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.     

The permeability coefficient of the wall of a villous membrane

Summary The equations hitherto used to correct the permeability coefficient for the unstirred layer influence are valid only for flat membranes. Therefore, appropriate equations for membranes with a villous surface (e.g., small intestine) have been derived. They take into account the non-linear concentration gradient in the intervillous part of the unstirred layer. Quantitative information about the geometry of the villous surface and the unstirred layer thickness are needed to calculate the permeability coefficient of the membrane wall (e.g., intestinal epithelium). The concentration of highly permeable substances drops sharply already in the upper part of the intervillous space, so that the tips of the villi function as effective absorbing area. The intervillous concentration gradient of a substance with a low permeability coefficient is so small, that such a substance is absorbed by the total surface area of the villous membrane. The effective absorbing area of substances with intermediate permeability coefficient lies between the described limits.     

Determination of kinetic parameters of a carrier-mediated transport in the perfused intestine by two-dimensional laminar flow model: effects of the unstirred water layer

The kinetic parameters of a carrier-mediated transport for D-glucose and for taurocholate were determined from rat in situ intestinal single perfusion experiments. The true parameters were obtained by the two-dimensional laminar flow model, in which the solute concentration at the aqueous-intestinal membrane interface can be calculated numerically without assuming the aqueous diffusion layer, discriminating the effects of the unstirred water layer. The true Michaelis constant was 4.5 mM for D-glucose and 1.5 mM for taurocholate. The true maximal transport velocity was 3.4 nmol/s per cm2 for D-glucose and 0.29 nmol/s per cm2 for taurocholate. The apparent Michaelis constant was raised by the factor of 6.6 for D-glucose and 3.6 for taurocholate due to the effects of the unstirred water layer. The maximal transport velocity was relatively unaffected by the unstirred water layer in both compounds. The values of the effective (operational) thickness of the unstirred water layer were compatible with those reported previously by employing various experimental methods. The kinetic parameters obtained in vitro everted sacs, for comparison, almost coincided with the true ones in situ. Therefore, the two-dimensional laminar flow model is shown to be valid not only for determining the kinetic parameters of a carrier-mediated transport in situ but also for predicting the absorption rate in situ from the uptake rate in vitro.     

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.     

Unstirred water layers in rabbit intestine: effects of pectin

Pectins have been shown to affect the absorption of several different nutrients in clinical studies; however, the mechanisms for decreased absorption have not been defined. A possibility not studied with regards to pectin, but previously demonstrated to be important in absorption, is the effect of change in the unstirred water layer. As the unstirred water layer increases in thickness, the rate of absorption decreases for certain nutrients. The effect of pectin on the unstirred water layer in the lumen of rabbit jejunum was examined by previously described techniques. It was observed that: (1) increases in pectin concentration resulted in an increased thickness of the unstirred water layer; (2) for any stir rate, the addition of pectin increased the thickness of the unstirred water layer; and (3) stir rate is inversely related to the thickness of the unstirred water layer. It was concluded from these results that pectin increases the thickness of the unstirred water layer in rabbit jejunum. This mechanism may explain, in part, the reduction of the rate of absorption of certain nutrients seen following pectin ingestion.     

Computer model of unstirred layer and intracellular pH changes. Determinants of unstirred layer pH

Transmembrane acid–base fluxes affect the intracellular pH and unstirred layer pH around a superfused biological preparation. In this paper the factors influencing the unstirred layer pH and its gradient are studied. An analytical expression of the unstirred layer pH gradient in steady state is derived as a function of simultaneous transmembrane fluxes of (weak) acids and bases with the dehydration reaction of carbonic acid in equilibrium. Also a multicompartment computer model is described consisting of the extracellular bulk compartment, different unstirred layer compartments and the intracellular compartment. With this model also transient changes and the influence of carbonic anhydrase (CA) can be studied. The analytical expression and simulations with the multicompartment model demonstrate that in steady state the unstirred layer pH and its gradient are influenced by the size and type of transmembrane flux of acids and bases, their dissociation constant and diffusion coefficient, the concentration, diffusion coefficient and type of mobile buffers and the activity and location of CA. Similar principles contribute to the amplitude of the unstirred layer pH transients. According to these models an immobile buffer does not influence the steady-state pH, but reduces the amplitude of pH transients especially when these are fast. The unstirred layer pH provides useful information about transmembrane acid–base fluxes. This paper gives more insight how the unstirred layer pH and its transients can be interpreted. Methodological issues are discussed.     

