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Chemical relaxation of cyclic enzyme reactions
J. Theoret. Biol. (1968) 21, 387-397
Chemical Relaxation
of Cyclic Enzyme Reactions
I. General Kinetic Treatment of Three-step Mechanisms G. H. CZERLINSKI
Northwestern University Medical School, Chicago, Illinois 60611, U.S.A. (Received 26 February 1968, and in revisedform 24 June 1968) One two-step and three three-step mechanisms are considered, which are particularly related to enzymatic reactions. Thus, they contain at leasttwo
bimolecular steps, proceeding in opposite direction. The relaxation times for the slowest relaxation process are derived under the experimentally important assumption that the considered relaxation time is “sufficiently separated” from faster ones. 1. Introduction
Although the concept of chemical relaxation has been known since 1954 (Eigen, 1954) general kinetic treatments of the chemical relaxation of cyclic reaction systems have barely been reported?. On the other hand, such treatments are of considerable interest to the biochemist, who wants to use chemical relaxation techniques. The two-step cyclic mechanisms have been discussed recently (Czerlinski, 1966), but enzyme reactions rarely proceed according to such simple mechanisms. Quite recently, n-step-mechanisms have been treated (Hammes & Schimmel, 1967); but the system is there handled as a linear, sequential one, and also for “buffering” only. A word on the concept of “buffering” seems appropriate at this point. “Conventionally”, a buffered concentration is kept constant by the very high concentrations of the buffering constituents. If for instance X denotes the buffered species, B one of the buffering constituents, and BX the other one, buffering of X is accomplished by the condition c$, c& 9 any other concentration coupled to component X directly or indirectly. However, buffering could as well be accomplished, if B and BX are omitted and ci is maintained much larger than the concentration of any other reactioncomponent. This condition is frequently used in Part III, as this is a rather t The concept of a cyclic reaction is to be understood here in terms of the mechanistic
scheme:the reaction “starts”
with a free enzyme form, proceeds through two or more reaction steps and finally arrives again at the same enzyme form. In a particular case, the average-enzyme-molecule may cycle through the reaction sequence many times (its property as catalyst). The term “cyclic” should not be equated with the term “periodic” (referring to oscillatory changes in some concentrations). 387
388
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common feature, if X represents the substrate S in enzyme reactions. If conventional buffering is present, c, could also be substituted by c; and c& (plus a dissociation constant). However, any relaxation time, involving cx kinetically, can then no longer be derived by using just X as interacting component: buffer constituent BX may transfer part X much faster than free X enters into the reaction (“buffer-catalysis”). To unify the treatment, Part III only uses ci % any other concentration; one may label this kind as “self-buffering” in contrast to “conventional buffering”. One may consider “temporal buffering” as a further kind: in non-equilibrium reactions, c, may be large enough during a defined time interval to be then considered as selfbuffering (e.g. initial stages in enzyme reactions with c> large). “Temporal buffering” could become important in the computation of relaxation times for temperature jump experiments, combined with rapid mixing. Cyclic systems have been treated more extensively under the buffering condition, which will be totally avoided here in Parts I and II. It is felt that a thorough derivation for non-buffered systems should be given first, before various buffering conditions are discussed in Part III. There, extensive comparison with the literature will also be conducted. For all the derivations below, only one assumption is made in addition to the ones generally applicable to chemical relaxation: The relaxation time under consideration is “sufficiently separated” from the surrounding ones. The feasibility of this assumption was discussed in detail earlier (Czerlinski, 1964). If the separation is not large, but the two relaxation times can still be evaluated with reasonable ease, the derived equations may be used in zeroapproximation. More refined (and much more complex) equations may later on be used in a computer-based evaluation. The most representative two-step mechanism will be discussed first, as the structures of the equations are needed for later comparisons. In Parts I and II, all relations will be expressed in terms of equilibrium concentrations. Relations between equilibrium concentrations and analytical concentrations will only be derived for the mentioned two-step mechanism as a representative case for cycles of any size, but containing only two bimolecular steps. Only bimolecular steps of the type A +B = AB will be considered (and thus no single-step exchange-reactions). Although three such steps can be combined in various ways, there are only two basic types; only one of them is useful for enzyme-kinetics (the other one may be considered a special type of polymerization). From that basic type one can derive two more, depending upon which bimolecular step is “made” monomolecular (the theoretically third type represents again some unusual type of polymerization). All these three basic types will be treated in this paper. Part II considers the kinetics of a selected number of cyclic four-step mechanisms.
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2. The Two-step Cycle Only one basic type is considered here, represented by the general symbolism Y‘S
Yl
y3
k4
kz
j (1)
k, -I?-
Y2 It-3
The components Y, and Y3 may be considered (free) substrate and product. The components Y4 and Y, represent free and bound enzyme respectively?. The analytical concentrations of “enzyme” and “substrate” are then given by c; = cq+z2 (2) c; =
(3)
z,+c,+c,.
The symbol Zi represents the equilibrium concentration of the ith component. The equilibrium constants of the two steps may then be written:
K,,3+~s3. 3
c2
(5)
The differential equations for the two steps can be written in various ways. Whenever possible, one should consider changes of components, which belong to only one step. That way the above system gives the two differential equations : dc,/dt = k,c,- k,c,c, (6) dc,/dt = k,c, - k,c,c,.
