Steady State Enzyme Kinetics
• Steady-state reactions are easy to measure because the rate of the reaction is constant for relatively long periods of time.
• Steady-state rates are those which are most relevant to metabolic levels.
• The analysis of steady state kinetics can only provide limited information on the kinetic mechanism.
The simplest reaction scheme is:
where,
k1 is the forward rate constant for substrate binding
k-1 is the reverse rate constant for substrate binding.
k2 is the catalytic rate constant (containing terms related to the transition state).
The ES complex is also called the "Michaelis complex".
The velocity of the reaction is:
v = d[P]/dt = k2[ES]
The change in [ES] as a function of time is:
d[ES]/dt = k1[E][S] - k-1[ES] - k2[ES]
During the steady state: d[ES]/dt = 0
0 = k1[E][S] - k-1[ES] - k2[ES]
The goal is to relate this equation to readily measurable experimental parameters, such as:
• The total amount of enzyme: ET = [E] + [ES]
• The concentration of substrate: [S]
• The measured steady state velocity (v = k2 [ES])
We do not have a suitable way to measure [E], so the total enzyme concentration will be used in its place:
[E] = ET - [ES]
[ES](k-1 + k-2) = k1[S](ET -[ES])
[ES](k-1 + k2) = k1 ET[ES] - k1[ES][S]
[ES](k-1 + k2 + k[S]) = k1 ET[S]
[ES] = k1ET[S]/(k-1 + k2 + k1 [S])
The velocity of the reaction:
v = k2[ES]
= k1k2ET[S]/{k-1 + k2 + k1[S]}
= k2ET[S]/{(k-1 + k2)/ k1+ [S]}
Defining a few terms:
The Vmax or maximal velocity: Vmax = k2ET
This is the highest reaction rate that can be attained because all (i.e. ET) of the enzyme is saturated with substrate.
The KM or Michaelis constant: KM = (k-1 + k2)/ k1
This is the substrate concentration that gives a reaction velocity equal to 1/2 of Vmax. Note that in the case of slow reaction kinetics (k2<<k-1), the KM is also the dissociation constant for substrate binding.
The use of these definitions gives the Michaelis-Menten Equation
