The effect of auxiliary conditions on intestinal unstirred layer diffusion modelled by numerical simulation.

Estimation of intestinal unstirred layer thickness usually involves inducing transmural potential difference changes by altering the content of the solution used to perfuse the small intestine. Osmotically active solutes, such as mannitol, when added to the luminal solution diffuse across the unstirred water layer (UWL) and induce osmotically dependent changes in potential difference. As an alternative procedure, the sodium ion in the luminal fluid can be replaced by another ion. As the sodium ion diffuses out of the UWL, the change in concentration next to the intestinal membrane alters the transmural potential difference. In both cases, UWL thickness is calculated from the time course of the potential difference changes, using a solution to the diffusion equation. The diffusion equation solution which allows the calculation of intestinal unstirred layer thickness was examined by simulation, using the method of numerical solutions. This process readily allows examination of the time course of diffusion under various imposed circumstances. The existing model for diffusion across the unstirred layer is based on auxiliary conditions which are unlikely to be fulfilled in the same intestine. The present simulation additionally incorporated the effects of membrane permeability, fluid absorption and less than instantaneous bulk phase concentration change. Simulation indicated that changes within the physiologically relevant range in the chosen auxiliary conditions (with the real unstirred layer length kept constant) can alter estimates of the apparent half-time. Consequently, changes in parameters unassociated with the unstirred layer would be misconstrued as alterations in unstirred layer thickness.     

Effect of unstirred layers on binding and reaction kinetics at a membrane surface

The effects of solution unstirred layers on the time course of chemical reactions and transport processes at a membrane surface are determined. A set of equations which describes non-steady-state diffusion through an unstirred layer coupled with chemical reaction at a membrane surface or transport through a membrane is developed. A numerical solution to the equations is obtained by uncoupling diffusive and chemical processes in an iterative manner. The diffusive process is solved by the Crank-Nicolson method; the chemical process is solved by integrating the differential equations describing the kinetics. Diffusive processes in one dimension, in three dimensions, and in the presence of an arbitrary potential near the membrane surface are solved. General characteristics of the calculated reaction time course are discussed using surface binding and membrane transport examples. Small, neglected, unstirred layers are shown to sometimes yield erroneous values of rate parameters for a surface reaction and to simulate competitive reaction kinetics. Experimental approaches for measuring unstirred layer thickness are reviewed.     

Donnan potential and surface potential of a charged membrane.

A model is presented for the electrical potential distribution across a charged biological membrane that is in equilibrium with an electrolyte solution. We assume that a membrane has charged surface layers of thickness d on both surfaces of the membrane, where the fixed charges are distributed at a uniform density N within the layers, and that these charged layers are permeable to electrolyte ions. This model unites two different concepts, that is, the Donnan potential and the surface potential (or the Gouy-Chapman double-layer potential). Namely, the present model leads to the Donnan potential when d much greater than 1/k' (k' is the Debye-Hückel parameter of the surface charge layer) and to the surface potential as d----0, keeping the product Nd constant. The potential distribution depends significantly on the thickness d of the surface charge layer when d less than or approximately equal to 1/k'.     

A method of calculating the liposome membrane permeability using dialysis method

A correct mode of calculating liposome membrane permeability determined with dialysis is proposed. The liposome membrane permeability is calculated with regard for the ion passage through two diffusion barriers: liposomal membrane and cellophane membrane. The asymmetrical ion distribution under equilibrium conditions is shown. The asymmetry is due to the formation of unstirred layers near the membrane. The equilibrium ion concentration in unstirred layers and the measured average bulk concentration in solution are different. The formula for calculating liposome membrane permeability that takes the mentioned factors into account is suggested.     

Biological and artificial ion exchangers: Electrical measurements with glass microelectrodes

Summary Biological (stratum corneum) and artificial (cation-exchange resin beads, Bio-Rad AG 50W-X2) ion exchangers were impaled by glass microelectrodes filled with KCl solution. The electrical potential difference recorded in these structures in reference to the external bathing medium was shown to be dependent on the KCl concentration of both the external and the microelectrode filling solutions. The potentials were interpreted on the grounds of the fixed charge theory of membrane potentials as a consequence of two phase boundary potentials (Donnan potentials), one at the matrix-external solution interface and the other at the matrix-microelectrode solution interface. The contribution of a diffusion component for the recorded potential was considered.     

