Hindered diffusion in agarose gels: test of effective medium model.

The diffusivities of uncharged macromolecules in gels (D) are typically lower than in free solution (D infinity), because of a combination of hydrodynamic and steric factors. To examine these factors, we measured D and D infinity for dilute solutions of several fluorescein-labeled macromolecules, using an image-based fluorescence recovery after photobleaching technique. Test macromolecules with Stokes-Einstein radii (rs) of 2.1-6.2 nm, including three globular proteins (bovine serum albumin, ovalbumin, lactalbumin) and four narrow fractions of Ficoll, were studied in agarose gels with agarose volume fractions (phi) of 0.038-0.073. The gels were characterized by measuring the hydraulic permeability of supported agarose membranes, allowing calculation of the Darcy permeability (kappa) for each gel sample. It was found that kappa, which is a measure of the intrinsic hydraulic conductance of the gel, decreased by an order of magnitude as phi was increased over the range indicated. The diffusivity ratio D/D infinity, which varied from 0.20 to 0.63, decreased with increases in rs or phi. Thus as expected, diffusional hindrances were the most severe for large macromolecules and/or relatively concentrated gels. According to a recently proposed theory for hindered diffusion through fibrous media, the diffusivity ratio is given by the product of a hydrodynamic factor (F) and a steric factor (S). The functional form is D/D infinity = F(rs/k1/2) S(f), where f = [(rs+rf)/rf]2 phi and rf is the fiber radius. Values of D/D infinity calculated from this effective medium theory, without use of adjustable parameters, were in much better agreement with the measured values than were predictions based on other approaches. The strengths and limitations of the effective medium theory for predicting diffusivities in gels are discussed.     

Structural mechanisms of acute VEGF effect on microvessel permeability

To investigate the ultrastructural mechanisms of acute microvessel hyperpermeability by vascular endothelial growth factor (VEGF), we combined a mathematical model (J Biomech Eng 116: 502-513, 1994) with experimental data of the effect of VEGF on microvessel hydraulic conductivity (L(p)) and permeability of various-sized solutes. We examined the effect of VEGF on microvessel permeability to a small solute (sodium fluorescein, Stokes radius 0.45 nm), an intermediate solute (alpha-lactalbumin, Stokes radius 2.01 nm), and a large solute [albumin (BSA), Stokes radius 3.5 nm]. Exposure to 1 nM VEGF transiently increased apparent permeability to 2.3, 3.3, and 6.2 times their baseline values for sodium fluorescein, alpha-lactalbumin, and BSA, respectively, within 30 s, and all returned to control within 2 min. On the basis of L(p) (DO Bates and FE Curry. Am J Physiol Heart Circ Physiol 271: H2520-H2528, 1996) and permeability data, the prediction from the model suggested that the most likely structural changes in the interendothelial cleft induced by VEGF would be a approximately 2.5-fold increase in its opening width and partial degradation of the surface glycocalyx.     

Analysis of the introduction and removal of glycerol in rabbit kidneys using a Krogh cylinder model

In 1982, Rubinsky and Cravalho described a Krogh cylinder model for the analysis of cryoprotectant transport in a perfused organ. By application of the Kedem-Katchalsky equations, changes in tissue volume caused by movements of water and solute were used to predict changes in capillary radius (Cryobiology 19, 70-82, 1982). We have now measured the changes in vascular resistance that are produced when sucrose or glycerol is introduced into the perfusate flowing through rabbit kidneys at 10 degrees C, and have analyzed these data by means of the Rubinsky-Cravalho semiempirical model. The sucrose data provided an estimate of hydraulic conductivity and the dimensions of the Krogh tissue units. Three rates of addition of glycerol, 10, 30, and 90 mM/min to a final concentration of 3 M, were studied. The vascular resistance fell to approximately 40% of its initial value (radius approximately 128% of initial value) with all three rates of addition, and then returned toward its normal value while the glycerol concentration was still increasing. This behavior could be explained either by a sudden change in solute permeability at that capillary radius, or by an inverse dependence of reflection coefficient upon solute concentration. Evidence is presented that favors the latter interpretation. The best fits for the apparent hydraulic conductivity and apparent solute permeability for glycerol are 1 X 10(-6) cm/sec atm and 6 X 10(-8) cm/sec, respectively, with the reflection coefficient falling from 1.0 when the glycerol concentration is zero to 0.1 when it is 3 M. The model is used to predict tissue concentrations of glycerol throughout each experiment.     

