BLBS102-c08 BLBS102-Simpson March 21, 2012 12:8 Trim: 276mm X 219mm Printer Name: Yet to Come
172 Part 2: Biotechnology and Enzymologyslow formation of product and release of free enzyme. It assumes
thatk 2 k− 1 in the following equation, andKmis the value of
[S] whenv=Vmax/2:E+Sk 1
k− 1ESk 2
−→E+P.Briggs and Haldane (1924) proposed another enzyme kinetic
model, called the steady state model; steady state refers to a
period of time when the rate of formation of ES complex is the
same as rate of formation of product and release of enzyme. The
equation is commonly the same as that of Michaelis and Menten,
but it does not requirek 2 k− 1 , andKm=(k− 1 +k 2 )/k 1 , be-
cause [ES] now is dependent on the rate of formation of the
ES complex (k 1 ) and the rate of dissociation of the ES com-
plex (k− 1 andk 2 ). Only under the condition thatk 2 k− 1 will
Ks=k− 1 /k 1 and be equivalent toKm, the dissociation constant
of the ES complex{Ks=k− 1 /k 1 =([E]−[ES])[S]/[ES]},oth-
erwiseKm>Ks.
TheKmis the substrate concentration at half of the maximum
rate of enzymatic reaction, and it is equal to the substrate con-
centration at which half of the enzyme active sites are saturated
by the substrate in the steady state. So,Kmis not a useful mea-
sure of ES binding affinity in certain conditions whenKmis not
equivalent toKs. Instead, the specific constant (kcat/Km) can be
substituted as a measure of substrate binding; it represents the
catalytic efficiency of the enzyme. Thekcat,the catalytic con-
stant, represents conversion of the maximum number of reactant
molecules to product per enzyme active site per unit time, or the
number of turnover events occurring per unit time; it also rep-
resents the turnover rate whose unit is the reciprocal of time. In
the Michaelis-Menten approach, the dissociation of the EP com-
plex is slow, and thekcatcontribution to this rate constant will
be equal to the dissociation constant. However, in the Briggs-
Haldane approach, the dissociation rate of ES complex is fast,
and thekcatis equal tok 2. In addition, the values ofkcat/Kmare
used not only to compare the efficiencies of different enzymes
but also to compare the efficiencies of different substrates for a
specified enzyme.
Deviations of expected hyperbolic reaction are occasionally
found due to such factors as experimental artifacts, substrate
inhibition, existence of two or more enzymes competing for
the same substrate, and the cooperativity. They show nonhyper-
bolic behavior that cannot fit well into the Michaelis-Menten
approach. For instance, a second molecule binds to the ES com-
plex and forms an inactive ternary complex that usually occurs
at high substrate concentrations and is noticed when rate values
are lower than expected. It leads to breakdown of the Michaelis-
Menten equation. It is not only the Michaelis-Menten plot that
is changed to show a nonhyperbolic behavior; it is also the
Lineweaver–Burk plot that is altered to reveal the nonlinearity
of the curve (see later).
A second example of occasionally occurring nonhyperbolic
reactions is the existence of more than one enzyme in a reac-
tion that competes for the same substrate; the conditions usually
are realized when crude or partially purified samples are used.
Moreover, the conformational change in the enzymes may be
induced by ligand binding for the regulation of their activities,as discussed earlier in this chapter; many of these enzymes are
composed of multimeric subunits and multiple ligand-binding
sites in cases that do not obey the Michaelis-Menten equation.
The sigmoid, instead of the hyperbolic, curve is observed; this
condition, in which the binding of a ligand molecule at one site of
an enzyme may influence the affinities of other sites, is known as
cooperativity. The increase and decrease of affinities of enzyme
binding sites are the effects of positive and negative cooperativ-
ity, respectively. The number of potential substrate binding sites
on the enzymes can be quantified by the Hill coefficient,h,and
the degree of cooperativity can be measured by the Hill equation
(see page 164). More detailed illustrations and explanations of
deviations from Michaelis-Menten behavior can be found in the
literatures (Bell and Bell 1988, Cornish-Bowden 1995).Data PresentationUntransformed GraphicsThe values ofKmandVmaxcan be determined graphically using
nonlinear plots by measuring the initial rate at various substrate
concentrations and then transforming them into different kinetic
plots. First, the substrate stock concentration can be chosen at a
reasonably high level; then, a two-fold (stringently) or five- or
ten-fold (roughly) serial dilution can be made from this stock.
After the data obtained has been evaluated, a Michaelis-Menten
plot of ratevias a function of substrate concentration [S] is
subsequently drawn, and the values of bothKmandVmaxcan
be estimated. Through the plot, one can check if a hyperbolic
curve is observed, and if the values estimated are meaningful.
Both values will appear to be infinite if the range of substrate
concentrations is low; on the other hand, although the value
ofVmaxcan be approximately estimated, that ofKmcannot be
determined if the range of substrate concentrations is too high.
Generally, substrate concentrations covering 20–80% ofVmax,
which corresponds to a substrate concentration of 0.25–5.0Km,
is appreciated.Lineweaver–Burk PlotsThough nonlinear plots are useful in determining the values of
KmandVmax, transformed, linearized plots are valuable in deter-
mining the kinetics of multisubstrate enzymes and the interaction
between enzymes and inhibitors. One of the best known plots
is the double-reciprocal or Lineweaver–Burk plot (Lineweaver
and Burk 1934). Inverting both sides of the Michaelis-Menten
equation gives the Lineweaver–Burk plot: 1/v=(Km/Vmax)
(1/[S])+ 1 /Vmax(Fig. 8.3). The equation is for a linear curve
when one plots 1/vagainst 1/[S] with a slope ofKm/Vmaxand a
yintercept of 1/Vmax. Thus, the kinetic values can also be deter-
mined from the slope and intercept values of the plot. However,
caution should be taken due to small experimental errors that
may be amplified by taking the reciprocal (the distorting effect),
especially in the measurement ofvvalues at low substrate con-
centrations. The problem can be solved by preparing a series of
substrate concentrations evenly spaced along the 1/[S] axis, that