TrnasFac2G.proj
18jun08
JBBnotes:
The key feature of the model is countertransport facilitation. See Default parameter set where the transport of solute A from V1 to V2 drives solute B from V2 to V! so that B1 exceeds B2 even though B is being formed from A only in V2. It eventually equilibrates so that B1 = B2 and A is all used up.
"Default" parameter set: PLOTPAGES: Concn.t and MM
Shows transport of A1 to A2 and reaction in V2 from A to B.
This is a six state transporter model for 2 solutes in competition. Two solute species compete for the transporter site on either side of a membrane between two mixing chambers. In chamber 2 A is reacted to form B in an enzymatic reaction approximated by a Michaelis Menten expression, and without.any accounting for binding of substrate or product to the enzyme.
MODEL VERIFICATION: Total Mass is conserved: Substrate in solution is totalled as SubstrateV, and substrate bound to transporter as SubstrateM, for membrane bound. Total transporter conservation is forced through the equation for T2.
The MM reaction is at 50% of maximum at the Km, shown on the PLOTGPAGE labeled MM.
ObligCC.par: PLOTPAGE: Concn.t
A1=10, B2=6. kT12=kT21=0
Flux occurs until TA1=TA2 and TB1=TB2, and then stops,
leaving gradients in both A and B, but not in either of TA nor TB.
LeakObligCC.par: PLOTPAGE: Concn.t.
Same as ObligCC.par but kT12 = kT21 = 100. This is not a solute leak but a normal non-obligatory transporter configuration.
Get gradual equilibrium in both A and B.
(This can be speeded up by increasing Ttot.)
CCfacil.par Same plot Concn.t:
Addition of B to side 2 enhances flux of A from 1->2 by delivering more T to side 1 that would be the case without it. Conditions are: kT12 = KT21 = low;
KTB21=kTB12 = higher than KT12.
The stored par set with kTB > kTA >kT and with concns A10 and B20 near the Kd
causes an overshoot in A2 so that it becomes higher than A1, while at the same time during the first 10 seconds, B1 and B2 are converging. Note in Plot 2 that the total transporter bound is over 50% at the end of 10 seconds, and this accounts for a significant fraction of the total A and B.
Fig 8-11 is for the Tranpsort Book chapter.
Init Vel: Similar to Fig 8-3 and Initvel.tiff but with slow transporter binding.the apparent Km is shifter to higher concentrations.
With par set "Fig8.9 and konA1=0.1, then the ends of the trajectories are fitted by Rmax = 4.1e-4 and AppKm = 0.04 mM
konA1= 0.01 Rmax = 5.8e-5 and AppKm = 0.3 mM
konA1= 1 = 8.3e-4 and = 0.024 mM
konA1= 100 = 9.0e-4 = 0.017 mM though Kd= 0.01 mM
Because the curve for Rmas does not show as dots unles the volume is made so small that several points are plotted it is necessary to make V1 small to see the points at higher A1. This creates a problem at the lower A1__init since A1 diminshes strongly when V1 is small, so that the trajectories of A1 are slanted toward lower concenrations aa dA2/dt rises.
FIX: the plot routine to show the dots at the chosen size.
Scientific conclusion:
Lowering konA1 reduces AppKm as well as lowering Rmax.
/* MODEL NUMBER: 0010
MODEL NAME: Transp2sol.Comp2
SHORT DESCRIPTION: Facilitating Transporter for 2 competing solutes including binding steps.
Shows countertransport facilitation/inhibition Enymatic conversion in V2.
*/
import nsrunit; unit conversion on;
math Transp2sol.Comp2 {realDomain t sec; t.min = 0; t.max = 1e4; t.delta = 1;
real V1 = 1 ml/g, //Volume 1
V2 = 1 ml/g, //Volume 2
Surf= 1 cm^2/g, //Surface area for exchange
Ttot= 0.1 umol/cm^2;//Transporter conc per unit surf area
real KdA1 = 100 uM, KdA2 = 100 uM,//Equilib dissoc const on each side, solute A
KdB1 = 10 uM, KdB2 = 10 uM, //Equilib dissoc const on each side, solute B
konA1 = 1 uM^(-1)*s^(-1), konA2 = 1 uM^(-1)*s^(-1), //on rates, solute A
konB1 = 1 uM^(-1)*s^(-1), konB2 = 1 uM^(-1)*s^(-1), //on rates, solute B
koffA1= KdA1*konA1, koffA2= KdA2*konA2, //off rates s^(-1) solute A
koffB1= KdB1*konB1, koffB2= KdB2*konB2, //off rates s^(-1) solute B
kT12 = 3 s^(-1), kT21 = 3 s^(-1), //porter flip rate 1->2 & 2->1
kTA12 = 100 s^(-1), kTA21 = 100 s^(-1), //TA flip rates
kTB12 = 100 s^(-1), kTB21 = 100 s^(-1), //TB flip rates
KmA2 = 0.4 mM, //Km for consump of A in V2
GmaxA2= 0.1 umol/(g*min), // consump A, Vmax for A to B reaction
SoV1 = Surf/V1, SoV2 = Surf/V2; // surface to volume ratios
real TestA = kTA12*kT21*konA1*koffA2 /kTA21*kT12*koffA1*konA2, //NE 1? where's ATP?
TestB = kTB12*kT21*konB1*koffB2 /kTB21*kT12*koffB1*konB2; //NE 1? where's ATP?
