I
William T. Yap, B a r b a r a F. Howell a n d Robert S c h a f f e r Bioorganic Standards Section Analytical Chemistry Division National Bureau OI Standards Washington. D.C. 20234
Determination of the Kinetic Constants a TWO-SubshateEnzymatic Reaction 1
I
Eneyme-catalyzed reactions are a fascinating and important c l a s of chemical reactions. In our efforts t o develov Standard Reference Materials for clinical laboratories, we have made numerous measurements o n t h e kinetics of t h e reaction of pyruvate with reduced nicotinamide adenine dinucleotide (NADH), catalyzed b y t h e enzyme lactate dehydrogenase (LDH). This enzyme has been studied by many workers, e.g. ( I ) a n d (2).I n this article we would like t o present a method which we found helpful i n analyzing t h e portion of our kinetic results which involves t h e initial velocities a s functions of substrate concentrations. This analysis provides a n illustration a n d a n accurate method of determination of t h e kinetic constants in a two-substrate enzymatic reaction. The Kinetic Constants Let us begin with a simple one-substrate (A) reaction, catalyzed by the enzyme E, giving the product P k!
k3
h2
k.
A+E=X=P+E
(1)
l/[pyruvate]+0.02/ [NADH]
+
Figure 1. Plot of I/@ versus r = 1/[A] cl[B]. The concenbatlons of me substrates are in mmolell, inverse initial velocities In min, and c = 0.02.
Under steady-state conditions, the velocity of disappearance of A can be written as follows ( 4 ) where the k's are reaction rate constants as indicated in ean. . (11: . . . E.. is the torn1 mz\meconeentratlm whirh is taken to include both i.: and theenryme-rrnrtnnt complex ?(.and th~aymholr[ I desieatecuncentration. Moreover, a t an early enough stage of the reaction when the concentration of P is essentially zero, the expression for this velocity simplifies to
This is the well known Michaelis-Menten eouation. Here uo is the initial velurity,and \'and K are kineticconstants, where V = k X , is the maximum vrlocity and K = \h2 k 11,h , is thr \lirhaelrseonstant. Now for an enzyme catalyzed reaction involving the two substrates, A and B ~~
~
A+B+E+X+ProductstE
(4)
the initial velocity becomes a function of the concentrations of both AandB. However, when [B] is held constant, and the initialconcentrations of A are varied in aseries of m eaoeriments. then the initial velocities, uO;;, as a function of [Ali, againtake the form of eqn. (3), i.e.
vO..-V,[Ali
" - Kj + [A];
i=1,2,3,,,rn
(5)
But now the apparent kinetic constants, V; and K;, are functions of [B], where the subscript j denotes the jthconstant value of [B]. We define KA,the Michaelis constant of A as the value of the apparent K;, for IBI; ., m. Similarlv. we can interchanee " the roles of A and B and the same nnalpls applies. Thus in the same manner, we defme KH,the Michaelis constant for H as thevalueof theapparent K, for [A]; m.
-
l/uO= KnIV[A]
+ 1IV
~
+
.
