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.

Prime factors

We use Voiculescu’s free probability theory to prove the existence of prime factors, hence answering a longstanding problem in the theory of von Neumann algebras.

On representations of finite type

Representations of the (infinite) canonical anticommutation relations and the associated operator algebra, the fermion algebra, are studied. A “coupling constant” (in (0,1]) is defined for primary states of “finite type” of that algebra. Primary, faithful states of finite type with arbitrary coupling are constructed and classified. Their physical significance for quantum thermodynamical systems at high temperatures is discussed. The scope of this study is broadened to include a large class of operator algebras sharing some of the structural properties of the fermion algebra.

Fourier transform infrared spectroscopy reveals a rigid α-helical assembly for the tetrameric Streptomyces lividans K+ channel

The structure of the tetrameric K+ channel from Streptomyces lividans in a lipid bilayer environment was studied by polarized attenuated total reflection Fourier transform infrared spectroscopy. The channel displays approximately 43% α-helical and 25% β-sheet content. In addition, H/D exchange experiments show that only 43% of the backbone amide protons are exchangeable with solvent. On average, the α-helices are tilted 33° normal to the membrane surface. The results are discussed in relationship to the lactose permease of Escherichia coli, a membrane transport protein.

Mutant G protein α subunit activated by Gβγ: A model for receptor activation?

How receptors catalyze exchange of GTP for GDP bound to the Gα subunit of trimeric G proteins is not known. One proposal is that the receptor uses the G protein's βγ heterodimer as a lever, tilting it to pull open the guanine nucleotide binding pocket of Gα. To test this possibility, we designed a mutant Gα that would bind to βγ in the tilted conformation. To do so, we excised a helical turn (four residues) from the N-terminal region of αs, the α subunit of GS, the stimulatory regulator of adenylyl cyclase. In the presence, but not in the absence, of transiently expressed β1 and γ2, this mutant (αsΔ), markedly stimulated cAMP accumulation. This effect depended on the ability of the coexpressed β protein to interact normally with the lip of the nucleotide binding pocket of αsΔ. We substituted alanine for an aspartate in β1 that binds to a lysine (K206) in the lip of the α subunit's nucleotide binding pocket. Coexpressed with αsΔ and γ2, this mutant, β1-D228A, elevated cAMP much less than did β1-wild type; it did bind to αsΔ normally, however, as indicated by its unimpaired ability to target αsΔ to the plasma membrane. We conclude that βγ can activate αs and that this effect probably involves both a tilt of βγ relative to αs and interaction of β with the lip of the nucleotide binding pocket. We speculate that receptors use a similar mechanism to activate trimeric G proteins.

Quantum groups with invariant integrals

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

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.

Cosmic microwave background theory

A long-standing goal of theorists has been to constrain cosmological parameters that define the structure formation theory from cosmic microwave background (CMB) anisotropy experiments and large-scale structure (LSS) observations. The status and future promise of this enterprise is described. Current band-powers in ℓ-space are consistent with a ΔT flat in frequency and broadly follow inflation-based expectations. That the levels are ∼(10−5)2 provides strong support for the gravitational instability theory, while the Far Infrared Absolute Spectrophotometer (FIRAS) constraints on energy injection rule out cosmic explosions as a dominant source of LSS. Band-powers at ℓ ≳ 100 suggest that the universe could not have re-ionized too early. To get the LSS of Cosmic Background Explorer (COBE)-normalized fluctuations right provides encouraging support that the initial fluctuation spectrum was not far off the scale invariant form that inflation models prefer: e.g., for tilted Λ cold dark matter sequences of fixed 13-Gyr age (with the Hubble constant H0 marginalized), ns = 1.17 ± 0.3 for Differential Microwave Radiometer (DMR) only; 1.15 ± 0.08 for DMR plus the SK95 experiment; 1.00 ± 0.04 for DMR plus all smaller angle experiments; 1.00 ± 0.05 when LSS constraints are included as well. The CMB alone currently gives weak constraints on Λ and moderate constraints on Ωtot, but theoretical forecasts of future long duration balloon and satellite experiments are shown which predict percent-level accuracy among a large fraction of the 10+ parameters characterizing the cosmic structure formation theory, at least if it is an inflation variant.

Corneal collagen fibril structure in three dimensions: Structural insights into fibril assembly, mechanical properties, and tissue organization

The ability of the cornea to transmit light while being mechanically resilient is directly attributable to the formation of an extracellular matrix containing orthogonal sheets of collagen fibrils. The detailed structure of the fibrils and how this structure underpins the mechanical properties and organization of the cornea is understood poorly. In this study, we used automated electron tomography to study the three-dimensional organization of molecules in corneal collagen fibrils. The reconstructions show that the collagen molecules in the 36-nm diameter collagen fibrils are organized into microfibrils (≈4-nm diameter) that are tilted by ≈15° to the fibril long axis in a right-handed helix. An unexpected finding was that the microfibrils exhibit a constant-tilt angle independent of radial position within the fibril. This feature suggests that microfibrils in concentric layers are not always parallel to each other and cannot retain the same neighbors between layers. Analysis of the lateral structure shows that the microfibrils exhibit regions of order and disorder within the 67-nm axial repeat of collagen fibrils. Furthermore, the microfibrils are ordered at three specific regions of the axial repeat of collagen fibrils that correspond to the N- and C-telopeptides and the d-band of the gap zone. The reconstructions also show macromolecules binding to the fibril surface at sites that correspond precisely to where the microfibrils are most orderly.

