Eur. J. Biochem. 51,437-447 (1975)
Computer Simulation Study of Hexokinase I1 from Ehrlich Ascites Cells Lillian GARFINKEL Moore School of Electrical Engineering, William Pepper Laboratory, Department of Pathology, and Department of Biophysics University of Pennsylvania, Philadelphia (Received July 9/September 24, 1974)
A study of the mechanism of hexokinase I1 from ascites cells and the effects of its binding to mitochondrial membranes has been carried out by computer simulation. This is based on experimental data of Kosow and Rose and of Gumaa and McLean, and the theoretical methods of Cleland. For the soluble enzyme the mechanism is random with ternary product-inhibition complexes; when bound to mitochondria, the mechanism becomes ordered-on, random-off, as the binding of ATP to the free enzymes becomes negligibly slow. The requirements of experimental data for mechanistic studies are discussed.
This study was undertaken as part of a continuing stimulation study of glycolysis [l - 31. Its original aim was to elucidate the transient behaviour of hexokinase where the enzyme appears to be insensitive to glucose 6-phosphate inhibition at early time intervals. Further investigations has required attention to the mechanism of the enzyme, and therefore to relevant initial velocity studies, as well as its effective physiological regulation. Hexokinase is of considerable interest as the first enzyme of glycolysis. Its literature is very large, almost overwhelming: there were over 500 entries in the 1967- 72 five-year Index of Chemical Abstracts alone. There are a number of studies of mechanism, but many were probably made with isozyme mixtures using steady-state methods of analysis which apply to single enzymes. An important source of disagreement is the actual concentration of the reactive species of ATP, its chelate with Mg2+.Morrison and Heyde have pointed out that this is a function of type of buffer, ionic strength, pH, and the presence of other enzymes [4]. This question becomes even more important since the availability of Mg2+ in vivo appears to be an important control [5,6].
different relative proportions (which may vary with physiological conditions, such as insulin level). Type I1 is the isozyme most subject to such variations; it is also less stable than other isozymes under laboratory conditions, being sensitive to proteases, to heating [7] and to freezing [8]. Dog heart hexokinase undergoes successive changes in kinetic properties during storage [9]. Stability of the enzyme in storage requires the presence of 10 mM glucose [7]; in its absence rat heart hexokinase lost 80% of its activity within 24 h [81. b) Another unusual feature of mammalian hexokinase is its irreversibility, which interferes with both initial velocity studies and radioactive exchange studies [lo]. This necessitates using inductive reasoning in attempts to synthesize mechanisms from experiments which furnish only a part of the necessary data. c) Hexokinase association with both the inner and outer mitochondrial membranes is well known in a variety of tissues, and has been widely studied as a novel metabolic control mechanism. It was first reported for ascites hexokinase by Rose and Warms [ll],who expressed the relation between soluble and bound hexokinase as : Binding sites
Relevant Known Properties of the Enzyme
a) Hexokinase from a number of tissues contains isozymes of differing kinetic properties and also Enzyme. Hexokinase (EC 2.7.1.1)
Eur. J. Biochem. 51 (1975)
+ soluble enzyme + Mg” e Complex-bound enzyme.
Hexokinase activators, such as inorganic phosphate, favored binding while inhibitors such as glucose 6-phosphate and unchelated ATP favored
Simulation of Hexokinase
438
solubilization. This is very similar to the allosteric model of Rabin [12], if the bound and soluble forms have different kinetic properties. Results of studies on mitochondrial hexokinase have not been unanimous. A number of authors have reported differing effects of mitochondrial binding on the kinetic properties of the enzyme [S, 13- 151. Although glucose 6-phosphate is the strongest known inhibitor of mammalian hexokinase, it does not appear to be operating at early time intervals when ascites cells are incubated with glucose [16- 181; however, ADP appears to be controlling hexokinase here [17,19]. The finding that glucose 6-phosphate is inhibitory in a cell-free extract suggested to Gumaa and McLean that a change in hexokinase binding might be occurring. Further evidence for this view was the close agreement between the 18-s half-life for release of mitochondrially bound hexokinase reported by Rose and Warms [ll], and the 15-s interval for the onset of “normal” inhibitory behavior by glucose 6-phosphate [20]. The results of studies of supposedly mitochondrially bound hexokinase are often contradictory, partly because the actual binding state of the enzyme at the time of measurement may be unclear, and possibly because the enzyme bound to mitochondria may be exposed to concentrations of ATP and glucose different from those in solution. As an extreme illustration, hexokinase I is described as “latent”, since it is normally inaccessible to its substrates in the cytosol [21,22]. (At times the presence of this isozyme may interfere in studies of hexokinase 11.) d) The “allosterism” of hexokinase is based on the non-competitive inhibition of glucose by glucose 6-phosphate, the specificity of glucose 6-phosphate as an inhibitor, and the weak inhibition by fructose 6-phosphate and mannose 6-phosphate [23]. It has recently been found (M. J. Achs and D. Garfinkel, personal communication) that more than one molecule of glucose 6-phosphate may be involved in inhibiting heart hexokinase, and that M$+ and inorganic phosphate play complex roles in regulating it. Kosow and Rose have recently reported negative co-operativity towards ADP inhibition [24]. However, under most conditions hexokinase behavior is relatively straightforward, and brain hexokinase is not dissociated by agents which dissociate other allosteric enzymes into subunits. e) Some enzymes show delayed response to rapid changes in ligand concentration. Frieden [25,26] has named these enzymes hysteretic, suggested the delay as involving protein - protein interaction (which is slower than interaction between proteins and small molecules), and discussed their possible significance in metabolic control. The most recent thinking about the
insensitivity of mammalian hexokinase to inhibitors, especially glucose 6-phosphate, has centered on hysteretic effects. A preliminary study of glycolysis in ascites cells subjected to a temperature jump (B. Chance, P. K. Maitra and I. Y. Lee, unpublished) indicates that hysteresis is important in vivo. f) The mechanism of hexokinase is still undefined, despite many studies, both because of its unusual properties, and the lack of agreement on how to define an enzyme mechanism, which is discussed below.
METHODS Computer Programs Computations were performed on the PDP-6 and PDP-10 computers of the University of Pennsylvania Medical School Computer Facility, using two programs that give solutions to steady-state initial-velocity problems very rapidly and economically. These are based on matrix inversion [27] and rate laws [28]. The first program is especially convenient, since computed results are actually displayed as doublereciprocal plots together with the experimental data. Required input for these programs consists of chemical reactions, rate constants, substrate and modifier concentrations, and enzyme activities; it is easy to make many trials in order to fit the data. The hyperbolic curve-fitting program of Cleland [29] was used on the data for the soluble enzyme to determine the kinetic constants in the uninhibited reactions, and to determine if any other inhibition patterns than those reported were feasible. When this study was close to completion, the models were subjected to sensitivity tests : individual rate constants were varied one at a time. This was necessary because the final rate laws were too complicated for paper-and-pencil manipulation. Experimental Data Base The experimental data on which this simulation was based were the initial velocity studies of Kosow and Rose for hexokinase I1 from Ehrlich ascites cells in both the soluble and mitochondrially bound states [30,31]. This work indicated a sequential mechanism with random substrate addition, since competitive inhibitors of one substrate were usually noncompetitive inhibitors of the other. Inhibition constants in both the soluble and mitochondrially bound states were similar, but K, values for both substrates were slightly lower when mitochondrially bound. After the model had been constructed on the basis of this published work, it was compared to the raw Eur. J. Biochem. 51 (1975)
L. Garfinkel
439
laboratory data, which Dr Rose then kindly made available, and found generally to fit it much better. The inhibition patterns observed by these workers plus the finding by others that mannose competes with glucose are tabulated in Table 1. The finding for anhydroglucitol 6-phosphate vs ATP in the bound form is dubious, as it is based very heavily on extrapolation. In general, both Ki slope and Ki intercept were similar for both enzyme forms. This inhibition pattern seems to indicate a random mechanism. Theoretical Methodology The simplest and most familiar method for analyzing kinetic enzyme data is the double-reciprocal Lineweaver-Burk plot. A problem with this method is the human tendency to assign more weight to the inherently less accurate points far from the axes when drawing the resulting lines. The fact that both V and K, (determined from their intercepts along the axes) are essentially determined by extrapolation may obscure the fit to the data. Inhibitions are considered competitive, uncompetitive, or non-competitive according to how these double-reciprocal lines do (or do not) intersect. This is the basis of a number of schemes for determining reaction mechanisms : Reiner, Alberty and Cleland have all developed these methods. This study has relied heavily on the work of Cleland [32]. Often there is information other than that from steadystate kinetics (e.g., direct-binding studies) which require explanation. Thus the only generalization possible is that a given mechanism should explain as many facts as possible.
