Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices

We consider Seiberg electric-magnetic dualities for 4d $\mathcal{N}=1$ SYM theories with SO(N) gauge group. For all such known theories we construct superconformal indices (SCIs) in terms of elliptic hypergeometric integrals. Equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals. In particular, we describe a number of new elliptic beta integrals associated with the s-confining theories with the spinor matter fields. Reductions of some dualities from SP(2N) to SO(2N) or SO(2N+1) gauge groups are described. Interrelation of SCIs and the Witten anomaly is briefly discussed. Possible applications of the elliptic hypergeometric integrals to a two-parameter deformation of 2d conformal field theory and related matrix models are indicated. Connections of the reduced SCIs with the state integrals of the knot theory, generalized AGT duality for (3+3)d theories, and a 2d vortex partition function are described.Comment: Latex, 58 pages; paper shortened, to appear in Commun. Math. Phy

Kinetic mechanisms of cholesterol oxidase from Streptomyces hygroscopicus and Brevibacterium sterolicum

The kinetic properties of two cholesterol oxidases, one from Brevibacterium sterolicum (BCO) the other from Streptomyces hygroscopicus (SCO) were investigated. BCO works via a ping-pong mechanism, whereas the catalytic pathway of SCO is sequential. The turnover numbers at infinite cholesterol and oxygen concentrations are 202 s−1 and 105 s−1 for SCO and BCO, respectively. The rates of flavin reduction extrapolated to saturating substrate concentration, under anaerobic conditions, are 235 s−1 for BCO and 232 s−1 for SCO (in the presence of 1% Thesit and 10% 2-propanol). With reduced SCO the rate of Δ5-6→Δ4-5 isomerization of the intermediate 5-cholesten-3-one to final product is slow (0.3 s−1). With oxidized SCO and BCO the rate of isomerization is much faster (≈ 300 s−1), thus it is not rate-limiting for catalysis. The kinetic behaviour of both reduced COs towards oxygen is unusual in that they exhibit apparent saturation with increasing oxygen concentrations (extrapolated rates ≈ 250 s−1 and 1.3 s−1, for BCO and SCO, respectively): too slow to account for catalysis. For BCO the kinetic data are compatible with a step preceding the reaction with oxygen, involving interconversion of reactive and nonreactive forms of the enzyme. We suggest that the presence of micelles in the reaction medium, due to the necessary presence of detergents to solubilize the substrate, influence the availability or reactivity of oxygen towards the enzyme. The rate of re-oxidation of SCO in the presence of product is also too slow to account for catalysis, probably due to the impossibility of producing quantitatively the reduced enzyme product complexes