Enzyme Kinetics

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Enzyme Kinetics 
Owen Moe and Richard Cornelius 
Lebanon Valley College, Annville. PA 17003 
Nature prepares innumerable catalysts to accelerate the 
myriad reactions that occur in biological systems. These 
catalysts, which are all different types of protein molecules, 
are called enzvmes. Tbev are essential for nearlv even, reac- 
tion that occurs in nature. Even a reaction as simple as the 
dissociation of carbonic acid to carbon dioxide and water, 
requires an enzyme called carbonic anhydrase in order to 
oroceed as raoidlv as needed. 
Enzymes are ihe most effective and versatile catalysts 
known. Thev differ from common chemical catalysts in that 
they 
1. are incredibly efficient, giving catalytic enhancements of up to 
101%ver corresponding uncatalyzed reactions, 
2. can show a high degree of specificity, discriminating between 
reactants having close structural similarities, including optical 
isomers, 
3. are subject, in some cases, to sensitive regulation of the rate of 
catalysis by cellular metabolites, and 
4. show a remarkablediversity in the types ofreactionscatalyzed, 
even though the enzymes rhemseives are quite similar in their 
chemical composition and general structure. 
In biochemical terminology, the reactants in enzyme-cata- 
lyzed reactions are referred to assubstrates. The location on 
the enzyme surface a t which catalysis occurs is called the 
actiue site. Compounds that increase or decrease the rates of 
enzyme-catalyzed reactions are actiuators and inhibitors, 
respertively. A recent artirle in this Journal ( 1 ) discussed 
the general structural aspects of enzvmes and described the 
mechanisms by which they catalyze reactions. 
Since the salient feature of enzvmes is their remarkable 
ability to accelerate reactions, it is hardly surprising that one 
of the most important experimental methods of studying 
enzymes is that of enzyme kinetics. The field of enzyme 
kinetics has the goals of measuring, analyzing, and interpret- 
ing rates of enzyme-catalyzed reactions. The mathematics 
necessary to deal quantitatively with enzyme kinetics may 
stand in the way of high school and college students gaining a 
recognition of how useful kinetic studies can be. In this 
paper we hope to convey an appreciation of enzyme kinetic 
analysis by using a practical and intuitive approach. 
Enzyme Assays 
Enzvme kinetic studies require accurate and reliable 
methods to measure experimekally the rates of enzyme- 
catalvzed reactions. Rate determinations are accomplished 
t h ~ o ; ~ h the use of enzyme adsajs, which measure thechange 
in the ronrentration of either a substrate or a product of a 
reaction as a function of time. An assay depends on the 
availability of a chemical or physical method for the deter- 
mination of a substrate or oroduct. Manv assavs are soectro- 
photometric, but other 'techniques include fluorescence 
measurements. monitorine the fate of radioactivelv labelled - 
compounds, and electrochemical detection. Consider, for ex- 
ample, the hydrolysis of the phosphate ester p-nitrophenyl- 
phosphate @NPP) to produce p-nitrophenoxide ion (pNP), 
inorganic phosphate (Pi), and hydrogen ion. This reaction is 
catalyzed in a basic medium by the enzyme alkaline phos- 
phatase (2): 
Several experimental approaches are available as,assays 
for this enzvme. One method involves measurinn the concen- - 
tration of a product, inorganic phosphate, using a spectro- 
photometric analysis for phosphate (3). This assay consists 
of removing an aliquot from the reaction mixture, inactivat- 
inn the enzvme to prevent further catalysis, adding a molyb- 
date-basedreage& to form a blue complex withthe phos- 
phate, and measuring the absorbance of the phosphomolyb- 
date complex at 690 nm. The concentration of the phosphate 
produced is calculated from the absorbance of the phospho- 
molvbdate comolex through the use of a standard curve. - 
This fixed-time assay requires the removal of several ali- 
q u o t ~ for analysis a t intervals after initiation of the reaction. 
