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.

Complex formation by the human RAD51C and XRCC3 recombination repair proteins

In vertebrates, the RAD51 protein is required for genetic recombination, DNA repair, and cellular proliferation. Five paralogs of RAD51, known as RAD51B, RAD51C, RAD51D, XRCC2, and XRCC3, have been identified and also shown to be required for recombination and genome stability. At the present time, however, very little is known about their biochemical properties or precise biological functions. As a first step toward understanding the roles of the RAD51 paralogs in recombination, the human RAD51C and XRCC3 proteins were overexpressed and purified from baculovirus-infected insect cells. The two proteins copurify as a complex, a property that reflects their endogenous association observed in HeLa cells. Purified RAD51C–XRCC3 complex binds single-stranded, but not duplex DNA, to form protein–DNA networks that have been visualized by electron microscopy.

The architecture of the human Rad54–DNA complex provides evidence for protein translocation along DNA

Proper maintenance and duplication of the genome require accurate recombination between homologous DNA molecules. In eukaryotic cells, the Rad51 protein mediates pairing between homologous DNA molecules. This reaction is assisted by the Rad54 protein. To gain insight into how Rad54 functions, we studied the interaction of the human Rad54 (hRad54) protein with double-stranded DNA. We have recently shown that binding of hRad54 to DNA induces a change in DNA topology. To determine whether this change was caused by a protein-constrained change in twist, a protein-constrained change in writhe, or the introduction of unconstrained plectonemic supercoils, we investigated the hRad54–DNA complex by scanning force microscopy. The architecture of the observed complexes suggests that movement of the hRad54 protein complex along the DNA helix generates unconstrained plectonemic supercoils. We discuss how hRad54-induced superhelical stress in the target DNA may function to facilitate homologous DNA pairing by the hRad51 protein directly. In addition, the induction of supercoiling by hRad54 could stimulate recombination indirectly by displacing histones and/or other proteins packaging the DNA into chromatin. This function of DNA translocating motors might be of general importance in chromatin metabolism.

p300 is a component of an estrogen receptor coactivator complex.

The estrogen receptor (ER) is a ligand-dependent transcription factor that regulates expression of target genes in response to estrogen in concert with other cellular signaling pathways. This suggests that the mechanism by which ER transmits an activating signal to the general transcription machinery may include factors that integrate these diverse signals. We have previously characterized the estrogen receptor-associated protein, ERAP160, as a factor that complexes with ER in an agonist-dependent manner. We have now found that the transcriptional coactivator p300 associates with agonist bound ER and augments ligand-dependent activation by ER. Our studies show that an ER coactivator complex involves a direct hormone-dependent interaction between ER and ERAP160, resulting in the recruitment of p300. In addition, antibodies directed against the cloned steroid receptor coactivator 1 (SRC1) recognize ERAP160. The known role of p300 in multiple signal transduction pathways, including those involving the second messenger cAMP, suggests p300 functions as a point of integration between ER and these other pathways.

Selective response of ternary complex factor Sap1a to different mitogen-activated protein kinase subgroups.

Mitogenic and stres signals results in the activation of extracellular signal-regulated kinases (ERKs) and stress-activated protein kinase/c-Jun N-terminal kinases (SAPK/JNKs), respectively, which are two subgroups of the mitogen-activated protein kinases. A nuclear target of mitogen-activated protein (MAP) kinases is the ternary complex factor Elk-1, which underlies its involvement in the regulation of c-fos gene expression by mitogenic and stress signals. A second ternary complex factor, Sap1a, is coexpressed with Elk-1 in several cell types and shares attributes of Elk-1, the significance of which is not clear. Here we show that Sap1a is phosphorylated efficiently by ERKs but not by SAPK/JNKs. Serum response factor-dependent ternary complex formation by Sap1a is stimulated by ERK phosphorylation but not by SAPK/JNKs. Moreover, Sap1a-mediated transcription is activated by mitogenic signals but not by cell stress. These results suggest that Sap1a and Elk-1 have distinct physiological functions.

Mice with disrupted GM2/GD2 synthase gene lack complex gangliosides but exhibit only subtle defects in their nervous system.

Gangliosides, sialic acid-containing glycosphingolipids, are abundant in the vertebrate (mammalian) nervous system. Their composition is spatially and developmentally regulated, and gangliosides have been widely believed to lay essential roles in establishment of the nervous system, especially in neuritogenesis and synaptogenesis. However, this has never been tested directly. Here we report the generation of mice with a disrupted beta 1,4-N-acetylgalactosaminyltransferase (GM2/GD2 synthase; EC 2.4.1.92) gene. The mice lacked all complex gangliosides. Nevertheless, they did not show any major histological defects in their nervous systems or in gross behavior. Just a slight reduction in the neural conduction velocity from the tibial nerve to the somatosensory cortex, but not to the lumbar spine, was detected. These findings suggest that complex gangliosides are required in neuronal functions but not in the morphogenesis and organogenesis of the brain. The higher levels of GM3 and GD3 expressed in the brains of these mutant mice may be able to compensate for the lack of complex gangliosides.