RESULTS The model chosen to represent the data is given in Table 2. The mechanism for the soluble enzyme was random bi-bi. Product release rather than conversion of ternary complex was made the limiting step on the basis of radioactive studies [31]. Gulbinsky and Cleland [33] have shown that the presence of linear double-reciprocal lines does not require conversion of ternary complexes to be the limiting step. The observed inhibition patterns could not be matched by any of the reaction mechanisms given by Cleland, if the non-varied substrates were present in saturating concentrations. The formation of ternary dead-end complexes was invoked in an attempt to explain the observed inhibition patterns. A typical chemical equation is : ADP
+ Enz . Glc = Enz . Glc . ADP
where Enz . Glc . ADP denotes the ternary dead-end complex of hexokinase, glucose and ADP. For such Eur. J. Biochem. 51 (1975)
Table 1. Nature of the inhibition of various inhibitors against ATP and glucose for soluble and bound hexokinase II N = Non-competitive; C = competitive Inhibitor
Soluble
ADP AMP Glucose 6-phosphate Mannose Anhydroglucitol 6-phosphate
Bound
ATP
glucose ATP
N
N
C C
C
N N N
N
N N (C)
-
-
C
N
N
N
N C
glucose
Table 2. Ascites hexokinase model Even-numbered reactions are the reverse of the preceding odd-numbered reactions. Enz = hexokinase, Glc = glucose, Glc-6-P = glucose 6-phosphate, anGlc-6-P = anhydroglucitol 6-phosphate. The rate constants are either first order (min-') or second order (1 mol-' min-') Chemical equations
Rate constants soluble
bound
Enz
3 x105 2.2 x 103
2.5 x lo7 2.2 x 103
. ATP
I xi05 1.3 x 104 2.5 x lo6 1 . 9 104 ~
1 x106 4.4 x 103 1 XlOO I .9 x lo4
1.8 x 107 4 x103
5 x108 4 xi03
2 xi05 I xi03
2 xi05 i xi03
1.5 x 104 1 XI06 2 xi03 2 xi05
1.5 x 104 1 XI06 2 xi03 2 xi05
2.4 x 103 I xi05 8.5 x 10' 5 xi05
2.4 x 103 I x105 8.5 x 10' 3 xi05
1 xi07 1.4 x 104
I xi07 1 . 4 104 ~
5 XI06 8 XI02
5 xl06 8 xl@
1 XI08 2 xi03 4.5 x 105 I xi03
1 XI08 2 xi03 4.5 x 105 I xi03
4.5 x 105 1 xi03
4.5 x 105 I xi03
+ Glc = Enz . Glc Enz . Glc + ATP = Enz . Glc + ATP = Enz . ATP Enz . ATP + Glc = Enz . Glc
Enz
. ATP
Enz . Glc . ATP . ADP
Enz . Glc-6-P
=
Enz . Glc-6-P . ADP . Glc-6-P + ADP
=
Enz
Enz . Glc-6-P = Glc-6-P Enz . Glc-6-P . ADP . ADP + Glc-6-P Enz . ADP
=
ADP
ADP + Enz . Glc . ADP
=
+ Enz
Enz
+ Enz
=
Enz . Glc
Enz . Glc + anGlc-6-P . Glc . anGlc-6-P
=
Enz
Enz + anGlc-6-P = Enz . anGlc-6-P
+ AMP = Enz . AMP Enz . Glc + AMP = Enz . Glc
Enz
. AMP
440
Simulation of Hexokinase 1600 1
1000 800
1
I
0
- 4 - 2
2
4
6
8
1
0 2 4 I/[Glucose] (rnM-’)
1 /[Glucose] (rnM-’)
Fig. 1. Reciprocal plot ojanhydroglucitol6-phosphateinhibition vs glucose for the soluble enzyme. (0) Published data, (W, A) raw data, all from three different experiments. Concentrations as reported by Kosow and Rose. ATP-Mg was 5 mM (hexokinase activity was 5 x unit or 0.03 mM for the simulated curve; velocity units are dA,,/min) [30]
6
- 6 - 4 - 2
0
8
Fig.3. Reciprocal plot of ADP inhibition vs glucose for the soluble enzyme. (0)Published data, (D) raw data. Concentrations as reported by Kosow and Rose (ATP-Mg was 5 mM; hexokinase “activity” was 0.038 mM for the simulated curves) ~301
1 400 r
1000
-
5
‘Oo0 800
-0.6 -0.4 -0.2
0
0.2 0.4
0.6
0.8
1 .C
l / [ A T P ] (mM-’)
-0.6
-0.4
-0.2
I 0
0.2
0.4
0.6
0.8
1.0
l l [ A T P ] (rnM-’)
Fig. 2. Reciprocalplot of anhydroglucitol6-phosphate inhibition vs A T P for the soluble enzyme. (0)Published data, (El) raw data. Concentrations as reported by Kosow and Rose (glucose was 1 mM, ATP-Mg was varied between 1 and 4 mM; hexokinase “activity” was 0.032 mM for simulated curves) [30]
Fig.4. Reciprocal plot of ADP vs ATP for the soluble enzyme. (0) Published data, (0)raw data. Concentrations as given by Kosow and Rose (glucose 1 mM ; hexokinase “activity” 0.053 mM for simulated curves) [30]
an inhibition the Ki is the quotient association rate/ dissociation rate, e.g., K,,/K,, for Enz . Glc . ADP and similarly for the others. The random mechanism was chosen in part because those experimental studies which indicate an ordered mechanism disagree as to which substrate adds first. It is also based on the non-competitive
inhibition of mannose when ATP is the varied substrate. If this inhibition were uncompetitive instead and the unvaried substrate non-saturating, an ordered bi-bi or Theorell-Chance mechanism would be possible. Likewise, the present data for the bound enzyme are consistent with an iso-ordered bi-bi mechanism if anhydroglucitol 6-phosphate is assumed to behave Eur. J. Biochem. 51 (1975)
441
L. Garfinkel
2 000
1400 1800
1200 1600 1400
-
:. 1m 0 -1 \ t 0
q 2
l0OC
800
2
-
4
-
2 0 2 4 l/[Glucose](mM-’)
6
8
10
0
0
Fig. 5 . Reciprocal plot of AMP inhibition vs glucose for the soluble enzyme. (0) Published data, raw data. Concentrations as reported by Kosow and Rose. (ATP-Mg was 5 mM ; hexokinase “activity” 0.038 mM for simulated curve.) [301
(a)
1800 1600
t1
I
1400 -
/
- 1200 -
c ._
E
b 1000 -
600
/
8
/
400
1 / [ ATP] (rnM-’)
Fig. 6. Reciprocalplot of AMP inhibition vs ATP for the soluble enzyme. (0) Published data, (Kl) raw data. Concentrations as reported by Kosow and Rose (glucose was 1 mM; hexokinase “activity” was 0.03 mM for simulated curve) [30]
like the natural product glucose 6-phosphate ;however, this is the least reliable inhibitor pattern. The model presented here is probably not unique. Too many of the reaction steps cannot be defined Eur. J. Biochem. 51 (1975)
/
/ t ,
- 4 - 2
*0°
I
0
2 4 1/ [Glucose] (rnM-’)
6
8
10
Fig. I . Reciprocal plot of anhydroglucitol6-phosphateinhibition vs glucose for the bound enzyme. (0) Published data, (17) raw data. Concentrations as given by Kosow and Rose (ATP-Mg was 5 mM; hexokinase “activity” was 0.018 mM) [30]. Soluble enzyme activities range from 26-40% of total. Deviant point at 0.1 mM glucose, no inhibitor (25 % soluble), and 0.1 mM glucose and 0.038 mM anhydroglucitol 6-phosphate)
from the work that has thus far been done, primarily because of the irreversibility of the enzyme (ie.,there are few radioactive experiments). However, the overall features of this model are probably correct, and the undefined steps (no. 11- 18, Table 2) are not significant. There appear to be insufficient published data to define these steps uniquely for any mammalian hexokinase. Although it was difficult to fit the published experimental data, the fit to the raw experimental data was much better. The fit of the model to the data (both published and unpublished) is shown in Fig. 1- 6 for the soluble enzyme and Fig. 7- 12 for the bound enzyme. The two sets of published data for anhydroglucitol 6-phosphate as inhibitor with glucose and with Mg-ATP as the varied substrate (Fig. 13 and 14) were found to be inconsistent with each other, and the value of rate constant 23 (Table 3) needed to fit them varied by a factor of 200. Since these experiments were run at different substrate concentrations, the double-reciprocal lines of Fig. 13 and 14 were extrapolated slightly to yield velocities at comparable
442
Simulation of Hexokinase
-
1100
2200
1 GOO
2000
900
I800
800
1 600
700
--
c .-
E
- *0 T d 1
/
1 400
c
t
600
1200
0 -, \f o
0
T
500
5 1000
.
