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.

Evolution of an enzyme active site: The structure of a new crystal form of muconate lactonizing enzyme compared with mandelate racemase and enolase

Muconate lactonizing enzyme (MLE), a component of the β-ketoadipate pathway of Pseudomonas putida, is a member of a family of related enzymes (the “enolase superfamily”) that catalyze the abstraction of the α-proton of a carboxylic acid in the context of different overall reactions. New untwinned crystal forms of MLE were obtained, one of which diffracts to better than 2.0-Å resolution. The packing of the octameric enzyme in this crystal form is unusual, because the asymmetric unit contains three subunits. The structure of MLE presented here contains no bound metal ion, but is very similar to a recently determined Mn2+-bound structure. Thus, absence of the metal ion does not perturb the structure of the active site. The structures of enolase, mandelate racemase, and MLE were superimposed. A comparison of metal ligands suggests that enolase may retain some characteristics of the ancestor of this enzyme family. Comparison of other residues involved in catalysis indicates two unusual patterns of conservation: (i) that the position of catalytic atoms remains constant, although the residues that contain them are located at different points in the protein fold; and (ii) that the positions of catalytic residues in the protein scaffold are conserved, whereas their identities and roles in catalysis vary.

Biogeographic range expansion into South America by Coccidioides immitis mirrors New World patterns of human migration

Long-distance population dispersal leaves its characteristic signature in genomes, namely, reduced diversity and increased linkage between genetic markers. This signature enables historical patterns of range expansion to be traced. Herein, we use microsatellite loci from the human pathogen Coccidioides immitis to show that genetic diversity in this fungus is geographically partitioned throughout North America. In contrast, analyses of South American C. immitis show that this population is genetically depauperate and was founded from a single North American population centered in Texas. Variances of allele distributions show that South American C. immitis have undergone rapid population growth, consistent with an epidemic increase in postcolonization population size. Herein, we estimate the introduction into South America to have occurred within the last 9,000–140,000 years. This range increase parallels that of Homo sapiens. Because of known associations between Amerindians and this fungus, we suggest that the colonization of South America by C. immitis represents a relatively recent and rapid codispersal of a host and its pathogen.

A new isoleucine substitution of Val-20 in transthyretin tetramers selectively impairs dimer-dimer contacts and causes systemic amyloidosis.

The most frequent form of inherited amyloidoses is associated with mutations in the transthyretin (TTR) gene coding for 127-amino acid residues of four identical, noncovalently linked subunits that form a pair of dimers in the plasma protein complex. Amyloid fibrils containing the variant and to a lesser extent the wild-type form of the TTR molecule are deposited in various organs, including peripheral nerves and the myocardium, with polyneuropathy and cardiomyopathy as major clinical manifestations. So far, more than 40 distinct amino acid substitutions distributed throughout the TTR sequence over 30 positions have been found to be correlated with an increased amyloidogenicity of TTR. Most of these amyloidogenic amino acid substitutions are suspected to alter the conformation and stability of the monomer. Here we identify and characterize by protein and DNA analysis a novel amyloidogenic Val-20 to Ile mutation in a German three-generation family. The index patient suffered from severe amyloid cardiomyopathy at the age of 60. Conformational stability and unfolding behavior of the Ile-20 monomer in urea gradients was found to be almost indistinguishable from that of wild-type TTR. In contrast, tetramer stability was significantly reduced in agreement with the expected change in the interactions between the two opposing dimers via the side chain of Ile-20. Our observations provide strong evidence for the view that amyloidogenic amino acid substitutions in TTR facilitate the conversion of tetrameric TTR complexes into those conformational intermediates of the TTR folding pathway that have an intrinsic amyloidogenic potential.

The new dysmorphology: application of insights from basic developmental biology to the understanding of human birth defects.

Information obtained from studies of developmental and cellular processes in lower organisms is beginning to make significant contributions to the understanding of the pathogenesis of human birth defects, and it is now becoming possible to treat birth defects as inborn errors of development. Mutations in genes for transcription factors, receptors, cell adhesion molecules, intercellular junctions, molecules involved in signal transduction, growth factors, structural proteins, enzymes, and transporters have been identified in genetically caused human malformations and dysplasias. The identification of these mutations and the analysis of their developmental effects have been greatly facilitated by the existence of natural or engineered models in the mouse and even of related mutations in Drosophila, and in some instances a remarkable conservation of function in development has been observed, even between widely separated species.