Regulatory Behavior of Monomeric Enzymes

Transcription

1 Eur. J. Biochem. 49, (974) Regulatory Behavior of Monomeric Enzymes. The Mnemonical Enzyme Concept Jacques RICARD, Jean-Claude MEUNIER, and Jean BUC Laboratoire de Physiologie Cellulaire Vegetale du Centre National de la Recherche Scientifique, Universite d Aix-Marseille I (Received May 4/July 2, 974) A new concept, that of a mnemonical transition, is proposed to explain the departure from Michaelian behavior of monomeric enzymes following ordered reaction mechanisms. The concept integrates three simple ideas : the free enzyme occurs under two conformational states in equilibrium ; the collision of any of these forms with the first substrate induces the same third new configuration required for proper substrate binding; the collision of only one of these enzyme forms with the last product stabilizes that form without any new conformational change. This whole set of definitions is equivalent to assuming that the free enzyme which is released after catalysis, is in a conformation different from the initial one. The enzyme can be said to recall for a while the configuration stabilized by the last product before relapsing to the initial conformation. The non-hyperbolic behavior is thus the consequence of the cooperation of two different conformations of the free enzyme to the overall reaction process. The reciprocal steady-state rate equations have been established and thoroughly discussed both for one-substrate, one-product and two-substrate, two-product mnemonical enzymes. The departure from Michaelian behavior does not appear as a consequence of a slow conformational transition, but is defined in a simpler way by the relative values of the activation free energies of conformation changes required for substrate binding on the two enzyme forms. A two-substrate, two-product enzyme following an ordered reaction mechanism and exhibiting the mnemonical transition has a very distinctive kinetic behavior. The curvature of the primary plots is observed with regard to the first substrate only, and is independant of the concentration of the second substrate as well as that of the first product. The enzyme is not inhibited by an excess of the substrate and the primary plots are either concave up or down. The slopes and the intercepts of the straight lines obtained in double reciprocal plots with the second substrate should give, when these are replotted against the reciprocal of the first substrate concentration, a straight line and a curve, respectively. The cooperation of the enzyme conformations to the overall reaction process can be either positive or negative. Since the reciprocal plots cannot exhibit an extreme, the extent of that cooperation can be measured by the numerical value of the second derivative of the reciprocal rate equation. The extent of cooperation between the free enzyme forms is highly controlled by the concentration of the last product. If the cooperation was already negative, the product strengthens that cooperation. If, on the other hand, the cooperation was positive, the product decreases or even reverses that cooperation. A very general property of one-sited mnemonical enzymes is that cooperation between enzyme forms is only kinetic and does not appear in the substrate binding isotherms. It is well known that the random binding of sub- of asymptotic or degeneracy conditions of the recipstrates on a rigid (or nearly rigid) monomeric enzyme rocal-rate equation [,6-8, it is unlikely these can, in theory, generate deviations from Michaelis- deviations will be found, and the complicated velocity Menten behavior [l- 5. However, it seems that functions often lead to a kinetic behavior which is these deviations have not been observed so far for either purely or approximately Michaelian, at least any real enzyme. Because of the common occurrence within the error-limits of rate measurements.

