Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem. Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense. An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented. (C) 2003 Elsevier Science Ltd. All rights reserved.

Caveolin interacts with the angiotensin II type 1 receptor during exocytic transport but not at the plasma membrane

The mechanisms involved in angiotensin II type 1 receptor (AT(1)-R) trafficking and membrane localization are largely unknown. In this study, we examined the role of caveolin in these processes. Electron microscopy of plasma membrane sheets shows that the AT(1)-R is not concentrated in caveolae but is clustered in cholesterol-independent microdomains; upon activation, it partially redistributes to lipid rafts. Despite the lack of AT(1)-R in caveolae, AT(1)-R. caveolin complexes are readily detectable in cells co-expressing both proteins. This interaction requires an intact caveolin scaffolding domain because mutant caveolins that lack a functional caveolin scaffolding domain do not interact with AT(1)-R. Expression of an N-terminally truncated caveolin-3, CavDGV, that localizes to lipid bodies, or a point mutant, Cav3-P104L, that accumulates in the Golgi mislocalizes AT(1)-R to lipid bodies and Golgi, respectively. Mislocalization results in aberrant maturation and surface expression of AT(1)-R, effects that are not reversed by supplementing cells with cholesterol. Similarly mutation of aromatic residues in the caveolin-binding site abrogates AT(1)-R cell surface expression. In cells lacking caveolin-1 or caveolin-3, AT(1)-R does not traffic to the cell surface unless caveolin is ectopically expressed. This observation is recapitulated in caveolin-1 null mice that have a 55% reduction in renal AT(1)-R levels compared with controls. Taken together our results indicate that a direct interaction with caveolin is required to traffic the AT(1)-R through the exocytic pathway, but this does not result in AT(1)-R sequestration in caveolae. Caveolin therefore acts as a molecular chaperone rather than a plasma membrane scaffold for AT(1)-R.

Synaptic vesicle transport and synaptic membrane transporter sites in excitatory amino acid nerve terminals in Alzheimer disease

The apparent L-[H-3]glutamate uptake rate (v') was measured in synaptic vesicles isolated from cerebral cortex synaptosomes prepared from autopsied Alzheimer and non-Alzheimer dementia cases, and age-matched controls. The initial synaptosome preparations exhibited similar densities of D-[H-3]aspartate membrane binding sites (B-MAX values) in the three groups. In control brain the temporal cortex D-[H-3]aspartate B-MAX was 132% of that in motor cortex, parallel with the L- [H-3]glutamate v' values (temporal = 139% of motor; NS). Unlike D- [H-3]aspartate B-MAX values, L- [H-3]glutamate v' values were markedly and selectively lower in Alzheimer brain preparations than in controls, particularly in temporal cortex. The difference could not be attributed to differential effects of autopsy interval or age at death. Non-Alzheimer dementia cases resembled controls. The selective loss of vesicular glutamate transport is consistent with a dysfunction in the recycling of transmitter glutamate.

Abnormal extracellular matrix protein transport associated with increased apoptosis of vascular smooth muscle cells in Marfan syndrome and bicuspid aortic valve thoracic aortic aneurysm

Background - Marfan syndrome (MS) is a genetic disorder caused by a mutation in the fibrillin gene FBN1. Bicuspid aortic valve (BAV) is a congenital heart malformation of unknown cause. Both conditions are associated with ascending aortic aneurysm and premature death. This study examined the relationship among the secretion of extracellular matrix proteins fibrillin, fibronectin, tenascin, and vascular smooth muscle cell (VSMC) apoptosis. The role of matrix metalloproteinase (MMP)- 2 in VSMC apoptosis was studied in MS aneurysm. Methods and Results - Aneurysm tissue was obtained from patients undergoing surgery ( MS: 4 M, 1 F, age 27 - 45 years; BAV: 3 M, 2 F, age 28 - 65 years). Normal aorta from subjects with nonaneurysm disease was also collected ( 4 M, 1 F, age 23 - 93 years). MS and BAV aneurysm histology showed areas of cystic medial necrosis (CMN) without inflammatory infiltrate. Immunohistochemical study of cultured MS and BAV VSMC showed intracellular accumulation and reduction of extracellular distribution of fibrillin, fibronectin, and tenascin. Western blot showed no increase in expression of fibrillin, fibronectin, or tenascin in MS or BAV VSMC and increased expression of MMP-2 in MS VSMCs. There was 4-fold increase in loss of cultured VSMC incubated in serum-free medium for 24 hours in both MS ( 27 +/- 8%) and BAV ( 32 +/- 14%) compared with control ( 7 +/- 5%). Conclusions - In MS and BAV there is alteration in both the amount and quality of secreted proteins and an increased degree of VSMC apoptosis. Up-regulation of MMP-2 might play a role in VSMC apoptosis in MS VSMC. The findings suggest the presence of a fundamental cellular abnormality in BAV thoracic aorta, possibly of genetic origin.

