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.

Functionality of major histocompatibility complex class II molecules in mice doubly deficient for invariant chain and H-2M complexes

By combining two previously generated null mutations, Ii° and M°, we produced mice lacking the invariant chain and H-2M complexes, both required for normal cell-surface expression of major histocompatibility complex class II molecules loaded with the usual diverse array of peptides. As expected, the maturation and transport of class II molecules, their expression at the cell surface, and their capacity to present antigens were quite similar for cells from Ii°M° double-mutant mice and from animals carrying just the Ii° mutation. More surprising were certain features of the CD4+ T cell repertoire selected in Ii°M° mice: many fewer cells were selected than in Ii+M° animals, and these had been purged of self-reactive specificities, unlike their counterparts in Ii+M° animals. These findings suggest (i) that the peptides carried by class II molecules on stromal cells lacking H-2M complexes may almost all derive from invariant chain and (ii) that H-2M complexes edit the peptide array displayed on thymic stromal cells in the absence of invariant chain, showing that it can edit, in vivo, peptides other than CLIP.

Invariant size–frequency distributions along a latitudinal gradient in marine bivalves

In the most extensive analysis of body size in marine invertebrates to date, we show that the size–frequency distributions of northeastern Pacific bivalves at the provincial level are surprisingly invariant in modal and median size as well as size range, despite a 4-fold change in species richness from the tropics to the Arctic. The modal sizes and shapes of these size–frequency distributions are consistent with the predictions of an energetic model previously applied to terrestrial mammals and birds. However, analyses of the Miocene–Recent history of body sizes within 82 molluscan genera show little support for the expectation that the modal size is an evolutionary attractor over geological time.

In vitro roles of invariant helix–turn–helix motif residue R383 in σ54 (σN)

In vitro DNA-binding and transcription properties of σ54 proteins with the invariant Arg383 in the putative helix–turn–helix motif of the DNA-binding domain substituted by lysine or alanine are described. We show that R383 contributes to maintaining stable holoenzyme–promoter complexes in which limited DNA opening downstream of the –12 GC element has occurred. Unlike wild-type σ54, holoenzymes assembled with the R383A or R383K mutants could not form activator-independent, heparin-stable complexes on heteroduplex Sinorhizobium meliloti nifH DNA mismatched next to the GC. Using longer sequences of heteroduplex DNA, heparin-stable complexes formed with the R383K and, to a lesser extent, R383A mutant holoenzymes, but only when the activator and a hydrolysable nucleotide was added and the DNA was opened to include the –1 site. Although R383 appears inessential for polymerase isomerisation, it makes a significant contribution to maintaining the holoenzyme in a stable complex when melting is initiating next to the GC element. Strikingly, Cys383-tethered FeBABE footprinting of promoter DNA strongly suggests that R383 is not proximal to promoter DNA in the closed complex. This indicates that R383 is not part of the regulatory centre in the σ54 holoenzyme, which includes the –12 promoter region elements. R383 contributes to several properties, including core RNA polymerase binding and to the in vivo stability of σ54.

Invariant scaling relationships for interspecific plant biomass production rates and body size

The allometric relationships for plant annualized biomass production (“growth”) rates, different measures of body size (dry weight and length), and photosynthetic biomass (or pigment concentration) per plant (or cell) are reported for multicellular and unicellular plants representing three algal phyla; aquatic ferns; aquatic and terrestrial herbaceous dicots; and arborescent monocots, dicots, and conifers. Annualized rates of growth G scale as the 3/4-power of body mass M over 20 orders of magnitude of M (i.e., G ∝ M3/4); plant body length L (i.e., cell length or plant height) scales, on average, as the 1/4-power of M over 22 orders of magnitude of M (i.e., L ∝ M1/4); and photosynthetic biomass Mp scales as the 3/4-power of nonphotosynthetic biomass Mn (i.e., Mp ∝ Mn3/4). Because these scaling relationships are indifferent to phylogenetic affiliation and habitat, they have far-reaching ecological and evolutionary implications (e.g., net primary productivity is predicted to be largely insensitive to community species composition or geological age).

