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.

Human milk lactoferrin inactivates two putative colonization factors expressed by Haemophilus influenzae

Haemophilus influenzae is a major cause of otitis media and other respiratory tract disease in children. The pathogenesis of disease begins with colonization of the upper respiratory mucosa, a process that involves evasion of local immune mechanisms and adherence to epithelial cells. Several studies have demonstrated that human milk is protective against H. influenzae colonization and disease. In the present study, we examined the effect of human milk on the H. influenzae IgA1 protease and Hap adhesin, two autotransported proteins that are presumed to facilitate colonization. Our results demonstrated that human milk lactoferrin efficiently extracted the IgA1 protease preprotein from the bacterial outer membrane. In addition, lactoferrin specifically degraded the Hap adhesin and abolished Hap-mediated adherence. Extraction of IgA1 protease and degradation of Hap were localized to the N-lobe of the bilobed lactoferrin molecule and were inhibited by serine protease inhibitors, suggesting that the lactoferrin N-lobe may contain serine protease activity. Additional experiments revealed no effect of lactoferrin on the H. influenzae P2, P5, and P6 outer-membrane proteins, which are distinguished from IgA1 protease and Hap by the lack of an N-terminal passenger domain or an extracellular linker region. These results suggest that human milk lactoferrin may attenuate the pathogenic potential of H. influenzae by selectively inactivating IgA1 protease and Hap, thereby interfering with colonization. Future studies should examine the therapeutic potential of lactoferrin, perhaps as a supplement in infant formulas.

Minimum size limit for useful locomotion by free-swimming microbes

Formulas are derived for the effect of size on a free-swimming microbe’s ability to follow chemical, light, or temperature stimuli or to disperse in random directions. The four main assumptions are as follows: (i) the organisms can be modeled as spheres, (ii) the power available to the organism for swimming is proportional to its volume, (iii) the noise in measuring a signal limits determination of the direction of a stimulus, and (iv) the time available to determine stimulus direction or to swim a straight path is limited by rotational diffusion caused by Brownian motion. In all cases, it is found that there is a sharp size limit below which locomotion has no apparent benefit. This size limit is estimated to most probably be about 0.6 μm diameter and is relatively insensitive to assumed values of the other parameters. A review of existing descriptions of free-floating bacteria reveals that the smallest of 97 motile genera has a mean length of 0.8 μm, whereas 18 of 94 nonmotile genera are smaller. Similar calculations have led to the conclusion that a minimum size also exists for use of pheromones in mate location, although this size limit is about three orders of magnitude larger. In both cases, the application of well-established physical laws and biological generalities has demonstrated that a common feature of animal behavior is of no use to small free-swimming organisms.

Somatic and germ cell parasitism in a colonial ascidian: Possible role for a highly polymorphic allorecognition system

A colonial protochordate, Botryllus schlosseri, undergoes a natural transplantation reaction in the wild that results alternatively in colony fusion (chimera formation) or inflammatory rejection. A single, highly polymorphic histocompatibility locus (called Fu/HC) is responsible for rejection versus fusion. Gonads are seeded and gametogenesis can occur in colonies well after fusion, and involves circulating germ-line progenitors. Buss proposed that colonial organisms might develop self/non-self histocompatibility systems to limit the possibility of interindividual germ cell “parasitism” (GCP) to histocompatible kin [Buss, L. W. (1982) Proc. Natl. Acad. Sci. USA 79, 5337–5341 and Buss, L. W. (1987) The Evolution of Individuality (Princeton Univ. Press, Princeton]. Here we demonstrate in laboratory and field experiments that both somatic cell and (more importantly) germ-line parasitism are a common occurrence in fused chimeras. These experiments support the tenet in Buss’s hypothesis that germ cell and somatic cell parasitism can occur in fused chimeras and that a somatic appearance may mask the winner of a gametic war. They also provide an interesting challenge to develop formulas that describe the inheritance of competing germ lines rather than competing individuals. The fact that fused B. schlosseri have higher rates of GCP than unfused colonies additionally provides a rational explanation for the generation and maintenance of a high degree of Fu/HC polymorphism, largely limiting GCP to sibling offspring.

The RESID Database of protein structure modifications and the NRL-3D Sequence–Structure Database

The RESID Database is a comprehensive collection of annotations and structures for protein post-translational modifications including N-terminal, C-terminal and peptide chain cross-link modifications. The RESID Database includes systematic and frequently observed alternate names, Chemical Abstracts Service registry numbers, atomic formulas and weights, enzyme activities, taxonomic range, keywords, literature citations with database cross-references, structural diagrams and molecular models. The NRL-3D Sequence–Structure Database is derived from the three-dimensional structure of proteins deposited with the Research Collaboratory for Structural Bioinformatics Protein Data Bank. The NRL-3D Database includes standardized and frequently observed alternate names, sources, keywords, literature citations, experimental conditions and searchable sequences from model coordinates. These databases are freely accessible through the National Cancer Institute–Frederick Advanced Biomedical Computing Center at these web sites: http://www.ncifcrf.gov/RESID, http://www.ncifcrf.gov/ NRL-3D; or at these National Biomedical Research Foundation Protein Information Resource web sites: http://pir.georgetown.edu/pirwww/dbinfo/resid.html, http://pir.georgetown.edu/pirwww/dbinfo/nrl3d.html

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.

Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(φ) of a two-dimensional modular Galois representation φ. We start with the p-adic Galois representation φ0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(φ0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let φ denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting φ0, and the characteristic power series of the Selmer group Sel(ad(φ)) is given by a p-adic L-function interpolating L(1, ad(φk)) for weight k + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(φ) ⊗ ν−1), where ν is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.