NOTES:
Situation 1: (Load parameter set Par1)
Only B is permeating: it equilibrates across the membrane, reaching 0.5 mM on both sides.
A does not cross the membrane, so A2 remains at zero, but A1 decreases and reaches a steady state at A1 < 1mM. Why?
Calculate the amount of A in V1. (qA1 = V1*A1). How much A is lost from V1.
What is the pressure difference acorss the membrane at t = 100 sec? What to you expect it to be to cause the observed pressure difference?
SITUATION 2: (Load parameter set Par2)
Rge only difference is to let A permeate. Everything equilibrates, after the inital deviations in pressure and concentrations.
Change PermA to equal PermB. Before you run predict the shape of the pressure-time and volume-time curves.
Next, put B20 = 0. Stiffen the chambers by making the two Elast = 9135.
What do you expect? What is the final pressure difference? How close is it to your expectations from the Van't Hoff expression?
JSim v1.1
/*
MODEL NUMBER: 0273
MODEL NAME: Osm.Uncoupled1
SHORT DESCRIPTION: Uncoupled, independent fluxes of water and of 2 solutes,
across a membrane separating 2 stirred tanks of equal elasticity.
*/
import nsrunit; unit conversion on;
math OSM.UncoupledA {realDomain t sec; t.min=0; t.max=100; t.delta=0.1;
//PARAMETERS:
real PermA = 0 cm/s, PermB = 0.1 cm/s, // Permeab for solutes 1 & 2
Am = 1 cm^2, // Surface area of membrane
Lp = 0.02 cm * s^-1 * mmHg^-1, // Hydraulic conductivity
p1zero= 0 mmHg, p2zero= 0 mmHg, // Initial pressures in V1 V2
V1zero= 1 mL, V2zero= 1 mL, // Initial volumes in V1 V2
RT = 19.347*10^6 mmHg*cm^3*mol^(-1), //at 37C
Elast1 = 135.9508 mmHg/ml, //elasticity of V1, rhoHg=13.59508 g/ml
Elast2 = 135.9508 mmHg/ml, //elasticity of V2
A10 = 1 mM, A20 = 0 mM, B10 = 0 mM, B20 = 1 mM, // Initial concns at t=0.
Atot = A10*V1zero + A20*V2zero, Btot = B10*V1zero + B20*V2zero; // Conserve?
//VARIABLES:
real A1(t) mM, A2(t) mM, B1(t) mM, B2(t) mM, // concentrations
qA1(t) mole, qB1(t) mole, // amts of A,B in V1
V1(t) mL, V2(t) mL, // volumes of V1 and V2
p1(t) mmHg, p2(t) mmHg; // Hydrostat press on sides 1 and 2
// INIT COND:
when (t=t.min) {V1 = V1zero; V2 = V2zero; qA1=A10*V1zero; qB1=B10*V1zero; }
// ODEs: volume and solute flux equations: fluxes are from 1 to 2
V1:t = -Lp*Am*(p1-p2 - RT*(A1-A2 + B1-B2)); // water conductivity *area* pressure
V2:t = -V1:t; // assumes conservation of water volume
qA1:t = -Am*PermA*(A1-A2); // Quantity of A exchanging
qB1:t = -Am*PermB*(B1-B2); // Quantity of B exchanging
A1 = qA1/V1; B1 = qB1/V1; // Calculate Concns in V1
A2 = (Atot - qA1)/V2; B2 = (Btot - qB1)/V2; // Calculate Concns in V2
p1 = p1zero + (V1-V1zero)*Elast1; //Elast = 1/(column area * rho)
p2 = p2zero + (V2-V2zero)*Elast2; //p2 is pressure. mmHg/areabase*height2
}
/*
DETAILED DESCRIPTION:
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 coefficienr, Pf cm/s,
Pf = Lp* Vw / RT
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, andthere is no solute advection..
The system is composed of two volumes of pressure-dependent size, yet
stirred instantaneously continually. The pressure/volume relationship is
expressed via the elasticity of the chambers, Elast, the slope of the
pressure/volume relationship. An equivalent structure is to use rigid chambers
from each of which there rises narrow columns of fluid to heights h1 and h2.
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 raises 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 (h2-h1)*rho cm H2O, where rho is the fluid
density. g/ml. The linear chamber elastance used in this model, 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 parameter set: par1 PermA = 0, PermB > 0.1. See Notes.
Situation 2 = Model parameter set: par2 PermB > 0.
See Notes tab for more discussion.
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: Uncoupled Osmotic permeation, compartmental, tutorial, passive permeation,
Vant Hoff, stirred tanks
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: 12/26/09
Revised by : BEJ Date: 12/29/09
Revision: 1) 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.
*/