Partitioning and diffusion of large molecules in fibrous structures

This paper consists of three parts. In the first part we present diffusion and partition coefficients of proteins in agarose that were measured with gel permeation chromatography. In the second part we present a partition model which includes the effects of solute concentration and of cosolutes. In the third part we modify and extend Ogston’s diffusion equation to account for the effects of solute and fiber flexibility, solute concentration and cosolutes. We find good agreement between the proposed models and data from the literature.     

Effect of protein adsorption on the transport characteristics of asymmetric ultrafiltration membranes.

The effects of bovine serum albumin adsorption on the transport characteristics of asymmetric poly(ether sulfone) ultrafiltration membranes were determined using polydisperse dextrans with gel permeation chromatography. Actual dextran sieving coefficients were evaluated from observed sieving data for both the clean and preadsorbed membranes using a stagnant film model. The flux dependence of the actual dextran sieving coefficients was used to evaluate the intrinsic membrane hindrance factors for convective (i.e., sieving) and diffusive transport for the different molecular weight dextrans using classical membrane transport theory. Protein adsorption caused a reduction in both dextran sieving and diffusion, with the magnitude of the reduction a function of the dextran molecular weight and pore size. The effects of adsorption on the specific pore area and the membrane porosity were then determined using a recent model for solute transport through asymmetric ultrafiltration membranes. The data indicate that protein adsorption occurs preferentially in the larger membrane pores, causing a greater reduction in solute sieving compared to the membrane hydraulic permeability and porosity than would be predicted on the basis of either a simple pore blockage or pore constriction model.     

Application of a fiber-matrix model to transport in renal tubules

The effects of tight junction structure on water and solute fluxes across proximal tubular epithelium were examined with fiber-matrix equations previously derived by Curry and Michel (1980. Microvascular Research. 20:96-99). Using plausible estimates of tight junction fiber length and width the model predicts solute (Ps) and water permeability (Lp) coefficients that agree with the measured values. When fiber-matrix and pore models were compared for physiologically relevant ranges of matrix void fraction (80-98%) and pore radii (0-20 A), the fiber-matrix model predicted a 10-fold higher Lp/Ps ratio. Lp/Ps was most sensitive to small changes in tight junction structure when void fractions exceeded 90%. Void fractions of 96.5% and 97.1% predicted previously measured values for Lp and solute permeabilities in rat and rabbit proximal tubules. These values are consistent with void fractions and permeabilities of artificial membranes. The fiber-matrix tight junction model was incorporated into a model of reabsorption from the rat proximal tubule developed by Weinstein (1984). American Journal of Physiology. 247:F848-F862.) A void fraction of 98% predicted the experimental results for isosmotic reabsorption driven by active transport. Changing void fraction over the range of 97-99% produced a 50-75% change in predicted volume reabsorption with active transport. According to the fiber-matrix model: (a) solute permeabilities alone cannot be used to predict Lp, (b) previously measured solute permeabilities in the proximal tubule are compatible with significant water reabsorption through a water-permeable tight junction, and (c) hydraulic and solute permeabilities may be sensitive to small changes in tight junction fiber length and diameter or ionic strength within the tight junction.     

A model relating root permeability to flux and potentials: application to existing data from soybean and other plants

A model that relates hydraulic permeability to water flux and to gradients in pressure potential and solute potential was tested using soybean (Glycine max) plants. Water flux was varied by additions of polyethylene glycol 6,000 around one portion of a divided root system and by changing the light intensity and CO₂ concentration around the plants. The data are compatible with the model only if the hydraulic permeability varies with flux; however, the data were insufficient for rigorous testing. Three sets of published data fit the model only if hydraulic permeability varies. Evidence originally presented as involving constant hydraulic permeability is shown, rather, to require variable hydraulic permeability.     

