Enzyme Memory. 1. A Transient Kinetic Study of Wheat-Germ Hexokinase LI

Transcription

1 Eur. J. Biochem. 80, (1977) Enzyme Memory 1. A Transient Kinetic Study of Wheat-Germ Hexokinase LI Jacques RICARD, Jean BUC, and Jean-Claude MEUNIER Laboratoire de Physiologie Cellulaire Vegetale AssociC au Centre National de la Recherche Scientifique, Unite d Enseignement et de Recherche Scientifique de Luminy, Universite d Aix-Marseille 11 (Received March 17, 1977) When wheat germ hexokinase LI is rapidly mixed with its substrates in a stopped-flow apparatus, the reaction proceeds and exhibits a slow burst. The length of the negative induction time, z, is independent of enzyme concentration but depends on glucose and magnesium-atp concentration. At low and high glucose concentrations, the burst disappears but becomes maximum at intermediate hexose concentrations. The approach to steady state can be considered and treated as a relaxation spectrum. Two relaxations can be detected experimentally: a fast (0.2 s) and a slow process (100 s). The slow relaxation, studied at low glucose concentration, is very similar to the free enzyme relaxation which takes part in the mnemonical transition. This relaxation has been studied in further detail. When the reciprocal of the slow relaxation time,,i2, is plotted against glucose concentration, one obtains a curve concave down, that reaches a plateau for high glucose concentrations. This result is taken to mean that glucose binding and enzyme transconformation occur as two distinct steps and that the binding step is faster than the transconformation process. The burst obtained at the start of the reaction is interpreted as indicative of a progressive shift from an enzyme species able to bind glucose easily and to undergo the proper conformation change, to an other form less able to bind glucose in a productive way. This shift would not be spontaneous but would be forced by the reaction process itself until the dynamic equilibrium that occurs under steady state is reached. Evidence in favor of enzyme memory has been obtained from pre-mixing experiments with glucose 6-phosphate. If the enzyme, already premixed with glucose 6-phosphate in one of the drive syringes of the stopped-flow apparatus, is mixed with the others reactants, the reaction exhibits a slow lag instead of a burst. This transient would correspond to the shift from a weakly reactive enzyme form to a strongly reactive one. Thus, as expected with the concept of mnemonical transition, the free enzyme must occur under two forms having different reactivities for glucose and glucose 6-phosphate. In recent years several research groups have shown [I - 71 that a monomeric enzyme can exhibit atypical kinetics and have a behavior similar to that of an allosteric enzyme. Although the random binding of substrates on a rigid, or nearly rigid, monomeric enzyme can, in theory, generate this type of behavior, it seems unlikely that the sigmoidal or the anticooperative kinetics, of a monomeric one-sited enzyme can often be explained in that way. A more likely explanation of these effects seems to be found in the existence of conformational transitions which occur far from pseudo-equilibrium conditions. The so- Enzyme. Hexokinase (EC ) called mnemonical transition [5] represents an interesting possibility among these conformational transitions. This type of conformational transition requires the existence of a sort of enzyme memory, that is when the last product is released from the active site, the enzyme recalls for a while the conformation stabilized by that product. The departure from Michaelian behavior would then be the consequence of a cooperation between two different conformation states of the enzyme. However, this concept of enzyme memory has been reached on an indirect basis by studying steadystate kinetic properties of an enzyme. It is thus of great interest to aquire more direct evidence for the existence

