Twist fields, the elliptic genus, and hidden symmetry

We combine infinite dimensional analysis (in particular a priori estimates and twist positivity) with classical geometric structures, supersymmetry, and noncommutative geometry. We establish the existence of a family of examples of two-dimensional, twist quantum fields. We evaluate the elliptic genus in these examples. We demonstrate a hidden SL(2,ℤ) symmetry of the elliptic genus, as suggested by Witten.

Analysis of the relation between the sequence and secondary and three-dimensional structures of immunoglobulin molecules.

Methods of structural and statistical analysis of the relation between the sequence and secondary and three-dimensional structures are developed. About 5000 secondary structures of immunoglobulin molecules from the Kabat data base were predicted. Two statistical analyses of amino acids reveal 47 universal positions in strands and loops. Eight universally conservative positions out of the 47 are singled out because they contain the same amino acid in > 90% of all chains. The remaining 39 positions, which we term universally alternative positions, were divided into five groups: hydrophobic, charged and polar, aromatic, hydrophilic, and Gly-Ala, corresponding to the residues that occupied them in almost all chains. The analysis of residue-residue contacts shows that the 47 universal positions can be distinguished by the number and types of contacts. The calculations of contact maps in the 29 antibody structures revealed that residues in 24 of these 47 positions have contacts only with residues of antiparallel beta-strands in the same beta-sheet and residues in the remaining 23 positions always have far-away contacts with residues from other beta-sheets as well. In addition, residues in 6 of the 47 universal positions are also involved in interactions with residues of the other variable or constant domains.

Membrane-bound state of the colicin E1 channel domain as an extended two-dimensional helical array

Atomic level structures have been determined for the soluble forms of several colicins and toxins, but the structural changes that occur after membrane binding have not been well characterized. Changes occurring in the transition from the soluble to membrane-bound state of the C-terminal 190-residue channel polypeptide of colicin E1 (P190) bound to anionic membranes are described. In the membrane-bound state, the α-helical content increases from 60–64% to 80–90%, with a concomitant increase in the average length of the helical segments from 12 to 16 or 17 residues, close to the length required to span the membrane bilayer in the open channel state. The average distance between helical segments is increased and interhelix interactions are weakened, as shown by a major loss of tertiary structure interactions, decreased efficiency of fluorescence resonance energy transfer from an energy donor on helix V of P190 to an acceptor on helix IX, and decreased resonance energy transfer at higher temperatures, not observed in soluble P190, implying freedom of motion of helical segments. Weaker interactions are also shown by a calorimetric thermal transition of low cooperativity, and the extended nature of the helical array is shown by a 3- to 4-fold increase in the average area subtended per molecule to 4,200 Å2 on the membrane surface. The latter, with analysis of the heat capacity changes, implies the absence of a developed hydrophobic core in the membrane-bound P190. The membrane interfacial layer thus serves to promote formation of a highly helical extended two-dimensional flexible net. The properties of the membrane-bound state of the colicin channel domain (i.e., hydrophobic anchor, lengthened and loosely coupled α-helices, and close association with the membrane interfacial layer) are plausible structural features for the state that is a prerequisite for voltage gating, formation of transmembrane helices, and channel opening.

Two-dimensional IR correlation spectroscopy: Sequential events in the unfolding process of the λ Cro-V55C repressor protein

A question often posed in protein folding/unfolding studies is whether the process is fully cooperative or whether it contains sequential elements. To address this question, one needs tools capable of resolving different events. It seems that, at least in certain cases, two-dimensional (2D) IR correlation spectroscopy can provide answers to this question. To illustrate this point, we have turned to the Cro-V55C dimer of the λ Cro repressor, a protein known to undergo thermal unfolding in two discrete steps through a stable equilibrium intermediate. The secondary structure of this intermediate is compatible with that of a partially unfolded protein and involves a reorganization of the N terminus, whereas the antiparallel β-ribbon formed by the C-terminal part of each subunit remains largely intact. To establish whether the unfolding process involves sequential events, we have performed a 2D correlation analysis of IR spectra recorded over the temperature range of 20–95°C. The 2D IR correlation analysis indeed provides evidence for a sequential formation of the stable intermediate, which is created in three (closely related) steps. A first step entails the unfolding of the short N-terminal β-strand, followed by the unfolding of the α-helices in a second step, and the third step comprises the reorganization of the remaining β-sheet and of some unordered segments in the protein. The complete unfolding of the stable intermediate at higher temperatures also undergoes sequential events that ultimately end with the breaking of the H bonds between the two β-strands at the dimer interface.

