Michaelis-Menten Kinetics Chrisantha Fernando University of Birmingham October, 2006 Chrisantha Fernando University of Birmingham October, 2006

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.

Simplifying 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 E total = 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 E total = E + ES.

From the full ODEs  We have he equation…  Substituting assumption 2 (E t = E + ES), and 1.  We have he equation…  Substituting assumption 2 (E t = E + ES), and 1.

Rearranging.. Since rate = k 2 [ES] And dividing by k 1

Inhibition Competitive Un-competitive Non-competitive Substrate Inhibition

Cooperativity and the Hill Equation

Monod-Wyman-Changeux Kinetics

Applications  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 Network Gene b Gene a Gene d Gene c A AB B D C

Transcription Regulation  Bintu et al PTO.

Etc…