Transp2sol.Comp2F

Facilitating Transporter for 2 competing solutes including binding steps. Shows countertransport facilitation/inhibition Enzymatic conversion in V2.

Model number: 0011

Run Model:

Help running a JSim model.

(Java model applet may take 10-20 seconds to load.)

Figure

The input function Ain(t) = 1 mM, a constant. Solute A is transported from V1 to V2 by a transporter that can bind to either A or B. The transporter can flip from side to side in the unbound form T, or in the bound forms, TA and TB. In V2, A is converted to B by a saturable Michaelis-Menten process.

When the rate of flipping as unbound transporter, T1 and T2, is less than the rate for the bound transporter, TA1, TB1, TA2, and TB2, then the presence of B in V2 facilitates the transfer of A from V1 to V2 because it makes more unbound T1 on side 1 than if there were no B in V2. This can be seen by the increase in transmembrane extraction of A after 35-45 seconds when B2 has built up enough TB2 and TB1, which then delivers B to V1 giving more unbound T1 so A1 diminishes.

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 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 totaled 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.

Equations

${\it Consump_{A_2}}={\frac {G_{maxA_2}} {K_{mA_2} +A_2}}$
${\it SoV_1}={\frac {{\it Surf}}{{\it V_1}}} \,\,\,\,\,\,\, {\it SoV_2}={\frac {{\it Surf}}{{\it V_2}}}$
${\it K_{dA_1}}={\frac {{\it k_{offA_1}}}{{\it k_{onA_1}}}} \,\,\,\,\,\,\, {\it K_{dA_2}}={\frac {{\it k_{offA_2}}}{{\it k_{onA_2}}}}$
${\it K_{dB_1}}={\frac {{\it k_{offB_1}}}{{\it k_{onB_1}}}} \,\,\,\,\,\,\, {\it K_{dB_2}}={\frac {{\it k_{offB_2}}}{{\it k_{onB_2}}}}$

Ordinary Differential Equations

${\frac {d {\it A_1}}{dt}} = {\frac {-{\it Flow}\cdot L\cdot \left( A_{in}-A_1 \right) }{{\it V_1}}} +{ \it SoV_1}\cdot \left( {\it k_{offA_1}}\cdot {\it TA_1} -{\it k_{onA_1}}\cdot {\it A_1}\cdot { \it T_1} \right)$

${\frac {d {\it A_2} }{dt}} = {\frac { -{\it Consump_{A_2}} \cdot {\it A_2}}{{\it V_2}} } +{ \it SoV_2}\cdot \left( {\it k_{offA_2}}\cdot {\it TA_2} -{\it k_{onA_2}}\cdot {\it A_2}\cdot { \it T_2} \right)$

${\frac {d {\it B_1}}{dt}} = {\frac {-{\it Flow}\cdot L\cdot \left( B_{in}-B_1 \right) {{\it V_1}}}} +{ \it SoV_1}\cdot \left( {\it k_{offB_1}}\cdot {\it TB_1} -{\it k_{onB_1}}\cdot {\it B_1}\cdot { \it T_1} \right)$

${\frac {d {\it B_2} }{dt}} = {\frac { {\it Consump_{A_2}} \cdot {\it A_2}}{{\it V_2}}} +{ \it SoV_2}\cdot \left( {\it k_{offB_2}}\cdot {\it TB_2} -{\it k_{onB_2}}\cdot {\it B_2}\cdot { \it T_2} \right)$

${\frac {d {\it T_1} }{dt}} = { \left( {\it k_{offA_1}}\cdot {\it TA_1} -{\it k_{onA_1}}\cdot {\it A_1}\cdot { \it T_1} \right) +{ \left( {\it k_{offB_1}}\cdot {\it TB_1} -{\it k_{onB_1}}\cdot {\it B_1}\cdot { \it T_1} \right) + \left({\it kT_{21}}\cdot {\it T_2}-{\it kT_{12}}\cdot {\it T_1}\right)$

${\frac {d {\it T_2} }{dt}} = { \left( {\it k_{offA_2}}\cdot {\it TA_2} -{\it k_{onA_2}}\cdot {\it A_2}\cdot { \it T_2} \right) +{ \left( {\it k_{offB_2}}\cdot {\it TB_2} -{\it k_{onB_2}}\cdot {\it B_2}\cdot { \it T_2} \right) - \left({\it kT_{21}}\cdot {\it T_2}-{\it kT_{12}}\cdot {\it T_1}\right)$

