Abstract
The properties of nonlinear equations describing the solute and solvent transport across a simplified Patlak-Goldstein-Hoffman model (two membranes in series without unstirred layers) are investigated both analytically and numerically. The analysis shows that the principal coefficients measured in transport experiments in the presence of active transport are dependent on the experimental conditions. These ‘apparent’ system parameters are extensions of the corresponding parameters determined both in passive systems and in the linear Kedem-Katchalsky theory. Moreover, they are related to the local phenomenological coefficients of the single membranes of the array. Several relationships between measurable quantities and the local system parameters are indicated, allowing the planning of experiments aimed at the measurement of the latter. Data in the literature have been used to check the proposed volume flow equation.
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Literature
Axel, L. 1976. “Flow Limits of Kedem-Katchalsky Equations for Fluid Flux.”Bull. math. Biol. 38, 671–677.
Bentzel, C. J., M. Davies, W. N. Scott, M. Zatzman and A. K. Solomon. 1968. “Osmotic Volume Flow in the Proximal Tubule of Necturus Kidney.”J. gen. Physiol. 51, 517–533.
Bresler, E. H. and R. P. Wendt. 1969a. “Diffusive and Convective Flow Across Membranes: Irreversible Thermodynamic Approach.”Science 163, 944–945.
— and —. 1969b. “Reply to Manning.”Science 166, 1438.
Brodsky, W. A. and T. P. Schilb. 1965. “Osmotic Properties of Isolated Turtle Bladder.”Am. J. Physiol. 208, 46–57.
Caplan, S. R. and D. C. Mikulecky. 1966. “Transport Processes in Membranes.” inIon Exchange, Ed. J. A. Marinsky. Vol. I, pp. 1–64, New York: Marcel Dekker.
Celentano, F., G. Monticelli and M. N. Orsenigo. 1978. “Non Linear Volume Flow Dependence on Osmotic Pressure Difference in Frog Skin.”J. Physiol., Paris 74, 365–367.
Curran, P. F. 1960. “Na, Cl and Water Transport by Rat Ileumin Vitro.”J. gen. Physiol. 43, 1137–1148.
— and J. R. McIntosh. 1962. “A Model System for Biological Water Transport.”Nature, Lond. 193, 347–348.
Dainty, J. 1963. “The Polar Permeability of Plant Cell Membranes to Water.”Protoplasma 57, 220–228.
Diamond, J. M. 1964. “The Mechanism of Isotonic Water Transport.”J. gen. Physiol. 48, 15–42.
— 1966. “Non Linear Osmosis.”J. Physiol., Lond. 183, 58–82.
— 1978. “Solute-linked Water Transport in Epithelia.” InMembrane Transport Processes, Ed. J. F. Hoffman, pp. 257–276. New York: Raven Press.
— and W. H. Bossert. 1967. “Standing Gradient Osmotic Flow. A Mechanism for Coupling of Water and Solute Transport in Epithelia.”J. gen. Physiol. 50, 2061–2083.
Fischbarg, J. 1978. “Fluid Transport by Corneal Endothelium.” InComparative Physiology: Water, Ions and Fluid Mechanics, Ed. K. Schmidt-Nielsen, L. Bolis and S. H. P. Maddrell, pp. 21–39. Cambridge: University Press.
—, C. R. Warshavsky and J. J. Lim. 1977. “Pathways for Hydraulically and Osmotically Induced Water Flows across Epithelia.”Nature, Lond. 266, 71–74.
Hakim, A. A. and N. Lifson. 1969. “Effects of Pressure on Water and Solute Transport by Dog Intestinal Mucosain Vitro.”Am. J. Physiol. 216, 276–284.
Hays, R. M. 1968. “A New Proposal for the Action of Vasopressin, Based on Studies of a Complex Synthetic Membrane.”J. gen. Physiol. 51, 385–398.
Hill, A. E. 1975a. “Solute-Solvent Coupling in Epithelia: a Critical Examination of the Standing-gradient. Flow Theory.”Proc. R. Soc. Lond. Biol. Sci. 190, 99–114.
—. 1975b “Solute-Solvent Coupling in Epithelia: Contribution of the Junctional Pathway to Fluid Production.”Proc. R. Soc. Lond. Biol. Sci. 191, 537–547.
Kedem, O. and A. Katchalsky. 1958. “Thermodynamic Analysis of the Permeability of Biological Memvranes to Non-electrolytes.”Biochim. Biophys. Acta 27, 229–246.
— and —. 1963. “Permeability of Composite Membranes. Part 3. Series Array of Elements.”Trans. Faraday Soc. 59, 1941–1953.
Koefoed-Johnsen, V. and H. H. Ussing. 1958. “The Nature of the Frog Skin Potential.”Acta physiol. scand. 42, 298–308.
Levitt, D. G. 1975. “General Continuum Analysis of Transport through Pores.”Biophys. J. 15, 533–551.
Lichtenstein, N. S. and A. Leaf. 1966. “Evidence for a Double Series Permeability Barrier at the Mucosal Surface of the Toad Bladder.”Ann. N.Y. Acad. Sci. 137, 556–565.
