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# Comp2FlowMMExchangeReaction

Two compartment model with flow, Michaelis-Menten type exchanger, and a reversible reaction in non-flowing compartment converting C to B.

Model number: 0249

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## Description

This is a two compartmental (plasma and parenchymal cell (pc) exchange model with flow in the plasma compartment. The compartments are instantaneously well mixed. Substances C and B reversibly convert to each other by a first order process in the parenchymal cell compartment. This model has a Michaelis-Menten (MM) type of transporter of the cell membrane between the capillary plasma and the parenchymal cell, expressed in terms of Km and PSmax, for which the denominators include terms for the solute on both sides of the meembrane, as if the binding site is accessible simultaneously from either side. In this sense the transporter is acting more like a channel that a ferry boat.

• Understanding the Comp2FlowMMExchangeReaction Model. (PDF file)

## Equations

#### Ordinary Differential Equations

$\large {\frac {d}{dt}}{\it C_p} \left( t \right) ={\frac {{\it F_p}\cdot \left( { \it C_{in}}-{\it C_p} \right) }{{\it V_p}}}+{\frac {{\it PS_{maxC}}\cdot{\it C_{pc}} }{{\it V_p}\cdot \left( {\it K_{mc}}+{\it C_{pc}}+{\it C_{p}} \right) }}-{\frac {{\it PS_{maxC} }\cdot{\it C_p}}{{\it V_p}\cdot \left( {\it K_{mc}}+{\it C_p}+{\it C_{pc}} \right) }}$

$\large {\frac {d}{dt}}{\it B_p} \left( t \right) =-{\frac {{\it F_p}\cdot{\it B_p}} {{\it V_p}}}+{\frac {{\it PS_{maxB}}\,{\it B_{pc}}}{{\it V_p}\cdot \left( {\it K_{mB}} +{\it B_{pc}}+{\it B_{c}} \right) }}-{\frac {{\it PS_{maxB}}\cdot{\it B_p}}{{\it V_p}\cdot \left( {\it K_{mB}}+{\it B_p}+{\it B_{pc}} \right) }}$

$\large {\frac {d}{dt}}{\it C_{pc}} \left( t \right) =-{\it G_{c2b}}\cdot{\it C_{pc}} +{ \it G_{b2c}}\cdot{\it B_{pc}}-{\frac {{\it PS_{maxC}}\cdot{\it C_{pc}}}{{\it V_{pc}}\cdot \left( {\it K_{mC}}+{\it C_{pc}}+{\it C_{p}} \right) }}+{\frac {{\it PS_{maxC}}\cdot{\it C_p}} {{\it V_{pc}}\cdot \left( {\it K_{mC}}+{\it C_p}+{\it C_{pc}} \right) }}$

$\large {\frac {d}{dt}}{\it B_{pc}} \left( t \right) ={\it G_{c2b}}\cdot{\it C_{pc}} -{\it G_{b2c}}\cdot{\it B_{pc}}-{\frac {{\it PS_{maxB}}\cdot{\it B_{pc}}}{{\it V_{pc}}\cdot \left( { \it K_{mB}}+{\it B_{pc}}+{\it B_{c}} \right) }}+{\frac {{\it PS_{maxB}}\cdot{\it B_p}}{{\it V_{pc}}\cdot \left( {\it K_{mB}}+{\it B_p}+{\it B_pc} \right) }}$

#### Initial Conditions

$\large {\it C_p} \left( 0 \right) ={\it C_p{0}}$$\large {\it C_{pc}} \left( 0 \right) ={\it C_{pc}0}$$\large {\it B_p} \left( 0 \right) ={\it B_p{0}}$ ,  and   $\large {\it B_{pc}} \left( 0 \right) ={\it B_{pc}0}$ .

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

 Wilfred D Stein. Transport and Diffusion across Cell Membranes, 2nd Ed.



## Related Models

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## Key Terms

Course, compartment, compartmental, tutorial, exchange, multiple compartments, flux, steady state, reaction, conversion, flow, plasma, parenchymal cell, Michaelis-Menten, transporter