New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

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.

p-adic L functions and trivial zeroes

The following is adapted from the notes for the lecture. It announces results and conjectures about values of the p-adic L function of the symmetric square of an elliptic curve.

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.

Distinct functions and cooperative interaction of the subunits of the transporter associated with antigen processing (TAP)

The ATP-binding cassette (ABC) transporter TAP translocates peptides from the cytosol to awaiting MHC class I molecules in the endoplasmic reticulum. TAP is made up of the TAP1 and TAP2 polypeptides, which each possess a nucleotide binding domain (NBD). However, the role of ATP in peptide binding and translocation is poorly understood. We present biochemical and functional evidence that the NBDs of TAP1 and TAP2 are non-equivalent. Photolabeling experiments with 8-azido-ATP demonstrate a cooperative interaction between the two NBDs that can be stimulated by peptide. The substitution of key lysine residues in the Walker A motifs of TAP1 and TAP2 suggests that TAP1-mediated ATP hydrolysis is not essential for peptide translocation but that TAP2-mediated ATP hydrolysis is critical, not only for translocation, but for peptide binding.

Functions of FKBP12 and Mitochondrial Cyclophilin Active Site Residues In Vitro and In Vivo in Saccharomyces cerevisiae

Cyclophilin and FK506 binding protein (FKBP) accelerate cis–trans peptidyl-prolyl isomerization and bind to and mediate the effects of the immunosuppressants cyclosporin A and FK506. The normal cellular functions of these proteins, however, are unknown. We altered the active sites of FKBP12 and mitochondrial cyclophilin from the yeast Saccharomyces cerevisiae by introducing mutations previously reported to inactivate these enzymes. Surprisingly, most of these mutant enzymes were biologically active in vivo. In accord with previous reports, all of the mutant enzymes had little or no detectable prolyl isomerase activity in the standard peptide substrate-chymotrypsin coupled in vitro assay. However, in a variation of this assay in which the protease is omitted, the mutant enzymes exhibited substantial levels of prolyl isomerase activity (5–20% of wild-type), revealing that these mutations confer sensitivity to protease digestion and that the classic in vitro assay for prolyl isomerase activity may be misleading. In addition, the mutant enzymes exhibited near wild-type activity with two protein substrates, dihydrofolate reductase and ribonuclease T1, whose folding is accelerated by prolyl isomerases. Thus, a number of cyclophilin and FKBP12 “active-site” mutants previously identified are largely active but protease sensitive, in accord with our findings that these mutants display wild-type functions in vivo. One mitochondrial cyclophilin mutant (R73A), and also the wild-type human FKBP12 enzyme, catalyze protein folding in vitro but lack biological activity in vivo in yeast. Our findings provide evidence that both prolyl isomerase activity and other structural features are linked to FKBP and cyclophilin in vivo functions and suggest caution in the use of these active-site mutations to study FKBP and cyclophilin functions.

Chaperone activity and structure of monomeric polypeptide binding domains of GroEL

The chaperonin GroEL is a large complex composed of 14 identical 57-kDa subunits that requires ATP and GroES for some of its activities. We find that a monomeric polypeptide corresponding to residues 191 to 345 has the activity of the tetradecamer both in facilitating the refolding of rhodanese and cyclophilin A in the absence of ATP and in catalyzing the unfolding of native barnase. Its crystal structure, solved at 2.5 Å resolution, shows a well-ordered domain with the same fold as in intact GroEL. We have thus isolated the active site of the complex allosteric molecular chaperone, which functions as a “minichaperone.” This has mechanistic implications: the presence of a central cavity in the GroEL complex is not essential for those representative activities in vitro, and neither are the allosteric properties. The function of the allosteric behavior on the binding of GroES and ATP must be to regulate the affinity of the protein for its various substrates in vivo, where the cavity may also be required for special functions.

Shk1, a homolog of the Saccharomyces cerevisiae Ste20 and mammalian p65PAK protein kinases, is a component of a Ras/Cdc42 signaling module in the fission yeast Schizosaccharomyces pombe.

We describe a protein kinase, Shk1, from the fission yeast Schizosaccharomyces pombe, which is structurally related to the Saccharomyces cerevisiae Ste20 and mammalian p65PAK protein kinases. We provide genetic evidence for physical and functional interaction between Shk1 and the Cdc42 GTP-binding protein required for normal cell morphology and mating in S. pombe. We further show that expression of the STE20 gene complements the shk1 null mutation and that Shk1 is capable of signaling to the pheromone-responsive mitogen-activated protein kinase cascade in S. cerevisiae. Our results lead us to propose that signaling modules composed of small GTP-binding proteins and protein kinases related to Shk1, Ste20, and p65PAK, are highly conserved in evolution and participate in both cytoskeletal functions and mitogen-activated protein kinase signaling pathways.

