### Derivation of the Michaelis-Menten Equation for Enzyme Kinetics

The BioFitWeb progress curve fitting tool fits time-course data to the Michaelis-Menten equation, which gives the velocity v of the reaction in terms of the substrate concentration [S]:

### (1)

The use of this equation implies several assumptions, which require a bit of explanation. The reaction scheme below is often referred to as the Michaelis-Menten reaction:

### (2)

In this scheme, an enzyme combines with a substrate in a reversible interaction to form an enzyme-substrate complex (ES, in equation 2). This complex persists for a short time and then either dissociates into enzyme and substrate, or the substrate is changed into a product P. The product in this scheme is assumed to have no affinity for the enzyme, so it detaches. Thus the second part of the reaction is irreversible formation of product accompanied by regeneration of the active enzyme.

To derive equation 1 from the reaction scheme 2 we start from an expression for the velocity v which follows from scheme 2, which is:

### (3)

Here we assume that the overall reaction velocity is limited by the product formation step, so that the overall rate is given by the rate of that step, i.e. the product of the concentration of enzyme-substrate complex and the rate constant k2.

Now we obtain more useful expressions for both k2 and [ES], but we must make a couple more assumptions to do so.

We now assume that the reaction is in a state where the amount of complex [ES] is nearly constant, with the rate of formation equal to the rates of dissociation - both backward and forward. These rates are expressed as in equation 3 and we now assert that

### (4)

Some rearrangement gives us equation 5:

### (5)

We simplify equation 5 by combining the three rate constants into one, which is called Km, the Michaelis-Menten rate constant, thus:

### (6)

This allows us to rewrite equation 5 this way:

### (7)

Now we want to eliminate [E] from this expression. To do this we use the fact that the total amount of enzyme in the system doesn't change, since it is a catalyst and is always regenerated after each reaction. (This 'fact' is actually another assumption, one which we will not discuss here.) We express that fact in equation 8, where [E0] is the total (or equivalently initial) enzyme concentration:

### (8)

When equation 8 is solved for [E], we can eliminate [E] from equation 7 by substitution:

### (9)

Now rearrange and solve equation 9 for [ES] to obtain an expression which depends only on [S]:

### (10)

When equation 10 is substituted into equation 3, we are almost finished:

### (11)

We now come to the last important assumption necessary to arrive at equation 1. Observe that the maximum rate for reaction 2 occurs when the greatest possible proportion of the enzyme is combined with substrate all the time, which occurs when [S] >> [E].  If we assume that this is the case, then we can write

### (12)

just as we did to obtain equation 3. Substituting equation 12 into equation 11 produces the desired result - equation 1, the Michaelis-Menten equation:

The last three assumptions are collectively known as the quasi-steady-state assumption, or QSSA.