Rapid Equilibrium-Ordered Enzyme Mechanisms Thomas R. Chauncey, Rebecca Jarabak, and John Westley' Department of Biochemistry. The University of Chicago. 920 East 58th Street, Chicago, IL 60637 Numerous renorts have described euzvme-catalvzed reactions in which there is obligatory order uf substrate rumplex formation in addition to evidence that saturation with the second sul~stratecauses the steady-state velocity to become independent of the concentration of the leading suhstrate (I -11). Other reports have desvril)ed enzymes that require activution by complex formati~mwith an essentiul mrtal ion or other activator prior to substrate equilihrat~on.a process with similar kinetic hehnvinr although the lending reactant in these cases is not a suhstrate (12-18). All of the reaction sequences that generatesuch hehaviur have been called rapid equilihrium-ordered merhanisms, and this category thwrforc i n c l ~ ~ dseveral rs forms nf somrwhat different ~.haracter. Some of the experimenti~lpapers cited ahove contain theoretical discussions of initial velocity behavior and the inhihirim oatterns of nrodurts and dead-end inhihitors in such systems. Treatments that are primarily theoretical include Albertv's orieinal consideration of this tvoe of svstem (19) and contri6utioni from Cleland (20,21), ~ e g e l ( 2 2 jFrieden , (231, Fromm (24,25), and Purich et al. (26). Much of the discussion since Frieden's presentation (23) in 1976 has centered on the nrohlem of distineuishine eauilihrium-ordered from eauiiihrium-random mechanism when the former involve deadend complexes with a substrate or product. The overall theoretical development of equilibrium-ordered systems seems reasonahly complete hut rather scattered. Recent involvement of this laboratory in studies of enzymes that yield equilihrium-ordered hehavior ( 6 , I l ) has prompted an effort to clarify the distinctions that can and cannot he made within this category on the basis of steady-state data. Characteristic Initial Velocity behavlor
I t is customary (22, 23) to consider rapid equilihriumordered hisuhstrate mechanisms in terms of the five-step reaction sequence shown in Figure 1.The reaction begins with the rapid-equilibrium formation of successive hinary and ternarv enzvme-substrate comnlexes. The sole rate-limitinr step i s the transformation of ihe ternary enzyme-substrate com~lex.The later discharees of successive oroducts are ranid enough to permit an equilihrium that is essentially undisturbed hv the steadv-state interconvetsion of the ternarv complexes. Such a mechanism implies an initial velocity eauation that describes the characteristic steadv-state kinetic behavior of equilihrium-ordered systems (eqn. ill). In double reciorocal form the rate eauation orovides a term containing the concentration of the second sibstrate, a cross term containing the concentrations of hoth substrates, and the usual concentration-independent term. However, there is no separate term for the concentration of the leading suhstrate, and it is this feature that relates to the distinctive hehavior of such mechanisms:
where k+a is rate limiting2. However. the above hehavior is not unioue to such "fullscale" equilihrium-ordered formal mechanisms, and equivalent kinetic hehavior is uhtained whm nnlv the formatlcm nf the binary complex with the leading suhstrate is a t equilibrium and the remainder of the reaction sequence is in the 656
Journal of Chemical Education
Figure 1. General rapid equiltbrium-ordered bisubstrate mechanism. A and 6. substrates: X and Y, products; E. enzyme. Reaction (3)Is the sole rate-limiting sten in both directions.
Figure 2. Minimal eq!dlibriumdrder bisubshate mechanism wim dead-end inhialtors. 0,and Q2. inhibitors. Either reaction (1) alone or both reanions(1)and (2) may be at equilibrium.
steady state. For example, the formal mechanism shown in Figwe 2, in the absence of the inhibitors indicated in reactions 4 and 5, produces initial velocity eqn. (2) when only reaction 1 is considered to he a t equilibrium, and eqn. (3) when hoth reactions 1and 2 are a t equilihrium. Eoluo =
k-2 + k+s k-2 + h+3 k+zh+3 [Bl Klk+&+z IAIIBI 1 1
Eoluo =
+
K2k+3 IBI
+-k1+ ~
+ K ~ K z k + a[AIIBI+-k 1t 3
(2) (3)
Equation (3) is identical to eqn. (1) for the full-scale equilibrium mechanism and eqn. (2) is of the same form. On the other hand, the rate equation for the same reaction scheme with only reaction 2 a t equilihrium does not yield comparable hehavior. The corresponding rate equation (eqn. (4)) contains separate terms for hoth substrates and is thus indistinguishable in form
' Author to whom reprint requests should be addressed.
Rate equations for full-scale equilibrium systems were derived algebraically from the equilibrium expressions,while those for full-scale steady-state systems were obtained by the King and Altman procedure (27).and those fw combined equilibrium and steadystatesystems were obtained by the methods of Cha (28).
from the rate equation for a full-scale steady-state mechanism.
