10.4. Kinetics of real enzymes and machines[[Student version, January 17, 2003]] 381
ThusKMhas the units of a concentration;vmaxis a rate of change of concentration. In terms of
these quantities, Equation 10.18 becomes theMichaelis–Menten(orMM)rule:
v=vmax
cS
KM+cS. Michaelis–Menten rule (10.20)The MM rule displays saturation kinetics. At very low substrate concentration the reaction
velocity is proportional tocS,aswemight have expected from na ̈ıve one-step kinetics (Section 8.2.3
on page 269). At higher concentration, however, the extra delay in escaping from the enzyme-
substrate complex starts to modify that result: vcontinues to increase with increasingcS,but
never exceedsvmax.
Let us pause to interpret the two constantsvmaxandKMdescribing a particular enzyme. The
maximum turnover numbervmax/cE,defined in Section 10.1.2, reflects the intrinsic speed of the
enzyme. According to Equation 10.19, this quantity just equalsk 2 ,which is indeed a property of
asingle enzyme molecule. To interpretKM,wefirst notice that whencS=KMthen the reaction
velocity is just one half of its maximum. Suppose the enzyme binds substrate rapidly relative to
the rate of catalysis and the rate of substrate dissociation (that is, supposek 1 is large). Then even
alowvalue ofcSwill suffice to keep the enzyme fully occupied, or in other wordsKMwill be small.
The explicit formula (Equation 10.19) confirms this intuition.
The Lineweaver–Burk plot Our very reductionist model of a catalyzed reaction has yielded
atestable result: We claim to predict the full dependence ofvuponcS,afunction,using only
twophenomenological fitting parameters,vmaxandKM.Analgebraic rearrangement of the result
shows how to test whether a given experimental dataset follows the Michaelis–Menten rule. Instead
of graphingvas a function ofcS,consider graphing the reciprocal 1/vas a function of 1/cS(a
“Lineweaver–Burk plot”). Equation 10.20 then becomes
1
v=^1
vmax(
1+KM
cS)
. (10.21)
That is, the MM rule predicts that 1/vshould be a linear function of 1/cS,with slopeKM/vmax
and intercept 1/vmax.
Remarkably, many enzyme-mediated reactions really do obey the MM rule, even though few are
so simple as to satisfy our assumptions literally.
Example Pancreatic carboxypeptidase cleaves amino acid residues from one end of a polypep-
tide. The table below gives the initial reaction velocity versuscSfor this reaction
for a model system, a peptide of just two amino acids :Substrate concentration (mM) 2.5 5.0 10.0 15.0 20.0
Initial velocity (mMs−^1 ) 0.024 0.036 0.053 0.060 0.064
FindKMandvmaxbythe Lineweaver–Burk method and verify that this reaction
obeys the MM rule.
Solution: The graph in Figure 10.20b is indeed a straight line, as expected from
the MM rule. Its slope equals 75s,and the intercept is 12mM−^1 s.From the above
formulas, then,vmax=0. 085 mMs−^1 andKM=6. 4 mM.The key to the great generality of the MM rule is that some of the assumptions we made were not
necessary. Problem 10.6 illustrates the general fact thatanyone-dimensional device (that is, one