2 582 Enzyme Memory. 1. of the conformational transition involved in enzyme memory. In this paper we present such evidence based on the study of the transient kinetics of the reaction catalyzed by wheat germ hexokinase LI. MATERIALS AND METHODS Wheat germ hexokinase LI was isolated and purified to homogeneity as previously described [8]. Its molecular weight is All the kinetic measurements were done in a Durrum-Gibson (model D 110) stopped-flow apparatus, equipped with a 20-mm light-path cuvette. The temperature was maintained at 30 "C by circulating thermostated water. The stopped-flow device was connected with a data analyzer and a mini-computer. The reaction was followed by the changing ph indicator (cresol red) method of Darrow and Colowick [9]. The buffer used was 5 mm glycyl-glycine ph 8.5. Under the experimental conditions used the absorbance change at 574 nm is proportional to the production of H'. The data analyzer was a Zoomax device (SocietC d'electronique Industrielle et NuclCaire, Paris, France) connected with a Novelec (Paris) data sampler and analog digital converter. This device allowed sampling, storing and averaging of numeric data. The output of memory contents was effected on a teletype. Each point corresponded to the average of about experimental values. The mathematical treatment of these data was effected with a Wang 2200 computer with extended memory. The analysis of transient kinetic data in terms of relaxation spectrum necessitates prior determination of the steady-state appearance of products. The steady-state progress curve was obtained by linear least-squares analysis [lo]. A computer program was written which allows one to obtain the optimized least-squares fit. The goodness of fit was obtained from this program by the sum of squares of residuals, which should be minimum, and the distribution of residuals which should be normally distributed with a zero mean [ll, 121. THEORY From a steady-state kinetic study of wheat germ hexokinase LI it was previously concluded [6] that the enzyme binds its substrates and releases its products following either a compulsory, or a preferred order pathway. This pathway is shown in Fig. 1A where G, MA, MA' and G' are glucose, MgATP2-, MgADP-, and glucose 6-phosphate, respectively. If the concentration of magnesium-atp is not too low, and if the rates of interconversion of enzyme species El, EZ and Eg are slow with respect to all the other steps, the enzyme species E3, Eq and Es will be in a pseudo-steady state at the start of the reaction, whereas the free enzyme and the enzyme. glucose complex will still be under pre-steady-state conditions (see Appendix). Therefore a model of Fig. 1A can be formally written as shown in Fig.1B. This is equivalent to assuming that during the transient phase the concentrations of all the enzyme. ligand(s) complexes except the EG complex, are very low. In Fig.1 it is implied that glucose binding and transconformation occur in one step only. This is, indeed, a limiting case. If binding and transconformation occur as two distinct steps, and if binding is fast while the transconformation is slow, the model in Fig. 1 B can be expressed as shown in Fig. 2A, which in turn is formally equivalent to the model of Fig. 2B. A B IE IL E2 / MA E3 E4 Fig. 1. The rnnemonical model /or itheat germ hexokinase LI. Glucose binding and transconformation are assumed to occur in one step. (A) Model for ordered binding of substrates and ordered release of products. (B) Simplified model which postulates that enzyme forms E3, E4 and Es are at very low concentrations. G, MA, G', MA' represent glucose, magnesium-atp, glucose 6-phosphate and magnesium-adp, respectively.

3 J. Ricard, J. Buc, and J.-C. Meunier 583 A B + G' Fig. 2. The mnemonical modeljor wheat germ hexokinase LI. Glucose binding and induced transconformation occur as two distinct steps. (A) Simplified model which postulates that substrate binding is faster than transconformation. (B) Formal representation following Cha [I31 of the previous model The f' coefficients are the fractional concentrations factors of Cha [I31 and are expressed as type of equation for the models of both Fig.1B and Fig. 2A, namely f1= 1 If the affinity constants K1 and K6 of glucose for the enzyme are not too high, and if the glucose concentration is not too high, the fractional concentration factors reduce to The rate equations for the appearance of any product P can be integrated (see Appendix) if glucose concentration is in large excess with respect to the total concentration of enzyme. One gets the same where a, v,, $1 and $2 are groupings of the rate constants given in the Appendix. Indeed, these parameters have different values depending on whether model 1 B or 2A applies. When the steady state is approached, the two exponential terms vanish and the progress curve is then linear with time, us being the steady-state rate. The approach to steady-state product appearance, AP, can obviously be considered as a relaxation spectrum AP = $1 e-"" + $2 e-"", (4) where $1 and $2 are the relaxation amplitudes, 21 and A2 the reciprocal of the relaxation