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Sungman Chaand Steady Stateor Combined Assumptions of Equilibriumunder the Rapid Equilibrium AssumptionEquations for Enzyme-catalyzed ReactionsA Simple Method for Derivation of RateENZYMOLOGY:

1968, 243:820-825.J. Biol. Chem.

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THE JOURNAL cm BIOLOGICAL CHEMISTR YVol. 243, No. 4, Issue of February 25, pp. 820-825, 1968

hinted in U.S.A.

A Simple Method for Derivation of Rate Equations forEnzyme-catalyzed Reactions under the RapidEquilibrium Assumption or CombinedAssumptions of Equilibrium andSteadv State*J

(Received for publication, July 28, 1967)SUNGMAN CH.~From the Division of Biological and Medical Sciences, Byown Unive rsity, Providence, Rhode Island 02912

SUMMARYA method is described by which rate equations for enzymic

reactions under the equilibrium assumptions may be derivedby inspecting the reaction pathways without solving simul-taneous equations in the conventional way. Also describedis a simple graphic method which combines the above tech-nique with King and Altmans method. By the latter method,rate equations may be derived with minimal eff ort for com-plicated enzymic mechanisms in which some steps areconsiderably faster than the rest so that the enzymic speciesconnected by those rapid steps may be at a near equilibriumwhile the over-all reaction is in a steady state. Rate equa-tions are derived, for the purpose of illustration, for thereversible single substrate-single modifier system, thepartial equilibrium ping-pong bi-bi mechanism, the generalmechanism for two-substrate system, and a complicatedthree-substrate mechanism.

The method of King and Altman (1) has been successfu llyused for the derivation of steady state rate equations for en-zymic reactions, and the method can be further simplified by theapplicat,ion of the theory of graphs as shown by Volkenstein andGoldstein (2). The steady state treatment of complicated reac-tion mechanisms, however, leads to equations and constants socomplex that the basic kinetic properties of the mechanism maybe obscured. For this reason the equations derived on the basisof rapid equilibrium assumptions are sometimes more usefulthan the steady state equations. The purpose of this paper isto describe a simplified method to derive rate equations under

* This study was supported in part, by United States PublicHealth Service Research Grant GM 14093 from the NationalInstitute for General Medical Sciences.

the equilibrium assumptions or the combination of steady stateand equilibrium assumptions.T H E O R Y

When an enzyme-catalyzed reaction consists of more than onestep, and one or more portions of the whole reaction are signifi-cantly faster than the over-all reaction, the partial reactions insuch segments must reach near equilibrium as the over-all reac-tion reaches a steady state. A term, rapid equilibrium segment,will be used to designate such a portion of the pathway which isconnected by one or more rapid steps without. including anyslow steps. In certain mechanisms, there may be only one rapidequilibrium segment and all the enzyme species are included inthe segment. These special cases are known as rapid equilib-rium mechanisms (3), and the simplest example of this kind is,of course, the classic Michaelis-Menten mechanism (4). Ingeneral, however, there may be some enzyme species which areconnected to others only through slow steps, or there may be twoor more rapid equilibrium segments isolated by one or more slowsteps. This latter case will be called in this paper the partialequilibrium mechanism.

Another term, fractional concentration factor is defined forthe purpose of this paper as the fraction of the total enzyme in arapid equilibrium segment occurring as the species in question.More speci fical ly, the fractional concentration factor ( ji) for theith enzyme species (Ei) belonging to a rapid equilibrium segmentconsisting of n enzyme species may be defined as

(1)

where Ni is the numerator of ji, and D, the denominator, iscommon to the expressions of all ji, and D = C Ni sinceCji = 1.Each numerator term Ni can be expressed in terms of the

820

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Issue of February 25, 1968 S. Cha 821concentration variables (of ligands) and rate constants (k).N; is the product of all rate constants and concentration variablesalong the pathway from an arbitrarily chosen reference enzymespecies, El, to the ith enzyme species, Ei, in that direction, di-vided by the product of all rate constants and concentrationvariables in the reverse direction. The term for the referenceenzyme species, N1, is always 1. For all other enzyme speciesthe corresponding Ni term may be expressed as a fraction con-sisting of the concentration variables and dissociation constantsor equilibrium constants. If a step invo lves a binding of aligand to an enzyme species, the concentration variable (of theligand) appears in the numerator of Ni and the dissociationconstant in the denominator of Ni; but i f the step invo lves adissociation, the concentration variable appears in the denomi-nator and the dissociation constant in the numerator; and if thestep is unimolecular in both directions, the equilibrium constantdefined in the forward direction (from El to Ei) appears in thenumerator. When there is more than one pathway throughwhich El may be converted to Eiy any one and only one of thesepathways may be used for the evaluation of N i.l

