Michaelis-Menten KineticsMichaelis-Menten Kinetics

Chrisantha Fernando

University of Birmingham

October, 2006

Chrisantha Fernando

University of Birmingham

October, 2006

How fast does an Enzymatic Reaction Run?

How fast does an Enzymatic Reaction Run?

Assume a reaction of the form…

k1 is the forward rate of ES complex formation, and k1’ is the backward rate.

We want an expression for the forward rate, i.e. k2[ES], but this cannot be solved analytically.

Assume a reaction of the form…

k1 is the forward rate of ES complex formation, and k1’ is the backward rate.

We want an expression for the forward rate, i.e. k2[ES], but this cannot be solved analytically.

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E +S k1/k−1← → ⏐ ⏐ ES k2 ⏐ → ⏐ E +P

Simplifying AssumptionsSimplifying Assumptions

Assume that k2 << k1 and k1’, i.e. that there is a quasi-equilibrium of the state of ES, I.e. assume that there is a quasi-steady state of the ES complex, i.e. d[ES]/dt = 0

If, initially [E] << [S], then this means that Etotal = E + ES.

Assume that k2 << k1 and k1’, i.e. that there is a quasi-equilibrium of the state of ES, I.e. assume that there is a quasi-steady state of the ES complex, i.e. d[ES]/dt = 0

If, initially [E] << [S], then this means that Etotal = E + ES.

From the full ODEsFrom the full ODEs

We have he equation…

Substituting assumption 2 (Et = E + ES), and 1.

We have he equation…

Substituting assumption 2 (Et = E + ES), and 1. €

d[ES]

dt= k1[E][S]− (k1 '−k2 )[ES]

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0 = k1(Etot −ES)S −(k1 '−k2 )ES

Rearranging..Rearranging..

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ES =k1EtotS

k1S+ k1' + k2

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ES =k2EtotS

S+k1

' + k2

k1

Since rate = k2[ES]And dividing by k1

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ES =VmaxS

S+Km;Vmax = k2Etot ,Km =

k1' + k2

k1

InhibitionInhibition

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v =VmaxS

Kmi3 +SCompetitive

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v =VmaxS

Km +S.i4Un-competitive

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v =VmaxS

(Km +S).i4Non-competitive

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v =VmaxS

(Km +S(1+S /K I ))SubstrateInhibition

Cooperativity and the Hill Equation

Cooperativity and the Hill Equation

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v =VmaxKBS

n

1+KBSn

Monod-Wyman-Changeux Kinetics

Monod-Wyman-Changeux Kinetics

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V =VmaxKRS

(1+KRS)

1

(1+L

(1+KRS)n)

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L'= L(1+K I I )

n

(1+KAA)n

ApplicationsApplications

Regulation of the Lac Operon in Cellerator, see CSN webpage…

Transcription Factor Regulation, see next page…

Regulation of the Lac Operon in Cellerator, see CSN webpage…

Transcription Factor Regulation, see next page…

TF NetworkTF Network

Gene b

Gene a

Gene d

Gene c

A

ABB

DC

Transcription RegulationTranscription Regulation

Bintu et al 2005. PTO. Bintu et al 2005. PTO.

Etc…