Three-dimensional cryoelectron microscopy of dimeric kinesin and ncd motor domains on microtubules.

Kinesin and ncd motor proteins are homologous in sequence yet move in opposite directions along microtubules. We have previously shown that monomeric kinesin and ncd bind in the same orientation on equivalent sites relative to the ends of tubulin sheets of known polarity. We now report cryoelectron microscope images of 16-protofilament microtubules decorated with both single- and double-headed kinesin and double-headed ncd. Three-dimensional density maps and difference maps show that, in adenosine 5'-[beta,gamma-imido]triphosphate, both dimeric motors bind tightly to microtubules via one head, leaving the other free, though apparently in a fixed position. The attached heads of dimers bind to tubulin in the same way as single kinesin heads. The second heads are connected to the tops of the first but, whereas the second kinesin head is closely associated with the first, pairs of ncd heads are splayed apart. There is also a distinct difference in orientation: the second kinesin head is tilted toward the microtubule plus end, while the second head of ncd points toward the minus end.

Projection structure of frog rhodopsin in two crystal forms.

Rhodopsin is the G protein-coupled receptor that upon light activation triggers the visual transduction cascade. Rod cell outer segment disc membranes were isolated from dark-adapted frog retinas and were extracted with Tween detergents to obtain two-dimensional rhodopsin crystals for electron crystallography. When Tween 80 was used, tubular structures with a p2 lattice (a = 32 A, b = 83 A, gamma = 91 degrees) were formed. The use of a Tween 80/Tween 20 mixture favored the formation of larger p22(1)2(1) lattices (a = 40 A, b = 146 A, gamma = 90 degrees). Micrographs from frozen hydrated frog rhodopsin crystals were processed, and projection structures to 7-A resolution for the p22(1)2(1) form and to 6-A resolution for the p2 form were calculated. The maps of frog rhodopsin in both crystal forms are very similar to the 9-A map obtained previously for bovine rhodopsin and show that the arrangement of the helices is the same. In a tentative topographic model, helices 4, 6, and 7 are nearly perpendicular to the plane of the membrane. In the higher-resolution projection maps of frog rhodopsin, helix 5 looks more tilted than it appeared previously. The quality of the two frog rhodopsin crystals suggests that they would be suitable to obtain a three-dimensional structure in which all helices would be resolved.

Calicheamicin-DNA complexes: warhead alignment and saccharide recognition of the minor groove.

The solution structures of calicheamicin gamma 1I, its cycloaromatized analog (calicheamicin epsilon), and its aryl tetrasaccharide complexed to a common DNA hairpin duplex have been determined by NMR and distance-refined molecular dynamics computations. Sequence specificity is associated with carbohydrate-DNA recognition that places the aryl tetrasaccharide component of all three ligands in similar orientations in the minor groove at the d(T-C-C-T).d(A-G-G-A) segment. The complementary fit of the ligands and the DNA minor groove binding site creates numerous van der Waals contacts as well as hydrogen bonding interactions. Notable are the iodine and sulfur atoms of calicheamicin that hydrogen bond with the exposed amino proton of the 5'- and 3'-guanines, respectively, of the d(A-G-G-A) segment. The sequence-specific carbohydrate binding orients the enediyne aglycone of calicheamicin gamma 1I such that its C3 and C6 proradical centers are adjacent to the cleavage sites. While the enediyne aglycone of calicheamicin gamma 1I is tilted relative to the helix axis and spans the minor groove, the cycloaromatized aglycone is aligned approximately parallel to the helix axis in the respective complexes. Specific localized conformational perturbations in the DNA have been identified from imino proton complexation shifts and changes in specific sugar pucker patterns on complex formation. The helical parameters for the carbohydrate binding site are comparable with corresponding values in B-DNA fibers while a widening of the groove is observed at the adjacent aglycone binding site.

The structure of Ascaris hemoglobin domain I at 2.2 A resolution: molecular features of oxygen avidity.

The perienteric hemoglobin of the parasitic nematode Ascaris has an exceptionally high affinity for oxygen. It is an octameric protein containing two similar heme-binding domains per subunit, but recombinant constructs expressing a single, monomeric heme-binding domain (domain 1; D1) retain full oxygen avidity. We have solved the crystal structure of D1 at 2.2 A resolution. Analysis of the structure reveals a characteristic globin fold and illuminates molecular features involved in oxygen avidity of Ascaris perienteric hemoglobin. A strong hydrogen bond between tyrosine at position 10 in the B helix (tyrosine-B10) and the distal oxygen of the ligand, combined with a weak hydrogen bond between glutamine-E7 and the proximal oxygen, grips the ligand in the binding pocket. A third hydrogen bond between these two amino acids appears to stabilize the structure. The B helix of D1 is displaced laterally by 2.5 A when compared with sperm whale myoglobin. This shifts the tyrosine-B10 hydroxyl far enough from liganded oxygen to form a strong hydrogen bond without steric hindrance. Changes in the F helix compared with myoglobin contribute to a tilted heme that may also be important for oxygen affinity.