and plot llu':, versus I , , . Here c isan arbitrary constant whose value is chosen to make the plotting convenient and easy to read. (This is in analogy to the well known Zimm plot ( 5 ) in light scattering in macromolecular solutions.) Least square regression lines are then drawn for sets of points with the same [Ali; and extrapolating these lines to l/[Blj = Ogives the locus of inverse initialvelocity as afunetion of l/[A]; a t infinite concentration of B. These points are shown as triangles in Figure 1.Then a least square regression line is drawn through these extrapolated points. The equation for this line is
-
Determination of t h e Kinetic Constants-The Double Reciprocal Plot Experimentally, we measured the initial velodties of disappearance of B (in our case NADH) at n initial concentrations of B and rn initial concentrations of A (i.e., pyruvste). Thus we have a matrix of rn X n values of uO;,, that is, one value for each pair of [Ali, [B]; for i = 1,2 .. .m, and j = 1,2. ..n. We define a variable zi; = l/[A]; t e/[B]j 254 1 Journal of Chemical Education
whirh ir a rearranged form of eqn. (31.Thisline intersects the ordinate ax~sat1/V, the maximumvelocity.TheslopeollhislineirK~ll':thus the .Viehaelis constant for A, KA,is obtained i n n the ratio of the slope to the intercept. Similarly,extrapolating the least square lines through sets of points of the same [B], to l/[A]; = 0 gives the locus of inverse velocity as a function of l/[B]; at infinite [A];. These are the circles in Figure 1;and Wle regression line through these extrapolated points should intersect the ordinate axis again a t 1/V. This line has a slope equal to K&V; and thus Ke, the Michaelis constant for B, is obtained from the ratio of this slope to the intercept. The changes of the slopes of the two families of lines as functions of [A] or [B] depend on the mechanism of the reaction. For instance, for the Pine Pone mechanism the olot of 11" versus z eives two sets of oarallel fines. while for a seauekalmeehanism. th;alonea of~~~~the ~~~, ~~~~~~~~, lines in each family will vary monotonically with the concentration of the sulrstrate held constant, asshown in Figure 2. Thedopeaof the line. with constant [B] aregiven by K A ( t~ K , A K ~ , I K A [ B ] I I V , ~ ~ ~ those with constant [A] by K d l + Kd[A])IVc. Here K;Aii another kinetic constant, the so-called inhibition constant. Thus bv olottine either set of slopes versus the corresponding r e c i p d conc&tratioi one could obtain K;A from the slope of the resulting line. Further discussion on mechanisms can he found in references (3)and (4). ~
~
~
~~
~~~~
7~~
Results and Dlscusslon The reaction of NADH with pyruvate, catalyzed by LDH NADH
+ pyruvate + H+ 2NAD+ + ladate
has been measured in our laboratory. A typical set of the inverse initial velocity as a function of concentration of NADH, [B], and pyruvate,
[A], is given in the table. Figure 1gives the plot of the inverse initial velocity as function of z. From this plot, the Michaelis constant for NADH is determined to be KNADH= 27.57 pmolefl and for pyruvate, K,,, = 0.528 mmoleil. The maximum velocity for this series of experiments is V = 36.3 pmole I-' min-'; and as indicated before, V is a function of the concentration and activity of the enzyme used in the ex~eriments. The determination of there kinetic constant-, involwa eatrapolations into the hypothetical regionsofv~ryhigh substrate ronrentrations where other effects such as inhibition become important, as some of our experiments have indicated; thus the situation there is considerably more complex than the simple kinetic picture presented here. However, in the range of concentrations of experimental interest, the simple model presented seems t o be adequate and thus the determined kinetic constants are meaningful characterizations of the materials as substrates. One of the procedures (3.4.6) used in the analysis of two-substrate enzymatic reartmns 1s the ~uccesfiiveapplication of reciprocal plots, i.c, plotting inverse velocity versus mwrse [Al at ronstant[R], then vlottinc the interrevwand s l o ~ eof s these lines versus m w n e IRI. Due experimental uncertaintie', two different sets of kinetic constants could be obtained depending on which of the two substrate concentrations was utilized first in the series of plottings. The simple modifications presented here of plotting llu" versus 2 have the advantage of putting all the data conveniently on one plot and the beauty of treating both substrates symmetrically.
.~~ ~
-
+
Figure 2. Theoretical plot of 1/@ versus z = 1/[A] 2/[B] for a sequential two-subshatesreaction: KA = 3.K. = ~ . K ( = A 4 and V = 1.
Valuer of Inverse Initial Velocity lmin), as a Function of NADH and Pyruvate Concentrations (mmole/ll
INADHI Ipvruvatel
0.0412
0.0314
~~~
~~
Literature Cited J.,snd Kaplan. N. O . i n "Advances in Enzyrnology."Vnl. 37,iEdi!or Meirfer. A ) , Interscience. 1973. I21 Schwcrt,C.,J.R i d Chrm., 244,1285(1969). 131 Cldand, W . W.,Riochem. Riophva. Ana, 67.1SSi19631. (41 Pb,wman, K. M.. "Enzyme Kinetics? McCrsw-Hill,New York. 1972. (51 Z1mm.B. H.. J. Chem Ph.sa.. 16.1099 11948). 161 Mahlor. H. R.,andCordes. E. H.."Biolqical Chemistry,"Harperand Row Publishers. New York. 1966. Chap. 6. (11 Eve-.
0.0205
0.0165
Volume 54, Number 4, April 1977 / 255