times. Since the sum and the product of the two A are equal to the groupings of rate constants Q and d2 given in the Appendix, the reciprocal of the relaxation times can be obtained as the roots of the quadratic equation L2 - LdI + dz = 0 (5) that is dl? Id: - 4d2 A1,2 = - 2 If the rate of the conformational transition involved in enzyme memory (interconversion between enzyme forms El and E6) is slow, and if the concentration of MgATP2- is not too high, it can be shown (see Appendix) that the expression of the two 2 becomes, in the case of model 1 B : (6) 22 = kl k6 [GI2 + (kl ka [MA] + kl k-6 + k-1 k6) [GI + (k5 + k-5) (k-1 + k-6 + ka [MA]) (kl + k6) [GI + k-i + k-6 + ka [MA] (8)

4 584 Enzyme Memory. 1. A : --I _. B. "..._ _... I0 -- r..c.-''. C."' ' N Fig. 3. Possible vnrialioti of 1.2 against tlrr c.oricenlrurion of magnrsiuui-a TP and gl~rco.se (computer ourputs). The dotted lines simulate Eqn (9). The rate and binding constants have the following values: k; = lo4, kl, = 20, k; = kl, = 5 x k; = klb = k, = 100, Ki = 0.1, K6 = 100. The fixed values of [glucose] (when [MgATP' 1 is varied) and [MgATP2-](when [glucose] is varied) are both equal to Units are arbitrary A;' is the 'fast' relaxation time, whereas AT1 is the the relaxation of the free enzyme involved in the 'slow' relaxation time. mnemonical transition : If the model of Fig.2 applies the two A can also be calculated easily (see Appendix), but the analytical /22 = k5 + k-5 (model 1B) expression of Al is so complicated that it cannot be (12) A2 = k; + kls (model 2A). used experimentally. A2 can be expressed as A2 = a Ki K6 [GIz + (b Ki + c K6) [GI + d e KI Ks [GI2 + CfKi + g K6) [GI + h. The parameters a, b,... h are grouping of rate constants given in the Appendix (Table 2). (9) If model 2A applies, and if glucose concentration is not too high in such a way that Eqn (2) can be considered as reasonable approximations, the expression of the reciprocal of the relaxation time is considerably simplified and reduces to k[k8kikg[gl2 +(kikika[ma]+kiki6ki +klik$k6)[g]+(k;+kis)(kii +ki, +k,[ma]) A2 =. (13) (k; Ki + k8 K6) [GI + kii + ki, + k, [MA] It is noteworthy that Eqns (8) and (9) allow one to decide whether binding of glucose and transconformation occur in one or in two steps (Fig. 1 and 2). If binding and transconformation occur in one step, as postulated in Fig.1, 122 should reach an oblique asymptote, (A,)", when glucose concentration is increased If, on the other hand, binding and transconformation occur as two distinct steps, as shown in Fig.2, A2 should reach a plateau for very high glucose concentrations. This asymptotic value could be written as kik; + kilki + k;ki6 + kik, [MA] (")" = k; + k$ + kil + kih + k, [MA]. (11) By plotting the value of A2 against glucose concentration, it would then be possible to select either model 1 B or model 2A on experimental grounds. Moreover, for low glucose concentrations no matter which of the two models 1B or 2A applies, A2 will describe Since the slow relaxation of the enzyme is certainly an important aspect of the molecular dynamics of enzyme memory it is of interest to analyze the variation of A2 with respect to glucose concentration. This can be done easily with the 'simplified' Eqn (13). If, as already assumed, the rate constants k; and ki5 of the interconversion between the two free enzyme forms are very small with respect to all the other rate constants, the first derivative, with respect to [GI, is always positive. Therefore, Eqn (1 3) cannot exhibit an extreme. Similarly, the second derivative, with respect to [GI, of Eqn (13) is always negative. Therefore, a plot of A2 against glucose concentration should be concave downwards. In Fig.3 this type of plot is shown. For glucose concentrations that are not too high, in such a way that Eqn (13) applies, the reciprocal of the 'slow' relaxation time varies hyperbolically with magnesium-atp concentration. This hyperbolic function will continuously increase if

