Effects of spatial variation in membrane diffusibility and solubility on the lateral transport of membrane components.

There exist many examples of membrane components (e.g. receptors) accumulating in special domains of cell membranes. We analyze how certain variations in lateral diffusibility and solubility of the membrane would increase the efficiency of transport to these regions. A theorem is derived to show that the mean-time-of capture, tc, for particles diffusing to a trap from an annular region surrounding it, is intermediate to the tc values that correspond to the minimum and maximum diffusion coefficients that obtain in this region. An analytical solution for tc as a function of the gradient of diffusivity surrounding a trap is derived for circular geometry. Since local diffusion coefficients can be increased dramatically by reducing the concentration of intra-membrane particles and/or allowing them to form aggregates, such mechanisms could greatly enhance the diffusion-limited transport of particular membrane components to a trap (e.g. coated pit). If the trap is surrounded by an annular region in which the probe particles' partition function is increased, say, by the local segregation of certain phospholipids, tc is shown to vary inversely with the logarithm of the relative partition function. We provide some conjectural examples to illustrate the magnitude of the effects which heterogeneities in diffusibility and solubility may have in biological membranes.     

The diffusion transit time; A simple derivation

The mean first passage time for free diffusion can be derived directly by solving a simple analogue steady state problem. In this problem the diffusion starting region is considered as a time independent source of diffusing particles and the diffusion target assumes the behaviour of a perfectly absorbing sink. It is shown here that the transit time between the source and the sink, which in this particular problem is equal to the ratio between the holdup of the system and the total flux, is identical to the Brownian movement concept of the mean first passage time for free diffusion. This established identity considerably facilitates the derivation and investigation of the timing of diffusion in complicated structures such as those commonly found in living organisms.     

Measuring and Overcoming Limits of the Saffman-Delbrück Model for Soap Film Viscosities

We observe tracer particles diffusing in soap films to measure the two-dimensional (2D) viscous properties of the films. Saffman-Delbrück type models relate the single-particle diffusivity to parameters of the film (such as thickness h) for thin films, but the relation breaks down for thicker films. Notably, the diffusivity is faster than expected for thicker films, with the crossover at h/d = 5.2 ± 0.9 using the tracer particle diameter d. This indicates a crossover from purely 2D diffusion to diffusion that is more three-dimensional. We demonstrate that measuring the correlations of particle pairs as a function of their separation overcomes the limitations of the Saffman-Delbrück model and allows one to measure the viscosity of a soap film for any thickness.     

The Effect of Antisite Disorder and Particle Size on Li Intercalation Kinetics in Monoclinic LiMnBO3

In materials containing 1D lithium diffusion channels, cation disorder can strongly affect lithium intercalation processes. This work presents a model to explain the unusual transport properties of monoclinic LiMnBO₃, a material determined by scanning electron microscopy and synchrotron X‐ray diffraction to contain a wide particle size distribution and Mn/Li antisite disorder. First‐principles calculations indicate that Mn occupying Li sites obstruct the 1D lithium diffusion channel along the [001] direction. While channel blockage by the antisites significantly lowers Li mobility in large particles, Li kinetics in small particles and particle surfaces are found to be less sensitive to the presence of antisite disorder. Thus, in an electrode containing a large particle size distribution, smaller particles have higher Li mobility, and the measured Li diffusivity as determined by potentiostatic intermittent titration test varies as a function of particle size. The Li capacity in monoclinic LiMnBO₃ is kinetically controlled by the fraction of large particles with antisite disorder, but is not intrinsically limited. These results strongly suggest that particle nanosizing will significantly enhance the electrochemical performance of LiMnBO₃.     

The role of polymer matrix structure and interparticle interactions in diffusion-limited drug release.

A lattice random-walk model is used to simulate diffusion in a porous polymer. This model may be useful for the practical design of drug-release systems. Both interacting and noninteracting particles (random walkers) were allowed to diffuse through a pore with a single exit hole. It was found that the specific interactions among the diffusing particles have little influence on the overall release rate. Diffusion through more complicated structures was investigated by simulating the diffusion of particles through two pores connected by a constricted channel whose length and width were varied. The overall rate of release was found to be proportional to the width of the constricted channel. When the length of the channel was greater than or equal to the length of the pore, the rate of release was also inversely proportional to the channel length. From a practical standpoint, release rates can be decreased (and times for release increased) by one or two orders of magnitude by decreasing the width and expanding the length of the interconnecting channels in the polymer matrix.     

