The size of the unstirred layer as a function of the solute diffusion coefficient.

By monitoring the concentration distribution of several solutes that are diffusing at the same time under given mixing conditions, it was established that the unstirred layer (USL) has no clearly defined boundary. For the cases of solute permeation and water movement across planar bilayer lipid membranes, respectively, experiments carried out with double-barreled microelectrodes have shown that the thickness of the USL depends on which species is diffusing. Small molecules with a larger diffusion coefficient encounter an apparently thicker USL than larger molecules with a smaller diffusion coefficient. The ratio of the USL thicknesses of two different substances is equal to the third root of the ratio of the respective diffusion coefficients. This experimental finding is in good agreement with theoretical predictions from the theory of physicochemical hydrodynamics.     

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.     

Role of unstirred layer in intestinal absorption of phenylalanine in vivo

The appearance rate of l- and d-phenylalanine in the venous blood of rat jejunal loops in vivo is increased up to 60% if the intraluminal solution is mixed more efficiently by the simultaneous perfusion of air. The effect decreases as the luminal concentration is increased to 100 mmol/1. Thus, the apparent Michaelis constants are by 50% lower in the case of the reduced unstirred layer (26 to 17 for l- and 9 to 6 mmol/1 for d-phenylalanine).The enhancement of the absorption and the reduction of the Michaelis constants can be attributed to the reduction of the effective unstirred layer thickness by about 400–500 μm.     

Direct measurement of the thickness of the unstirred diffusion layer outside immobilized biocatalysts

The concentration gradient of O₂ outside cells and oxygen-consuming particles and fibres with immobilized glucose oxidase (EC 1.1.3.4) has been directly measured with oxygen microelectrodes. Measurements have been performed in systems with different relative velocities between particles and solvent. In systems with a relative velocity equal to zero (unstirred systems), the thickness of the diffusion layer was found to be approximately the radius of the particles, and much greater than the radius of the fibres. This is in agreement with the results predicted by theoretical analysis. The thickness of the diffusion layer was not negligible, even at the highest relative velocity (1.5 x 10⁻⁴ ms⁻¹ used in this study.     

A method for estimating the diffusion resistance of the unstirred layer of microorganisms

A general enzyme-mediated flux equation which incorporates diffusion resistance was derived. By fitting the model to phosphate flux data from a blue-green alga, Anacystis nidulans, diffusion resistance, the half-saturation constant and maximal uptake velocity were estimated. Diffusion resistance for this organism was estimated to be 29 s / cm with a relative precision of about 10%. The parameters were also estimated by graphical analysis of a Woolf plot, but this method was less precise and tended to yield biased estimates.     

Kinetic constants for intestinal transport of four monosaccharides determined under conditions of variable effective resistance of the unstirred water layer

Summary Theoretical considerations have suggested that variations in the resistance of the unstirred water layer (UWL) have a profound effect on the kinetic constants of intestinal transport. In this study, a previously validatedin vitro technique was employed to determine the unidirectional flux rate of glucose, galactose, 3-O-methyl glucose and fructose into the rabbit jejunum under carefully-defined conditions of stirring of the bulk phase known to yield different values for the effective resistance of the UWL. For each monosaccharide, uptake is much greater when the resistance of the UWL is low than when high. The maximal transport rate,J _d ^m , of glucose was half as large as theJ _d ^m of galactose and 3-O-methyl glucose (3-O-MG), and was twice as great as theJ _d ^m of fructose. The apparent affinity constant,K _m ^* ,of glucose is less than that of fructose, which was lower than theK _m ^* of galactose and 3-O-MG. The use of the Lineweaver-Burk double reciprocal plot is associated with an overestimation of bothJ _d ^m andK _m ^* .This discrepancy between the true and apparent values of the kinetic constants is much greater for lower than for higher values ofJ _d ^m andK _m ^* ;variations in the resistance of the unstirred layer influences the magnitude and direction of the discrepancy. The apparent passive permeability coefficient is similar for each sugar, but because of the different values ofJ _d ^m , passive permeation contributes relatively more to the uptake of glucose and fructose than of galactose or 3-O-MG. Under conditions of high unstirred layer resistance, differences in uptake rates of the sugars are due to differences in theirJ _d ^m rather than theirK _m ^* .Kinetic analysis is compatible with the suggestion that the glucose carriers are predominantly near the tip of the villus, whereas those for galactose and 3-O-MG are located along the entire villus and theK _m ^* of their carriers at the tip is lower than theirK _m ^* towards the base of the villus. It is proposed that there are multiple or heterogeneous intestinal carriers for glucose, galactose and 3-O-methyl glucose in the jejunum of the rabbit.Abbreviations Used in this Paper C ₁ Concentration of the probe molecule in the bulk phase - C ₂ Concentration of the probe molecule at the aqueous-membrane interface - d Effective thickness of the intestinal unstirred water layer - D Free diffusion coefficient of the probe molecule     