5 J. Ricard, J. Buc, and J.-C. Meunier i t Fig. 4. Possible variation o/ the induction time T against glucose and magnesium-a TP Concentrations (computer outputs). The dotted curves (computeroutputs)simulateeqn(15).thevaluesoftherateand bindingconstantsare:k, = l,,kl~ =0.2,k; = 10-3,kis = 5x 10-6,kB = 1, kis = k, = 100, Ki = 1, Kg = 0.1. The fixed values of [MgATP2-](when [glucose] is varied) and [glucose](when [MgATPZ-] is varied) are both equal to Units are arbitrary which is a most likely condition. This type of variation is shown in Fig. 3. The Eqn (3) expressing product appearance with time predicts that the progress curve can have a burst or a lag at the start of the reaction. The induction time z is defined as the extrapolation of the straight part of the progress curve on the time axis, whereas the burst or the lag is measured by extrapolating the linear part of the progress curve on the ordinate axis. When a is positive (z negative) the reaction exhibits a lag. The expression of z can be written as for the model of Fig. 2. The expression of the v and 6 are to be found in the Appendix. A possible type of variation of the induction time against glucose concentration is shown in Fig.4. On the other hand the variation of the induction time versus magnesium- ATP is a hyperbola. RESULTS The spectrophotometric method [9] which is used to measure reaction rates gives consistent and reproducible results. This can be checked by plotting the values of the initial steady-state rate vs measured with the stopped-flow technique, against enzyme concentration (Fig.5). In that case one obtains a straigth line passing through the origin which is obviously what would be expected. Similarly, a plot of the reciprocal of the initial steady-state rate against the reciprocal of glucose concentration is concave downwards, whereas a reciprocal plot with respect to MgATPZ- is a straight line (Fig. 6). These results are similar to those obtained [6] with more con- wi41 / OO *t./ o 1.5 [El0 (PM) Fig. 5. Variation of the initiul.y~~w/~~-.\/o/c rate versus hexokinase concentration. The rate is measured with the stopped-flow technique (see under Materials and Methods). The glucose and magnesium- ATP concentrations are 50 mm and 5 mm, respectively ventional techniques (coupled assay [14] or ph-stat [151). If wheat germ hexokinase LI is rapidly mixed, in a stopped-flow apparatus with glucose, magnesium- ATP, and cresol red, the reaction proceeds and exhibits a slow burst (Fig. 7). The value of the negative induction time z is independent of the enzyme concentration (Fig. 7) but sharply depends on glucose concentration. At low and high glucose concentrations the values of the z are small (Fig. 8A). The maximum negative value of z is obtained for an intermediate glucose concentration. When z is plotted versus MgATP2- concentration one obtains a curve compatible with a hyperbola (Fig. 8 B). Two relaxations are detectable during the approach to the steady state (Fig. 9), a fast process (A; Y 0.2 s)

6 586 Enzyme Memory. 1. and a 'slow' process (22' R 100 s). The 'fast' relaxation varies slightly with glucose concentration, but this variation has not been studied in further detail. The 'slow' process, which takes account of the slow isomerization of the enzyme involved in the molecular IA 1.Q memory, varies with both glucose and MgATP2- concentrations. As expected from the theory, a plot of h against glucose concentration is concave down, whereas the plot versus MgATP2- concentration is compatible with a hyperbola. These conclusions are exemplified in Fig.10. It is obvious from the results of Fig. 10 that 22 reaches a horizontal asymptote when the concentration of glucose is increased. As shown under the previous section, this result tentatively suggests that glucose binding and induced transconformation occur in two detectable steps as shown in Fig. 2. From the values of A2 at low glucose concentration, one can estimate the lower limit of the relaxation time of the free enzyme involved in the molecular memory. One obtains 1 2 l/[glucose] (rnm-') l/[mgatp*-] (rnm-') Fig. 6. Examples of' the primary Lineweaver-Burk plots obtained with the stopped$ow, technique. (A) Plot against l/[glucose]. The magnesium-atp concentration is 5 mm. (B) Plot against 1: [MgATP*-]. The glucose concentration is 50 mm which is a surprisingly high value, An interesting prediction of the enzyme memory concept is that the hexokinase LI retains for a time the conformation stabilized by glucose 6-phosphate before relapsing to the initial conformation. Moreover, the fact that the enzyme exhibits a negative cooperation under steady-state conditions implies that the conformation normally stabilized by glucose 6-phosphate reacts only weakly with glucose. The burst would then correspond to a shift from an enzyme species highly reactive with the substrate to another species having only a slight reactivity with that substrate. Thus, by pre-mixing the enzyme with glucose 6-phosphate before mixing with all the other reactants, one should get the appearance of a lag. This is exactly what is obtained (Fig. 11). It is noteworthy that in [Elo (PI) o 1.5 B Y) - I? * * 0,1 : 2! : : t (min) Fig. 7. The burst qfwheat germ hexokinase LI. (A) Pre-steady-state kinetics of wheat germ hexokinase (computer outputs). The dotted line corresponds to the steady-state progress curve, calculated from the experimental results. The stars are the successive means of the experimental results. The open circles are the theoretical values computed from Equation (3). The concentrations of glucose and magnesium-atp are 10 mm and 5 mm, respectively. The duration of the negative induction time (z) is - 90 s and the value of the slow relaxation time (AT') is 183 s. The ph of the reaction medium is 8.5. The enzyme concentration is 1.8 pm. (B) Variation of the negative induction time with respect to enzyme concentration. The concentration of glucose and magnesium-atp are 50 mm and 5 mm, respectively. The ph of the reaction medium is 8.5

7 ... J. Ricard, J. Buc, and J.-C. Meunier 587 [Glucose] (M) 0 0 A [MqATP'.] (M) Fig. 8. Variations of the negative induction time!ri against glucose and MgATF concentrations. (A) Variation of the negative induction time against glucose concentration. The stars are means experimental values. The dotted curve (computer outputs) is a non-linear least-squares MgATPZ- concentra- fit, havingforequationr = (-34.2 [GI [GI [GI + 4 x 10-2)/([G] [GIZ + 8 x lo-' [GI + tion is 5 mm. The ph is 8.5. (B) Variation of the negative induction time against MgATPZ- concentration. The stars are the means of experimental values. The dotted curve (computer outputs) is a non-linear least-squares fit, having for equation r = ( [MA] ) ([MA] + 3.4~ Glucose concentration is 50 mm. The ph is : : 51.2: ; T a., Y * %..a- Q *..'d.. +()& : i2 4;.g.$'..o *,-.-.. o 0.4.""' 1:...,... >....,...,...,. ~.~.,..,...,...,...,... ~ a...,... i... I I... ~,.~~~,~~.~.t.~~~.i..~~l f (s) f (min) Fig.9. The two relaxations of'the transient hiwticsjbr wheat germ hexokinase LI. (A) The fast relaxation. For a time interval of some seconds; the absorbances at 574 nm are deduced from the corresponding values taken on the linear extrapolation of the straight part of the curve. The da values are proportional to ([P]/[E]o) - (a + $2) + $2 12t and allows the calculation of both $1 and (see Appendix). The open circles correspond to the theoretical values (computer outputs) of the first exponential (first relaxation). The stars are means of exprimental values. Glucose and MgATPZ- concentrations are both 50 mm. The value of the il is 4.7 s-'. (B) The slow relaxation. For a time interval of several minutes the absorbances at 574 nm are deduced from the corresponding values taken on the linear extrapolation of the straight part of the curve (steady state). The AA values are proportional to ([P]/[EIo)- (a + 0.t) and allow the calculation of both +Z and 12 (see Appendix). The open circles correspond to the theoretical values (computer outputs) of the second exponential (second relaxation). The stars are means of experimental values. Glucose and MgATPZ- concentrations are 10 mm and 5 mm, respectively. The value of I2 is 8.9 x s-' this experiment the appearance of the lag depends only on the order of mixing the reactants. The final steady-state rate is identical in the two cases (Fig. 11). If glucose 6-phosphate is not pre-mixed with the enzyme in one of the drive syringes of the stoppedflow apparatus as shown in Fig. 11 A, the burst is lowered or even supressed, by glucose 6-phosphate, but never reversed into a lag. The appearance of a lag specifically after pre-mixing the enzyme with glucose 6-phosphate clearly gives direct evidence in favor of the existence of enzyme memory. DISCUSSION The results presented in this paper clearly show the existence of a 'slow' burst at the start of the reaction catalyzed by wheat germ hexokinase LI. This result is to be compared with that obtained by Neet and coworkers [3] on yeast hexokinase. This burst is interpreted using the assumption that the free enzyme exists is equilibrium between two conformations (the circle and the rhombus, Fig. 1 and 2). On the basis of classical steady-state study of hexokinase LI it

