A computer program for fitting and statistically analyzing initial rate data applied to bovine hexokinase type III isozyme

ARCHIVES

OF

BIOCHEMISTRY

AND

A Computer Program Data Applied DONALD Department

BIOPHYSICS

170, 587-600

(197%

for Fitting and Statistically Analyzing Initial to Bovine Hexokinase Type III lsozymel

B. SIANO, of Biochemistry

JUDITH

W. ZYSKIND,

and Biophysics, Received

Iowa

March

AND

State

HERBERT

University,

Ames,

Rate

J. FROMM Iowa

50010

27, 1975

Initial rate kinetics were used to study the kinetic mechanism of bovine liver Type III hexokinase (EC 3.7.1.1). The kinetic data were tit to initial rate equations with a computer program written in OMNITAB language. The correct weights to be used in the fitting of the kinetic data were obtained by replicate sample analysis and the effect of various weighting schemes upon the kinetic parameters are shown. The computer program can be changed to evaluate a variety of mathematical models by changing a single command statement in the OMNITAB program. It was concluded from experiments with the dead-end inhibitors for substrates, N-acetylglucosamine and ATP’-, that the kinetic mechanism is Random Bi Bi. Product inhibition experiments suggest that although the kinetic mechanism is random it may not involve rapid equilibration of enyzme and substrate interaction steps prior to the interconversion of the productive ternary complexes. Bovine liver Type III hexokinase appears to be inhibited by inorganic orthophosphate in the physiological range found in calf liver. This effect appears to be a result of the fact that phosphate is a competitive inhibitor of ATP.

It is now well established that hexokinase exists in multiple isozymic forms in mammalian tissue (1, 2). Two of the isozymes, the brain (I) and muscle (II) types, have been well characterized kinetically by a number of investigators. Brain hexokinase has been purified to homogeneity (3, 4) and is believed to exhibit a Random Bi Bi kinetic mechanism of action (5-7). The muscle isozyme 03) and glucokinase (9) are also believed to exhibit this same kinetic mechanism. One characteristic common to the hexokinase isozymes, with the exception of glucokinase (101, is their marked inhibition by micromolar concentrations of glucose-gP. It is currently held that this inhibition by product is responsible in large measure for the regulation and control of mamma-

lian hexokinases (8, 11). A very interesting feature of hexokinase regulation is associated with the observation that the glucose6-P inhibited brain hexokinase isozyme is activated by orthophosphate (12), whereas glucose-6-P inhibition of the muscle phosphotransferase is increased by this anion (8). Very little information is currently available on liver hexokinase (Type III). The enzyme is present in a variety of mammalian tissues with liver, lung, and spleen showing the greatest activity (13). Type III hexokinase has been partially purified by Gonzalez et al. (1) and Grossbard and Schimke (2). The only feature of liver hexokinase that appears to distinguish it from other hexokinases is its inhibition by glucose (2). In this report kinetic evidence is presented which indicates that hexokinase Type III, like all other mammalian hexokinases studied, exhibits a Random Bi Bi kinetic mechanism. The initial rate data were treated by computer with a very simple model fitting

1 This research was supported in part by Research Grant NS 10546 from the National Institutes of Health, United States Public Health Service and by Grant GB 33400 from the National Science Foundation. Journal Paper J-8222 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa 50010, Project 2004. 587 Copyright All rights

0 1975 by Academic Press, of reproduction in any form

Inc. reserved.

588

SIANO,

ZYSKIND,

program written in the OMNITAB language. The OMNITAB program permits a statistical evaluation of the various kinetic parameters, and it can be used to determine their sensitivity to different weighting schemes. A priori, there is no reason to weight double reciprocal plots of initial rate data to the l/(velocityj4 as is routinely done at the present time. This paper describes how one must weight initial rate data when making double reciprocal plots. EXPERIMENTAL