. -
-->
400
800
600
,A:: -0.6 -04 -0.2 0
0.2 0.4
0.6
0.8
1
.O
-0.6 -0.4 -0.2
0.2 0.4 0.6
0.8 1.0
l / [ A T P ] (rnM-’)
l / [ A T P ] (KIM-’)
Fig. 8. Reciprocal of anhydroglucitol6-phosphate inhibition vs. ATP for the bound enzyme. (0)Published data, (D) raw data. Concentrations as reported by Kosow and Rose (glucose was 1 mM ; hexokinase “activity” was 0.03 mM) [30]. Soluble activities are about 20 % of the total except for 0.4 mM Glc, no inhibitor, and 0.4 mM Glc, 0.056 mM anGlc-6-P
Fig.10. Reciprocal plot of ADP inhibition vs ATP ,for the bound enzyme. (0)Published data, (m) raw data. Concentrations as reported by Kosow and Rose (glucose was 1 mM; hexokinase “activity” 0.015 mM). Bound activities are 75 of total except for points at 1.5 mM ATP, 1 and 3 mM ADP) [30]
Table 3. Values of rate constants for inhibitions needed to ,fit data in Fig. 13 andfinal model The rate constants are either first order (min-’) or second order (1 mol-’ min-’)
I
1600
Chemical equations
Rate constants
(21) Enz . Glc + anGlc-6-P = Enz . Glc . anGlc-6-P (22) (23) Enz + anGlc-6-P = Enz . anGlc-6-P (24)
Fig. 13
final model
1x10~ 8x10’
5~ 106 8x102
1 XI08 2 x 103
1x108 2 x 103
I
- 4 - 2
0
2
4
6
8 1 0
l/[Glucose](RIM-’)
Fig. 9. Reciprocal plot of ADP inhibition vs glucose ,for the bound enzyme. (0)Published data, raw data. Concentrations as reported by Kosow and Rose (ATP-Mg was 5 mM; hexokinase “activity” was 0.021 mM). Soluble activities were generally 40% of total activity, except for a deviant point at 0.25 mM glucose, 4 mM ADP [30]
(m)
concentrations of substrates and inhibitors, and the results plotted in Fig. 15. Since substrate and inhibitor concentrations are now equal, the velocities should all be proportional to the stated activity of the enzyme samples independent of any assumed mechanism. Fig. 15 shows that they are not. Eur. J. Biochem. 51 (1975)
443
L. Garfinkel 1800
2000
1 600
0
1800 O
/
1 400
/
1600
1200 1400 L .-
t
F - 1200
1000
-3 ,w
E
T
72 l00C 0
80C \
a
c
Y
b
:80C
/
0 0
0
I . 3
2
4
6
J
8
1
0
-4
-2
0
2
4
6
8
10
l / [Glucose](rnM-’)
l/[Glucos~](mM-’)
Fig. 11. Reciprocal plot of A M P inhibition vs glucose for the bound enzyme. (0) Published data, (m) raw data. Concentrations as given by Kosow and Rose (ATP-Mg was 5 m M , hexokinase “activity” was 0.018 mM). Soluble activities range from 15 to 50 % in this experiment [30]
Fig. 13. Reciprocal plot of published data on anhydroglucitol 6-phosphate inhibition vs. glucose for soluble enzyme. Experimental data were fitted to a model that was incompatible with Fig. 14 [30]
1200
-
1400 r
1
I4O0
1
1000 -
c
L
0
-0.6 -0.4 -0.2
0
0.2 0.4
0.6 0.8
1 .0
l / [ A T P ] (rnM-’)
Fig. 12. Reciprocal plot o j A M P inhibition vs A T P for bound enzyme. (0) Published data, (m) raw data. Concentrations given by Kosow and Rose (glucose was 1 m M ; hexokinase “activity” was 0.045 mM). Soluble activities were 18 to 25 of the total [30] Eur. J. Biochem. 51 (1975)
-0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1.0
l / [ A T P ] (rnM-’)
Fig. 14. Reciprocal plot of published data on anhydroglucitol 6-phosphate inhibition vs A T P f o r soluble enzyme. (0)Published data. (a) Incompatible model that fitted the data of Fig. 13 and 1
Simulation of Hexokinase
444
with glucose 6-phosphate off last. Thus will be discussed further in connection with the sensitivity tests. While this inhibition may be non-linear at other concentration ranges, it appears to be normal for the range reported here [31,35]. AMP is a fairly weak inhibitor, and probably has little effect in situ. The data can be matched fairly well even though the type of inhibition differs some200 I what from that postulated by the experimenters (Fig. 5 and 6). 0I Interpretation of all of the data for the bound 0 0.01 0 02 0.03 0.04 0.05 0.06 0.08 is complicated by the inability to mainenzyme form [Anhydroglucitol 6-phosphate] [ m M ) tain all of the enzyme in the membrane-bound state. Fig. 15. Replot ofexperimental data from Fig.13 and 14 (extraThere was always some enzyme in solution ; to correct polated to 1 mM glucose and 5 mM A TP). (m) Fig. 13, (0) for this the experimenters removed the soluble enzyme Fig. 14 by centrifugation, assayed it, and then determined the total amount of mitochondria1 enzyme activity. Unfortunately the amount of soluble enzyme found ranged from 10 to 50% of the total activity, and This inconsistency may be caused by a loss of attempts at correcting for this caused large deviations. enzyme activity in the absence of glucose as a stabilizer To avoid these, raw data were used whenever large (as discussed above) of about 12 % while the inhibitor differences in the amount of soluble enzyme within a experiments were performed. This explanation is set of determinations were found; otherwise, the supported by the report of Hammes and Kochavi [34] model reported is based on the published data. that their yeast hexokinase samples lost about 10 % A source of error in the published data was that activity in one day. Furthermore, inspection of the the points(s) at the end of the range of concentration original laboratory notes showed similar deviations studied were sometimes deviant, distorting the resultbetween assays of different aliquots of the same ing curves drawn through the set of points. enzyme preparation. Simulation of the bound enzyme started with the When the original data became available, it was model for the soluble enzyme, which was matched found that data from two unpublished experiments successfully to the bound glucose and ATP data. This with anhydroglucitol 6-phosphate fitted the model process required changing a number of rate constants, appreciably better (Fig. 1). Although the experimenters especially reducing the rate of binding of ATP to the did not have a model to use as a criterion for accepting free enzyme to a negligibly small value. There is a or rejecting sets of experimental data with poor precedent for this: Wedler and Boyer [36] recently reproducibility, other criteria are possible. One would reported a case where an inhibitor of glutamine have been to use overlapping concentration ranges so synthetase completely blocked one of its partial the results would have been directly comparable. reactions. This makes the mechanism ordered for Another would be to use that set of data which is binding of substrates, but random for release of prodmost nearly an average of those obtained. In these ucts. experiments, three apparent inhibition constants were obtained. The raw data which agree best with the Sensitivity Tests model have a value for Ki of 0.05 mM, which approximates the mean of the three values (0.022, 0.05 and Since the mechanism chosen for hexokinase results 0.07 mM). in a rate law too complicated for pencil-and-paper To characterize the anhydroglucitol 6-phosphate manipulation, information about its behaviour was inhibition correctly would require more experimental sought through sensitivity tests. A single rate conpoints than were previously measured, preferably stant was varied, each time by a factor of 10 until a every possible combination of the concentrations region of insensitivity or direct proportionality to originally reported for each substrate. velocity was found. Logarithmic plots of the results The fit between model and experiment for ADP were made by means of line-printer graphs which inhibition is shown in F i g 3 and 4. Tracer studies usually yielded smooth curves. showed exchange of 32Pbetween ADP and ATP but The form of these graphs is either sigmoid or not between anhydroglucitol 6-phosphate and ATP, hyperbolic, with insensitive regions. The rate constants suggesting a preferred pathway for product release that yield hyperbolic curves are more important as 1
1
Eur. J. Biochem. 51 (1975)
L. Garfinkel
they are rate determining in their regions of direct variation. These hyperbolic rate constants are those for interconversion of ternary complexes, product release, inhibition, and (strangely) the reaction for dissociation of substrates from ternary complexes. Rate constants become rate limiting when they change the apparent type of inhibition. Similar findings have been reported by Walter in a slightly different context [371. One result is that in the presence of both products the sensitivity pattern of rate constants for product release changed from hyperbolic to sigmoid. This would seem to indicate that release of a product cannot be rate limiting when both products are present. This conclusion also diagrees with the interpretation given by Kosow and Rose [31] as to their inability to detect exchange between anhydroglucitol. 6-phosphate and ATP. Anhydroglucitol 6-phosphate may simply be an allosteric effector, with no effect through the reverse reaction. The regions of insensitivity are interesting from a theoretical as well as a practical point of view. The presence of regions where changing rate constants will have no effect on velocities would in certain cases permit a wide variety of insensitive values. Although theoretically the reactions around the alternate pathways of a random mechanism should be in equilibrium, the insensitive values would seem to negate that requirement. In fact it was sometimes found that a given rate constant ceased to have any significance when made very small and that this effect could be duplicated by deleting that reaction from the model.
