CONC PLOT PAGE:
Figure 1: Solute Concentrations: Default Parameter set
Figure 2: Transporter Concentrations: Default Parameter set
Shows transport of A1 to A2 and reaction in V2 converting
A to B in an enzymatic process modeled using Michaelis
Menten reaction kinetics. This is a six state transporter model for 2 solutes
competing for the transporter.
All concentrations are plotted as functions of time.
The key feature of the model is countertransport
facilitation. The transport of solute A from V1 to V2
drives solute B from V2 to V1 so that B1 exceeds B2
even though B is being formed from A only in V2. Steady
state equilibration has B2>B1 (B1 is being removed by flow.)
Multiply t.max and t.delta both by 100 to see the steady state
equilibrium.
AMOUNT PLOT PAGE:
MODEL VERIFICATION: Conservation of mass of both solute and transporter
is verified by the following calculations for each.
Solute Mass Balance
SubstrateT(t) = V1*(A1+B1)+V2*(A2+V2) + Surf*(TA1+TB1+TA2+TB2) (solid red)
/t

Substrate(t) =  Flow (Ain(t')  A1(t') + Bin(t')  B1(t') ) dt'

/
0
(dashed purple)
Transporter(t) = Ttot (green dashed line)
= Surf*(T1(t)+TA1(t)+TB1(t)+T2(t)+TA2(t)+TB2(t) ) (solid blue)
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 nonobligatory 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.
20dec08:
The addition of Flow to bring in A or B is controlled by setting the inputs Ain and Bin. When, during a prolonged infusion of Ain, B is introduced
as set by fgen_2 as a pulse. one sees that the flux of A is inhibited.This transient inhibition will be seen later in the "bolus sweep" multiple indicator dilution technique: the transient raising of nontracer mother substance accompanying a bolus of tracer, with concentrations ranging from below the Km to above the Km allows efficient estimation of the Km and minimizing physiological changes in response to the solute.
// MODEL NUMBER: 0011
// MODEL NAME: Transp2sol.Comp2F
// SHORT DESCRIPTION: Facilitating Transporter for 2 competing solutes including
// binding steps. Shows countertransport facilitation/inhibition. Substrate A is
// converted to B in region 2.
import nsrunit; unit conversion on;
math Transp2sol.Comp2F {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
Flow= 1 ml/(g*min),//Flow into V1
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?
real 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
ConsumpA2(t) ml*min^(1)*g^(1), // Consumption rate
SubstrateV(t) umol/g, SubstrateM(t) umol/g, SubstrateTot(t) umol/g; //Totals
extern real Ain(t) mM; extern real Bin(t) mM;
// 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
A1:t = SoV1*(koffA1*TA1  konA1*A1*T1) + Flow*(Ain  A1)/V1;
ConsumpA2 = GmaxA2 /(KmA2 + A2);
A2:t = SoV2*(koffA2*TA2  konA2*A2*T2)  ConsumpA2*A2 / V2;
B1:t = SoV1*(koffB1*TB1  konB1*B1*T1) + Flow*(Bin  B1)/V1;
B2:t = SoV2*(koffB2*TB2  konB2*B2*T2) + ConsumpA2*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.
// 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;
real Substrate(t) umol/g;
when(t=t.min) Substrate=SubstrateTot;
Substrate:t = Flow*(AinA1+BinB1);
} // end
/* DIAGRAM
___________________________________________
Flow V1 [A1] [B1]  Flow
> konA1 /^ ^\ konB1 >
Ain  //koffA1 koffB1\\  Aout
Bin  <// \\>  Bout
TA1<>T1<>TB1
 / / / 
S kTA21kTA12 kT21kT12 kTB21kTB12 
 / / / 
TA2<>T2<>TB2
V2 <\\ //> 
 \\kdA2 koffB2// 
 kbA2 \\ Enz // konB2 
 [A2] > [B2] 

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 steadystate.
If these ratios deviate from 1, the model will run to a steadystate
net concentration gradient when the inflowing concentrations are constant.
This would be the case if the transporter is coupled to a energy source,
which is not explicitly modeled here.
DETAILED DESCRIPTION:
Transp2sol.Comp2F 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 compartment 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; With the initial substrate concentrations high
compared to the transporter binding affinities, KdA or KdB, most of the
transporter is bound and the system behaves as an obligatory countertransporter,
though the gradient eventually dissipates.
MODEL VERIFICATION: Total Mass is conserved: Substrate in solution is
totaled as SubstrateV, and substrate bound to transporter as SubstrateM,
for membrane bound.
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.
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, Flow, Tutorial
REFERENCES:
Klingenberg M. Membrane protein oligomeric structure and transport function. Nature 290: 449454, 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: 109183, 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: H1502H1511, 2000.
Foster DM and Jacquez JA. An analysis of the adequacy of the asymmetric carrier model for sugar transport.
Biochim Biophys Acta 436: 210221, 1976.
REVISION HISTORY:
Original Author : JBB Date: Jun/07
Revised by : JBB Date: 20/Dec/08
Revision: 1) Revised from Transp2sol.Comp2 to include Input via Flow, and washout
Revised by : BEJ Date: 16/12/09
Revision: 1) Update format of comments
COPYRIGHT AND REQUEST FOR ACKNOWLEDGMENT OF USE:
Copyright (C) 19992009 University of Washington. From the National Simulation Resource,
Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 981955061.
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.
*/