A simulated plot of the results of such afixed-time assay is 
shown in Figure 1. The increase in phosphate concentration 
with time is initially nearly linear but becomes nonlinear as 
the reaction approaches equilibrium. The initial rate of the 
reaction, defined as the slope at zero time, is indicated in the 
figure. Kinetic analysis of enzymatic activity typically 
makes use of initial rates of reaction. 
Another assay for the alkaline phosphatase reaction (and - - 
the one most frequently used) consists of measuring the 
absorbance of the p-nitrophenoxide product, which has an 
absorption maximum a t 410 nm (4). This continuous assay 
can be carried out in a spectrophotometer cell by adding 
Flgwe 1. Plot01 ~)n~enhal lonof #nOIwnlC phospbte VS. time for aflxedtlme 
assay (squares) show n5 nitial rate lsolid line). 
Volume 65 Number 2 February 1988 137 
enzvme to the reaction mixture, mixing the solution, placing 
thecell in the spectrophotometer, andcontinuously iecord: 
ina the absorbance. Beer's law and the molar absorptivity of 
I 1 Modal 1 
the p-nitrophenoxide ion are used to relate the absorbance 
to the concentration of product. 
Another pnslihle continuous assay would use a pH-stat to 
add standard base to the reaction mixture, maintaining the 
initial nH bv neutralizine the hvdronium ion oroduced. Con- . - 
tinuous assays allow cofection-of all the data for the initial 
rate determination in a single experimental step. Fixed-time 
assays are generally used only when no adequate continuous 
method exists. 
Klneilc Models and Rate Laws 
Kinetic data can provide information about two impor- 
tant aspects of enzymatic catalysis: 
1. how the rate of catalysis depends on the concentrations of the 
substrates (and products) of the reaction, and 
2. the magnitude of the maximal rate of catalysis that can occur 
at a single active site of the enzyme. 
The concentration dependence of the reaction is, as we shall 
see later, related to the equilibrium constant for the disso- 
ciation of the enzyme-substrate complex. The maximal rate 
of reaction provides a direct. auantitative assessment of . . 
catalytic actkity. 
The experimental design most often used in kinetic stud- 
ies is that of measu~in~ini t ia l reaction rates over a wide 
ranee of suhstrate concentrations, keeping the enzyme con- 
ceniration constant. Analysis and interpretation of these 
kinetic measurements require rate laws, which express the 
mathematical relationshi& between the initial rates and the 
substrate concentrations. Rate laws are based on and de- 
rived from kinetic models. which consist of oostulated se- 
quences of steps in the catalytic process. he derivation of 
rate laws from kinetic models is essential to enzyme kinetics, 
and methods for carrying out these derivations are well es- 
tablished (5, 6). Carrying out such derivations would he 
beyond the scope of this paper; we will focus here instead on 
the basic assumotions that underlie the derivations. 
In order to understand the concept of a kinetic model, 
consider the reaction catalyzed by the enzyme glucose-6- 
phosphate isomerase (7): 
[O,PO-CHT(CH(OH)l,CH0l2-_ - 
G6P 
[03PO-CH,(CH(OH)I,CO-CH,0H]z~ 
F6P 
(3) 
This enzyme catalyzes the isomerization of an aldose sug- 
ar-phosphate, glucose-&phosphate (G6P), to a ketose sugar- 
phosphate, fructose-&phosphate (F6P). The simplest possi- 
ble sequence of steps that we are able to postulate for this 
single-substrate reaction we will call Model 1, 
where E reoresents the enzvme. This model savs that, in the 
presence of the enzyme, the substrate and are inter- 
converted a t rates that depend on the forward and reverse 
rate constants, kl and kz. This model makes no attempt to 
define any role for the enzyme catalyst. The rate law for this 
model, which is derived under the assumptions of a constant 
enzyme concentration and a product (F6P) concentration 
equal to zero, yields a simple first-orderdependence on the 
concentration of G6P: 
initial rote = k,[E],[G6P] = k'[GGP] (4) 
In this rate law klf is a composite rate constant that is 
equal to k, [Elt, where [Elt is the constant, total enzyme 
Figure 2. Plot of obselved initial rate vs. concentration for glu-6-phas- 
phate isomerase (solid line) and rate predicted by Model 1 (dashed line). 
