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.