Apparent 2-D diffusivity in a ruffled cell membrane

Most biological cell membranes have a microtopology that increases their surface area, including a highly ruffled surface in the case of leukocytes. Thus, molecular membrane diffusivities as measured by fluorescence recovery after photobleaching or other methods are decreased when projected onto a plane. We use a two-dimensional crested cycloid as a parameterized surface to simulate the random-walk diffusion of a molecule within a ruffled membrane. The apparent 2-D diffusivity was then calculated when the ruffled membrane is projected onto a plane. It is shown that the apparent diffusivity decreases as a function of the membrane area, to the -1.4 power.     

The position of regenerating cambia: auxin/sucrose ratio and the gradient induction hypothesis.

To account for the positions in which vascular cambia regenerate in wound callus, a gradient induction hypothesis was proposed in 1961 in terms of gradients in 'some factor as yet unknown'. It now seems likely that the gradient is based on morphogen diffusion between source and sink on opposite sides of existing cambia, with morphogen diffusing into the adjoining wound callus. It is specifically proposed that there are two morphogens, auxin diffusing centrifugally and sucrose diffusing centripetally. The cambium then regenerates along a path where the ratio of auxin to sucrose concentration is similar to that at the original cambium, and its orientation (as regards xylem and phloem formation) is determined by the direction of the gradient in this ratio. These proposals are supported by published evidence on auxin and sucrose concentration gradients across the cambium, and on their sources, movements, and known effects on vascular differentiation. Simulations of the proposed positional control system predict patterns of cambial regeneration and orientation corresponding to those observed in four different types of wound and graft.     

Diffusional anisotropy in collagenous tissues: fluorescence imaging of continuous point photobleaching

Molecular transport in avascular collagenous tissues such as articular cartilage occurs primarily via diffusion. The presence of ordered structures in the extracellular matrix may influence the local transport of macromolecules, leading to anisotropic diffusion depending on the relative size of the molecule and that of extracellular matrix structures. Here we present what we believe is a novel photobleaching technique for measuring the anisotropic diffusivity of macromolecules in collagenous tissues. We hypothesized that macromolecular diffusion is anisotropic in collagenous tissues, depending on molecular size and the local organization of the collagen structure. A theoretical model and experimental protocol for fluorescence imaging of continuous point photobleaching was developed to measure diffusional anisotropy. Significant anisotropy was observed in highly ordered collagenous tissues such as ligament, with diffusivity ratios >2 along the fiber direction compared to the perpendicular direction. In less-ordered tissues such as articular cartilage, diffusional anisotropy was dependent on site in the tissue and size of the diffusing molecule. Anisotropic diffusion was also dependent on the size of the diffusing molecule, with greatest anisotropy observed for larger molecules. These findings suggest that diffusional transport of macromolecules is anisotropic in collagenous tissues, with higher rates of diffusion along primary orientation of collagen fibers.     

Dynamics of single potassium channel proteins in the plasma membrane of migrating cells

Cell migration is an important physiological process among others controlled by ion channel activity. Calcium-activated potassium channels (K(Ca)3.1) are required for optimal cell migration. Previously, we identified single human (h)K(Ca)3.1 channel proteins in the plasma membrane by means of quantum dot (QD) labeling. In the present study, we tracked single-channel proteins during migration to classify their dynamics in the plasma membrane of MDCK-F cells. Single hK(Ca)3.1 channels were visualized with QD- or Alexa488-conjugated antibodies and tracked at the basal cell membrane using time-lapse total internal reflection fluorescence (TIRF) microscopy. Analysis of the trajectories allowed the classification of channel dynamics. Channel tracks were compared with those of free QD-conjugated antibodies. The size of the label has a pronounced effect on hK(Ca)3.1 channel diffusion. QD-labeled channels have a (sub)diffusion coefficient D(QDbound) = 0.067 microm(2)/s(alpha), whereas that of Alexa488-labeled channels is D(Alexa) = 0.139 microm(2)/s. Free QD-conjugated antibodies move much faster: D(QDfree) = 2.163 microm(2)/s(alpha). Plotting the mean squared distances (msd) covered by hK(Ca)3.1 channels as a function of time points to the mode of diffusion. Alexa488-labeled channels diffuse normally, whereas the QD-label renders hK(Ca)3.1 channel diffusion anomalous. Free QD-labeled antibodies also diffuse anomalously. Hence, QDs slow down diffusion of hK(Ca)3.1 channels and change the mode of diffusion. These results, referring to the role of label size and properties of the extracellular environment, suggest that the pericellular glycocalyx has an important impact on labels used for single molecule tracking. Thus tracking fluorescent particles within the glycocalyx opens up a possibility to characterize the pericellular nanoenvironment.     