A unified model of flames as gasdynamic discontinuities

Viewed on a hydrodynamic scale, flames in experiments are often thin so that they may be described as gasdynamic discontinuities separating the dense cold fresh mixture from the light hot burned products. The original model of a flame as a gasdynamic discontinuity was due to Darrieus and to Landau. In addition to the fluid dynamical equations, the model consists of a flame speed relation describing the evolution of the discontinuity surface, and jump conditions across the surface which relate the fluid variables on the two sides of the surface. The Darrieus-Landau model predicts, in contrast to observations, that a uniformly propagating planar flame is absolutely unstable and that the strength of the instability grows with increasing perturbation wavenumber so that there is no high-wavenumber cutoff of the instability. The model was modified by Markstein to exhibit a high-wavenumber cutoff if a phenomenological constant in the model has an appropriate sign. Both models are postulated, rather than derived from first principles, and both ignore the flame structure, which depends on chemical kinetics and transport processes within the flame. At present, there are two models which have been derived, rather than postulated, and which are valid in two non-overlapping regions of parameter space. Sivashinsky derived a generalization of the Darrieus-Landau model which is valid for Lewis numbers (ratio of thermal diffusivity to mass diffusivity of the deficient reaction component) bounded away from unity. Matalon & Matkowsky derived a model valid for Lewis numbers close to unity. Each model has its own advantages and disadvantages. Under appropriate conditions the Matalon-Matkowsky model exhibits a high-wavenumber cutoff of the Darrieus-Landau instability. However, since the Lewis numbers considered lie too close to unity, the Matalon-Matkowsky model does not capture the pulsating instability. The Sivashinsky model does capture the pulsating instability, but does not exhibit its high-wavenumber cutoff. In this paper, we derive a model consisting of a new flame speed relation and new jump conditions, which is valid for arbitrary Lewis numbers. It captures the pulsating instability and exhibits the high-wavenumber cutoff of all instabilities. The flame speed relation includes the effect of short wavelengths, not previously considered, which leads to stabilizing transverse surface diffusion terms.

Intracellular sorting and transport of proteins

The secretory and endocytic pathways of eukaryotic organelles consist of multiple compartments, each with a unique set of proteins and lipids. Specific transport mechanisms are required to direct molecules to defined locations and to ensure that the identity, and hence function, of individual compartments are maintained. The localisation of proteins to specific membranes is complex and involves multiple interactions. The recent dramatic advances in understanding the molecular mechanisms of membrane transport has been due to the application of a multi-disciplinary approach, intergrating membrane biology, genetics, imaging, protein and lipid biochemistry and structural biology. The aim of this review is to summarise the general principles of protein sorting in the secretory and endocytic pathways and to highlight the dynamic nature of these processes. The molecular mechanisms involved in this transport along the secretory and endocytic pathways are discussed along with the signals responsible for targeting proteins to different intracellular locations. (C) 2003 Elsevier Science Ltd. All rights reserved.

Selecting methods to solve multi-well master equations

There are several competing methods commonly used to solve energy grained master equations describing gas-phase reactive systems. When it comes to selecting an appropriate method for any particular problem, there is little guidance in the literature. In this paper we directly compare several variants of spectral and numerical integration methods from the point of view of computer time required to calculate the solution and the range of temperature and pressure conditions under which the methods are successful. The test case used in the comparison is an important reaction in combustion chemistry and incorporates reversible and irreversible bimolecular reaction steps as well as isomerizations between multiple unimolecular species. While the numerical integration of the ODE with a stiff ODE integrator is not the fastest method overall, it is the fastest method applicable to all conditions.

Vps20p and Vta1p interact with Vps4p and function in multivesicular body sorting and endosomal transport in Saccharomyces cerevisiae

Vps4p (End13p) is an AAA-family ATPase that functions in membrane transport through endosomes, sorting of soluble vacuolar proteins to the vacuole, and multivesicular body (MVB) sorting of membrane proteins to the vacuole lumen. In a yeast two-hybrid screen with Vps4p as bait we isolated VPS20 (YMR077c) and the novel open reading frame YLA181c, for which the name VTA1 has recently been assigned (Saccharomyces Genome Database). Vps4p directly binds Vps20p and Vta1p in vitro and binding is not dependent on ATP-conversely, Vps4p binding to Vps20p is partially sensitive to ATP hydrolysis. Both ATP binding [Vps4p-(K179A)] and ATP hydrolysis [Vps4p-(E233Q)] mutant proteins exhibit enhanced binding to Vps20p and Vta1p in vitro. The Vps4p-Vps20p interaction involves the coiled-coil domain of each protein, whereas the Vps4p-Vta1p interaction involves the (non-coiled-coil) C-terminus of each protein. Deletion of either VPS20 (vps20Delta) or VTA1 (vta1Delta) leads to similar class E Vps(-) phenotypes resembling those of vps4Delta, including carboxypeptidase Y (CPY) secretion, a block in ubiquitin-dependent MVB sorting, and a delay in both post-internalisation endocytic transport and biosynthetic transport to the vacuole. The vacuole resident membrane protein Sna3p (whose MVB sorting is ubiquitin-independent) does not appear to exit the class E compartment or reach the vacuole in cells lacking Vps20p, Vta1p or Vps4p, in contrast to other proteins whose delivery to the vacuole is only delayed. We propose that Vps20p and Vta1p regulate Vps4p function in vivo.

