On the two-dimensional theory of incompressible flow over inlets

The linearized solution for the two-dimensional flow over an inlet of general form has been derived, assuming incompressible potential flow. With this theory suction forces at sharp inlet lips can be estimated and ideal inlets can be designed.

Two-dimensional isostatic meshes in the finite element method

In a Finite Element (FE) analysis of elastic solids several items are usually considered, namely, type and shape of the elements, number of nodes per element, node positions, FE mesh, total number of degrees of freedom (dot) among others. In this paper a method to improve a given FE mesh used for a particular analysis is described. For the improvement criterion different objective functions have been chosen (Total potential energy and Average quadratic error) and the number of nodes and dof's of the new mesh remain constant and equal to the initial FE mesh. In order to find the mesh producing the minimum of the selected objective function the steepest descent gradient technique has been applied as optimization algorithm. However this efficient technique has the drawback that demands a large computation power. Extensive application of this methodology to different 2-D elasticity problems leads to the conclusion that isometric isostatic meshes (ii-meshes) produce better results than the standard reasonably initial regular meshes used in practice. This conclusion seems to be independent on the objective function used for comparison. These ii-meshes are obtained by placing FE nodes along the isostatic lines, i.e. curves tangent at each point to the principal direction lines of the elastic problem to be solved and they should be regularly spaced in order to build regular elements. That means ii-meshes are usually obtained by iteration, i.e. with the initial FE mesh the elastic analysis is carried out. By using the obtained results of this analysis the net of isostatic lines can be drawn and in a first trial an ii-mesh can be built. This first ii-mesh can be improved, if it necessary, by analyzing again the problem and generate after the FE analysis the new and improved ii-mesh. Typically, after two first tentative ii-meshes it is sufficient to produce good FE results from the elastic analysis. Several example of this procedure are presented.

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 freezing in the liquid–vapor interface of a dilute Pb:Ga alloy

We report the results of x-ray reflectivity and grazing incidence x-ray diffraction studies of the liquid–vapor interface of a dilute alloy of Pb in Ga over the temperature range of 23–76°C. Our data show that the liquid–vapor interface of this alloy is stratified for several atomic diameters into the bulk liquid and that a monolayer of Pb forms the outermost stratum of the interface. Over the temperature range of 23–56°C, the monolayer of Pb is in an ordered hexagonal phase. At about 58°C, this monolayer undergoes a first-order transition to a hexatic phase, which remains stable to 76°C. An analogy between the observed transition and the first-order melting transition in a one-component classical plasma is suggested.

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.

Attenuated T2 relaxation by mutual cancellation of dipole–dipole coupling and chemical shift anisotropy indicates an avenue to NMR structures of very large biological macromolecules in solution

Fast transverse relaxation of 1H, 15N, and 13C by dipole-dipole coupling (DD) and chemical shift anisotropy (CSA) modulated by rotational molecular motions has a dominant impact on the size limit for biomacromolecular structures that can be studied by NMR spectroscopy in solution. Transverse relaxation-optimized spectroscopy (TROSY) is an approach for suppression of transverse relaxation in multidimensional NMR experiments, which is based on constructive use of interference between DD coupling and CSA. For example, a TROSY-type two-dimensional 1H,15N-correlation experiment with a uniformly 15N-labeled protein in a DNA complex of molecular mass 17 kDa at a 1H frequency of 750 MHz showed that 15N relaxation during 15N chemical shift evolution and 1HN relaxation during signal acquisition both are significantly reduced by mutual compensation of the DD and CSA interactions. The reduction of the linewidths when compared with a conventional two-dimensional 1H,15N-correlation experiment was 60% and 40%, respectively, and the residual linewidths were 5 Hz for 15N and 15 Hz for 1HN at 4°C. Because the ratio of the DD and CSA relaxation rates is nearly independent of the molecular size, a similar percentagewise reduction of the overall transverse relaxation rates is expected for larger proteins. For a 15N-labeled protein of 150 kDa at 750 MHz and 20°C one predicts residual linewidths of 10 Hz for 15N and 45 Hz for 1HN, and for the corresponding uniformly 15N,2H-labeled protein the residual linewidths are predicted to be smaller than 5 Hz and 15 Hz, respectively. The TROSY principle should benefit a variety of multidimensional solution NMR experiments, especially with future use of yet somewhat higher polarizing magnetic fields than are presently available, and thus largely eliminate one of the key factors that limit work with larger molecules.

