Computer Simulation Studies with Yeast Hexokinase and Additional Evidence for the Random Bi Bi Mechanism*

SUMMARY The mechanism of action of yeast hexolrinase was studied by computer simulation of the steady state random Bi Bi mechanism. The initial rate, product inhibition, and competitive inhibition studies were found to be consistent with the enzymic mechanism being steady state random but approximated by the rapid equilibrium assumption. Results of studies on both the forward and reverse reaction are presented and much of the seemingly conflicting data in the literature regarding the mechanism appears to be resolved. Investigation of the kinetic mechanism of yeast hexokinase (


Investigation
of the kinetic mechanism of yeast hexokinase (ATP :n-hexose 6-phosphotransferase; EC 2.7.1.1) has been an area of active research in recent years.
A number of new kinetic techniques have been applied to the study of yeast hexokinase in order to elucidate its mechanism of action. Studies of competitive inhibition (l-3), product inhibition (l-6), isotope exchange at equilibrium (7), alternative substrates (1,2,8), alternativeprcducts (l), and isotope competition (8) were all used during their development to investigate the hexokinase reaction. All initial rate studies, as well as the isotope exchange studies have led to the conclusion that the mechanism involves formation of a ternary complex of enzyme-sugar-nucleotide, but there is considerable disagreement as to whether the substrates bind in a random or ordered sequence. Hammes and Kochavi (4) suggested in 1962 from initial rate studies that the reaction mechanism was ordered with glucose the obligatory first substrate. Fromm and !&ewe (1) proposed in 1962 from similar initial rate studies, that the mechanism was random with all steps in the reaction sequence in rapid equilibrium relative to the interconversion of the ternary complexes. Ottolenghi has since indicated that Hammes' data are also consistent with a random mechanism (9). Subsequent support for the random mechanism involved studies with alternative substrates (1, 2, 8), competitive inhibitors (l-3), isotope competition (8), isotope exchange (7) fluorescence quenching of hexokinase (2), as well as experiments on the ATPase activity of yeast hexokinase (10, 11). Other binding and kinetic experiments (5, 12) have been int'erpreted as supporting the ordered mechanism proposed by Hammes and Kochavi (4). Some of the data in these studies are different from those obtained in our investigations, as well as studies in other laboratories, and these discrepancies will be discussed in this report. This paper will present a reinterpretation of the yeast hexokinase mechanism in light of the steady state random pathway of enzyme and substrate interaction which will hopefully conclude the controversy concerning the mechanism.
EXPERIMENTAL PROCEDURE

Materials
Crystalline yeast hexokinase (specific activity 200 e.u. per mg) from Calbiochem was chromatographed as described by Schulze,Gazith,and Gooding (13) and found to exist as the S forms of t'he enzyme.
The P-I and P-II forms of hexokinase were prepared as described by Schulae et al. (13) and had specific activities of 200 and 600, respectively.
Uniformly labeled [i4C]glucose was obtained from Schwarz BioResearch and had a specific activity of 250 mCi of r4C per mM.
[i4C]Glucose g-phosphate was prepared from the glucose by use of hexokinase with no added cold glucose and an excess of ATP. The

Applicability
of Steady State Random Mechanism to Yeast Hexokinase Reaction-The isotope exchange studies on hexokinase are consistent with the yeast hexokinase mechanism being random (7) but the ATP f ADP exchange is approximately twice the glucose = glucose 6-phosphate exchange. Fromm, Silverstein, and Boyer (7) suggested that this result was caused by the mechanism being not truly rapid equilibrium random and that release of glucose g-phosphate also influenced the velocity of the reaction. Cleland (17) has also pointed this out and suggested that a steady state mechanism may appear to approximate a rapid equilibrium one in initial rate studies and with product inhibition.
A similar situation is observed with Escherichia coli galactokinase which was studied in Cleland's laboratory and was shown to be random from initial rate studies (18).