Anomalous diffusion due to binding: a Monte Carlo study.

In classical diffusion, the mean-square displacement increases linearly with time. But in the presence of obstacles or binding sites, anomalous diffusion may occur, in which the mean-square displacement is proportional to a nonintegral power of time for some or all times. Anomalous diffusion is discussed for various models of binding, including an obstruction/binding model in which immobile membrane proteins are represented by obstacles that bind diffusing particles in nearest-neighbor sites. The classification of binding models is considered, including the distinction between valley and mountain models and the distinction between singular and nonsingular distributions of binding energies. Anomalous diffusion is sensitive to the initial conditions of the measurement. In valley models, diffusion is anomalous if the diffusing particles start at random positions but normal if the particles start at thermal equilibrium positions. Thermal equilibration leads to normal diffusion, or to diffusion as normal as the obstacles allow.     

Ion-specific diffusion rates through transmembrane protein channels. A molecular dynamics study

The migration of different alkali metal cations through a transmembrane model channel is simulated by means of the molecular dynamics technique. The parameters of the model are chosen in close relation to the gramicidin A channel. Coulomb- and van der Waals-type potentials between the ions and flexible carbonyl groups of the pore-forming molecule are used to describe the ion channel interaction. The diffusion properties of the ions are obtained from three-dimensional trajectory calculations. The diffusion rates for the different ions Li+, Na+, K+ and Rb+ are affected not only by the mass of the particles but also very strongly by their size. The latter effect is more pronounced for rigid channels, i.e., for binding vibrational frequencies of the CO groups with v greater than 400 cm-1. In this range the selectivity sequence for the diffusion rates is the inverse of that expected from normal rate theory but agrees with that found in experiments for gramicidin A.     

Relationship between statistical properties of single ionic channel recordings and the thermodynamic state of the channels

Recordings of the electric conductivity of a single ionic channel usually exhibit two levels of conductance: a zero and a finite level. The channel may, however, be in a few states which have the same conductivity level, and the distribution of dwell time durations at this conductivity level is thus not monoexponential. It is shown that the joint probability p(tc,to) of the occurrence of a time interval tc during which the channel is not conducting, immediately followed by a time interval to during which the channel is conducting may or may not be equal to the joint probability pr(tc,to) of the occurrence of a non-conducting interval tc preceded by a conducting interval to. If the interconversions between the various states in which the channel can exist obey detailed balance, i.e., if the channel behaves like a system at thermodynamic equilibrium, then p(tc,to) = pr(tc,to). This should help to reveal whether irreversible processes, like metabolic reactions or flows of substances across the membrane, are coupled to the gating process of the ionic channels.     

Carousel effect in alveolar models

Experimental work over the past decade has shown that recirculation in alveoli substantially increases the transport of particles. We have previously shown that, for nondiffusing passive particles, this can be understood with the aid of Moffatt's famous corner flow model. Without wall motion, passive particles recirculate in a regular fashion and no chaos exists; however, wall motion produces extensive chaotic flow. Aerosols typically do not follow this flow as they are fundamentally different from fluid particles. Here, we construct a simple model to study diffusing particles in the presence of recirculation. We assume that all particles are passive, that is to say that they do not significantly alter the underlying flow. In particular, we consider particles with high Peclet number and neglect inertial effects. We modify the Lagrangian system for corner eddies to accommodate diffusing particles. Particle transport is governed by Langevin equations. Ensembles of diffusing particles are tracked by numerical integration. We show that transport of diffusing particles is enhanced by sufficiently strong underlying recirculation through a mechanism that we call the "carousel effect." However, as the corner is approached, the recirculation rapidly decreases in intensity, favoring motion by diffusion. Far from the corner's apex, recirculation dominates. For real alveoli, the model indicates that sufficiently strong recirculation can enhance transport of diffusing particles through the carousel effect.     

