# Integrated Michaelis-Menten equation

Athel athel at ir2cbm.cnrs-mrs.fr
Sun Apr 13 10:49:51 EST 1997

```Claus Lundegaard asked:

"As I am not a great matematician, I wondered if anybody has derived the
integrated Michaelis-Menten eq. with the product formation isolated. I
need this for getting a Km value from a computational fit to a progress
curve."

I'm not sure what you mean by "with the product formation isolated", but
if you mean the integrated form of the MichaelisMenten equation
allowing for product inhibition then many people have done this, but as
this was a much more popular topic in the 1950s than it is now much of
the literature is old. Carl Niemann wrote numerous papers on this and
related subjects, and for more recent work Betty Boeker's papers from
the early 1980s are a good source.

If product inhibition is competitive, then the integrated
Michaelis-Menten equation has EXACTLY the same form as it does if there
is no inhibition, i.e.

Vapp.t = p + Kmapp.ln(pinf/(pinf - p))

where t, p and pinf are time, product concentration and final product
concentration respectively. The constants Vapp and Kmapp can have just
about ANY values (including negative or infinite under easily attainable
conditions) and  SHOULD NOT be treated as approximations to V and Km.
For example,

Kmapp = Km.(1 + pinf/Kp)/(1 - Km/Kp)

where Kp is the competitive inhibition constant for product inhibition.
It is the denominator term (1 - Km/Kp) that wreaks havoc with any
attempt to treat Kmapp as an approximation to Km. It is not uncommon for
Kp to be smaller than Km, for example for reactions with NADH as a
product. If pinf = 2Km (as an example), then Kp must be at least 61Km
(i.e. product inhibition must be virtually undetectable by ordinary
criteria), for Kmapp to be within 5% of Km.

Note that all this means that IT IS IMPOSSIBLE to deduce whether
competitive product inhibition is present from the shape of the progress
curve (unless it is strong enough to make Kmapp negative). If the
inhibition is not strictly competitive then in principle there are
deviations from the equation as written above, but in practice they are
unlikely to be detectable.

These points are discussed in my paper in Biochem. J. 149, 305-312
(1975), and in more general fashion on pp. 44-47 of my book
"Fundamentals of Enzyme Kinetics"
(http://ir2lcb.cnrs-mrs.fr/lcbpage/athel/fek.html).

Athel