylated zeta and not by peptide fragments containing a single phosphorylated ITAM. Taken together, these data indicate that the catalytic activity of Zap-70 is most likely activated by a trans-phosphorylation mechanism.

The role of CD4-Lck in T-cell receptor antagonism: evidence for negative signaling.

Small changes in the complex between a peptide and a molecule of the major histocompatibility complex generate ligands able to partially activate (partial agonist) or even inhibit (antagonist) T-cell functions. T-cell receptor engagement of antagonist complex results in a partial zeta chain phosphorylation without activation of the associated ZAP-70 kinase. Herein we show that, despite a strong inhibition of both inositol phospholipid hydrolysis and extracellular increasing antagonist concentrations increased the activity of the CD4-Lck kinase. Addition of anti-CD4 antibody to culture medium prevented inhibitory effects induced by antagonist ligand. We propose that CD4-Lck activation triggered by antagonist complexes may act in a dominant negative mode, thus overriding stimulatory signals coming from agonist ligand. These findings identify a new T-cell signaling profile that may explain the ability of some T-cell receptor variant ligands to inhibit specific biological activities or trigger alternative activation programs.

Ceramide-binding and activation defines protein kinase c-Raf as a ceramide-activated protein kinase.

Interleukin 1 is the prototype of an inflammatory cytokine, and evidence suggests that it uses the sphingomyelin pathway and ceramide production to trigger mitogen-activated protein kinase (MAPK) activation and subsequent gene expression required for acute inflammatory processes. To identify downstream signaling targets of ceramide, a radioiodinated photoaffinity labeling analog of ceramide ([125I] 3-trifluoromethyl-3-(m-iodophenyl)diazirine-ceramide) was employed. It is observed that ceramide specifically binds to and activates protein kinase c-Raf, leading to a subsequent activation of the MAPK cascade. Ceramide does not bind to any other member of the MAPK module nor does it bind to protein kinase C-zeta. These data identify protein kinase c-Raf as a specific molecular target for interleukin 1 beta-stimulated ceramide formation and demonstrate that ceramide is a lipid cofactor participating in regulation of c-Raf activity.

Nonclathrin coat protein gamma, a subunit of coatomer, binds to the cytoplasmic dilysine motif of membrane proteins of the early secretory pathway.

Coatomer, a cytosolic heterooligomeric protein complex that consists of seven subunits [alpha-, beta-, beta'-, gamma-, delta-, epsilon-, and zeta-COP (nonclathrin coat protein)], has been shown to interact with dilysine motifs typically found in the cytoplasmic domains of various endoplasmic-reticulum-resident membrane proteins [Cosson, P. & Letourneur, F. (1994) Science 263, 1629-1631]. We have used a photo-cross-linking approach to identify the site of coatomer that is involved in binding to the dilysine motifs. An octapeptide corresponding to the C-terminal tail of Wbp1p, a component of the yeast N-oligosaccharyltransferase complex, has been synthesized with a photoreactive phenylalanine at position -5 and was radioactively labeled with [125I]iodine at a tyrosine residue introduced at the N terminus of the peptide. Photolysis of isolated coatomer in the presence of this peptide and immunoprecipitation of coatomer from photo-cross-linked cell lysates reveal that gamma-COP is the predominantly labeled protein. From these results, we conclude that coatomer is able to bind to the cytoplasmic dilysine motifs of membrane proteins of the early secretory pathway via its gamma-COP subunit, whose complete cDNA-derived amino acid sequence is also presented.

p56Lck and p59Fyn regulate CD28 binding to phosphatidylinositol 3-kinase, growth factor receptor-bound protein GRB-2, and T cell-specific protein-tyrosine kinase ITK: implications for T-cell costimulation.

T-cell activation requires cooperative signals generated by the T-cell antigen receptor zeta-chain complex (TCR zeta-CD3) and the costimulatory antigen CD28. CD28 interacts with three intracellular proteins-phosphatidylinositol 3-kinase (PI 3-kinase), T cell-specific protein-tyrosine kinase ITK (formerly TSK or EMT), and the complex between growth factor receptor-bound protein 2 and son of sevenless guanine nucleotide exchange protein (GRB-2-SOS). PI 3-kinase and GRB-2 bind to the CD28 phosphotyrosine-based Tyr-Met-Asn-Met motif by means of intrinsic Src-homology 2 (SH2) domains. The requirement for tyrosine phosphorylation of the Tyr-Met-Asn-Met motif for SH2 domain binding implicates an intervening protein-tyrosine kinase in the recruitment of PI 3-kinase and GRB-2 by CD28. Candidate kinases include p56Lck, p59Fyn, zeta-chain-associated 70-kDa protein (ZAP-70), and ITK. In this study, we demonstrate in coexpression studies that p56Lck and p59Fyn phosphorylate CD28 primarily at Tyr-191 of the Tyr-Met-Asn-Met motif, inducing a 3- to 8-fold increase in p85 (subunit of PI 3-kinase) and GRB-2 SH2 binding to CD28. Phosphatase digestion of CD28 eliminated binding. In contrast to Src kinases, ZAP-70 and ITK failed to induce these events. Further, ITK binding to CD28 was dependent on the presence of p56Lck and is thus likely to act downstream of p56Lck/p59Fyn in a signaling cascade. p56Lck is therefore likely to be a central switch in T-cell activation, with the dual function of regulating CD28-mediated costimulation as well as TCR-CD3-CD4 signaling.