The ATP F! ADP exchange was approximately 24 times the galactose = galactose-phosphate exchange and it was concluded that, although the rapid equilibrium assumption was clearly not applicable to this enzyme, the steady state mechanism was well described by the rapid equilibrium assumption.
By computer simulation, Wratten and Cleland (19) have suggested several cases in which the reciprocal plots for a steady state random mechanism will appear linear. The random mechanism for a two-substrate enzyme is shown in Scheme I below.
The steady state rate equation for this mechanism has been derived and has the following form KIAB + KzAB2 + K,A2B ' = K, + K,A + KGB + KvAB + KsA2 + K9B2 (1) Experimental values for the various dissociations were obtained from the kinetic data of DeLaFuente and Sols (6) for pH 7.0 and a value of 10 was taken for the ratio of kg: klO. This value is close to that calculated from the Haldane equation and is consistent with the equilibrium constant.
To calculate a relative velocity, the other unimolecular rate constants were given values relative to ks and the bimolecular rate constants are defined by the assumed dissociation constants. Cleland has suggested that, in view of the different rates of exchange, the rate constants for the dissociation of the nucleotides from the ternary complexes are at least 4 to 5 times the rate constant for the interconversion of the ternary complex and also the rate constants for the dissociation of the sugars from the ternary complexes (18). With the rate constants given in Fig. 1, the theoretical reciprocal plot depicted in Fig. 1 is obtained. It was found that as the outer rate constants (k2, kd, klS, kl7) were made larger than kg, the lines became more linear.

Velocity
(v) is the relative number calculated by the computer.
it was found that when the dissociation constants were varied independent of each other, the calculated plots also simulated the experimental situation.
Linear plots are also obtained if either kg or kir and kir are made much lower than the other rate constants.
It appears that even when the assumptions made above to reduce the complicated rate equation to Equation 2 are not satisfied, but the values are about equal, the reciprocal plots will remain linear.
Contrary to a previous report from this laboratory (l), it was found during this study that ADP is a noncompetitive inhibitor of ATP, a finding consistent with results from other laboratories (4,5,20).
In order to determine that such an effect is consistent with the steady state mechanism, the steady state equation describing product inhibition of hexokinase was derived. The formation of an abortive complex, EBC, in the presence of product C was assumed.
The postulated interactions are The total rate equation has 45 numerator and 672 denominator terms which include many squared and cubed concentration terms.
fiy assuming that k2 > k5B, kd > k7A, k16 > BIB, k17 > k14C, and kq > k& the rate equation may be reduced to the following form2: 2The presence of a substrat,e term in the identities can serve only t,o lower the product of the rate constant times substrate, i.e. it is unlikely that ksB > kg since B will rarely approach 1 M. Thus, it is expected that kz > khB will exhibit a greater inequality than kz > kg.
A double reciprocal plot of the calculated relative initial velocity of yeast hexokinase versus the reciprocal of ATP concentration at different assumed levels of ADP. Glucose was assumed constant at 0.2 mM and ATP was varied from 0.1 to 1.0 mM. ADP concentrations were 0 (l), 2.5 (2), 5 (3), and 10 rnht (4). The rate constants were the same as in Fig. 1 with the addition of k19, 100,000 M-I set-r; kzo, 500 see-I; kn, 20,000 bP set-*; kzt, 100 set-r. ATP xas assumed constant at 0.2 mM and glucose was varied from 0.1 to 1.0 mM. ADP concentrations were 0 (l), 2.5 (2), 5 (3), and 10 mu (4). The assumed rate constants were the same as for Fig. 2.
Once again the K's represent various combinations of rate con stants.
The total rate equation was programmed as described for the initial rate simulation and representat#ive plots are depicted in Figs. 2 and 3. It can be seen that the product is a noncotnpetitive inhibitor of both substrates and this is t'rue whether C is ADP or glucose 6-phosphate.
Only when the unimolecular rate constants for the abortive complex formation were much smaller than the outer rate constants, would the inhibition approach competitive.
Various combinations of dissociation constants and rate constants were tested and found to generally give similar results to those shown.
