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# Progress3.Enz

Three substrates converted by two 1st order reversible enzymatic reactions.

Model number: 0268

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## Description

Two reversible enzyme catalyzed reactions convert A to B and B to C. There are two enzymes, E1 and E2. The intermediate complexes are AE1 and CE2. The "on" rate constants are given and the "off" rate constants are given as the product of the "on" rate constants multiplying dissociation constants.

## Equations

#### Off Rate Equations

$\large {\it k_{offA}}={\it k_{onA}}\cdot {\it KA}$ ,   $\large {\it k_{offBE_1}}={\it k_{onBE_1}}\cdot {\it KB_1}$
$\large {\it k_{offBE_2}}={\it k_{onBE_2}}\cdot {\it KB_2}$ ,   $\large {\it k_{offC}}={\it k_{onC}}\cdot {\it KC}$ .

#### Ordinary Differential Equations

$\large {\frac {d}{dt}}A=-{\it k_{onA}}\cdot A \cdot {\it E_1}+{\it k_{offA}}\cdot {\it AE_1}$

$\large {\frac {d}{dt}}{\it AE_1}= {\it k_{onA}}\cdot A \cdot{\it E_1}-{\it k_{offA}}\cdot {\it AE_1} +{\it k_{onBE_1}}\cdot B \cdot {\it E_1}- {\it k_{offBE_1}}\cdot {\it AE_1}$

$\large {\frac {d}{dt}}B=-{\it k_{onBE_1}}\cdot B \cdot {\it E_1}+{\it k_{offBE_1}}\cdot {\it AE_1}-{\it k_{onBE_2}}\cdot B \cdot {\it E_2}+{ \it k_{offBE_2}}\cdot {\it CE_2}$

$\large {\frac {d}{dt}}{\it CE_2}={\it k_{onBE_2}}\cdot B \cdot {\it E_2}-{\it k_{offBE_2}}\cdot {\it CE_2} +{\it k_{onC}}\cdot C \cdot {\it E_2 }-{\it k_{offC}}\cdot {\it CE_2}$

$\large {\frac {d}{dt}}C=-{\it k_{onC}}\cdot C \cdot {\it E_2}+{\it k_{offC}}\,{\it CE_2}$

#### Mass Balance Equations for Enzymes

$\large {\it E_1}={\it E_{1tot}}-{\it AE_1}$ ,   and   $\large {\it E_2}={\it E_{2tot}}-{\it CE_2}$ .

#### Initial Conditions

$\large {\it A} \left( 0 \right) ={\it A_0}$ ,   $\large {\it AE_1} \left( 0 \right) ={\it AE_{10}}$ ,   $\large {\it B} \left( 0 \right) ={\it B_0}$ ,   $\large {\it CE_2} \left( 0 \right) ={\it CE_{20}}$ ,   and   $\large {\it C} \left( 0 \right) ={\it C_0}$ .

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

    Bassingthwaighte JB.: Enzymes and Metabolic Reactions, Chapter 10 in
"Transport and Reactions in Biological Systems"

Lehninger AL. Biochemistry. New York: Worth Publishers, Inc., 1975.



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Pharmacology:

## Key Terms

Progress curves, reversible reactions, equilibrium dissociation first order enzyme kinetics, Competitive Binding, tutorial