Note on comp1.react:
1. With G = zero, Flow =0.01 ml/sec and V1 = .05 ml, the mean
transit time through the mixing chamber is 5 seconds.
With CinA = 0, and A10 = 1 mM, the concn should fall to 1/e times
the initial concentraion in 5 sec.
The solution shows that A(t=5sec) = 0.36787944; 1/e = 0.367879
so the solutioon seems good to six figures at least.
2. With an input CinA = 0 mM, G = 0.01 ml/sec, Rev = 0 ml/sec,
and V1=0.05 ml, A falls as it is converted to B. With Flow set to 0,
the time to A10*(1/e) is 5.0 sec. as before, and the fraction of A
converted to B is 1-1/e = 0.632121.
3.With an input CinA = 1 mM, G = 0.01 ml/sec, Rev = 0.01 ml/sec,
and V1=0.05 ml, and with Flow set to 0.01 ml/sec, A falls as it
is converted to B. It levels off at A = 0.6666666 mM and B levels off
at 0.3333333 mM. Calculate to see if this is correct.
(Hint: In steady state, the derivatives = zero, so set A:t and B:t
to zero and solve the two simultaneous equations for A and B.
The answer is 2/3 and 1/3.) Did your computation depend on the value
of V1? (No it didn't.) Does the form of the solution depend on V1?
JSim v1.1
import nsrunit;
unit conversion on;
// MODEL NUMBER: 0041
// MODEL NAME: Anderson_JC_2007_fig4
// SHORT DESCRIPTION: Tracer added after tracee and binding site have equilibrated.
// Figure 4 of "Tracers in Physiological Systems Modeling".
math comp1.react {
realDomain t sec; t.min=0; t.max=30; t.delta=0.1; // independent variable
real C1(t) mM,C2(t) mM, // concn of C in compartment 1 & 2
B(t) mM, CB(t) mM, // concn of binder B in compartment 1
Qtest(t) mole, // check on continuing mass balance
C10 = 1 mM, // initial concn C in compartment 1
C20 = 0 mM, // initial concn C in compartment 2
PS = 1 ml/sec, // PS product
kon = 0.01 1/(mM*sec), // rate of C + B -> CB
kd = 1 mM, // equilibrium dissociation const
koff = kd*kon, // rate of dissociation of CB
Btot =.1 mM, // total vinding substance
V1 = 1 ml, // volume of compartment 1
V2 = 1 ml, // volume of compartment 1
Rate = 0.1 s^-1; // rate of tracer addition, scaling CinA(t)
extern real CinA(t) mM; // created using function generator, a delayed step fn.
// Initial conditions
when(t=t.min) {C1=C10; C2=C20; CB = 0;}
real C1tr(t) pM, C2tr(t) pM, CBtr(t) pM, fluxtr1(t); //tracers, tr, for C
real C1tr0 = 0 pM, C2tr0 = 0 pM;
real C1T(t) pM, C1T0 = 0 pM, C2T(t) pM, C2T0 = 0 pM; //tracer, T, for inst, binding
when(t=t.min) {C1tr = C1tr0; C2tr=C2tr0; CBtr = 0;
C1T = C1T0; C2T = C2T0; } //
// ODE's from mass balance
// Non-tracer xolute equations:
C1:t = -(PS/V1)*(C1-C2) - kon*C1*B + koff*CB;
C2:t = (PS/V2)*(C1-C2);
B = Btot - CB - CBtr;
CB:t = kon*C1*B - koff*CB;
//Conservation expession: C1:t +(V2/V1)*C2:t + CB:t = 0.
Qtest = V1*C1 + V2*C2 + V1*CB;
C1tr:t = -(PS/V1)*(C1tr-C2tr) - kon*C1tr*B + koff*CBtr + Rate*CinA;
C2tr:t = (PS/V2)*(C1tr-C2tr);
//B = Btot - CB -CBtr;
CBtr:t = kon*C1tr*B - koff*CBtr; //bound reacer
fluxtr1 = (C1tr/C1)*C1:t;
// Equations assuming instantaneous binding instead of slow binding for tracer:
C1T:t = -PS/(V1*(1+ Btot/kd))*(C1T-C2T);
C2T:t = (PS/V2)*(C1T-C2T);
}
/*
DETAILED DESCRIPTION:
Two compartment model with constant volumes, V1 and V2,
B is an uncomplexed binding site which reacts, first order, with C1 to form CB
The reverse dissociation reaction converts CB to C1 and B.
The dissociation const Kd = koff/kon and at equilibrium = C1*B/CB.
Reproduces figure 4 of Anderson JC 2007 paper.
SHORTCOMINGS/GENERAL COMMENTS:
- Look at Loops page of project to see how changes in kon and PS affect concentrations
of C1 and C2 for tracer.
KEY WORDS: tracer, tracee, metabolic physiologic modeling, lumped compartmental versus
spatially distributed systems, capillary-tissue exchange, membrane transporters, enzyme
reactions, steady state versus transient states.
REFERENCES:
Anderson JC and Bassingthwaighte JB: "Tracers in physiological systems modeling".
In: "Mathematical Modeling in Nutrition and Agriculture". Proc 9th International Conf on Mathematical
Modeling in Nutrition, Roanoke, VA, August 14-17, 2006, edited by Mark D. Hanigan JN and Casey L Marsteller.
Virginia Polytechnic Institute and State University, Blacksburg, VA 2007, pp 125-159.
REVISION HISTORY:
Original Author : JBB Date: 08/12/08
Revised by : BEJ Date: 05/26/2009
Revision: 1) Update comments and format
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 one reprint
to the address given above.
*/