The simulations also predict that the effect of competitive in hibitors are the same as suggested by the rapid equilibrium assumption.
That is, a competitive inhibitor of one substrate will be noncompetitive relative to the other substrate. can be assumed to have a steady state mechanism but the equilibrium assumption will generally describe the initial velocity results observed.
Another conclusion that can be drawn from the computer simulations concerns the effect of pentoses on hexokinase.
We have recently shown (11) that lyxose induces substrate inhibition with ATP at much lower concentrations of ATP than is seen in the absence of lyxose.
This effect was studied in both the hexokinase reaction and the lyxose activation of the ATPase activity of hexokinase.
It was suggested that lyxose induces a conformational change that allows 2 moles of ATP to bind per mole of enzyme. It was found that lyxose binds both to free enzyme and the enzyme-ATP complex and it was concluded that the lyxose effect supported the random mechanism.
The computer simulation shows that, when the outer rate constants, k,, and kd, are set equal to kg, substrate inhibition at concentration levels similar to those seen in the lyxose studies could be observed.
The suggestion would be that lyxose causes a conformational change in the enzyme which changes the relative magnitudes of different rate constants, and that this conformation is relatively stable with subsequent replacement of lyxose by glucose. Such an interpretation is consistent with the effect of lyxose on the binding of ADP observed by Womack and Colowick (21). ADP binds to the enzyme very tightly in the presence of lyxose and the tight binding persisted even after the lyxose was exhaustively dialyzed away. The untreated free enzyme has a low affinity for BDP, and Colowick,Womack,and NeJsen (12) have also shown that only 1 mole of ADP is bound per mole of enzyme in the presence of lyxose.
Thus, this interpretation is somewhat more attractive than the suggestion that 2 moles of ATP are bound to explain the substrate inhibition induced by lyxose.
Another situation that could apply to controlled enzymes from the simulation studies is that, when the outer rate constants are smaller than kg, substrate activation is predicted. It might be inferred that a shift in rate constants caused by binding of another ligand might have some influence in changing the response of an enzyme with a random mechanism to a @type response making it much more susceptible to regulation.
Kinetic Studies oj Forward Reaction of Yeast Hexokinase-It has been suggested (22) that the presence of different degraded isozymes present in the crystalline preparations of yeast hexokinase (23) could give rise to the different kinetic results seen in different laboratories.
To test this possibility, a complete study of the forward reaction of hexokinase was made with both the com- Competitive Noncompetitive Noncompetitive mercial crystalline S enzyme and with the P-I enzyme prepared as described by Schulze et al. (13).
Initial rate studies gave only quantitative differences in the Km values and the same effect was found for inhibition by AMP, ADP, ATP4-, N-acetylglucosamine, lyxose, and glucose 6-phosphate, as well as AMP in the back reaction.
The experiments involving the forward reaction have been published previously (l-4, 6), although under somewhat different conditions, and the data will not be presented here. The type of inhibition effected by various inhibitors used in this study is summarized in Table I. Some laboratories have published results at variance with some of these data.
The discrepancies will be considered under "Discussion." Kinetic Studies of Back Reaction-In order to gain additional support for the random mechanism, the effect of AMP on the back reaction and an isotope competition experiment were carried out with hexokinase.
The back reaction has been shown to be sequential by Fromm et al. (7) and DeLaFuente and Sols (6). The reaction conditions previously worked out for the initial rate studies (7) were used in these experiments, which were made at ' pH 6.5.
AMP is shown in Fig. 4 to be a competitive inhibitor of ADP in the back reaction.
This effect is similar to that observed in the forward direction.
If the reverse reaction were ordered with glucose 6-phosphate as the first substrate on the enzyme, the inhibition by AMP relative to glucose 6-phosphate would be uncompetitive (1). If, however, the reaction were random, it would be noncompetitive (l), as is shown in Fig. 5. It should be emphasized that the concentration of glucose 6-phosphate in this competitive experiment was well below its K,, a situation which should reveal any binding of AMP to the glucose 6-phosphate site which would give rise to noncompetitive inhibition in an ordered mechanism.