concentration. The rate law predicts that a plot of the mea- 
sured initial rates versus the concentrations of G6P sub- 
strate used to eenerate those rates will he linear over the 
entire range O ~ Y G G P concentration. This predicted behavior 
is shown as the dashed line in Fieure 2. The actual ex~er i - 
mentally measured initial rates for this enzyme reaction (8) 
show a very different type of response to substrate concen- 
tration as indicated by the solid line in Figure 2. Thus, 
Model 1 is not able to predict the observed concentration 
dependence of the initial rate and cannot serve as a useful 
description of the steps involved in this enzyme reaction. 
The experimental initial rate data (solid line in Fig. 2) 
show substrate saturation a t higher concentrations of suh- 
strate. At these concentrations the rate of reaction levels off ~~ ~~~~ ~ ~~~~~ ~~ ~ 
even though the suhstrate concentration continues to in- 
crease. This hehavior sueeests a saturation of enzvme active -- 
sites by the substrate. 
We can modifv Model 1 bv addine a steo involvine the 
binding (complex formation? of t h e k 6 ~ kbstrate to the 
enzvme to form the catalvticallv active E(G6P) com~lex. 
~ h k modification produceb ~ o d k l 2 , 
*1 *J 
E+WP+E(GGP)=E+KP 
*2 4 
The rate constants k, and k4 govern the rates of formation 
of the complex, E(G6P). from suhstrate and product, respec- 
tivelv. The rate constants kl and kl reflect the rates of 
E ( G ~ P ) breakdown. Note tha; the rate constant k 3 includes 
both the step forcatalvticconversionof G6P into F61'on the 
enzyme surface and the step involving release of F6P from 
the enzvme. Model 2 postulates that the substrate must hind 
to the enzyme beforecatalysis canoccur; hut it does not offer 
any explanation of how catalysis actually occurs. 
~heber ivat ion of the ratelaw for this model is suhstan- 
tially more complicated than the one for Model 1. Therefore, 
in addition to the orevious assum~tions of a constant con- 
centration of enzyie and an initiafproduct concentration of 
zero. the followine additional sim~lifvine assum~tions. 
which are basic to enzyme kinetics, are usedu(@: 
1. The rates that are measured are true initiol rates. When this 
assumption holds, the experimental rate measurement corre- 
s~onds to a known substrate concentration, equal to the initial ~. 
suhstrate concentration in the reaction mixture. 
2. The total concentration of substrate in the reaction mixture is 
always much greater than the total concentration of enzyme: 
IG6P1 >> IEL. This condition is almost alwavs true since en- . , . .. 
zymes are >uch efliclent catalysts that nanomolsr ccmcentra- 
llunsor less are urually sufficient for use in rare mearuremenls. 
I'ndcr thls assurnptlon, the cunrentratron of free substrate 
138 Journal of Chemical Education 
(i.e., not complexed with the enzyme) is closely approximated 
bv the total. known substrate concentration. 
3. The steody-state assumption: The concentration of the en- 
zyme-substrate complex, [E(GGP)], rapidly reaches and main- 
tains a constant steadv-state level in which its rates of forma- 
tion and breakdown are equal. 
Using these assumptions i t is possible to derive the follow- 
ing rate law for the above model (5,6): 
initial rate = u = 
V[GGP] 
K , + [GGP] 
In this rate law, which is written in the form that is gener- 
ally used in biochemistry, the initial rate is given the symbol 
u, and is called the initial velocity. Vis the maximal uelocity 
and is equal to k3[E],. TheconstantK, is called the Michael- 
is constant and is equal to (kz + k3)lkl. 
