Addison Aulf
Cornell College Mt. Vernon, Iowa 52314
An Introduction to Enzyme Kinetics
The purpose of this paper is to present a simple, intuitive, yet accurate introduction to some of the fundamental concepts of enzyme kinetics. The experimental definitions of competitive, uncompetitive, and noncompetitive inhibition will be given, and a simple mechanistic model will be presented for each case. While these models may be too simple to adequately represent most actual enzymatic reactions, they correctly indicate possible modes of action of enzyme inhibitors.
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Characteristics of Enzyme Kinetics
Most organic reactions show a linear dependence of the rate upon the concentrations of the reactants. For example, the kinetics of many nucleophilic substitution reactions show that the rate is directly proportional to both the concentration of the nucleophile and the concentration of the substance undergoing substitution ( S d kinetics). In other reactions the rate of the reaction may be independent of the concentration of one of the reactants, and many other nucleophilic substitution reactions show this type of behavior (SN1kinetics). In contrast, however, to most organic reactions, enzyme catalyzed reactions usually show a nonlinear dependence of the initial rate of reaction on the concentration of a reactant, or substrate, S . This characteristic variation of initial rate with substrate concentration, [S], is shown in Figure la. The equation for a line of this form, a hyperbola, is given by
1 rate
(1)
Thus, as in the case of a straight line, two parameters determine the line, and a plot of experimentally determined values of the initial rate of an enzymatic reaction versus [S] will yield information only as to the values for the two parameters. The physical significance of the parameter a can be seen by considering eqn. (1) a t high S concentrations. At high [S], eqn. (1) reduces to: rate = a, since the value of the fraction in the parenthesis approaches one as [S] increases. The parameter a is thus seen to be the maximum initial rate of reaction which can he attained by having S present a t high concentration. The value of a can be estimated from the intercept of the upper horizontal dotted line of Figure 2. The physical significance of the parameter b can be understood by considering eqn. (1) when b and [S] are equal. When this condition is met, eqn. (1) reduces to: rate = a(1/2). Thus b equals that concentration of S a t which the rate is half its maximum possible value. The smaller b, the more rapidly the rate of the reaction will rise toward a with increasing [S]; b is a measure of the sensitivity of the rate to changes in [S]. The value of b can be estimated from the intercept of the vertical dotted line of Figure 2.
Figure 1. a, Characteristic variation of initial rate with substrate concentration. No inhibition. b, Double reciprocal plat of la.
shown by Lineweaver and Burk, taking the reciprocal of both sides of eqn. (1) gives b 1 rate Thus one can see that a plot of (llrate) versus (l/[S]) will give a straight line whose intercept on the (ljmte) axis will be l/a, and whose slope will be bla. The value of a is then the reciprocal of the intercept on the ordinate, and the value of b is the value of the slope times a. Figure l b shows the double reciprocal plot which corresponds to
Lineweaver-Burk or Double Reciprocal Plots
At this point, we will consider an alternative method of obtaining values of the parameters a and b fmm values of initial rate of reaction a t different concentrations of S . As
[SI Figure 2. Determination of thevalue of parameter a.
Volurne51, Nurnber6. June 1974 / 381
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Fieure l a . The advantaee of analvzine the data in terms of; "double reciprocal" plot is that it is easier to accuratelv fit and extra~olatea straight - line than a curved line. So far, we have not said a word about a possible enzymatic mechanism, or model, which could account for the kmd of kinetic behavior summarized in eqn. (1) and represented in Figure 1. Indeed, to do so would have been putting the cart before the horse, or the interpretation hefore the experiment.
fi in ewe aver-^+'&
A Model Wilhout lnhibition
A particularly simple model for an enzymatic reaction whose kinetics can be expressed by eqn. (1) is E
S-P
If the reaction were ta occur according to the scheme
be present as E, and the rate therefore will he half maximal. In addition, if b is large, it means that either k-I, or k,, or both are large relative to kl (it means that ES tends to break down more rapidly to give E than it is formed from E) and so it will take a higher concentration of S to provide a particular steady-state concentration of ES. In other words, the larger b, the more slowly the rate will increase with increasing [S]. In most discussions of enzyme kinetics, the expressions for a and b which are derived on the basis of this mechanism are given special names: kp[Eo] is called V,.,, and (k_l k.,)/k~ is called KM, the Michaelis constant. It is important to realize that KM does not generally equal k - ~ / k ~or, KI, the equilibrium dissociation constant for the combination of E with S. KM and KI will he equal if and only if k, K M Interpretation: Presence of I reduces the concentration of the enzyme species which can bind S. n~~~~
not bind free S. This lowers the concentration of the enzyme in the various species present when I is absent, and does it in such a way that the addition of S cannot overcome the effect. NoncompetitiveInhibition Experimental: 1) Intercept of Lineweaver-Burk plot increased. 2) Slope of Lineweaver-Burk plot increased. 3) Effect of I cannot be overcome by addition of S; (maximum initial rate = a < Vmaz). 4) Apparent Michaelis constant may he larger than, smaller than, or equal toKnr. Interpretation: Joint occurrence of the two actions described for competitive and uncompetitive inhibition.
Literature some of the iiteretur. which I h n d most "Saf"1 and interrating includes the fol-
Uneompetitive Inhibition Experimental: 1) Intercept of Lineweaver-Burk plot increased. 2) Slope of Lineweaver-Burk plot unchanged. 3) Effect of I cannot be overcome by addition of S; (maximum initial rate = a < Vmax). 4) Apparent Michaelis constant = b < KM Interpretation: I combines with an enzyme species which does
386 / Journal of Chemical Education
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A. G..Dlsrvasions Pornday Soc., No. 20. p. 161 (1955).(Activation and InhihitionofE"zyrnes1. Alherty, R.A..Adoon. inENymol., I?. l(1956J.(EnwmeKinetiea). King. E. L., and Altman, C., J Phys. Chem., 60. 1375 (19561.(A Sehemstie Method of ~ e t i v i n g t h e R a Lt ~a w s fmEnryme-Catalysed ReastionsJ. Omton,
Cldand, W. W., "The Enzymn," (Editor Boyc., P. D.), 3rd Ed., Academic P.es, NealYork, 1910, pp. 1-65. (Steady State Kinetics). Plowman. Kent, "Enzyme Kinetic$/ MeCraw-Hili Bmk Co., New Ymk, 1913.