Stathmin interaction with a putative kinase and coiled-coil-forming protein domains.

Stathmin is a ubiquitous, cytosolic 19-kDa protein, which is phosphorylated on up to four sites in response to many regulatory signals within cells. Its molecular characterization indicates a functional organization including an N-terminal regulatory domain that bears the phosphorylation sites, linked to a putative alpha-helical binding domain predicted to participate in coiled-coil, protein-protein interactions. We therefore proposed that stathmin may play the role of a relay integrating diverse intracellular regulatory pathways; its action on various target proteins would be a function of its combined phosphorylation state. To search for such target proteins, we used the two-hybrid screen in yeast, with stathmin as a "bait." We isolated and characterized four cDNAs encoding protein domains that interact with stathmin in vivo. One of the corresponding proteins was identified as BiP, a member of the hsp70 heat-shock protein family. Another is a previously unidentified, putative serine/threonine kinase, KIS, which might be regulated by stathmin or, more likely, be part of the kinases controlling its phosphorylation state. Finally, two clones code for subdomains of two proteins, CC1 and CC2, predicted to form alpha-helices participating in coiled-coil interacting structures. Their isolation by interaction screening further supports our model for the regulatory function of stathmin through coiled-coil interactions with diverse downstream targets via its presumed alpha-helical binding domain. The molecular and biological characterization of KIS, CC1, and CC2 proteins will give further insights into the molecular functions and mechanisms of action of stathmin as a relay of integrated intracellular regulatory pathways.

Dendritic cell ontogeny: A human dendritic cell lineage of myeloid origin

Dendritic cells (DC) have been thought to represent a family of closely related cells with similar functions and developmental pathways. The best-characterized precursors are the epidermal Langerhans cells, which migrate to lymphoid organs and become activated DC in response to inflammatory stimuli. Here, we demonstrate that a large subset of DC in the T cell-dependent areas of human lymphoid organs are nonactivated cells and belong to a separate lineage that can be identified by high levels of the interleukin 3 receptor α chain (IL-3Rαhi). The CD34+IL-3Rαhi DC progenitors are of myeloid origin and are distinct from those that give rise to Langerhans cells in vitro. The IL-3Rαhi DC furthermore appear to migrate to lymphoid organs independently of inflammatory stimuli or foreign antigens. Thus, DC are heterogeneous with regard to function and ontogeny.

Receptor-mediated Endocytosis in the Caenorhabditis elegans Oocyte

The Caenorhabditis elegans oocyte is a highly amenable system for forward and reverse genetic analysis of receptor-mediated endocytosis. We describe the use of transgenic strains expressing a vitellogenin::green fluorescent protein (YP170::GFP) fusion to monitor yolk endocytosis by the C. elegans oocyte in vivo. This YP170::GFP reporter was used to assay the functions of C. elegans predicted proteins homologous to vertebrate endocytosis factors using RNA-mediated interference. We show that the basic components and pathways of endocytic trafficking are conserved between C. elegans and vertebrates, and that this system can be used to test the endocytic functions of any new gene. We also used the YP170::GFP assay to identify rme (receptor-mediated endocytosis) mutants. We describe a new member of the low-density lipoprotein receptor superfamily, RME-2, identified in our screens for endocytosis defective mutants. We show that RME-2 is the C. elegans yolk receptor.

The multivariate L1-median and associated data depth

This paper gives three related results: (i) a new, simple, fast, monotonically converging algorithm for deriving the L1-median of a data cloud in ℝd, a problem that can be traced to Fermat and has fascinated applied mathematicians for over three centuries; (ii) a new general definition for depth functions, as functions of multivariate medians, so that different definitions of medians will, correspondingly, give rise to different dept functions; and (iii) a simple closed-form formula of the L1-depth function for a given data cloud in ℝd.

