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134 Enzyme Competitive Inhibition. Graphical Determination of Ki and Presentation of Data in Comparative Studies PAOLO ASCENZI, a MARIA GRAZIA ASCENZI b and GINO AMICONI a

CNR, Center for Molecular Biology Department of Biochemical Sciences University of Rome "La Sapienza" Piazzale Aldo Moro 5 00185 Rome, Italy and b Department of Mathematics The City University of New York at Brooklyn College Brooklyn, New York 11210, USA

1). Although such a graph is reminiscent of the replot of the slope (ie, KamPP/V) of a series of Lineweaver-Burk graphs (obtained in the presence of different inhibitor concentrations) versus [/],2 analysis of data according to Eqn (3) appears to be convenient for comparative studies. In fact, the presentation of experimental data (ie K~pp values) in relative terms to K m (ie /~mPP/Km) allows us to handle, in a single graph, very different K app values obtained under a wide variety of conditions. This is an advantage which is also gained with respect to the treatment of enzyme competitive inhibition according to the classical graphical method of Dixon. 3 In particular, 10 8

In true competitive inhibition, substrate and inhibitor binding to the free enzyme are mutually exclusive and rates for substrate and inhibitor association and dissociation are never limiting in the process. Therefore, a competitive inhibitor acts only to increase the Michaelis constant for the substrate. Using the steady-state assumption, enzyme competitive inhibition may be described according to the well-known reaction scheme

E+S + I

K•P ~ E.S

V

4 2

°o" 4" 8" i2' le' 2o 11)

E+P

5 IMI

E

g i 1l

IE e

E.I where E is the enzyme, S is the substrate, E.S represents the enzyme.substrate and/or enzyme.product transient(s) involved in the reaction, P is the reaction product(s),/(-ampp is the apparent Michaelis constant (obtained in the presence of constant inhibitor concentration, [/]), V is the maximum velocity (independent of [/]) and Ki is the inhibition dissociation constant of the enzyme.inhibitor complex (E-/). The dependence of the initial velocity (v) on substrate concentration, [S], in the presence of constant inhibitor concentration, may be described by the Michaelis-Menten equation ~

[s])

(1)

~ mpp = K m (1 + [/]/Ki)

(2)

v = v.[s]/(ledp

+

where /~mpp, defined as

replaces Kin, the 'true' Michaelis constant (obtained in the absence of the inhibitor). Eqn (2) may be transformed as follows /~mPP/Km = [/].K71 + 1

(3)

Therefore, in pure competitive inhibition systems, a plot of I~mPP/Kmversus [/] gives a straight line of slope gi -1 (Fig BIOCHEMICAL EDUCATION 15(3) 1987

4 2 0 0 . . 2 . . .4 . . 6 8 ' 10 [I] xlO4 (M)

Figure 1 (Top panel) Plot of K~P/Km versus [I] describing the inhibitory effect of p-aminobenzamidine (circles," K71 = 1.2 × 105 M-l), benzamidine (squares; K~ 1 = 5.3 × 104 M -1) and phenylguanidine (triangles, K7 1 = 1.4 × 104 M -1) on the hydrolysis o f N-et-carbobenzoxy-L-lysine pnitrophenyl ester (open symbols; K m = 1.0 x 10 -4 M ) and N-ot-carbobenzoxy-L-tyrosine p-nitrophenyl ester (filled symbols; K,n = 2.3 x 10 - 4 M ) catalyzed by bovine [3trypsin. (Bottom panel) Plot of K~P/Km versus [I] describing the inhibitory effect of benzamidine on the hydrolysis of N-ot-carbobenzoxy-L-lysine p-nitrophenyl ester catalyzed by bovine B-trypsin (circles; Km = 1.0 × 10 -4 M ; K7 1 = 5.3 × 104 M - l ) , bovine factor Xa (squares; Km = 3.0 × 10 -5 M," K~-1 = 5.9 × 105 M -1) and porcine pancreatic f3-kallikrein-B (triangles; Km = 1.8 × 10 - 4 M ; K7 1 = 1.7 × 105 M-l). Data were obtained at p H 6.8 (I = 0.1 M, phosphate buffer) and T = 21°C. Experimental procedures have been reported elsewhere. 4-6 For details see text.