The effect of cell size and shape on the resistance of unstirred layers to solute diffusion

The conventional flat plate model used to relate diffusion resistance to the thickness of the unstirred layer was shown to be inappropriate for microorganisms. Alternate models were developed, and they predict that diffusion resistance decreases as cell size decreases and that it depends on the shape of the cell.     

Dependence of the apparent thickness of the unstirred layer at the intestinal mucosa on nutrient concentration

The thickness of the intestinal unstirred layer determined with glycine is shown to increase from 247 ± 12 to 319 ± 14 μm (p < 0.001) as its concentration is raised from 5 to 40 mM; with glucose raised from 10 to 40 mM, the apparent value increases from 316 ± 13 to 380 ± 26 μm, because at higher concentrations the nutrients penetrate deeper to the bases of intestinal villi. The weak positive correlation between the apparent layer thickness (assayed with glucose, maltose, sucrose, alanine, leucine, and Gly-Ala in rats) and the parameter characterizing the rate of nutrient uptake (maximal short-circuit current response to nutrient) contradicts the assertion that the transporters are distributed uniformly along the villus height but because of high absorption in the upper part the nutrients do not reach the deeper transporters. It appears that the transporter distribution along the villi is nonuniform. The described method may serve to study the topography of nutrient transporters.     

A new diffusion chamber system for the determination of drug permeability coefficients across the human intestinal epithelium that are independent of the unstirred water layer.

A new method for determining permeability coefficients, that are independent of the unstirred water layer (UWL), has been developed. The method was used to determine the cellular permeability coefficient of the rapidly absorbed drug testosterone in monolayers of the human intestinal epithelial cell line, Caco-2. Using a new diffusion cell with an effective stirring system based on a gas lift, the cellular permeability coefficient for testosterone was (1.98 +/- 0.13).10(-4) cm/s which is 3.5-times higher than the permeability coefficient obtained in the unstirred system. The thickness of the UWL obtained with the well stirred diffusion cell was 52 +/- 4 microns. This value is much lower than those previously reported in various well stirred in vitro models. The calculated cellular permeability of testosterone was 13-23-times lower than that for an UWL of the same thickness as the epithelial cell (17-30 microns). We conclude that the permeability of the epithelial monolayer must be included in calculations of the thickness of the UWL.     

The development of osmotic flow through an unstirred layer

We investigate the errors involved in estimating the osmotic permeability of a semi-permeable membrane, from the measured osmotic flow and the difference in concentration of osmotically active solute across it, without taking account of the unstirred layer in the solution next to the membrane. In the problem solved, this layer is represented as a region of thickness δ at the far side of which a solute concentration C_b is imposed for time . The initial diffusion of solute towards the membrane causes the concentration at the membrane C_m to rise, generating an osmotic flow of water, J, whose convective effect opposes the diffusion. The problem is made non-linear by the dependence of J on C_m. Ultimately a steady state is set up, in which C_m is less than C_b. The solution is shown to depend on a single parameter β, equal to (L_pRT) δ C_b/D, where L_pRT is the osmotic permeability of the membrane and D is the diffusivity of solute. Solution of the steady state leads to a prediction of C_m/C_b as a function of β, and analysis of the decay of transient terms leads to a prediction of the decay time π, also as a function of β. Numerical data for membranes with a wide range of osmotic permeabilites, and for a reasonable range of solute, i.e. sucrose, concentrations, suggest that values of β can range from 0.001 or below to 7.5 or above. The former value implies negligible error in neglecting the unstirred layer, while the latter implies a 79%. error. For β = 0.1 and for δ = 2 × 10⁻⁴ m, π is predicted to be around 74 s. This decreases as β increases (for fixed δ); for values of β above about 27, the decay of transients is no longer monotonic but takes the form of damped oscillations.     