Essential role for complex N-glycans in forming an organized layer of bronchial epithelium.

Mice lacking the complex subset of N-glycans due to inactivation of the Mgat1 gene die at mid-gestation, making it difficult to identify specific biological functions for this class of cell surface carbohydrates. To circumvent this embryonic lethality and to uncover tissue-specific functions for complex N-glycans, WW6 embryonic stem cells with inactivated Mgat1 alleles were tracked in chimeric embryos. The Mgat1 gene encodes N-acetylglucosaminyltransferase I (Glc-NAc-TI; EC 2.4.1.101), the transferase that initiates the synthesis of complex N-glycans. WW6 cells carry an inert beta-globin transgene that allows their identification in chimeras by DNA-DNA in situ hybridization. Independent Mgat1-/- and Mgat1+/- mutant WW6 isolates contributed like parent WW6 cells to the tissues of embryonic day (E) 10.5 to E16.5 chimeras. However, a cell type-specific difference was observed in lung. Homozygous null Mgat1-/- WW6 cells did not contribute to the epithelial layer in more than 99% bronchi. This deficiency was corrected by transfection of a Mgat1 transgene. Interestingly, heterozygous Mgat1+/- WW6 cells were also deficient in populating the layer of bronchial epithelium. Furthermore, examination of lung bud in E9.5 Mgat1-/- mutant embryos showed complete absence of an organized epithelial cell layer in the bronchus. Thus, complex N-glycans are required to form a morphologically recognizable bronchial epithelium, revealing an in vivo, cell type-specific function for this class of N-glycans.

Identification of novel genes in Drosophila reveals the complex regulation of early gene activity in the mesoderm.

Two zygotic genes, twist and snail, are indispensable for the correct establishment of the mesoderm primordium in the early Drosophila embryo. They are also needed for morphogenesis and differentiation of the mesoderm. Both genes code for transcription factors with different, albeit complementary, functions. Therefore, to understand the early development of the mesoderm, it will be necessary to identify and study the genes regulated by twist and snail. We have searched for downstream genes using a subtractive cDNA library enriched in sequences expressed in the mesoderm. We have isolated sequences that correspond to 13 novel early mesoderm genes. These novel genes show a variety of expression patterns and also differ in their dependence on twist and snail functions. This indicates that the regulation of early gene activity in the mesoderm is more complex than previously thought.

Evidence for a preformed transducer complex organized by the B cell antigen receptor.

The B cell antigen receptor (BCR) consists of the membrane-bound immunoglobulin (mIg) molecule and the Ig-alpha/Ig-beta heterodimer, which functions as signaling subunit of the receptor. Stimulation of the BCR activates protein tyrosine kinases (PTKs) that phosphorylate a number of substrate proteins, including the Ig-alpha/Ig-beta heterodimer of the BCR itself. How the PTKs become activated after BCR engagement is not known at present. Here, we show that BCR-negative J558L cells treated with the protein tyrosine phosphatase inhibitor pervanadate/H2O2 display only a weak substrate phosphorylation. However, in BCR-positive transfectants of J558L, treatment with pervanadate/H2O2 induces a strong phosphorylation of several substrate proteins. Treatment with pervanadate/H2O2 does not result in receptor crosslinking, yet the pattern of protein phosphorylation is similar to that observed after BCR stimulation by antigen. The response requires cellular integrity because tyrosine phosphorylation of most substrates is not visible in cell lysates. Cells that express a BCR containing an Ig-alpha subunit with a mutated immunoreceptor tyrosine-based activation motif display a delayed response. The data suggest that, once expressed on the surface, the BCR organizes protein tyrosine phosphatases, PTKs, and their substrates into a transducer complex that can be activated by pervanadate/H202 in the absence of BCR crosslinking. Assembly of this preformed complex seems to be a prerequisite for BCR-mediated signal transduction.

A structural model for a homeotic protein-extradenticle-DNA complex accounts for the choice of HOX protein in the heterodimer.

The genes of the homeotic complex (HOX) encode DNA binding homeodomain proteins that control developmental fates by differentially regulating the transcription of downstream target genes. Despite their unique in vivo functions, disparate HOX proteins often bind to very similar DNA sequences in vitro. Thus, a critical question is how HOX proteins select the correct sets of target genes in vivo. The homeodomain proteins encoded by the Drosophila extradenticle gene and its mammalian homologues, the pbx genes, contribute to HOX specificity by cooperatively binding to DNA with HOX proteins. For example, the HOX protein labial cooperatively binds with extradenticle protein to a 20-bp oligonucleotide that is sufficient to direct a labial-like expression pattern in Drosophila embryos. Here we have analyzed the protein-DNA interactions that are important for forming the labial-extradenticle-DNA complex. The data suggest a model in which labial and extradenticle, separated by only 4 bp, bind this DNA as a heterodimer in a head-to-tail orientation. We have confirmed several aspects of this model by characterizing extradenticle-HOX binding to mutant oligonucleotides. Most importantly, mutations in base pairs predicted to contact the HOX N-terminal arm resulted in a change in HOX preference in the heterodimer, from labial to Ultrabithorax. These results demonstrate that extradenticle prefers to bind cooperatively with different HOX proteins depending on subtle differences in the heterodimer binding site.