DISCUSSION The fit of this model to the data does not constitute proposed model in all detail. It does indicate that the overall features of the model are correct, and that the effects of the inhibitors when of the substrates is varied are consistent. Without a model this is very difficult to observe with the range of concentrations used. As has been discussed by Cleland [38], the effect of an inhibitor is a function not only of the equilibrium constant for the formation of its complex with enzyme of enzyme . substrate intermediate, but also of the kinetic constants of the uninhibited reaction. Although this has been shown rigorously for an ordered bi-bi reaction, it probably applies to other mechanisms as well, and seems to be applicable here. Mitochondria1 binding appears to have no effect on the inhibitor reactions ; the reported differences are mostly due to questionable corrections. The Eur. J. Biochem. 51 (1975)
445
observed insensitivity to glucose 6-phosphate inhibition could be due to transient properties such as the delayed response reported by Kosow and Rose [39] for mitochondrially bound hexokinase 11. There appears to be a slightly decrease in the Michaelis constants of both substrates, which may be due to the presence of macromolecules, as similar effects are observed with yeast hexokinase bound to polymer molecules [40]. The explanation offered for the increased activity of the immobilized enzyme is that the ionic charges on the polymer render the local concentration of substrate in the vicinity of the enzyme higher than in the bulk solution. Possibly a similar situation exists with membrane-bound hexokinase, especially for Mg-ATP, in addition to metabolic effects of the mitochondria. Although double-reciprocal plots have often been criticized, they continue to be used because of their convenience. The other linearizations which have been proposed have not been applied, although Webb [41] recommends using six types of graphs on a single set of data. Procedures for correcting raw experimental data and deriving optimal lines through the corrected values have been developed [42] and programmed for computer use [29,43]. Recent developments involve direct fitting to curves which have not been linearized [43- 451. The most sophisticated of these programs fits data to a model, and shows when that model has been validated according to previously established criteria [46]. This large variety of procedures indicates a lack of accepted standards in analyzing experimental data. The traditional view that any statistical analysis of experimental data is unnecessary still persists [47]. As application of these sophisticated techniques relies heavily on computers, their use many not always be feasible. Nevertheless, the quality of data can be improved if the traditional methods are applied prudently. In order to apply Cleland’s rules to enzyme kinetics data, data under both unsaturating and saturating concentrations are needed [32]. It is not accidental that the very complete study of Noat et al. on yeast hexokinase included analog computer simulation [48]. Methods that rely heavily on extrapolation should be avoided, since these tend to obscure the fit to data. Cleland has recommended using data from a large number of experiments in evaluating constants [43]. This is preferable to attempting to draw conclusions from the number of measurements that can be made in one working day. Definitions of terms should be consistent: four or five different definitions of “K,” were encountered in doing this work. Obtaining reliable data when working with enzymes is difficult, especially with an enzyme as
446
unstable as hexokinase 11. Reich has stressed the difficulty of obtaining samples of uniform properties [49]. Enzyme samples should be checked quite often for loss or change of activity when there is known to be instability. Some replication of experimental results is desirable so that inconsistencies are apparent simply from comparing results. Ottaway has stressed the need for replication when fitting data to mathematical models [44]. Although the proper experimental strategy for performing enzyme kinetic experiments has not been defined, it is desirable to guard against systematic errors by making measurements in a random sequence of concentrations. In the present experimental study the surface defined by reactant concentrations was not complete because of the missing concentration. Few of these experiments meet Cleland’s criteria of concentrations giving between 1/6 and 5/6 of the maximal velocity [43]. The ultimate answer to this problem may be to perform the necessary experiments with rapid automatic devices which are being used in clinical laboratories. Their development and application has been described [50]. The increased number of experiments possible should make it easier to determine enzyme mechanisms more adequately. Fortunately, computers and the necessary programs are becoming more widely available. A number of experimental enzymologists have been using computer simulation programs to fit experimental data to model, and to establish the feasibility of assumed mechanisms. When an enzyme mechanism is being studied, the investigator has to formulate a conceptual model and obtain values for kinetic constants. It then becomes straightforward to formulate the model so that a computer can determine if the experimental data are compatible both the assumed model and each other. Supported by grants GM16501 and RR15 of the National Institutes of Health. The author wishes to thank Dr 1. A. Rose, Institute for Cancer Research, Philadelphia, Pa 19111, U.S.A. for making original laboratory data available.
REFERENCES 1. Garfinkel, D. & Hess, B. (1964) J . Biol. Chem. 239, 971 -983. 2. Garfinkel, D. (1966) J . Biol. Chem. 241,286-294. 3. Garfinkel, D., Frenkel, R. & Garfinkel, L. (1968) Comput. Biomed. Res. 2, 68 - 91. 4. Morrison, J. F. & Heyde, E. (1972) Annu. Rev. Biochem. 41,29 - 54. 5. Achs, M. J. & Garfinkel, D. (1973) in Proc. ZFAC Symposium on Dynamics and Control in Physiological Systems, Rochester, N. Y., pp. 19-21.
Simulation of Hexokinase 6. Garner, P. & Rosett, T. (1973) FEBS Lett. 34, 243-246. 7. Grossbard, L. & Schimke, R.T. (1966) J . Biol. Chem. 241, 3546- 3560. 8. England, P. J. & Randle, P. J. (1967) Biochem. J . 105, 907 - 920. 9. Mayer, S. E., Mayfield, A. C. & Haas, J. A. (1966) Mol. Pharmacol. 2, 393-405. 10. Rose, I. A. & Rose, Z. B. (1969) Compr. Biochem. 17, 93-166. 11. Rose, 1. A. & Warms, J. V. B. (1967) J . Biol. Chem. 242, 1635-1645. 12. Rabin, B. R. (1967) Biochem. J . 102, 22C-23C. 13. Newsholme, E. A,, Rolleston, F. S. & Taylor, K. (1968) Biochem. J . 106, 193-201. 14. Tuttle, J. F. & Wilson, J. E. (1970) Biochim. Biophys. Acts, 212, 185- 188. 15. Purich, D. L. & Fromm, H. J. (1971) J . Biol. Chem. 246, 3456 - 3463. 16. Lonberg-Holm, K. (1959) Biochim. Biophys. Acta, 35, 464- 472. 17. Garfinkel, D. & Hess, B. (1964) J . Biol. Chem. 239, 971 -983. 18. Gumaa, K. A. & McLean, P. (1969) Biochem. J . 115, 1009-1029. 19. Coe, E. L. (1970) Biochem. Biophys. Res. Commun. 38, 1105- 1112. 20. Gumaa, K. A. & McLean, P. (1969) Biochem. Biophys. Res. Commun. 35,824-831. 21. Wilson, J. E. (1967) Biochem. Biophys. Res. Commun. 28, 123- 127. 22. Katzen, H. M., Soderman, D. D. & Wiley, C. E. (1970) J. Biol. Chem. 245,408 1- 4096. 23. Crane, R. K. & Sols, A. (1954) J . Biol. Chem. 210, 597606. 24. Kosow, D. P. & Rose, I. A. (1970) Fed. Proc. 29,399. 25. Frieden, C. (1970) J. Biol. Chem. 245,5788- 5799. 26. Frieden, C. (1971) Annu. Rev. Biochem. 40,653-696, 27. Green, S. B. & Garfinkel, D. (1970) Comput. and Biomed. Res. 3, 166- 173. 28. Rhoads, D. G. & Pring, M. (1968) J . Theor. Biol. 20, 297-313. 29. Cleland, W. W. (1963) Nature (Lond.) 198, 463-465. 30. Kosow, D. P. & Rose, 1. A. (1968) J . Biol. Chem. 243, 3623 - 3630. 31. Kosow, D. P. & Rose, I. A. (1970) J. Biol. Chem. 245, 198- 204. 32. Cleland, W. W. (1963) Biochim. Biophys. Acta, 67, 104137. 33. Gulbinsky, J. S. & Cleland, W. W. (1968) Biochemistry, 7,566- 575. 34. Hammes, G. G. & Kochavi, D. (1962) J . Am. Chem. SOC. 84,2069-2073. 35. Ning, J., Purich, D. L. & Fromm, H. J. (1969) J. Biol. Chem. 244,3840- 3846. 36. Wedler, F. C. & Boyer, P. D. (1973) J . Theor. Bid. 28, 539-558. 37. Walter, C. (1967) J . Theor. Biol. 15, 1-33. 38. Cleland, W. W. (1963) Biochim. Biophys. Acta, 67, 173187. 39. Kosow, D. P. & Rose, I. A. (1972) Biochem. Biophys. Res. Commun. 48, 376-383. 40. Mosbach, K. & Mattiasson, B. (1970) Acta Chem. Scand. 24,2093 - 2100. 41. Webb, J. L. (1963) Enzyme and Metabolic Inhibitors, vol. 1, pp. 150- 180, Academic Press, New York.
Eur. J. Biochem. 51 (1975)
L. Garfinkel 42. Wilkinson, G. N. (1961) Biochem. J . 80,324-332. 43. Cleland, W. W. (1967) Adv. Enzymol. 29, 1- 32. 44. Hess, B., Chance, E. M., Busse, H. & Wurster, B. (1972) in Analysis and Simulation of Biochemical Systems (Hemker, H. C. & Hess, B., eds) pp. 119-143, NorthHolland, Amsterdam. 45. Ottaway, J. H. (1973) Biochem. J . 134, 729-736.
447 46. Reich, J. G., Wangermann, G., Falck, M. & Rohde, K. (1972) Eur. J . Biochem. 26, 368-379. 47. Dixon, M. & Webb, E. C. (1964) Enzymes, 2nd edn, p. 70, Academic Press, New York. 48. Noat, G., Ricard, J., Borel, M. & Got, C. (1968) Eur. J. Biochem. 5, 55 - 70. 49. Reich, J. G. (1 970) FEBS Lett. 9, 245 - 251. 50. Anderson, N. G. (1969) Clin. Chim. Acta, 25, 321-330.
