Homogeneous structures and rigidity of isoparametric submanifolds in Hilbert space

We study isoparametric submanifolds of rank at least two in a separable Hilbert space, which are known to be homogeneous by the main result in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149-181], and with such a submanifold M and a point x in M we associate a canonical homogeneous structure I" (x) (a certain bilinear map defined on a subspace of T (x) M x T (x) M). We prove that I" (x) , together with the second fundamental form alpha (x) , encodes all the information about M, and we deduce from this the rigidity result that M is completely determined by alpha (x) and (Delta alpha) (x) , thereby making such submanifolds accessible to classification. As an essential step, we show that the one-parameter groups of isometries constructed in [E. Heintze and X. Liu, Ann. of Math. (2), 149 (1999), 149-181] to prove their homogeneity induce smooth and hence everywhere defined Killing fields, implying the continuity of I" (this result also seems to close a gap in [U. Christ, J. Differential Geom., 62 (2002), 1-15]). Here an important tool is the introduction of affine root systems of isoparametric submanifolds.

Crystal structures of Leptospira interrogans FAD-containing ferredoxin-NADP+ reductase and its complex with NADP+

Abstract Background Ferredoxin-NADP(H) reductases (FNRs) are flavoenzymes that catalyze the electron transfer between NADP(H) and the proteins ferredoxin or flavodoxin. A number of structural features distinguish plant and bacterial FNRs, one of which is the mode of the cofactor FAD binding. Leptospira interrogans is a spirochaete parasitic bacterium capable of infecting humans and mammals in general. Leptospira interrogans FNR (LepFNR) displays low sequence identity with plant (34% with Zea mays) and bacterial (31% with Escherichia coli) FNRs. However, LepFNR contains all consensus sequences that define the plastidic class FNRs. Results The crystal structures of the FAD-containing LepFNR and the complex of the enzyme with NADP+, were solved and compared to known FNRs. The comparison reveals significant structural similarities of the enzyme with the plastidic type FNRs and differences with the bacterial enzymes. Our small angle X-ray scattering experiments show that LepFNR is a monomeric enzyme. Moreover, our biochemical data demonstrate that the LepFNR has an enzymatic activity similar to those reported for the plastidic enzymes and that is significantly different from bacterial flavoenzymes, which display lower turnover rates. Conclusion LepFNR is the first plastidic type FNR found in bacteria and, despite of its low sequence similarity with plastidic FNRs still displays high catalytic turnover rates. The typical structural and biochemical characteristics of plant FNRs unveiled for LepFNR support a notion of a putative lateral gene transfer which presumably offers Leptospira interrogans evolutionary advantages. The wealth of structural information about LepFNR provides a molecular basis for advanced drugs developments against leptospirosis.