Questions:
(1) What are the equilibrium concentrations if Psa and Psb
are non-zero, and at least one of G1 or G2 is non-zero?
(2) Set any parameters except the rate parameters so that
B2(0) = B2(long time). Show analytically that this
result is correct.
Figure 1: Concentration : Default parameter set
Concentrations of A1 (black solid line), A2 (black dashed
line), B1 (red solid line), and B2 (red dashed line) are
plotted as a function of time. The equilibrium concentration
for either B (G1 and G2 not both 0) or for A (G1 and G2 both 0)
is plotted (green dashed line).
Without changing any of the rate parameters, make B2's
initial value the same as the final equilibrium value.
What equation must you solve to get this result? (See below
for answer).
Figure 2: RUN LOOPS: Default parameter set
The B1 concentration is plotted as a function of time when
G1=0 and G2=1 (Solid black line), and when G1=1 and G2=0
(dashed line) as a function of time.
Case 1: G1=0, G2=1. B1 rises slowly because A1 must first
cross the barrier (A1->A2), then be converted to B2 (A2->B2),and then cross the barrier again (B2->B1).
Case 2: G1=1, G2=0. B1 rises quickly because A is directly
converted to B (A1->B1), and A1 is initially set to a high
value.
Where the reaction takes place usually makes a big
difference in the shapes of the concentration curves in
multi-compartment problems when a substance reacts to
become a different substance.
Answers:
(1) The equilibrium concentrations for B1 and B2 are equal
and given by
( V1*(A10+B10)+V2*(A20+B20) )/( V1 + V2 ).
(2) To have B2(initial) = B2(t = infinity), solve
V1*(A10+B10)+V2*(A20+B20) = (V1+V2)*B20
i.e. the total amount of material in the system initially
(RHS) is the total amount of material in the system finally
(LHS). For this problem, at equilibrium, B1=B2, that is
the final amount of material is
V1*B1+V2*B2=V1*B2+V2*B2
=(V1+V2)*B2
=(V1+V2)*B20.
Therefore
V1 A10 + V1 B10 + V2 A20
B20= ------------------------
V1
Set A10=1 mM, B10=0.5 mM, A20=2.0 mM. Set B20 to 5.7857143,
and run model for 600 seconds and check final value for
B2.
/* MODEL NUMBER: 0246
MODEL NAME: Comp2ExchangeReaction
SHORT DESCRIPTION: Two comparment model with two substances,
A and B, and where A may be irreversibly converted to B.
*/
import nsrunit; unit conversion on;
math Comp2ExchangeReaction {
// INDEPENDENT VARIABLE
realDomain t sec; t.min=0.0; t.max=60; t.delta = 0.1;
// PARAMETERS
real V1 = 0.07 ml/g, // volume of compartment 1
V2 = 0.15 ml/g, // volume of compartment 2
PSa = 1 ml/(g*min), // Permeability Surface area product
// for exchange of A between two compartments
PSb = 1 ml/(g*min), // Permeability Surface area product
// for exchange of A between two compartments
G1 = 0.0 ml/(g*min), // Conversion rate of A to B in compartment 1
G2 = 1.0 ml/(g*min), // Conversion rate of A to B in compartment 2
A10 = 1 mM, // initial concentration of A in compartment 1
A20 = 0 mM, // initial concentration of A in compartment 2
B10 = 0 mM, // initial concentration of B in compartment 1
B20 = 0 mM; // initial concentration of B in compartment 2
// VARIABLES
real A1(t) mM, // concentration of A in compartment 1
A2(t) mM, // concentration of A in compartment 2
B1(t) mM, // concentration of B in compartment 1
B2(t) mM, // concentration of B in compartment 2
Equilibrium(t) mM; // Equilibrium concentration for B if G>0
// else for A if G=0
// INITIAL CONDITIONS
when (t=t.min) {A1 = A10; A2 = A20; B1 = B10; B2 = B20; }
//ORDINARY DIFFERENTIAL EQUATIONS
A1:t = (PSa/V1)*(A2-A1)-(G1/V1)*A1;
B1:t = (PSb/V1)*(B2-B1)+(G1/V1)*A1;
A2:t = (PSa/V2)*(A1-A2)-(G2/V2)*A2;
B2:t = (PSb/V2)*(B1-B2)+(G2/V2)*A2;
Equilibrium = (V1*(A10+B10)+V2*(A20+B20) )/(V1+V2); //Mass balance check
} // END OF MODEL
/*
FIGURE:
+-----------+ +-----------+
| G1 | PSa | G2 |
| A1---->B1 |<-------->| A2---->B2 |
| V1 | PSb | V2 |
+-----------+ +-----------+
DETAILED DESCRIPTION:
This is a two compartment model for two substances, A
and B. Both substances can passively move from one
compartment to the other. A is irreversibly converted to
B in either or both compartments.
KEY WORDS:
Course, compartment, compartmental, tutorial, exchange,
multiple compartments, flux, steady state, reaction,
conversion
REFERENCES: None.
REVISION HISTORY:
JBB: Equilibrium(t) changed to a function of time. 6apr2012
JSim SOFTWARE COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
JSim software was developed with support from NIH grants HL088516,
and HL073598. Please cite these grants in any publication for which
this software is used and send one reprint of published abstracts or
articles to the address given below. Academic use is unrestricted.
Software may be copied so long as this copyright notice is included.
Copyright (C) 1999-2012 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/