2 96 The Mnemonical Enzyme Concept However, it has been pointed out as early as 967 by Rabin [9] that a monomeric flexible one-substrate enzyme, that is an enzyme exhibiting conformation changes upon substrate binding, can possess a non- Michaelian, or a so-called regulatory, behavior. The basic idea of the Rabin scheme is that the enzyme conformation which is released at the end of the catalytic step is different from the initial one. Deviations from Michaelian behavior, thus appear as a consequence of a different affinity of two conformations of the monomeric enzyme for the substrate. The term enzyme memory has even been coined by Whitehead [lo] to express the idea that the enzyme recalls for a while the conformation stabilized by the substrate. A more general reaction scheme based on conformation changes of the enzyme and of the various enzyme forms has been recently analyzed and discussed by Ainslie et a/. [ll]. We propose to represent by the term mnemonical enzyme, an enzyme exhibiting memory phenomena. Moreover, it has been experimentally found that some monomeric enzymes exhibit a non-michaelian behavior [2,3]. Although no clear-cut kinetic interpretation of these facts has been presented so far, there is little doubt that these effects are due to conformational changes of the enzyme during the reaction. The main goal of the present paper is to discuss the kinetic and the thermodynamic grounds of this regulatory behavior with the monomeric one-substrate and two-substrate mnemonical enzymes. The accompanying paper [4] will apply these theoretical developments to the case of a real mnemonical enzyme : a wheat germ hexokinase. THEORY Molecular Basis of Enzyme Memoisy : The Mnemonical Transition One of the basic ideas of the present paper is to consider that the non-michaelian behavior of some monomeric enzymes is the immediate consequence of a special type of conformational transition, which we call the mnemonical transition. Three requirements (Fig. ) have to be met to define such a transition : a) the single-site enzyme should exist in two conformational states (the circle and the rhombus) in equilibrium ; b) the collision of a ligand, L, with either of these enzyme forms should induce a new conformation (the square) which is required for proper ligand binding ; Fig.. The mnernonical transition c) another ligand, M, should be bound competitively, at the same site, on one enzyme form only, without any further conformation change, thus stabilizing that protein configuration (the rhombus). This situation is depicted in Fig.. The mnemonical transition combines induced-fit phenomena (steps 2 and 3) with the occurence of a pre-equilibrium between two enzyme forms shifted by the binding of a ligand (steps and 4). If the mnemonical transition is inserted into a reaction scheme in such a way that L is the first substrate and M the last product, it clearly appears to be a generalization of the idea of enzyme memory put forward by Rabin [9] and Whitehead [lo]. If, for instance, it is assumed that the preequilibrium is shifted towards one of the enzyme configurations (the circle) it can be noted that the free enzyme, which is released on product desorption, is in a conformation different from the initial one. Thus the enzyme appears to recall for a while the configuration stabilized by the last product before relapsing to the initial conformation. Whereas the non-hyperbolic kinetic behavior of polymeric enzymes is the immediate consequence of a cooperativity between the active sites, the atypical behavior of mnemonical enzymes is due to a cooperation of two different conformations (the circle and the rhombus) of the free enzyme to the overall reaction process. As for interaction between sites, the cooperation between different conformations of the same enzyme molecule can be either positive or negative. This point will be demonstrated and discussed at length in the present paper. It is important to note that the existence, in the mnemonical transition, of both induced-fit and pre-equilibrium between two enzyme forms is imposed by the principle of microscopic reversibility. A scheme in which the enzyme, at the start of the reaction, would be in a conformation regenerated by an irreversible step would not be in accord with known thermodynamic principles.

3 J. Ricard, J.-C. Meunier, and J. Buc 97 Table. Explicit formulation of E9n () Coefficient Equation the kinetic parameters of I El- k k Fig. 2. The mnemonical transition,for a one-substrate, oneproduct monomeric enzyme (model I) The Concept of Enzyme Memory for One-Substrate, One-Product Monomeric Enzymes The kinetic scheme of a one-substrate, oneproduct monomeric enzyme exhibiting the mnemonica transition is presented in Fig. 2. In this scheme, the step corresponding to the release of the product has been represented as irreversible. This does not mean that the principle of microscopic reversibility does not apply, but simply that the kinetics of the reaction is analyzed under initial steady-state conditions, in such a way that the concentration of the product can be neglected. For that model, the important idea is that the regulatory behavior is due to a memory of the enzyme that keeps in mind for a while the conformation stabilized by the product. The initial steady-state rate equation corresponding to model I can be written as c( P = (k4 + k-4) [(k- + k-3) (k + k,) + kk,] = (klk4 + k3k--4) (k + k + k,) + (k + k,) (kik-3 + k3k-) f kkikz Y = klk3 (k + k + k,) 6 = kkz (klk4 + k3k-4) & with = kk,k,k, EP - 6Y = kklkzk3 [(klk-3 + k3k-) (k + k2) + kkikz] (3) is, of necessity, positive. The curvature of the plots is defined by the sign of the second derivative Obviously, if k3 > k, the second derivative will be positive and the curve will be concave upwards. If k3 < k, the reciprocal plots will be concave downwards. The equations of the asymptote (when l/[s] + m) and of the tangent (when l/[s] + 0) can be easily derived, as well as the coordinates of their intersection (Table 2). Some shapes of the plots obtained by computer simulation are presented in Fig. 3. For a reciprocal rate equation exhibiting no extreme, the extent of cooperation, r, of the two conformation states to the overall reaction process can then be measured by the value of the second derivative for /[A] + 0. From Eqn (4), one gets where [Elo is the total enzyme concentration. The coefficients a, p, y, 6, E are groupings of rate constants that can be obtained by writing down the reciprocal rate equation in explicit form. The explicit formulation of these kinetic coefficients is to be found in Table. The reciprocal plots cannot exhibit any extreme because the first derivative If the specific reciprocal reaction velocity, [EIo/u, is measured in seconds, the extent of cooperation, r, will be expressed in sm2. It is worth noting that the extent of cooperation between the two enzyme forms depends only on three rate constants k,, k3, k4, but not on the equilibrium constant between the two enzyme conformations. This way of measuring the cooperation between two different conformations of the same enzyme molecule is analogous to that previously used by Teipel and Koshland [5] to estimate the cooperativity, that is the interaction between sites of a polymeric enzyme.

