// MODEL NUMBER: 0274 MODEL NAME: Osm.Uncoupled2 SHORT DESCRIPTION: Uncoupled, independent fluxes of water and of 2 solutes, across a membrane separating 2 stirred tanks equipped with columns above each to provide observable column heights as measures of their pressures. Lp = 0.02 cm/(s*mmHg), //Hydraulic conductivity = filtration coeff // which translates to Pf,cm/s, by multiplying Lp by RT/Vw= 1.02*1e6 mmHg at 20 degC // so Pf =2.031e4 cm/s. // Mlekody, Moore Levitt JGP 81: 213, 1983 give Pf = 2e-2 cm/s at 23C in RBC. . Here Pf = 2.15*1e4 cm/s @37C. Note the the surface to volume ratio here is 1 cm^2 / 1 ml whereas that for RBC is much higher/ MODEL VERIFICATION: Total Mass is conserved. See Aerr and Berr: Check ro see if V1 + V2 is constant. DETAILED DESCRIPTION: This model is the same as Osm.Uncoupled1.proj except for using column heights instead of elasticity. It also allows temperature changes. Uncoupled, independent fluxes of water and of 2 solutes, A and B, across a membrane separating 2 stirred tanks. Solute activities are assumed unity so concentrations = thermodynamic activity. The model describes a situation similar to that for the simplest expressions of Kedem and Katchalsky (1958) but omits all interactions between solutes and between water and any solute. One can think of the solutes passing though the membrane by passive permeation with permeability coefficients PermA and Perm B, and the water passing through aqueous pores with filtration coefficient or hydraulic conductivity, Lp. The aqueous pores do not permit solute passage. Lp is the same as the traditional filtration coefficient Kf. Lp translates to a conventional permeability for water filtration, Pf cm/s, Pf = Lp*RT/Vw where RT = 19.347*10^6 mmHg*cm^3*mol^(-1) at 37C, Vw is the partial molar volume of water, 18 ml/mol or the concentration of water in water is 55.55 M The driving forces are the pressure difference for water flux and the concentration for the solute fluxes. The pressure difference across the membrane is the hydrostatic pressure difference minus the osmotic pressure difference. The osmotic pressure is given by Van't Hoff's Eq: p_osm = a.C.RT, where p_osm is the osmotic pressure, mmHg, "a" is the activity coefficient, assumed in this model to equal unity, C is concentration, M, and RT is the Gas Constant times Temperature Kelvin. In this model the solute doesn't permeate the aqueous pore so there is no consideration of a reflection coefficient, or rather it is assumed to be unity. Thus solute concentration in the pore water is zero, and there is no solute advection.. The system is composed of two volumes of pressure-dependent size, yet stirred instantaneously continually. The slope of the pressure/volume relationship is linear and defined by the height of a column of fluid above the rigid chambers The narrow columns of fluid have heights h1 and h2. The pressure is rho*grav*h1 in chamber 1, where rho is fluid density, grav is acceleration due to gravity; The fluid in the columns is considered to be instantaneously mixed with that in the chamber from which it rises. Fluid or volume flux, Jv, from side 1 to side 2 causes a difference in the column heighta between the two sides by Base*(h2-h1) = Jv, where Base = area of the base of the column, and the pressure difference rises to rho* grav*(h2-h1) cm H2O, where rho is the fluid density. g/ml, in the narrow colums.. The model OsmUncoupledA.proj uses an analogous linear chamber elastance, Elast mmHg/ml, gives an equivalent measure for flexible chambers, assuming a linear relationship between the pressure change and the volume change. (1 mmHg = 13.59 cm H2O.) Notes: Situation 1:= Model default par. . PermA = 0, PermB > 0.1. See Notes. Situation 2 = Model par2 PermB > 0. SHORTCOMINGS/GENERAL COMMENTS: ASSUMPTIONS: 1. Compartmental assumptions apply to the solutions on either side of the membrane. These are: Instantaneously stirred tank. No concentration gradients. No diffusion limitation for reactions. KEY WORDS: Two ideal solutes, compartment, water and solute exchamge uncoupled, passive transmembrane exchanes independent, tutorial REFERENCES: Katchalsky A and Curran PF. Nonequilibrium Thermodynamics in Biophysics. Cambridge, MA: Harvard University Press, 1965. Kedem O and Katchalsky A. Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. Biochim Biophys Acta 27: 229-246, 1958. Stein WD. The Movement of Molecules across Cell Membranes. New York: Academic Press, 1967. Stein WD. Transport and Diffusion across Cell Membranes. Orlando, Florida: Academic Press Inc., 1986. REVISION HISTORY: Original Author : JBB Date: 26/Dec/09 Revised by : BEJ Date: 29/Dec/09 Revision: Update format of comments. COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE: Copyright (C) 1999-2009 University of Washington. From the National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061. Academic use is unrestricted. Software may be copied so long as this copyright notice is included. This software was developed with support from NIH grant HL073598. Please cite this grant in any publication for which this software is used and send an email with the citation and, if possible, a PDF file of the paper to: staff@bioeng.washington.edu.

Hydraulic conductivity, Kf

Y(t):t uses Pf, not Lp

Permeab for solutes 1 & 2

T degK

Permeab for solutes 1 & 2

Gas Constant in mmHg/(mM*degK)

Initial pressures in V1 V2

press p = rho*grav*height; height = VolChange/BaseArea

qA2(t) mole, qB2(t) mole, // amts of A,B in V1

Initial volumes in V1 V2

Conservation is assumed for A and B to reduce ODEs

concentrations

concentrations

calculate RT for TempC

Initial pressures in V1 V2

volumes of V1 and V2

volumes of V1 and V2

Initial concns at t=0.

Initial volumes in V1 V2

deg Centigrade

Surface area of membrane

Initial concns at t=0.

water partial molar volume

density -> cm H2O height to mmHg (rhoHg = 13.59508 g/ml)

water permeability of membrane

qA2(t) mole, qB2(t) mole, // amts of A,B in V1

Initial concns at t=0.

Hydrostat press in V1 and V2

Hydrostat press in V1 and V2

concentrations

concentrations

1.0746*1e6 at 37C. Water activity = pressure*Vw/RT

area of base of volume 2, to calc p2.

area of base of volume 1, to calc p1.

Initial concns at t=0.

Initial volumes in V1 V2

Initial volumes in V1 V2

Initial volumes in V1 V2

water conductivity *area* pressure

when PermA = 0 and permB finite