Although the simplest form of enzyme, such as the free enzyme,is usually chosen as the reference enzyme species (E1 in theabove), any enzyme species in the same rapid equilibrium seg-ment may be chosen as the reference. For example, the frac-tional concentration factors in a rapid equilibrium segment

kz hAEB- EdEA- kaB kz (2)may be evaluated by choosing E as the reference, thusfE = (E) 10-0 + @CA) + (EB) = l , h(A) I b(B)

kz ka(3)

h(A)(Ed kzfEA = (E) + @A) + (EB) = 1, k,(A) I h(B)kz ka(4)(4

K,=1+((A)+(&)Ka Kh

h(B)(EB) h___ =fEB = (E) + (EA) + (EB) k,(A)1+7 + $E)

(5)(B)AKh-

where K, and Kb are the dissociation constants of EA and EB,respective ly. Alternatively , EB may be chosen as the reference,1 The proof of this rule is available on request.

thenKh

fE = Kh(B) KhcAl =1

(A) (B) (6)+(B)+(U)K, +K,+z

with the same result as before. Note that the step EB + Edinvolves the dissociation of B and the binding of A.Derivation of Rate Equation for Rapid Equilibrium Mechanisms-The over-all reaction velocity may be defined as the net flusthrough the rate-limiting step (algebraic sum if alternativepathways exist), and the derivation procedure reduces to theevaluation of the fractional concentration facto rs of the speciesinvolved in the rate-limiting steps, which can be done by in-specting the reaction diagram as will be shown in the examples.Derivat icm of Rate Equations for Partial Equilibrium Mech.-anises-When there is at least one enzyme species or a rapidequilibrium segment which is connected to other enzyme speciesonly by slow steps, the following method may be used to derivethe rate equation.1. Draw a reaction pathway diagram that includes all theenzyme species and the interconversion steps with the rateconstants and concentration terms, e.g. (S) and (P).

2. Assign a new symbol to each rapid equilibrium segment,X, X, etc. For each slow step by which any enzyme species inthe rapid equilibrium segment (X) is converted to other speciesoutside the segment, mul tiply the rate constant (ki) for the stepby a fractional concentration factor ( fi), so that the rate of fluxthrough this step can be represented as kifi(X), where ki fi is anapparent rate constant. For reaction in the reverse directionalong this step, the rate constant is unchanged, unless the stepconnects two rapid equilibrium segments, in which case bothrate constants are multiplied by their respective f ractional con-centration factors. Draw a new reaction pathway diagramemploying new symbols.

3. Apply King and Altmans method to the above newlyconstructed reaction scheme. Regard each rapid equilibriumsegment as if it were an enzyme species, and use the apparentrate constants, e.g. fik l, in place of the rate constant, kl.

4. Finally , the fractional concentration factors expressed interms of concentrations and dissociation constants are substitutedinto the equation derived in the previous step.

EXAMPLES

Example 1. Reversible Single Substrate-Single MoclifLer MechanismRapid Equilibrium Case-The single substrate-single modifiermechanism has already attracted the attention of many kinet-

ic&s, among whom are Segal, Kachmar, and Boyer (5), Bottsand Morales (6), King (7), Ari&ns, van Rossum, and Simonis(8), and Frieden (9). The mechanism involves six enzyme-containing species and nine interconversion steps

E+S h-----ES KG = kz/kl (7)kzkaES-----EPkg Ksi, = ks/k4 (8)ksEl--E+Pkg Keg = k&s (9)

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s22 Derivation of Enzyme Rate Equations Vol. 243, No. 4hE+M- - MEIts