(7)
t A system with only two forms of enzyme, Y, and Y4 in mechanism (1), corresponds to the simplest kind, as also originally employed by Michaelis and Menton (in the reversible extensions). Mechanistically, a single form of bound enzyme seems unrealistic, as one would expect at least the two forms ES and EP (to useconventionalterminology).However, singleformscould certainlyhe present,for instancein enzymaticallycatalyzedcistrans-isomerizations. Unfortunately, kinetics would give then no information, as to whetherY, corresponds to ES or to EP; useof the unspecificY, is thereforepreferrable. Evenif ES andEP are actuallypresent,their separateexistencecouldnot be detectedby chemicalrelaxationmethods,if their equilibriumratio is eithervery large,or very small. This would he kinetically equivalentto writing only ES or EP (anduseof Y, leavesthe choiceopen).Finally, evenif the concentrations of ES andEP aresimilar,onemay not be able to differentiatethem kinetically in the caseof immeasurably fast interconversion. In summary,the point is defendedto incorporateonly thosecomponents in a reaction scheme,which may he distinguished by observation.By the way, this discussion for Y, appliesaswell for Y,, althoughfree enzymeis not commonlyconsidered asappearingin two forms. T.B. 26
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The concentration-symbol c without a bar indicates time-dependent concentrations. Any Ci may then be written as consisting of two parts: a timedependent one, vanishing to zero for t+ oc), and a time-independent one, the equilibrium concentration. Thus: ci
= Zi + ACi.
(8)
Under the given condition, Aci may also be larger than Z, in certain time ranges. Equation (8) may now be inserted into equations (6) and (7). One obtains then also the terms k, Acr AC, and k, AC, AC,. These terms can be neglected, if Ei 9 Aci. (9) This is the basic condition for chemical relaxation. Remembering also that dzi/dt s 0, the original differential equations become then simply (the dot indicating differentiation with respect to t): AC1 = k,Ac,-k,&Ac4-k,Z,Ac, At, = k,Ac, - k&AC,k&A+
(10) (11) The two differential equations contain four unknown Aci and the system can therefore not be solved as such. Equations (2) and (3) are also valid at any time. One may then differentiate them, convert differentials to differences and omit At; this gives: 0 = Ac,+Ac, 0 = Ac,+Ac,+Ac,.
(12) (13)
These equations may now be used to eliminate AcZ and AC, from equations (10) and (11): A& = - k,Ac, - k,Ac, - k,Z,Ac, - kl?rAcJ - k,i;,Ac, (14) At, = - k,Ac, - k4Ac3 - k&AC, -k&A+ - k&AC,. (1% The system of two linear homogeneous differential equations may be exactly solved quite easily (Eigen, 1954; Czerlinski, 1966; Hammes & Schimmel, 1967). To show the principle employed throughout, the simplified treatment will be demonstrated here. The simplification involves “sufficient separation” of neighboring relaxation time constants. This condition is equivalent to [kz -t- k&i
+ Q]
P [k, + k&s
+ Z,>].
(16)
The left side represents then directly the relaxation time of the fast process. Because of the symmetry of the cycle, only one case needs to be treated. Equation (16) means also pre-equilibration of the step involving Yi. This means for the time range of the slow process: Ahc, = 0. Ai, # 0, (17)
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One may then solve equation (14) for the pre-equilibrated AC, = Insertion
K2,t
+h
K,,,+Z,+E,
391
I
Acl, giving
AC
(18)
3’
of equation (18) into equation (15) results in
BE, = -AC,
k,
[(
1-
K2,1+
21
K2,1+cl+Cb
+ k3zb+
>I .
(19)
This equation can directly be solved; the expression in brackets represents the inverse of the slow relaxation time for system (l), r2. Upon simplification of this expression one obtains (20)
This equation does not agree with equation (3-158) of Czerlinski (1966); transfer errors seem to have occurred there (aside from a difference in indexing of system (3-157) of Czerlinski (1966) compared to the above system (1)). A detailed discussion of equation (20) is postponed until part III, where equation (20) is also compared with various others, to be derived in parts I and II. The equilibrium concentrations for system (1) can easily be expressed in terms of analytical concentrations, although system (1) is actually composed of two bimolecular steps. But the “feedback-character” by the cyclic sequencing “degenerates” the system to one with an easily solvable quadratic equation. This becomes easily apparent after dividing equation (4) by equation (5) : (21) K&G,3 = GIC3. Substituting equation (21) into equation (3) gives c; = Z2 + E,(l + K.t,3/K&. (22) The dissociation constant K,,, from equation (4) is now expressed in terms of & only, employing equations (2) and (22): K 2,l =
~~CC~-C~+~,(~+K,,,/K,,,)I ’
c;-h(1+K4,3/K2,1)
One obtains an expression, quadratic in ~~: K&--G&,I +&,3) = GtC-c;)+%l the solution of which is @A%,2
+&,3+&,1 = - G-4 W+ K,,,IK,,,)
+&,J&,d,
(23)
(24)
4WL +&,I) 1 + [c;-c;+K~,~+K~,~]~
’ (25)
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However, a solution with negative concentrations ment of (25) gives 4CxK4,3+KZJ
L-c~-c~+~,,,+~,,112
is not useful. RearrangeG-cY+K4,3+&,1 2U
+K,,,I&,,)* (26)