The determination of membrane transport parameters with the cell pressure probe: theory suggests that unstirred layers have significant impact

A simulation model was written to compute the time-kinetics of turgor pressure, P, change in Chara corallina during cell pressure probe experiments. The model allowed for the contribution of a membrane plus zero, one, or two unstirred layers of any desired thickness. The hypothesis that a cell with an unstirred layer is a composite membrane that will follow the same kind of kinetics with or without unstirred layers was tested. Typical ‘osmotic pulse’ experiments yield biphasic curves with minimum or maximum pressures, P_min(max), at time t_min(max) and a solute exponential decay with halftime . These observed data were then used to compute composite membrane properties, namely the parameters L_p = the hydraulic conductance, σ = reflection coefficient and P_s = solute permeability using theoretical equations. Using the simulation model, it was possible to fit an experimental data set to the same values of P_min(max), t_min(max) and incorporating different, likely values of unstirred layer thickness, where each thickness requires a unique set of plasmalemma membrane values of L_p, σ and P_s. We conclude that it is not possible to compute plasmalemma membrane properties from cell pressure probe experiments without independent knowledge of the unstirred layer thickness.     

Intestinal absorption of free oleic acid in the unanesthetized rat: evidence for a saturable component?

The intestinal absorption of free oleic acid at low intraluminal concentrations and the influence of luminal factors on its absorption were studied in the unanesthetized rat. The relationship between oleic acid concentration (30-2500 microM) and its rate of absorptions fitted best to a rectangular hyperbola (y = x/(2.19 + 0.0015x), r = 0.94). Oleic acid's rate of absorption increased as the hydrogen ion and sodium taurocholate concentrations were increased or as the thickness and resistance of the unstirred water layer were diminished or following the addition of lysolecithin. The additions of the artificial detergent Tween-80, or lecithin and linoleic, linolenic and arachidonic acids to the perfusate decreased oleic acid's rate of absorption. It was concluded that oleic acid absorption in this range of concentrations displays apparent saturation kinetics which are due to unstirred layer effects, limited aqueous solubility of oleic acid and possible saturation of cytosol fatty acid binding proteins. Factors which increase oleic acid's protonated concentration or diminish the unstirred layer resistance, enhance its absorption rate, while factors which enhance its micellar solubility or interfere with its transfer out of the cell membrane decrease its overall rate of absorption.     

Derivation of the equations that describe the effects of unstirred water layers on the kinetic parameters of active transport processes in the intestine

Unidirectional flux of solutes into the intestinal mucosal cells is determined by the rate of movement of these molecules across both an unstirred water layer and the microvillus membrane of the epithelial cell. Therefore, an equation is derived in this paper that describes the velocity of active transport as a function of the characteristics of both the transport carrier in the membrane and the resistance of the overlying unstirred water layer. Using this equation a series of curves are presented that depict the effect on the kinetics of active transport of varying the thickness (d) or surface area (S_w) of the unstirred water layer, the free diffusion coefficient (D) of the solute, the distribution of active transport sites along the villus ($\text{?}{}_{\mathrm{<mtext>n</mtext>}}$), the maximal transport velocity (J^m_d) and the true Michaelis constant (K_m). These theoretical curves illustrate the serious limitations inherent in interpretation of previously published data dealing with active transport processes in the intestine.     

