Figures and Explanations
Figure 1: Concentration A1, A2 (LOOPS): Default
Parameter set.
This is a plot of A1 and A2 as function of time.
(1) Explore influence of time step size. Does
it make a difference which solver you use?
(2) Explore changes in initial concentration (Loop over
A10, the initial concentration of A1. At what value of
A10 does the output curve for A2 not change very much.
(Hint: Look at Figure 2: Binding konA1*A1*T1*SoV1, the
binding of A1 with the transporter.)
(3) What is the effect of increasing and decreasing the
rate of binding (konA1 and konA2) without changing KdA1 and
KdA2?
(4) What is the effect of increasing and decreasing the rate
of return of T2 to T1 (kT21)?
(5) Make kT12/kT21 assymetrical, e.g. set kT21=kT12*2.
What is the effect?
(6) Set the flow to 1 ml/sec. Explain the bends in the
concentration curve for A1.
Figure 3: Initial Velocity: InitVel parameter set.
Plotting A2/t.max vs. log(A1/ ( 1 mM) ) gives a set
of vertical lines over a range of different initial
concentrations for A1. A2/t.max is effective the
derivative of A2 with respect to time. saturate at high values
for the initial concentration of A1 (parameter A10).
The upper boundary of the vertical lines is the derivative
of A2 with respect to time. Since this set of runs has
t.max= 1 sec, we could also plot A2/t.max as an approximation
to the derivative.
Load the InitVel parameter set. Go to the Loops GUI
and change @*10 to @*1.4142. Change # of times to 41 and
run loops. The derivative A:t has a maximum value for
A10> 1 mM. We calculate
Vmax = koffA2*Ttot/2*SoV2 and
Km = (koffA1+koffA2)/konA1 .
Vmax (green solid line) and Vmax/2 (red solid line)
are plotted. The orange line is Km. Note that Km and
Vmax/2 intersect the upper boundary of the initial
velocity curves.
As long as the flipping rates are "fast" compared to the
on and off rates, this is reasonable. "Fast" is defined as
2 to 3 orders of magnitude faster. When the flip rates are
"slow" and unequal, the
Figure 4: Effects of site saturation on side 2: Satur8
parameter set
Use CVODE as the solver. Run for 1.0e04 seconds with
t.delta = 0.25 seconds. The conditions are Kd = 0.1 mM,
A1(0) = 100*Kd = 10 mM, A2(0) = 0 mM, V1 = 1 ml, V2 = 0.1 ml.
The transport rate is low. V2 is smaller than V1 so the
concentration A2 builds up faster than A1 is depleted.
A1 (dashed red curve) and A2 (dashed blue curve) are
plotted as functions of time.
V1*dA1/dt is the net flux out of V1 (solid Red curve).
V2*dA2/dt is the net flux into V2 (solid Blue curve).
(1) V2*dA2/dt has local minimum at approximately 250 seconds and a
local maximum at 3250 seconds. Explain both.
(2) What events cause the solid red and blue curves to differ?
(3) Does The flux out of V1 converge with the flux into
V2?
(4) What will be the final fluxes at time=infinity?
(5) Can the concentration of A2 become greater than A1?
/* MODEL NUMBER: 0008
MODEL NAME: Transp1sol.Comp2F
SHORT DESCRIPTION: Models a two compartment with flow, 1 solute, T1-T2 (facilitated
4-state transporter. Includes binding steps and transmembrane flip
rates for free and occupied transporters.
*/
import nsrunit; unit conversion on;
math Transp1solComp2F {
// INDEPENDENT VARIABLE
realDomain t sec; t.min = 0; t.max = 30; t.delta = 0.1;
//PARAMETERS
real Flow = 1 ml/min, //Flow into and out of compartment 1
V1 = 1 ml, //Volume 1
V2 = 1 ml, //Volume 2
Surf = 1 cm^2, //Surface area for exchange
Ttot = 0.1 umol/cm^2, //Total transporter concentration
KdA1 = 10 uM, //Equilibrium dissociation constant on side 1
// KdA1=koffA1/konA1
KdA2 = 10 uM, //Equilibrium dissociation constant on side 2
// KdA1=koffA2/konA2
konA1 = 1 uM^(-1)*s^(-1), //Binding rate side 1, solute A1
konA2 = 1 uM^(-1)*s^(-1), //Binding rate side 2, solute A2
koffA1 = KdA1*konA1, //Dissociation rate s^(-1) side 1, solute A1
koffA2 = KdA2*konA2, //Dissociation rate s^(-1) side 2, solute A2
kT12 = 100 s^(-1), //Free transporter flip rate 1->2
kTA12 = 100 s^(-1), //TA flip rate 1->2
kT21 = 100 s^(-1), //Free transporter return rate 2->1
A10 = 1 mM,
A20 = 0 mM,
kTA21 = 100 s^(-1), //TA return rate 2->1