Further confirmation of the back reaction mechanism was obtained from an isotope competition experiment. The theory for this technique and its application to the forward reaction has been presented previously (8). The experiments involve use of a radioactive substrate and observation of the effect of various levels of unlabeled substrate on the production of labeled product. The unlabeled substrate is a competitive inhibitor of the same labeled substrate while inhibition relative to the other substrate gives dBerent results for the different sequential mechanisms. For an ordered mechanism, the inhibition caused by the second substrate will be nonlinear relative to the isotope of the first substrate.
With the random mechanism inhibition by either substrate wJ1 be noncompetitive relative to the other substrate. The results of an isotope competition experiment with [14C]glucase 6-phosphate are shown in Fig. 6. As predicted for a random mechanism, the inhibition relative to ADP is noncompetitive. The inhibition studies on the back reaction are consistent with either the random mechanism or with ADP being the first substrate in an ordered mechanism.
For an ordered mechanism to be consistent with the suggested data, the mechanism would have to be one in which the product of the substrate which adds to the enzyme first also dissociates first from the enzyme. This is unlikely (7), since it would require the exchange rates for both pairs of substrates and products to be depressed as the concentration of the other pair was raised.
This decrease in exchange for both pairs is unique for this mechanism since in the normal ordered mechanism, the exchange of B s D is not depressed with increasing concentrations of A and C. Pre-equilibrium Isotope Exchange with Yeast Hexokinase- Kosow and Rose (20)  was allowed to proceed to approximately 20% completion and it was found that a 22'3S exchange of ADP into ATP relative to the total reaction occurred. It was suggested that ADP reacted with the enzyme-glucose 6-phosphate complex to give this result.
From ADP inhibition experiments they suggested that the maximum contribution of the pathway involving dissociation of glucose 6-phosphate from the ternary complex was 36% of the VmaX.
If the release of glucose 6-phosphate from the ternary complex were not rate limiting and the release of ADP from the binary complex was relatively slow, an exchange of glucose 6-phosphate into ATP or glucose should be observed similar to that described for the ADP $ ATP exchange. Kosow and Rose (20) incubated [32P]glucose 6-phosphate with glucose, MgATP and hexokinase P-II.
Pyruvate kinase and P-enolpyruvate were added to keep the concentration of ADP very low. No incorporation of the 32P into ATP could be detected.
Under the experimental conditions described by Kosow and Rose (20), with the omission of the ATP-regenerating system, we studied the effect of adding 5 m&f ADP to the reaction mixture. Under conditions when approximately 20% of the glucose was phosphorylated, only a 0.02% conversion of glucose 6-phosphate was observed. This observation is in contrast to the suggestion of Kosow and Rose (20) that the back reaction is significant relative to the forward reaction at pH 8.0. Although the use of the Haldane relationship for this pH gives a (V,,,, forward) to (V,,,, reverse) ratio of 104, the presence of ATP and glucose at essentially saturating levels in their experiments would make the ratio much greater. The suggestion made from the isotope exchange studies that glucose 6-phosphate release affected the V,,, of the reaction (7) is quite consistent with the effect observed by Kosow and Rose (20).
It is reasonable to infer from the isotope data that the release of glucose 6-phosphate is slow relative to the release by guest on July 9, 2020 http://www.jbc.org/ Downloaded from 6616 Kinetics of Yeast Hexokinase Vol. 246, so. a1 of ADP. Thus, when ADP was present, it could react with enzyme-glucose 6-phosphate causing the exchange observed. The concentration of enzyme-ADP, on the other hand, would be small enough in the absence of added ADP to prevent the exchange involving glucose B-phosphate. Their suggestion of a preferred pathway is the same as was proposed from the isotope exchange study by Fromm et al. (7).
