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# TranspMM.2sided.Comp2F

Comparison of 1-sided and 2 sided Michaelis-Menten transporters in a two compartment model with flow.

Model number: 0018

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## Description

Figure 1: Concentrations of A1, A2, B1, and B2 are plotted as functions of time. The normalized values of PS (PSa/PSmax and PSb/PSmax) are also plotted as functions of time.

Two types of a saturable Michaelis-Menten transporter are considered in this two compartment model with flow--a one sided transporter for solute A and a two-sided transporter for solute B. The model for solute A is cis- side driven. Concentration of A in V1, A1, determines the fractional saturation, PSa/PSmax, where PSmax is Vmax/Km, and PSa = PSmax/(1 + A1/Km). The model for solute B is cis-trans driven. Concentration of B in both V1 and V2, B1 and B2 respectively, determine the fractional saturation, where PSmax is Vmax/Km and PSb = PSmax/(1 + B1/Km + B2/Km).

## Equations

#### One sided Michaelis-Menten Transporter

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

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

#### Ordinary Differential Equations

$\large {\frac {d}{dt}}{\it A_1} \left( t \right) = {\frac {{\it Flow}\, \left( {\it A_{in}}-{\it A_1} \left( t \right) \right) }{{\it V_1}}} + {\frac {{\it PS_a}\, \left( { \it A_2} \left( t \right) -{\it A_1} \left( t \right) \right) }{{\it V_1 }}}$
$\large {\frac {d}{dt}}{\it A_2} \left( t \right) = {\frac {{\it PS_a}\, \left( {\it A_1} \left( t \right) -{\it A_2} \left( t \right) \right) }{{\it V_2}}} -{\frac {{\it G2}\,{\it A_2} \left( t \right) }{{\it V_2}}}$

#### Initial Conditions

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

#### Two sided Michaelis-Menten Transporter

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

$\large {\it PS_b}={\frac {{\it PS_{max}}} { {1}+{\it B_1} \left( t \right)/K_m + {\it B_2} \left{ t \right)/K_m }$

#### Ordinary Differential Equations

$\large {\frac {d}{dt}}{\it B_1} \left( t \right) = {\frac {{\it Flow}\, \left( {\it A_{in}}-{\it B_1} \left( t \right) \right) }{{\it V_1}}} + {\frac {{\it PS_b}\, \left( { \it B_2} \left( t \right) -{\it B_1} \left( t \right) \right) }{{\it V_1 }}}$
$\large {\frac {d}{dt}}{\it B_2} \left( t \right) = {\frac {{\it PS_b}\, \left( {\it B_1} \left( t \right) -{\it B_2} \left( t \right) \right) }{{\it V_2}}} -{\frac {{\it G2}\,{\it B_2} \left( t \right) }{{\it V_2}}}$

#### Initial Conditions

$\large {\it B_1} \left( 0 \right) ={\it B_{10}}$
$\large {\it B_2} \left( 0 \right) ={\it B_{20}}$

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: 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.



None.

## Key Terms

Compartment, Michaelis-Menten, MM, transporter, 1 sided, 2 sided, initial velocity