Free diffusion

In the absence of restrictions on the diffusion, i.e., for free diffusion, the Eqs. (17)-(19) must be true for every arbitrary value of z. Equation (17)  

Substituting Eq. (25) into Eq. (23) we find the expression for the propagator of free diffusion in the normal mode representation  

The well-known Gaussian expression for free diffusion is found by evaluating the standard integral in Eq. (26)  

In many treatments of free diffusion the propagator is immediately written as a Gaussian function with the argument that it fulfils the diffusion equation. Equation (27) shows the relation with the normal mode solution of the diffusion equation. For diffusion in a bounded region the propagator is no 

The normal mode expression for the propagator Eq. (16) is perfect for working out the propagator for diffusion in a rectangular box with reflecting walls between the x -planes at z = 0 and z = a. One of the boundary conditions here is (reflecting walls)  

Although there is a natural tendency toward equilibrium of the solute concentration on both sides of the membrane, such an equilibrium is rare in a living system, and selective permeability of the plasma membrane therefore assures the required distribution of metabolically important material inside and outside the cell. Kinetic studies of solute transport often permit characterization of the type of transmembrane movement involved (Neame and Richards, 1972). As outlined by Csaky (1965), a given substance can cross the cell membrane in several different ways free diffusion, diffusion through pores, pinocytosis, and carrier-mediated transport. 

Uninfluenced by other factors, all solutes will diffuse down a concentration gradient. In the same way, solutes will move across a membrane from an area of high concentration to one of lower concentration in an attempt to equilibrate. The rate of diffusion should then behave according to Pick s first law of diffusion, dnidt = -DA(dcldx), where dni dt is the number of molecules per unit time crossing area A in the interface at a concentration difference of dc over the distance dx. The cell membrane, however, acts as a selectively permeable mechanism owing to its lipid layers. In general, the rate of permeation of a substance is related to its lipid solubility, and the membrane thus acts as a solvent for nonpolar molecules. 

Penetration of a substance is measured by the permeability coefficient, P, which could be converted to a measurable diffusional coefficient, D, if Pick s law applied strictly. In the more complex situation of a membrane barrier, Kedem and Katchalsky (1958, 1961) have shown that under rigidly controlled conditions there exist at least three parameters which must be considered when characterizing the behavior of a membrane toward a particular solute (1) the interaction between membrane and solvent (2) the interaction between solute and membrane and (3) the interaction between solute and solvent. The reflection coefficient, (T, measures relative rates of solute and solvent permeabilities in the system (Staverman, 1952) and is therefore a measure of semipermeability. Lp is the mechanical coefficient of filtration or pressure filtration coefficient, and co is the solute mobility or solute diffusional coefficient. In the case of living membranes, conditions such as volume flow, osmotic gradients, and cell volume can be manipulated in order to measure the phenomenological coefficients cr, o , and Lp. Detailed discussions of the theories, methods, and problems involved in such 

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In this regime the applied force completely overwhelms the binding potential and the ligand is subject to free diffusion. The mean free passage time in this regime is equal to Td and is on the order of 25 ns. 

Furthermore, assuming a constant deposition rate J (particles per area and time) during MBE, we can define a further length scale, namely the free diffusion length or the capture length... 

This length is apparently related to the capture time by the relation Pi J Tc and il A physical meaning of the free diffusion length 4 is that the maximum size of a stable adsorbed two-dimensional nucleus on a facet cannot essentially exceed this free diffusion length. If the nucleus is smaller, all atoms depositing on the surface can still find the path to the boundary of a nucleus in order to be incorporated there. If the nucleus is larger, a new nucleus can develop on its surface. 

Mitochondria are surrounded by a simple outer membrane and a more complex inner membrane (Figure 21.1). The space between the inner and outer membranes is referred to as the intermembrane space. Several enzymes that utilize ATP (such as creatine kinase and adenylate kinase) are found in the intermembrane space. The smooth outer membrane is about 30 to 40% lipid and 60 to 70% protein, and has a relatively high concentration of phos-phatidylinositol. The outer membrane contains significant amounts of porin —a transmembrane protein, rich in /3-sheets, that forms large channels across the membrane, permitting free diffusion of molecules with molecular weights of about 10,000 or less. Apparently, the outer membrane functions mainly to... 

The receptor compartment is defined as the aqueous volume containing the receptor and cellular system. It is assumed that free diffusion leads to ready access to this compartment (i.e., that the concentration within this compartment is the free concentration of drug at the receptor). However, there are factors that can cause differences between the experimentally accessible liquid compartment and the actual receptor compartment. One obvious potential problem is limited solubility of the drug being added to the medium. The assumption is made tacitly that the dissolved drug in the stock solution, when added to the medium bathing the pharmacological preparation, will stay in solution. There are cases where this may not be a valid assumption. 

In systems comprised of cells in culture, there is no formal architecture (such as might be encountered in a whole tissue) that would hinder free diffusion. Such... 