Nonelectrolyte diffusion across lipid bilayer systems

The permeability coefficients of a homologous series of amides from formamide through valeramide have been measured in spherical bilayers prepared by the method described by Jung. They do not depend directly on the water:ether partition coefficient which increases regularly with chain length. Instead there is a minimum at acetamide. This has been ascribed to the effect of steric hindrance on diffusion within the bilayer which increases with solute molar volume. This factor is of the same magnitude, though opposite in sign to the effect of lipid solubility, thus accounting for the minimum. The resistance to passage across the interface has been compared to the resistance to diffusion within the membrane. As the solute chain length increases the interface becomes more important, until for valeramide it comprises about 90% of the total resistance. Interface resistance is also important in urea permeation, causing urea to permeate much more slowly than an amide of comparable size, after allowance is made for the difference in the water:ether partition coefficient. Amide permeation coefficients have been compared with relative liposome permeation data measured by the rate of liposome swelling. The ratios of the two measures of permeation vary between 3 and 16 for the homologous amides. The apparent enthalpy of liposome permeation has been measured and found to be in the neighborhood of 12 kcal mol-1 essentially independent of chain length. Comparison of the bilayer permeability coefficients with those of red cells shows that red cell permeation by the lipophilic solutes resembles that of the bilayers, whereas permeation by the hydrophilic solutes differs significantly.     

A mixture theory analysis for passive transport in osmotic loading of cells

The theory of mixtures is applied to the analysis of the passive response of cells to osmotic loading with neutrally charged solutes. The formulation, which is derived for multiple solute species, incorporates partition coefficients for the solutes in the cytoplasm relative to the external solution, and accounts for cell membrane tension. The mixture formulation provides an explicit dependence of the hydraulic conductivity of the cell membrane on the concentration of permeating solutes. The resulting equations are shown to reduce to the classical equations of Kedem and Katchalsky in the limit when the membrane tension is equal to zero and the solute partition coefficient in the cytoplasm is equal to unity. Numerical simulations demonstrate that the concentration-dependence of the hydraulic conductivity is not negligible; the volume response to osmotic loading is very sensitive to the partition coefficient of the solute in the cytoplasm, which controls the magnitude of cell volume recovery; and the volume response is sensitive to the magnitude of cell membrane tension. Deviations of the Boyle-van't Hoff response from a straight line under hypo-osmotic loading may be indicative of cell membrane tension.     

Permeation of protons, potassium ions, and small polar molecules through phospholipid bilayers as a function of membrane thickness.

Two mechanisms have been proposed to account for solute permeation of lipid bilayers. Partitioning into the hydrophobic phase of the bilayer, followed by diffusion, is accepted by many for the permeation of water and other small neutral solutes, but transient pores have also been proposed to account for both water and ionic solute permeation. These two mechanisms make distinctively different predictions about the permeability coefficient as a function of bilayer thickness. Whereas the solubility-diffusion mechanism predicts only a modest variation related to bilayer thickness, the pore model predicts an exponential relationship. To test these models, we measured the permeability of phospholipid bilayers to protons, potassium ions, water, urea, and glycerol. Bilayers were prepared as liposomes, and thickness was varied systematically by using unsaturated lipids with chain lengths ranging from 14 to 24 carbon atoms. The permeability coefficient of water and neutral polar solutes displayed a modest dependence on bilayer thickness, with an approximately linear fivefold decrease as the carbon number varied from 14 to 24 atoms. In contrast, the permeability to protons and potassium ions decreased sharply by two orders of magnitude between 14 and 18 carbon atoms, and leveled off, when the chain length was further extended to 24 carbon atoms. The results for water and the neutral permeating solutes are best explained by the solubility-diffusion mechanism. The results for protons and potassium ions in shorter-chain lipids are consistent with the transient pore model, but better fit the theoretical line predicted by the solubility-diffusion model at longer chain lengths.     

On the Distribution of a Permeable Solute during Poiseuille Flow in Capillary Tubes

Equations are derived describing the dispersion of a permeable solute during Poiseuille flow in a capillary model. It is shown that for the normal range of physiological parameters such as capillary radius, capillary length, blood flow, permeability coefficients, and diffusion constants, the center of mass of a bolus of solute moves at a speed very close to the mean speed of flow and that the solute leaves the capillary with an exponential time course depending on the permeability but not on the diffusion constant. There is no appreciable difference in the dispersion of the solute or in its rate of permeation from the capillary whether one considers piston flow or Poiseuille flow. A bolus of arbitrary radial shape tends to become radially uniform very close to the arterial end of the capillary.     