Electrical excitability of artificial enzyme membranes. IV. Theoretical approach of the membrane potential of synthetic proteinic films

This paper deals with the theoretical approach of the membrane potential of artificial proteinic film. Programming techniques using finite difference simulations for the steady state and transient solutions of the Nernst-Planck and Poisson equations were used and solved by the collocation and corrector methods. This approach allows one to calculate the membrane potential without any discontinuity between the Donnan and the diffusion potentials, the thickness of the boundary layers being automatically determined by the intrinsic properties of the solution and of the membrane. The theoretical results are compared with experimental potentials measured on proteinic artificial films.     

Measurements of local pH changes near bilayer lipid membrane by means of a pH microelectrode and a protonophore-dependent membrane potential. Comparison of the methods.

Shifts of pH near the bilayer lipid membrane (BLM) were measured in the absence of pH difference between bulk solutions by two methods, i.e. pH microelectrode and membrane potential recordings in the presence of a protonophore. A quantitative agreement of the results of both methods was obtained. The kinetics of the generation of potential induced by the addition of ammonium chloride was accounted for by the time of the diffusion through the unstirred layers. The thickness of the unstirred BLM layers was determined in the experiment.     

Effects of unstirred layers or transport number discontinuities on the transient and steady-state current-voltage relationships of membranes

The effects of current-induced electrolyte accumulation and depletion on the electrical properties of a two-layered membrane system have been examined. The membrane consisted of a charged, ion permselective layer and an uncharged, non-selective layer. The model was designed to reveal the properties of membranes possessing long pores with ionic charges at one end or of ion-selective membranes bounded by highly unstirred aqueous layers. Electrolyte concentration profiles in the inert layer and their time-dependent changes were obtained from solutions of the diffusion equation under the condition of constant current. The profiles were then used to calculate the voltage developed across the membrane at various times after the current is switched on. The theoretical results are presented in the form of $\text{i-V}$ curves with reduced coordinates that can be used to obtain time-current-voltage relationships for membranes of the type considered having any thickness of the non-selective layer and bathed in any concentration of any 1:1 electrolyte. Experimental results on a model composite membrane were in good agreement with calculations that assume that ion transport occurs only under the influence of electrical potential and concentration gradients, suggesting that in such systems, the combined effects of convection, osmosis, electro-osmosis, and concentration-dependence of diffusion coefficients, activity coefficients, and transference numbers are small. Voltage fluctuations in the form of periodic spikes were observed experimentally at the limiting current density (the current density at which the electrolyte concentration at one surface of the selective layer goes to 0). These phenomena were not seen when the current was in the direction leading to accumulation of electrolyte in the non-selective (unstirred) layer. Such composite membranes can exhibit S-shaped and N-shaped $\text{i-V}$ curves under ramp-voltage and ramp-current clamps, respectively.     

Membrane potential and Donnan potential

The Nernst-Planck-Poisson equations for the potential profile across a membrane are exactly solved without recourse to the assumption of constant field within the membrane. It is assumed that the membrane core of thickness dc is covered by a surface layer of thickness ds in which the membrane-fixed charges are distributed at a uniform density N. The membrane boundary potentials as well as the diffusion potentials contribute to the membrane potential. It is shown that for ds greater or similar 1/k, k being the Debye-Hückel parameter, the potential in the membrane surface layer except in the region very near the membrane/solution boundary is effectively equal to the Donnan potential and that its contribution to the membrane potential becomes dominant as N increases. For low N, on the other hand, the membrane potential arises mostly from the diffusion potential.     

Potassium ion accumulation at the external surface of the nodal membrane in frog myelinated fibers.

