Figure 1: Concentrations: Default parameter set.
The concentrations A1(t) (solid red), and A2(t) ( solid blue),
B1(t)(dashed orange), and B2(t) dashed green are plotted as
functions of time.
Questions:
(1) Why are the initial slopes of A1 and B1 equal? A2 and B2?
(2) After about 300 seconds, the output curves have the same slop.
This is apparent on a log-log plot. What is the slop (analytic answer
required.) Why?
Figure 2: PSa/PSmax vs. A2; PSb vs B1+B2: Default parameter set
PSa/PSmax = 1/(1+A1/Km). When A1=Km, PS/PSmax=0.5.
Therefore the where 0.5 intersects the red line (appears
as dashed red and blue) a vertical line drawn through that
point will intersect the A1-axis at A1=Km.
PSb/PBmax will lay on top of the original curve when plotted
against the sum of B1 + B2.
Figure3: Velocity vs. Concentration (LOOPS): InitVel parameter set.
Load the InitVel parameter page and run LOOPS.
The quantity,
VmaxEstB = V2*B2/t.max is plotted (Black Lines) against . The uppermost
point of each nearly vertical line is an estimate of the velocity
of the transporter between 0 and 1 second. Note that V1 has been
set to a very large value so there is essentially no back flux.
V2*B2/t.max when t.max is 1 second, is effectively
V2*dB2/dt = PSeffective*(B1-B2).
B2<<B1 during the first second. As the Michaelis-Menten transporter
becomes saturated (increasing values of B10 ranging over 6 orders of
magnitude in LOOPS), the uppermost points of the vertical lines
V2*dB2/dt asymptotically approach Vmax (Green Line). Half
of the value, Vmax/2 (Red line) intersects the upper outline of
these vertical lines where B1+B2 = 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>0.6*Km and A1<1.4*Km) VmaxEst else 0 vs. B1+B2 (curve 4).
Figure 4: Exponential Fit to A1, A2, B1 and B2: Default Parameter Set
Plotting log(Concentrations/(1 mM)) vs. time shows that the concentrations are
exponentially decaying at the same rate. The equation for the rate curve is
C*exp(-k*t). Adjust C and k so that it fits the tail of the curves.
What is this rate analytically? Derive the analytic expression. (Hint:
Use dimensional analysis, i.e., what combination of parameters yield an
inverse time scale?) See what values are needed when you change parameters one
at a time.
/* MODEL NUMBER: 0018
MODEL NAME: TranspMM.2sided.Comp2F
SHORT DESCRIPTION: Comparison of 1-sided and 2 sided Michaelis-Menten
transporters in a two compartment model with flow.
*/
import nsrunit; unit conversion on;
math TranspMM_2sided_Comp2F {
// INDEPENDENT VARIABLE
realDomain t sec; t.min = 0; t.max = 1000; t.delta = 0.01;
// PARAMETERS
real Flow = 0.02 ml/(g*s), // Flow into and out of compartment 1
V1 = 1 ml/g, // Compartment 1 volume
V2 = 1 ml/g, // Compartment 2 volume
Vmax = 1 umol/(g*s), // Vmax is maximum flux at 100% saturation.
Km = 1 mM, // Equilibrium dissociation constant
PSmax ml/(g*s), // Equals Vmax/Km
G2 = 0 ml/(g*s), // Gulosity, first order consumption in V2
A10 = 1 mM, // Initial concentration of A in compartment 1
A20 = 0 mM, // Initial concentration of A in compartment 2
B10 = 1 mM, // Initial concentration of B in compartment 1
B20 = 0 mM; // Initial concentration of B in compartment 2
extern real Ain(t) mM; // Inflowing concentration of A or B into compartment 1
PSmax = Vmax/Km;
// DEPENDENT VARIABLES
real A1(t) mM, // Concentration of A in compartment 1
A2(t) mM, // Concentration of A in compartment 2
PSa(t) ml/(g*s); // Exchange rate of A between compartments
PSa = PSmax/(1 + A1/Km);
real B1(t) mM, // Concentration of B in compartment 1
B2(t) mM, // Concentration of B in compartment 2
PSb(t) ml/(g*s); // Exchange rate of B between compartments
PSb = PSmax/(1 + B1/Km + B2/Km);
// INITIAL CONDITIONS
when(t=t.min) {A1 = A10; A2 = A20;
B1 = B10; B2 = B20;}
// ODES FOR 1-SIDED VERSION
A1:t = Flow/V1*(Ain-A1) -PSa*(A1-A2)/V1;
A2:t = PSa*(A1-A2)/V2 - G2*A2/V2;
// ODES FOR 2-SIDED VERSION
B1:t = Flow/V1*(Ain -B1) -PSb*(B1 - B2)/V1;
B2:t = PSb*(B1 - B2)/V2 - G2*B2/V2;
// INITIAL VELOCITY
real VmaxEstB(t) umol/(g*sec);
VmaxEstB= V2*B2/t.max;
// EXPONENTIAL FIT
real C = 1 mM, k = 0.01 s^(-1), ExpFit(t) mM;
ExpFit = C*exp(-k*t);
}
/*
DIAGRAMS:
A 1 sided transporter for solute A.
+----------------+
|V1 A1(t) |
Flow*Ain ----> ---->Flow*A1
| PS | PS = PSmax/(1 + A1/Km)
+-------|--------+
| v |
| A2(t) G2(t)|
|V2 |
+----------------+
A 2 sided transporter for solute B.
+----------------+
|V1 B1(t) |
Flow*Ain ----> ---->Flow*A1
| PS | PS = PSmax/(1 + B1/Km + B2/Km)
+-------|--------+
| v |
| B2(t) G2(t)|
|V2 |
+----------------+
DETAILED DESCRIPTION:
Two types of a saturable Michaelis-Menten transporter are considered
in this two compartment model with flow--a one sided transporter
for solute A and a two-sided transporter for solute B.
The model for solute A is cis- side driven. Concentration of A in V1, A1,
determines the fractional saturation, PSa/PSmax, where PSmax is Vmax/Km, and
PSa = PSmax/(1 + A1/Km).
The model for solute B is cis-trans driven. Concentration of B in both
V1 and V2, B1 and B2 respectively, determine the fractional saturation, where
PSmax is Vmax/Km and
PSb = PSmax/(1 + B1/Km + B2/Km).
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: Compartment, Michaelis-Menten, MM, transporter, 1 sided,
2 sided, initial velocity
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: Nov/08
Revised by : GMR Date: June/2010
Revision: REvised to standard 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 an email
with the citation and, if possible, a PDF file of the paper to: staff@bioeng.washington.edu.
*/