Orientation in operator algebras

A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics.

The problem of the spreading of a liquid film along a solid surface: A new mathematical formulation

A new mathematical model is proposed for the spreading of a liquid film on a solid surface. The model is based on the standard lubrication approximation for gently sloping films (with the no-slip condition for the fluid at the solid surface) in the major part of the film where it is not too thin. In the remaining and relatively small regions near the contact lines it is assumed that the so-called autonomy principle holds—i.e., given the material components, the external conditions, and the velocity of the contact lines along the surface, the behavior of the fluid is identical for all films. The resulting mathematical model is formulated as a free boundary problem for the classical fourth-order equation for the film thickness. A class of self-similar solutions to this free boundary problem is considered.

Contributions of distinct quaternary contacts to cooperative operator binding by Mnt repressor

Mnt, a tetrameric repressor encoded by bacteriophage P22, uses N-domain dimers to contact each half of its operator site. Experiments with a double mutant and structural homology with the P22 Arc repressor suggest that contacts made by Arg-28 and stabilized by Glu-33 are largely responsible for dimer–dimer cooperativity in Mnt. These dimer–dimer contacts are energetically more important for operator binding than solution tetramerization, which is mediated by an independent C-terminal coiled-coil domain. Indeed, once one dimer of the Mnt tetramer contacts an operator half site, binding of the second dimer occurs with an effective concentration much lower than that expected if both dimers were flexibly tethered. These results suggest that binding of the second dimer introduces some strain into the protein–DNA complex, a mechanism that could serve to limit the affinity of operator binding and to prevent strong binding of the Mnt tetramer to nonoperator sites.

The operator algebra approach to quantum groups

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.

On the coefficients of the characteristic series of the U-operator

A conceptual proof is given of the fact that the coefficients of the characteristic series of the U-operator acting on families of overconvegent modular forms lie in the Iwasawa algebra.

Operator binding by lambda repressor heterodimers with one or two N-terminal arms.

The first 6 amino acids (NH2-Ser1-Thr2-Lys3-Lys4-Lys5-Pro6) of bacteriophage lambda cI repressor form a flexible arm that wraps around the operator DNA. Homodimeric lambda repressor has two arms. To determine whether both arms are necessary or only one arm is sufficient for operator binding, we constructed heterodimeric repressors with two, one, or no arms by fusing the DNA binding domain of lambda repressor to leucine zippers from Fos and Jun. Although only one arm is visible in the cocrystal structure of the N-domain-operator complex, our results indicate that both arms are required for optimal operator binding and normal site discrimination.

Binding of purified multiple antibiotic-resistance repressor protein (MarR) to mar operator sequences.

Elevated expression of the marORAB multiple antibiotic-resistance operon enhances the resistance of Escherichia coli to various medically significant antibiotics. Transcription of the operon is repressed in vivo by the marR-encoded protein, MarR, and derepressed by salicylate and certain antibiotics. The possibility that repression results from MarR interacting with the marO operator-promoter region was studied in vitro using purified MarR and a DNA fragment containing marO. MarR formed at least two complexes with marO DNA, bound > 30-fold more tightly to it than to salmon sperm DNA, and protected two separate 21-bp sites within marO from digestion by DNase I. Site I abuts the downstream side of the putative -35 transcription-start signal and includes 4 bp of the -10 signal. Site II begins 13 bp downstream of site I, ending immediately before the first base pair of marR. Site II, approximately 80% homologous to site I, is not required for repression since a site II-deleted mutant (marO133) was repressed in trans by wild-type MarR. The absence of site II did not prevent MarR from complexing with the site I of marO133. Salicylate bound to MarR (Kd approximately 0.5 mM) and weakened the interaction of MarR with sites I and II. Thus, repression of the mar operon, which curbs the antibiotic resistance of E. coli, correlates with the formation of MarR-site I complexes. Salicylate appears to induce the mar operon by binding to MarR and inhibiting complex formation, whereas tetracycline and chloramphenicol, which neither bind MarR nor inhibit complex formation, must induce by an indirect mechanism.