Potassium accumulation associated with outward membrane potassium current was investigated experimentally in myelinated fibers and analyzed in terms of two models-three-compartment and diffusion in an unstirred layer. In the myelinated fibers, as in squid giant axons, the three-compartment model satisfactorily describes potassium accumulation. Within this framework the average space thickness, theta, in frog was 5,900 +/- 700 A, while the permeability coefficient of the external barrier, PK, was (1.5 +/- 0.1) X 10(-2) cm/s. The model of ionic diffusion in an unstirred aqueous layer adjacent to the axolemma, as an alternative explanation for ion accumulation, was also consistent with the experimental data, provided that D, the diffusion constant, was (1.8 +/- 0.2) X 10(-6) cm/s and l, the unstirred layer thickness, was 1.4 +/- 0.1 micron, i.e., similar to the depth of the nodal gap. An empirical equation relating the extent of potassium accumulation to the amplitude and duration of depolarization is given.     

Diffusion from a Circular Stoma through a Boundary Layer: A FIELD-THEORETICAL ANALYSIS

The case of diffusion of a gas from a single circular stoma through an unstirred boundary layer of finite thickness into a perfectly stirred atmosphere free of convective effects is examined theoretically, with the gas assumed to be at constant concentration across the stoma. The analysis employs a mathematical solution to an analogous problem in electrostatic physics previously obtained by Kuz'min (1972 Sov Phys Tech Phys 17: 473-476). The diffusion flux is shown to be no more than 1% greater than that into a perfectly unstirred atmosphere if the boundary layer is thicker than 40 times the stomatal radius. Under the conditions assumed, for realistic boundary-layer and stomatal dimensions, taking the diffusion flux through the boundary layer to be linear with the stomatal radius would usually involve no significant error. This result may indicate that the principal effect of wind velocity on mass exchange between leaf and atmosphere may be exerted through influencing convection outside the boundary layer rather than through determining the thickness of that layer.     

The use of phospholipid-impregnated millipore filters for recording nonelectrogenic cation flows in the presence of Men+/nH+ exchangers

It has been shown that with a cation (K+, Na+, Ca2+) concentration gradient on a Millipore filter impregnated with a decane solution of phospholipid, in the presence of a Men+/nH+ exchanger (nigericin, monensin, A23187), addition of a protonophore induces the formation of an electric potential positively charged on the side where the concentration of the cation is lower. The formation of the potential is induced by the hydrogen ion concentration gradient in the filter and in the unstirred layers as a result of the Men+/nH+ exchange. In such a system, with a pH gradient on the filter in the presence of monensin and valinomycin, a potential is generated with the plus on the side of the lower concentration of hydrogen. The effect is the result of the formation of a potassium ion concentration gradient in the unstirred layers in the course of the K+/H+ exchange. It is concluded that phospholipid-impregnated filters can be used for search and identification of electroneutral membrane ionophores of the Men+/nH+ exchanger type.     

Electrostatic interaction between ion-penetrable multi-layered membranes

A theory of the double layer interaction regulated by the Donnan potential between two ion-penetrable membranes in an electrolyte solution developed previously by Ohshima and Kondo is extended to the case in which the membranes consist of many layers having different thickness and densities of membrane-fixed charges. The interaction force is found to be determined mainly by the contributions from layers located within the depth of 1/kappa (kappa, Debye-Hückel parameter) from the membrane surface. It is also predicted that the interaction force may alter its sign with changing electrolyte concentration.     

Diffusion of charged ions in mucus gel: effect of net charge

The interposition of a neutral starch gel greatly retarded bulk ionic movement by free flow. A mucus (charged) gel preparation of identical concentration and thickness further retarded ionic diffusion. The findings suggest that the charges in the mucus matrix may exert an ionic exclusion effect (Donnan Exclusion), thus retarding other ionic diffusion. We speculate that a mucus layer may physiologically behave as an ion exchange gel column and modify the traffic of charged ions through it.     