L. Garfinkel, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A. 19174
Eur. J. Biochem. 51 (1975)
Computer Simulation Study of Hexokinase I1 from Ehrlich Ascites Cells Lillian GARFINKEL Moore School of Electrical Engineering, William Pepper Laboratory, Department of Pathology, and Department of Biophysics University of Pennsylvania, Philadelphia (Received July 9/September 24, 1974)
A study of the mechanism of hexokinase I1 from ascites cells and the effects of its binding to mitochondrial membranes has been carried out by computer simulation. This is based on experimental data of Kosow and Rose and of Gumaa and McLean, and the theoretical methods of Cleland. For the soluble enzyme the mechanism is random with ternary product-inhibition complexes; when bound to mitochondria, the mechanism becomes ordered-on, random-off, as the binding of ATP to the free enzymes becomes negligibly slow. The requirements of experimental data for mechanistic studies are discussed.
This study was undertaken as part of a continuing stimulation study of glycolysis [l - 31. Its original aim was to elucidate the transient behaviour of hexokinase where the enzyme appears to be insensitive to glucose 6-phosphate inhibition at early time intervals. Further investigations has required attention to the mechanism of the enzyme, and therefore to relevant initial velocity studies, as well as its effective physiological regulation. Hexokinase is of considerable interest as the first enzyme of glycolysis. Its literature is very large, almost overwhelming: there were over 500 entries in the 1967- 72 five-year Index of Chemical Abstracts alone. There are a number of studies of mechanism, but many were probably made with isozyme mixtures using steady-state methods of analysis which apply to single enzymes. An important source of disagreement is the actual concentration of the reactive species of ATP, its chelate with Mg2+.Morrison and Heyde have pointed out that this is a function of type of buffer, ionic strength, pH, and the presence of other enzymes [4]. This question becomes even more important since the availability of Mg2+ in vivo appears to be an important control [5,6].
different relative proportions (which may vary with physiological conditions, such as insulin level). Type I1 is the isozyme most subject to such variations; it is also less stable than other isozymes under laboratory conditions, being sensitive to proteases, to heating [7] and to freezing [8]. Dog heart hexokinase undergoes successive changes in kinetic properties during storage [9]. Stability of the enzyme in storage requires the presence of 10 mM glucose [7]; in its absence rat heart hexokinase lost 80% of its activity within 24 h [81. b) Another unusual feature of mammalian hexokinase is its irreversibility, which interferes with both initial velocity studies and radioactive exchange studies [lo]. This necessitates using inductive reasoning in attempts to synthesize mechanisms from experiments which furnish only a part of the necessary data. c) Hexokinase association with both the inner and outer mitochondrial membranes is well known in a variety of tissues, and has been widely studied as a novel metabolic control mechanism. It was first reported for ascites hexokinase by Rose and Warms [ll],who expressed the relation between soluble and bound hexokinase as : Binding sites
Relevant Known Properties of the Enzyme
a) Hexokinase from a number of tissues contains isozymes of differing kinetic properties and also Enzyme. Hexokinase (EC 2.7.1.1)
Eur. J. Biochem. 51 (1975)
+ soluble enzyme + Mg” e Complex-bound enzyme.
Hexokinase activators, such as inorganic phosphate, favored binding while inhibitors such as glucose 6-phosphate and unchelated ATP favored
Simulation of Hexokinase
438
solubilization. This is very similar to the allosteric model of Rabin [12], if the bound and soluble forms have different kinetic properties. Results of studies on mitochondrial hexokinase have not been unanimous. A number of authors have reported differing effects of mitochondrial binding on the kinetic properties of the enzyme [S, 13- 151. Although glucose 6-phosphate is the strongest known inhibitor of mammalian hexokinase, it does not appear to be operating at early time intervals when ascites cells are incubated with glucose [16- 181; however, ADP appears to be controlling hexokinase here [17,19]. The finding that glucose 6-phosphate is inhibitory in a cell-free extract suggested to Gumaa and McLean that a change in hexokinase binding might be occurring. Further evidence for this view was the close agreement between the 18-s half-life for release of mitochondrially bound hexokinase reported by Rose and Warms [ll], and the 15-s interval for the onset of “normal” inhibitory behavior by glucose 6-phosphate [20]. The results of studies of supposedly mitochondrially bound hexokinase are often contradictory, partly because the actual binding state of the enzyme at the time of measurement may be unclear, and possibly because the enzyme bound to mitochondria may be exposed to concentrations of ATP and glucose different from those in solution. As an extreme illustration, hexokinase I is described as “latent”, since it is normally inaccessible to its substrates in the cytosol [21,22]. (At times the presence of this isozyme may interfere in studies of hexokinase 11.) d) The “allosterism” of hexokinase is based on the non-competitive inhibition of glucose by glucose 6-phosphate, the specificity of glucose 6-phosphate as an inhibitor, and the weak inhibition by fructose 6-phosphate and mannose 6-phosphate [23]. It has recently been found (M. J. Achs and D. Garfinkel, personal communication) that more than one molecule of glucose 6-phosphate may be involved in inhibiting heart hexokinase, and that M$+ and inorganic phosphate play complex roles in regulating it. Kosow and Rose have recently reported negative co-operativity towards ADP inhibition [24]. However, under most conditions hexokinase behavior is relatively straightforward, and brain hexokinase is not dissociated by agents which dissociate other allosteric enzymes into subunits. e) Some enzymes show delayed response to rapid changes in ligand concentration. Frieden [25,26] has named these enzymes hysteretic, suggested the delay as involving protein - protein interaction (which is slower than interaction between proteins and small molecules), and discussed their possible significance in metabolic control. The most recent thinking about the
insensitivity of mammalian hexokinase to inhibitors, especially glucose 6-phosphate, has centered on hysteretic effects. A preliminary study of glycolysis in ascites cells subjected to a temperature jump (B. Chance, P. K. Maitra and I. Y. Lee, unpublished) indicates that hysteresis is important in vivo. f) The mechanism of hexokinase is still undefined, despite many studies, both because of its unusual properties, and the lack of agreement on how to define an enzyme mechanism, which is discussed below.
METHODS Computer Programs Computations were performed on the PDP-6 and PDP-10 computers of the University of Pennsylvania Medical School Computer Facility, using two programs that give solutions to steady-state initial-velocity problems very rapidly and economically. These are based on matrix inversion [27] and rate laws [28]. The first program is especially convenient, since computed results are actually displayed as doublereciprocal plots together with the experimental data. Required input for these programs consists of chemical reactions, rate constants, substrate and modifier concentrations, and enzyme activities; it is easy to make many trials in order to fit the data. The hyperbolic curve-fitting program of Cleland [29] was used on the data for the soluble enzyme to determine the kinetic constants in the uninhibited reactions, and to determine if any other inhibition patterns than those reported were feasible. When this study was close to completion, the models were subjected to sensitivity tests : individual rate constants were varied one at a time. This was necessary because the final rate laws were too complicated for paper-and-pencil manipulation. Experimental Data Base The experimental data on which this simulation was based were the initial velocity studies of Kosow and Rose for hexokinase I1 from Ehrlich ascites cells in both the soluble and mitochondrially bound states [30,31]. This work indicated a sequential mechanism with random substrate addition, since competitive inhibitors of one substrate were usually noncompetitive inhibitors of the other. Inhibition constants in both the soluble and mitochondrially bound states were similar, but K, values for both substrates were slightly lower when mitochondrially bound. After the model had been constructed on the basis of this published work, it was compared to the raw Eur. J. Biochem. 51 (1975)
L. Garfinkel
439
laboratory data, which Dr Rose then kindly made available, and found generally to fit it much better. The inhibition patterns observed by these workers plus the finding by others that mannose competes with glucose are tabulated in Table 1. The finding for anhydroglucitol 6-phosphate vs ATP in the bound form is dubious, as it is based very heavily on extrapolation. In general, both Ki slope and Ki intercept were similar for both enzyme forms. This inhibition pattern seems to indicate a random mechanism. Theoretical Methodology The simplest and most familiar method for analyzing kinetic enzyme data is the double-reciprocal Lineweaver-Burk plot. A problem with this method is the human tendency to assign more weight to the inherently less accurate points far from the axes when drawing the resulting lines. The fact that both V and K, (determined from their intercepts along the axes) are essentially determined by extrapolation may obscure the fit to the data. Inhibitions are considered competitive, uncompetitive, or non-competitive according to how these double-reciprocal lines do (or do not) intersect. This is the basis of a number of schemes for determining reaction mechanisms : Reiner, Alberty and Cleland have all developed these methods. This study has relied heavily on the work of Cleland [32]. Often there is information other than that from steadystate kinetics (e.g., direct-binding studies) which require explanation. Thus the only generalization possible is that a given mechanism should explain as many facts as possible.
