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# TranspMM.1sided.Comp2F

Model for two compartments with flow, 1 solute, 1 sided Michaelis-Menten transporter.

Model number: 0015

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## Description

Figure 4: Exponential Fit to A1 and A2: Default Parameter Set Plotting log(A1) and log(A2) vs. time shows that the concentrations are exponentially decaying at the same rate. Derive an analytic expression for this rate.

Compartmental models are based on mass balance equations. A compartment has a volume, V, and a time-varying concentration of a substance, A(t). An underlying assumption of compartmental models is that the material in the compartment is instantaneously well mixed. Compartmental models are also call well-stirred tank models.

This is a compartmental model for facilitated exchange between two chambers separated by a membrane. It is an open model (with inflow and outflow), with volumes V1 and V2 for each compartment, time dependent concentrations A1(t) and A2(t) respectively, and an exchange coefficient PS. G2 is for Gulosity, the first order consumption of the solute in V2. This model assumes instantaneous solute binding to a Michaelis-Menten type transporter, with only a single site available from the V1 side of the membrane. Fluxes are set by the concentration of A1(t) in V1. A1 determines the fractional saturation, PS/PSmax.

See model TranspMM.2sided.Comp2F.proj to compare with a transporter binding on either side of the membrane

## Equations

#### One sided Michaelis-Menten Transporter

$\large {\it PS_{max}}={\frac {{\it V_{max}}}{{\it K_m}}}$

$\large {\it PS}={\frac {{\it PS_{max}}}{{1}+{\it A_1} \left( t \right)/K_m }}$

#### Ordinary Differential Equations

$\large {\frac {d}{dt}}{\it A_1} \left( t \right) = {\frac {{\it Flow}\, \left( {\it Ain}-{\it A1} \right) }{{\it V1}}}+ {\frac {{\it PS}\, \left( { \it A_2} \left( t \right) -{\it A_1} \left( t \right) \right) }{{\it V1 }}}$
$\large {\frac {d}{dt}}{\it A_2} \left( t \right) = {\frac {{\it PS}\, \left( {\it A_1} \left( t \right) -{\it A_2} \left( t \right) \right) }{{\it V2}}} -{\frac {{\it G2}\,{\it A_2} \left( t \right) }{{\it V2}}}$

#### Initial Conditions

$\large {\it A_1} \left( 0 \right) ={\it A_{10}}$
$\large {\it A_2} \left( 0 \right) ={\it A_{20}}$

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:

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.

## 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

Compartmental, Exchange, Mixing Chamber, MM, Vmax, Km, Michaelis-Menten, one sided, facilitated exchange