// MODEL NUMBER: 0159
/* MODEL NAME: Comp2.Binding
SHORT DESCRIPTION: Models two compartments with a single substance
passively exchanging between the two compartments, plus first order
binding to solute Z to form ZC, both of which are confined to V1.
*/
import nsrunit; unit conversion on;
math Comp2Binding {
// INDEPENDENT VARIABLE
realDomain t s; t.min=0; t.max=10; t.delta=0.1;
// PARAMETERS
real C10 = 1.5 mM, // Initial concentration in first compartment
C20 = 0.50 mM, // Initial concentration in second compartment
Z0 = 0.2 mM, // Init concn of free Z, a binding site, in V1
ZC0 = 0.2 mM, // Init concn of ZC in V1
KdZ = 1 mM, // Equil dissoc const for C binding to Z
Ztot = 1 mM, // Total amount: Ztot = Z + ZC, should be constant
konZ = 1 1/(mM*s), // on-binding rate for C to Z
koffZ= KdZ*konZ, // off- or unbinding rate for C from CZ
V1 = 0.05 ml, // Volume of first compartment
// units are milliliters per gram of tissue
V2 = 0.15 ml, // Volume of second compartment
PS = 5 ml/s; // Exchange rate
// VARIABLES
real C1(t) mM, // Time dependent concentration in first compartment
C2(t) mM, // Time dependent concentration in second compartment
Z(t) mM, // Time dependent binding site concn in V1
ZC(t) mM; // Time dependent bound ZC concn in V1
// INITIAL CONDITIONS
when(t=t.min) { C1 = C10; C2 = C20; ZC = 0; }
// ORDINARY DIFFERENTIAL EQUATIONS
C1:t= (PS/V1)*(C2-C1) - konZ*C1*Z + koffZ*ZC;
C2:t=-(PS/V2)*(C2-C1);
ZC:t = konZ*C1*Z - koffZ*ZC;
Z = Ztot - ZC; // forcing mass balance for Z
/* ANALYTIC SOLUTIONS from Maple(TM) */
real analyticC1(t) mM,
analyticC2(t) mM,
Exponential(t) dimensionless;
Exponential = exp(-PS*(V1+V2)*t/(V1*V2));
analyticC1(t) = (1/(V1+V2)) * (C20*V2+C10*V1
+ V2*(C10-C20)*Exponential);
analyticC2(t) = (1/(V1+V2)) * (C20*V2+C10*V1
- V1*(C10-C20)*Exponential);
//Mass Check
real Caverage(t) mM, MassTotC(t) umol;
MassTotC = V1*(C1 +ZC) + V2*C2;
Caverage = (V1*(C1 +ZC) + V2*C2)/(V1+V2);
} // END OF MODEL
/*
FIGURE:
+----------+ +----------+
| C1 | PS | C2 |
|Z+C1<->ZC |<-------->| |
| V1 | | V2 |
+----------+ +----------+
DETAILED DESCRIPTION:
This is a model of two compartments, one with volume V1
and initial concentration C10, and the other with volume
V2 and concentration C20. The two compartments exchange
material at a rate PS. PS is the product of the permeability
multiplying the surface area. The exchange is passive and
equal in both directions in this model.
In this newer version, built upon model #245 in the JSim library
at www.physiome.org, is a first order binding site Z in V1. It still
contains the analyticC1 and ..C2 from the parent model the time
course in the absence of a binding site, and which track the solutions
for C1 and C2 whenthere is no binding substance, Ztot= 0.
Par Set KdDown: With Kd for binding = 10 mM and Ztot =10 mM,
and initial values of C1 and C2 at 20 and 2 mM, there is a rapid
early rise in the bound form ZC (blue) in the first 0.3 s, after which it
drops toward its final steady state level. C1 and C2 take smooth routes
to the final equilibrium level, necessarily below the level reached in
the absence of the binding site (black and red lines).
Running the loops, lowering the Kd for the site from 100 to 10^-3 mM:
at Kd = 100 mM, the affinity is low, and the model solutions for C1 and C2
are just a little lower than for the result with Ztot=0, But with higher
affinities, Kds down to 1e-3, the final concentrations are lowered to about
half as high, but no further. No further because ZC almost = Ztot,
i.e. all the binding sites are filled. To lower the concentrations more
requires increasing Ztot. Why just from 10 mM to 6.5 mM? Because with the
inital conditions of Kd =10 mM and C1 = 20 mM and C2 = 2 mM,i.e. there was
not enough binder, Z, to soak up even half of C1 when more C entered V1
from V2 via the PS = 0.1 ml/sec.
ParSet ZtotUp: Using parameter values as for Par Set KdDown,
the effect of increasing the binding capacity is readily predicable;
The early peak in ZC around 0.3 sec remains evident at all levels, a
consequence of the combination of PS, the V's and the kon. The dot-dot
curves are for the highest Ztot, = 40 mM (blue) and the lowest equilibrium
concentration of unbound C1 (black dot-dot at about 3.7 mM), and the lowest
C2 (red dot-dot) equilibrating with C1 at 3.7 mM. Summary, higher Ztot has
little effect on the time to equilibrate, but lowers the free equilibirum
concentrations unidirectionally with the increase.
KEY WORDS:
Course, compartment, compartmental, tutorial, exchange,
multiple compartments, flux, steady state, binding kinetics, buffering
REFERENCES: None.
REVISION HISTORY:
Author JBB,190823: Plagerizing the basic model #245 of a 2-compartment system
to add the binding solute.
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2019 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.
When citing JSim please use this reference: Butterworth E, Jardine BE, Raymond GM, Neal ML, Bassingthwaighte JB.
JSim, an open-source modeling system for data analysis [v3; ref status: indexed, http://f1000r.es/3n0]
F1000Research 2014, 2:288 (doi: 10.12688/f1000research.2-288.v3)
This software was developed with support from NIH grants HL088516 and HL073598, NIBIB grant BE08417,
the Cardiac Energy Grid HL122199 (PI: J.B. Bassingthwaighte), and the Virtual Physiological Rat program
GM094503 (PI: D.A.Beard). Please cite these grants 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@physiome.org.
*/