RESULTS The model chosen to represent the data is given in Table 2. The mechanism for the soluble enzyme was random bi-bi. Product release rather than conversion of ternary complex was made the limiting step on the basis of radioactive studies [31]. Gulbinsky and Cleland [33] have shown that the presence of linear double-reciprocal lines does not require conversion of ternary complexes to be the limiting step. The observed inhibition patterns could not be matched by any of the reaction mechanisms given by Cleland, if the non-varied substrates were present in saturating concentrations. The formation of ternary dead-end complexes was invoked in an attempt to explain the observed inhibition patterns. A typical chemical equation is : ADP
+ Enz . Glc = Enz . Glc . ADP
where Enz . Glc . ADP denotes the ternary dead-end complex of hexokinase, glucose and ADP. For such Eur. J. Biochem. 51 (1975)
Table 1. Nature of the inhibition of various inhibitors against ATP and glucose for soluble and bound hexokinase II N = Non-competitive; C = competitive Inhibitor
Soluble
ADP AMP Glucose 6-phosphate Mannose Anhydroglucitol 6-phosphate
Bound
ATP
glucose ATP
N
N
C C
C
N N N
N
N N (C)
-
-
C
N
N
N
N C
glucose
Table 2. Ascites hexokinase model Even-numbered reactions are the reverse of the preceding odd-numbered reactions. Enz = hexokinase, Glc = glucose, Glc-6-P = glucose 6-phosphate, anGlc-6-P = anhydroglucitol 6-phosphate. The rate constants are either first order (min-') or second order (1 mol-' min-') Chemical equations
Rate constants soluble
bound
Enz
3 x105 2.2 x 103
2.5 x lo7 2.2 x 103
. ATP
I xi05 1.3 x 104 2.5 x lo6 1 . 9 104 ~
1 x106 4.4 x 103 1 XlOO I .9 x lo4
1.8 x 107 4 x103
5 x108 4 xi03
2 xi05 I xi03
2 xi05 i xi03
1.5 x 104 1 XI06 2 xi03 2 xi05
1.5 x 104 1 XI06 2 xi03 2 xi05
2.4 x 103 I xi05 8.5 x 10' 5 xi05
2.4 x 103 I x105 8.5 x 10' 3 xi05
1 xi07 1.4 x 104
I xi07 1 . 4 104 ~
5 XI06 8 XI02
5 xl06 8 xl@
1 XI08 2 xi03 4.5 x 105 I xi03
1 XI08 2 xi03 4.5 x 105 I xi03
4.5 x 105 1 xi03
4.5 x 105 I xi03
+ Glc = Enz . Glc Enz . Glc + ATP = Enz . Glc + ATP = Enz . ATP Enz . ATP + Glc = Enz . Glc
Enz
. ATP
Enz . Glc . ATP . ADP
Enz . Glc-6-P
=
Enz . Glc-6-P . ADP . Glc-6-P + ADP
=
Enz
Enz . Glc-6-P = Glc-6-P Enz . Glc-6-P . ADP . ADP + Glc-6-P Enz . ADP
=
ADP
ADP + Enz . Glc . ADP
=
+ Enz
Enz
+ Enz
=
Enz . Glc
Enz . Glc + anGlc-6-P . Glc . anGlc-6-P
=
Enz
Enz + anGlc-6-P = Enz . anGlc-6-P
+ AMP = Enz . AMP Enz . Glc + AMP = Enz . Glc
Enz
. AMP
440
Simulation of Hexokinase 1600 1
1000 800
1
I
0
- 4 - 2
2
4
6
8
1
0 2 4 I/[Glucose] (rnM-’)
1 /[Glucose] (rnM-’)
Fig. 1. Reciprocal plot ojanhydroglucitol6-phosphateinhibition vs glucose for the soluble enzyme. (0) Published data, (W, A) raw data, all from three different experiments. Concentrations as reported by Kosow and Rose. ATP-Mg was 5 mM (hexokinase activity was 5 x unit or 0.03 mM for the simulated curve; velocity units are dA,,/min) [30]
6
- 6 - 4 - 2
0
8
Fig.3. Reciprocal plot of ADP inhibition vs glucose for the soluble enzyme. (0)Published data, (D) raw data. Concentrations as reported by Kosow and Rose (ATP-Mg was 5 mM; hexokinase “activity” was 0.038 mM for the simulated curves) ~301
1 400 r
1000
-
5
‘Oo0 800
-0.6 -0.4 -0.2
0
0.2 0.4
0.6
0.8
1 .C
l / [ A T P ] (mM-’)
-0.6
-0.4
-0.2
I 0
0.2
0.4
0.6
0.8
1.0
l l [ A T P ] (rnM-’)
Fig. 2. Reciprocalplot of anhydroglucitol6-phosphate inhibition vs A T P for the soluble enzyme. (0)Published data, (El) raw data. Concentrations as reported by Kosow and Rose (glucose was 1 mM, ATP-Mg was varied between 1 and 4 mM; hexokinase “activity” was 0.032 mM for simulated curves) [30]
Fig.4. Reciprocal plot of ADP vs ATP for the soluble enzyme. (0) Published data, (0)raw data. Concentrations as given by Kosow and Rose (glucose 1 mM ; hexokinase “activity” 0.053 mM for simulated curves) [30]
an inhibition the Ki is the quotient association rate/ dissociation rate, e.g., K,,/K,, for Enz . Glc . ADP and similarly for the others. The random mechanism was chosen in part because those experimental studies which indicate an ordered mechanism disagree as to which substrate adds first. It is also based on the non-competitive
inhibition of mannose when ATP is the varied substrate. If this inhibition were uncompetitive instead and the unvaried substrate non-saturating, an ordered bi-bi or Theorell-Chance mechanism would be possible. Likewise, the present data for the bound enzyme are consistent with an iso-ordered bi-bi mechanism if anhydroglucitol 6-phosphate is assumed to behave Eur. J. Biochem. 51 (1975)
441
L. Garfinkel
2 000
1400 1800
1200 1600 1400
-
:. 1m 0 -1 \ t 0
q 2
l0OC
800
2
-
4
-
2 0 2 4 l/[Glucose](mM-’)
6
8
10
0
0
Fig. 5 . Reciprocal plot of AMP inhibition vs glucose for the soluble enzyme. (0) Published data, raw data. Concentrations as reported by Kosow and Rose. (ATP-Mg was 5 mM ; hexokinase “activity” 0.038 mM for simulated curve.) [301
(a)
1800 1600
t1
I
1400 -
/
- 1200 -
c ._
E
b 1000 -
600
/
8
/
400
1 / [ ATP] (rnM-’)
Fig. 6. Reciprocalplot of AMP inhibition vs ATP for the soluble enzyme. (0) Published data, (Kl) raw data. Concentrations as reported by Kosow and Rose (glucose was 1 mM; hexokinase “activity” was 0.03 mM for simulated curve) [30]
like the natural product glucose 6-phosphate ;however, this is the least reliable inhibitor pattern. The model presented here is probably not unique. Too many of the reaction steps cannot be defined Eur. J. Biochem. 51 (1975)
/
/ t ,
- 4 - 2
*0°
I
0
2 4 1/ [Glucose] (rnM-’)
6
8
10
Fig. I . Reciprocal plot of anhydroglucitol6-phosphateinhibition vs glucose for the bound enzyme. (0) Published data, (17) raw data. Concentrations as given by Kosow and Rose (ATP-Mg was 5 mM; hexokinase “activity” was 0.018 mM) [30]. Soluble enzyme activities range from 26-40% of total. Deviant point at 0.1 mM glucose, no inhibitor (25 % soluble), and 0.1 mM glucose and 0.038 mM anhydroglucitol 6-phosphate)
from the work that has thus far been done, primarily because of the irreversibility of the enzyme (ie.,there are few radioactive experiments). However, the overall features of this model are probably correct, and the undefined steps (no. 11- 18, Table 2) are not significant. There appear to be insufficient published data to define these steps uniquely for any mammalian hexokinase. Although it was difficult to fit the published experimental data, the fit to the raw experimental data was much better. The fit of the model to the data (both published and unpublished) is shown in Fig. 1- 6 for the soluble enzyme and Fig. 7- 12 for the bound enzyme. The two sets of published data for anhydroglucitol 6-phosphate as inhibitor with glucose and with Mg-ATP as the varied substrate (Fig. 13 and 14) were found to be inconsistent with each other, and the value of rate constant 23 (Table 3) needed to fit them varied by a factor of 200. Since these experiments were run at different substrate concentrations, the double-reciprocal lines of Fig. 13 and 14 were extrapolated slightly to yield velocities at comparable
442
Simulation of Hexokinase
-
1100
2200
1 GOO
2000
900
I800
800
1 600
700
--
c .-
E
- *0 T d 1
/
1 400
c
t
600
1200
0 -, \f o
0
T
500
5 1000
.