Linking genome and proteome by mass spectrometry: Large-scale identification of yeast proteins from two dimensional gels

The function of many of the uncharacterized open reading frames discovered by genomic sequencing can be determined at the level of expressed gene products, the proteome. However, identifying the cognate gene from minute amounts of protein has been one of the major problems in molecular biology. Using yeast as an example, we demonstrate here that mass spectrometric protein identification is a general solution to this problem given a completely sequenced genome. As a first screen, our strategy uses automated laser desorption ionization mass spectrometry of the peptide mixtures produced by in-gel tryptic digestion of a protein. Up to 90% of proteins are identified by searching sequence data bases by lists of peptide masses obtained with high accuracy. The remaining proteins are identified by partially sequencing several peptides of the unseparated mixture by nanoelectrospray tandem mass spectrometry followed by data base searching with multiple peptide sequence tags. In blind trials, the method led to unambiguous identification in all cases. In the largest individual protein identification project to date, a total of 150 gel spots—many of them at subpicomole amounts—were successfully analyzed, greatly enlarging a yeast two-dimensional gel data base. More than 32 proteins were novel and matched to previously uncharacterized open reading frames in the yeast genome. This study establishes that mass spectrometry provides the required throughput, the certainty of identification, and the general applicability to serve as the method of choice to connect genome and proteome.

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.

Two-dimensional confocal images of organization, density, and gating of focal Ca2+ release sites in rat cardiac myocytes

In cardiac myocytes Ca2+ cross-signaling between Ca2+ channels and ryanodine receptors takes place by exchange of Ca2+ signals in microdomains surrounding dyadic junctions, allowing first the activation and then the inactivation of the two Ca2+-transporting proteins. To explore the details of Ca2+ signaling between the two sets of receptors we measured the two-dimensional cellular distribution of Ca2+ at 240 Hz by using a novel confocal imaging technique. Ca2+ channel-triggered Ca2+ transients could be resolved into dynamic “Ca2+ stripes” composed of hundreds of discrete focal Ca2+ releases, appearing as bright fluorescence spots (radius ≅ 0.5 μm) at reproducible sites, which often coincided with t-tubules as visualized with fluorescent staining of the cell membrane. Focal Ca2+ releases triggered stochastically by Ca2+ current (ICa) changed little in duration (≅7 ms) and size (≅100,000 Ca ions) between −40 and +60 mV, but their frequency of activation and first latency mirrored the kinetics and voltage dependence of ICa. The resolution of 0.95 ± 0.13 reproducible focal Ca2+ release sites per μm3 in highly Ca2+-buffered cells, where diffusion of Ca2+ is limited to 50 nm, suggests the presence of about one independent, functional Ca2+ release site per half sarcomere. The density and distribution of Ca2+ release sites suggest they correspond to dyadic junctions. The abrupt onset and termination of focal Ca2+ releases indicate that the cluster of ryanodine receptors in individual dyadic junctions may operate in a coordinated fashion.

Behavior of several two-dimensional fluid equations in singular scenarios

We give conditions that rule out formation of sharp fronts for certain two-dimensional incompressible flows. We show that a necessary condition of having a sharp front is that the flow has to have uncontrolled velocity growth. In the case of the quasi-geostrophic equation and two-dimensional Euler equation, we obtain estimates on the formation of semi-uniform fronts.

Two-dimensional protein crystallization via metal-ion coordination by naturally occurring surface histidines.

A powerful and potentially general approach to the targeting and crystallization of proteins on lipid interfaces through coordination of surface histidine residues to lipid-chelated divalent metal ions is presented. This approach, which should be applicable to the crystallization of a wide range of naturally occurring or engineered proteins, is illustrated here by the crystallization of streptavidin on a monolayer of an iminodiacetate-Cu(II) lipid spread at the air-water interface. This method allows control of the protein orientation at interfaces, which is significant for the facile production of highly ordered protein arrays and for electron density mapping in structural analysis of two-dimensional crystals. Binding of native streptavidin to the iminodiacetate-Cu lipids occurs via His-87, located on the protein surface near the biotin binding pocket. The two-dimensional streptavidin crystals show a previously undescribed microscopic shape that differs from that of crystals formed beneath biotinylated lipids.

Three-dimensional structure of a cysteine-rich repeat from the low-density lipoprotein receptor.