135 analysis of results according to Eqn (3) allows us to compare directly (i) the inhibitory effect of compounds of the same class on the enzyme activity detected with substrates showing different K m values and (ii) the inhibitory action of agents on the catalytic activity of various enzymes of the same family characterized by different Km values for substrates (Fig 1). Thus, in simple systems, such as various proteinases and low molecular weight inhibitors, the comparison of data involves only one constant, K~-1 (ie the value of the slope of the straight line(s) in plots of/~mPP/Km versus [I]) (Fig 1). Thus, although an exact quantitative approach to inhibitor binding thermodynamics is obtained only by numerical procedures, the proposed graphical method finds its usefulness in comparative studies, and provides a simple way to assess the inhibition parameter, Ki, as well as to test the goodness-of-fit of the data to the enzyme competitive inhibition model. Since students have previously learned that the value of Ki is equivalent to the inhibitor concentration that doubles the slope (ie KamPP/V) of Lineweaver-Burk plots (in fact, Ki does not correspond to the [/] that yields 50% inhibition), the proposed graph is strictly related to this useful notion in enzymology.

The overall reaction is described by the following equation: C, = Clot" e -kI~''' + Cstow " e -ks'°-''

(1)

where Ct is the residual activity at time t, Cfa~t and Cslow, kraft and kstow are the amplitudes and first-order rate constants of the fast and the slow phases, respectively.

\® I / rr

,s-o

3

- "" -..

~ "

'1- .........

"-..

fo L

~

T

"

t t,. f,,st

~"

""

TIME

t i/2 s l o w

TIME

References 1Michaelis, L and Menten, M L (1913) Biochem Z 49, 333-369 2Cleland, W W (1970) in The Enzymes (edited by Boyer, P D), Vol 2, pp 1-65, AcademicPress, New York and London 3Dixon, M (1953) Biochem J 55, 170-171 4Robison, D J, Furie, B, Furie, B C and Bing, D H (1980) J Biol Chem 255, 2014-2021 5Menegatti, E, Guarneri, M, Ferroni, R, Bolognesi,M, Ascenzi, P and Antonini, E (1982) FEBS Letters 141, 33-36 6Ascenzi, P, Menegatti,E, Guarneri, M, Bolognesi,M and Amiconi,G (1984) Biochim Biophys Acta 789, 99-103

Figure 1 constants reaction. abscissa. observed

Semilog plot for the determination of the rate of fast (A) and slow (B) phases of a biphasic The slow phase has been extrapolated to the The inset shows the log difference between the values and extrapolated values versus time

Extrapolating the slow phase to zero time so that it cuts the abscissa at points S, the antilog of S gives the amplitude of the slow phase. Now % Cfast "[- % Cslow = 1 0 0 ,

An Easy Method to Determine the Kinetic Parameters of Biphasic Reactions

hence % Cfa~t can be calculated from Eqn (2). In order to get the half-life of the slow phase, subtract 0.3 from S and read off the value corresponding to it on the ordinate (tv.low). The rate can be calculated from the simple equation:

A M KAYASTHA and A K GUPTA Department of Biochemistry Banaras Hindu University Varanasi-221005, India Enzyme kinetic experiment data (for example % residual activity versus time) often shows biphasic kinetics. Here we describe a very simple method for the determination of the rate constants of biphasic reactions. Consider a hypothetical case: first of all we plot log % residual activity versus time. This typical plot (Fig 1) shows two distinct phases, fast (A) and slow (B). Phase B shows a slow exponential process. Extrapolation of the slow phase to zero time and correction for the same shows that the fast phase also exhibits first-order kinetics (inset).

B I O C H E M I C A L E D U C A T I O N 15(3) 1987

(2)

k = 0.69/t,a

(3)

Now plot the log of the difference between observed and extrapolated values of the residual activity of fast phase versus time (inset). Let us assume that the straight line cuts the abscissa at point f. Subtract 0.3 from this value and read off the value corresponding to it on the ordinate. This is the half life of the fast phase (t,~a~t) and using Eqn (3) kraft can be determined. We do not think that this method is in any sense original (see reference), but if used by biology students might save their time for calculations.

Reference Ray, W J and Koshland, D E (1961) Brookhaven Symp Biol 13, 139