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.

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.

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.