ES+ M ks- MESk 10EP+M hl- MEP

k 12

Kno = b/k? (10) converted to the other species, any pathway may be used torepresent the fractional concentration of the latter species. Forinstance in the above equation, (MES)/(E,) was evaluated byKm8 = km/b (11) the pathway, E + ME -+ MES, but it could be done by E --fES --t MES aswell, then the numerator term of the fraction

becomes (8) (M) /&;KK;;~~ instead of (M) (8) /KmoKm;. ButKmp = knlku (1% these two apparently different terms are actually identical be-

cause K,,K;;;, = K;,K,;.ME+ S k----% MESk 14

kMES 15 MEPk 16kuMEP--ME+Pkm

E + B hw EB k9 kwhere S is the substrate, P is the product, and M is the modifier. p,EQ& E+Q (20)ks k13 hzAlso shown are the definitions of the dissociation constant or theequilibrium constant of each step. suppose two steps, EA = EP and EB = EQ, are much slower,As repeatedly pointed out by the above mentioned authors, compared to the rest, then the reaction may be represented by

Kms = ku/kn (1% Example 2. Partial Equilibrium Ping-Pung Bi-Bi MechanismSimple Casewith Two Rapid Equilibrium Segments-In the

Km,, = ks/kle (14) ping-pong bi-bi mechanismE-I-A h k,- EA - ks- - EP - E + P (19)K,i, = kn/kls (15) h kd -x-

the steady state treatment of this mechanism leads to a rateequation too complicated to be of much practical value. How-ever, if assumptions are made that the isomerization steps,ES e EP and MES s MEP, are the slow ones, and all othersteps are in rapid equilibrium, the rate equation may be writtendown by inspecting the following reaction pathway diagramwithout solving any simultaneous equations.

i . . . (slow) : .DIAGRAM II

4A

DIAGRAM Ik,oXo

DIAGRAM IIaSince the veloc ity may be expressed as the net flux throughthe rate-limiting steps,

oro = k,(ES) + ku(MES) - kd(EP) - k&fEP) (16)

v = [ksja + kujn - kJa - k&lUL) (17)where h fib , f4 , and fi6 represent (ES) l(EJ, (MES) l(&),(EP) /(E,) and (MEP) /(EJ, respectively. Expressing thesefactors in terms of concentration variables and dissociation con-stantsV= (18)Note that the denominator of the above equation is the sum ofthe numerators of all the fractional concentration factors.The firs t term 1 represents E, and the rest represent ES, ME,MES (via ME), MEP (via ME), and EP, respectively, in thatorder. As mentioned above, when there are alternative path-ways through which the reference species (E in this case) may be

X

Diagram II or IIa where X and X represent the rapid equilib-rium segments, EA = E ti EQ, and EP + E G EB, respec-tively, and

LX) = (El + @A) + (E&l (21)(X) = (E) + (EB) + (EP) (22)

and the fractional concentration factors may be expressed ask,(A)

WA)f3=(x)= kzMA)lf- k&Q)k) +k 11= k2kn + klkn(A) + k&la(Q)

ka(P)

(23)

(EP)f4 = (xl) = ksh(B)1+- k@)ks +ksk&,(P)

= kck, + ksk,(B) + k&&Y

(24)

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Issue of February 25, 1968 S. Chak,(B)

(EB)=ix.)=1+

Itsk,(B) kdp)k+k-8 a

k&(B)= k&g + kjk,(B) + k6kdP)

(EQ)h(Q)

f10 = 0 = 1 +kn

k,(A) kd&)kl+T- 11

k&d&)

(25) E\ AGXY\ HEEGX EGY

\/ EGDIAGRAM I II

(26) For the purpose of this paper, this mechanism may be repre-sented by Diagram IIIa. The derivation of full steady state= ktkll + klkn(A) + kzkd&)

~. ._ _. ______._.Xi EB EPAn application of King and Altmans method to Diagram IIayields

(27)

823

Substituting Equations 23 to 26 into Equation 27, and rearrang-ing

(E,)[k,k,ksk,k,k,,(A) (B) - kzk4ksk sk,okdP)(Q )l = k&k5kskll(A) + ktksk,kskn(B)

+ klksk,ktdka + ks)(A)(B) + kzk,k,ksk ll(P)+ kzkSk8k10kdQ) + k3k6ksk12@+ kd (P) (&I (28)+ k&&&n &a + kd (A) (PI+ k?ksk Tklz(ks + kd (B) (Q)

This equation may be compared to EquaCon 29 derived by thetotal s teady state treatment of the same mechanism.(EJ[k,ksk,k,ksk,,(A)(B) - k,kaksksklokdP)(Q)l

= kJiaka(kska + k*kll + kskd(A)+k,ks kn(kzkr + ktks + kzkd(B)

+ [klk,kgkll(ka + k, + k5) + klkakekT(ks + klo + kdl. (A)(B) + kzkaks(ksho + k&n + k&d U) (29)+ ksklOklz(kzkc + kzks + kakd(Q) + [kskak1okd kz + ka+ kd) + kzkaksk ln(ks + ks + kd](P)(Q) + klkdk3+ kd)(kaklo + kskll + k&d(A)(P) + kddks+ klo ) (kzk, + k,ks + k3 ks) (B) (Q)

It can be easily shown that Equation 29 reduces to Equation 28,as it should, when kS, kd, ks, and klo are made much smaller thanthe other unimolecular rate constants, kz, kj, ks, and kll.