Comparative assessment of the resistance of the unstirred water layer to solute transport between two different intestinal perfusion systems

The resistance of the unstirred water layer to solute transport was estimated in two different intestinal single-pass perfusion systems for a comparative study, using D-glucose as a model compound. One is a well established perfusion system in anesthetized rats as a standard (system A). The other is the one in unanesthetized rats for comparison (system B). It was demonstrated that in system B as well as in system A the resistance of the unstirred water layer to D-glucose transport should be taken into account and this resistance, accordingly, the effective thickness of the unstirred water layer (delta) which is assumed to be in proportion to its resistance, could be described as a function of the perfusion rate by using a film model. The delta decreased with increasing perfusion rate and was larger in system A than in system B at each perfusion rate; 785 microns in system A versus 319 microns in system B at the perfusion rate of 0.16 ml/min and 337 microns versus 184 micron at that of 2.95 ml/min. Thus in system B the effective thickness, accordingly, the resistance, of the unstirred water layer was reduced to about 50% of that in system A, but the resistance of the unstirred water layer could still account for 85% of the total resistance at the maximum as far as D-glucose absorption was concerned, while 93% in system A. These results suggest that, compared with perfusion experiments in anesthetized rats (system A), the resistance of the unstirred water layer is reduced but cannot be left out of consideration even if perfusion experiments are performed in unanesthetized rats (system B). And the lower resistance of the unstirred water layer in system B was attributed to a turbulent flow in contrary to a laminar flow in system A.     

Determinants of intestinal mucosal uptake of short- and medium-chain fatty acids and alcohols

Uptake rates across the jejunal brush border have been measured for water-soluble fatty acids and alcohols and analyzed to determine the relative roles of the unstirred water layer and the lipid cell membrane as determinants of the intestinal absorptive process. Initial studies involving measurement of time courses of electrical transients developed across the intestine exposed to poorly permeant solute molecules showed no anomalous discrimination of probe molecules of different size or charge. This finding suggests that the diffusion barrier in the intestine can be considered as an unstirred water layer. Next, uptake rates of fatty acid were found to be linear with respect to concentration of the test solute, demonstrated no competitive inhibition or contralateral stimulation, had low temperature dependency, and were insensitive to metabolic inhibition, indicating that uptake proceeds by passive diffusion. Passive permeability coefficients, *P, varied from 22 +/- 1.4 to 395 +/- 9.2 nmoles.min(-1).100 mg(-1).mm(-1) for the saturated fatty acids 2:0 through 12:0 and from 119 +/- 3.3 to 581 +/- 45.2 for the saturated alcohols 6:0 through 10:0. Vigorous stirring of the bulk buffer solution enhanced *P values in direct proportion to chain length while the presence of bile acid micelles depressed apparent permeability coefficients in proportion to fatty acid chain length. These results demonstrate that uptake of short-chain fatty acid monomers is rate limited by the lipid cell membrane but diffusion through the unstirred water layer becomes increasingly rate limiting as the chain length increases. It is also possible to conclude from these data that diffusion through the unstirred water layer becomes totally rate limiting for uptake of long-chain fatty acid monomers of physiological importance.     

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.     

Thiamine transport in thiamine-deficient rats. Role of the unstirred water layer.

As part of a systematic study of alcoholism and thiamine absorption, the effect of diet-induced thiamine deficiency and the role of the unstirred water layer on the thiamine transport were investigated. Using 3H-labeled dextran as a marker of adherent mucosal volume, jejunal uptake of 14C-labeled thiamine hydrochloride was measured, in vitro, in thiamine-deficient rats and pair-fed controls. Uptake of low thiamine concentrations (0.2 and 0.5 muM) was greater in the thiamine-deficient rats than in the controls. In contrast, uptake rates for high thiamine concentrations (20 and 50 muM) were similar in both groups. While Jmax was unaltered, Km was decreased in thiamine deficiency, suggesting a decrease in unstirred water layer thickness. Accordingly, the thickness of the water layer was measured in both groups of animals and correlated with Jmax and Km under unstirred and stirred conditions. Without stirring, there was no difference in Jmax between the two groups. In contrast, both Km and the water layer were reduced in the thiamine-deficient rats. With stirring, Jmax was not affected, but both Km and the water layer thickness were reduced to similar values in both groups. Reversal of thiamine deficiency resulted in the return of thiamine uptake and the unstirred water layer thickness to control values. These data support the concept of a dual system of thiamine transport and emphasize the role of the unstirred water layer as an important determinant of transport kinetics not only under physiologic situations but also in diet-induced rat thiamine deficiency, a model for a clinical patholigical state. The decrease in the unstirred water layer thickness in thiamine deficiency may be also viewed as a possible adaptive mechanism to facilitate absorption of meager supplies of thiamine.     