For c? < [c~+K,,,+K,,,]. Certainly the associated condition may not always hold. And if it is not fulfilled, the positive root of equation (25) is valid, giving
c;--c:--&,,-&,I
”
=
2(1
+K@~,I)
For: c; > [ci+K,,,+K,,,]. (27) The limiting case has been excluded for both equations (26) and (27). One derives from either equation:
For: ci = [ci+K,,,+K2,J Two other limiting cases are derivable for the two extreme values of cy (with ci kept finite). Expansion of the square roots in equations (26) and (27) is then allowed, resulting for equation (26) in El
--,
(c~+K,,,+K,,,)(l+K,,,IK,,,)
(29)
For: cO,+O and one obtains likewise from equation (27)
For: c;l --t co. This same equation (30) may also be derived from first principles, as equation (3) simplifies then to c; & z1+z3. (31) Equation (31) may then be directly combined with equation (21) to give first K 2,1 K 493
f% &-----. cy-Cl
(32)
The limiting equation (32) can directly be rearranged to give equation (30). A quite similar derivation of this type was demonstrated recently (Czerlinski, 1967). There, a graphical representation was also given, demonstrating clearly the smooth transitions through the various ranges of validity. The derivation of ‘statics’ (representing the relations between equilibrium and
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analytical concentrations) has been given so explicitly, as it is applicable much more generally to cyclic systems of any size but only two bimolecular steps in opposite direction. Any additional concentration in a monomolecular step is fully expressed by the relevant equilibrium constant. Equation (21) needs then only to be extended to contain the various equilibrium constants for monomolecular steps. Although such a treatment could easily be included in the later derivations, it is omitted, as there are various ways of substitution and as the derivations would become even more extensive than they already are. The main goal of the derivations in parts I and II is, to give the background for part III, where the derived expressions for relaxation times will be extensively discussed. 3. Three-step Cycles The selected case with three bimolecular steps is given by the scheme Y,
(33)
The three differential equations may directly be written down and immediately in terms of the Aci: A& = kgAcg - k&AC7 -k&AC‘+, (34) Ai1 = k,Ac,- k,C,Ac,- k,Z,Ac,, (35) A& = k4Acz-- k&AC, - k&A+ (36) One visualizes from scheme (33) immediately the following relations among the ACi, valid for the slowest relaxation process: Ac,+Ac,+Ac, = 0, (37) AC, = -AC,-AC,, (38) AC, = AC, -AC,. (39) There are six equations and six unknown ACi. An exact solution may thus be obtained. First, Ac2, AC, and AC, in equations (34) to (36) are replaced by employing equations (37) to (39). The basic assumption for simplified treatment was that the slowest process is “sufficiently separated” from the faster ones. One is then left with three possibilities, which cannot be converted into each other.
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(1) k, and k6 determine slowest process. One obtains for its relaxation time the complex expression 23
~SW,,l +z, +Q+c&,3 (K,,,+2;1+~~)(K4,3+~3+~~)-KKZ,1KQ,3
-l =k,
+ k5’4
E,(K,,,
+z3
+E5)+K,,,E,
+k5E7(Kz,~+El+E4)(K4,3+E3+E5)-KZ,1K4,3-
(40)
Concerning its straight forward derivation, a few remarks are made in the Appendix under (1). The relation of this equation to others will be discussed in part III, as well as simplification upon buffering the system. The relaxation time rr is trivial, while t2 is equivalent to equation (3-61) of Czerlinski (1966). (2) k, and k4 determine the slowest process. One obtains: 23
-l =k
+ Z1+ C4 + z,) 4(K6,s+~4+~7)(K2,1+~4++l)-~l~7 z4CK6.5
Z4(K6,5+E1
+C4+Z,)
(K~,5+~4+~7)(KZ,1+E4+~i)-C1C,
>
* (41)
The derivation is quite similar to the one for the previous equation, but is nevertheless briefly discussed in the Appendix (2). The relaxation time z2 corresponds to equation (3-63) of Czerlinski (1966). (3) The rate constants k, and k, determine the slowest process. One obtains 73
-1=
E5(Ks,5
+
E4
+ + c7)
?,a?,
(K4.3+E3+Es)(Ks,s+E4+E7)-K6,5~3k2+ ?4W4,3 W4.3
+ E3
+ + + + + c3
Es)
es)W,,,
K6,5E,
E4 + 27)
. (42) -K,,,c,
The derivation is similar to the one for equations (40) and (41) and not separately treated in the Appendix. There are now two possible expressions for rZ, depending upon, which one of the two steps is slower. The two expressions are represented by equations (3.66) and (3.68) of Czerlinski (1966). In scheme (33) one may now set Y, E 0. One obtains then a three-stepcycle with only two bimolecular steps:
&k, ‘l
k2
f---
(43) Y3
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While equations (35) and (36) are still valid, equation (34) has to be altered to Ack = k,Ac, - k,Ac,. w The prime indicates that only one step is considered. This can be done, if this step is isolated in time; the prime indicates then the restriction to a timerange of validity. Otherwise, the prime has to be understood as indicating the difference among differential equations. Concerning the relation among the Aci (considering only slowest processes), equations (37) and (38) remain valid, while equation (39) simply vanishes. There are then five unknown Aci and five equations, allowing the exact solution. To obtain a better insight into the structure of the expressions, the simplifying assumption of separation in time will again be made. Instead of three cases, one now needs to differentiate only two cases because of the symmetry of system (43). (I) The step with k, and k, is slowest. One arrives at (compare Appendix (3)) : r3