It is interesting to consider the facile reversibility of yeast hexokinase relative to the "irreversibility" of mammalian hexokinases (24). At pH 7.6 the ratio (V,,,, forward) to V,,,,, reverse) is approximately 13,200 based upon the Haldane relationship with brain hexokinase. It has recently been shown that at elevated levels of Mg2+, demonstrable reversibility of the brain hexokinase reaction can be achieved (25). This effect is clearly not at variance with the concept of microscopic reversibility and indicates that demonstration of reversibility with an enzyme is related not only to the equilibrium constant for a reaction but also to the Michaelis constants.

DISCUSSION
Although the evidence supporting the random mechanism for yeast hexokinase is considerable, several authors have argued against this mechanism for various reasons. In this discussion we will consider these arguments and show that the results of almost all studies are consistent with a limiting steady state random mechanism for yeast hexokinase. Cohn reported in 1963 (26) that an enhancement of the proton relaxation rate of water was observed with yeast hexokinase and MnADP only in the presence of glucose. This effect has been often quoted as support for the ordered mechanism, but as we suggested in 1964 (2) and as Mlldvan and Cohn have pointed out in a recent review (27), lack of enhancement does not rule out binding.
It only implies that if binding occurred it was at an external site on the enzyme and the motion of the water in the coordination sphere of the metal ion was not altered upon binding.
Cohn also reported that when better defined preparations of hexokinase are used, no definitive results could be obtained in the enhancement studies (27). Another argument against the random mechanism has been results obtained from the elegant binding studies done by Colowick et al. (12). The free enzyme was found to have a low affinity for nucleotides while it had an appreciable affinity for glucose.
The K, for ATP was about 0.2 mM and the dissociation constant was about 5 mM. The K, for the ATPase activity of hexokinase was also 5 InM (10, 28).
For a rapid equilibrium random Bi Bi mechanism, the Kdiss for either substrate should equal the K,,, when the initial rate reciprocal plots intersect on the l/S axis, which is the case for most studies with yeast hexokinase (1,5,6). The discrepancy in the experimental values was cited as evidence against the random mechanism.
The simulations of the steady state random mechanism of Cleland and Wratten (19) involved the mechanism of Scheme II.

ExEA& k,
EAB -E + PRODUCTS SCHEME II When ks and kg are not nnwh larger than kg, :ts was concluded from the exchange data and the computer simulations presented under "Results," the dissociation constant for the EA complex (Ki,) is kz(kc + kg + kg)/kT(kc + k,) and the K, for il is (kc + k,)/kb. For the rapid equilibrium mechanism, Ki,, = X-p/k, am1 K, = ks/kT. Therefore, Ki, in the steady state mechanism is not simply the dissociation constant obtained from binding studies, but is a complex constant that depends on the rate tollstants associated with the binding of A to EB. Similar interpretation can be made concerning the K, for ATP in the ATPase reaction depicted in Scheme III.
In this case the Km for ATP is simply k2/kl and should equal the binding constant as it does experimentally (23).
E+ADPtPi SCHEME III Studies of Gazith et al. (23) also show that the binding coustant for glucose is higher than the K,, determined kinetically. Similar arguments can be made as were made for the ATPbinding observations. The intersection of the reciprocal plots on the I/S axis would appear to be a fortuity since it implies that, (ks + k9)(k6 + kg) 2 kz(ks + kg + k,) and not' that, the substrates bind independently of each other. The binding data in which the presence of lyxose greatly stimulates the binding of nucleotides (12) are also consistent with this idea. It is re:t8011able to assume that binding of either substrate enhances the binding of the other substrate.
This has been shown to be the case for both creatine kinase (29) and galactokinase (18), whic~h both exhibit random mechanisms.
Another criticism has been advanced by Koat et al. (5) regarding the isotope exchange at equilibrium studies. They have suggested that the ATP and ADP concentrations were not raised to a high enough level to see a depression of the eschange that would be expected for an ordered mechanism.
The fact that the ATP and ADP concentrat,ions were nearly saturating would make this suggestion unlikely as was pointed out earlier (3).
