Steady-state diffusion of non-electrolytes through epithelial brush borders

Weak acid permeability of a villous membrane: formic acid transport across rat proximal tubule

Chloride/formate exchange, in parallel with Na+/H+ exchange and nonionic diffusion of H2CO2, has been proposed as a mechanism of electroneutral transcellular Cl- reabsorption by the proximal tubule. However, the measured brush border H2CO2 permeability of the rat proximal tubule is at least an order of magnitude too low to support sufficient H2CO2 recycling. To investigate the possibility that an unstirred layer within the brush border might depress the measured H2CO2 permeability, we constructed a mathematical model of a villous membrane. Axial fluxes along villous and intervillous spaces were specified by Nernst-Planck diffusion equations. Model parameters were set to achieve agreement with ion and water fluxes measured in the rat proximal tubule. The equations were solved numerically to generate steady-state concentration profiles in the villous and intervillous spaces. An apparent brush border H2CO2 permeability was determined by perturbing luminal [H2CO2] and calculating the change in H2CO2 flux. Overall, the ratio of apparent brush border H2CO2 permeability to cell membrane H2CO2 permeability was greater than 90%. Contributing to the small decrease in apparent permeability are finite diffusion coefficients, folding of the membrane, and acidification of the luminal solution. An approximate analysis of this system shows the critical parameters of brush border formate transport to be the actual membrane H2CO2 permeability, and the diffusion coefficients of HCO2- and HCO3-. Nevertheless, decreasing the diffusion coefficients by one order of magnitude failed to depress apparent brush border H2CO2 permeability by more than an additional 25%. We conclude that although permeability is systematically underestimated across a villous membrane, unstirred layer effects in the brush border are still too small to resolve the discrepancy between the measured value of H2CO2 permeability and the value needed to allow recycling.

Effect of villosity and distension on the absorptive and secretory flux in the small intestine

The effects of villosity and distension on the absorptive and secretory flux in the small intestine were investigated theoretically in a simplified model. In the case of low epithelial permeability, villosity increases both fluxes by surface enlargement, but in the case of high epithelial permeability, this occurred only if the intervillous spaces are very narrow. Otherwise, the flux is reduced due to the intervillous diffusion resistance, which is more effective than the enlargement of the surface area in that case. Distension increases the fluxes due to the additional surface exposed, by opening the intervillous spaces. In the case of low epithelial permeability this increase exceeds that expected from the enlargement of the smooth inner cylindrical surface area. In the case of high epithelial permeability, however, the increase of the fluxes exceeds surface enlargement only in the first phase, just after opening the intervillous spaces. Otherwise, the increase of the flux is less, since the hindrance by the intervillous diffusion resistance is more effective than the increase of the smooth inner cylindrical surface area. In the intervillous spaces the concentration gradient is non-linear with the steepest slope at the entrance due to the permeation through the lateral surfaces of the villi. The gradient approaches linearity in the center of broad intervillous spaces and becomes steeper when the width decreases and the epithelial permeability increases. In rat small intestine broad intervillous spaces are formed at the front sides of the trapezoidal villi by the predominant circular distension. The diffusion resistance in these spaces and the increase of the supravillous diffusion resistance weaken the increase of the absorptive and secretory flux by distension, especially in the case of high epithelial permeability.

Flow through brushborders and similar protuberant wall structures

Longitudinal flow through channels with protuberant wall structures such as brushborders were studied both experimentally and theoretically. The experiments were performed using tubes with synthetic internal brushborders, scaled to resemble the geometry and flow in proximal tubules of the kidney. Fractional flow rates in the brushborder were deduced from transit times of dye traces through the central core and total flow rate. The measured ratios of brushborder to core flow QB/QC were found to be independent of Reynolds number over the range Re = 0.01 - 0.2. They agreed reasonably well with theoretical predictions based on the Kozeny-Carman equation for flow through arrays of parallel cylinders. The predictions can be extended to arbitrary geometries and turbulent flow conditions by appropriate modifications. Extrapolation of the results to the proximal tubule of the kidney indicate that brushborder flowrates there vary from 0.003 to 0.09% of total flow. Any axial transport in these structures would be predominantly diffusive in nature, and would be highly unlikely to affect radial brushborder gradients and transmural flux.

The permeability coefficient of the wall of a villous membrane

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 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.

Mechanism of inhibition of the proximal tubular isotonic fluid absorption by polylysine and other cationic polyamino acids

The present study was initiated with the hope of clarifying the role of negative charges in the luminal brush border membrane in the overall process of trans-epithelial isotonic sodium and water absorption. Using micropuncture techniques, cationic polyamino acids such as polylysine (mol wt 100,000, 17,000 and 1,500-5,000, 1 mg/ml), tetralysine, polyornithine (mol wt 100,000, 1mg/ml), polyethyleneimine (2 mg/ml), polymyxin B (2 mg/ml), protamine sulfate (25 mg/ml) and histone (0.5 mg/ml) were perfused through the segments of rat kidney proximal tubule for 30 sec to 2 min. The rate of isotonic fluid absorption was measured before and after each perfusion with the Gertz's split drop method using Ringer's solution as a shrinking drop. Polylysine 100,000 and 17,000 and polyornithine were the most potent, inhibiting isotonic reabsorption by 93%. The sequence of inhibitory effect was: polylysine 100,000 congruent to polyornithine 100,000 congruent to polylysine 17,000 greater than polyethyleneimine greater than polylysine 1,500-5,000 congruent to polymyxin B greater than protamine sulfate congruent to histone. In contrast, tetralysine (2 mg/ml) showed no inhibitory effect. Electrical potential difference (p.d.) of the proximal tubular cells was destroyed within 10 sec of luminal perfusion with polylysine 100,000 (1 mg/ml). Simultaneously with the drop in p.d., electrical resistance of the luminal brush border membrane was nearly totally eliminated, whereas transepithelial input resistance remained unaltered. Furthermore, trypan blue dye was taken up by polylysine 100,000-perfused tubular cells but not by normal cells. Expanding drop analysis (mannitol solution as a split drop) was performed as a screening test to examine if the permeability for water and sodium in the lateral paracellular pathway is altered by polylysine 100,000. No significant difference was observed in the velocity of split drop expansion between untreated and polylysine-perfused tubules. A lower concentration of polylysine 100,000 (0.1 mg/ml) showed a much less inhibitory effect on fluid absorption and on cell p.d. These observations indicate that the strong inhibition on proximal tubular fluid absorption exerted by polylysine and perhaps also by other cationic polyamino acids is due not to modification of membrane negative charges but to the lysis of tubular cells by these polycations.

The sodium-alanine interaction in rabbit ileum. Effect of sodium on alanine fluxes

The model of the interaction between Na and alanine at the mucosal border of rabbit ileum has been tested further by examining the efflux of alanine from the cells toward the mucosal solution. Alanine efflux shows a tendency toward saturation as cellular alanine concentration increases and is influenced by cellular Na concentration. A decrease in cell Na concentration causes an increase in the apparent Michaelis constant with little change in maximal efflux rate. Studies on strips of mucosa treated with ouabain or cyanide showed that the direction of net alanine transfer between the cells and the medium is determined by the direction of the Na concentration difference. The cells extrude alanine against a concentration difference when cell [Na] exceeds medium [Na] and accumulate alanine when cell [Na] is less than medium [Na]. The observations are consistent with the model previously suggested involving a transport site that combines with and translocates both Na and alanine, and with the concept that the Na concentration difference between mucosal solution and cytoplasm provides at least part of the energy for active transport of alanine.