Equilibrium Binding and Steady-State Enzyme Kinetics An Approach Emphasizing Their Commonality H. Brian Dunford University of Alberta. Edmonton, Alberta, Canada T6G 2G2
Two commonly recurring requirements in chemistry and molecular biology are the measurement of equilibrium hinding (or dissociation) constants and the experimental determination of the two parameters which govern steady-state enzyme kinetics. The major purpose of this article is to show that these two apparently unrelated problems have a great deal in common. The equations and their error analysis can be cast in identical forms. Thus, if the solution of one problem is taught, the solution of the other is also. Various methods of data analysis are evaluated. Steady-State Enzyme Kinetics For a single-substrate, enzyme-catalyzed reaction the simplest model is k1
brat
E+X+EX-+E+Praduct
(1)
b-,
where X is substrate. The steady-state approximation may be applied to EX and the conservation relation (2) [Eln = [El + [EX1 is valid; [El0 is the total enzyme concentration. It follows that the velocity of the reaction u (M s-') is given by
leading to a rectangular hyperbolic plot (Fig. 1).In eqn. (3), kcat (s-1) is the turnover number and K = (k-I + k,,t)lkl is the Michaelis constant in M units. The maximum velocity (for [XI m ) is kmt [Elo. For any smaller value of u only afraction of the maximum velocity f is attained
-
Therefore eqn. (3) may be expressed as
Fgure I Reclang.lar nyperoo c p 01 of lend)-slale enzyme kmalocaala( en ord "ale) For e q d br%m ornamg data .se tnc rlgnl oramale and replace K,, by (LX Erra I m.Is n Kare s n o w o, me d a m nes mese Imlsarc somewha1 deceptive for small (X) because of the steepness of the curve.
centration of enzyme [Elo,is eqn. (2). For weak hinding and large enough [XI .
-
[Xln = 1x1 If only EX absorbs light of a given wavelength then A
= CEX
[EX]
(9)
where A is the absorbance and E E X the molar absorptivity. (The units of r ~ in x eqn. (5) are M-1 since unit path length is included.) If [EX] [El0 then
-
A 0 = €EX [El0
For steady-state enzyme kinetics the Michaelis constant K is not a dissociation constant; and the more elaborate the model the more obvious this statement becomes. Nevertheless, K has an important property of a dissociation constant: the larger the value of K, the larger [XI must become in order that u kmt [Elo.
(8)
(10)
and CEX may be determined independently. However, for weak hinding both K and r ~ are x unknown until the performance of accurate experiments and data analysis. The above equations may be combined into the form
+
A plot of AI[EIoversus [XI is a rectangular hyperbola as shown in Figure 1.The fraction of total enzyme which has X bound to it, f , is
Equilibrium Binding The system E+X=EX may represent, for example, binding of ligand or inhibitor by an enzyme or formation of a charge-transfer complex. The dissociation constant K (in molar units M ) is
The conservation relation, expressed in terms of total con-
Substitution of eqn. (12) into eqn. (11) leads to eqn. (5),which was derived for steady-state kinetics. The Common Equation Equation (51, which leads to a rectangular hyperbolic plot whether the data are for equilibrium hinding or steady-state enzyme kinetics, can he expressed conveniently in at least four other ways to obtain K: Volume 61 Number 2 February 1984
129
Table 1.
Type of Analysis of the Common Equations for Steady-stale Enzyme Kinetics and Equilibrium Bindinga Form of Plot
Type of leastsquares analysis
fversus [XI (Fig. 1)
rectangular hyperbola
nonlinear
1 1 -versus -(Fig. 2) f [XI
straight line
[XI -versus f
[XI (Fig. 3)
f -versus
f(Fig. 4)
Form of equation
I=-1
Plot
1
Names assac~atedwith Equilibrium
...
Deranleau-Neurath (3
linear
Lineweaver-Burk (3
Benesi-Hildebrand (4)
straight line
linear
Hanes ( 5 )
straight line
linear
Eadie-Hofstee (7-9)
sigmoidal
nonlinear
K
+-[XI
l-
+1 [XI
f
Kinetics
-Ixl- (X) + K f f
-= - 1 K
[XI
+1 K
...
1x1
f versus log [XI (Fig. 5 )
...
con (61 Scatchard ( 10)
Bjerrum ( 1)
f = ~ractionol maximum velocity.
