Global Analysis of the Flows of Fluids with Pressure-Dependent Viscosities

 To describe the flows of fluids over a wide range of pressures, it is necessary to take into account the fact that the viscosity of the fluid depends on the pressure. That the viscosity depends on the pressure has been verified by numerous careful experiments. While the existence of solutions local-in-time to the equations governing the flows of such fluids are available for small, special data and rather unrealistic dependence of the viscosity on the pressure, no global existence results are in place. Our interest here is to establish the existence of weak solutions for spatially periodic three-dimensional flows that are global in time, for a large class of physically meaningful viscosity-pressure relationships. (Accepted May 1, 2002) Published online November 15, 2002 Communicated by S. S. ANTMAN     

Single-site mutation and secondary structure stability: an isodesmic reaction approach. The case of unnatural amino acid mutagenesis Ala-->Lac

A method is described to evaluate backbone interactions in proteins via computational unnatural amino acid mutagenesis. Several N-acetyl polyalanyl amides (AcA(n)NH(2)) were optimized in the representative helical (3(10)-, 4(13)-, and a "hybrid" kappa-helix, n = 7, 9, 10, 14) and hairpin (two- and three-stranded antiparallel beta-sheets with type I turns betaalphaalphaepsilon, n = 6, 9, 10) conformations, and extended conformers of N-acetyl polyalanyl methylamides (n = 2, 3) were used to derive multistranded beta-sheet fragments. Subsequently, each residue of every model structure was substituted, one at a time, with l-lactic acid. The resulting mutant structures were again optimized, and group-transfer energies DeltaE(GT) were obtained as heats of the isodesmic reactions: AcA(n)NHR + AcOMe --> AcA(x)LacA(y)NHR + AcNHMe (R = H, CH(3)). These group-transfer energies correlate with the degree of charge polarization of the substituted peptide linkages as measured by the difference Deltae in H and O Mulliken populations in HN-C=O and with the H-bond distances in the "wild-type" structures. A good correlation obtains for the HF/3-21G and B3LYP/6-31G* group-transfer energies. The destabilization effects are interpreted in terms of loss of interstrand and intrastrand H-bonds, decrease in Lewis basicity of the C=O group, and O...O repulsion. On the basis of several comparisons of Ala --> Lac DeltaE(GT)'s with heats of the NH --> CH(2) substitutions, the latter contribution is estimated (B3LYP/6-31G*) to range between 1.5 and 2.4 kcal mol(-1), a figure close to the recent experimental DeltaDeltaG(o) value of 2.6 kcal mol(-1) (McComas, C. C.; Crowley, B. M.; Boger, D. L. J. Am.Chem. Soc. 2003, 125, 9314). The partitioning yields the following maximum values of the electronic association energy of H-bonds in the examined sample of model structures (B3LYP/6-31G* estimates): 3(10)-helix D(e) = -1.7 kcal mol(-1), alpha-helix D(e) = -3.8 kcal mol(-1), beta-sheet D(e) = -6.1 kcal mol(-1). The premise of experimental evaluations of the backbone-backbone H-bonding that Ala --> Lac substitution in proteins is isosteric (e.g., Koh, J. T.; Cornish, V. W.; Schultz, P. G. Biochemistry 1997, 36, 11314) is often but not always corroborated. Examination of the integrity of H-bonding pattern and phi(i), psi(i) distribution identified several mutants with significant distortions of the "wild-type" structure resulting inter alia from the transitions between i, i + 3 and i, i + 4 H-bonding in helices, observed previously in the crystallographic studies of depsipeptides (Ohyama, T.; Oku, H.; Hiroki, A.; Maekawa, Y.; Yoshida, M.; Katakai, R. Biopolymers 2000, 54, 375; Karle, I. L.; Das, C.; Balaram, P. Biopolymers 2001, 59, 276). Thus, the isodesmic reaction approach provides a simple way to gauge how conformation of the polypeptide chain and dimensions of the H-bonding network affect the strength of backbone-backbone C=O...HN bonds. The results indicate that the stabilization provided by such interactions increases on going from 3(10)-helix to alpha-helix to beta-sheet.     

Coordination number of percolating clusters

The coordination number of percolating clusters and their backbones, on the square and simple cubic lattices, is investigated numerically. At the percolation threshold, the average coordination number of the clusters is 2.239 ± 0.012 and 2.079 ± 0.016 for d = 2 and d = 3, respectively. The corresponding numbers for the backbones are 2.402 ± 0.016 and 2.246 ± 0.034.     

Boundary conditions at a fluid-solid interface

We study the boundary conditions at a fluid-solid interface using molecular dynamics simulations covering a broad range of fluid-solid interactions and fluid densities and both simple and chain-molecule fluids. The slip length is shown to be independent of the type of flow, but rather is related to the fluid organization near the solid, as governed by the fluid-solid molecular interactions.     

Hidden valley structure of Ising spin glasses

The hidden valley structure of the phase space of short-range spin glasses is investigated. In the spirit of a similar investigation for liquids by Stillinger and Weber, valleys are defined as the set of states which by steepest descent lead to the same metastable state. The lifetime of valleys and the size of spin clusters involved in intervalley transitions on thermal Monte Carlo trajectories are calculated as a function of temperature.     

Guided modes of integrated optical guides. A mathematical study

A waveguide in integrated optics is defined by its refractiveindex. The guide is assumed to be invariant in the propagationdirection while in the transverse direction it is supposed tobe a compact perturbation of an unbounded stratified medium.We are interested in the modes guided by this device, whichare waves with a transverse energy confined in a neighbourhoodof the perturbation. Our goal is to analyse the existence of such guided modes. Underthe assumptions of weak guidance the problem reduces to a two-dimensionaleigenvalue problem for a scalar field. The associated operatoris unbounded, selfadjoint, and bounded from below. Its spectrumconsists of the discrete spectrum corresponding to the guidedmodes and of the essential spectrum corresponding to the radiationmodes. We present existence results of guided modes and an asymptoticstudy at high frequencies, which shows that contrarily to thecase of optical fibers, the number of guided modes can remainbounded. The major tools are the min-max principle and comparisonof results between different eigenvalue problems. The originalityof the present study lies in the stratified character of theunbounded reference medium.     

Mechanical unfolding of ubiquitin molecules

Mechanical stretching of ubiquitin and of its several repeats are studied through molecular-dynamics simulations. A Go-type model [H. Abe and N. Go, Biopolymers 20, 1013 (1981)] with a realistic contact map and with Lennard-Jones contact interactions is used. The model qualitatively reproduces the experimentally observed differences between force-extension patterns obtained on polyubiquitins stretched by various linkages. The terminal-to-terminal stretching of polyubiquitin results in peak forces similar to those measured for titin-based polyproteins and of a magnitude that matches measurements. Consistent with the experimental measurements, the simulated peak forces depend on the pulling speed logarithmically when thermal fluctuations are explicitly introduced. These results validate the application of topology-based models in the study of the mechanical stretching of proteins.