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# TranspMM.2sided.Comp2

Comparison of 1-sided and 2 sided Michaelis-Menten transporters in a two compartment model without flow.

Model number: 0017

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## Description

Figure3: Determining Vmax and Km from initial velocity: The quantity, VmaxEstB = V2*B2/t.max is plotted (Black Lines) against B1+B2. The uppermost point of each nearly vertical line is an estimate of the velocity of the transporter between 0 and 1 second. VmaxEstB equals approximately V2*dB2/dt = PSeffective*(B1-B2).

As the Michaelis-Menten transporter becomes saturated (increasing values of B1+B2 ranging over 6 orders of magnitude), the uppermost points of the vertical lines asymptotically approach Vmax (Green Line). Half of the value, Vmax/2 (Red line) intersects the upper outline of these vertical lines where B1+B2 = Km (dashed Blue line).

Two types of a saturable Michaelis-Menten transporter are considered in this two compartment model without 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

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

#### 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

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

#### 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.



## 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

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

## Model History

Get Model history in CVS.

## 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.