OZONE-INDUCED MEMBRANE PERMEABILITY CHANGES

The effect of 0.5 ppm ozone for 0.5-1 hr on plant cell membrane permeability was ascertained. Permeabilities to both water and solutes were estimated by measuring leaf disc weight changes and following tritiated water and ⁸⁶Rb fluxes. Measurements were made immediately after ozone exposure and 24 hr after exposure. The reflection coefficient, σ, an index of solute permeability, decreased in ozone-treated primary leaves of pinto bean (Phaseolus vulgaris). The latter indicates an increase in membrane solute permeability or internal solute leakage. Water and THO flux estimates both indicated a decrease in membrane permeability to water; both the hydraulic conductivity (L_p) and the water diffusional coefficient (L_D) apparently decreased, an anomaly which is discussed. These data indicate that ozone has a direct effect on membrane function by altering permeability characteristics. We assume from these data that cell membranes are primary target sites for ozone injury.     

Hindered convection of macromolecules in hydrogels

Hindered convection of macromolecules in gels was studied by measuring the sieving coefficient (theta) of narrow fractions of Ficoll (Stokes-Einstein radius, r(s) = 2.7-5.9 nm) in agarose and agarose-dextran membranes, along with the Darcy permeability (kappa). To provide a wide range of kappa, varying amounts of dextran (volume fractions < or = 0.011) were covalently attached to agarose gels with volume fractions of 0.040 or 0.080. As expected, theta decreased with increasing r(s) or with increasing concentrations of either agarose or dextran. For each molecular size, theta plotted as a function of kappa fell on a single curve for all gel compositions studied. The dependence of theta on kappa and r(s) was predicted well by a hydrodynamic theory based on flow normal to the axes of equally spaced, parallel fibers. Values of the convective hindrance factor (K(c), the ratio of solute to fluid velocity), calculated from Theta and previous equilibrium partitioning data, were unexpectedly large; although K(c) < or = 1.1 in the fiber theory, its apparent value ranged generally from 1.5 to 3. This seemingly anomalous result was explained on the basis of membrane heterogeneity. Convective hindrances in the synthetic gels were quite similar to those in glomerular basement membrane, when compared on the basis of similar solid volume fractions and values of kappa. Overall, the results suggest that convective hindrances can be predicted fairly well from a knowledge of kappa, even in synthetic or biological gels of complex composition.     

Determination of Solute Permeability in Chara Internodes by a Turgor Minimum Method : Effects of External pH

The analysis of Sha'afi et al. (Sha'afi, Rich, Mickulecky, Solomon 1970 J Gen Physiol 55: 427-450) for determining solute permeability in red blood cells has been modified and applied to turgid plant cells. Following the addition of permeating solute to the external medium, a biphasic response of cell turgor can be measured with the pressure probe in isolated internodes of Chara corallina. After an initial decrease in turgor due to water flow (water phase), turgor increases due to the uptake of the solute (solute phase) until the original turgor is reattained. From the pressure/time course in the neighborhood of the minimum turgor, the permeability of the osmotic solute can be determined. The data obtained by the minimum method for rapidly permeating (ethanol, methanol) and slowly permeating (formamide, dimethylformamide) solutes are similar to those calculated from the half-time of pressure changes during the solute phase and to data obtained by other workers using radioactive tracers. The methods employing the pressure probe were applied to examine the effect of high pH (up to pH 11) on the membrane permeability. There appeared to be no effect of high pH on the permeability coefficients, reflection coefficients, and hydraulic conductivity.     

Transport of water and solutes across bullfrog alveolar epithelium

Water and solute transport properties of the alveolar epithelium of isolated bullfrog lungs were studied. Lungs from Rana catesbeiana were removed and mounted in an Ussing chamber. Unstirred layers on both sides of the tissue were estimated from the time courses of dilution potential development, and the measured transport parameters were corrected for the effect of the unstirred layers. Spontaneous potential difference, short-circuit current, tissue resistance, instantaneous voltage-current relationships, diffusional permeabilities of water and hydrophilic solutes, and hydraulic conductivities were determined. The hydraulic conductivity obtained from hydrostatically driven water flow anomalously decreased with time, and was initially 100 -1,000 times higher than osmotically determined hydraulic conductivity. The equivalent pore radius of the bullfrog alveolar epithelium was estimated to be 0.8-0.9 nm. We conclude that the alveolar epithelium is extremely tight, presenting a major barrier to water and solute flow. This high resistance to water and solute flow may be helpful in maintaining the alveolar lumen relatively free of fluid under normal physiological conditions.     

Equilibrium gel permeation: a single-photon counting spectrophotometer for studies of protein interaction.