Size effects on diffusion processes within agarose gels

To investigate diffusion processes in agarose gel, nanoparticles with sizes in the range between 1 and 140 nm have been tested by means of fluorescence correlation spectroscopy. Understanding the diffusion properties in agarose gels is interesting, because such gels are good models for microbial biofilms and cells cytoplasm. The fluorescence correlation spectroscopy technique is very useful for such investigations due to its high sensitivity and selectivity, its excellent spatial resolution compared to the pore size of the gel, and its ability to probe a wide range of sizes of diffusing nanoparticles. The largest hydrodynamic radius (R(c)) of trapped particles that displayed local mobility was estimated to be 70 nm for a 1.5% agarose gel. The results showed that diffusion of particles in agarose gel is anomalous, with a diverging fractal dimension of diffusion when the large particles become entrapped in the pores of the gel. The latter situation occurs when the reduced size (R(A)/R(c)) of the diffusing particle, A, is >0.4. Variations of the fractal exponent of diffusion (d(w)) with the reduced particle size were in agreement with three-dimensional Monte Carlo simulations in porous media. Nonetheless, a systematic offset of d(w) was observed in real systems and was attributed to weak nonelastic interactions between the diffusing particles and polymer fibers, which was not considered in the Monte Carlo simulations.     

Diffusion of sucrose and dextran through agar gel membranes

Mass transfer limitations severely impede the performance of bioreactions involving large molecules by gel-entrapped microorganisms. This paper describes a quantitative investigation of such diffusional limitations in agar gel membranes. Sucrose and commercial dextran fractions with (weight-average) molecular weights ranging from 10,000 to 2,000,000 Da were used as standard diffusants. For all tested solutes but sucrose, the values of the agar/water partition coefficients highlighted steric hindrance at the entrance of the membrane pores. The effective diffusivity of sucrose in agar was similar to that in water. All dextran fractions, however, displayed restricted diffusion in the agar membranes. Their effective diffusivities were a decreasing function of the agar content of the gel membrane (0.5, 1.0, or 1.5% w/v). The effective diffusivity in a given membrane decreased as the molecular weight of the diffusing molecule increased. T500 (ucbar|M_w = 470,000 Da) and ucbar|M_w = 1,950,000 Da) fractions were unable to diffuse through 1.0 or 1.5% agar membranes. The diffusion data did not agree with the classical (Renkin) model for a hard sphere diffusing through a cylindrical pore. These results are discussed in terms of gel and diffusant characteristics.     

Single and Multicomponent Gas Phase Diffusion in a Porous Media: Modeling and Laboratory Measurements