Systematic derivation of partition functions for ligand binding to two-dimensional lattices

The Ising problem consists in finding the analytical solution of the partition function of a lattice once the interaction geometry among its elements is specified. No general analytical solution is available for this problem, except for the one-dimensional case. Using site-specific thermodynamics, it is shown that the partition function for ligand binding to a two-dimensional lattice can be obtained from those of one-dimensional lattices with known solution. The complexity of the lattice is reduced recursively by application of a contact transformation that involves a relatively small number of steps. The transformation implemented in a computer code solves the partition function of the lattice by operating on the connectivity matrix of the graph associated with it. This provides a powerful new approach to the Ising problem, and enables a systematic analysis of two-dimensional lattices that model many biologically relevant phenomena. Application of this approach to finite two-dimensional lattices with positive cooperativity indicates that the binding capacity per site diverges as Na (N = number of sites in the lattice) and experiences a phase-transition-like discontinuity in the thermodynamic limit N → ∞. The zeroes of the partition function tend to distribute on a slightly distorted unit circle in complex plane and approach the positive real axis already for a 5×5 square lattice. When the lattice has negative cooperativity, its properties mimic those of a system composed of two classes of independent sites with the apparent population of low-affinity binding sites increasing with the size of the lattice, thereby accounting for a phenomenon encountered in many ligand-receptor interactions.

Nuclear Pore Complex Number and Distribution throughout the Saccharomyces cerevisiae Cell Cycle by Three-Dimensional Reconstruction from Electron Micrographs of Nuclear Envelopes

The number of nuclear pore complexes (NPCs) in individual nuclei of the yeast Saccharomyces cerevisiae was determined by computer-aided reconstruction of entire nuclei from electron micrographs of serially sectioned cells. Nuclei of 32 haploid cells at various points in the cell cycle were modeled and found to contain between 65 and 182 NPCs. Morphological markers, such as cell shape and nuclear shape, were used to determine the cell cycle stage of the cell being examined. NPC number was correlated with cell cycle stage to reveal that the number of NPCs increases steadily, beginning in G1-phase, suggesting that NPC assembly occurs continuously throughout the cell cycle. However, the accumulation of nuclear envelope observed during the cell cycle, indicated by nuclear surface area, is not continuous at the same rate, such that the density of NPCs per unit area of nuclear envelope peaks in apparent S-phase cells. Analysis of the nuclear envelope reconstructions also revealed no preferred NPC-to-NPC distance. However, NPCs were found in large clusters over regions of the nuclear envelope. Interestingly, clusters of NPCs were most pronounced in early mitotic nuclei and were found to be associated with the spindle pole bodies, but the functional significance of this association is unknown.

Three-dimensional modeling of and ligand docking to vitamin D receptor ligand binding domain

The ligand binding domain of the human vitamin D receptor (VDR) was modeled based on the crystal structure of the retinoic acid receptor. The ligand binding pocket of our VDR model is spacious at the helix 11 site and confined at the β-turn site. The ligand 1α,25-dihydroxyvitamin D3 was assumed to be anchored in the ligand binding pocket with its side chain heading to helix 11 (site 2) and the A-ring toward the β-turn (site 1). Three residues forming hydrogen bonds with the functionally important 1α- and 25-hydroxyl groups of 1α,25-dihydroxyvitamin D3 were identified and confirmed by mutational analysis: the 1α-hydroxyl group is forming pincer-type hydrogen bonds with S237 and R274 and the 25-hydroxyl group is interacting with H397. Docking potential for various ligands to the VDR model was examined, and the results are in good agreement with our previous three-dimensional structure-function theory.