1/f Membrane noise generated by diffusion processes in unstirred solution layers.

A mathematical treatment is given for 1/f noise observed in the ion transport through membranes. It is shown that this noise can be generated by current or voltage fluctuations which occur after step changes of the membrane permeability. Due to diffusion polarization in the unstirred solution layers near the membrane these fluctuations exhibit a 1 square root of t time course which produces noise with a 1/f frequency dependence. The spectral density of 1/f noise is calculated for porous membranes with random switches between a finite and zero pore permeability. A wide frequency range and a magnitude of 1/f noise are obtained which are compatible with experimental data of 1/f noise reported for nerve membranes.     

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.     

Effect of an unstirred layer on the membrane permeability of anandamide

To study the effect of an unstirred layer (UL), we have investigated the exchange efflux kinetics of anandamide at 0 degrees C, pH 7.3, from albumin-free as well as from albumin-filled human red blood cell ghosts to media of various BSA concentrations ([BSA](o)). The rate constant (k(m)) of unidirectional flux from the outer membrane leaflet to BSA in the medium increased with the square root of [BSA](o) in accordance with the existence of a UL, which is a water layer adjacent to the membrane that is not subject to the same gross mixing that takes place in the rest of the medium. From k(m), it is possible to calculate the rate constant of anandamide dissociation from BSA (k(1)) if we know the membrane binding of anandamide, the equilibrium dissociation constant of BSA-anandamide complexes, and the diffusion constant of anandamide. We estimated k(1) to be 3.33 +/- 0.27 s(-1). The net flux of [(3)H]anandamide is balanced by an equal and opposite movement of nonradioactive anandamide in exchange efflux experiments. This means that our results are also valid for uptake. We show that for anandamide with rapid membrane translocation, UL causes a significant resistance to cellular uptake. Depicting the rate of anandamide uptake as a function of equilibrium water phase concentrations results in a parabolic uptake dependence. Such apparent "saturation kinetics" is often interpreted as indicating the involvement of transport proteins. The validity of such an interpretation is discussed.     

The unstirred water layer as a site of control of apolipoprotein B secretion

Apolipoprotein B-100 (apoB) is a constituent of low density lipoproteins that has been implicated in the development of coronary artery disease (Editorial (1988) Lancet 1, 1141-1142). It is produced primarily in the liver, but mechanisms of secretory control are unclear. We examined the possibility that rapid reuptake of newly secreted lipoproteins regulates the net output of apoB by cultured liver cells. Polyclonal blocking antibodies to the low density lipoprotein receptor markedly increased apoB output, and varying the width of the unstirred water layer around the cells also changed apoB output, consistent with local reuptake. Labeled apoB added to the bulk fluid phase of the incubation media was not detectably taken up, implying that re-uptake is predominantly local. We conclude that a major site of apoB secretory control resides in the unstirred water layer, external to the cell. Because many cells secrete products for which they possess receptors, local re-uptake from the unstirred water layer may be a general mechanism for secretory control.     

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.     

Aging: its influence on the intestinal unstirred water layer thickness, surface area, and resistance in the unanesthetized rat

The intestinal absorption of some nutrients changes with aging. As the unstirred water layer (UWL) is an important rate limiting step in the absorption of nutrients in general and of lipid soluble nutrients in particular, we investigated possible changes in the UWL dimensions in the aging rat in vivo. We measured the thickness (d) of the UWL using rapid changes in the luminal sodium concentration to induce changes in the transmucosal potential differences. We assessed the surface area (Sw) and resistance (d/SwD) of the UWL at varying flow rates by using increasingly lipophilic medium chain saturated alcohols as probes. At high UWL resistance, d decreased from 318 to 268 microns between 1 and 29 months of age. As the animals aged, Sw changed from 114 to 106 cm2/100 cm and from 262 to 214 cm2/100 cm at low and high flow rates, respectively, using dodecanol as a probe. The resistance of the UWL (d/SwD) remained relatively stable at all ages studied. These experiments demonstrate that age-related changes in absorption are dependent on the aqueous diffusion coefficient and degree of lipid solubility of the specific nutrients. At low UWL resistance, absorption of compounds with higher diffusion coefficients and greater aqueous solubility is decreased with aging. In contrast, previous studies have demonstrated that the absorption of nutrients with low diffusion coefficients and high lipid solubility increases with aging especially when the resistance of the UWL is high.