4 98 The Mnemonical Enzyme Concept Table 2. Explicit,jormulatiorzs of asymptote, ([E]Jv)~, tangent, ([E]o/v)r, and asvmptote-tangent intersection (Is, I,),for an onesubstrate enzyme I, is the abscissa and I, is the ordinate of the intersection Asymptote or tangent Equation Table 3. Rate constantsfbr Fig. 3 Units are arbitrary Rate Michaelian Positive Negative constant behavior cooperation cooperation o o3 o3 I o3 0-3 IO-~ I IL 6 07 [A Fig. 3. Some possible shapes (computer outputs) oj the reciprocal plots jor a one-substrate, one-product mnemonical enzyme. () Michaelian behavior; (2) positive cooperation; (3) negative cooperation. The values of the rate constants (in arbitrary units) are given in Table 3. Simulation is effected with a Wang 2200 computer The fact that cooperation between the two enzymes forms is positive or negative, rests on simple thermodynamic conditions. The free energy of activation, linked with the phenomenological rate constant, kl, can be split up [6] into two components: a) a binding component, AG?, that is the free energy of activation, without taking into account the conformation change ; b) a transconformation component, AG;, that is the activation free energy for conformational change apart from binding effects. Neither of these contributions can have a negative value, and one can write In the same way, one has AGT = AG? + AG:. AG,~ = AG? + AG;, where AG?, is the free energy of activation for the other conformation change (transconformation of the rhombus into the square in Fig. 2) apart from binding effects. Here again, this energy contribution cannot have a negative value. By comparing Eqns (6) and (7), it is obvious that if (6) (7)

5 J. Ricard, J.-C. Meunier, and J. Buc 99 and the cooperation between the two enzyme forms will be positive, leading in turn to an upward curvature of the plots. If on the other hand then AG;, > AG; (0) k3 kl ( ) and the cooperation between the enzyme conformations will be negative, leading to a downward curvature. In the same way, the extent of cooperation between the two enzyme forms is associated with the respective values of AG? and AG:,. A theoretical plot of r against the difference (AG: - dg,f,), is presented in Fig. 4. The Concept of Enzyme Memory,for a Two-Substrate, Two-Product Monomeric Enzyme Following an Ordered Mechunism A two-substrate, two-product monomeric enzyme following a compulsory order mechanism and exhibiting a mnemonical transition is presented in Fig. 5. In the absence of any product, the reciprocal rate equation corresponding to model I can be written under either of the two following forms -.I -2.0 Fig. 4. Effkcts of activation free energies of conformation changes on the extent cooperation of a one-substrate, oneproduct mnemonical enzyme (computer outputs). The values of the rate constants are: k, lo3; k,, lo6; k,, lo3; k4, lo6; k, ; k-3, k-4, I. k, varies from 550 up to 800 and k-, from 0.5 up to.8. The numerical values are given in arbitrary units. Simulation is effected with a Wang 2200 computer The coefficients a, P, y, 6, e are defined in Table 4. While rate Eqn (3) predicts linear plots [E],/v versus l/[b] (at fixed concentrations of A), the plots [E],/v versus /[A] (at fixed concentrations of B) should, as a general rule, exhibit deviations from linearity [Eqn (2)]. This is illustrated in Fig. 6. It is to be noted that the primary plots against /[B] do not intersect at the same point. The equations of the asymptotes (for /[A] -+ m) and of the tangents (for /[A] -+ 0) of the plots [E],/v against l/[a] are given in Table 5. In the same Table are given the coordinates of some noteworthy intersection points. Moreover, if the kinetic coefficients of rate Eqn (3) are replotted against /[A], the k k Fig. 5. The mnemonical transition for a two-substrate, twoproduct monomeric enzyme (model II) r