The water permeability of toad urinary bladder. I. Permeability of barriers in series with the luminal membrane

Antidiuretic hormone (ADH) induces a large increase in the water permeability of the luminal membrane of toad urinary bladder. Measured values of the diffusional water permeability coefficient, Pd(w), are spuriously low, however, because of barriers within the tissue, in series with the luminal membrane, that impede diffusion. We have now determined the water permeability coefficient of these series barriers in fully stretched bladders and find it to be approximately 6.3 X 10(- 4) cm/s. This is equivalent to an unstirred aqueous layer of approximately 400 microns. On the other hand, the permeability coefficient of the bladder to a lipophilic molecule, hexanol, is approximately 9.0 X 10(-4) cm/s. This is equivalent to an unstirred aqueous layer of only 100 microns. The much smaller hindrance to hexanol diffusion than to water diffusion by the series barriers implies a lipophilic component to the barriers. We suggest that membrane-enclosed organelles may be so tightly packed within the cytoplasm of granular epithelial cells that they offer a substantial impediment to diffusion of water through the cell. Alternatively, the lipophilic component of the barrier could be the plasma membranes of the basal cells, which cover most of the basement membrane and thereby may restrict water transport to the narrow spaces between basal and granular cells.     

Nitric oxide scavenging by red blood cells as a function of hematocrit and oxygenation

The reaction rate between nitric oxide and intraerythrocytic hemoglobin plays a major role in nitric oxide bioavailability and modulates homeostatic vascular function. It has previously been demonstrated that the encapsulation of hemoglobin in red blood cells restricts its ability to scavenge nitric oxide. This effect has been attributed to either factors intrinsic to the red blood cell such as a physical membrane barrier or factors external to the red blood cell such as the formation of an unstirred layer around the cell. We have performed measurements of the uptake rate of nitric oxide by red blood cells under oxygenated and deoxygenated conditions at different hematocrit percentages. Our studies include stopped-flow measurements where both the unstirred layer and physical barrier potentially participate, as well as competition experiments where the potential contribution of the unstirred layer is limited. We find that deoxygenated erythrocytes scavenge nitric oxide faster than oxygenated cells and that the rate of nitric oxide scavenging for oxygenated red blood cells increases as the hematocrit is raised from 15% to 50%. Our results 1) confirm the critical biological phenomenon that hemoglobin compartmentalization within the erythrocyte reduces reaction rates with nitric oxide, 2) show that extra-erythocytic diffusional barriers mediate most of this effect, and 3) provide novel evidence that an oxygen-dependent intrinsic property of the red blood cell contributes to this barrier activity, albeit to a lesser extent. These observations may have important physiological implications within the microvasculature and for pathophysiological disruption of nitric oxide homeostasis in diseases.     

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.     

The determination of the kinetic parameters of a carrier mediated transport process in the presence of an unstirred water layer

A procedure is described, based on the Eadie-Hofstee plot, from which it is possible to determine the Michaelis constant and the maximum velocity of a membrane bound transport process that is separated from the substrate in the bulk solution by an unstirred layer. This can be done without knowing the magnitude of the latter provided that its effective thickness can be varied by altering the rate of stirring. Further, if the affinity of the carrier for the substrate is sufficiently large, then, it is possible to determine the rate constant and the effective concentration of the membrane bound carrier.