When a small column or flow cell packed with gel particles is completely saturated with a solution containing molecular species of interest, the average cross-sectional area occupied by the solute (partition cross section) is conveniently and precisely determined by direct optical scanning. For a mixture of interacting solutes this equilibrium gel permeation measurement yields the weight average of the species partition cross sections and the variation of this quantity with solute concentration permits determination of the solute interaction parameters (stoichiometry, equilibrium constants). We have developed a computer-controlled single-photon counting spectrophotometer for these measurements. The instrument exhibits high precision over a wide range of optical density. With counting times in the range of 10-1000 s the standard deviations on optical densities of protein solutions measured at 220 nm are typically 0.0006 at 1 OD, 0.002 at 2 OD, 0.005 at 4 OD. Beer's law tests show that deviations from linearity are less than these precision limits. Partition cross-section measurements for proteins can be made with an accuracy of better than 0.001 and information can be obtained with protein solutions at least as low as 1 mug/ml.     

Transport parameters in the human red cell membrane: solute-membrane interactions of amides and ureas

We have studied the permeability of a series of hydrophilic amides and ureas through the red cell membrane by determining the three phenomenological coefficients which describe solute-membrane interaction: the hydraulic permeability (Lp), the phenomenological permeability coefficient (omega i) and the reflection coefficient (sigma i). In 55 experiments on nine solutes, we have determined that the reflection coefficient (after a small correction for solute permeation by membrane dissolution) is significantly less than 1.0 (P less than 0.003, t-test), which provides very strong evidence that solute and water fluxes are coupled as they cross the red cell membrane. It is proposed that the aqueous channel is a tripartite assembly, comprising H-bond exchange regions at both faces of the membrane, joined by a narrower sieve-specific region which crosses the lipid. The solutes bind to the H-bond exchange regions to exchange their solvation shell with the H-bonds of the channel; the existence of these regions is confirmed by the finding that the permeation of all the amides and ureas requires binding to well-characterized sites with Km values of 0.1-0.5 M. The sieve-specific regions provide the steric restraints which govern the passage of the solutes according to their size; their existence is shown by the findings that: (1) the reflection coefficient (actually the function [1-corrected sigma i]) is linearly dependent upon the solute molecular diameter; and (2) the permeability coefficient is linearly dependent upon solute molar volume. These several observations, taken together, provide strong arguments which lead to the conclusion that the amides and urea cross the red cell membrane in an aqueous pore.     

Explaining glomerular pores with fiber matrices. A visualization study based on computer modeling.

The extracellular space of the glomerular capillary wall is occupied by a complex meshwork of fibrous molecules. Little is understood about how the size, shape, and charge recognition properties of glomerular ultrafiltration arise from this space-filling fiber matrix. We studied the problem of size recognition by visualizing the void volume accessible to hard spheres in computer-generated three-dimensional homogeneous random fiber matrices. The spatial organization of the void volume followed a complex "blob-and-throat" pattern in which circumscribed cavities of free space within the matrix ("blobs") were joined to adjacent cavities by narrower throats of void space. For sufficiently small solutes, chains of blobs and throats traversed the matrix, providing pathways for trans-matrix permeation. The matrices showed threshold or gating properties with respect to permeation: solutes whose radius exceeded a critical value, at which a throat on the last connected trans-matrix pathway pinched off, could not cross, whereas smaller solutes had nonzero permeability. The thresholds may give the glomerular fiber matrix porelike response properties and explain why pore models have been such a useful means of treating permselectivity.     

Molecular-sieve chromatography and electrophoresis in polyacrylamide gels

1. The absolute electrophoretic mobilities of eight proteins have been measured at pH8.76, I 0.05, in polyacrylamide gels of 20 different compositions at 10 degrees C. 2. The partition coefficients of these proteins have been determined chromatographically under the same conditions by using columns of granulated polyacrylamide gel prepared simultaneously. 3. The electrophoretic mobilities are an exponential function of the gel concentrations when the latter are corrected for water uptake. The constants of this function have been determined by curvefitting methods. They have been shown to be related to the free solution mobility and to the mean molecular radius respectively. 4. The reduced mobilities have been shown to be a linear function of the partition coefficients by statistical analyses. 5. The physical significance of the relation between electrophoretic mobility and chromatographic phase distribution in gel media is discussed in the context of these results.