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?
// MODEL NUMBER: 0039
// MODEL NAME: Anderson_JC_2007_fig2
// SHORT DESCRIPTION: Equilibrium conc for increasing kon for two compartment model.
// Figure 2 of "Tracers in Physiological Systems Modeling".
JSim v1.1
import nsrunit;
unit conversion on;
math comp2.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
C10 = 1 mM, // initial concn C in compartment 1
C20 = 0 mM, // initial concn C in compartment 2
CB0 = 0 mM, // initial concn CB in compartment 1
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
extern real CinA(t);
// Initial conditions
when(t=t.min) {C1=C10; C2=C20; CB = CB0;}
real C1tr(t) pM, C2tr(t) pM, CBtr(t) pM, fluxtr1(t);//tracers for C
real C1tr0 = 1 pM, C2tr0 = 0 pM;
when(t=t.min) {C1tr = C1tr0; C2tr=C2tr0; CBtr = 0;
} //
// 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.
C1tr:t = -(PS/V1)*(C1tr-C2tr) - kon*C1tr*B + koff*CBtr;
C2tr:t = (PS/V2)*(C1tr-C2tr);
//B = Btot - CB -CBtr;
CBtr:t = kon*C1tr*B - koff*CBtr; //bound reacer
fluxtr1 = (C1tr/C1)*C1:t;
}
/* Template on how to solve quadratic for instantaneous binding
km1 = 0.142 1/s, //Dissoc rate for fa-alb site 1
Kd1 = 3e-5 mM, // Equilib dissoc const for alb-fa, 30nM
kp1 = km1 / Kd1, //rate of binding of F to A
CAt = 0.6 mM, // High Albumin Conc Source
CFt = 0.54 mM, // High Fatty Acid Conc Source
real solpa, solpb, solp1, solp2, CAF1, CAF2, CF1, CF2,
CA1, CA2, CAF0, CA0, CF0;
real CA(t,x) mM, CF(t,x) mM, CAF(t,x) mM;
real AFR(t) mM, AR(t) mM, R(t) mM;
// Solve quadratic for concentrations of A, AF and F in source solution
solpa = (0.5)*(CFt+CAt+Kd1);
solpb = (0.5)*(sqrt(CFt^2-2*CAt*CFt+2*CFt*Kd1+CAt^2+2*CAt*Kd1+Kd1^2));
solp1 = solpa+solpb;
solp2 = solpa-solpb;
CAF1 = solp1;
CAF2 = solp2;
CF1 = CFt-solp1;
CF2 = CFt-solp2;
CA1 = CAt-solp1;
CA2 = CAt-solp2;
CAF0 = if ( CAF1 > 0 and CF1 > 0 and CA1 > 0 ) CAF1
else CAF2;
CA0 = if ( CAF1 > 0 and CF1 > 0 and CA1 > 0 ) CA1
else CA2;
CF0 = if ( CAF1 > 0 and CF1 > 0 and CA1 > 0 ) CF1
else CF2;
*/
/*
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
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
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.
*/