8 Enzyme Memory. 3. Fig. 10. Variation oj the reciprocal 41 the slow relaxation time A2 against glucose and magnesium-atp concentr.ufion. (A) Plot of i.2 against glucose concentration. The stars are means of experimental results. The dotted line is a non-linear least-squares fit of the results having for equation A2 = (3.155 x lo-' [GI ~ [GI x 10-8)/([G]Z x lo-' [GI + 2.4~ The MgATP2- concentration is 5 mm. (B) Plot of AZ against magnesium-atp experimental values. The stars are means of experimental values. The dotted line is a nonlinear least-square fit having for equation 1.2 = (9.9 x - A ** 25 i * i 25 I * :.* [GI + 3 x 10-4)/[G] x lo-'). The glucose concentration is 50 mm **. ;f...%....*. t 20 f - o > 7 020: w. * "7' - w ; *a.. -.;.*'.- *)' - i /= 9-15;.;i 7 15; ' p? 0 : 1 3 ; I /-. A* L : -.*.*.: *: * :..:... &... ~...,... - B,#* >T? :.m -0 - ;*- Y 0 *? - I * 2.: - 10 fly... 2'.. **'.'..**:.' #* :.' "4" /* :.:. 5: *&.:' N,:: :,*- : *-... f. 0 -&...*;..., ~.~...,... ~..~..~..*.~~...' i.. ~ ~ ~ ~ ~ ~ i ~ ~ ~ ~ ~ ~ ~ ~ ~.. t (rnin) t (rnin) Fig. 11. Pre-mixing experiments with wheat germ /zxokinase LL (A) Control experiment. The enzyme (final concentration 0.55 pm) is in the drive syringe 1. Glucose (final concentration 10 mm), MgATP' (final concentrations 5 mm), glucose 6-phosphate (final concentration 0.1 M) and the dye are in the drive syringe 2. The initial steady-state rate us is 6.60 x lo-' s-' and the T value is ~ s. (B) Pre-mixing experiment. The enzyme (final concentration 0.55 pm) and glucose 6-phosphate (final concentration 0.3 M) are in the drive syringe 1. Glucose (final concentration 10 mm), MgATPZ- (final concentration 5 mm) and the dye are in the drive syringe 2. The initial steady state rate z', is 6.59 x lo-' s-' and the T value is s. In both control and pre-mixing experiments, the starts are means ofexperimental results. The dotted line is the initial steady-state progress curve was previously concluded [6] that glucose can be bound to these two forms. The simplest scheme in accord with this idea was thus model 1, in which binding and transconformation were supposed to occur in one step. With this assumption, the departure from Michaelian behavior is then expressed by a parameter, r, defined as [5,6] The present pre-steady-state study has shown that a fast glucose binding step is followed by a slow inducedconformation change, as represented in Fig. 2. The parameter r has to be modified to take account of the existence of these two steps. Under moderate glucose concentrations it can be easily shown that the new expression of r is then The burst would correspond to a progressive shift from an enzyme form (circle) able both to bind glucose easily and to undergo the proper conformation change (k;ki high) to another form (the rhombus) less able to bind glucose in a productive way (k& low). However, this shift would not be spontaneous but would be forced by the reaction process itself until the dynamic steady-state equilibrium between

9 J. Ricard, J. Buc, and J.-C. Meunier 589 the two enzyme conformations is reached. Following this interpretation, the approach to steady state would be a much faster process than a spontaneous equilibration between the two free enzyme forms. Since the value of the negative induction time, z, is independent of the enzyme concentration, the amplitude of the burst, as expressed by extrapolating the linear part of the progress curve on the ordinate axis, is strongly enzyme-concentration-dependent. Therefore, it is not surprising that the slow burst has escaped detection at very low enzyme concentration 161. The approach to steady state can be adequately described by two relaxations of which the slower one at low glucose concentration describes the conformational shift of the free enzyme. Probably the more convincing evidence in favor of enzyme memory comes from pre-mixing experiments with glucose 6-phosphate. If the burst corresponds to a shift from an active (ki Kl high) to a less active (k$ K6 low) conformation, and if this less active conformation has a strong affinity for glucose 0-phosphate, one should trap this conformation by 1 )I cinixing the enzyme plus glucose 6-phosphate in one of the drive syringes of the stopped-flow apparatus. The transient phase should correspond to the shift from a weakly reactive form (the rhombus) to a strongly reactive one (the circle). The transient should then be a lag [16,17]. As expected, a lag is obtained in these conditions (Fig. 11). Moreover, it is of interest to speculate on enzyme memory as a possible device involved in enzyme catalysis and its regulation. It has been emphasized that during a catalytic process the enzyme has a much higher affinity for its transition state than for the substrate or the product [ If the free enzyme occurs in one conformation state only, the optimal catalysis in both directions is obtained if the active site has a structure