The low-density lipoprotein (LDL) receptor plays a central role in mammalian cholesterol metabolism, clearing lipoproteins which bear apolipoproteins E and B-100 from plasma. Mutations in this molecule are associated with familial hypercholesterolemia, a condition which leads to an elevated plasma cholesterol concentration and accelerated atherosclerosis. The N-terminal segment of the LDL receptor contains a heptad of cysteine-rich repeats that bind the lipoproteins. Similar repeats are present in related receptors, including the very low-density lipoprotein receptor and the LDL receptor-related protein/alpha 2-macroglobulin receptor, and in proteins which are functionally unrelated, such as the C9 component of complement. The first repeat of the human LDL receptor has been expressed in Escherichia coli as a glutathione S-transferase fusion protein, and the cleaved and purified receptor module has been shown to fold to a single, fully oxidized form that is recognized by the monoclonal antibody IgG-C7 in the presence of calcium ions. The three-dimensional structure of this module has been determined by two-dimensional NMR spectroscopy and shown to consist of a beta-hairpin structure, followed by a series of beta turns. Many of the side chains of the acidic residues, including the highly conserved Ser-Asp-Glu triad, are clustered on one face of the module. To our knowledge, this structure has not previously been described in any other protein and may represent a structural paradigm both for the other modules in the LDL receptor and for the homologous domains of several other proteins. Calcium ions had only minor effects on the CD spectrum and no effect on the 1H NMR spectrum of the repeat, suggesting that they induce no significant conformational change.

Experimental evidence for hydrogen-bonded network proton transfer in bacteriorhodopsin shown by Fourier-transform infrared spectroscopy using azide as catalyst.

Experimental evidence for proton transfer via a hydrogen-bonded network in a membrane protein is presented. Bacteriorhodopsin's proton transfer mechanism on the proton uptake pathway between Asp-96 and the Schiff base in the M-to-N transition was determined. The slowdown of this transfer by removal of the proton donor in the Asp-96-->Asn mutant can be accelerated again by addition of small weak acid anions such as azide. Fourier-transform infrared experiments show in the Asp-96-->Asn mutant a transient protonation of azide bound to the protein in the M-to-N transition and, due to the addition of azide, restoration of the IR continuum band changes as seen in wild-type bR during proton pumping. The continuum band changes indicate fast proton transfer on the uptake pathway in a hydrogen-bonded network for wild-type bR and the Asp-96-->Asn mutant with azide. Since azide is able to catalyze proton transfer steps also in several kinetically defective bR mutants and in other membrane proteins, our finding might point to a general element of proton transfer mechanisms in proteins.

Rapid mass spectrometric peptide sequencing and mass matching for characterization of human melanoma proteins isolated by two-dimensional PAGE.

We report a general mass spectrometric approach for the rapid identification and characterization of proteins isolated by preparative two-dimensional polyacrylamide gel electrophoresis. This method possesses the inherent power to detect and structurally characterize covalent modifications. Absolute sensitivities of matrix-assisted laser desorption ionization and high-energy collision-induced dissociation tandem mass spectrometry are exploited to determine the mass and sequence of subpicomole sample quantities of tryptic peptides. These data permit mass matching and sequence homology searching of computerized peptide mass and protein sequence data bases for known proteins and design of oligonucleotide probes for cloning unknown proteins. We have identified 11 proteins in lysates of human A375 melanoma cells, including: alpha-enolase, cytokeratin, stathmin, protein disulfide isomerase, tropomyosin, Cu/Zn superoxide dismutase, nucleoside diphosphate kinase A, galaptin, and triosephosphate isomerase. We have characterized several posttranslational modifications and chemical modifications that may result from electrophoresis or subsequent sample processing steps. Detection of comigrating and covalently modified proteins illustrates the necessity of peptide sequencing and the advantages of tandem mass spectrometry to reliably and unambiguously establish the identity of each protein. This technology paves the way for studies of cell-type dependent gene expression and studies of large suites of cellular proteins with unprecedented speed and rigor to provide information complementary to the ongoing Human Genome Project.

Finite temperature dynamics of vortices in the two dimensional anisotropic Heisenberg model

We study the effects of finite temperature on the dynamics of non-planar vortices in the classical, two-dimensional anisotropic Heisenberg model with XY- or easy-plane symmetry. To this end, we analyze a generalized Landau-Lifshitz equation including additive white noise and Gilbert damping. Using a collective variable theory with no adjustable parameters we derive an equation of motion for the vortices with stochastic forces which are shown to represent white noise with an effective diffusion constant linearly dependent on temperature. We solve these stochastic equations of motion by means of a Green's function formalism and obtain the mean vortex trajectory and its variance. We find a non-standard time dependence for the variance of the components perpendicular to the driving force. We compare the analytical results with Langevin dynamics simulations and find a good agreement up to temperatures of the order of 25% of the Kosterlitz-Thouless transition temperature. Finally, we discuss the reasons why our approach is not appropriate for higher temperatures as well as the discreteness effects observed in the numerical simulations.