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# Transp1sol.Comp2

Models a two compartment, 1 solute, T1-T2 (facilitated 4-state transporter. Includes binding steps and transmembrane flip rates for free and occupied transporters.

Model number: 0007

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## Description

The model is a saturable four state transporter with a binding site on both sides of the membrane for 1 solute in a two compartment model. The binding site undergoes a conformational change, flipping from side 1 to side 2 and back again when it is empty (T1<->T2) or filled (TA1<->TA2).

## Equations

#### 4 state transporter between two compartments without flow

$\large {\it SoV_1}={\frac {{\it Surf}}{{\it V_1}}} \,\,\,\,\,\,\, {\it SoV_2}={\frac {{\it Surf}}{{\it V_2}}}$

$\large {\it K_{dA_1}}={\frac {{\it k_{offA_1}}}{{\it k_{onA_1}}}} \,\,\,\,\,\,\, {\it K_{dA_2}}={\frac {{\it k_{offA_2}}}{{\it k_{onA_2}}}}$

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

$\large {\frac {d}{dt}}{\it A_2} \left( t \right) =\, {\it k_{offA_2}}\cdot {\it TA_2}\cdot { \it SoV_2}-{\it k_{onA_2}}\cdot {\it A_2}\cdot {\it T_2}\cdot {\it SoV_2}$

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

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

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

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

where
Surf is the surface area of the membrane,
V1 is the volume of compartment 1,
V2 is the volume of compartment 2,
KdA1 and KdA2 are equilibrium dissociation constants,
konA1 and konA2 are the on rate constants,
koffA1 and koffA2 are the off rate constants,
KT12 is the flip rate for the empty transporter from side 1 to side 2,
KT21 is the flip rate for the empty transporter from side 2 to side 1,
KTA12 is the flip rate for the filled transporter from side 1 to side 2,   and
KTA21 is the flip rate for the filled transporter from side 2 to side 1.

WARNING: An additional thermodynamic constraint is not included in the model. For a passive transporter, the transport rate constants should satisfy the following constraint:

$\large {\frac {{\it kTA_{12}}\cdot {\it kT_{21}}\cdot {\it k_{onA_1}}\cdot {\it k_{offA_2}}} {{\it kTA_{21} }\cdot {\it kT_{12}}\cdot {\it k_{offA_1}}\cdot {\it k_{onA_2}}}}=1$

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.

## 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):

Transporter models from Compartment Tutorial (mostly passive exchange): Two Compartment Michaelis-Menten (MM) Transporter Models: Two Compartment 2-sided Facilitated Transporter (T1-T2) Models:

## Key Terms

two compartment, facilitated transporter, binding constants, single site, noncompetitive binding, four state transporter, tutorial