These equations indicate that mechanisms in which the binding of the leading suhstrate is a t equilibrium can be distinguished from those lacking this feature on the basis of initial velocity patterns. Only when the leading suhstrate is at equilibrium do the primary double reciprocal plots for the second suhstrate intrrwrt onthe ordinateand does the linrar secondary plot uf the slopes of the double rec~proralplols for the leadinr suhstrate extranolate thn~uehthe oririn. tluwever. distinctiok that relate tdequilibriu& in the bynding of thd second suhstrate or in subseauent reactions of the catalvtic cycle cannot be made on thisbasis. B Figure 3. Deadend inhibition in the presence 01 inhibition by the second substrate. Symbols are as in Figures 1 and 2.
Dead-End lnhlbltion The use of dead-end inhibitors that are competitive with the substrates also fails to yield any formal distinction between full-scale equilibrium systems and those that have only the leading suhstrate a t equilihrium. Equation (5)is the rate equation for the scheme in Figure 2 when both suhstrates equilibrate and eqn. (6) corresponds to the same scheme when only the first substrate equilibrates. I +KdQd 1 + KdQzl+ 1 K I K z ~ + ~ [ A I [ B Kzk+dBI I k+3 (k-2 + k+3)(1+ K d Q d ) (k-2 + k+3)(1+ KdQzl) Eoluo = KL~+&+~[AI[BI k+&+dBl
Eoluo =
+
(5)
+
1
+-
(6)
k+3 In both of these systems, the inhibitor $1 binds to the same enzyme form as the leading substrate and is kinetically competitive with both substrates. This behavior is also seen in equilibrium-random mechanisms.3 A similar dead-end inhibitor that functions in an ordered system with concurrent inhibition by the second suhstrate (Fig. 3) will also fail to yield a conclusion about second-substrate equilibrium. This is clear from the identical form of eqns. (7) and (8),which are the rate equations for this mechanism in full-scale equilihrium and leading-substrate equilibrium, respectively.
3
-Products Figure 4. Activation of enzyme by binding an activator. M, activator. In lhis mechanism. Us assocIatian/dissociation equilibrium of the activator i$ pesumad to m C U r only with free enzyme, and Kis the association equilibrium constant €3
fm m is reaction.
'
+ ( k - P + k+3)(1 + KdQl) + L(8)
K~k+zk+dAI[Bl k+3 As noted previoulsv bv Frieden (23). these rate equations for the full-&ale equifihrjum-ordered system also contain sepa. ratr terms for the leading substrate. This is aconsequence of inhihition by thesecond suhsrratr, and such systems may fail to he identified as equilibrium-ordered merhanims at all.
lnhlbltion by Single Products The last product released in a full-scale equilibrium-ordered mechanism (Y in Fig. 1) serves as an inhibitor competitive with both substrates. Similarly, for all of the mechanisms that yield the characteristic equilibrium-ordered kinetic hehavior, the concentration of the last product as inhibitor appears only
The inhibitor Q2 shows only ihe behavior expected for an inhibitor in that position in any ordered steady-state mechanism: competitive with 6, uncompetitive with respect to A.
on the IAI[HI cross term in the double rcriprural rare equation. This is true whether both suhetrate hindinn reartions art. at equilibrium or only the binding of the leading substrate is a t equilibrium. The inhibition behavior of the first product released is more complicated. As Frieden (23) has noted, the fust product alone (X in Fig. 1)is not an inhibitor in the full-scale equilibriumordered system. This product is an inhibitor neither in the more limited equilibrium-ordered systems where only reactions (1)and (5) in Finure 1are a t equilibrium nor in those schemes in which onlya single ternary complex is considered kinetically significant (Fig. 2). On the other hand, X becomes an inhibitor in all of these systems whenever release of the final product occurs as a steady-state reaction (i.e., when (E-Y) is a kineticallv significant intermediate). The concentration of X then appear&in the [B], the [A][B],and the concentration-iude~endentterms of the rate eauation, and the presence of X thudaffects both slopes and intkrcepts of the double reciprocal plots for both substrates. An Activator a s Leadlng.Reactant 0rdert:d b~suhstratemechanisms that involw generation of the active form of the enzyme by complexinn the free enzyme with a metal ion or other actkator ( ~ i g4 . also display rate hehavior that resembles that of a full-scale rapid equilibrium-ordered mechanism. If all the reactions prior to the rate-limiting conversion of (E.M.A.B) are essentially a t equilibrium, the rate equation has.no term containing the Volume 62 Number 8
August 1965
657
concentration of the activator (M) alone or the concentration of the leading substrate (A) alone (eqn. (9)). Slopes of double reciprocal plots for both of these reaction components approach zero as the concentration of the second substrate approaches saturation.