test = kTA12*kT21*konA1*koffA2/(kTA21*kT12*koffA1*konA2);
//for Passive transport test= 1
extern real Ain(t) mM; // Inflow concentration from function generator
real SoV1 = Surf/V1;
real SoV2 = Surf/V2;
// DEPENDENT VARIABLES
real A1(t) mM, // A concentration in V1
A2(t) mM, // A concentration in V2
TA1(t) umol/cm^2, // Transporter complex with A1
TA2(t) umol/cm^2, // Transporter complex with A2
T1(t) umol/cm^2, // Free transporter on side 1
T2(t) umol/cm^2; // Free transporter on side 2
// INITIAL CONDITIONS
/* Note that the actual initial conditions depend
on solving The steady state equations at time =o. */
when(t=t.min) {A1 = A10; A2 = A20; }
when(t=t.min) {TA1 = 0; TA2 = 0; T1 = 0.5*Ttot; T2=0.5*Ttot;}
// ODEs
A1:t = koffA1*TA1*SoV1 - konA1*A1*T1*SoV1+Flow/V1*(Ain-A1);
A2:t = koffA2*TA2*SoV2 - konA2*A2*T2*SoV2;
T1:t = -konA1*A1*T1 + koffA1*TA1 - kT12*T1 + kT21*T2;
TA1:t = konA1*A1*T1 - koffA1*TA1 - kTA12*TA1 + kTA21*TA2;
TA2:t = konA2*A2*T2 - koffA2*TA2 + kTA12*TA1 - kTA21*TA2;
T2:t = -konA2*A2*T2 + koffA2*TA2 + kT12*T1 - kT21*T2;
/* Fit of Michaelis-Menten equation to upper outline of
InitVel parameter set plot run with 41 loops*/
real Vmax mM/sec, Km mM;
Vmax = koffA2*Ttot/2*SoV2;
Km = (koffA1+koffA2)/(konA1);
realDomain logMMA1 dimensionless; logMMA1.min=-3.1; logMMA1.max=3.1;
logMMA1.delta =0.01;
real MMA1(logMMA1) mM;
real dMMA2dt(logMMA1);
MMA1=(1 mM)*10.0^logMMA1;
dMMA2dt = Vmax*MMA1/(Km+MMA1);
/* CALCULATE EFFECTIVE PS
PS12 = (dA1/dt*V1-Flow/V1*(Ain-A1) )/(A2-A1)
PS21 = dA2/dt*V2/(A1-A2)
*/
real PS12(t) ml/min, PS21(t) ml/min;
PS12 = (koffA1*TA1*SoV1 - konA1*A1*T1*SoV1 -Flow/V1*(Ain-A1))*V1/(A2-A1);
PS21 = (koffA2*TA2*SoV2 - konA2*A2*T2*SoV2)*V2/(A1-A2);
}
/*
FIGURE:
This is a four state transporter model with parallel linear transport.
+-------------------------------+
|V1 [A1] |
Flow*Ain --> konA1 /^ -->Flow*A1
| //koffA1 |
| <-// |
+-----------TA1---------->T1----+
| /|| /|| |
|Surf kTA21 ||kTA12 kT21 ||kT12|
| ||/ ||/ |
+-----------TA2---------->T2<---+
|V2 <-\\ |
| \\koffA2 |
| konA2 \\-> |
| [A2] |
+-------------------------------+
DETAILED DESCRIPTION:
The model is a saturable four state transporter with a binding site on both sides
of the membrane for 1 solute in a two compartment model with flow into and out of
the first compartment. The binding site undergoes a conformational change, flipping
from side 1 to side 2 and back again when it is empty (T1<->T2) or filled (TA1<->TA2).
SHORTCOMINGS/GENERAL COMMENTS:
WARNING: An additional thermodynamic constraint is not included in the model.
For a passive transporter, the transport rate constants should satisfy
the following constraint:
kTA12*kT21*konA1*koffA2 kTA12*kT21*kdA2
------------------------ = 1 = test = ---------------.
kTA21*kT12*koffA1*konA2 kTA21*kT12*kdA1
This constraint ensures that the model runs to equilibrium at steady-state. If these ratios
deviate from 1, the model is assumed to be coupled to an energy source, as if it were active
transport, and creates a transmembrane concentration gradient. An energy source coupled
to a transporter is not explicitly model here.
KEY WORDS: two compartment, facilitated transporter, binding constants, single site,
noncompetitive binding, four state transporter, tutorial, flow
REFERENCES:
Klingenberg M. Membrane protein oligomeric structure and transport function. Nature 290: 449-454, 1981.
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.
Wilbrandt W and Rosenberg T. The concept of carrier transport and its corollaries in pharmacology.
Pharmacol Rev 13: 109-183, 1961.
Schwartz LM, Bukowski TR, Ploger JD, and Bassingthwaighte JB. Endothelial adenosin transporter
characterization in perfused guinea pig hearts. Am J Physiol Heart Circ Physiol 279: H1502-H1511, 2000.
Foster DM and Jacquez JA. An analysis of the adequacy of the asymmetric carrier model for sugar
transport. Biochim Biophys Acta 436: 210-221, 1976.
REVISION HISTORY:
Orig Author: J Bassingthwaighte Date: 2008
Modified 23 nov08 by J Bassingthwaighte
Modified by BEJ Date: Jan 2009
Revised by : BEJ Date: 16/Dec/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.
*/