mG,1+
-I = k,+k,-
El + C4) + K4.324
(45)
c,(K,,,+~,+~,)+Kz.lZ,’
This equation is rather simple in structure. Only one r2 may be derived (because of symmetry), in structure identical to equation (3-61) of Czerlinski (1966). (2) The step with k, and k, is slowest. One arrives now at (compare Appendix (4)) : (46) As the remaining two steps are not symmetric to each-other, there are two different possible structures for r2, either according to equation (3-57) or to equation (3-58) of Czerlinski (1966). In scheme (33) one may alternatively (to the simplification for scheme (43)) set Y3 = 0. This leads to another three-step cycle with only two bimolecular steps : y4
Yl
Y7 k, ks k, 77 y2-
ks (47)
k4
ys k, For this scheme, equations (34) and (35) remain valid, while equation (36) has to be replaced by AI?; = k,Ac, - k,Ac,. (48)
396
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The prime indicates again that only one step is considered; see below equation (44) for further details. And with reference to the relations among the Aci, only equations (37) and (39) remain valid; equation (38) simply “vanishes”. There are thus again five unknown Aci and five equations: the system is exactly described. Separation of relaxation processes is again assumed for simplicity. One may then distinguish two cases. (1) The slowest step is determined by k3 and k,+. One obtains then (compare Appendix (5)) : 7;’ = k,+k,
f&,+z,+c$+2, K,,,+c1+c,+Z,’
(49)
The second slowest relaxation time corresponds to equation (3.63) of Czerlinski (1966). (2) The slowest step is determined by k, and k,. One obtains then (in a manner similar to the one discussed in Appendix (5)): k, + k&(1
+ K,,,)
There are now two possible expressions for the second slowest relaxation process, either according to equation (3-50) or to equation (3-53) of Czerlinski (1966). Although scheme (33) could also be simplified by setting Y1 = 0 the resulting scheme would be of little interest and will therefore not be discussed here. A detailed discussion of the various expressions for the slowest relaxation times follows in part III. The largest part of this work was performed while at Cornell University, Ithaca, N.Y. Financial support by the National Science Foundation is gratefully acknowledged (Grant GB-6693). Requests for reprints should be directed to the author at 4-065 Ward Bldg., 303 E. Chicago Ave., Chicago, Illinois 60611, U.S.A. REFERENCES G. (1964). J. Theoret. Biul. 7,463. G. (1966).“Chemical Relaxation, An Introduction to Theory and Application of StepwisePerturbation”. New York: Dekker. CZERLINSKI, G. (1967).J. Theoret. Biol. 17, 343. CZERLINSKI, CZERLINSKI,
EIGEN, M. (1954). Discuss. Faraday HAMMES, G. G. & SCHIMMEL, P. R.
Sot. 17,463.
(1967).J. phys. Chem. 71, 917. Appendix
(1) To obtain equation (40), the three differential equations are first written in terms of AC,, AC, and AC,, as explained after equation (39). Ai, # 0 for the time-range of the slowest process, where At, = 0 and Ai3 = 0
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according to assumption. These latter two equations are then solved for AC, and AC, and in terms of AC, only. The results are then inserted into the equation for AE, # 0, leading upon rearrangement to equation (40) for the coefficient of - AC,. The method is similar to the one, outlined for the twostep cycle above. (2) Equation (41) may be obtained as described in Appendix (l), but it is now AE, # 0, AC, = 0, AC, = 0. These latter two equations are now solved for AC, and AC, and solely in terms of AC,. The resulting expressions for AC, and AC, are then substituted in the equation for AE, (containing Acr, AC, and AC, beforehand). The negative coefficient of AC, is simplified to give equation (41). (3) Equation (45) is obtained by expressing first AE,, AE, and AC!; in terms of AC,, AC, and AC, only (by use of equations (37) and (38)). As only AE: # 0, AC, and AC, are derived from AC, = 0 and A\c, = 0 and expressed only in terms of AC,. These expressions for AC, and AC, are then used in the equation for ALi # 0. The negative coefficient of AC, leads upon simplification to equation (45). (4) Equation (46) is derived quite similar to the description in Appendix (3) but it is now AE, # 0, AC?, = 0, AL; = 0. Expressions for AC, and AC, (in terms of AC, only) are again derived from the latter two equations (written fully in terms of Acl, Acs, and AC,). These expressions are substituted in the equation for At, # 0 leading finally to the expression for r3 in equation (46). (5) To derive equation (49), equations (37) and (39) are first used to express the three differential equations in terms of AC,, AC, and AC,. The basic assumption allows AL, = 0 and AL, = 0, which in turn allows us to express AC, and AC, in terms of AC, only. These expressions are then used for substitution into At, # 0 (not equation (48), but the one after AC, replacement, containing then only Acl, AC, and AC,). The negative coefficient of AC, gives equation (49) after rearrangements.
