Simplified Treatment of Two-Substrate Enzyme Kinetics Allan G. Splittgerber Gustavus Adolphus College, St. Peter, MN 56082
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The teachine of enzvme kinetics in introductorv biochemistry courses has not received much attention in the chemical education literature, although a t least one excellent intro( I ) . Most ductory article has appeared in THIS JOURNAL textbooks devote some attention to reaction kinetics in general and also cover those enzyme cases in which the enzyme converts a single substrate to products. Those cases where two or more substrates are converted are not so well treated, probably because of space limitations. However, inasmuch as a large fraction of enzymes are multisubstrate, it would appear that some treatment of the subiect would not be wasted. There are several hooks-and articles which consider multisubstrate kinetics (2-6); however, most of these are not suitable as material for supplementing the textbooks. These sources enter much more deeply into the subject than is necessary or possihle in a general course. Other difficulties in treating the subject are limitations on class time and mathematical comnlexitv. Where a course must cover a wide ranee of material, there h a y remain little time for a detailed consideration of enzvme kinetics at anv level. Also, the derivations of the necessaty hathematical relaiionships for niultisubstrate cases are tedious. time consumina, and ~ossiblvincom~rehensible to the a&age student. ~ ithese r reasons, ventures into this area are seldom a t t e m ~ t e din undergraduate This paper attempts to show that the extension from single substrate to multisubstrate kinetics, particularly two-substrate cases, need not necessarily require the investment of a great deal of additional class t i m e . - ~ o s tif, not all, of the comolexities in mathematics may he avoided by the invocation of a-few easily memorized rules. The mechanisms of multisubstrate cases and the effects of dead-end and product inhil~itor*may he discuxsed in trrms d t h w e rules. I.'inally, hecause it may happen that thwestudenrs whuare must likrly to avoid studyingkinetin in the physical chemistry course are also most likely to enroll in biochemistry, it would seem that the extra treatment in the biochemistry course would be beneficial. The Single Substrate Case The enzyme-catalyzed reaction sequence may he written as hl
k3
12
k1
E+S+ES+E+P
This equation has the general form - numerator2[Pl " = constantnumerator~[S] + (coefficient S)[S] + (coefficient P)[P]
(5)
where the numerator (num), coefficient (coef), and constant terms contain combinations of rate constants. I t is often useful to express the rate equation in terms of kinetic constants rather than rate constants. Multiplication of eqn. (4) by (numz)l(coef S)(coef P ) yields
Kinetic constants in this equation are defined as follows: V1 = num2/coef S = k&; VZ = num2/coef P = kzE,; K, = constlcoef S = (kz k3)lkl; Kp = constlcoef P = (kz kdlk4; and K?, = numlInum2 = klk31k2k4. VI is the maximum velocity m the forward direction for the enzyme catalyzed reaction in the case where IP1 = 0 and substrate S is at a level sufficient to maintain ali enzyme in the ES form. Vz is the maximum velocity in the reverse direction, K, the Michaelis s constant for the forward direction, K , the ~ i c h a e l i constant for the reverse reaction, and K,, the equilibrium constant for the reaction. The above equation may also be written as
+
+
VlISl Vz[Pl KP+ [PI + (KdKd[Sl K, [S] + (K./KJ[PI = rate of forward reaetmn -rote ofreuerse reaction
U =
+
(7)
If initial rate studies are done under conditions of negligible product concentration, the t&ms in [PI drop out and the eouation reduces to the more familiar Michaelis-Menten eduation (7). Accumulation Of product increases the denominator of the forward rate expression, showing inhibition by the product. Where Vz[P] is negligible compared to Vl[S] in eqn. (71, the right hand side of eqn. (7) may be dropped and the equation inverted, resulting in
(1)
where E is enzyme, S substrate, ES enzyme-substrate complex, and P product. The step with rate constant k4 allows for reaction of product with enzyme. Following the usual procedures, differential equations for the rate of change of [El, [ES], [S], and [PI with time are written. The steady state principle is applied, i.e., d[ES]ldt and d[E]ldt are set to zero. Finally, use of the conservation equation E, = [El [ES], where E, is total enzyme concentration, allows concentrations of E and ES to he found.
+
The rate equation for the enzyme-catalyzed reaction is then found to be
.A plot of I I versus I 151 f t m wvious cmcentratiuna ctiS will result in ,I st~:iigh[line o r k ~ p v( K . \ ' I ) t 1 r IF'] K,,! and intt.nYD1 I \',. Uliiereut iixed levrl.; d 1' will i h i n r e the slupe of thk plot but not the intercept. Under conditions wh& product is present but not to the extent that the reaction proceeds with a significant rate in the reveise direction, the product inhibition is shown by the changing slope and fixed intercept of the plot to be of the competitive type, i.e., P competes with S for enzyme E. A more realistic reaction sequence may be written by adding an enzyme-product [EP] complex to the previous reaction sequence. However, this more complex case reduces to the same form as eqn. (8).
Volume 60
Number 8
August 1983
651
Two Substrate Cases
For pedagogical purposes, the consideration of two bisubstrate cases is probably sufficient. In the nomenclature introduced by Cleland (8) these are the ordered sequential bi bi and the ping pong bi bi mechanisms. Bi hi mechanisms are those in which two substrates hind to the enzyme and two products are released. In sequential mechanisms the substrates bind before release of products whereas in the ping pong mechanisms there is alternating addition of substrate and release of product. The sequential mechanism is represented by
in which all enzyme intermediates are shown. Substrates are denoted by A, B, C, . . . ,while products are P, Q, R, etc. The representation for the ping pong bi bi mechanism is
in which alternate substrate addition and product release is shown. To show individual rate constants, a closed form representation mav be written for either mechanism. For the sequential mechanism this is
Figure 1. Reciprocal plots, sequential ordered bi bi mechanism
(the variable substrate) to determine a series of reaction velocities. These velocities are plotted as l/u versus l/[A], which yields a straight line with slope (K$V1)(1+ Ki&dK.[B]) and intercept ( l / V l ) ( l + Kb[B]). If [B] is raised, a second straight line with smaller intercept results. If several values of [B] are tried, a family of lines will result which intersect to the left of t h e y axis, as shown in Figure 1. For the ping pong hi bi mechanism, the rate constant representation is
from which the initial velocity rate law is found to be
from which the complete rate equation may be found by standard methods (9). Generally, only initial velocity rate equations are desired (no products present initially) so that all the [PI and [Q] terms may he dropped. The initial velocity equation for the sequential mechanism then becomes
This equation, in terms of kinetic constants, is
and the reciprocal is which is of the form If B is the changing fixed substrate and A the variable snbstrate, a plot of l / u versus l/[A] yields a straight line with slope K,/V1 and intercept (11V1)(1 Ki,/[B]). As [B] is raised, the intercept is lowered but the slope remains fixed (Fig. 2). Figures 1and 2 show the differences in experimentalline patterns for the two mechanisms, which may he used to distinguish between them.
+
This equation may be written in terms of kinetic constants by dividing through by coef AB which yields
+
where Vl = num~lcoefAB = h5h7Etl(h5 h7). Ka = (coef B)/(coef AB) = h6k7/hl(h5 h7),Ki, = (coef A)/(coef AB) = h7(h4+ h5)lh3(hS h7), and Ki, = k2/hl. The inverse of this equation, the most useful form, is
+
+
Writing the equation in this way corresponds to [B] being held constant (changingfixed substrate) while substrate A is varied 652
Journal of Chemical Education
Cleland's Rules '1'" prewnt the foregoing material on bisuhstmtt kinetics in 11(,11111 would rewire a lorre ani