Finally, a plot off versus log [XI may be employed (I). All of the forms of eqn. (5) (eqns. (13-15)) may be cast into the appropriate form for steady-state kinetics or equilibrium binding by substituting for f using eqn. (4) or (12). All of the methods are summarized in Table 1and Figures 1-5. The parameters used to construct Figures 1-5 are listed in Table 2. A more detailed discussion of errors is given below. However, at this point it is irnportmr ro realize-thnt experimental uointsshould br ilirvrnlq spnced as posaihle oi a function uf f i n order to obtain maximum a c c ~ r &in- the ~ analysis (11). There are several useful criteria to determine which is the best form to plot. The computer age has changed one of these criTable 2.
Parameter Values in Figures 1-5
Kinetics K = 10+M [Elo = M ...k = to4 SC
Equilibria K= [E]o = &Y
lo-'
M M
= 10' W '
Figure 2. Lineweaver-Burk plot of steady-state enzyme kinetic results (len ordinatel. oiot of eouilibrium bindino data use riaht ordinate ~For Benesi-Hildebrand ~ , nno rep ace k,,, of t.:, Nole how the exper mental polnls tunich are spawd l a r.y evcnly as a 1.m on of 0 are jammeo togelnw near m e ordmale ~
130
teria. No longer is it necessary to cast the data into the form of a straight line equation because analysis of a curve is now just as easy. Therefore, the rectangular hyperbola, analyzed directly (see ref. (2) for an example involving equilibrium binding), and the semilog plot both deserve a higher priority. Of the two curves, the experimental points are better spaced along the curve in the semilog,plot (Fig. 5 compared to Fig. I),and the errors are shown mure clearly. The semilog plot is symmetric with the value of K occurring a t the inflection point, but there is no intrinsic change in the rectangular hyperbolic plot in the region of [XI = K. The subject of fitting polynomial equations to curves has been reviewed exhaustively (12), and thus its specifics will not be discussed here. To a certain extent in the Hanes (Scott) plots and particularly in the Lineweaver-Burk (Benesi-Hildehrand) plots (Figs. 2 and 3) the experimental points aie spaced unevenly. The plots are open-ended along both axes. Therefore, select data might make a superb-looking but virtually useless linear plot. The negative abscissa intercepts in Figures 2 and 3, which are also shown in many textbooks, desetve comment. They serve one purpose only: to point out that there would be advantages to having experimental points lie along the hypotenuse of a triangle. The hypotenuse (slope) and its two ends (two intercepts) would define both parameters to be determined and the self-consistency of their values. This does occur in the Eadie-Hofstee (Scatchard) plots in Figure 4. However, the negative intercepts in Figures 2 and 3 do not contribute to the analysis nor to its self-consistency,
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Journal of Chemical Education
Figure 3. Hams plot of steady-state kinetic data (lefl ordinate). Scon plat of equilibrium binding data (right ordinate and kc., replaced by k).
Table 3.
Assessment of the Relative Merits of Various Types of Plots of Steady-Sfate Kinetic and Equilibrium Binding Data
Type of Plot
Rating (on 6-pi.
Points well
Errors accurately and clearly
scale)
spaced
displayed
Entire accessible range of exptl. pts. displayed along
'
Clear visual repre~entation of one or both of K and kcat(fd
one axis
two axes
one
+
both
4
+ +
+
+ +
-
Rectangular hyperbola (Deranieau-Neurath)
3
t
+I?)
t
-
+ + -
-
Hanes (Scon)
2
+I?)
-
-
-
t
-
6
Eadie-Hotstee (Scatchard) Semilog (Bierrum)
1
Lineweaver-Burk (Benesi-Hildebrand) ' A "f"sign indicates that
t
-
the type of plot exhibits the given characteristic: a
+
-
sign, that ttdoes not.