Chemical Relaxation
of Cyclic Enzyme Reactions
I. General Kinetic Treatment of Three-step Mechanisms G. H. CZERLINSKI
Northwestern University Medical School, Chicago, Illinois 60611, U.S.A. (Received 26 February 1968, and in revisedform 24 June 1968) One two-step and three three-step mechanisms are considered, which are particularly related to enzymatic reactions. Thus, they contain at leasttwo
bimolecular steps, proceeding in opposite direction. The relaxation times for the slowest relaxation process are derived under the experimentally important assumption that the considered relaxation time is “sufficiently separated” from faster ones. 1. Introduction
Although the concept of chemical relaxation has been known since 1954 (Eigen, 1954) general kinetic treatments of the chemical relaxation of cyclic reaction systems have barely been reported?. On the other hand, such treatments are of considerable interest to the biochemist, who wants to use chemical relaxation techniques. The two-step cyclic mechanisms have been discussed recently (Czerlinski, 1966), but enzyme reactions rarely proceed according to such simple mechanisms. Quite recently, n-step-mechanisms have been treated (Hammes & Schimmel, 1967); but the system is there handled as a linear, sequential one, and also for “buffering” only. A word on the concept of “buffering” seems appropriate at this point. “Conventionally”, a buffered concentration is kept constant by the very high concentrations of the buffering constituents. If for instance X denotes the buffered species, B one of the buffering constituents, and BX the other one, buffering of X is accomplished by the condition c$, c& 9 any other concentration coupled to component X directly or indirectly. However, buffering could as well be accomplished, if B and BX are omitted and ci is maintained much larger than the concentration of any other reactioncomponent. This condition is frequently used in Part III, as this is a rather t The concept of a cyclic reaction is to be understood here in terms of the mechanistic
scheme:the reaction “starts”
with a free enzyme form, proceeds through two or more reaction steps and finally arrives again at the same enzyme form. In a particular case, the average-enzyme-molecule may cycle through the reaction sequence many times (its property as catalyst). The term “cyclic” should not be equated with the term “periodic” (referring to oscillatory changes in some concentrations). 387
388
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H.
CZERLINSKI
common feature, if X represents the substrate S in enzyme reactions. If conventional buffering is present, c, could also be substituted by c; and c& (plus a dissociation constant). However, any relaxation time, involving cx kinetically, can then no longer be derived by using just X as interacting component: buffer constituent BX may transfer part X much faster than free X enters into the reaction (“buffer-catalysis”). To unify the treatment, Part III only uses ci % any other concentration; one may label this kind as “self-buffering” in contrast to “conventional buffering”. One may consider “temporal buffering” as a further kind: in non-equilibrium reactions, c, may be large enough during a defined time interval to be then considered as selfbuffering (e.g. initial stages in enzyme reactions with c> large). “Temporal buffering” could become important in the computation of relaxation times for temperature jump experiments, combined with rapid mixing. Cyclic systems have been treated more extensively under the buffering condition, which will be totally avoided here in Parts I and II. It is felt that a thorough derivation for non-buffered systems should be given first, before various buffering conditions are discussed in Part III. There, extensive comparison with the literature will also be conducted. For all the derivations below, only one assumption is made in addition to the ones generally applicable to chemical relaxation: The relaxation time under consideration is “sufficiently separated” from the surrounding ones. The feasibility of this assumption was discussed in detail earlier (Czerlinski, 1964). If the separation is not large, but the two relaxation times can still be evaluated with reasonable ease, the derived equations may be used in zeroapproximation. More refined (and much more complex) equations may later on be used in a computer-based evaluation. The most representative two-step mechanism will be discussed first, as the structures of the equations are needed for later comparisons. In Parts I and II, all relations will be expressed in terms of equilibrium concentrations. Relations between equilibrium concentrations and analytical concentrations will only be derived for the mentioned two-step mechanism as a representative case for cycles of any size, but containing only two bimolecular steps. Only bimolecular steps of the type A +B = AB will be considered (and thus no single-step exchange-reactions). Although three such steps can be combined in various ways, there are only two basic types; only one of them is useful for enzyme-kinetics (the other one may be considered a special type of polymerization). From that basic type one can derive two more, depending upon which bimolecular step is “made” monomolecular (the theoretically third type represents again some unusual type of polymerization). All these three basic types will be treated in this paper. Part II considers the kinetics of a selected number of cyclic four-step mechanisms.
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2. The Two-step Cycle Only one basic type is considered here, represented by the general symbolism Y‘S
Yl
y3
k4
kz
j (1)
k, -I?-
Y2 It-3
The components Y, and Y3 may be considered (free) substrate and product. The components Y4 and Y, represent free and bound enzyme respectively?. The analytical concentrations of “enzyme” and “substrate” are then given by c; = cq+z2 (2) c; =
(3)
z,+c,+c,.
The symbol Zi represents the equilibrium concentration of the ith component. The equilibrium constants of the two steps may then be written:
K,,3+~s3. 3
c2
(5)
The differential equations for the two steps can be written in various ways. Whenever possible, one should consider changes of components, which belong to only one step. That way the above system gives the two differential equations : dc,/dt = k,c,- k,c,c, (6) dc,/dt = k,c, - k,c,c,.
