IGEM:IMPERIAL/2007/Projects/Modelling/Tutorial

Modelling: Tutorial

• Introduction
• Tutorial
• Notes
• References

Michaelis-Menton Kinetics

Introduction

Michaelis-Menten kinetics describes the kinetics of many enzymes.

Reaction at Steady State

Assumptions of Model:

• [Enzyme] is much less than the [Substrate]
• Enzyme is not allosteric.
• Reaction is in steady state
• Equilibrium favours product over transition complex.

Derivation of MM

Consider the following reaction:

k₁ and k₃ are forward rate constants. k₂ and k₄ are backward rate constants.

1. Rate of formation of ES: k₁[E][S] + k₄[E][P]

However, we assume k₃ ≫ k₄, i.e. the equilibrium favours product over transition complex, so we ignore the term with k₄ in it.

2. Rate of removal of ES: k₂[ES] + k₃[ES]

3. In the steady state, rate of formation of ES = rate of removal of ES:

k₁[E][S] = k₂[ES] + k₃[ES]

If we rearrange this to get all the rate constants on one side and all the concentration terms on the other, we can define a constant called Km, which is the Michaelis constant:

K_m = (k₂+k₃) / k₁ = [ES] / [E][S]

4. We can’t measure [E] (free enzyme) or [ES] easily, but we do know the total enzyme concentration, as that is what we will have added when we made up our enzyme solution.

[E] = [E]T – [ES]

If we substitute this into our definition of Km and rearrange it, we get:

[ES] = [E]T[S] / ([S]+K_m)

5. The overall (initial) reaction rate V is

V = k₃[ES]

If we substitute this into the last result, we get:

V = k₃[E]T[S] / ([S]+K_m)

6. When the enzyme is saturated, [ES]=[E]T, so we can define a maximum velocity, V_m:

V_m = k₃[E]T

And this gives us our final result: V = V_m [S] / K_m + [S]

MM Parameters

Michaelis-Menton Kinetics

V_m (or Vmax) is the maximum rate at which a certain mass of enzyme can work, and equals k₃ (the rate constant for [ES] degradation to product) times the total enzyme concentration [E]T. Fast enzymes have higher V_m. V_m is not a constant, because it is dependent on the amount of enzyme present in a solution, and on the volume of this solution. It is measured in mol s_-1 or more usually μmol min_-1.

K_m is (k₂+k₃) / k₁ = [E][S] / [ES], i.e. K_m is the K_eq for the non-productive dissociation of the ES complex back to E and S. A low K_m indicates the enzyme has a high affinity for its substrate. Therefore efficient enymes have lower K_m. K_m is a constant: it equals [S] at ½V_m, and is measured in mol L_-1, or more usually mM.