. -
-->
400
800
600
,A:: -0.6 -04 -0.2 0
0.2 0.4
0.6
0.8
1
.O
-0.6 -0.4 -0.2
0.2 0.4 0.6
0.8 1.0
l / [ A T P ] (rnM-’)
l / [ A T P ] (KIM-’)
Fig. 8. Reciprocal of anhydroglucitol6-phosphate inhibition vs. ATP for the bound enzyme. (0)Published data, (D) raw data. Concentrations as reported by Kosow and Rose (glucose was 1 mM ; hexokinase “activity” was 0.03 mM) [30]. Soluble activities are about 20 % of the total except for 0.4 mM Glc, no inhibitor, and 0.4 mM Glc, 0.056 mM anGlc-6-P
Fig.10. Reciprocal plot of ADP inhibition vs ATP ,for the bound enzyme. (0)Published data, (m) raw data. Concentrations as reported by Kosow and Rose (glucose was 1 mM; hexokinase “activity” 0.015 mM). Bound activities are 75 of total except for points at 1.5 mM ATP, 1 and 3 mM ADP) [30]
Table 3. Values of rate constants for inhibitions needed to ,fit data in Fig. 13 andfinal model The rate constants are either first order (min-’) or second order (1 mol-’ min-’)
I
1600
Chemical equations
Rate constants
(21) Enz . Glc + anGlc-6-P = Enz . Glc . anGlc-6-P (22) (23) Enz + anGlc-6-P = Enz . anGlc-6-P (24)
Fig. 13
final model
1x10~ 8x10’
5~ 106 8x102
1 XI08 2 x 103
1x108 2 x 103
I
- 4 - 2
0
2
4
6
8 1 0
l/[Glucose](RIM-’)
Fig. 9. Reciprocal plot of ADP inhibition vs glucose ,for the bound enzyme. (0)Published data, raw data. Concentrations as reported by Kosow and Rose (ATP-Mg was 5 mM; hexokinase “activity” was 0.021 mM). Soluble activities were generally 40% of total activity, except for a deviant point at 0.25 mM glucose, 4 mM ADP [30]
(m)
concentrations of substrates and inhibitors, and the results plotted in Fig. 15. Since substrate and inhibitor concentrations are now equal, the velocities should all be proportional to the stated activity of the enzyme samples independent of any assumed mechanism. Fig. 15 shows that they are not. Eur. J. Biochem. 51 (1975)
443
L. Garfinkel 1800
2000
1 600
0
1800 O
/
1 400
/
1600
1200 1400 L .-
t
F - 1200
1000
-3 ,w
E
T
72 l00C 0
80C \
a
c
Y
b
:80C
/
0 0
0
I . 3
2
4
6
J
8
1
0
-4
-2
0
2
4
6
8
10
l / [Glucose](rnM-’)
l/[Glucos~](mM-’)
Fig. 11. Reciprocal plot of A M P inhibition vs glucose for the bound enzyme. (0) Published data, (m) raw data. Concentrations as given by Kosow and Rose (ATP-Mg was 5 m M , hexokinase “activity” was 0.018 mM). Soluble activities range from 15 to 50 % in this experiment [30]
Fig. 13. Reciprocal plot of published data on anhydroglucitol 6-phosphate inhibition vs. glucose for soluble enzyme. Experimental data were fitted to a model that was incompatible with Fig. 14 [30]
1200
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Fig. 12. Reciprocal plot o j A M P inhibition vs A T P for bound enzyme. (0) Published data, (m) raw data. Concentrations given by Kosow and Rose (glucose was 1 m M ; hexokinase “activity” was 0.045 mM). Soluble activities were 18 to 25 of the total [30] Eur. J. Biochem. 51 (1975)
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Fig. 14. Reciprocal plot of published data on anhydroglucitol 6-phosphate inhibition vs A T P f o r soluble enzyme. (0)Published data. (a) Incompatible model that fitted the data of Fig. 13 and 1
Simulation of Hexokinase
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with glucose 6-phosphate off last. Thus will be discussed further in connection with the sensitivity tests. While this inhibition may be non-linear at other concentration ranges, it appears to be normal for the range reported here [31,35]. AMP is a fairly weak inhibitor, and probably has little effect in situ. The data can be matched fairly well even though the type of inhibition differs some200 I what from that postulated by the experimenters (Fig. 5 and 6). 0I Interpretation of all of the data for the bound 0 0.01 0 02 0.03 0.04 0.05 0.06 0.08 is complicated by the inability to mainenzyme form [Anhydroglucitol 6-phosphate] [ m M ) tain all of the enzyme in the membrane-bound state. Fig. 15. Replot ofexperimental data from Fig.13 and 14 (extraThere was always some enzyme in solution ; to correct polated to 1 mM glucose and 5 mM A TP). (m) Fig. 13, (0) for this the experimenters removed the soluble enzyme Fig. 14 by centrifugation, assayed it, and then determined the total amount of mitochondria1 enzyme activity. Unfortunately the amount of soluble enzyme found ranged from 10 to 50% of the total activity, and This inconsistency may be caused by a loss of attempts at correcting for this caused large deviations. enzyme activity in the absence of glucose as a stabilizer To avoid these, raw data were used whenever large (as discussed above) of about 12 % while the inhibitor differences in the amount of soluble enzyme within a experiments were performed. This explanation is set of determinations were found; otherwise, the supported by the report of Hammes and Kochavi [34] model reported is based on the published data. that their yeast hexokinase samples lost about 10 % A source of error in the published data was that activity in one day. Furthermore, inspection of the the points(s) at the end of the range of concentration original laboratory notes showed similar deviations studied were sometimes deviant, distorting the resultbetween assays of different aliquots of the same ing curves drawn through the set of points. enzyme preparation. Simulation of the bound enzyme started with the When the original data became available, it was model for the soluble enzyme, which was matched found that data from two unpublished experiments successfully to the bound glucose and ATP data. This with anhydroglucitol 6-phosphate fitted the model process required changing a number of rate constants, appreciably better (Fig. 1). Although the experimenters especially reducing the rate of binding of ATP to the did not have a model to use as a criterion for accepting free enzyme to a negligibly small value. There is a or rejecting sets of experimental data with poor precedent for this: Wedler and Boyer [36] recently reproducibility, other criteria are possible. One would reported a case where an inhibitor of glutamine have been to use overlapping concentration ranges so synthetase completely blocked one of its partial the results would have been directly comparable. reactions. This makes the mechanism ordered for Another would be to use that set of data which is binding of substrates, but random for release of prodmost nearly an average of those obtained. In these ucts. experiments, three apparent inhibition constants were obtained. The raw data which agree best with the Sensitivity Tests model have a value for Ki of 0.05 mM, which approximates the mean of the three values (0.022, 0.05 and Since the mechanism chosen for hexokinase results 0.07 mM). in a rate law too complicated for pencil-and-paper To characterize the anhydroglucitol 6-phosphate manipulation, information about its behaviour was inhibition correctly would require more experimental sought through sensitivity tests. A single rate conpoints than were previously measured, preferably stant was varied, each time by a factor of 10 until a every possible combination of the concentrations region of insensitivity or direct proportionality to originally reported for each substrate. velocity was found. Logarithmic plots of the results The fit between model and experiment for ADP were made by means of line-printer graphs which inhibition is shown in F i g 3 and 4. Tracer studies usually yielded smooth curves. showed exchange of 32Pbetween ADP and ATP but The form of these graphs is either sigmoid or not between anhydroglucitol 6-phosphate and ATP, hyperbolic, with insensitive regions. The rate constants suggesting a preferred pathway for product release that yield hyperbolic curves are more important as 1
1
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L. Garfinkel
they are rate determining in their regions of direct variation. These hyperbolic rate constants are those for interconversion of ternary complexes, product release, inhibition, and (strangely) the reaction for dissociation of substrates from ternary complexes. Rate constants become rate limiting when they change the apparent type of inhibition. Similar findings have been reported by Walter in a slightly different context [371. One result is that in the presence of both products the sensitivity pattern of rate constants for product release changed from hyperbolic to sigmoid. This would seem to indicate that release of a product cannot be rate limiting when both products are present. This conclusion also diagrees with the interpretation given by Kosow and Rose [31] as to their inability to detect exchange between anhydroglucitol. 6-phosphate and ATP. Anhydroglucitol 6-phosphate may simply be an allosteric effector, with no effect through the reverse reaction. The regions of insensitivity are interesting from a theoretical as well as a practical point of view. The presence of regions where changing rate constants will have no effect on velocities would in certain cases permit a wide variety of insensitive values. Although theoretically the reactions around the alternate pathways of a random mechanism should be in equilibrium, the insensitive values would seem to negate that requirement. In fact it was sometimes found that a given rate constant ceased to have any significance when made very small and that this effect could be duplicated by deleting that reaction from the model.