Consider what Model 2 and its associated rate law (eq 5) 
predict. At very low substrate concentrations where [G6P] 
> K,, eq 5 reduces to u = V. This equation corre- 
sponds to a zero-order rate law, and thus is able to predict 
the observed phenomenon of substrate saturation, as is 
shown in Figure 3. 
We thus see that the rate law given as eq 5 prediets a plot 
of initial velocity vs. substrate concentration, which has the 
form of a rectangular hwerbola (Fin. 3). This hyperbolic 
response shown b j glucu;~-6-l,hoipha~le isomerase is rharac- 
teristic of most enzymes. Model 2, the simplest kinetic mod- 
el that can predictthis observed rate behavior, is called the 
Michaelis-Menten model in honor of its early proponents. 
The Klnetic Constants, Vand K, 
The Michaelis-Menten rate law, given in its general form, 
requires only two parameters, V and K,, to express the 
relationship between initial velocity and substrate concen- 
tration, [S]: 
At a particular set of experimental conditions (pH, tem- 
perature, buffer concentration, enzyme concentration, etc.) 
values of V and K, can be experimentally evaluated to 
provide information about the enzyme under study. If we 
take a closer look at these two kinetic constants, we can 
understand more clearly what they tell us about an enzyme. 
Substrata 
S a t u r a t i o n 
R e g i o n 
F t s t ... / Ordar R e g i o n 
Figure 3. Plot of initial velocity vs. substrate concentration given by eq 5. 
Consider first the Michaelis constant, K,. From the form 
of eq 6 it can be seen that the units associated with K, must 
be those of concentration (M, mM, etc.) since it is being 
added to a concentration in the denominator. These units 
are consistent with both u and V having units of concentra- 
tion per unit time. If we consider the case for whichK, = [S], 
then eq 6 reduces to v = Vl2. This relationship tells us that 
the Michaelis constant has the value of the substrate con- 
centration that will produce a rate equal to one half of the 
maximal uelocitv. An enzvme havina a larae value for a 
.Michaelis constant will therefore reqnire higher concentra- 
tions of substrate to arhiere a riven frnction of its maximal 
velocity than will an enzyme wl'th a small value of K,. 
Figure 4 shows substrate saturation curves for two en- 
zymes that have identical maximal velocities but have Mi- 
chaelis constants differing by a factor of 10. At concentra- 
tions of substrate far below the saturation level (region A) 
marked differences appear between the rates of the two 
enzymes, while a t substrate concentrations much larger than 
the K, values of either enzyme (region B) the rates are 
nearlv eaual. , . 
A way of intuitively understanding the fundamental 
meaninr of the Michaelis constant comes from its definition 
in ~ o d i l 2 : K, = (k2 + k3)lk:. Thus, K, comprises two first- 
order rate constants (kz and k3) having units of s-', and one 
second-order rate constant, kt, having units of M-'s-l. In- 
sertion of these units into the K , expression yields the units 
of molar concentration deduced earlier. 
The expression for K, can be rewritten as two separate 
terms: 
K , = k,lk, + k31k, (7) 
The ratio k2/kl is equal to the dissociation constant, Kd, 
for the enzymesubstrate complex. The formation constant 
for the enzyme-substrate complex, generally called the 
binding constant, K, is the reciprocal of the dissociation 
constant, giving us the following relationships: 
Since k3 includes the rate constant for the catalytic con- 
version of substrate to product, this rate constant is often 
found to correspond to the rate-determining step in enzyme 
reactions. In such cases, the k31kl term in eq 8 is very small, 
and the expression for K, reduces to the actual dissociation 
constant,Kd. 
Because of its close relations hi^ to KA (and to the bindinn 
constant;K), the Michaelis const'ant can provide a quantita- 
tive measure of the strength of interaction between the sub- 
Figure 4. S~bsbBte SahlmiOn curves lor two enzymes having identical maxi- 
mal velocities but Michaelis constants differing by a factw of 10. 