complementary to the transition state. That situation allows an optimal use of the binding energy for an equal destabilization of the substrates and the products [21]. However, it has been pointed out [21] that if the enzyme has a structure complementary to that of the product, a better destabilization of the substrate is achieved and the initial rate of catalysis is increased, but the product then becomes an extremely strong inhibitor of the reaction. One can speculate that if an enzyme occurs under two different conformations, one complementary to the transition state and the other one complementary to the product, that would constitute a device allowing both an efficient catalysis by substrate destabilization and a precise control of the reaction rate by both the substrate and the product. This is precisely what is believed to occur with the mnemonical transition where the free enzyme is assumed to exist under two forms, one having presumably a strong affinity for the transition state (the circle) and the other one a strong affinity for the product (the rhombus). If El and E6 are these two free enzyme forms (the circle and the rhombus, respectively) the ratio [El]/ [E6] will decrease when glucose concentration is increased. This situation means that at low glucose concentration the free enzyme, being mostly under the El form, will be poorly inhibited by the product. Conversely, at high glucose concentration, the enzyme being mostly under the form E6 will be strongly inhibited by glucose 6-phosphate. The burst would then correspond to the appearance of the regulatory element (the rhombus conformation of the enzyme) which will allow control of the reaction rate by the product. It is obviously tempting to assume that this special device has been evolved for the control of metabolic pathways. However physiological studies are needed in order to prove the biological significance of the mnemonical enzyme concept. The x are either monomolecular rate constants or products of bimolecular rate constants by the corresponding concentration of a substrate [22]. If for a given time interval E3, E4, E5 are at steady state whereas El, E2, E6 are not, the differential Eqns (l ), can be rewritten as The xi6 is now the product of an apparent rate constant k, (3 ) by the concentration of magnesium-atp. This type of analytical formulation is equivalent to assuming that during the transient phase the concentration of the complexes [E3], [E4] and [E5] does not vary significantly and is negligible with respect to the total

10 590 Enzyme Memory. 1. enzyme concentration [El", in such a way that the conservation equation can be approximated to [E]o [El] + [E21 + [Efj]. (4') This simplification implies that the net rates of catalysis and of product desorption are faster than the net rates of substrate addition and of conformation change. Therefore, model 1 A can be formally represented by model 1 B. Now, if glucose binding and transconformation occur as two distinct steps, and if the first step is much Moreover [XI0 [Xfj (O)] = l+r By using the Laplace-Carson transformation, one can write faster than the second, models of Fig.1 have to be replaced by those of Fig. 2. The differential equations governing the system are now [A11 = - (kifi' + k 15fL5) [Xi] + klifli + k ig [&I [A,] = kifi' [Xi] -(klifli + klfj fifj + ka [MA]) [XZ] + kdf6 [x6] [A61 = kl5 f i 5 [xi] + (kifjfl6 The X are + k, [MA]) [xz] - ( ki8 + kdfb) [xfj]. [Xi] = [El] + [EI'] [X,] = [E21 [Xfj] = [E6] + [Ed] (5') (6') and the f' are the fractional concentrations factors ofcha [13]: fl1 = 1 where p is a Laplace-Carson operator and Xl, X2, X6 the transforms of the concentration [XI. The appearance of product P, either glucose 6-phosphate or magnesium-adp, can be described in the transient and initial steady-state phase by the equation The determinant A@) can be written as Since the above quadratic equation in parentheses has two distinct, real and negative roots -21 and -A2 one has Solving system (1 1 ') for Xz gives - x2 A2@) - nlp + n2 -.~ - 1x10 d(p) (P + 14 (p + A,). (15') The expression of di, d2, n1 and n2 is given in Table 1. By using the Heaviside's theorem, one can find the original of J?2, X2, and by inserting its expression into Eqn (12') and integrating with the boundary condition that t = 0 and [PI = 0, one gets fl, = 1 (7') Eqns (5') can be integrated if [GI 9 [Elo (or [XI,) and [MA] 9 [El0 (or [XI,). At the start of the reaction one has Clearly, depending on whether nld2 - n2d1 is positive or negative, the reaction will exhibit a burst or a lag. The determination of the two roots A1 and 22 can be easily effected if dl 9 4 d2. (17') It can be shown, when explicit formulation of d? and d2 are written, that expression (17') is very