${\frac {d {\it TA_1} }{dt}} = -{ \left( {\it k_{offA_1}}\cdot {\it TA_1} -{\it k_{onA_1}}\cdot {\it A_1}\cdot { \it T_1} \right) + \left({\it kTA_{21}}\cdot {\it TA_2}-{\it kTA_{12}}\cdot {\it TA_1}\right)$

${\frac {d {\it TA_2} }{dt}} = -{ \left( {\it k_{offA_2}}\cdot {\it TA_2} -{\it k_{onA_2}}\cdot {\it A_2}\cdot { \it T_2} \right) - \left({\it kTA_{21}}\cdot {\it TA_2}-{\it kTA_{12}}\cdot {\it TA_1}\right)$

${\frac {d {\it TB_1} }{dt}} = -{ \left( {\it k_{offB_1}}\cdot {\it TB_1} -{\it k_{onB_1}}\cdot {\it B_1}\cdot { \it T_1} \right) + \left({\it kTB_{21}}\cdot {\it TB_2}-{\it kTB_{12}}\cdot {\it TB_1}\right)$

${\frac {d {\it TB_2} }{dt}} = -{ \left( {\it k_{offB_2}}\cdot {\it TB_2} -{\it k_{onB_2}}\cdot {\it B_2}\cdot { \it T_2} \right) - \left({\it kTB_{21}}\cdot {\it TB_2}-{\it kTB_{12}}\cdot {\it TB_1}\right)$

Transporter Mass Conservation

${\it T_{total}}={ {\it T_1}+{\it T_2}+{\it TA_1}+{\it TA_2} +{\it TB_1}+{\it TB_2}$

Substrate Mass Conservation

$\int _{{\it t_{min}}}^{{\it t_{max}}}\!{\it Flow}\, \left( {\it A_{in}} -{\it A_{out}}+{\it B_{in}}-{\it B_{out}} \right) {dt} \,= \,\sum {{\it V_1}\cdot \left( {\it A_1} +{ \it B_1} \right) +{\it V_2}\cdot \left( {\it A_2}+{\it B_2} \right) +{\it Surf}\cdot \left( {\it TA_1}+{\it TA_2}+{\it TB_1}+{\it TB_2} \right) }$

WARNING: Thermodynamic constraint are not included in the model. For a passive transporter, the transport rate constants should satisfy the following constraints:

${\frac {{\it kTA_{12}}\cdot {\it kT_{21}}\cdot {\it k_{onA_1}}\cdot {\it k_{offA_2}}} {{\it kTA_{21} }\cdot {\it kT_{12}}\cdot {\it k_{offA_1}}\cdot {\it k_{onA_2}}}}=1\ \ and\ \ {\frac {{\it kTB_{12}}\cdot {\it kT_{21}}\cdot {\it k_{onB_1}}\cdot {\it k_{offB_2}}} {{\it kTB_{21} }\cdot {\it kT_{12}}\cdot {\it k_{offB_1}}\cdot {\it k_{onB_2}}}}=1$

The equations for this model may also be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.

Download JSim project file

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.

Related Models

Master Two Compartment Transporter Model (includes all cases):

TransComp2: 2 compartments, flow/no flow, 1 or 2 solutes, 2 types of reactions, 6 types of transporters

Transporter models from Compartment Tutorial (mostly passive exchange):

Two Compartment Michaelis-Menten (MM) Transporter Models:

Two Compartment 2-sided Facilitated Transporter (T1-T2) Models:

Two Region 2-sided Facilitated Transporter (T1-T2) Models:

Transp2sol.Distrib2F: 2 regions with flow, 2 solutes, both T1-T2 and passive transporters, Michaelis-Menten enzymatic reaction, Counter-Transport Faciliation

Distributed(always have flow)
PM1sd1solD2F
PM2sd1solD2F
PM4_1sd2solD2F
TT2sd1solD2F
TT2sd2solD2F

Key Terms

Two solutes, competing solutes, enzymatic reaction, transmembrane flip, countertransporter, six state transporter, Flow, tutorial

Model Feedback

We welcome comments and feedback for this model. Please use the button below to send comments:

Model History

Get Model history in CVS.

Acknowledgements

Please cite www.physiome.org in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

[This page was last modified 20Jul10, 9:00 am.]

Model development and archiving support at physiome.org provided by the following grants: NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.