Analysis of receptor signaling pathways by mass spectrometry: Identification of Vav-2 as a substrate of the epidermal and platelet-derived growth factor receptors

Oligomerization of receptor protein tyrosine kinases such as the epidermal growth factor receptor (EGFR) by their cognate ligands leads to activation of the receptor. Transphosphorylation of the receptor subunits is followed by the recruitment of signaling molecules containing src homology 2 (SH2) or phosphotyrosine interaction domains (PID). Additionally, several cytoplasmic proteins that may or may not associate with the receptor undergo tyrosine phosphorylation. To identify several components of the EGFR signaling pathway in a single step, we have immunoprecipitated molecules that are tyrosine phosphorylated in response to EGF and analyzed them by one-dimensional gel electrophoresis followed by mass spectrometry. Combining matrix-assisted laser desorption/ionization (MALDI) and nanoelectrospray tandem mass spectrometry (MS/MS) led to the identification of nine signaling molecules, seven of which had previously been implicated in EGFR signaling. Several of these molecules were identified from low femtomole levels of protein loaded onto the gel. We identified Vav-2, a recently discovered guanosine nucleotide exchange factor that is expressed ubiquitously, as a substrate of the EGFR. We demonstrate that Vav-2 is phosphorylated on tyrosine residues in response to EGF and associates with the EGFR in vivo. Binding of Vav-2 to the EGFR is mediated by the SH2 domain of Vav-2. In keeping with its ubiquitous expression, Vav-2 seems to be a general signaling molecule, since it also associates with the platelet-derived growth factor (PDGF) receptor and undergoes tyrosine phosphorylation in fibroblasts upon PDGF stimulation. The strategy suggested here can be used for routine identification of downstream components of cell surface receptors in mammalian cells.

Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

Molecular phylogenetic analysis of evolutionary trends in stonefly wing structure and locomotor behavior

Insects in the order Plecoptera (stoneflies) use a form of two-dimensional aerodynamic locomotion called surface skimming to move across water surfaces. Because their weight is supported by water, skimmers can achieve effective aerodynamic locomotion even with small wings and weak flight muscles. These mechanical features stimulated the hypothesis that surface skimming may have been an intermediate stage in the evolution of insect flight, which has perhaps been retained in certain modern stoneflies. Here we present a phylogeny of Plecoptera based on nucleotide sequence data from the small subunit rRNA (18S) gene. By mapping locomotor behavior and wing structural data onto the phylogeny, we distinguish between the competing hypotheses that skimming is a retained ancestral trait or, alternatively, a relatively recent loss of flight. Our results show that basal stoneflies are surface skimmers, and that various forms of surface skimming are distributed widely across the plecopteran phylogeny. Stonefly wings show evolutionary trends in the number of cross veins and the thickness of the cuticle of the longitudinal veins that are consistent with elaboration and diversification of flight-related traits. These data support the hypothesis that the first stoneflies were surface skimmers, and that wing structures important for aerial flight have become elaborated and more diverse during the radiation of modern stoneflies.

Structural analysis of the binding modes of minor groove ligands comprised of disubstituted benzenes

Two-dimensional homonuclear NMR was used to characterize synthetic DNA minor groove-binding ligands in complexes with oligonucleotides containing three different A-T binding sites. The three ligands studied have a C2 axis of symmetry and have the same general structural motif of a central para-substituted benzene ring flanked by two meta-substituted rings, giving the molecules a crescent shape. As with other ligands of this shape, specificity seems to arise from a tight fit in the narrow minor groove of the preferred A-T-rich sequences. We found that these ligands slide between binding subsites, behavior attributed to the fact that all of the amide protons in the ligand backbone cannot hydrogen bond to the minor groove simultaneously.

