Congruences between modular forms: Raising the level and dropping Euler factors

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida’s result accounts for congruences in terms of the value of an L-function, and Ribet’s result is related to the behavior of the period that appears there. Wiles’ theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at “nonminimal level” is obtained from one at “minimal level” by dropping Euler factors from the L-function.

X-ray structure of a vanadium-containing enzyme: chloroperoxidase from the fungus Curvularia inaequalis.

The chloroperoxidase (EC 1.11.1.-) from the fungus Curvularia inaequalis belongs to a class of vanadium enzymes that oxidize halides in the presence of hydrogen peroxide to the corresponding hypohalous acids. The 2.1 A crystal structure (R = 20%) of an azide chloroperoxidase complex reveals the geometry of the catalytic vanadium center. Azide coordinates directly to the metal center, resulting in a structure with azide, three nonprotein oxygens, and a histidine as ligands. In the native state vanadium will be bound as hydrogen vanadate(V) in a trigonal bipyramidal coordination with the metal coordinated to three oxygens in the equatorial plane, to the OH group at one apical position, and to the epsilon 2 nitrogen of a histidine at the other apical position. The protein fold is mainly alpha-helical with two four-helix bundles as main structural motifs and an overall structure different from other structures. The helices pack together to a compact molecule, which explains the high stability of the protein. An amino acid sequence comparison with vanadium-containing bromoperoxidase from the seaweed Ascophyllum nodosum shows high similarities in the regions of the metal binding site, with all hydrogen vanadate(V) interacting residues conserved except for lysine-353, which is an asparagine.

Isolation of HLA-DR1.(staphylococcal enterotoxin A)2 trimers in solution.

Mutational studies indicate that the superantigen staphylococcal enterotoxin A (SEA) has two separate binding sites for major histocompatibility complex (MHC) class II molecules. Direct evidence is provided here for the formation of SEA-MHC class II trimers in solution. Isoelectric focusing separated SEA-HLA-DR1 complexes into both dimers and HLA-DR1.SEA2 trimers. The molar ratio of components was determined by dual isotope labeling. The SEA mutant SEA-F47S, L48S, Y92A, which is deficient in MHC class II alpha-chain binding, formed only dimers with HLA-DR1, whereas a second SEA mutant, SEA-H225A, which lacks high-affinity MHC class II beta-chain binding was incapable of forming any complexes. Thus SEA binding to its MHC receptor is a two-step process involving initial beta-chain binding followed by cooperative binding of a second SEA molecule to the class II alpha chain.

Periodic variation in side-chain polarities of T-cell antigenic peptides correlates with their structure and activity.

We present an analysis that synthesizes information on the sequence, structure, and motifs of antigenic peptides, which previously appeared to be in conflict. Fourier analysis of T-cell antigenic peptides indicates a periodic variation in amino acid polarities of 3-3.6 residues per period, suggesting an amphipathic alpha-helical structure. However, the diffraction patterns of major histocompatibility complex (MHC) molecules indicate that their ligands are in an extended non-alpha-helical conformation. We present two mutually consistent structural explanations for the source of the alpha-helical periodicity, based on an observation that the side chains of MHC-bound peptides generally partition with hydrophobic (hydrophilic) side chains pointing into (out of) the cleft. First, an analysis of haplotype-dependent peptide motifs indicates that the locations of their defining residues tend to force a period 3-4 variation in hydrophobicity along the peptide sequence, in a manner consistent with the spacing of pockets in the MHC. Second, recent crystallographic determination of the structure of a peptide bound to a class II MHC molecule reveals an extended but regularly twisted peptide with a rotation angle of about 130 degrees. We show that similar structures with rotation angles of 100-130 degrees are energetically acceptable and also span the length of the MHC cleft. These results provide a sound physical chemical and structural basis for the existence of a haplotype-independent antigenic motif which can be particularly important in limiting the search time for antigenic peptides.