In this exercise we shall estimate the influence of transport limitations when testing an ammonia catalyst such as that described in Exercise 5.1 by estimating the effectiveness factor e. We are aware that the radius of the catalyst particles is essential so the fused and reduced catalyst is crushed into small particles. A fraction with a narrow distribution of = 0.2 mm is used for the experiment. We shall assume that the particles are ideally spherical. The effective diffusion constant is not easily accessible but we assume that it is approximately a factor of 100 lower than the free diffusion, which is in the proximity of 0.4 cm s . A test is then made with a stoichiometric mixture of N2/H2 at 4 bar under the assumption that the process is far from equilibrium and first order in nitrogen. The reaction is planned to run at 600 K, and from fundamental studies on a single crystal the TOP is roughly 0.05 per iron atom in the surface. From Exercise 5.1 we utilize that 1 g of reduced catalyst has a volume of 0.2 cm g , that the pore volume constitutes 0.1 cm g and that the total surface area, which we will assume is the pore area, is 29 m g , and that of this is the 18 m g- is the pure iron Fe(lOO) surface. Note that there is some dispute as to which are the active sites on iron (a dispute that we disregard here). 

The artificial lipid bilayer is often prepared via the vesicle-fusion method [8]. In the vesicle fusion process, immersing a solid substrate in a vesicle dispersion solution induces adsorption and rupture of the vesicles on the substrate, which yields a planar and continuous lipid bilayer structure (Figure 13.1) [9]. The Langmuir-Blodgett transfer process is also a useful method [10]. These artificial lipid bilayers can support various biomolecules [11-16]. However, we have to take care because some transmembrane proteins incorporated in these artificial lipid bilayers interact directly with the substrate surface due to a lack of sufficient space between the bilayer and the substrate. This alters the native properties of the proteins and prohibits free diffusion in the lipid bilayer [17[. To avoid this undesirable situation, polymer-supported bilayers [7, 18, 19] or tethered bilayers [20, 21] are used. 

For an artificial lipid bilayer of any size scale, it is a general feature that the bilayer acts as a two-dimensional fluid due to the presence of the water cushionlayer between the bilayer and the substrate. Due to this fluidic nature, molecules incorporated in the lipid bilayer show two-dimensional free diffusion. By applying any bias for controlling the diffusion dynamics, we can manipulate only the desired molecule within the artificial lipid bilayer, which leads to the development of a molecular separation system. 

In Figure 1.15, the propagator of free self-diffusion is shown. Actually, selfdiffusion was the first case where the propagator terminology was discussed in the NMR context by Karger (see also Chapter 3.1). The propagator of free diffusion is of... 

The in vitro system we have been using to study the transepithelial transport is cultured Madin-Darby canine kidney (MDCK) epithelial cells (11). When cultured on microporous polycarbonate filters (Transwell, Costar, Cambridge, MA), MDCK cells will develop into monolayers mimicking the mucosal epithelium (11). When these cells reach confluence, tight junctions will be established between the cells, and free diffusion of solutes across the cell monolayer will be markedly inhibited. Tight junction formation can be monitored by measuring the transepithelial electrical resistance (TEER) across the cell monolayers. In Figure 1, MDCK cells were seeded at 2 X 104 cells per well in Transwells (0.4 p pore size) as described previously. TEER and 14C-sucrose transport were measured daily. To determine 14C-sucrose... 

Fig. 2.13 Examples of liquid junctions (A) liquid junction with free diffusion is formed in a three-way cock which connects the solution under investigation with the salt bridge solution (B) liquid junction with restrained diffusion is formed in a ceramic plug which connects the salt bridge with the investigated solution...

In an early attempt, Mozumder (1968) used a prescribed diffusion approach to obtain the e-ion geminate recombination kinetics in the pure solvent. At any time t, the electron distribution function was assumed to be a gaussian corresponding to free diffusion, weighted by another function of t only. The latter function was found by substituting the entire distribution function in the Smoluchowski equation, for which an analytical solution was possible. The result may be expressed by... 

The long-time survival probability for the entire track approaches the limit P(t)/ = 1 + 0.6x-0 6, where T is a normalized time, much earlier than the P(r)/Pesc = 1 + (7TT)-0-5 predicted by the free diffusion theory (Bartczak and Hummel,1997). Notice that the T 06 dependence of the existence probability had been established eariler in the experiment of van den Ende et al.(1984). 

Next, the indicator dye needs a solvent to interact with the analyte. Pure crystalline indicator dyes might react at the surface but not all indicator would react due to hindered diffusion. Therefore, the indicator is dissolved in a polymer which allows free diffusion of the analyte to and from the indicator molecule. 

These authors interpret their data as the natural result of restrictions imposed on the free diffusion of the labeled receptor by encounters with other transmembrane proteins in the bilayer. However they consider that their data are incompatible with the hop and skip model based on spectrin mesh confinement. 

We emphasize that any utilization of Eq. (5) already rest upon a number of (often reasonable) assumptions. Equation (5) represents an ordinary deterministic differential equation, based on assumption of homogeneity, free diffusion, and random collision, and neglecting spatial [102] or stochastic effects [103]. While such assumptions are often vindicated for microorganisms, the application of Eq. (5) to other cell types, such as human or plant cells, sometimes mandates careful verification. 

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