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.

Rapid treadmilling of brain microtubules free of microtubule-associated proteins in vitro and its suppression by tau

We have determined the treadmilling rate of brain microtubules (MTs) free of MT-associated proteins (MAPs) at polymer mass steady state in vitro by using [3H]GTP-exchange. We developed buffer conditions that suppressed dynamic instability behavior by ≈10-fold to minimize the contribution of dynamic instability to total tubulin-GTP exchange. The MTs treadmilled rapidly under the suppressed dynamic instability conditions, at a minimum rate of 0.2 μm/min. Thus, rapid treadmilling is an intrinsic property of MAP-free MTs. Further, we show that tau, an axonal stabilizing MAP involved in Alzheimer’s disease, strongly suppresses the treadmilling rate. These results indicate that tau’s function in axons might involve suppression of axonal MT treadmilling. We describe mathematically how treadmilling and dynamic instability are mechanistically distinct MT behaviors. Finally, we present a model that explains how small changes in the critical tubulin subunit concentration at MT minus ends, caused by intrinsic differences in rate constants or regulatory proteins, could produce large changes in the treadmilling rate.