Notes on TranspMM.2sol2sided.Comp2 20dec08
It should be clear from perusing the TranspMM set of models that the Michaelis-Menten concepts for an enzymatgic reaction cannot be applied exactly to a transporter across a membrane.The representation of:
PS = PSmax/(1 + A1/KmA + A2/KmA +B1/KmB + B2/KmB)
is based on the reasonable assumption that A and B have distinct Kms
and that the affinity is the same for each solute on both sides of the membrane. As seen on the PS.vs.Conc Plot using the.
Parameter set: MM.par for plots Conc_t and PS.vs.Conc
Parameter Set Conct and Plotpage Cont_t: 20dec08
t.min = 0
t.max = 300
t.delta = .01
V1 = 1
V2 = 1
PSmax = 1
GA2 = 2
GB2 = .1
KmA = 1
KmB = 1
A1init = 50
A2init = 0
B1init = 1
B2init = 0
Shortcomings: (1) trans side binding is not accounted for separately,
(2) No accounting for capacitance for solute binding to transporter..
Conclusion: A simple MM tranpsorter cannot be correct, because:
1. Cis and trans sides not treated identically.
2. No capacitance in membrane, and thus no Barrer time lag.
3. No oppportunity for sidedness due to coupling with energy source..
Running the program:
1. ParSet MM: Plotpages: Conc_t and PS.vs.Conc
. On PlotPage "Conc_t". Initially A1 decreases linearly with time, as does B1.
Put A1(t) on a new plot page and make the axes semilog to check whether or not it is exactly exponential. What is the rate constant for the exponential? What should the rate constant be? How close is it to G2/(V1+V2)?
View Plotpage PS.vs.Conc:
The PS/PSmax for solutes A and B is (1 + (A1+A2)/KmA +(B!+B2)/KmB).Is this true when KmA is DIFFERENT from KmB?
The curve for PS/PSmax appears to have the expected form when the two Km's are the same, namely PS/PSmax =0.5 at A1 +A2+B1+B2 = Km, 0.909 at Concn = 0.1Km, 0.99 at concn = Km/100, 0.0909 at 10Km, etc
What happens when KmA is NOT eqault to KmB.Explore not only different Kms, but different initial concnetrations. When is the curve asymmetric? Under what circumstances does Ps/PSmax go through 50% at KmB rather than KmA?
2. Initial velocity experiments.Par set "Initvelpar", Loops run
t.min = 0
t.max = 1
t.delta = 0.1
V1 = 1000
V2 = 10
PSmax = 1
G2 = 0
Km = 0.1
A1init = 1E-6
A2init = 0
B1init = 1
B2init = 0
Parameter set:"InitVel" plotpage; 'InitVel"
Initial velocity experiments are based on the premise that when the trans-concentration is zero, the rate of rise of concentration on side 2, V2*dA2/dt, gives an estimate of the unidirectional flux from side 1, the cis side, to side 2, the trans side. The flux V2*dA2/dt = PSeff*A1.
This therefore provides a means to estimate PSeff; by using a series of experiments initiated with A1 at varied initial concentrations. The slope dA2/dt = PSeff(A10)/V2. In the upper panel of Plotpage "InitVel" the values of A2 versus time are plotted.
In the lower panel are plotted the sequence of values of A2 using the loop mode to provide a sequence of values of A2(t=0)from 1e-6 mM to 2 to the 21th times that, a million fold range. The last or highest-valued point at each A1 represents the last point calculated at t = 1 second, and so represents almost exactly dA2/dt times 1 second, which is PSeff/V2. A plot of these 1 second values is a plot of PSeff(A1)/V2 versus A1.
The plot shows that these 1 second values of A2 plateau at high concnetrations where PSeff/V2 = PSmax/V2. The value of A2 at which PSeff/V2 = PSmax/V2 = Km, the apparent Michaelis constant, in this case at A1 =1e-4 mM or 0,1 uM.
Review the Barrer time lag experiments in Chapter 6 Diffusion. The Barrer analysis showed that there was a delay related to the thickness of the membrane. It appears here that the time intercept for each line dA2/dt is at the origin. Why is there no time lag?
JSim v1.1 /* Model TranspMM.2sol2sided.Comp2:
MODEL NUMBER: 0023
MODEL NAME: TranspMM.2sol2sided.Comp2
SHORT DESCRIPTION: Michaelis-Menten type transport governed by solutes A and B.
*/
import nsrunit; unit uM = 1e-6 M; unit conversion on;
math TranspM2sol2sided_Comp2 { realDomain t sec; t.min = 0; t.max = 1000; t.delta = 0.01;
//PARAMETERS
real V1 = 1 ml/g, // Volume 1
V2 = 1 ml/g, // Volume 2
PSmax = 1 ml/(g*s), // PSmax is max PS at 0% saturation.
GA2 = 2 ml/(g*s), // Gulosity, consumes A in V2 forming B
GB2 = 0.1 ml/(g*s), // Gulosity, consumes A in V2
KmA = 1 mM, // Equilib dissoc const on each side
KmB = 1 mM, // Equilib dissoc const on each side
A1init = 1 mM, A2init = 0 mM, // Initial concns in V1 and V2
B1init = 1 mM, B2init = 0 mM;
// State variables
real A1(t) mM, A2(t) mM, // Solute A conc side 1, side 2
B1(t) mM, B2(t) mM, // Solute B conc side 1, side 2;
PS(t) ml/(g*s); // Permeability-surface area product
// initial conditions
when(t=t.min) {A1 = A1init; A2 = A2init; B1 = B1init; B2 = B2init;}
// ODEs:FOR TWO-SIDED INFLUENCES
PS = PSmax/(1+ (A1+A2)/KmA + (B1+B2)/KmB); // PS calc
V1*A1:t = - PS*(A1 - A2);
V2*A2:t = PS*(A1 - A2) - GA2*A2; //first order (proportional) reaction
V1*B1:t = - PS*(B1 - B2);
V2*B2:t = PS*(B1 - B2) + GA2*A2 - GB2*B2;
}
/*
// Diagram:
// __________________
// |V1 A1(t),B1(t) |
// | |
// | ^ PS | PS = PSmax/(1 + (A1+A2)/KmA + (B1+B2)/KmB)
// --------|--------|
// | v |
// | A2(t),B2(t) |
// |V2 A--->B |
// -----------------|
//
DETAILED DESCRIPTION:
A facilitated transporter kinetic model assuming instantaneous solute binding to transporter
of Michaelis-Menten type, single site available from both sides of the membrane.
This model contains two solutes in two compartments, giving concentrations A1, A2, B1, and B2.
Permeability-Surface Area product (PS) is determined by the concentration of A1 and B2.
PSmax is max PS at 0% saturation. There is Gulosity in V2. In V2, A is consumed, forming B. B
is also consumed. Two solutes A and B compete, but have different Km's.
PS is governed by concentrations of (A1+A2)/KmA and (B1 + B2)/KmB.
Reaction of solute A --> solute B unidirectionally in V2 only. Reaction to
degrade B2 to a solute not influencing the transport occurs also in V2.
SHORTCOMINGS/GENERAL COMMENTS:
- Specific inadequacies or next level steps
KEY WORDS: Gulosity, Michaelis-Menten, two sided, two compartment, two solutes, Permeability Surface,
Equilibrium Dissociation Constant, tutorial
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:
Original Author : JBB Date: 26/Dec/08
Revised by : BEJ Date: 16/Dec/09
Revision: 1) Update format of comments
2) Cgange units of V, G, Vmax
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.
*/