Food Biochemistry and Food Processing (2 edition)

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172 Part 2: Biotechnology and Enzymology

slow 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+S

k 1




k− 1

ES

k 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 Presentation

Untransformed Graphics

The 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 Plots

Though 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