// STATE VARIABLES
real A1(t) mM, A2(t) mM, B1(t) mM, B2(t) mM, // Solute concns
TA1(t) umol/cm^2, TA2(t) umol/cm^2, // TA concns
TB1(t) umol/cm^2, TB2(t) umol/cm^2, // TB concns
T1(t) umol/cm^2, T2(t) umol/cm^2, // Free transporter concns
GA2(t) ml*min^(-1)*g^(-1), // Consumption rate
SubstrateV(t) umol/g, SubstrateM(t) umol/g, SubstrateTot(t) umol/g; //Totals
// INITIAL CONDITIONS
when(t=t.min) {A1 = 10; A2 = 0; B1 = 0; B2 = 0;
TA1 = 0; TA2 = 0; TB1 = 0; TB2 = 0; T1 = 0.5*Ttot; }
// ODEs
GA2 = GmaxA2 /(KmA2 + A2);
A1:t = SoV1*(koffA1*TA1 - konA1*A1*T1);
A2:t = SoV2*(koffA2*TA2 - konA2*A2*T2) - GA2*A2 / V2;
B1:t = SoV1*(koffB1*TB1 - konB1*B1*T1);
B2:t = SoV2*(koffB2*TB2 - konB2*B2*T2) + GA2*A2 / V2;
T1:t = -(konA1*A1 + konB1*B1)*T1 + koffA1*TA1 + koffB1*TB1 - 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;
TB1:t = konB1*B1*T1 - koffB1*TB1 - kTB12*TB1 + kTB21*TB2;
TB2:t = konB2*B2*T2 - koffB2*TB2 + kTB12*TB1 - kTB21*TB2;
T2 = Ttot - TA1 - TA2 - TB1 - TB2 - T1; //Conservation of transporter.
// Check on Apparent Km:
real mult = 1, Rate(t) mM/s, Rmax mM/s, AppKm mM;
Rmax = 5e-4; AppKm = 1e-2;
Rate = Rmax*A1/(AppKm + A1);
// Mass conservation check for the closed system of V1 and V2: //Mass Balance?
SubstrateV = V1*(A1+B1) + V2*(A2+B2); //Substrate Amt in solution
SubstrateM = Surf*(TA1+TA2+TB1+TB2); //Substrate bound to transporter
SubstrateTot= SubstrateV + SubstrateM;
} // end
/*
FIGURE:
// ___________________________________________
// |V1 [A1] [B1] |
// | konA1 /^ ^\ konB1 |
// | //koffA1 koffB1\\ |
// | <-// \\-> |
// ----------TA1<-------->T1<------->TB1-------
// | /|| /|| /|| |
// |S kTA21||kTA12 kT21||kT12 kTB21||kTB12 |
// | ||/ ||/ ||/ |
// ----------TA2<-------->T2<-------->TB2------
// |V2 <-\\ //-> |
// | \\kdA2 koffB2// |
// | kbA2 \\ Enz // konB2 |
// | [A2] ------> [B2] |
// --------------------------------------------
DETAILED DESCRIPTION:
This model is a six state transporter model for 2 solutes in competition.
Two solute species compete for the transporter site on either side of a
membrane between two mixing chambers. In chamber 2, A is reacted to form B
in an enzymatic reaction approximated by a Michaelis Menten expression,
and without any accounting for binding of substrate or product to the
enzyme. When the rates of conformational state change for transmembrane
flipping of TA and TB are high compared to that for uncomplexed transporter T,
then the model behaves much like an obligatory countertransporter, exchanging
B for A across the membrane;
MODEL VERIFICATION: Total Mass is conserved: Substrate in solution is
totalled as SubstrateV, and substrate bound to transporter as SubstrateM,
for membrane bound. Total transporter conservation is forced through the
equation for T2.
// WARNING: An additional thermodynamic constraint is not included in the model.
// For a passive transporter, the transport rate constants should satisfy
// the following constraints:
//
// kTA12*kT21*konA1*koffA2
// ------------------------ = 1 (1) see TestA
// kTA21*kT12*koffA1*konA2
//
// kTB12*kT21*konB1*koffB2
// ------------------------ = 1 (2) see TestB
// kTB21*kT12*koffB1*konB2
//
// These constraints ensure that the model runs to equlibrium at steady-state.
// If these ratios deviate from 1, the model will run to a steady-state
// net concentration gradient. This would be the case if the transporter
// is coupled to a energy source, which is not explicitly modeled here.
SHORTCOMINGS/GENERAL COMMENTS:
ASSUMPTIONS:
1. Compartmental assumptions apply to the solutions on either side of the
membrane. These are: Instantaneously stirred tank. No concentration gradients.
No diffusion limitation for reactions.
2. Reactions are first order with fixed rates, not fractal.
KNOWN BUGS: These calculations are subject to numerical
round off error under certain conditions, such as when
kdA1/koffA1 >> kTA12/kTA21.
This occurs because the net flux (A1:t) is calculated as the
difference of two much larger unidirectional fluxes.
Calculations of state variables (concentrations) are accurate.
KEY WORDS: Two solutes, competing solutes, enzymatic reaction, transmembrane flip, countertransporter,
six state transporter, tutorial, Transp2sol, two compartment
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 : Update format of comments
Revised by : BEJ Date: 30/Mar/11 : Edit comments
Revised 19dec08 by J Bassingthwaighte
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 1999-2011 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.
*/