6 200 Table 4. Explicit formulation of the kinetic parameter of Eqns (2) - (9) and (25) - (35) The Mnemonical Enzyme Concept A 5.:" Kinetic parameter Equation El a2 a3 a4 a5...'...' 4,......'._.. 3 _... P Pz P3 P4 85 P6 7 Yz Y3 Y4 6 8 secondary plots thus obtained are defined by the following equations where and z are the slopes and the intercepts of the primary plots. Eqn (4) predicts curvilinear plots against /[A] (Fig. 7) but Eqn (5) does not. As a matter of fact, it can be shown, after some algebra, that Eqn (5) is equivalent to Fig. 6. Some possible shapes (computer outputs) of the reciprocal plots for a two-substrate, two-product mnemonical enzyme. (A) Reciprocal of the rate plotted against the reciprocal of the first substrate concentration, for different concentrations of the second substrate. The values of the rate constants are: k, lo3; kl, lo4; k2, lo6; k3, lo3; k4, lo3; k,, ; k6, ; k', ; k-l, 0; k-2, ; k-5, lo3; k-6r. The concentrations of the second substrate are: 0. (curve I ), 5.0 x IO-~ (curve 2), 2.0~ (curve 3), I x IO-~ (curve 4) and 5 x (curve 5). (B) Reciprocal of the rate plotted against the reciprocal of the second substrate concentration, for different concentrations of the first substrate. The values of the rate constants are the same as (A). The concentrations of the first substrate are: 0 (plot I), 5 (plot 2), (plot 3), 0.25 (plot 4) and (plot 5). The numerical values are given in arbitrary units. Simulation is effected with a Wang 2200 computer

7 ~ J. Ricard, J.-C. Meunier, and J. Buc 20 Table 5. Explicit formulations of asymptote, ([E]o/~)A, tangent, ([E]o/~)T, and asymptote-tangent-intersection (Is, I,) for a twosubstrate enzyme I, and I, are still the abscissa and the ordinate of the intersection Function Equation A (kk4 + k'k4 + k3k4 + kk3) (kik, + k-5k6)' + kkik3k4k5 (k, - k6) _ ~ kk,k4 (k,k, + k-5k6)2 + -~ kk3 + k-2k3 + k'k-2 kk2k3 [BI (;lo) z k'k, + k3k4 + kk4 + kk3 k'k-2 + k-2k3 + kk3 (kk3 + k-2k3 + k'k-2) (kik-6 + k-ik,) kk3h kk2k3 LB kklk2k3k6 [B 0 u) X '0030/ m a - (D " A c a - X N G! 2.0 B : which is obviously linear in /[A]. This conclusion is exemplified in the computer outputs of Fig. 7. In Table 6 are presented the equations of the asymptote and of the tangent of the curvilinear secondary plot as well as the coordinates of their intersection. The first derivative (with respect to l/[a]) of Eqn (2) is

8 202 The Mnemonical Enzyme Concept Table 6. E-xplicit formulations of asymptote, ia, tangent, it, and asymptote-tangent intersection (I,, I,,) of the curvilinear secondary plot Function Equation - IT - k3k4 + k'k4 + kk, + kk3 +- kk3k4 k6 "4 and the coefficients are still defined in Table 4. Since is of necessity positive, the first derivative in Eqn (7) is positive too, and the primary plots corresponding to Eqn (2) cannot exhibit any extreme. The second derivative (with respect to l/[a]) of Eqn (2) can be written now as Table 7. Rate constants for Fig. 8 The units are arbitrary Rate Michaelian Positive Negative constant behavior cooperation cooperation o o o o o4 06 o It will be noticed that this expression is identical to that of the second derivative (with respect to l/[a]) of Eqn (4), that is (?)''= Thus ([E]O/u)" and I" are either positive or negative following the respective values of k6 and k,. The situation is thus similar to that described with onesubstrate monomeric enzymes. As before the numerical value of the second derivative for /[A] -+ 0 can be taken as a measure of the extent of cooperation of enzyme forms. From Eqn (9) one gets This relation is analogous to Expression (5) and the extent of cooperation is dependant on three rate constants only. Here again the fact that the cooperation is positive or negative rests on very simple thermodynamic conditions. One has where AG?, AG; and AG?, are defined as before. Depending on the respective values of AGZ and AG,f,, the cooperation will be positive or negative. If