Mutagenesis of the peptidyltransferase center of 23S rRNA: the invariant U2449 is dispensable

U2449 is one of many invariant residues in the central loop of domain V of 23S rRNA, a region that constitutes part of the peptidyltransferase center of the ribosome. In Escherichia coli, this U is post-transcriptionally modified to dihydrouridine (D) and is the only D modification found in E.coli rRNAs. To analyze the role of this base and its modification in ribosomal function, all three base substitutions were constructed on a plasmid copy of the rrnB operon and assayed for their ability to support cell growth in a strain of E.coli lacking chromosomal rrn operons. Both purine substitution mutations were not viable. However, growth and antibiotic sensitivity of cells expressing only the mutant D2449C rRNA was indistinguishable from wild type. We conclude that while a pyrimidine is required at position 2449 for proper ribosomal function, the D modification is dispensable.

Essential requirement of an invariant V alpha 14 T cell antigen receptor expression in the development of natural killer T cells.

NK1.1+ T [natural killer (NK) T] cells express an invariant T cell antigen receptor alpha chain (TCR alpha) encoded by V alpha 14 and J alpha 281 segments in association with a limited number of V betas, predominantly V beta 8.2. Expression of the invariant V alpha 14/J alpha 281, but not V alpha 1, TCR in transgenic mice lacking endogenous TCR alpha expression blocks the development of conventional T alpha beta cells and leads to the preferential development of V alpha 14 NK T cells, suggesting a prerequisite role of invariant V alpha 14 TCR in NK T cell development. In V beta 8.2 but not B beta 3 transgenic mice, two NK T cells with different CD3 epsilon expressions, CD3 epsilon(dim) and CD3 epsilon(high), can be identified. CD3 epsilon(high) NK T cells express surface V alpha 14/V beta 8 TCR, indicating a mature cell type, whereas CD3 epsilon(dim) NK T cells express V beta 8 without V alpha 14 TCR and no significant CD3 epsilon expression (CD3 epsilon(dim)) on the cell surface. However, the latter are positive for recombination activating gene (RAG-1 and RAG-2) mRNA, which are only expressed in the precursor or immature T cell lineage, and also possess CD3 epsilon mRNA in their cytoplasm, suggesting that CD3 epsilon(dim) NK T cells are the precursor of V alpha 14 NK T cells.

The invariant system of coordinates of antibody molecules: prediction of the "standard" C alpha framework of VL and VH domains.

A new approach of comparing protein structures that does not involve the procedure of superposition is suggested. An invariant system of coordinates for immunoglobulin molecules that is based on the geometrical symmetry inherent to the variable domain light-chain (VL)-heavy-chain (VH) complex is described. The coordinates of the Calpha atoms in 22 immunoglobulin structures are calculated in the invariant system of coordinates. We found that 76 identical positions in this Calpha framework are symmetrical about the twofold axis. Comparison of the identical positions in these molecules allows us to select 96 positions in the light chains and 87 positions in the heavy chains whose Calpha atom coordinates are approximately the same. To check whether the average coordinates of Calpha atoms in these positions complies with the stereochemical requirements, we calculated Calpha-Calpha distances. Seventy-three positions of the light chains and 72 positions of the heavy chains satisfy the Calpha-Calpha distance criterion. The Calpha atoms in these positions are used for constructing the "standard" Calpha framework of VL and VH complexes. The average coordinates of Calpha atoms are presented.

Evolution of chlorophyll and bacteriochlorophyll: the problem of invariant sites in sequence analysis.