Solution structure of the Shc SH2 domain complexed with a tyrosine-phosphorylated peptide from the T-cell receptor.

She is a widely expressed adapter protein that plays an important role in signaling via a variety of cell surface receptors and has been implicated in coupling the stimulation of growth factor, cytokine, and antigen receptors to the Ras signaling pathway. She interacts with several tyrosine-phosphorylated receptors through its C-terminal SH2 domain, and one of the mechanisms of T-cell receptor-mediated Ras activation involves the interaction of the Shc SH2 domain with the tyrosine-phosphorylated zeta chain of the T-cell receptor. Here we describe a high-resolution NMR structure of the Shc SH2 domain complexed to a phosphopeptide (GHDGLpYQGLSTATK) corresponding to a portion of the zeta chain of the T-cell receptor. Although the overall architecture of the protein is similar to other SH2 domains, distinct structural differences were observed in the smaller beta-sheet, BG loop, (pY + 3) phosphopeptide-binding site, and relative position of the bound phosphopeptide.

Determinants of the phagocytic signal mediated by the type IIIA Fc gamma receptor, Fc gamma RIIIA: sequence requirements and interaction with protein-tyrosine kinases.

The Fc gamma receptor-associated gamma and zeta subunits contain a conserved cytoplasmic motif, termed the immunoglobulin gene tyrosine activation motif, which contains a pair of YXXL sequences. The tyrosine residues within these YXXL sequences have been shown to be required for transduction of a phagocytic signal. We have previously reported that the gamma subunit of the type IIIA Fc gamma receptor (Fc gamma RIIIA) is approximately 6 times more efficient in mediating phagocytosis than the zeta subunit of Fc gamma RIIIA. By exchanging regions of the cytoplasmic domains of the homologous gamma and zeta chains, we observed that the cytoplasmic area of the gamma chain bearing a pair of the conserved YXXL sequences is important in phagocytic signaling. Further specificity of phagocytic signaling is largely determined by the two internal XX amino acids in the YXXL sequences. In contrast, the flanking amino acids of the YXXL sequences including the seven intervening amino acids between the two YXXL sequences do not significantly affect the phagocytic signal. Furthermore, the protein-tyrosine kinase Syk, but not the related kinase ZAP-70, stimulated Fc gamma RIIIA-mediated phagocytosis. ZAP-70, however, increased phagocytosis when coexpressed with the Src family kinase Fyn. These data demonstrate the importance of the two specific amino acids within the gamma subunit YXXL cytoplasmic sequences in phagocytic signaling and explain the difference in phagocytic efficiency of the gamma and zeta chains. These results indicate the importance of Syk in Fc gamma RIIIA-mediated phagocytosis and demonstrate that ZAP-70 and syk differ in their requirement for a Src-related kinase in signal transduction.

Characterization of immature thymocyte lines derived from T-cell receptor or recombination activating gene 1 and p53 double mutant mice.

The T-cell receptor (TCR) beta chain is instrumental in the progression of thymocyte differentiation from the CD4-CD8- to the CD4+CD8+ stage. This differentiation step may involve cell surface expression of novel CD3-TCR complexes. To facilitate biochemical characterization of these complexes, we established cell lines from thymic lymphomas originating from mice carrying a mutation in the p53 gene on the one hand and a mutation in TCR-alpha, TCR-beta, or the recombination activating gene 1 (RAG-1) on the other hand. The cell lines were CD4+CD8+ and appeared to be monoclonal. A cell line derived from a RAG-1 x p53 double mutant thymic lymphoma expressed low levels of CD3-epsilon, -gamma, and -delta on the surface. TCR-alpha x p53 double mutant cell lines were found to express complexes consisting of TCR-beta chains associated with CD3-epsilon, -gamma, and -delta chains and CD3-zeta zeta dimers. These lines will be useful tools to study the molecular structure and signal transducing properties of partial CD3-TCR complexes expressed on the surface of immature thymocytes.

Transcriptional activation of human adult alpha-globin genes by hypersensitive site-40 enhancer: function of nuclear factor-binding motifs occupied in erythroid cells.

The developmental stage- and erythroid lineage-specific activation of the human embryonic zeta- and fetal/adult alpha-globin genes is controlled by an upstream regulatory element [hypersensitive site (HS)-40] with locus control region properties, a process mediated by multiple nuclear factor-DNA complexes. In vitro DNase I protection experiments of the two G+C-rich, adult alpha-globin promoters have revealed a number of binding sites for nuclear factors that are common to HeLa and K-562 extracts. However, genomic footprinting analysis has demonstrated that only a subset of these sites, clustered between -130 and +1, is occupied in an erythroid tissue-specific manner. The function of these in vivo-occupied motifs of the alpha-globin promoters, as well as those previously mapped in the HS-40 region, is assayed by site-directed mutagenesis and transient expression in embryonic/fetal erythroid K-562 cells. These studies, together with our expression data on the human embryonic zeta-globin promoter, provide a comprehensive view of the functional roles of individual nuclear factor-DNA complexes in the final stages of transcriptional activation of the human alpha-like globin promoters by the HS-40 element.