9 J. Ricard, J.-C. Meunier, and J. Buc 203 3, I.: 0.I... 2/-..., : : :....: L lo3/[ai Fig. 8. Cooperation?fa two-substrate, two-product mnemonical enzyme. () Michaelian behavior, (2) positive cooperation, (3) negative cooperation. The values of the rate constants (arbitrary units) are given in Table 7. The concentration of the second substrate (in arbitrary units) is 0. for all plots. Simulation is effected with a Wang 2200 computer -8 - Fig. 9. Ef ects of activation free energies of conformation changes on the extent of cooperation of a two-substrate, twoproduct mnemonical enzyme (computer outputs). The values of the rate constants are (in arbitrary units): k, lo3; k2, lo6; k3, lo3; k,, lo3; k,, lo6; k,, lo4; k, ; k-2, ; k-,, ; k-6, lo-. k,andk~,varyfrom6.6x03upto5.4x03.simulation is effected with a Wang 2200 computer the plots with regard to /[A] will be concave downwards (negative cooperation) ; if AC?, < AGT (24) the plots will be concave upwards (positive cooperation). Some possible different shapes of the reciprocal plots predicted by the mnemonical model I are presented in Fig. 8. In Fig. 9 is plotted the variation of the extent of cooperation, r, against AG,f - AG,f,. The Last Product as an Eflector of Cooperation between Enzyme Forms for c Two-Substrate, Two-Product Mnemonical Enzyme From Eqn (9) it is obvious that the second substrate B has no effect on the extent of cooperation. However, one may wonder what happens when one of the products is introduced into the reaction medium. The reciprocal rate equation in the presence of the product P takes either of the two forms

10 204 The Mnemonical Enzyme Concept The explicit formulation of the new parameters is still presented in Table 4. From Eqns (25) and (26) one notes that the plots of [E],/v against [PI should be straight lines and that product P should behave as a non-competitive inhibitor of substrate B. This point need not be discussed further. The first derivative (with respect to l/[a]) of rate Eqn (25) is [see Eqn (27)] Since [see Eqn (28)] is always positive, the reciprocal plots exhibit no extreme. Surprisingly, the second derivative (with respect to l/[a]) of rate Eqn (25) is still Expression (9). It thus appears that the first product has no effect on the cooperation between the free enzyme conformations. The situation is completely different if the initial steady-state rate is measured in the presence of the last product Q. The reciprocal rate equation takes either of the two forms [see Eqn (29) and (30)] The explicit formulation of the new kinetic parameters is still to be found in Table 4. The plots of [E],/u versus [Q] are obviously straight lines and the second product Q is non-competitive with regard to B. From Eqn (29) it appears that the curves intersect the ordinate axis at the same point, whatever the concentration of the product Q (Fig. 0). This intercept,, is expressed as The situation is obviously different if the inhibitor is the first product P [Eqn (25)]. Then the curves intersect the ordinate axis at different points, depending on the product concentration (Fig. 0). The intercept,, is now expressed as =- Y + Y2 [BI + Y3 [PI + Y4 [BI [PI. (32) E [BI This conclusion is exemplified in the computer outputs offig. 0. The first derivative (with respect to l/[a]) of rate Eqn (29) is [see Eqn (33)] and is always positive since [see Eqn (34)] The second derivative (with respect to l/[a]) is now [see Eqn (35)] II n II