An approach to verifying and debugging simulation models governed by ordinary differential equations: Part 1. Methodology for residual generation

For dynamic simulations to be credible, verification of the computer code must be an integral part of the modelling process. This two-part paper describes a novel approach to verification through program testing and debugging. In Part 1, a methodology is presented for detecting and isolating coding errors using back-to-back testing. Residuals are generated by comparing the output of two independent implementations, in response to identical inputs. The key feature of the methodology is that a specially modified observer is created using one of the implementations, so as to impose an error-dependent structure on these residuals. Each error can be associated with a fixed and known subspace, permitting errors to be isolated to specific equations in the code. It is shown that the geometric properties extend to multiple errors in either one of the two implementations. Copyright (C) 2003 John Wiley Sons, Ltd.

An approach to verifying and debugging simulation models governed by ordinary differential equations: Part 2. Residuals analysis and a case study

In Part 1 of this paper a methodology for back-to-back testing of simulation software was described. Residuals with error-dependent geometric properties were generated. A set of potential coding errors was enumerated, along with a corresponding set of feature matrices, which describe the geometric properties imposed on the residuals by each of the errors. In this part of the paper, an algorithm is developed to isolate the coding errors present by analysing the residuals. A set of errors is isolated when the subspace spanned by their combined feature matrices corresponds to that of the residuals. Individual feature matrices are compared to the residuals and classified as 'definite', 'possible' or 'impossible'. The status of 'possible' errors is resolved using a dynamic subset testing algorithm. To demonstrate and validate the testing methodology presented in Part 1 and the isolation algorithm presented in Part 2, a case study is presented using a model for biological wastewater treatment. Both single and simultaneous errors that are deliberately introduced into the simulation code are correctly detected and isolated. Copyright (C) 2003 John Wiley Sons, Ltd.

A new wavelet-based adaptive method for solving population balance equations

A new wavelet-based adaptive framework for solving population balance equations (PBEs) is proposed in this work. The technique is general, powerful and efficient without the need for prior assumptions about the characteristics of the processes. Because there are steeply varying number densities across a size range, a new strategy is developed to select the optimal order of resolution and the collocation points based on an interpolating wavelet transform (IWT). The proposed technique has been tested for size-independent agglomeration, agglomeration with a linear summation kernel and agglomeration with a nonlinear kernel. In all cases, the predicted and analytical particle size distributions (PSDs) are in excellent agreement. Further work on the solution of the general population balance equations with nucleation, growth and agglomeration and the solution of steady-state population balance equations will be presented in this framework. (C) 2002 Elsevier Science B.V. All rights reserved.

Wall mediated transport in confined spaces: Exact theory for low density

We present a theory for the transport of molecules adsorbed in slit and cylindrical nanopores at low density, considering the axial momentum gain of molecules oscillating between diffuse wall reflections. Good agreement with molecular dynamics simulations is obtained over a wide range of pore sizes, including the regime of single-file diffusion where fluid-fluid interactions are shown to have a negligible effect on the collective transport coefficient. We show that dispersive fluid-wall interactions considerably attenuate transport compared to classical hard sphere theory.

Molecular transport in nanopores

Simulation of the transport of methane in cylindrical silica mesopores have been performed using equilibrium and nonequilibrium molecular dynamics (NEMD) as well as dual control volume grand canonical molecular dynamics methods. It is demonstrated that all three techniques yield the same transport coefficient even in the presence of viscous flow. A modified locally averaged density model for viscous flow, combined with consideration of wall slip through a frictional condition, gives a convincing interpretation of the variation of the transport coefficient over a wide range of densities, and for various pore sizes and temperatures. Wall friction coefficients extracted from NEMD simulations are found to be consistent with momentum transfer arguments, and the approach is shown to be more meaningful than the classical slip length concept. (C) 2003 American Institute of Physics.

Analysis of multicomponent adsorption kinetics on activated carbon

An integrated mathematical model for the kinetics of multicomponent adsorption on microporous carbon was developed. Transport in this bidisperse solid is represented by balance equations in the macropore and micropore phases, in which gas-phase diffusion dominates the mass transfer in the macropores, with the phenomenological diffusivities represented by the generalized Maxwell-Stefan (GMS) formulation. Viscous flow also contributes to the macropore fluxes and is included in the MS expressions. Diffusion of the adsorbed phase controls the mass transfer in the micro ore phase, p which is also described in a similar way by the MS method. The adsorption isotherms are represented by a new heterogeneous modified vacancy solution theory formulation of adsorption, which has proved to be a robust method for adsorption on activated carbons. The model is applied to the coadsorption and codesorption of C2H6 and C3H8 on Ajax and Norit carbon, as well as the displacement on Ajax carbon. The effect of the viscous flow in the macropore phase is not significant for the cases studied. The model accurately predicts the overshoot behavior and rollup of C2H6 during coadsorption. The prediction for the heavier compound C3H8 is always satisfactory, though at higher C3H8 mole fraction, the overshoot extent of C2H6 is overpredicted, possibly due to neglect of heat effects.