Figure 1: Concentrations: Default parameter set.
The concentrations A1(t) (red) and A2(t) blue, are plotted as
functions of time.
Figure 2: PS (normalised) vs. log(A1/(1 mM) ): Default parameter set
PS/PSmax = 1/(1+A1/Km). When A1=Km, PS/PSmax=0.5.
Therefore the red line intersects the curve when A1=Km=
10^(-1) mM. Log10(Km/(1 mM)) = -1.
Figure 3: Velocity vs. Concentration (LOOPS): InitVel parameter set.
Load the InitVel parameter page and run LOOPS.
The quantity,
VmaxEst = V2*A2/t.max is plotted (Black Lines). The uppermost
point of each nearly vertical line is an estimate of the velocity
of the transporter between 0 and 1 second.
V2*A2/t.max when t.max is 1 second, is effectively
V2*dA2/dt = PSeffective*(A1-A2).
A2<<A1 during the first second. As the Michaelis-Menten transporter
becomes saturated (increasing values of fgen_1.Pulse1.amplitude
ranging over 6 orders of
magnitude in LOOPS), the uppermost points of the vertical lines
V2*dA2/dt asymptotically approach Vmax (Green Line). Half
of the value, Vmax/2 (Red line) intersects the upper outline of
these vertical lines where A2 = Km (Blue line, various textures
depending on value of Km). Vary both Vmax and Km (choose values of
Km which lie at the intersection of the black lines with the
horizontal axis. The plot of the blue line(s) is
if(A1>0.8*Km and A1<1.2*Km) VmaxEst else 0 vs. A1 (curve 4).
Figure 4: Exponential Fit to A1 and A2: Default Parameter Set
Plotting log(A1) and log(A2) vs. time shows that the concentrations are
exponentially decaying at the same rate. What is this rate analytically?
Derive the analytic expression. (Hint: Use dimensional analysis, i.e.,
what combination of parameters yield an inverse time scale?) Use C
(amplitude constant) and k rate constant to fit curves. See what values are
needed when you change parameters one at a time.
/* MODEL NUMBER: 0015
MODEL NAME: TranspMM.1sided.Comp2F
SHORT DESCRIPTION: Model for two compartments with flow, 1 solute, 1 sided
Michaelis-Menten transporter. */
import nsrunit; unit conversion on;
math TranspMM21idedComp2F {
// INDEPENDENT VARIABLE
realDomain t s; t.min = 0; t.max = 1000; t.delta = 0.1;
//PARAMETERS
real Flow = 0.016666 ml/(g*s), // Flow
V1 = 1 ml/g, // Volume 1
V2 = 1 ml/g, // Volume 2
Vmax = .01 umol/(g*s), // Vmax is max flux at 100% saturation.
Km = .01 mM, // Equilib dissoc const on each side
PSmax ml/(g*sec), // Maximum exchange rate
G2 = .02 ml/(g*s), // Gulosity, first order consumption in V2
A10 = 1 mM, // Initial concentration in V1
A20 = 0 mM; // Initial concentration in V2
extern real Ain(t) mM; // Inflow concentration of A into V1
PSmax = Vmax/Km;
// DEPENDENT VARIABLES
real A1(t) mM, // Solute A concentration on side 1
A2(t) mM, // Solute A concentration on side 2
PS(t) ml/(g*s); // Exchange rate
// INITIAL CONDITIONS
when(t=t.min) {A1 = A10; A2 = A20; }
// ALGEBRAIC AND ORDINARY DIFFERENTIAL EQUATIONS
PS = PSmax/(1+A1/Km); // Permeability-Surface are product (PS)
// depending only on concentration A1(t)
A1:t = Flow/V2*(Ain-A1) -PS*(A1-A2)/V1;
A2:t = PS*(A1-A2)/V2 - G2*A2/V2;
// INITIAL VELOCITY CALCULATION
real VmaxEst(t) umol/(g*s);
VmaxEst = V2*A2/t.max;
// EXPONENTIAL DECAY RATE
real C = 1 mM, k = 0.025 s^(-1);
/* Fit of Michaelis-Menten equation to upper outline of
InitVel parameter set plot run with 41 loops*/
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);
}
/*
FIGURE:
+----------------+
|V1 A1(t) |
Flow*Ain ----> ---->Flow*A1
| PS | PS = PSmax/(1 + A1/Km)
+-------|--------+
| v |
| A2(t) G2(t)|
|V2 |
+----------------+
DETAILED DESCRIPTION:
Compartmental models are based on mass balance equations.
A compartment has a volume, V, and a time-varying
concentration of a substance, A(t). An underlying
assumption of compartmental models is that the material
in the compartment is instantaneously well mixed.
Compartmental models are also call well-stirred tank
models.
This is a compartmental model for facilitated exchange
between two chambers separated by a membrane. It is an open
model (with inflow and outflow), with volumes V1 and V2 for each
compartment, time dependent concentrations A1(t) and A2(t)
respectively, and an exchange coefficient PS. G2 is for Gulosity,
the first order consumption of the solute in V2. This model assumes
instantaneous solute binding to a Michaelis-Menten type
transporter, with only a single site available from the V1
side of the membrane. Fluxes are set by the concentration of
A1(t) in V1. A1 determines the fractional saturation, PS/PSmax.
See model TranspMM.2sided.Comp2F.proj to compare with a
transporter binding on either side of the membrane
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS:
Compartmental, Exchange, Mixing Chamber, MM, Vmax, Km,
Michaelis-Menten, one sided, facilitated exchange
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.
REVISION HISTORY:
Original Author : JBB Date: 06/12/07
Revised by : JBB Date: 08/Dec/2008
Revision: 1) Terms to allow flowing input are commented out
Revised by : BEJ Date: 15/Dec/09
Revision: 1) Update comment format.
2) Updated units of G, Vmax, PS, V
Revised by : GR Date June/10
Revision: Revised for standard format
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) 1999-2009 University of Washington.
Contact Information:
The National Simulation Resource,
Director J. B. Bassingthwaighte,
Department of Bioengineering,
University of Washington, Seattle, WA
98195-5061
*/