Competing hypotheses seek to explain the evolution of oxygenic and anoxygenic processes of photosynthesis. Since chlorophyll is less reduced and precedes bacteriochlorophyll on the modern biosynthetic pathway, it has been proposed that chlorophyll preceded bacteriochlorophyll in its evolution. However, recent analyses of nucleotide sequences that encode chlorophyll and bacteriochlorophyll biosynthetic enzymes appear to provide support for an alternative hypothesis. This is that the evolution of bacteriochlorophyll occurred earlier than the evolution of chlorophyll. Here we demonstrate that the presence of invariant sites in sequence datasets leads to inconsistency in tree building (including maximum-likelihood methods). Homologous sequences with different biological functions often share invariant sites at the same nucleotide positions. However, different constraints can also result in additional invariant sites unique to the genes, which have specific and different biological functions. Consequently, the distribution of these sites can be uneven between the different types of homologous genes. The presence of invariant sites, shared by related biosynthetic genes as well as those unique to only some of these genes, has misled the recent evolutionary analysis of oxygenic and anoxygenic photosynthetic pigments. We evaluate an alternative scheme for the evolution of chlorophyll and bacteriochlorophyll.

Truncation of the class II beta-chain cytoplasmic domain influences the level of class II/invariant chain-derived peptide complexes.

Previous studies have established that antigen presenting cells (APC) expressing major histocompatibility complex class II beta chains with truncated cytoplasmic domains are impaired in their capacity to activate T cells. While it had been widely accepted that this impairment is due to a defect in class II cytoplasmic domain-dependent signal transduction, we recently generated transgenic mice expressing only truncated class II beta chains, and functional analyses of APC from these mice revealed signaling-independent defects in antigen presentation. Here, we demonstrate that T cells primed on such transgenic APC respond better to stimulation by APC expressing truncated beta chains than by wild-type APC. This finding suggests that APC expressing truncated class II beta chains are not inherently defective in their antigen presenting capacity but, rather, may differ from wild-type APC in the peptide antigens that they present. Indeed, analysis of the peptides bound to class II molecules isolated from normal and transgenic spleen cells revealed clear differences. Most notably, the level of class II-associated invariant chain-derived peptides (CLIP) is significantly reduced in cells expressing only truncated beta chains. Prior studies have established that CLIP and antigenic peptides compete for binding to class II molecules. Thus, our results suggest that the cytoplasmic domain of the class II beta chain affects antigen presentation by influencing the level of CLIP/class II complexes.

Invariant chain made in Escherichia coli has an exposed N-terminal segment that blocks antigen binding to HLA-DR1 and a trimeric C-terminal segment that binds empty HLA-DR1.

Invariant chain (Ii), a membrane glycoprotein, binds class II major histocompatibility complex (MHC) glycoproteins, probably via its class II-associated Ii peptide (CLIP) segment, and escorts them toward antigen-containing endosomal compartments. We find that a soluble, trimeric ectodomain of Ii expressed and purified from Escherichia coli blocks peptide binding to soluble HLA-DR1. Proteolysis indicates that Ii contains two structural domains. The C-terminal two-thirds forms an alpha-helical domain that trimerizes and interacts with empty HLA-DR1 molecules, augmenting rather than blocking peptide binding. The N-terminal one-third, which inhibits peptide binding, is proteolytically susceptible over its entire length. In the trimer, the N-terminal domains act independently with each CLIP segment exposed and free to bind an MHC class II molecule, while the C-terminal domains act as a trimeric unit.

Proteolysis of major histocompatibility complex class II-associated invariant chain is regulated by the alternatively spliced gene product, p41.

Invariant chain (Ii) is an intracellular type II transmembrane glycoprotein that is associated with major histocompatibility complex class II molecules during biosynthesis. Ii exists in two alternatively spliced forms, p31 and p41. Both p31 and p41 facilitate folding of class II molecules, promote egress from the endoplasmic reticulum, prevent premature peptide binding, and enhance localization to proteolytic endosomal compartments that are thought to be the sites for Ii degradation, antigen processing, and class II-peptide association. In spite of the dramatic and apparently equivalent effects that p31 and p41 have on class II biosynthesis, the ability of invariant chain to enhance antigen presentation to T cells is mostly restricted to p41. Here we show that degradation of Ii leads to the generation of a 12-kDa amino-terminal fragment that in p41-positive, but not in p31-positive, cells remains associated with class II molecules for an extended time. Interestingly, we find that coexpression of the two isoforms results in a change in the pattern of p31 degradation such that endosomal processing of p31 also leads to extended association of a similar 12-kDa fragment with class II molecules. These data raise the possibility that p41 may have the ability to impart its pattern of proteolytic processing on p31 molecules expressed in the same cells. This would enable a small number of p41 molecules to modify the post-translational transport and/or processing of an entire cohort of class II-Ii complexes in a manner that could account for the unique ability of p41 to enhance antigen presentation.

