Membranes finite

Dimova, R., Dietrich, C., Hadjiisky, A., Danov, K., and Pouligny, B. (1999) Falling ball viscosimetry of giant vesicle membranes finite-size effects. European Physical Journal B, 12, 589. 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

Let us proceed with the second interesting example concerning application of the ROZ equations. We would Hke first to mention that simple fluids confined to slits with permeable membranes have been studied by both computer simulations and theory, see, e.g.. Refs. 49-52. The simplest way is to visualize a permeable membrane as a barrier of finite height and width. To our best knowledge, no studies of a system containing multiple barriers of a more sophisticated geometry than the sHt-Hke have been undertaken so... 

ILs, on the other hand, are uniquely suited for use as solvents for gas separations. Since they are non-volatile, they cannot evaporate to cause contamination of the gas stream. This is important when selective solvents are used in conventional absorbers, or when they are used in supported liquid membranes. For conventional absorbers, the ability to separate one gas from another depends entirely on the relative solubilities (ratio of Henry s law constants) of the gases. In addition, ILs are particularly promising for supported liquid membranes, because they have the potential to be incredibly stable. Supported liquid membranes that incorporate conventional liquids eventually deteriorate because the liquid slowly evaporates. Moreover, this finite evaporation rate limits how thin one can make the membrane. This... 

Mixed potential resulting from an interfering redox reaction at membranes with finite electronic conductance... 

Some specific solutes diffuse down electrochemical gradients across membranes more rapidly than might be expected from their size, charge, or partition coefficients. This facilitated diffusion exhibits properties distinct from those of simple diffusion. The rate of facilitated diffusion, a uniport system, can be saturated ie, the number of sites involved in diffusion of the specific solutes appears finite. Many facihtated diffusion systems are stereospecific but, fike simple diffusion, require no metabolic energy. 

The first limitation is related to interference of the anode and the cathode. The finite permeability of the Nation membrane to fuel and oxygen results in crossover of fuel from the anode to the cathode, and oxygen crossover in the opposite direction. This may have a significant influence on electrode kinetics. 

To what extent does the SR contribute to the rise of [Ca2+]j that activates contraction In other words, what are the relative contributions of the SR and the surface membrane In contrast to the situation in striated muscle where inhibition of SR function abolishes most of contraction, there are several examples in smooth muscle of large amounts of force remaining under these conditions. The SR is an intracellular store of finite capacity. Release of Ca2+ from such a store is well suited to producing transient contractions. However, maintained contraction can be produced by steady state changes in Ca2+ fluxes across the surface membrane. Does the SR make different contributions during different phases of contraction ... 

The polycarbonate glazing is modeled as a simply supported plate subjected to nonlinear center deflections up to 15 times the pane thickness. Using the finite element solution of Moore (Reference 4), the resistance function is generated for each pane under consideration. Typically, the resistance is concave up, as illustrated for typical pane sizes in Figure 1. This occurs because membrane stresses induced by the stretching of the neutral axis of the pane become more pronounced as the ratio of the center pane deflection to the pane... 

Let us consider, for example, a flat symmetrical bilayer of which the area is large, so that end-effects can be ignored. Finite size effects are important, and will be discussed in the following section. The membrane is freely floating in solution, i.e. it is not supported by a frame. Combination of the first and second laws of thermodynamics gives for the difference of internal energy dl/of a bulk system with membranes with area A ... 

The finite size of the box has several important consequences. One of them is that the area of the membrane piece is only of the order of 100 nm2. It is expected that the membrane is, on this length scale, roughly flat, i.e. the area is small as compared with the persistence length for the bilayer. Interestingly, however, in recent simulations the first signs of fluctuations away from the flat bilayer structure (undulations) have reportedly been found by MD simulations [33],... 

The fourth term is a polarisation term. Here E(z) = di/z/dz is the electric field at position z. In previously published SCF results for charged bilayers, this last term is typically absent. It can be shown that the polarisation term is necessary to obtain accurate thermodynamic data. We note that all qualitative results of previous calculations remain valid and that, for example, properties such as the equilibrium membrane thickness are not affected significantly. The polarisation term represents relatively straightforward physics. If a (united) atom with a finite polarisability of erA is introduced from the bulk where the potential is zero to the coordinate z where a finite electric field exists, it will be polarised. The dipole that forms is proportional to the electric field and the relative dielectric permittivity of the (united) atom. The energy gain due to this is also proportional to the electric field, hence this term is proportional to the square of the electric field. The polarisation of the molecule also has an entropic consequence. It can be shown that the free energy effect for the polarisation, which should be included in the segment potential, is just half this value... 

Above, we argued that a portion of a finite-sized membrane can close upon itself to remove edge effects. In this way, vesicles are formed. The thermodynamic stability of vesicles is still a topic of hot debate in the literature, primarily because there are so many scenarios. The SCF analysis of vesicles leads to information on the mechanical parameters for a particular membrane system. 

The ideas of Overton are reflected in the classical solubility-diffusion model for transmembrane transport. In this model [125,126], the cell membrane and other membranes within the cell are considered as homogeneous phases with sharp boundaries. Transport phenomena are described by Fick s first law of diffusion, or, in the case of ion transport and a finite membrane potential, by the Nernst-Planck equation (see Chapter 3 of this volume). The driving force of the flux is the gradient of the (electro)chemical potential across the membrane. In the absence of electric fields, the chemical potential gradient is reduced to a concentration gradient. Since the membrane is assumed to be homogeneous, the... 

For the NaCI electrolysis, finite-gap operation proved initially to be the most promising way to realise good and reproducible cell voltages. Predominantly the reasons were a proper control of the membrane water content, the relatively simple retrofit of existing membrane electrolysers and the ongoing utilisation of the peripheral caustic equipment with the ODC technique. 

Membrane-like microstructures are generally several micrometers thick, while the lateral dimensions of the structures and the surrounding package are on the order of a few hundred micrometers. If the layered thin-film structure would be directly transferred to a 3-d geometry model, an enormous number of finite elements would be created, as the smallest structure size determines the mesh density. Averaging the structural information and properties over the different layers in the cross section of the membrane is a good method to avoid such problems. The membrane is, therefore, initially treated as a quasi-two-dimensional object. 

---