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, reaction 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-catalyzed 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 interpreting 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.
Several experimental approaches are available as,assays for this enzvme. One method involves measurinn- the concentration of a product, inorganic phosphate, using a spectrophotometric analysis for phosphate (3). This assay consists of removing an aliquot from the reaction mixture, inactivatinn the enzvme to prevent further catalysis, adding a molybdate-basedreage& to form a blue complex withthe phosphate, and measuring the absorbance of the phosphomolybdate complex at 690 nm. The concentration of the phosphate produced is calculated from the absorbance of the phosphomolvbdate comolex through the use of a standard curve. This fixed-time assay requires the removal of several aliq 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
-
Enzyme Assays
Enzvme kinetic studies require accurate and reliable methods to measure experimekally the rates of enzymecatalvzed reactions. Rate determinations are accomplished t h ~ o ; ~ the h 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 determination of a substrate or oroduct. Manv assavs are soectrophotometric, but other 'techniques include fluorescence measurements. monitorine the fate of radioactivelv labelled compounds, and electrochemical detection. Consider, for example, the hydrolysis of the phosphate ester p-nitrophenylphosphate @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 phosphatase (2):
-
Flgwe 1. Plot01 ~ ) n ~ e n h a l l o n o #nOIwnlC f phospbte VS. time for aflxedtlme assay (squares) show n5 nitial rate lsolid line).
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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 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. Continuous 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.
.
I
1
Modal 1
-
Klneilc Models and Rate Laws Kinetic data can provide information about two important aspects of enzymatic catalysis: 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. 1.
The concentration dependence of the reaction is, as we shall see later, related to the equilibrium constant for the dissociation 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 studies is that of m e a s u ~ i n ~ i n i t i areaction l rates over a wide ranee of suhstrate concentrations, keeping the enzyme conceniration 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 derived from kinetic models. which consist of oostulated sequences 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 established (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-6phosphate isomerase (7):
-
[O,PO-CHT(CH(OH)l,CH0l2-_ G6P [03PO-CH,(CH(OH)I,CO-CH,0H]z~ F6P
~~~~
~
~~~~~
~~
~
--
*2
(3)
(4)
In this rate law klf is a composite rate constant that is equal to k, [Elt, where [Elt is the constant, total enzyme Journal of Chemical Education
~~
*1
where E reoresents the enzvme. This model savs that, in the are interpresence of the enzyme, the substrate and 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-order dependence on the concentration of G6P:
138
concentration. The rate law predicts that a plot of the measured initial rates versus the concentrations of G6P substrate used to eenerate those rates will he linear over the entire range O~YGGP concentration. This predicted behavior is shown as the dashed line in Fieure 2. The actual e x ~ e r i mentally measured initial rates for this enzyme reaction (8) show a very different type of response to substrate concentration 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 suhstrate. At these concentrations the rate of reaction levels off even though the suhstrate concentration continues to increase. 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 modification k produceb ~ o d k l 2 , *J
E+WP+E(GGP)=E+KP
This enzyme catalyzes the isomerization of an aldose sugar-phosphate, glucose-&phosphate (G6P), to a ketose sugarphosphate, fructose-&phosphate (F6P). The simplest possible sequence of steps that we are able to postulate for this single-substrate reaction we will call Model 1,
initial rote = k,[E],[G6P] = k'[GGP]
Figure 2. Plot of obselved initial rate vs. concentration for glu-6-phasphate isomerase (solid line)and rate predicted by Model 1 (dashed line).
4
The rate constants k, and k4 govern the rates of formation of the complex, E(G6P). from suhstrate and product, respectivelv. The rate constants kl and kl reflect the rates of E ( G ~ Pbreakdown. ) 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. ~ h e b e r i v a t i o nof the ratelaw for this model is suhstantially more complicated than the one for Model 1.Therefore, in addition to the orevious assum~tionsof a constant concentration of e n z y i e 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 corres~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 enzymes are >uch efliclent catalysts that nanomolsr ccmcentrallunsor less are urually sufficient for use in rare mearuremenls. I'ndcr thls assurnptlon, the cunrentratron of free substrate
..
(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 enzyme-substrate complex, [E(GGP)],rapidly reaches and maintains a constant steadv-state level in which its rates of formation and breakdown are equal. Using these assumptions i t is possible to derive the following 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 generally 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 Michaelis 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 corresponds 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 isomerase is rharacresponse shown b j glucu;~-6-l,hoipha~le teristic of most enzymes. Model 2, the simplest kinetic model 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 concentration, [S]:
At a particular set of experimental conditions (pH, temperature, 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.
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 concentration 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 concentration 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 concentrations 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 enzymes that have identical maximal velocities but have Michaelis constants differing by a factor of 10. At concentrations 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 2K, : = (k2 k3)lk:. Thus, K, comprises two firstorder rate constants (kz and k3) having units of s-', and one second-order rate constant, kt, having units of M-'s-l. Insertion 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 conversion 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 quantitative measure of the strength of interaction between the sub-
Substrata
Saturation Region
/
F t s t Ordar R egion
Figure 3. Plot of initial velocity vs. substrate concentration given by eq 5.
Figure 4. S~bsbBteSahlmiOn curves lor two enzymes having identical maximal velocities but Michaelis constants differing by a factw of 10.
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139
strate and the enzyme active site (9). Recall that as the dissociation constant decreases (or conversely, as the binding constant increases), the strength of binding in the enzyme-substrate complex increases, and less substrate is needed to achieve saturation of the enzyme. Now, turning to the maximal velocity term in the Michaelis-Menten rate law, we see that it is defined as h,[E]t, the ~ r o d u cof t 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 enzyme 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 substrate to product a t a single active site on the enzyme surface, is obtained by dividing the catalytic constant by the number of active sites per enzyme molecule. Turnover numbers for enzyme reactions can approach 1,000,000 s-I ( l o ) , 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:
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 inhibitor is a c o m ~ o u n dthat can cause a decrease in the rate of catalysis, t h e magnitude of which is dependent on the concentrations 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 ~ d h a c ktnhib~lmn. pret,entsmetabolic pathways from produrmg to