PTGF-β, a type β transforming growth factor (TGF-β) superfamily member, is a p53 target gene that inhibits tumor cell growth via TGF-β signaling pathway

Identification and characterization of p53 target genes would lead to a better understanding of p53 functions and p53-mediated signaling pathways. Two putative p53 binding sites were identified in the promoter of a gene encoding PTGF-β, a type β transforming growth factor (TGF-β) superfamily member. Gel shift assay showed that p53 bound to both sites. Luciferase-coupled transactivation assay revealed that the gene promoter was activated in a p53 dose- as well as p53 binding site-dependent manner by wild-type p53 but not by several p53 mutants. The p53 binding and transactivation of the PTGF-β promoter was enhanced by etoposide, a p53 activator, and was largely blocked by a dominant negative p53 mutant. Furthermore, expression of endogenous PTGF-β was remarkably induced by etoposide in p53-positive, but not in p53-negative, cell lines. Finally, the conditioned medium collected from PTGF-β-overexpressing cells, but not from the control cells, suppressed tumor cell growth. Growth suppression was not, however, seen in cells that lack functional TGF-β receptors or Smad4, suggesting that PTGF-β acts through the TGF-β signaling pathway. Thus, PTGF-β, a secretory protein, is a p53 target that could mediate p53-induced growth suppression in autocrinal as well as paracrinal fashions. The finding made a vertical connection between p53 and TGF-β signaling pathways in controlling cell growth and implied a potential important role of p53 in inflammation regulation via PTGF-β.

The second STE12 homologue of Cryptococcus neoformans is MATa-specific and plays an important role in virulence

Cryptococcus neoformans STE12α, a homologue of Saccharomyces cerevisiae STE12, exists only in MATα strains. We identified another STE12 homologue, STE12a, which is MATa specific. As in the case with Δste12α, the mating efficiency for Δste12a was reduced significantly. The Δste12a strains surprisingly still mated with Δste12α strains. In MATα strains, STE12a functionally complemented STE12α for mating efficacy, haploid fruiting, and regulation of capsule size in the mouse brain. Furthermore, when STE12a was replaced with two copies of STE12α, the resulting MATa strain produced hyphae on filament agar. STE12a regulates mRNA levels of several genes that are important for virulence including CNLAC1 and CAP genes. STE12a also modulates enzyme activities of phospholipase and superoxide dismutase. Importantly, deletion of STE12a markedly reduced the virulence in mice, as is the case with STE12α. Brain smears of mice infected with the Δste12a strain showed yeast cells with a considerable reduction in capsule size compared with those infected with STE12a strains. When the disrupted locus of ste12a was replaced with a wild-type STE12a gene, both in vivo and in vitro mutant phenotypes were reversed. These results suggest that STE12a and STE12α have similar functions, and that the mating type of the cells influences the alleles to exert their biological effects. C. neoformans, thus, is the first fungal species that contains a mating-type-specific STE12 homologue in each mating type. Our results demonstrate that mating-type-specific genes are not only important for saprobic reproduction but also play an important role for survival of the organism in host tissue.

Drosophila phosphoinositide-dependent kinase-1 regulates apoptosis and growth via the phosphoinositide 3-kinase-dependent signaling pathway

Phosphoinositide-dependent kinase-1 (PDK-1) is a central mediator of the cell signaling between phosphoinositide 3-kinase (PI3K) and various intracellular serine/threonine kinases including Akt/protein kinase B (PKB), p70 S6 kinases, and protein kinase C. Recent studies with cell transfection experiments have implied that PDK-1 may be involved in various cell functions including cell growth and apoptosis. However, despite its pivotal role in cellular signalings, the in vivo functions of PDK-1 in a multicellular system have rarely been investigated. Here, we have isolated Drosophila PDK-1 (dPDK-1) mutants and characterized the in vivo roles of the kinase. Drosophila deficient in the dPDK-1 gene exhibited lethality and an apoptotic phenotype in the embryonic stage. Conversely, overexpression of dPDK-1 increased cell and organ size in a Drosophila PI3K-dependent manner. dPDK-1 not only could activate Drosophila Akt/PKB (Dakt1), but also substitute the in vivo functions of its mammalian ortholog to activate Akt/PKB. This functional interaction between dPDK-1 and Dakt1 was further confirmed through genetic analyses in Drosophila. On the other hand, cAMP-dependent protein kinase, which has been proposed as a possible target of dPDK-1, did not interact with dPDK-1. In conclusion, our findings provide direct evidence that dPDK-1 regulates cell growth and apoptosis during Drosophila development via the PI3K-dependent signaling pathway and demonstrate our Drosophila system to be a powerful tool for elucidating the in vivo functions and targets of PDK-1.