Example 3. General echanism for Two-Substrate SystemsCase of Rate-limiting Step within a Rapid Equilibrium Segment

-Wang and Hanes (10) pointed out that there are 784 terms,including cubic terms in some of the concentration variables, inthe full steady state rate equation for the following mechanism,which they considered to be the general mechanism for two-substrate systems.

: .._.. .._..kr 4. .-.-.........~..-----........-..............-.---.........--......

k, ks....--.....---.._...Xi K:, Mb ~....----........--.....-..-EP- - - EBDIAGRAM IIIa

rate equation for this mechanism will be extremely dif ficultfor obvious reasons. Partial equilibrium assumptions, however,simpli fy the equation. Assume that the binding and dissocia-tion of the reaction components are much faster than the actualgroup transfer reaction involving covalent linkages, then thereare three slow steps indicated by the single-lined double-headedarrows in Diagram IIIa, while the rest of the steps are assumedto be in rapid equilibrium. Now the reaction mechanism maybe represented by

kf,krfik,fi

K-2kzf, X X

ksfsksfs

DIAGRAM IIIbwhere X and X represent the two rapid equilibrium segments sothat

(X) = (E) + (EA) + (EB) + (EAB)+ (EP) + (EQ) + @P&l (30)

(X) = (E) + (EB) + (EP) (31)and the fractional concentration fact .ors are defined and evaluatedas follows

64) (B)WANf1=(, )= K&1 + $ + g + G g (3%0 21, + (;I I (Q) I (P)(Q)

Ki, Ki, K&i,

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824 Derivation of Enzyme Rate Equations Vol. 243, No. 4(P)(Q)

K,Ki, V-G) h(A) Bj k&3Q)>( h(B) k,(P)~ - K,Ki, __ __(EPQ)fi = (Xl

KiaKb K;a + K:,(A) u-3) (A)(B)l+jy+K+- &a& (33) +

k,ks(A) (B) kd&) (&Iza Lb

+ (p) + 9 + P)(Q) v= Ki,Kla - K&Ki, ,1K, K, f&Ki,

l I (A) I (B) + (A) U3 I 09 I (Q) I (P)(Q)Ki, Kib K,,Kb Ki, Kc K,Ki, (43)

f ks(B) k,(P)\-I-(-4 \ K:b T K:, /

(E-4fa=(x)= Ki,1++@+(A)(B)+ 1+ (B)+ (p)

(34) >(k&I) k.(Q)-+-Kib K:, Ki, Ki,Ki, Kia &a&+ (p) + (&) + 0) (&I It should be pointed out that the part o f velocity due to the path-

Ki, Ki, K&q way through E can be taken as k5(EB) - kc(EQ) instead ofk,(EA) - kd(EP).(P) Also note that, when the flux through E is negligible, i.e.-f Ki, k3 = k4 = ks = k6 = 0, Equation 38 reduces to that of the rapid1 = @PI equilibrium random bi-bi mechanism (3); and that, when the(X) (35) flux through the central complex, EAB = EPQ, is negligible,i.e. Kb = Kib = K, = Kip = m, the equation reduces to that

of Example 2 above.(B)

f. = (EH KibExample 4. Example of Mechanism with Enzyme Species Not

0 - =(Xj 1 I (B) I (P) (36)Included in Rapid Equilibrium Segment

Kit, _xp Another mechanism, for which the rate equation has beenderived by Cleland2 will be used for a further illustration. In(&I

this example, one of the four slow steps (E = EA) connects two- enzyme species, one of which does not belong to any rapid equi-

f6 = (E&j Kiq-=(Xj (A) (B) (A) (B) + K, + K;, + K;,Kh (37).1 . .- (PI (Q) 03 C&jL-L-L-

For Diagram IIIb , there are only two King and Altman patternsxwx xwhich yield

tm k4f4 + k5f5-=(Ed k,j3 + kdf4 + ksfb + kcf6 (38)