Gallbladder epithelial cell hydraulic water permeability and volume regulation

The hydraulic water permeability (Lp) of the cell membranes of Necturus gallbladder epithelial cells was estimated from the rate of change of cell volume after a change in the osmolality of the bathing solution. Cell volume was calculated from computer reconstruction of light microscopic images of epithelial cells obtained by the "optical slice" technique. The tissue was mounted in a miniature Ussing chamber designed to achieve optimal optical properties, rapid bath exchange, and negligible unstirred layer thickness. The control solution contained only 80% of the normal NaCl concentration, the remainder of the osmolality was made up by mannitol, a condition that did not significantly decrease the fluid absorption rate in gallbladder sac preparations. The osmotic gradient ranged from 11.5 to 41 mosmol and was achieved by the addition or removal of mannitol from the perfusion solutions. The Lp of the apical membrane of the cell was 1.0 X 10(-3) cm/s . osmol (Posm = 0.055 cm/s) and that of the basolateral membrane was 2.2 X 10(-3) cm/s . osmol (Posm = 0.12 cm/s). These values were sufficiently high so that normal fluid absorption by Necturus gallbladder could be accomplished by a 2.4-mosmol solute gradient across the apical membrane and a 1.1-mosmol gradient across the basolateral membrane. After the initial cell shrinkage or swelling resulting from the anisotonic mucosal or serosal medium, cell volume returned rapidly toward the control value despite the fact that one bathing solution remained anisotonic. This volume regulatory response was not influenced by serosal ouabain or reduction of bath NaCl concentration to 10 mM. Complete removal of mucosal perfusate NaCl abolished volume regulation after cell shrinkage. Estimates were also made of the reflection coefficient for NaCl and urea at the apical cell membrane and of the velocity of water flow across the cytoplasm.     

Osmosis in Cortical Collecting Tubules : A Theoretical and Experimental Analysis of the Osmotic Transient Phenomenon

This paper reports a theoretical analysis of osmotic transients and an experimental evaluation both of rapid time resolution of lumen to bath osmosis and of bidirectional steady-state osmosis in isolated rabbit cortical collecting tubules exposed to antidiuretic hormone (ADH). For the case of a membrane in series with unstirred layers, there may be considerable differences between initial and steady-state osmotic flows (i.e., the osmotic transient phenomenon), because the solute concentrations at the interfaces between membrane and unstirred layers may vary with time. A numerical solution of the equation of continuity provided a means for computing these time-dependent values, and, accordingly, the variation of osmotic flow with time for a given set of parameters including: P_f (cm s^–1), the osmotic water permeability coefficient, the bulk phase solute concentrations, the unstirred layer thickness on either side of the membrane, and the fractional areas available for volume flow in the unstirred layers. The analyses provide a quantitative frame of reference for evaluating osmotic transients observed in epithelia in series with asymmetrical unstirred layers and indicate that, for such epithelia, P_f determinations from steady-state osmotic flows may result in gross underestimates of osmotic water permeability. In earlier studies, we suggested that the discrepancy between the ADH-dependent values of P_f and P_DDw (cm s^–1, diffusional water permeability coefficient) was the consequence of cellular constraints to diffusion. In the present experiments, no transients were detectable 20–30 s after initiating ADH-dependent lumen to bath osmosis; and steady-state ADH-dependent osmotic flows from bath to lumen and lumen to bath were linear and symmetrical. An evaluation of these data in terms of the analytical model indicates: First, cellular constraints to diffusion in cortical collecting tubules could be rationalized in terms of a 25-fold reduction in the area of the cell layer available for water transport, possibly due in part to transcellular shunting of osmotic flow; and second, such cellular constraints resulted in relatively small, approximately 15%, underestimates of P_f.     

Ventilation and Its Effect on "Infinite Pool" Exchangers

SYNOPSIS. Unlike internal exchange surfaces, the skin contactsan "infinite pool" of air or water with which exchange of gases,water, ions, and other solutes may occur. Even though the "infinitepool" may be well mixed, an unstirred diffusion boundary layeris always present about the skin and may constitute a significantresistance to exchange. The thickness of the diffusion boundarylayer (as approximated by the fluid dynamic boundary layer)is related to the flow of the respiratory medium, viscosityand density of the medium, and the morphology of the exchangesurface. Oxygen microelectrode studies suggest that, in mostcircumstances, the diffusion boundary layer in water is at leastas thick as the blood-respiratory medium distance in amphibianskin. Accordingly, the movement of water about the skin {i.e.,skin ventilation) should have pronounced effects on cutaneousexchange, especially at low "free stream" velocities. Mountingphysiological evidence suggests that: (1) skin ventilation canaugment cutaneous gas exchange; and (2) some vertebrates activelyventilate their skins, especially in aquatic hypoxia. The ubiquityand significance of diffusion boundary layers are central toa general understanding of cutaneous exchange and all surface-mediatedexchange processes.     