Volume 65 Number 2 February 1988 139 
strate and the enzyme active site (9). Recall that as the 
dissociation constant decreases (or conversely, as the bind- 
ing constant increases), the strength of binding in the en- 
zyme-substrate complex increases, and less substrate is 
needed to achieve saturation of the enzyme. 
Now, turning to the maximal velocity term in the Michae- 
lis-Menten rate law, we see that it is defined as h,[E]t, the 
~ r o d u c t of the catalvtic rate constant and the total enzyme 
>oncentration. It is ;herefore the rate of rhe rractwn under 
condirions where the enz)rne rs complete/) saturated uith 
substrate. Dividing the maximal velocity by the total en- 
zyme concentration yields hs, which is called the catalytic 
constant and which is sometimes given the symbol kcat. The 
turnouer number, a rate constant for the conversion of sub- 
strate to product a t a single active site on the enzyme sur- 
face, is obtained by dividing the catalytic constant by the 
number of active sites per enzyme molecule. Turnover num- 
bers for enzyme reactions can approach 1,000,000 s-I ( lo ) , 
which means that conversions of substrate to products can 
occur on a microsecond time scale. 
Evaluation of Vand K, from Experimental Data 
For the determination of kinetic constants from kinetic 
data the hyperbolic Michaelis-Menten rate law (eq 6) is 
usually first transformed (eq 9) into alinear equation (eq 10) 
by taking the reciprocal of each side: 
This double reciprocal form of the Michaelis-Menten rate 
law predicts that a plot of llu vs. 1/[S] would be linear with 
an ordinate intercept of 1/V and a slope of KJV. Such a 
plot, called a Lineweauer-Burk Plot, is simulated in Figure 5 
for the glucose-6-phosphate isomerase reaction. Evaluation 
of V and K, from this type of plot is then easily carried out 
from these relationships: 
V = l/(ordinate intercept) 
K, = (slope)/(ordinate intercept) = (slope) (V) 
Lineweaver-Burk plots have traditionally formed the ba- 
sis of methods for analysis of initial rate kinetic data for 
enzyme reactions. Current approaches employ statistically 
more valid methods such as weighted linear and nonlinear 
least-squares fitting. Nonetheless, the Lineweaver-Burk 
plot remains the most common method of displaying initial 
rate kinetic data in biochemical textbooks and research oub- 
lications. 
I/" 
Figure 5. Llneweaver-Bulk a double reciprocal plot, which is linear if the 
Michattlls-Memen rate law applies. 
Enzyme lnhlbltlon 
One important application of kinetic measurements is the 
studv. both oualitatiue and ouantitatiue. of the effects of -. 
inhibitors on enzyme-catalyzed reactions. An enzyme inhib- 
itor is a com~ound that can cause a decrease in the rate of 
catalysis, t he magnitude of which is dependent on the con- 
centrations of both inhibitor and substrate. The current 
widespread interest inenzyme inhibition isdue in part ro the 
following areas where inhibitors play a central role (10, 11): 
1. Metabolism. Reeulation of the rates of metabolic oathwavs is " 
accurnpli,hed in large part thmugh inhibition of key enzymes 
try reaction produrts. Thrs effect, called / e~dhack tnhib~lmn. 
pret,ents metabolic pathways from produrmg to> [I], all the enzyme is driven into the form of the 
reactive ES complex, that is, [ES] >> [EI]. The type of plot 
shown in Figure 6 is diagnostic for the competitive inhibition 
mechanism. 
The competitive model is not, however, the only possiblemode of inhibition, and the others that exist can be identi- 
fied on the hasisof kineticstudieq. For example,nonrornpet- 
i t iceinhibilion occurs ifan inhibitor forms nonreactivecom- 
plexes with both the free enzyme, E, and the enzyme-sub- 
strate complex, E S (5, 6, 12). If the same series of 
experimentsthat are displaypd in Figure 6 were carried out 
for a noncompetitive inhihitor, the resultant plot would con- 
Gst of a familv of lines havinediffereot slooei and ordinate 
intercepts, with a single poin