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.

Magnesium-Chelatase from Developing Pea Leaves1 : Characterization of a Soluble Extract from Chloroplasts and Resolution into Three Required Protein Fractions

Mg-chelatase catalyzes the ATP-dependent insertion of Mg2+ into protoporphyrin-IX to form Mg-protoporphyrin-IX. This is the first step unique to chlorophyll synthesis, and it lies at the branch point for porphyrin utilization; the other branch leads to heme. Using the stromal fraction of pea (Pisum sativum L. cv Spring) chloroplasts, we have prepared Mg-chelatase in a highly active (1000 pmol 30 min−1 mg−1) and stable form. The reaction had a lag in the time course, which was overcome by preincubation with ATP. The concentration curves for ATP and Mg2+ were sigmoidal, with apparent Km values for Mg2+ and ATP of 14.3 and 0.35 mm, respectively. The Km for deuteroporphyrin was 8 nm. This Km is 300 times lower than the published porphyrin Km for ferrochelatase. The soluble extract was separated into three fractions by chromatography on blue agarose, followed by size-selective centrifugal ultrafiltration of the column flow-through. All three fractions were required for activity, clearly demonstrating that the plant Mg-chelatase requires at least three protein components. Additionally, only two of the components were required for activation; both were contained in the flow-through from the blue-agarose column.

Heat effects on DNA repair after ionising radiation: hyperthermia commonly increases the number of non-repaired double-strand breaks and structural rearrangements

After ionising radiation double-strand breaks (dsb) are lethal if not repaired or misrepaired. Cell killing is greatly enhanced by hyperthermia and it is questioned here whether heat not only affects dsb repair capacity but also fidelity in a chromosomal context. dsb repair experiments were designed so as to mainly score non-homologous end joining, while homologous recombination was largely precluded. Human male G0 fibroblasts were either preheated (45°C, 20 min) or not before X-irradiation. dsb induction and repair were measured by conventional gel electrophoresis and an assay combining restriction digestion using a rare cutting enzyme (NotI) and Southern hybridisation, which detects large chromosomal rearrangements (>100 kb). dsb induction rate in an X-chromosomal NotI fragment was 4.8 × 10–3 dsb/Gy/Mb. Similar values were found for the genome overall and also when cells were preheated. After 50 Gy, fibroblasts were competent to largely restore the original restriction fragment size. Five per cent of dsb remained non-rejoined and 14% were misrejoined. Correct restitution of restriction fragments occurred preferably during the first hour but continued at a slow rate for 12–16 h. In addition, dsb appeared to misrejoin throughout the entire repair period. After hyperthermia the fractions of non-rejoined and misrejoined dsb were similarly increased to 13 and 51%, respectively. It is suggested that heat increases the probability of dsb being incorrectly rejoined but it is not likely to interfere with one dsb repair pathway in particular.