DISCUSSION The fit of this model to the data does not constitute proposed model in all detail. It does indicate that the overall features of the model are correct, and that the effects of the inhibitors when of the substrates is varied are consistent. Without a model this is very difficult to observe with the range of concentrations used. As has been discussed by Cleland [38], the effect of an inhibitor is a function not only of the equilibrium constant for the formation of its complex with enzyme of enzyme . substrate intermediate, but also of the kinetic constants of the uninhibited reaction. Although this has been shown rigorously for an ordered bi-bi reaction, it probably applies to other mechanisms as well, and seems to be applicable here. Mitochondria1 binding appears to have no effect on the inhibitor reactions ; the reported differences are mostly due to questionable corrections. The Eur. J. Biochem. 51 (1975)
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observed insensitivity to glucose 6-phosphate inhibition could be due to transient properties such as the delayed response reported by Kosow and Rose [39] for mitochondrially bound hexokinase 11. There appears to be a slightly decrease in the Michaelis constants of both substrates, which may be due to the presence of macromolecules, as similar effects are observed with yeast hexokinase bound to polymer molecules [40]. The explanation offered for the increased activity of the immobilized enzyme is that the ionic charges on the polymer render the local concentration of substrate in the vicinity of the enzyme higher than in the bulk solution. Possibly a similar situation exists with membrane-bound hexokinase, especially for Mg-ATP, in addition to metabolic effects of the mitochondria. Although double-reciprocal plots have often been criticized, they continue to be used because of their convenience. The other linearizations which have been proposed have not been applied, although Webb [41] recommends using six types of graphs on a single set of data. Procedures for correcting raw experimental data and deriving optimal lines through the corrected values have been developed [42] and programmed for computer use [29,43]. Recent developments involve direct fitting to curves which have not been linearized [43- 451. The most sophisticated of these programs fits data to a model, and shows when that model has been validated according to previously established criteria [46]. This large variety of procedures indicates a lack of accepted standards in analyzing experimental data. The traditional view that any statistical analysis of experimental data is unnecessary still persists [47]. As application of these sophisticated techniques relies heavily on computers, their use many not always be feasible. Nevertheless, the quality of data can be improved if the traditional methods are applied prudently. In order to apply Cleland’s rules to enzyme kinetics data, data under both unsaturating and saturating concentrations are needed [32]. It is not accidental that the very complete study of Noat et al. on yeast hexokinase included analog computer simulation [48]. Methods that rely heavily on extrapolation should be avoided, since these tend to obscure the fit to data. Cleland has recommended using data from a large number of experiments in evaluating constants [43]. This is preferable to attempting to draw conclusions from the number of measurements that can be made in one working day. Definitions of terms should be consistent: four or five different definitions of “K,” were encountered in doing this work. Obtaining reliable data when working with enzymes is difficult, especially with an enzyme as
446
unstable as hexokinase 11. Reich has stressed the difficulty of obtaining samples of uniform properties [49]. Enzyme samples should be checked quite often for loss or change of activity when there is known to be instability. Some replication of experimental results is desirable so that inconsistencies are apparent simply from comparing results. Ottaway has stressed the need for replication when fitting data to mathematical models [44]. Although the proper experimental strategy for performing enzyme kinetic experiments has not been defined, it is desirable to guard against systematic errors by making measurements in a random sequence of concentrations. In the present experimental study the surface defined by reactant concentrations was not complete because of the missing concentration. Few of these experiments meet Cleland’s criteria of concentrations giving between 1/6 and 5/6 of the maximal velocity [43]. The ultimate answer to this problem may be to perform the necessary experiments with rapid automatic devices which are being used in clinical laboratories. Their development and application has been described [50]. The increased number of experiments possible should make it easier to determine enzyme mechanisms more adequately. Fortunately, computers and the necessary programs are becoming more widely available. A number of experimental enzymologists have been using computer simulation programs to fit experimental data to model, and to establish the feasibility of assumed mechanisms. When an enzyme mechanism is being studied, the investigator has to formulate a conceptual model and obtain values for kinetic constants. It then becomes straightforward to formulate the model so that a computer can determine if the experimental data are compatible both the assumed model and each other. Supported by grants GM16501 and RR15 of the National Institutes of Health. The author wishes to thank Dr 1. A. Rose, Institute for Cancer Research, Philadelphia, Pa 19111, U.S.A. for making original laboratory data available.
REFERENCES 1. Garfinkel, D. & Hess, B. (1964) J . Biol. Chem. 239, 971 -983. 2. Garfinkel, D. (1966) J . Biol. Chem. 241,286-294. 3. Garfinkel, D., Frenkel, R. & Garfinkel, L. (1968) Comput. Biomed. Res. 2, 68 - 91. 4. Morrison, J. F. & Heyde, E. (1972) Annu. Rev. Biochem. 41,29 - 54. 5. Achs, M. J. & Garfinkel, D. (1973) in Proc. ZFAC Symposium on Dynamics and Control in Physiological Systems, Rochester, N. Y., pp. 19-21.
Simulation of Hexokinase 6. Garner, P. & Rosett, T. (1973) FEBS Lett. 34, 243-246. 7. Grossbard, L. & Schimke, R.T. (1966) J . Biol. Chem. 241, 3546- 3560. 8. England, P. J. & Randle, P. J. (1967) Biochem. J . 105, 907 - 920. 9. Mayer, S. E., Mayfield, A. C. & Haas, J. A. (1966) Mol. Pharmacol. 2, 393-405. 10. Rose, I. A. & Rose, Z. B. (1969) Compr. Biochem. 17, 93-166. 11. Rose, 1. A. & Warms, J. V. B. (1967) J . Biol. Chem. 242, 1635-1645. 12. Rabin, B. R. (1967) Biochem. J . 102, 22C-23C. 13. Newsholme, E. A,, Rolleston, F. S. & Taylor, K. (1968) Biochem. J . 106, 193-201. 14. Tuttle, J. F. & Wilson, J. E. (1970) Biochim. Biophys. Acts, 212, 185- 188. 15. Purich, D. L. & Fromm, H. J. (1971) J . Biol. Chem. 246, 3456 - 3463. 16. Lonberg-Holm, K. (1959) Biochim. Biophys. Acta, 35, 464- 472. 17. Garfinkel, D. & Hess, B. (1964) J . Biol. Chem. 239, 971 -983. 18. Gumaa, K. A. & McLean, P. (1969) Biochem. J . 115, 1009-1029. 19. Coe, E. L. (1970) Biochem. Biophys. Res. Commun. 38, 1105- 1112. 20. Gumaa, K. A. & McLean, P. (1969) Biochem. Biophys. Res. Commun. 35,824-831. 21. Wilson, J. E. (1967) Biochem. Biophys. Res. Commun. 28, 123- 127. 22. Katzen, H. M., Soderman, D. D. & Wiley, C. E. (1970) J. Biol. Chem. 245,408 1- 4096. 23. Crane, R. K. & Sols, A. (1954) J . Biol. Chem. 210, 597606. 24. Kosow, D. P. & Rose, I. A. (1970) Fed. Proc. 29,399. 25. Frieden, C. (1970) J. Biol. Chem. 245,5788- 5799. 26. Frieden, C. (1971) Annu. Rev. Biochem. 40,653-696, 27. Green, S. B. & Garfinkel, D. (1970) Comput. and Biomed. Res. 3, 166- 173. 28. Rhoads, D. G. & Pring, M. (1968) J . Theor. Biol. 20, 297-313. 29. Cleland, W. W. (1963) Nature (Lond.) 198, 463-465. 30. Kosow, D. P. & Rose, 1. A. (1968) J . Biol. Chem. 243, 3623 - 3630. 31. Kosow, D. P. & Rose, I. A. (1970) J. Biol. Chem. 245, 198- 204. 32. Cleland, W. W. (1963) Biochim. Biophys. Acta, 67, 104137. 33. Gulbinsky, J. S. & Cleland, W. W. (1968) Biochemistry, 7,566- 575. 34. Hammes, G. G. & Kochavi, D. (1962) J . Am. Chem. SOC. 84,2069-2073. 35. Ning, J., Purich, D. L. & Fromm, H. J. (1969) J. Biol. Chem. 244,3840- 3846. 36. Wedler, F. C. & Boyer, P. D. (1973) J . Theor. Bid. 28, 539-558. 37. Walter, C. (1967) J . Theor. Biol. 15, 1-33. 38. Cleland, W. W. (1963) Biochim. Biophys. Acta, 67, 173187. 39. Kosow, D. P. & Rose, I. A. (1972) Biochem. Biophys. Res. Commun. 48, 376-383. 40. Mosbach, K. & Mattiasson, B. (1970) Acta Chem. Scand. 24,2093 - 2100. 41. Webb, J. L. (1963) Enzyme and Metabolic Inhibitors, vol. 1, pp. 150- 180, Academic Press, New York.
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L. Garfinkel 42. Wilkinson, G. N. (1961) Biochem. J . 80,324-332. 43. Cleland, W. W. (1967) Adv. Enzymol. 29, 1- 32. 44. Hess, B., Chance, E. M., Busse, H. & Wurster, B. (1972) in Analysis and Simulation of Biochemical Systems (Hemker, H. C. & Hess, B., eds) pp. 119-143, NorthHolland, Amsterdam. 45. Ottaway, J. H. (1973) Biochem. J . 134, 729-736.
447 46. Reich, J. G., Wangermann, G., Falck, M. & Rohde, K. (1972) Eur. J . Biochem. 26, 368-379. 47. Dixon, M. & Webb, E. C. (1964) Enzymes, 2nd edn, p. 70, Academic Press, New York. 48. Noat, G., Ricard, J., Borel, M. & Got, C. (1968) Eur. J. Biochem. 5, 55 - 70. 49. Reich, J. G. (1 970) FEBS Lett. 9, 245 - 251. 50. Anderson, N. G. (1969) Clin. Chim. Acta, 25, 321-330.
L. Garfinkel, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A. 19174
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