Weak acid transport across bilayer lipid membrane in the presence of buffers. Theoretical and experimental pH profiles in the unstirred layers.

This paper presents a simple model to describe experimental data on weak acid transport across planar bilayer lipid membrane separating two buffered solutions. The model takes into account multiple proton-transfer reactions occurring in the unstirred layers (ULs) adjacent to the membrane. Differential equations of the model are shown to be reduced to a set of nonlinear algebraic equations. Since the latter equations depend monotonically on unknown variables, they can be easily solved numerically, using bisection method. For the particular system studied experimentally (with acetate as the weak acid and TRIS+MES as the buffer mixture) pH profiles in the ULs are calculated from the model. These results are compared with experimental data obtained using pH microelectrode. The agreement between theoretical and experimental pH profiles is found to be satisfactory. The most pronounced deviations are observed at the UL/bulk solution boundary. To obtain a better correlation between the theoretical and experimental results, two other, less idealized models are considered. They take into account, respectively, (a) the electric field arising in the ULs from ion diffusion and (b) finiteness of the rates of proton-transfer reactions. However, both acetate membrane fluxes and pH profiles in the ULs computed from these models are found to be close to those of the simple model. One can thus conclude that the difference between experimental and theoretical pH profiles is due to the inconsistency of the generally accepted model of the "unstirred layer", assuming the existence of a strict boundary between the regions of "pure diffusion" and "ideal stirring".     

23Na-nuclear magnetic resonance study of ionophore-mediated cation exchange between two populations of liposomes.

A model system to observe and investigate the transfer of Na+ ions between different internal compartments in suspension of liposomes was developed, and the exchange was followed by nuclear magnetic resonance spectroscopy. The experiments were performed under conditions of a Donnan equilibrium. Quantitative analysis of this three-site transmembrane exchange system allowed us to distinguish between direct and indirect exchange between liposomes. It also disclosed a "confining" effect on the exchange between the two populations of liposomes. This confining effect may have been due to an electrostatic field in the presence of a membrane potential. Donnan potentials and ionic compositions at equilibrium for the three-compartment system were calculated numerically. The model system may be used to explore further the effects of membrane potentials, surface potentials, and ionic mobilities on ion transport in biological (model) systems in general.     

Diffusion of carbon dioxide through lipid bilayer membranes. Effects of carbonic anhydrase, bicarbonate, and unstirred layers

Diffusion of (14)C-labeled CO(2) was measured through lipid bilayer membranes composed of egg lecithin and cholesterol (1:1 mol ratio) dissolved in n-decane. The results indicate that CO(2), but not HCO(3-), crosses the membrane and that different steps in the transport process are rate limiting under different conditions. In one series of experiments we studied one-way fluxes between identical solutions at constant pCO(2) but differing [HCO(3-)] and pH. In the absence of carbonic anhydrase (CA) the diffusion of CO(2) through the aqueous unstirred layers is rate limiting because the uncatalyzed hydration-dehydration of CO(2) is too slow to permit the high [HCO(3-)] to facilitate tracer diffusion through the unstirred layers. Addition of CA (ca. 1 mg/ml) to both bathing solutions causes a 10-100-fold stimulation of the CO(2) flux, which is proportional to [HCO(3-)] over the pH range 7-8. In the presence of CA the hydration- dehydration reaction is so fast that CO(2) transport across the entire system is rate limited by diffusion of HCO(3-) through unstirred layers. However, in the presence of CA when the ratio [HCO(3-) + CO(3=)]:[CO(2)] more than 1,000 (pH 9-10) the CO(2) flux reaches a maximum value. Under these conditions the diffusion of CO(2) through the membrane becomes rate limiting, which allows us to estimate a permeability coefficient of the membrane to CO(2) of 0.35 cm s(-1). In a second series of experiments we studied the effects of CA and buffer concentration on the net flux of CO(2). CA stimulates the net CO(2) flux in well buffered, but no in unbuffered, solutions. The buffer provides a proton source on the upstream side of the membrane and proton sink on the downstream side, thus allowing HCO(3-) to facilitate the net transport of CO(2) through the unstirred layers.