Sequence analysis of the cis-regulatory regions of the bithorax complex of Drosophila.

The bithorax complex (BX-C) of Drosophila, one of two complexes that act as master regulators of the body plan of the fly, has now been entirely sequenced and comprises approximately 315,000 bp, only 1.4% of which codes for protein. Analysis of this sequence reveals significantly overrepresented DNA motifs of unknown, as well as known, functions in the non-protein-coding portion of the sequence. The following types of motifs in that portion are analyzed: (i) concatamers of mono-, di-, and trinucleotides; (ii) tightly clustered hexanucleotides (spaced < or = 5 bases apart); (iii) direct and reverse repeats longer than 20 bp; and (iv) a number of motifs known from biochemical studies to play a role in the regulation of the BX-C. The hexanucleotide AGATAC is remarkably overrepresented and is surmised to play a role in chromosome pairing. The positions of sites of highly overrepresented motifs are plotted for those that occur at more than five sites in the sequence, when < 0.5 case is expected. Expected values are based on a third-order Markov chain, which is the optimal order for representing the BXCALL sequence.

The choice of alternative 5' splice sites in influenza virus M1 mRNA is regulated by the viral polymerase complex.

The influenza virus M1 mRNA has two alternative 5' splice sites: a distal 5' splice site producing mRNA3 that has the coding potential for 9 amino acids and a proximal 5' splice site producing M2 mRNA encoding the essential M2 ion-channel protein. Only mRNA3 was made in uninfected cells transfected with DNA expressing M1 mRNA. Similarly, using nuclear extracts from uninfected cells, in vitro splicing of M1 mRNA yielded only mRNA3. Only when the mRNA3 5' splice site was inactivated by mutation was M2 mRNA made in uninfected cells and in uninfected cell extracts. In influenza virus-infected cells, M2 mRNA was made, but only after a delay, suggesting that newly synthesized viral gene product(s) were needed to activate the M2 5' splice site. We present strong evidence that these gene products are the complex of the three polymerase proteins, the same complex that functions in the transcription and replication of the viral genome. Gel shift experiments showed that the viral polymerase complex bound to the 5' end of the viral M1 mRNA in a sequence-specific and cap-dependent manner. During in vitro splicing catalyzed by uninfected cell extracts, the binding of the viral polymerase complex blocked the mRNA3 5' splice site, resulting in the switch to the M2 mRNA 5' splice site and the production of M2 mRNA.

Characterization of the VHL tumor suppressor gene product: localization, complex formation, and the effect of natural inactivating mutations.

The human VHL tumor suppressor gene has been implicated in the inherited disorder von Hippel-Lindau disease and in sporadic renal carcinoma. The homologous rat gene encodes a 185-amino acid protein that is 88% sequence identical to the aligned 213-amino acid human VHL gene product. When expressed in COS-7 cells, both the human and the rat VHL proteins showed predominant nuclear, nuclear and cytosolic, or predominant cytosolic VHL staining by immunofluorescence. A complicated pattern of cellular proteins was seen that could be specifically coimmunoprecipitated with the introduced VHL protein. A complex containing VHL and proteins of apparent molecular masses 16 and 9 kDa was the most consistently observed. Certain naturally occurring VHL missense mutations demonstrated either complete or partial loss of the p16-p9 complex. Thus, the VHL tumor suppressor gene product is a nuclear protein, perhaps capable of specifically translocating between the nucleus and the cytosol. It is likely that VHL executes its functions via formation of specific multiprotein complexes. Identification of these VHL-associated proteins will likely clarify the physiology of this tumor suppressor gene.

The glucocorticoid receptor type II complex is a target of the HIV-1 vpr gene product.

The vpr gene of human immunodeficiency virus type 1 (HIV-1) encodes a 15-kDa virion-associated protein that functions as a regulator of cellular processes linked to the HIV life cycle. We report the interaction of a 41-kDa cytosolic viral protein R interacting protein 1 (Rip-1) with Vpr in vitro. Rip-1 displays a wide tissue distribution, including relevant targets of HIV infection. Vpr protein induced nuclear translocation of Rip-1, as did glucocorticoid receptor (GR)-II-stimulating steroids. Importantly, Vpr and Rip-1 coimmunoprecipitated with the human GR as part of an activated receptor complex. Vpr complementation of a vpr mutant virus was also mimicked by GR-II-stimulating steroids. Vpr and GR-II actions were inhibited by mifepristone, a GR-II pathway inhibitor. Together these data directly link the activity of the vpr gene product to the glucocorticoid steroid pathway and provide a biochemical mechanism for the cellular and viral activity of Vpr, as well as suggest that a unique class of antivirals, which includes mifepristone (RU486), may influence HIV-1 replication.