Transp2sol.Comp2

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

Model number: 0010

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Description

Transp2sol.Comp2 is a six state transporter model for 2 solutes in which compete for the transporter site on both sides of a membrane between two instantaneously mixed compartments. In compartment 2, A is reacted to form B in an enzymatic reaction approximated by a Michaelis-Menten expression without any accounting for binding of solutes to the enzyme. When the rates of conformational state change for transmembrane flipping of transporters complexed with solutes, TA1, TA2, TB1 and TB2, are high compared to that for uncomplexed transporter, T1 and T2, then the model behaves much like an obligatory countertransporter, exchanging B for A across the membrane.

MODEL VERIFICATION:
Total Solute is conserved.

Substrate = V1*(A1+B1)+V2*(A2+B2) + Surf*(TA1+TA2+TB1+TB2); i

Total transporter is conserved.
Ttotal = Surf*(T1+TA1+TB1+T2+TA2+TB2);

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

${\frac {d {\it A_1}}{dt}} = { \it SoV_1}\cdot \left( {\it k_{offA_1}}\cdot {\it TA_1} -{\it k_{onA_1}}\cdot {\it A_1}\cdot { \it T_1} \right)$

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

${\frac {d {\it B_1}}{dt}} = { \it SoV_1}\cdot \left( {\it k_{offB_1}}\cdot {\it TB_1} -{\it k_{onB_1}}\cdot {\it B_1}\cdot { \it T_1} \right)$

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

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

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

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

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

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

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

Transporter Mass Conservation

${\it T_{total}}={\it Surf} \cdot \left({ {\it T_1}+{\it T_2}+{\it TA_1}+{\it TA_2} +{\it TB_1}+{\it TB_2} \right)$

Substrate Mass Conservation

${\it Substrate(t) } = \, {\it V_1}\cdot \left( {\it A_1}+{\it B_1} \right) +{\it V_2}\cdot \left( {\it A_2}+{\it B_2} \right) +{\it Surf}\cdot \left( {\it TA_1}+{\it TA_2}+{\it TB_1}+{\it TB_2} \right) }$

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.

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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.

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

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

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Key Terms

Two solutes, competing solutes, enzymatic reaction, transmembrane flip, countertransporter, six state transporter, tutorial, Transp2sol, two compartment

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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 22Jul10, 11:02 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.