(-m kafa + kGf6-=BJ k3fz + k4f4 + ksfs + kefe (39) DIaGRlM Iv

Note that the above two patterns do not include the closed loop librium segment. In Diagram IV where the single-lined arrowsbeginning and ending in the same rapid equilibrium segment represent the slow steps, the double-lined arrows the rapid steps,(X) in the left-hand loop of Diagram IIIb . and K1, K2, etc. , are the dissociation constants. Two rapid

The partial steady state rate may be set as equilibrium segments, X and X, are indicated by broken-linedrectangles. Now the mechanism may be represented by0 = kl(EAB) - k?(EPQ) + kz(EA) - kd(EP) F (40)

2r = k,fi(Xj - k&(X) + kafdX) - kdfa(X).(--E-y

(41) 2 if, iSubstituting Equations 38 and 39 into Equation 41,or

(E,)[(k lfi - kpfz)(k,f, + ksfd + (kafakbfs - bf&fdlv=kzf3 + kdf4 + ksfs + k cfe

(42)DIAGRAM IVa

i 1 .._._......_____.................................._....____......___:: \ \Ic3 E + products

Substituting the values of f (Equations 32 to 37) into Equation 42 * W. W. Cleland, personal commumcatron.

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Issue of February 25, 1968 S. Cha 825An application of King and Altmans method to this scheme gives

63 k&b f, + k, f-h, f7 + ka &, f7_=_(-Ed c (44)m kl (A) Wi(E,)=c (45)

(X) h(A)hjs-=U-L) c (46)

where the denominator, c is the sum of all three numerators.The velocity may be expressed as

ZJ = ks(EABC) + k,(FBC) = kjj,(X) + kTj,(X)Substituting Equations 45 and 46 into Equation 47

kkNk& + ksfdk&(Eo)' = h(A)(k& + kifd + k,f,(krf?+ k& + ksjdSubstituting the following three equationsWA)fz = f3 = (x) =

(B) (3(EABC)f5=(x)= KtK,m ((3

l+K,+Kn+(B) (ClKJG

(B)(C)(FBC)f6 = (x, = KZKTl + (B) + (c) + (B)(C)KS Ks KSKT

into Equation 48, the rate equation becomes

v=k,(A)(B)(C)

KsK1.

which may be expressed differently byploging the relationship, k&, = K&g

(Cl (B) ((3Tg+KlKa )Ik3)k&30jy-3

(47)

(48)

(49)

(50)

(51)

(52)

rearranging and em-

v xLan (53)7

1 + (B) + (c) + (B)(C) +(h + kd + kdB)(C)Kl K2 -->lKs k,(A) h(A)K&

9.10.11.12.13.

DISCUSSIONThe equilibrium rate equation, although it is only an approxi-

mation of the true kinetics, is simpler and more convenient to usethan the steady state equation, and it can off er valuable insightinto the reaction mechanism. The derivation of equilibriumrate equations by the conventional method amounts to solvingas many simultaneous linear equations as the number of enzymespecies, and the algebraic manipulation can be very cumbersomewhen there are many enzyme species. It is hoped that the useof the simple rules described in this paper m ay contribute to thederivation and the analysis of complicated enzymic mechanismssuch as kinetics of regulatory enzymes. Although considerableprogress has been made by investigators, notably by Monod,Wyman, and Changeux (11) and Koshland, Nbmethy, and Filmer(12) as reviewed by Atkinson (13), much more is desired to bedone.Rate equations fo r partial equilibrium mechanisms, of course,can be derived by imposing appropriate restrictions on the fullsteady state equations, as shown in Example 2. However,use of the present graphic method, which is a logical hybrid ofthe equilibrium treatment and King and Altmans schematicmethod for the steady state, saves a considerable effo rt for thederivation. With the present method and King and Altmansmethod or that of Volkenstein and Goldstein, it is now possibleto derive rate equations for almost any enzyme mechanism undervarious assumptions through a simple inspection of the reactionpathway or by a simple schematic method, without solvingsimultaneous equations.

Acknozoledgnzents-I wish to thank Professor R. E. Parks, Jr.,and Professor W. W. Cleland for their encouragement andadvice.

1.2.3.4.5.6.7.8.

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(1913).SEGAL, H. L., KACH~~AR, J. R., AXD BOYER, P. D., Enzymologia,

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