Chaperone activity and structure of monomeric polypeptide binding domains of���GroEL

The chaperonin GroEL is a large complex composed of 14 identical 57-kDa subunits that requires ATP and GroES for some of its activities. We find that a monomeric polypeptide corresponding to residues 191 to 345 has the activity of the tetradecamer both in facilitating the refolding of rhodanese and cyclophilin A in the absence of ATP and in catalyzing the unfolding of native barnase. Its crystal structure, solved at 2.5 ��� resolution, shows a well-ordered domain with the same fold as in intact GroEL. We have thus isolated the active site of the complex allosteric molecular chaperone, which functions as a ���minichaperone.��� This has mechanistic implications: the presence of a central cavity in the GroEL complex is not essential for those representative activities in vitro, and neither are the allosteric properties. The function of the allosteric behavior on the binding of GroES and ATP must be to regulate the affinity of the protein for its various substrates in vivo, where the cavity may also be required for special functions.

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 ��(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Tur��nian 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 ��-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. (Birkh��user 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 C��� nonterminating 6��5 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.

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.