Direct observation of disordered regions in the major histocompatibility complex class II-associated invariant chain.

Invariant chain (Ii) is a trimeric membrane protein which binds and stabilizes major histocompatibility complex class II heterodimers in the endoplasmic reticulum and lysosomal compartments of antigen-presenting cells. In concert with an intracellular class II-like molecule, HLA-DM, Ii seems to facilitate loading of conventional class II molecules with peptides before transport of the class II-peptide complex to the cell surface for recognition by T cells. The interaction of Ii with class II molecules is thought to be mediated in large part through a region of 24 amino acids (the class II-associated Ii peptide, CLIP) which binds as a cleaved moiety in the antigenic peptide-binding groove of class II molecules in HLA-DM-deficient cell lines. Here we use nuclear magnetic resonance techniques to demonstrate that a soluble recombinant Ii ectodomain contains significant disordered regions which probably include CLIP.

A general model of invariant chain association with class II major histocompatibility complex proteins.

The binding of invariant chain to major histocompatibility complex (MHC) proteins is an important step in processing of MHC class II proteins and in antigen presentation. The question of how invariant chain can bind to all MHC class II proteins is central to understanding these processes. We have employed molecular modeling to predict the structure of class II-associated invariant chain peptide (CLIP)-MHC protein complexes and to ask whether the predicted mode of association could be general across all MHC class II proteins. CLIP fits identically into the MHC class II alleles HLA-DR3, I-Ak, I-Au, and I-Ad, with a consistent pattern of hydrogen bonds, contacts, and hydrophobic burial and without bad contacts. Our model predicts the burial of CLIP residues Met-91 and Met-99 in the deep P1 and P9 anchor pockets and other detailed interactions, which we have compared with available data. The predicted pattern of I-A allele-specific effects on CLIP binding is very similar to that observed experimentally by alanine-scanning mutations of CLIP. Together, these results indicate that CLIP may bind in a single, general way across products of MHC class II alleles.

Structural features of the invariant chain fragment CLIP controlling rapid release from HLA-DR molecules and inhibition of peptide binding.

The invariant chain (Ii) prevents binding of ligands to major histocompatibility complex (MHC) class II molecules in the endoplasmic reticulum and during intracellular transport. Stepwise removal of the Ii in a trans-Golgi compartment renders MHC class II molecules accessible for peptide loading, with CLIP (class II-associated Ii peptides) as the final fragment to be released. Here we show that CLIP can be subdivided into distinct functional regions. The C-terminal segment (residues 92-105) of the CLIP-(81-105) fragment mediates inhibition of self- and antigenic peptide binding to HLA-DR2 molecules. In contrast, the N-terminal segment CLIP-(81-98) binds to the Staphylococcus aureus enterotoxin B contact site outside the peptide-binding groove on the alpha 1 domain and does not interfere with peptide binding. Its functional significance appears to lie in the contribution to CLIP removal: the dissociation of CLIP-(81-105) is characterized by a fast off-rate, which is accelerated at endosomal pH, whereas in the absence of the N-terminal CLIP-(81-91), the off-rate of C-terminal CLIP-(92-105) is slow and remains unaltered at low pH. Mechanistically, the N-terminal segment of CLIP seems to prevent tight interactions of CLIP side chains with specificity pockets in the peptide-binding groove that normally occurs during maturation of long-lived class II-peptide complexes.