11 J. Ricard, J. Buc, and J.-C. Meunier 591 Table 1. Expressions for dl, dz, nl and n2 n1 = Ki K6 (k; R + k$) [GI2 + (k; Ki R + k$ Kh) [GI (1 + Ki [GI) (1 + R) (1 + K6 [GI) which can be rearranged to The expressions of the v and 6 are to be found in Table 3. Obviously the expression of z simplifies if binding and transconformation occur in one step. Experimentally the determination of the two roots 21 and 22 can also be easily effected. Eqn (16 ) can be rewritten as likely to be fulfilled. This situation occurs if k; and kls have much lower values than all the other rate constants, and if the equilibria between the enzyme. glucose complexes are not completely shifted toward the form able to react with magnesium-atp. This last condition will be fulfilled if, for instance, k[/k:i = kijkl6 = 1. Indeed these conditions are sufficient, but by no means necessary to have inequation (18 ) satisfied. fd? - 4d2 = dl (1 - F). (18 ) If inequality (37 ) is satisfied the roots (the reciprocal of the relaxation times) can be approximated to : d? - d2 A1 = dl (19 ) dz 22=-. The A2 root describes the slow shift from one conformation of the free enzyme to the other, forced by the catalytic process itself. Its expression is Eqn (9). The parameters a, b,..., h are explicitely given in Table 2. The expression of the induction time z can be obtained from Eqn (16 ). One has 4 where a, us, $l and $2 are groupings of rate constants, v, is the initial steady-state rate, $1 and $2 the relaxation amplitudes. For a short time interval (some seconds, see Fig.9), the linear term (vst) is negligible with respect to the two exponential terms. Moreover, the slow exponential can be approximated to e- 1.>1 and Eqn (22 ) can be rewritten as =l-a2t (23 ) (a + $2) + $2 22 t = $1 e-. (24 ) Thus, by substrating from the values of [P]/[E]o the values of [(a + $2) - $2 1-2 t] that can be obtained experimentally, one can calculate both the amplitude and the time constant of the fast relaxation (see Fig. 9). Conversely, for a much longer time interval (several minutes, see Fig. 9), the first exponential can be cancelled and Eqn (22 ) becomes : As previously, by subtracting from [P]/[E]o the values of a + v,t which can be determined experi-

12 592 J. Ricard, J. Buc, and J.-C. Meunier: Enzyme Memory. 1. Table 3. Expressions for the parameters v and 6 mentally, one can calculate the amplitude and the time constant of the second relaxation. The skilful technical assistance of Mrs Mireille Riviere is gratefully acknowledged. The senior author (J. R.) wishes to thank Dr Grossman for carefully reading the manuscript, Drs Kirschner and Lazdunski for friendly discussions. This work has been done with the financial support of the Delegation Gendrale a la Recherche Scientifique et Technique to J. R. REFERENCES 1. Rabin, B. R. (1967) Biochem. J. 102, Ainslie, R. E., Shill, J. P. & Neet, K. E. (1972) J. Biol. Chem. 247, Shill, J. P. & Neet, K. E. (1975) J. Bid. Chem. 250, Rubsamen, H., Khandker, R. & Witzel, H. (1974) Hoppr- Seyler's 2. Physiol. Chem. 355, Ricard, J., Meunier, J.-C. & Buc, J. (1974) Eur. J. Biochem. 49, Meunier, J.-C., Buc, J., Navarro, A. & Ricard, J. (1974) Eur. f. Biochem. 49, Whitehead, E. P. (1970) pro^. Biophys. Mol. Biol. 21, Meunier, J.-C., Buc, J. & Ricard, J. (1971) FEBS Lett. 14, Darrow, R. A. & Colowick, S. P. (1962) Methods Enzymol. 5, Cleland, W. W. (1967) Adv. Enzymol. 29, Wong, J. T. F. (1975) Kinetics of' Enzyme Mechanisms, Academic Press, London 12. Mannervik, B., Gorna-Hall, B. & Bartfai, T. (1973) Eur. J. Biochem. 37, Cha, S. (1968) J. Bid. Chem. 243, Slein, M. W., Cori, G. T. & Cori, C. F. (1950) J. Biol. Chem. 186, Hammes, G. G. & Kochavi, D. (1962) J. Am. Chem. Soc. 84, Kirschner, K. (1971) f. Mol. Biol. 58, Kirschner, K., Gallego, E., Schuster, I. & Goodall, D. (1971) J. Mol. Bid. 58, Wolfenden, R. (1969) Nature (Lond.) 223, Secemski, I. I. & Lienhard, G. E. (1971) J. Am. Chem. Soc. 93, Lienhard, G. E., Secemski, I. I., Koehler, K. A. & Lindquist, R. N. (1972) Cold Spring Harbor Symp. Quant. Bid. 36, Jenks, W. P. (1975) Adv. Enzymol Wong, J. T. F. & Hanes, C. S. (1962) Can. J. Biochem. Physiol. 40, J. Ricard, J. Buc, and J.-C. Meunier, Labordtoire de Biochimie Vegetale, U.E.R. Scientifique de Luminy, Universite d' Aix-Marseille 11, 70 Route Leon-Lachamp, F Marseille-Cedex-2, France