In this screencast, we're going to derive the rate expression for Michaelis-Menten kinetics,
and what they correspond to is we have an enzyme that reacts with a substrate.
It's a reversible reaction to form an enzyme-substrate complex.
This complex then decomposes to reform the enzymes, the enzyme is a catalyst for the
reaction, and to make the product.
And we're going to analyze this by using the idea of rate-determining step.
So we're going to say that this second step is the rate-determining step, and we're also
going to say that we have a low concentration of our product P, so it's not going to enter
into the rate expression.
Well rate determining step says that if the second step is rate determining, the first
then is in quasi-equilibrium.
Clearly some reaction is taking place, but what we're assuming is that it's very close
to equilibrium, so we can write an equilibrium expression.
And what we're going to do with the convention is we're going to write an equilibrium expression
for the reverse reaction, so K_M is a dissociation constant, and so concentration of the two
reactants over the concentration of the product, so that's just the inverse of the equilibrium
constant for the reaction as written, or it's the equilibrium constant for the reverse reaction.
K_M is referred to as the dissociation constant.
The idea of rate determining step says the rate that we now form product is the rate
constant of this second step times the concentration of the complex, and we can get the concentration
of the complex from this equilibrium expression and from one other piece of information, and
that information is that the total amount of enzyme, total concentration of enzyme,
doesn't change.
Either it's free or it's part of this complex.
So we're going to take this equation plus this equation to get an expression for the
concentration of the enzyme-substrate complex, because this is not something we measure.
We want our rate expression in terms of things that we can measure.
If we make the substitution that the enzyme concentration, the free enzyme concentration
would just be K_M E*S complex over substrate.
So that's just rearranging this equation, and then the second term is not changed, but
what I have now I can factor out the enzyme-substrate concentration, let me do that.
So I've factored out, and I can rearrange this so I'm solving for the enzyme-substrate
concentration.
Now I have a term I can substitute back into my rate expression, so the rate becomes k,
I have E total, remember that's a constant, and the denominator looks like that.
So what is the normal convention, is we call this term Vmax, the maximum rate, K is a constant
at a given temperature, and E_T, total enzyme concentration in a given system is constant.
Now we typically rearrange this to put it in a slightly different form by multiplying
the numerator and denominator by the substrate concentration, so the numerator is Vmax times
the substrate concentration, denominator then is the substrate concentration plus this dissociation
constant.
This then is our final answer.
This is what the Michaelis-Menten rate expression looks like.
If we have a low concentration of S, then it's first order.
If we have a high concentration of S, then the concentration of S is large relative to
K_M, then it's going to be zero order.
So the idea, this is a fairly common type of kinetics that's observed for enzyme reactions.