Recall basic reaction model from last time:
For steady state:
d[ES]/dt= 0; Thus: 0 = d[ES]/dt= k_{1}[E][S]  k_{2}[ES]  k_{3}[ES].
Continuing we can now substitute for E (free enzyme), because hard to find, and gather constants:
{Note that if k_{2} >> k_{3} (that is the equil. of E+S with ES is rapid compared to breakdown of ES to P), then MM const = 1/(affinity)= the dissociation constant, but only in these special conditions.}
Now a couple of tricks: Solve for [ES]:
and recall that v_{i} = k_{3}[ES], and dividing both sides by k_{3}, v_{i}/k_{3} = [ES]
Substituting: and ,
But maximum possible velocity must = k_{3}[E_{t}] = V_{max}
So, Which is known as the MichaelisMenten Equation.
The rate constant (First order) for the breakdown of the [ES] complex, k_{cat} (k_{3}), is also known as the turnover number, that is the maximum number of substrate molecules processed/active site (moles substrate/mole active site): k_{cat}=V_{max} / [E]_{total}. Note that this is best determined under saturating conditions. At very low concentrations of [S] can find the secondorder rate constant for the conversion of E + S E + P:
v_{o} = (k_{cat} / K_{M})([E][S].
[Table 5.1]
Need in form: y = ax + b , so take reciprocals of both sides and have
. [Figure 5.6]
Other linear plots are also available, and are better in terms of statistics (LB one of worst, best quality points [high concentration] have least influence on slope, while low precision points [low concentration] are more spread out, and have a large moment, with a strong influence on the slope and the K_{M} intercept this is not as much of a concern now with computer statistical packages, but you still have to understand the statistics).
FYI  The EadieHofstee PlotOne common plot is shown below. Note that the data points are distributed much more evenly over the plot giving better statistics for the slope. In addition the value of K_{M} is obtained from the slope, giving better precision.

Look at three common and easily understood types. We will use Cleland Nomenclature. [Figure 5.7a, b]











Note that in the Ping Pong and Ordered sequential mechanisms the enzyme forms before & after are the same and thus Q is a competitive inhibitor of A (that is A can displace Q) where as in the random sequential P & Q will be noncompetitive since neither can be disiplaced by substrate because of the different forms allowed by the random binding. Such inhibition differences can be used to distinguish between the different mechanisms.
S & I are mutually exclusive, E can bind to one OR the other. [Figure 5.8a, b]
We can model this inhibition with chemical equations, keeping in mind that S & I are mutually exclusive, E can bind to one OR the other:
Model: ; and: ; where .
Pathway Diagrams 

© R. A. Paselk 2010;
Last modified 22 February 2013