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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.
€
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…