(7)
t A system with only two forms of enzyme, Y, and Y4 in mechanism (1), corresponds to the simplest kind, as also originally employed by Michaelis and Menton (in the reversible extensions). Mechanistically, a single form of bound enzyme seems unrealistic, as one would expect at least the two forms ES and EP (to useconventionalterminology).However, singleformscould certainlyhe present,for instancein enzymaticallycatalyzedcistrans-isomerizations. Unfortunately, kinetics would give then no information, as to whetherY, corresponds to ES or to EP; useof the unspecificY, is thereforepreferrable. Evenif ES andEP are actuallypresent,their separateexistencecouldnot be detectedby chemicalrelaxationmethods,if their equilibriumratio is eithervery large,or very small. This would he kinetically equivalentto writing only ES or EP (anduseof Y, leavesthe choiceopen).Finally, evenif the concentrations of ES andEP aresimilar,onemay not be able to differentiatethem kinetically in the caseof immeasurably fast interconversion. In summary,the point is defendedto incorporateonly thosecomponents in a reaction scheme,which may he distinguished by observation.By the way, this discussion for Y, appliesaswell for Y,, althoughfree enzymeis not commonlyconsidered asappearingin two forms. T.B. 26
390
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The concentration-symbol c without a bar indicates time-dependent concentrations. Any Ci may then be written as consisting of two parts: a timedependent one, vanishing to zero for t+ oc), and a time-independent one, the equilibrium concentration. Thus: ci
= Zi + ACi.
(8)
Under the given condition, Aci may also be larger than Z, in certain time ranges. Equation (8) may now be inserted into equations (6) and (7). One obtains then also the terms k, Acr AC, and k, AC, AC,. These terms can be neglected, if Ei 9 Aci. (9) This is the basic condition for chemical relaxation. Remembering also that dzi/dt s 0, the original differential equations become then simply (the dot indicating differentiation with respect to t): AC1 = k,Ac,-k,&Ac4-k,Z,Ac, At, = k,Ac, - k&AC,k&A+
(10) (11) The two differential equations contain four unknown Aci and the system can therefore not be solved as such. Equations (2) and (3) are also valid at any time. One may then differentiate them, convert differentials to differences and omit At; this gives: 0 = Ac,+Ac, 0 = Ac,+Ac,+Ac,.
(12) (13)
These equations may now be used to eliminate AcZ and AC, from equations (10) and (11): A& = - k,Ac, - k,Ac, - k,Z,Ac, - kl?rAcJ - k,i;,Ac, (14) At, = - k,Ac, - k4Ac3 - k&AC, -k&A+ - k&AC,. (1% The system of two linear homogeneous differential equations may be exactly solved quite easily (Eigen, 1954; Czerlinski, 1966; Hammes & Schimmel, 1967). To show the principle employed throughout, the simplified treatment will be demonstrated here. The simplification involves “sufficient separation” of neighboring relaxation time constants. This condition is equivalent to [kz -t- k&i
+ Q]
P [k, + k&s
+ Z,>].
(16)
The left side represents then directly the relaxation time of the fast process. Because of the symmetry of the cycle, only one case needs to be treated. Equation (16) means also pre-equilibration of the step involving Yi. This means for the time range of the slow process: Ahc, = 0. Ai, # 0, (17)
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One may then solve equation (14) for the pre-equilibrated AC, = Insertion
K2,t
+h
K,,,+Z,+E,
391
I
Acl, giving
AC
(18)
3’
of equation (18) into equation (15) results in
BE, = -AC,
k,
[(
1-
K2,1+
21
K2,1+cl+Cb
+ k3zb+
>I .
(19)
This equation can directly be solved; the expression in brackets represents the inverse of the slow relaxation time for system (l), r2. Upon simplification of this expression one obtains (20)
This equation does not agree with equation (3-158) of Czerlinski (1966); transfer errors seem to have occurred there (aside from a difference in indexing of system (3-157) of Czerlinski (1966) compared to the above system (1)). A detailed discussion of equation (20) is postponed until part III, where equation (20) is also compared with various others, to be derived in parts I and II. The equilibrium concentrations for system (1) can easily be expressed in terms of analytical concentrations, although system (1) is actually composed of two bimolecular steps. But the “feedback-character” by the cyclic sequencing “degenerates” the system to one with an easily solvable quadratic equation. This becomes easily apparent after dividing equation (4) by equation (5) : (21) K&G,3 = GIC3. Substituting equation (21) into equation (3) gives c; = Z2 + E,(l + K.t,3/K&. (22) The dissociation constant K,,, from equation (4) is now expressed in terms of & only, employing equations (2) and (22): K 2,l =
~~CC~-C~+~,(~+K,,,/K,,,)I ’
c;-h(1+K4,3/K2,1)
One obtains an expression, quadratic in ~~: K&--G&,I +&,3) = GtC-c;)+%l the solution of which is @A%,2
+&,3+&,1 = - G-4 W+ K,,,IK,,,)
+&,J&,d,
(23)
(24)
4WL +&,I) 1 + [c;-c;+K~,~+K~,~]~
’ (25)
392
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However, a solution with negative concentrations ment of (25) gives 4CxK4,3+KZJ
L-c~-c~+~,,,+~,,112
is not useful. RearrangeG-cY+K4,3+&,1 2U
+K,,,I&,,)* (26)