+-
1
(9)
k+a
Similarly, if the hinding reactions of the activator and the leading substrate, but not that of the second substrate, are at equilibrium, the rate equation (eqn. (10)) is formally identical to eqn. (9),and the initial velocity pattern cannot be used to distinguish between the two cases.
svstems constitute "ranid eauilibrium-orderedbehavior" onlv in the general sense that includes this range of possihiliries. Inhibition bv the first ~roductreleased can he used todetect steady-state, as distinct from rapid equilibrium, release of the last product. Double reciprocal rate equations for all leading-activator systems of the type considered here lack separate terms for activator concentration even when the activatorenzyme complex is not at rapid equilihrium. Separate terms for the concentration of the leadinpc substrate are also lacking in these equations whenever that substrate is at equilibrium with the activated enzyme, whether or not the second substrate is also at equilibrium. ~
~~
~
~
~~
~~
~~~
Acknowledgment This work was supported by research grants PCM 77-26861 and PCM-8116006 from the National Science Foundation and research grant GM-18939 and training grant GM-07183 from the National Institutes of Health.
Literature Cited However, if in such a system only the activator binding is a t equilibrium, while the binding of both substrates is at steady state, the behavior differs significantly (eqn. (11)).
There is now a separate term containing the concentration of the leading substrate, although still none for the activator concentration alone. For all simple systems of this sort involving nonsubstrate activators as leading reactants, the steady-state rate equations contain no separate activator terms, and the slopes of double reciprocal plots for such activators approach zero as the concentrations of the substrates approach saturation. Conclusions Ordered bisubstrate mechanisms in which only the binding reaction of the leading substrate is at equilibrium are not readily distinguished from full-scale equilibrium-ordered mechanisms in which both substrates are at equilibrium and the conversion of the ternary complex is the sole rate-limiting step. For this reason, the distinctive steady-state initial velocity patterns and inhibition responses common to these
658
Journal of Chemical Education
(1) Farley. J. R,C-, D. F., Yang. Y. H. J., and Segel, I. H., J B i d Ckm.. 251,4389 (1976). (2) Farlcy, J. R., N a k s m a . G.,C~i"~,D.,andSegel,I. H..Areh. Bioehem. Biophry. 1% 376 (1978). (3) Garben, D. L., Bimhim. Biophw. Act-, 523.62 (1978). (4) Lluis, C., a n d B 0 4 J., Bimhim. Biophys. Aefa.523,273 (1918). J. B i d Ckm..255.8941 (1978). (5) McClure. W. R., Ceeh, C. L., and Johnston, D. E., (6) Jsrabak, R..and Westley. J., Bioehomiatry. 19,9MI (1980). (7) Rife, J. E..and Cleknd, W. W., Bioehemisfry, 19.2321 (1980). (8) Grossman, G., and O'SuUiven, W. J., Aiutrol. J. Biol S c i , 34,269 (1981). (9) Nimmo, G. A..and Coggins, J. R.,Biochm J., 199.657 (1981). (10) Wiktomcaia. J. E.. Campm, K. L a n d Bonner, J.,Biochemiatry,Zo, 1464 (1981). (11) Chaunay.T.R., and Westley, J., J.Biol. Cham.,258,15037 (1983). (12) Chung, S. I., and Folk, J . E.. J B i d Chem., 245,681 (1970). (13) Morrison. J. F..snd Ebner, K. E.. J. Biol Chpm.. 216.3977 (1971). (14) Geren, C. R.. G e m . L. M., and Ebncr, K. E., Biochem. Biophys. Re=. Commn., I, 139 119751. 0 5 ) ~ u b i n ; ~M., . Zahler, W. L., and Randall, D. D., Arch. Bioehem. Biophys., 188. 70 11978). (16) Nielaen, N. C., Adee, A., aod Stumpf, P. K., Arch. Bimhem. Biophys., 192, 446 (1979). (17) Rauahc1.F. M..Rawding,C. J..Aodsrson,P. M., andvillafrsnes, J. J.,Biochemhtry, 18.5562 (1979). (IS) Knight, W. B., Fitts, S. W.. and Dunaway-Mariano, D.. Bimhrmirfry, 20. 4079 (1981). (19) Alberty, R. A,, J. Amer. C k m . Soc., 75,1928 (1953). (201 Cleland. W. W.. in "The Enzymes: (Editor: Boyer. P. D.). 3rd sd.,i\cademiePms. New York. 1970. Vol. 2, pp. 1 4 5 . (21) Cleland. W. W..Adu. Emymol., 45,273 (1977). (22) Segel. I. H., ' " E n w eKinetin: John Wiley and Sons, New York, 1975, pp. 3 2 e 329.
. ., ...,....,. (27) King, E. L.,snd Altman, C., J. Phys. Ckm.. 60,1375 (1956). (28) Che, S., J Biol. Chem., 243.820 (19681.