11 J. Ricard, J.-C. Meunier, and J. Buc 205 A. > c lij L 8- I I/[Al 6- < PA I/[4 Fig. 0. Effects of' the two products on the reciprocal of the rate of'a two-substrate mnemonical enzyme. (A) Effect of the first product on the reciprocal plots. The values of the rate constants are: k, lo3; k,, 04; k,, lo6; kj, lo3; k4, lo3; k,, ; k6, ; k', ; k-,, 0; k-,, ; k-3r lo6; k-,, lo3; k-6r. The concentrations of the product P are 0.0 (curve I), 0. (curve 2), 0.2 (curve 3), 0.4 (curve 4) and 0.6 (curve 5). The concentration of the second substrate B is The numerical values are given in arbitrary units. (B) Effect of the second product on the reciprocal plots. The values of the rate constants are: k, lo3; k,, : k2, lo6; k3, lo3; k4, 0'; k5, 0'; k,, 0'; k', ; k_l, ; k-', ; k-,, lo4; k-5, lo-'; k-6, The concentrations of the product Q are 2 (curve I), 4 (curve 2),6 (curve 3), 8 (curve 4) and 0 (curve 5). The concentration of the second substrate B is still The numerical values are given in arbitrary units. Simulation is effected with a Wang 2200 computer J C - 5 El i-, It is thus obvious that the product Q does effect the extent of cooperation. From Eqn (35) it appears that cooperation, in the presence of Q, is positive when ki > ki ( + K4 [QI) (36)

12 206 The Mnemonical Enzyme Concept L -LO Fig.. Reversal of cooperdon b.v the lust product (computer outputs). The values of the rate constants are: k, lo3; k,, I; k2,0h;k3,lo3;k,,02;k,,02;k,,02;k',;k~,,l;k~2,; k-,, 04; k-,, k-,, The concentration of the second substrate is lo-,. The concentrations of the second product Q are (curve I), 0.99 (curve 2), 2 (curve 3) and 4 (curve 4). The numerical values are given in arbitrary units. The simulation is effected with a Wang 2200 computer -8OL Fig. 2. Ejfect of' the lust product on the extent of' Cooperation (computer outputs). The values of the rate constants are: k, lo3; kl, ; k2, 0,; k3, lo3; k,, 0'; k,, 0'; k,, lo2; k', ; k-,, ; k.2, ; k-,, lo4; k-5, k-,, lo-'. The concentration of the second substrate is The numerical values are given in arbitrary units. Simulation is effected with a Wang 2200 computer (with K4 = k-,/k,) and negative when (37) Thus when the cooperation, in the absence of any product, is positive (k6 > kl), the product Q will decrease, or even reverse, that cooperation. When the cooperation is negative (k, < kl), in the absence of any product, Q will strengthen this negative cooperation. Lastly, when the cooperation is zero (k, = k,) in the absence of product, Q will induce a negative cooperation. The curves of Fig. obtained by computer simulation exemplify these conclusions. The last product Q clearly behaves as an effector of cooperation between enzyme forms. This effect is presented in Fig. 2. In the presence of product Q, the extent of cooperation, r, is Kinetic Symptomatology of Enzyme Memory Probably, the most obvious general property of monomeric enzymes exhibiting memory phenomena is that cooperation is only kinetic and does not appear in the substrate-binding isotherms. Moreover, a two- substrate, two-product mnemonical enzyme has very distinctive kinetic attributes. a) The enzyme has to follow a compulsory order mechanism, and the deviations from Michaelian behavior should be observed only with regard to the first Substrate (A). b) The extent of cooperation between enzyme forms expressed with regard to the leading substrate (A) has to be independant of the concentration of the second substrate (B) and of the first product (P). c) A mnemonical enzyme is not inhibited by an excess of the substrate. d) Probably, the most decisive criterion for twosubstrate enzyme memory comes from the analysis of secondary plots. The slopes and the intercepts of the straight lines obtained by plotting [El& versus /[B] should give, when replotted against l/[a], a straight line and a curve, respectively. e) The reciprocal steady-state rate of the reaction catalyzed by a mnemonical enzyme has to be proportional to the concentrations of the products P and Q. Moreover, both products should be noncompetitive with regard to the second substrate (B). Lastly, whatever the concentration of the second product Q, all the reciprocal plots against /[A] should intersect the ordinate at the same point. f) The last product Q has to induce, or to increase, a negative cooperation between enzyme forms with regard to the first substrate.