For c? < [c~+K,,,+K,,,]. Certainly the associated condition may not always hold. And if it is not fulfilled, the positive root of equation (25) is valid, giving
c;--c:--&,,-&,I
”
=
2(1
+K@~,I)
For: c; > [ci+K,,,+K,,,]. (27) The limiting case has been excluded for both equations (26) and (27). One derives from either equation:
For: ci = [ci+K,,,+K2,J Two other limiting cases are derivable for the two extreme values of cy (with ci kept finite). Expansion of the square roots in equations (26) and (27) is then allowed, resulting for equation (26) in El
--,
(c~+K,,,+K,,,)(l+K,,,IK,,,)
(29)
For: cO,+O and one obtains likewise from equation (27)
For: c;l --t co. This same equation (30) may also be derived from first principles, as equation (3) simplifies then to c; & z1+z3. (31) Equation (31) may then be directly combined with equation (21) to give first K 2,1 K 493
f% &-----. cy-Cl
(32)
The limiting equation (32) can directly be rearranged to give equation (30). A quite similar derivation of this type was demonstrated recently (Czerlinski, 1967). There, a graphical representation was also given, demonstrating clearly the smooth transitions through the various ranges of validity. The derivation of ‘statics’ (representing the relations between equilibrium and
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analytical concentrations) has been given so explicitly, as it is applicable much more generally to cyclic systems of any size but only two bimolecular steps in opposite direction. Any additional concentration in a monomolecular step is fully expressed by the relevant equilibrium constant. Equation (21) needs then only to be extended to contain the various equilibrium constants for monomolecular steps. Although such a treatment could easily be included in the later derivations, it is omitted, as there are various ways of substitution and as the derivations would become even more extensive than they already are. The main goal of the derivations in parts I and II is, to give the background for part III, where the derived expressions for relaxation times will be extensively discussed. 3. Three-step Cycles The selected case with three bimolecular steps is given by the scheme Y,
(33)
The three differential equations may directly be written down and immediately in terms of the Aci: A& = kgAcg - k&AC7 -k&AC‘+, (34) Ai1 = k,Ac,- k,C,Ac,- k,Z,Ac,, (35) A& = k4Acz-- k&AC, - k&A+ (36) One visualizes from scheme (33) immediately the following relations among the ACi, valid for the slowest relaxation process: Ac,+Ac,+Ac, = 0, (37) AC, = -AC,-AC,, (38) AC, = AC, -AC,. (39) There are six equations and six unknown ACi. An exact solution may thus be obtained. First, Ac2, AC, and AC, in equations (34) to (36) are replaced by employing equations (37) to (39). The basic assumption for simplified treatment was that the slowest process is “sufficiently separated” from the faster ones. One is then left with three possibilities, which cannot be converted into each other.
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(1) k, and k6 determine slowest process. One obtains for its relaxation time the complex expression 23
~SW,,l +z, +Q+c&,3 (K,,,+2;1+~~)(K4,3+~3+~~)-KKZ,1KQ,3
-l =k,
+ k5’4
E,(K,,,
+z3
+E5)+K,,,E,
+k5E7(Kz,~+El+E4)(K4,3+E3+E5)-KZ,1K4,3-
(40)
Concerning its straight forward derivation, a few remarks are made in the Appendix under (1). The relation of this equation to others will be discussed in part III, as well as simplification upon buffering the system. The relaxation time rr is trivial, while t2 is equivalent to equation (3-61) of Czerlinski (1966). (2) k, and k4 determine the slowest process. One obtains: 23
-l =k
+ Z1+ C4 + z,) 4(K6,s+~4+~7)(K2,1+~4++l)-~l~7 z4CK6.5
Z4(K6,5+E1
+C4+Z,)
(K~,5+~4+~7)(KZ,1+E4+~i)-C1C,
>
* (41)
The derivation is quite similar to the one for the previous equation, but is nevertheless briefly discussed in the Appendix (2). The relaxation time z2 corresponds to equation (3-63) of Czerlinski (1966). (3) The rate constants k, and k, determine the slowest process. One obtains 73
-1=
E5(Ks,5
+
E4
+ + c7)
?,a?,
(K4.3+E3+Es)(Ks,s+E4+E7)-K6,5~3k2+ ?4W4,3 W4.3
+ E3
+ + + + + c3
Es)
es)W,,,
K6,5E,
E4 + 27)
. (42) -K,,,c,
The derivation is similar to the one for equations (40) and (41) and not separately treated in the Appendix. There are now two possible expressions for rZ, depending upon, which one of the two steps is slower. The two expressions are represented by equations (3.66) and (3.68) of Czerlinski (1966). In scheme (33) one may now set Y, E 0. One obtains then a three-stepcycle with only two bimolecular steps:
&k, ‘l
k2
f---
(43) Y3
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While equations (35) and (36) are still valid, equation (34) has to be altered to Ack = k,Ac, - k,Ac,. w The prime indicates that only one step is considered. This can be done, if this step is isolated in time; the prime indicates then the restriction to a timerange of validity. Otherwise, the prime has to be understood as indicating the difference among differential equations. Concerning the relation among the Aci (considering only slowest processes), equations (37) and (38) remain valid, while equation (39) simply vanishes. There are then five unknown Aci and five equations, allowing the exact solution. To obtain a better insight into the structure of the expressions, the simplifying assumption of separation in time will again be made. Instead of three cases, one now needs to differentiate only two cases because of the symmetry of system (43). (I) The step with k, and k, is slowest. One arrives at (compare Appendix (3)) : r3
