Enzyme Kinetics

Enzymes (which are large protein molecules) are nature's catalysts. The vast majority of chemical reactions that keep living things alive are much too slow (without a catalyst) to sustain life. (This is so even though some of the reactions are highly thermodynamically favored.) An example of this is the oxidation of a sugar - say glucose - to give water, carbon dioxide and energy. You can leave glucose open to the air for years without any appreciable oxidation, yet this is one of the reactions that provides the energy to walk and run in daily life. There are diseases caused by the failure of the body to produce a specific enzyme. (For example, phenylketonuria is a disease which arises from the absence of a single enzyme, phenylalanine hydroxylase.)

The Michaelis-Menten mechanism for the catalysis of biological chemical reactions is one of the most important chemical reaction mechanisms in biochemistry. (Maud Menten graduated from the University of Toronto, but she was unable to obtain a university position in Canada because of the exclusion of women from Canadian universities at that time. As a consequence she did her work the United States.)

The Michaelis-Menten mechanism for enzyme kinetics is:

.               (1)
E is the enzyme, S is the "substrate" (the molecule on which the enzyme does its work), and ES is an enzyme-substrate complex. (It is presumed that the substrate must somehow bind to the enzyme before the enzyme can do its work.)

We analyze this mechanism as usual. First, we define the reaction rate as the rate of formation of product and write the kinetic equation implied by this mechanism,

.               (2)
The enzyme-substrate complex, ES, is a transient species so we set up an equation for its rate of change and apply the steady state approximation,
.               (3)
Solve for [ES],
,               (4)
and substitute it into the equation for the rate,
.               (5)
We might think that we are finished, but there is a complication and some new notation to introduce. First we introduce the Michaelis-Menten constant, KM,
,               (6)
so that the rate becomes,
.               (7)
Now we must deal with the difficulty that [E] is the concentration of free (uncomplexed) enzyme and this is usually not known. What is known is the total enzyme concentration, [E]o, but
               (8)
from which we obtain,
.               (9)
The rate becomes, then,
               (10).


Define the reaction velocity as v = Rate. So,

.               (11)
Note that the reaction velocity, v, is zero when [S] is zero and that the reaction velocity increases as we increase [S]. The reaction velocity reaches a maximum when [S] becomes very large. Define the maximum velocity, vmax, as,
,               (12)
then
.               (13)
Note that the kinetics of the reaction are characterized by two parameters, vmax and KM.  These are the parameters that are usually given in the literature in studies of the kinetics of biochemical reactions.

In order to deal with experimental data we write,

               (14)
In an experiment one measures v as a function of [S]. If we plot 1/v against 1/[S] we should get a straight line with slope, KM/vmax and intercept 1/vmax. This gives us both parameters,
.               (15)

Enzyme With Inhibitor

Recall the basic Michaelis-Menten mechanism,

.               (1)
There are several possibilities for an inhibitor, I, to interfere with this reaction:
.               (16)
In words, the inhibitor binds with the enzyme to the exclusion of the substrate.
.               (17)
In words, the inhibitor binds to the enzyme-substrate complex and alters the action of the enzyme on the substrate. You can have 1) or 2) or both. We will only work out the first case. The procedure is the same as for the uninhibited Michaelis-Menten mechanism except for an additional term in the expression for the total enzyme concentration and a new transient, EI. The rate is still
,               (18)
and we apply the steady state approximation to ES, which leads to
,               (19)
and the same rate expression
.               (20)
Use the same definition of KM,
,               (21)
which leads to,
.               (22)
But the enzyme-inhibitor complex is also a transient,
.               (23)
giving
.               (24)
Now the total enzyme concentration has an extra term
               (25a, b)
leading to
               (26)
and the rate is
.               (27)
vmax is still
               (28)
so the reaction velocity becomes
               (29)
and then
.               (30)
We still plot 1/v against 1/[S] and the intercept is 1/vmax, but the slope is
,               (31)
and
.               (32)
We must do several different experiments at different [I] to get KM and k'1/k'-1. Note that you can only get the ratio k'1/k'-1 and not the individual k's.

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Copyright 2004, W. R. Salzman
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Last updated 09 Jul 04
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