Pseudoknots in prion protein mRNAs confirmed by comparative sequence analysis and pattern searching

The human prion gene contains five copies of a 24 nt repeat that is highly conserved among species. An analysis of folding free energies of the human prion mRNA, in particular in the repeat region, suggested biased codon selection and the presence of RNA patterns. In particular, pseudoknots, similar to the one predicted by Wills in the human prion mRNA, were identified in the repeat region of all available prion mRNAs available in GenBank, but not those of birds and the red slider turtle. An alignment of these mRNAs, which share low sequence homology, shows several co-variations that maintain the pseudoknot pattern. The presence of pseudoknots in yeast Sup35p and Rnq1 suggests acquisition in the prokaryotic era. Computer generated three-dimensional structures of the human prion pseudoknot highlight protein and RNA interaction domains, which suggest a possible effect in prion protein translation. The role of pseudoknots in prion diseases is discussed as individuals with extra copies of the 24 nt repeat develop the familial form of Creutzfeldt–Jakob disease.

Phylogenetic analysis of tmRNA genes within a bacterial subgroup reveals a specific structural signature

Bacterial tmRNA mediates a trans-translation reaction, which permits the recycling of stalled ribosomes and probably also contributes to the regulated expression of a subset of genes. Its action results in the addition of a small number of C-terminal amino acids to protein whose synthesis had stalled and these constitute a proteolytic recognition tag for the degradation of these incompletely synthesized proteins. Previous work has identified pseudoknots and stem–loops that are widely conserved in divergent bacteria. In the present work an alignment of tmRNA gene sequences within 13 β-proteobacteria reveals an additional sub-structure specific for this bacterial group. This sub-structure is in pseudoknot Pk2, and consists of one to two additional stem–loop(s) capped by stable GNRA tetraloop(s). Three-dimensional models of tmRNA pseudoknot 2 (Pk2) containing various topological versions of the additional sub-structure suggest that the sub-structures likely point away from the core of the RNA, containing both the tRNA and the mRNA domains. A putative tertiary interaction has also been identified.

Proteomic analysis of the bacterial cell cycle

A global approach was used to analyze protein synthesis and stability during the cell cycle of the bacterium Caulobacter crescentus. Approximately one-fourth (979) of the estimated C. crescentus gene products were detected by two-dimensional gel electrophoresis, 144 of which showed differential cell cycle expression patterns. Eighty-one of these proteins were identified by mass spectrometry and were assigned to a wide variety of functional groups. Pattern analysis revealed that coexpression groups were functionally clustered. A total of 48 proteins were rapidly degraded in the course of one cell cycle. More than half of these unstable proteins were also found to be synthesized in a cell cycle-dependent manner, establishing a strong correlation between rapid protein turnover and the periodicity of the bacterial cell cycle. This is, to our knowledge, the first evidence for a global role of proteolysis in bacterial cell cycle control.

Molecular Cloning and Expression Analysis of the Mitochondrial Pyruvate Dehydrogenase from Maize1

Four cDNAs, one encoding an α-subunit and three encoding β-subunits of the mitochondrial pyruvate dehydrogenase, were isolated from maize (Zea mays L.) libraries. The deduced amino acid sequences of both α- and β-subunits are approximately 80% identical with Arabidopsis and pea (Pisum sativum L.) homologs. The mature N terminus was determined for the β-subunit by microsequencing the protein purified from etiolated maize shoot mitochondria and was resolved by two-dimensional gel electrophoresis. This single isoelectric species comprised multiple isoforms. Both α- and β-subunits are encoded by multigene families in maize, as determined by Southern-blot analyses. RNA transcripts for both α- and β-subunits were more abundant in roots than in young leaves or etiolated shoots. Pyruvate dehydrogenase activity was also higher in roots (5-fold) compared with etiolated shoots and leaves. Both subunits were present at similar levels in all tissues examined, indicating coordinated gene regulation. The protein levels were highest in heterotrophic organs and in pollen, which contained about 2-fold more protein than any other organ examined. The relative abundance of these proteins in nonphotosynthetic tissues may reflect a high cellular content of mitochondria, a high level of respiratory activity, or an extra plastidial requirement for acetate.