13 J. Ricard, J.-C. Meunier, and J. Buc 207 Probably neither of the above conditions is sufficientper se to establish the existence of a mnemonical model. As will be seen in the accompanying paper [ 4, some random mechanisms involving two substrates and a rigid enzyme can give rise, under certain simplifying assumptions, to some of the above conditions. However, the whole set of criteria certainly can be taken as a very strong argument in favor of the validity of the enzyme memory concept. DISCUSSION For polymeric enzymes, deviations from Michaelis- Menten behavior appear as a consequence of subunit interactions [7-2. For mnemonical enzymes consisting of one polypeptide chain it is a consequence of a cooperation between different conformations of the protein. This cooperation of two enzyme forms occurs within a certain type of conformational transition : the mnemonical transition. This concept applies to single-site enzymes and integrates the classical ideas of induced-fit and of allosteric transition with exclusive binding, already developed for polymeric enzyme [7,8]. The mnemonical models developed in the present paper are thus very different from those derived by Frieden [22] and Ainslie et al. [ll] which rest on the postulated existence of a pre-equilibrium between two enzymes forms having different affinities for the substrate and exhibiting no conformation change upon ligand binding or release. Induced-fit does not appear as a component of such a reaction scheme. An important idea put forward by both Frieden [22] and Ainslie et af. [ll] is that deviation from Michaelian behavior is the obligatory consequence of conformation changes that are slow with regard to the other reaction steps. For instance, Ainslie et al. [ll] strongly emphazise the view that linearity of the reciprocal plots is lost if the rates of the conformational transitions are of about the same order of magnitude as the other processes. Such a situation does not occur with the mnemonical models, and the regulatory behavior does not require that the conformational transitions be slow. Nothing needs to be postulated as to the relative magnitude of the various reaction steps. This clearly appears from the results obtained by computer simulation and discussed in the present paper. Since the concept of mnemonical transition implies the existence of an induced-fit, it can be used to analyse in the simplest way the regulatory behavior of twosubstrate, two-product monomeric enzymes following an ordered mechanism. It is obvious, on the other hand, that the slow isomerization model proposed by # 0 AgT Agfl > Agf Ag? < Ag? Negative cooperation Positive cooperation dgtl # = dg; Michaelian behaviour Fig. 3. Thermodynamic grounds of cooperationfor mnemoniral enzymes Ainslie et af. [Ill cannot be applied as such to compulsory-order reaction sequences, for that model does not include induced-fit phenomena which often are a component of ordered mechanism. However, from a phenomenological viewpoint, it can be reduced to the corresponding mnemonical model, on neglecting various reaction pathways and assuming that some conformation changes are so rapid that a binding step followed by a transconformation can be approximately considered as an induced-fit process. Obviously, this assumption is inconsistent with the basic idea [ll] that isomerizations of enzyme forms have to be slow. The inclusion, in the mnemonical models, of the forward and the backward reactions is clearly in accord with the principle of microscopic reversibility. The existence of closed loops in the reaction scheme implies some constraints in the corresponding rate constants. These constraints have been introduced into the rate equations and the computer simulations. The obtained equations are thus rigorous and consistent with classical thermodynamic principles. The regulatory behavior of single-site mnemonical enzymes is expressed by simple thermodynamic conditions. The cooperation between the two enzyme forms can be positive, zero, or negative depending on the respective values of the energy contributions AG? and AG?,, as summarized in Fig. 3. When considering the mnemonical transition (Fig. l), the extent of cooperation between the two free enzyme forms (the circle and the rhombus), that controls the curvature of the primary plots, is expressed by the value of r (in sm2). For any mnemonical monomeric enzyme and whatever the reaction mechanism, the extent of cooperation between the free enzyme forms, in the absence of any product, can be