The rectangular hyperbola and semilog plots (Figs. 1and 5) show the full accessible range of experiments along one axis, but only the Eadie-Hofstee (Scatchard) plots (Fig. 4) do so dong both axes. These considerations are tabulated in Table 3 which includes an evaluation of the relative merits of the different types of plots on an arbitrary scale. Error Analysis
In the error analvsis to follow we shall concentrate w o n K. Errors in EEX and kc,, become small provided f can be made sufficientlv laree. Errors in 1x1should also be nealieihle. - ~ q u a t i o k(1;) may he rearranged to K = [XI (llf - I)
-
t
(16)
off, K , and [XI are interrelated, for example through eqn. (16). Therefore, a plot of dKIK as a function of log [XI is also readilv obtained as shown in Fimre 6. In addition, this plot is symmctrlt: and is ot'grratvr practical \.slue jinrr log [XIis obtained orior r u data analviis: i is only ohtainrd after. The m i n i m ~ ~ e r r occurs or wheie [XI = K.. Each value of dKIK is a single-point determination. In other words, if only one experiment were performed with a 2% error in f then the percent error in calculating K can he read from Figure 6. Of course the percent error in K can he reduced by performing many experiments. A good rule-of-thumb for these experiments can he deduced from Figure 6: vary [XI from K110 to 10K. One wants to cover as broad a range of [XI
The differential of eqn. (13) is
In error analysis the - sign becomes f.Division by K and substitution of eqn. (16) leads to
For a constant error in f , one can readily show that a plot of dKIK versus f is symm&ic with a minimum value a t f = 0.5 (see (13)and Figure 1 of ref. ( I I ) , which is based on a more complicated butessentially equivalent equation). The values
log [XI Figure 5. Semilog plot of steady-state kinetic data (lefl ordinate). Bjerrum plot of equilibrium binding results (right ordinate and k,, replaced by 6 ~ ~ ) .
[El,
x 10-3, s-1
Fiaure 4. EadieHofstae olot of steady-state kinetic ex~eriments(lett ordinatel. scatchard plot of equilibrium binding data (right o r d i n k k,, replaced by, 6 and v replaced by A in the abscissa).
Figure 6. Relative errors in Kas a function of log [XI. Inset: dKlKversus (X). It is assumed that the error in f is constant at 2 % (eqn.(17)).
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Number 2 February 1984
131
as possible, but there is little value in experiments performed outside these limits. This rule-of-thumb is comparable to one which states that 75%of the range off should be covered (11, 14). In all of the plots (Figs. 1 3 ) we have shown the single-point error limits in K corresponding to 2%error in f. The error bars are vertical for Figures 1-3 and 5 and are not shown. In Figure 4, errors in f influence x - and y-values of each point equally so the error bars are symmetric with respect to the origin (11). The maximum y -axis value in Figures 1and 5 is k,.t (or €EX) which is proportional to f . Therefore, assignment of errors on a fixed scale to individual points is readily performed. The same applies to Figure 4 along both axes which leads to the error symmetry described above. Error bars in the two reciprocal plots (Figs. 2 and 3) are misleading since the percentage error scale varies with the y-value. Extensions The equilibrium binding analysis is readily extended to NMR chemical shift data where f = 6/60 (11). If both E and EX absorb light A = CE[E]
+ ~EX[EX]
(19)
The value of r~ is determined readily in the absence of [XI.
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Journal of Chemical Education
Withintroduction of the conservation relation (eqn. (3))it can be shown that A should be replaced by [A - EE [Elo]in Figures 1-5. . ..
Adherence to the equations for Figures 1-5 is evidence for Michaelis-Menten kinetics or 1:l binding. The theory may be extended for effects such as positive or negative cooperativity, allostery, and multiple-binding sites. I t is assumed in this paper that pH, ionic strength, and temperature are controlled accurately; otherwise apparent complications are introduced into simple systems. Llteralure Clled (1) Bjerrum, J. "Metal Amminc Formation in Aquwvs Soiutiona," P. H a m and Sona, Copenhagen, 1941, PP. 1-298. (2) Deranleau, D. A,, and Neurath. H.,Biuchemiatry, 5.1413 (1966). (3) Lineweaver, H.,andBurk, D., J Amer. Chsm. Sae.,56.658(1934). (4) Beneri, H. A.,snd Hildebrsnd,J. H., J. Amer Cham. S o ~ ? l , 2 7 0 3(1949). (5) Hanes, C. S.,Biocham. J.,26,1406(1932). (6) Sc0tt.R. L.,Rec. tmo. ehim., 75,787 l1956). (7) Esdie, G.S . , J B i d Chem., 14665 11942). (8) Hofitee,B. H. J . , L i r n r a , 116.329 (1952). (9) The Eadie-Hofsteeplotulas originally proposed by Wooifin 1932. Seo Diron,M.,and Webb, E. C. "The Enzymes," 2nd ed., Longmans, London, 1964, p. 69. (101 Scatrhsrd.G..Ann. N Y Acod. S e i , 51,660 (1949). (11) Deran1eau.D.A.. J. Amer. Cham. Sac.,91,4044 (1969). (12) Birge. R.T..Rsu. Mod. Phya.. 19.296 (1947). (131 Weber, G.in "Molecular Biophysics," IEdims: Pullman, B., and Wcisobluth, M.I. Academic Pless, New Yark, 1965, pp. 369396. 114) Person,W. B., J. Amar. Chom. Soc.87.167 (1965).