A more reasonable criticism of the exchange studies would have been the suggestion of Hanes and Wong (30)  They suggest that an active ternary complex is formed only when ATP binds to an enzyme-glucose complex. The abortive ternary complex is used to explain the lack of depression of exchange, but this assumption is tenuous since the formation of the abortive ternary complex will not contribute to the exchange, except to depress it (31).
The kinetic results summarized in Table I have been shown to be consistent with the random mechanism, although the type of inhibition by glucose B-phosphate, ATP4-, and AMP has been disputed.
ATP4-has been shown to be a competitive inhibitor of MgATP by Kosow and Rose (20), Hammes and Kochavi (4), Bohensack and Hofmann (32), and in studies with both Km and high levels of glucose in this laboratory.
These studies involved widely varying conditions and assay methods with both the S and P forms of the enzyme. Noat et al. (33) suggest, however, that STP4-i,s not competitive with MgATP at K, levels of glucose, but is at saturating levels of glucose.
In our studies at similar concentrations of glucose we obtained competitive inhibition. The small deviation from competitive inhibition seen by Noat et al. (33) may be attributed to the experimental error inherent in their use of a pH-stat to determine initial velocities (3). Hammes and Kochavi (4) suggested that ATI'd-is an uncompetitive inhibitor relative to glucose, a result that would support the ordered mechanism, but their data show that when the magnesium level is constant, the lines of the reciprocal l/glucose plots are intersecting at different levels of excess ATP. This noncompetitive inhibition is consistent with studies of Noat et al. (33), as well as the results of the present study. Noat et al. (33) have also indicated that AMP is a competitive inhibitor of ATP only at high concentrations of glucose. This effect was not observed in the present investigations and further evidence against this possibility was found in the study of AMP inhibition of ADP in the back reaction. It was competitive although the glucose 6-phosphate concentration was well below its K,.
A reasonable explanation of the results of Noat et al.
(4) 6617 (33) involves the level of free magnesium they introduce in their AMP experiments.
They have shown that free magnesium is an inhibitor of the reaction, competitive with ATP and noncompetitive with glucose. The inhibition by magnesium can be explained by considering it eit,her to be a product of the reaction or by its preventing the binding of ATP by complex formation. As glucose and ATP concentrations are increased, the inhibition by excess magnesium will be less apparent.
As their data show and from experiments in this laboratory, when the free magnesium is greater than from 3 to 5 mM, considerable inhibition results.
In the experimenm with AMP they added magnesium in equal molar concentration to the added AMP. The low affinity for magnesium by AMP (15) means that at most only 50% of the magnesium was chelated by AMP. From 5 to 20 mM AMP and from 5 to 20 mM magnesium was used in these experiments.
The study of AMP inhibition is thus comphcated by two types of inhibition which readily explains their data. The reduction of the inhibition by higher concentrations of glucose is consistent with glucose reducing the amount of inhibition by magnesium as is also shown by the results of Noat et al. (33).
The study of ADP inhibition of yeast hexokinase by Kosow and Rose (20) also involved use of inhibitory levels of Mg2+. The nonlinear intercept effects they observed in double reciprocal plots with inhibition relative to MgATP are difficult to interpret since they involve relatively high levels of magnesium and ADP.
The fact that the different inhibited reciprocal inhibition plots do not intersect at a common point indicates that more than a single inhibitory effect is being observed. The inhibition by glucose B-phosphate has given considerably different results. We, along with DeLaFuente and Sols (6) have found it to be a noncompetitive inhibitor of both glucose and ATP, but Hammes and Kochavi (4) and Noat et al. (5) have suggested it to be competitive relative to glucose and noncompetitive with ATP. The noncompetitive inhibition is consistent with the steady state mechanism particularly since the phosphorylated product should bind at both substrate sites. Wettermark,Borglund,and Brolin (34) have shown that when glucose is held extremely high, 220 IIlM, glucose 6-phosphate is a by guest on July 9, 2020 http://www.jbc.org/