Subsurface vapor migration of volatile chemicals may impact ambient and indoor air quality, increasing the importance to investigate the fate and transport of these chemicals. This project involved both modeling and experimental work to study the vapor phase transport behavior of single, binary, and tertiary component systems present in the gas phase. The experimental phase resulted in the development of a diffusion cell for measuring vapor phase transport. Three organic compounds (toluene, cumene, and isooctane) common to petroleum-based products were selected. The objective of this research project was to evaluate how the rate of a component diffusing alone in a stagnant gas mixture compares to the rate of the same component when diffusing in the presence of multiple diffusing species. The equipment was first validated by measuring the unobstructed gas phase diffusion fluxes for each organic compound. The diffusion coefficients were then calculated from the experimentally measured diffusive fluxes using Fick's Law at 20 and 25°C and compared to the respective literature values. The experimental/literature (E/L) ratio was calculated for toluene, cumene, and isooctane. The range of the average E/L ratio for the single component data sets is 0.93 to 1.05. The validation data provided the baseline for extending the research to multicomponent data. The multi-component systems research was characterized as either binary systems or a three-component system. The binary systems were either isooctane/tolu-ene or isooctane/cumene. The three-component system consisted of a mixture of all three compounds. For both temperatures and all compounds the flux rate decreased for any single component due to the dilution effect by incorporation into a mixture. Applying Fick's Law to calculate the effective diffusion coefficient for each compound that corresponded to the resulting concentration gradient by the mixture, an enhancement in the diffusive flux of each individual species was observed. This enhancement can be explained by a compositional coupling of each component to all others which results in a total vapor phase mass flux comprised of both diffusive and pseudo-advective mass transport. This pseudo-advective component is attributed to simultaneous diffusion of other species in the presence of the one of interest. Since this research project incorporated a mixture of toluene and cumene present in a background carrier solvent of isooctane, by calculating the ratios Dexp(3-component)/ Dexp(2-component) and Dexp(2-compo-nent)/Dexp(single component), an estimate is obtained of the enhancement effect due to the advective component of simultaneously diffusing chemicals. The diffusivity ratios for the three-component system compared to the dual component system ranged from 0.8 to 3.7. The diffusivity ratios for individual compounds were for 1.5-3.7 cumene, 0.8-1.2 for toluene, and 1.0-1.2 for isooctane. The diffusivity ratios for the dual component system to the single component systems ranged from 0.8 to 4.0. The range of diffusivity ratios for individual compounds were for 2.0-4.0 for cumene, 0.8-1.6 for toluene, and 1.1-1.4 for isooctane. A ratio greater than 1.0 indicated an enhancement effect on the molecular diffusion rate due to the presence of one or more additional diffusing chemical species present. The majority of fate and transport models are based on single component behavior modeled by Fick's Law using the pure gas phase diffusion coefficient. The enhancement of the individual diffusive flux in a multicomponent mixture observed in this study and accounted for by pseudo-advec-tive mass transport results in an under-prediction of the actual multicomponent diffusive fluxes. It is recommended that a more rigorous diffusion equation such as the Stefan-Maxwell equation be considered for incorporation into vapor phase transport models when modeling multicomponent/ contaminant systems.     

Diffusivity of oxygen in aerobic granules

This work for the first time estimated apparent oxygen diffusivity (D(app)) of two types of aerobic granules, acetate-fed and phenol-fed, by probing the dissolved oxygen (DO) level at the granule center with a sudden change in the DO of the bulk liquid. With a high enough flow velocity across the granule to minimize the effects of external mass transfer resistance, the diffusivity coefficients of the two types of granules were estimated with reference to a one-dimensional diffusion model. The carbon source has a considerable effect on the granule diameter (d) and the oxygen diffusivity. The diffusivity coefficients were noted 1.24-2.28 x 10(-9) m2/s of 1.28-2.50 mm acetate-fed granules, and 2.50-7.65 x 10(-10) m2/s of 0.42-0.78 mm phenol-fed granules. Oxygen diffusivity declined with decreasing granule diameter, in particular, the diffusivity of acetate-fed granules is proportional to the size, whereas the diffusivity of phenol-fed granules is proportional to the square of granule diameter. The existence of large pores in granule, evidenced by FISH-CLSM imaging, was proposed to correspond to the noted size-dependent oxygen diffusivity. The phenol-fed granules exhibited a higher excellular polymer (ECP) content than the acetate-fed granules, hence yielding a lower oxygen diffusivity.     

Restricted diffusion in biophysical systems: Theory

Theoretical results are presented on measurements of restricted diffusion in biophysical systems by the pulsed gradient spin echo nuclear magnetic resonance (PGSENMR) technique. A Fokker-Planck equation is developed to describe restricted diffusion, and it is shown that only two basic types of penetrable diffusion barriers exist, those in which the diffusing particles are partially excluded from the barrier region because of an increased free energy, and those in which the diffusing particles are not excluded but experience increased viscosity in the region. The Fokker-Planck equation is used to obtain expressions for the spin echo amplitude in PGSENMR experiments, and it is shown that for restricted diffusion the average diffusion coefficient measured in these experiments over short intervals is larger than that measured over long intervals. The possibility of distinguishing between the two types of barriers is considered. The experimental parameters required for intracellular restricted diffusion measurements are discussed, and it is shown that the interpretation of PGSENMR results in animal tissues should include the possibility of penetrable barriers rather than just the impenetrable barriers of previous PGSENMR calculations.