Propagating structure of Alzheimer’s β-amyloid(10–35) is parallel β-sheet with residues in exact register

The pathognomonic plaques of Alzheimer’s disease are composed primarily of the 39- to 43-aa β-amyloid (Aβ) peptide. Crosslinking of Aβ peptides by tissue transglutaminase (tTg) indicates that Gln15 of one peptide is proximate to Lys16 of another in aggregated Aβ. Here we report how the fibril structure is resolved by mapping interstrand distances in this core region of the Aβ peptide chain with solid-state NMR. Isotopic substitution provides the source points for measuring distances in aggregated Aβ. Peptides containing a single carbonyl 13C label at Gln15, Lys16, Leu17, or Val18 were synthesized and evaluated by NMR dipolar recoupling methods for the measurement of interpeptide distances to a resolution of 0.2 Å. Analysis of these data establish that this central core of Aβ consists of a parallel β-sheet structure in which identical residues on adjacent chains are aligned directly, i.e., in register. Our data, in conjunction with existing structural data, establish that the Aβ fibril is a hydrogen-bonded, parallel β-sheet defining the long axis of the Aβ fibril propagation.

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.