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Chemistry andDifferentialEquationsWhat happens whenequations in the real worldare not linear?

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Importance

Chemistsoccasionally runtests on chemical

kinetics. Currently, many

chemists arelooking into the

effectiveness ofcatalysts, toincrease the yieldof ethanol

production or

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Current Applications

Supposedly 90% ofcommerciallyproduced

chemical productsinvolve catalystsduringmanufacturing.

Petroleum refining,biodiesel, and fuelcells all rely onsome sort of

catalytic reaction.

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Simple Chemical Kinetics

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Michaelis-Menten Kinetics

Givens:

K1 K2

E + S ESE+P

K-1

From the law of

mass action: d[S]/dt=-k1[E][S]+k-1[ES]

d[E]/dt=-k1[E][S]+k-1[ES]

+k2[ES]

d[ES]/dt=k1[E][S]-k-1[ES]-

Difficulties System of

equations is notlinear.

Law ofSuperpositiondoes not apply.

Ramifications Much more difficult

to solve.

Variation ofparameters and

undetermined

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Michaelis-Menten Kinetics

Assumptions Reaction is in Equilibrium

Assume a Steady-State

Assume that the maximum reaction rateoccurs when the enzyme is completely used

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Equilibrium Reaction

Assumptions:The ratio of

substrate (S) toenzyme (E) isvery large.

All enzymes bindas muchsubstrate as

possible. No enzymes are

damaged andlose their ability

to react.

Implications: Reaction rate at

time t=0 ispractically zeroso V0 = K2[ES],

where V0 is the

production rate attime t=0.

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Steady-State Assumption

Assumptions:The concentration

of the enzyme-substrateintermediate (ES)is constant.

Implications:This is only true

when theformation anddeformation ratesfor ES are equalor:

K1 [E][S] = (K-1 +

K2) [ES]

Also, [E] = [ET]

[ES] where [ET] is

the total enzymeconcentration.

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Maximum Rate Assumption

Assumptions: Maximum rate is

obtained whenthe enzyme issaturated withsubstrate.

Implications Maximum rate is

achieved if [ES] =[E

T].

[ES] = enzymesubstrateconcentration

[ET] = total

enzymeconcentration

[E] = freeenzymeconcentration

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Putting It All Together

Givens: K1 K2 E + S ESE+P K-1

From the law of

mass action: d[S]/dt=-k1[E][S]+k-1[ES]

(1)

d[E]/dt=-k1[E][S]+k-1[ES]+k2[ES] (2)

d[ES]/dt=k1[E][S]-k-1[ES]-

Implications:

V0 = K2[ES] (5)

K1 [E][S] = (K-1 +

K2) [ES] (6) Maximum rate is

achieved if [ES] =[ET] (7)

[E] = [ET] [ES] (8)

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Putting It All Together

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Practicality of Michaelis-Menten

It is much easier to regulate and monitor thesubstrate concentration than it is to monitor therate of product formation.

The theory allows chemists to find optimum

substrate concentrations quickly and allowsthem to avoid solving nonlinear differentialequations.

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Conclusion

Differential equations are applied to kineticsmodels in chemical reactions.

Carefully placed assumptions and conditions allowfor the simplification of complex systems.

More complicated and real-world chemicalreactions may require one to solve nonlineardifferential equations.

Linear differential equations are the tip of the

iceberg when it comes to the world ofdifferential equations.

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Sources:

Lehninger Principles of Biochemistry

http://www.engr.ucr.edu/news/2010news.html

http://en.wikipedia.org/wiki/Catalyst#cite_note-9

http://www.lawrencepumps.com/industries/petroleum.htm

http://www.uwsp.edu/chemistry/tzamis/chem10609home.

Notes of Jeremy Lai

http://www.engr.ucr.edu/news/2010news.htmlhttp://en.wikipedia.org/wiki/Catalyst#cite_note-9http://www.lawrencepumps.com/industries/petroleum.htmhttp://www.lawrencepumps.com/industries/petroleum.htmhttp://www.uwsp.edu/chemistry/tzamis/chem10609home.htmlhttp://www.uwsp.edu/chemistry/tzamis/chem10609home.htmlhttp://www.uwsp.edu/chemistry/tzamis/chem10609home.htmlhttp://www.uwsp.edu/chemistry/tzamis/chem10609home.htmlhttp://www.lawrencepumps.com/industries/petroleum.htmhttp://www.lawrencepumps.com/industries/petroleum.htmhttp://en.wikipedia.org/wiki/Catalyst#cite_note-9http://www.engr.ucr.edu/news/2010news.htmlhttp://www.engr.ucr.edu/news/2010news.htmlhttp://www.engr.ucr.edu/news/2010news.html