14 208 J. Ricard, J.-C. Meunier, and J. Buc:The Mnemonical Enzyme Concept expressed as (39) where k, and kt (Fig. ) are the rate constants pertaining to the binding of the ligand L on the free enzyme forms, and kt (Fig. ) the rate constants of isomerization of the conformation (the rhombus) stabilized by the last product, to the initial one (the circle). It is noteworthy that the equilibrium constant between the two conformations of the free enzyme is not involved in the extent of cooperation r. The regulatory behavior of the enzyme is thus clearly the consequence of a memory but not that of a pre-equilibrium between the two free enzyme forms. The extent of cooperation within the mnemonical transition of a two-substrate, two-product enzyme is highly controlled by concentration of the last product. Increasing the concentration of that product makes the cooperation of the two enzyme forms more negative. This effect can be used to decrease or even to reverse a positive cooperation, to strengthen a negative cooperation or to induce the appearance of negative cooperation in a Michaelian enzyme. Thus the last product should play a fundamental role in the modulation of cooperation. When a two-substrate, two-product monomeric enzyme follows a compulsory order mechanism this situation usually implies the existence of induced-fit of the protein and could give rise in turn to memory phenomena. According to this view, the Michaelian behavior is then a particular case of cooperation and can be obtained in either of the two following cases: a) The case where AG; % AG?, ; b) the case where the steady-state concentration of one enzyme conformation (for instance the rhombus conformation in models I and ) is very low. Monomeric enzymes exhibiting an ordered binding of substrates and an ordered release of products are thus potentially good candidates for the role of regulatory enzymes. The major interest of a theoretical model is to give rise to predictions that can be checked experimentally. Thus the model has to be as simple as possible. We feel this is precisely the case for the mnemonical model. As already shown, it is possible to derive from it some simple predictions that can all be tested by appropriate experiments which can be used for the identification of the mnemonical transi- tion. A concrete exemple of such a diagnosis is to be found in the accompanying paper [4]. Thanks are due to Mrs Grossmann for carefully reading the manuscript. The senior author (Jacques Ricard) wishes to thank Dr H. Witzel for making available a preprint of his article [23]. Note Added in Proof. After submission of the manuscript a paper by Rubsamen et a/. [23] appeared and showed conclusively the existence of sigmoidal kinetics of ribonuclease I. The experimental results were interpreted on the basis of ligand induced shifts of conformation equilibria. REFERENCES. Dalziel, K. (958) Trans. Faraday Soc. 54, Sweeny, J. R. & Fisher, J. R. (968) Biochemistry, 7, Wong, J. T. F. & Hanes, C. J. (962) Can. J. Biochem. Physiol. 40, Ingraham, L. L. & Makower, B. (954) J. Phvs. Chem. 58, Ferdinand, W. (966) Biochem. J. 98, Cleland, W. W. (970) in The Enzymes (Boyer, P. D., ed.) vol. 2, pp. - 65, Academic Press, New York. 7. Pettersson, G. (969) Acta Chem. Scand. 23, Pettersson, G. & Nylen, V. (972) Acta Chem. Scand. 26, Rabin, B. R. (967) Biochem. J. 02, 22c-23c. 0. Whitehead, E. (970) Progress Biophys. 2, Ainslie, R. E., Schill, J. R. & Neet, K. E. (972) J. Biol. Chern. 247, Kosow, D. P. & Rose, I. A. (97) J. Biol. Chem. 246, Witzel, H. (967) Hoppe-Seyler s 2. Physiol. Chem. 348, Meunier, J. C., Buc, J., Navdrro, A. & Ricard, J. (974) Eur. J. Biochem. 49, Teipel, J. & Koshland, D. E. (969) Biochemistry, 8, Ricard, J., Mouttet, C. & Nari, J. (974) Ew. J. Biochem. 4, Monod, J., Wymann, J. & Changcux, J. P. (965) J. Mol. Biol. 2, Koshland, D. E., Jr, Nemcthy, G. & Filmer, D. (966) Biochemistry, 5, Koshland, D. E., Jr (969) Curr. Top. Cell. Regul., Koshland, D. E., Jr (970) in The Enzymes (Boyer, P. D.: ed.) vol., 3rd edition, pp , Academic Press, New York. 22. Kirschner, K. (968) Curr. Top. Microbiol. Immunol. 44, Frieden, C. (970) J. Bid. Chem. 245, Rubsamen, H., Rhandkcr, R. & Witzel, H. (974) Hoppe- Seyler s Z. Physiol. Chem. 355, J. Ricard. J.-C. Meunier, and J. Buc, Laboratoirc de Biochimie Vegetale, U.E.R. de Luminy, Universitk d Aix-Marseille, 70 Route Leon-Lachamp, F Marseille-Cedex-2, France