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# Comp1Reaction

Models single compartment with reversible reaction C becoming D and D becoming C with rate constants Gc2d and Gd2c. Uses non-physiological units.

Model number: 0243

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## Description

In a single compartment, C is converted to D with rate constant Gc2d, and D is converted back to C with rate constant Gd2c. This is an "in vitro" beaker experiment. Numeric and analytic solutions are given. The concentrations at long time are calculated.

Further reading:

## Equations

#### Ordinary Differential Equations

$\large {\frac {d}{dt}}C \left( t \right) =-{\frac {{\it G_{c2d}}\ \cdot C \left( t \right) }{V}}+{\frac {{\it G_{d2c}}\cdot {D} \left( t \right) }{V}}$
$\large {\frac {d}{dt}} {D} \left( t \right) =+{\frac {{\it G_{c2d}}\cdot C \left( t \right) }{V}}-{\frac {{\it G_{d2c}}\cdot {D} \left( t \right) }{V}}$

#### Initial Conditions

$\large {\it C} \left( 0 \right) ={\it C_0}$ ,   $\large {\it D} \left( 0 \right) ={\it D_0}$ .

#### Analytic Solutions

$\large {\it C_{analytic}}= \frac {\left( {\it G_{d2c}}\, \left( {\it D_0}+{\it C_0} \right) + \left( {\it G_{c2d}}\,{\it C_0}-{\it D_0}\,{\it G_{d2c}} \right) {e^{-{ \frac { \left( {\it G_{c2d}}+{\it G_{d2c}} \right) t}{V}}}} \right) } { \left( {\it G_{c2d}}+{\it G_{d2c}} \right) }$
$\large {\it D_{analytic}}= \frac {\left( {\it G_{c2d}}\, \left( {\it D_0}+{\it C_0} \right) + \left( {\it G_{c2d}}\,{\it C_0}-{\it D_0}\,{\it G_{d2c}} \right) {e^{-{ \frac { \left( {\it G_{c2d}}+{\it G_{d2c}} \right) t}{V}}}} \right) } { \left( {\it G_{c2d}}+{\it G_{d2c}} \right) }$

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

 None.



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## Key Terms

Course, compartment, compartmental, tutorial, conversion, reaction, flux

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## 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.

[This page was last modified 14Mar18, 3:17 pm.]

Model development and archiving support at physiome.org provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.