Lifson, N. and M. B. Visscher. 1947. “Osmosis in Living Systems.” InMedical Physics, Ed. O. Glasser, Vol. 1, 869–892. Chicago, IL: The Year Book Medical Publishers.
Loeschke, K., C. J. Bentzel and T. Z. Csaky. 1970. “Asymmetry of Osmotic Flow in Frog Intestine: Functional and Structural Correlation.”Am. J. Physiol. 218, 1723–1731.
Lucké, B. and M. McCutcheon. 1932. “The Living Cell as an Osmotic System.”Physiol. Rev. 12, 68–139.
MacRobbie, E. A. and H. H. Ussing. 1961. “Osmotic Behaviour of the Epithelial Cells of Frog Skin.”Acta physiol. scand. 53, 348–365.
Meyer, J., F. Sauer and D. Woermann. 1974. “Coupling of Mass Transfer and Chemical Reaction across an Asymmetric Sandwich Membrane.” InMembrane Transport in Plants, Ed. U. Zimmermann and J. Dainty, pp. 28–35. Berlin: Springer Verlag.
Michaels, D. W. and D. Dennis. 1973. “Asymmetric Phospholipid Bilayer Membranes: Formation and Electrical Characterization.”Biochem. biophys. Commun. 51, 357–363.
Mikulecky, D. C. 1977. “A Simple Network Thermodynamic Method for Series-Parallel Coupled Flows: II. The Non-linear Theory, with Applications to Coupled Solute and Volume Flow in a Series Membrane.”J. theor. Biol. 69, 511–541.
— and S. R. Thomas. 1978. “A simple Network Thermodynamic Method for Series-Parallel Coupled Flows: III. Application to Coupled Solute and Volume Flows Through Epithelial Membranes.”J. theor. Biol. 73, 697–710.
Monticelli. G. and F. Celentano. 1980. “Transport Coefficients in a Non Linear Series Membranes Array.” InDevelopments in Biophysical Research. Ed. A. Borsellino, P. Omodeo, R. Strom, A. Vecli and E. Wanke, pp. 157–164. New York: Plenum Press.
— and — and G. Torelli. 1975. “Sodium Chloride Reflection Coefficient in Rabbit Gall Bladder.”Biochim. biophys. Acta 401, 41–50.
Ogilvie, J. T., J. R. McIntosh and P. F. Curran. 1963. “Volume Flow in a Seriesmembrane System.”Biochim. biophys. Acta 66, 441–444.
Patlak, C. S., D. A. Goldstein and J. F. Hoffman. 1963. “The Flow of Solute and Solvent Across a Two Membranes System.”J. theor. Biol. 5, 426–442.
Przestalski, S. and M. Kargol. 1976. “Composite Membranes.” InBiophysics of Membrane Transport, Ed. J. Kuczera and S. Przestalski, pp. 53–69. Wroclaw, Poland: Agricultural University.
Richardson, I. W. 1972. “Multiple Membrane Systems as Biological Models.”J. Membrane Biol. 8, 219–236.
Ripoche, P., J. Bourguet and M. Parisi. 1973. “The Effect of Hypertonic Media on Water Permeability of Frog Urinary Bladder.”J. gen. Physiol. 61, 110–124.
Sackin, H. and E. L. Boulpaep. 1975. “Models for Coupling of Salt and Water Transport. Proximal Tubular Reabsorption inNecturus Kidney.”J. gen. Physiol. 66, 671–733.
Schafer, J. A., S. L. Troutman and T. E. Andreoli. 1974. “Osmosis in Cortical Collecting Tubules—ADH Independent Osmotic Flow Rectification.”J. gen. Physiol. 64, 228–240.
Sha'afi, R. I., G. T. Rich, D. C. Mikulecky and A. K. Solomon. 1970. “Determination of Urea Permeability in Red Cells by Minimum Method.”J. gen. Physiol. 55, 427–450.
Tomicki, B. and F. Ludwików. 1971. “Further Properties of the Two-membrane System Pressure in the Middle Compartment.”Studia biophys. 26, 21–30.
Urakabe, S., J. S. Handler and J. Orloff. 1970. “Effect of Hypertonicity on Permeability Properties of the Toad Bladder.”Am. J. Physiol. 218, 1179–1187.
Weinstein, A. M. and J. L. Stephenson. 1979. “Electrolyte Transport across a Simple Epithelium. Steady-state and Transient Analysis.”Biophys. J. 27, 165–186.
— and —. 1981. “Models of Coupled Salt and Water Transport across Leaky Epithelia.”J. Membrane Biol. 60, 1–20.
Whitlock, R. T. and H. O. Wheeler. 1964. “Coupled Transport of Solute and Water across Rabbit Gallbladder Epithelium.”J. clin. Invest. 48, 2249–2265.
Wright, E. M., A. P. Smulders and J. McD. Tormey. 1972. “The Role of the Lateral Intercellular Spaces and Solute Polarization Effects in the Passive Flow of Water across the Rabbit Gallbladder.”J. Membrane Biol. 7, 198–219.
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Monticelli, G., Celentano, F. Further properties of the two-membrane model. Bltn Mathcal Biology 45, 1073–1096 (1983). https://doi.org/10.1007/BF02458831
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DOI: https://doi.org/10.1007/BF02458831