mG,1+
-I = k,+k,-
El + C4) + K4.324
(45)
c,(K,,,+~,+~,)+Kz.lZ,’
This equation is rather simple in structure. Only one r2 may be derived (because of symmetry), in structure identical to equation (3-61) of Czerlinski (1966). (2) The step with k, and k, is slowest. One arrives now at (compare Appendix (4)) : (46) As the remaining two steps are not symmetric to each-other, there are two different possible structures for r2, either according to equation (3-57) or to equation (3-58) of Czerlinski (1966). In scheme (33) one may alternatively (to the simplification for scheme (43)) set Y3 = 0. This leads to another three-step cycle with only two bimolecular steps : y4
Yl
Y7 k, ks k, 77 y2-
ks (47)
k4
ys k, For this scheme, equations (34) and (35) remain valid, while equation (36) has to be replaced by AI?; = k,Ac, - k,Ac,. (48)
396
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The prime indicates again that only one step is considered; see below equation (44) for further details. And with reference to the relations among the Aci, only equations (37) and (39) remain valid; equation (38) simply “vanishes”. There are thus again five unknown Aci and five equations: the system is exactly described. Separation of relaxation processes is again assumed for simplicity. One may then distinguish two cases. (1) The slowest step is determined by k3 and k,+. One obtains then (compare Appendix (5)) : 7;’ = k,+k,
f&,+z,+c$+2, K,,,+c1+c,+Z,’
(49)
The second slowest relaxation time corresponds to equation (3.63) of Czerlinski (1966). (2) The slowest step is determined by k, and k,. One obtains then (in a manner similar to the one discussed in Appendix (5)): k, + k&(1
+ K,,,)
There are now two possible expressions for the second slowest relaxation process, either according to equation (3-50) or to equation (3-53) of Czerlinski (1966). Although scheme (33) could also be simplified by setting Y1 = 0 the resulting scheme would be of little interest and will therefore not be discussed here. A detailed discussion of the various expressions for the slowest relaxation times follows in part III. The largest part of this work was performed while at Cornell University, Ithaca, N.Y. Financial support by the National Science Foundation is gratefully acknowledged (Grant GB-6693). Requests for reprints should be directed to the author at 4-065 Ward Bldg., 303 E. Chicago Ave., Chicago, Illinois 60611, U.S.A. REFERENCES G. (1964). J. Theoret. Biul. 7,463. G. (1966).“Chemical Relaxation, An Introduction to Theory and Application of StepwisePerturbation”. New York: Dekker. CZERLINSKI, G. (1967).J. Theoret. Biol. 17, 343. CZERLINSKI, CZERLINSKI,
EIGEN, M. (1954). Discuss. Faraday HAMMES, G. G. & SCHIMMEL, P. R.
Sot. 17,463.
(1967).J. phys. Chem. 71, 917. Appendix
(1) To obtain equation (40), the three differential equations are first written in terms of AC,, AC, and AC,, as explained after equation (39). Ai, # 0 for the time-range of the slowest process, where At, = 0 and Ai3 = 0
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according to assumption. These latter two equations are then solved for AC, and AC, and in terms of AC, only. The results are then inserted into the equation for AE, # 0, leading upon rearrangement to equation (40) for the coefficient of - AC,. The method is similar to the one, outlined for the twostep cycle above. (2) Equation (41) may be obtained as described in Appendix (l), but it is now AE, # 0, AC, = 0, AC, = 0. These latter two equations are now solved for AC, and AC, and solely in terms of AC,. The resulting expressions for AC, and AC, are then substituted in the equation for AE, (containing Acr, AC, and AC, beforehand). The negative coefficient of AC, is simplified to give equation (41). (3) Equation (45) is obtained by expressing first AE,, AE, and AC!; in terms of AC,, AC, and AC, only (by use of equations (37) and (38)). As only AE: # 0, AC, and AC, are derived from AC, = 0 and A\c, = 0 and expressed only in terms of AC,. These expressions for AC, and AC, are then used in the equation for ALi # 0. The negative coefficient of AC, leads upon simplification to equation (45). (4) Equation (46) is derived quite similar to the description in Appendix (3) but it is now AE, # 0, AC?, = 0, AL; = 0. Expressions for AC, and AC, (in terms of AC, only) are again derived from the latter two equations (written fully in terms of Acl, Acs, and AC,). These expressions are substituted in the equation for At, # 0 leading finally to the expression for r3 in equation (46). (5) To derive equation (49), equations (37) and (39) are first used to express the three differential equations in terms of AC,, AC, and AC,. The basic assumption allows AL, = 0 and AL, = 0, which in turn allows us to express AC, and AC, in terms of AC, only. These expressions are then used for substitution into At, # 0 (not equation (48), but the one after AC, replacement, containing then only Acl, AC, and AC,). The negative coefficient of AC, gives equation (49) after rearrangements.