PROCEDURE

Materials. Glucose-6-P dehydrogenase was purchased from Boehringer Mannheim Corporation. ADP, &glucose, N-acetylglucosamine, p-mercaptoethanol, and HEPES (N-2-hydroxyethyl-piperazineN’-2-ethanesulfonic acid) were products of Calbiochem. ATP, glucose 6-phosphate, and TPN+ were purchased from Sigma. Uniformly labeled D [“Clglucose (specific radioactivity, 162 mCi/mole) was supplied by Schwa&Mann. Microgranular DEAE cellulose (DE521 was supplied by Reeve Angel. Deionized water was used for the preparation of reagents. All other reagents were of the highest purity available commercially. Methods. Bovine liver hexokinase Type III isozyme was purified by the procedure for rat liver hexokinase Type III isozyme used by Gonzalez et al. (1) as modified by Groasbard and Schimke (2) with the following changes. The source of the enzyme was bovine liver taken from a freshly slaughtered 18-moold male Shorthorn. Homogenization was carried out in a Waring Blendor. Desalting was accomplished by dialysis against standard phosphate-glucose buffer, pH 7.0 (10 rnrd potassium phosphate, 5 mM &mercaptoethanol, 5 mM Na,EDTA, and 10 mM glucose. The DEAE-cellulose column chromatography step used in the isolation of bovine liver hexokinase Type III isozyme was performed as described by Grossbard and Schimke (2). Only two peaks of hexokinase activity, glucokinase and hexokinase Type III isozyme, were observed in the elution profile of bovine liver extract on DEAE-cellulose. Grossbard and Schimke (2) observed all four hexokinase activities in their elution profile of rat liver extract on DEAE-cellulose. Another difference between the two elution profiles is that the glucokinase and hexokinase Type III isozyme in bovine liver were eluted from the DEAE-cellulose column at different KC1 concentrations than those observed by Grossbard and Schimke (2) for rat liver. We observed that the peak fractions of the substrateinhibited Type III hexokinase activity from bovine liver were eluted at approximately 0.135 M KC1

AND

FROMM

while Grossbard and Schimke in their rat liver preparation (21 found this activity eluting at 0.27 M KCl. These alterations in elution patterns may reflect differences between the rat and bovine enzymes. The preparation was not concentrated by ultrafiltration aRer the hydroxyapatite chromatography step nor was solid n-glucose added. The preparation had no detectable myokinase activity under the experimental conditions employed in the present study. The enzyme preparation had a specific activity of 0.023 unitslmg. A unit of activity is defined as 1 pmole of glucose-6-P produced/min at 28°C in a total reaction volume of 3.1 ml under the standard assay conditions. The standard assay contained 0.1 mM glucose, 3 mM ATP, 5 mM MgCl,, 0.12 mM TPN+, 20 mM HEPES, pH 7.7,5 mM pmercaptoethanol, and substantial excess glucose-6-P dehydrogenaee (1.7 units) in a total volume of 3.1 ml. The partially purified enzyme which was stored in standard phosphate-glucose buffer, pH 7.0, suffered a slow loss of activity over a 3-mo period. For initial rate studies the enzyme was diluted appropriately from storage into 20 mM HEPES buffer pH 7.0 containing 5 mM B-mercaptoethanol. The substrates and products of the hexokinase reaction were assayed enzymatically as previously described (14). Nucleotide solutions were assayed using molar extinction coefficients presented elsewhere (15). Initial velocity measurements were made in a model 15 Cary recording spectrophotometer (O-O.1 slide wire) as described previously (16) using 5-cm cells. Initial rate measurements made in the presence of glucose-6-P were carried out by following the conversion of n-[“Clglucose to [“Clglucose-6-P as outlined elsewhere (17). The stability constant of the MgATP’complex in 20 rnM HEPES, pH 7.7, was taken to be 100,000 M-’ (18). All initial rate experiments were carried out in 20 mM HEPES, pH 7.7. Initial velocities are expressed as molar product per minute. The inorganic phosphate (PJ content of calf liver tissue was determined as follows. A 2-3 g liver biopsy specimen was removed from the animal after anesthetizing the skin tissue in the rib cage region. The tissue was blotted quickly and immersed in liquid nitrogen. The tissue was prepared for Pi analysis according to the suggestions of Lowry and Passonneau (19) using the neutralization procedure with KOH, imidazole base, and KCl. P, was determined by the method of Herries (20). A value of 6 rnrw Pi was obtained for calf liver assuming that 75% of the tissue is water. This value is obviously an upper limit number. Data processing. The data were analyzed according the the usual method of weighted least squares by a computer program written in the OMNITAB language. This language, developed by the National Bureau of Standards (211, is well suited to the analysis of kinetic data because of its extreme simplicity

KINETIC

STUDIES

OF

BOVINE

in programming (considerably easier than FORTRAN, though not so general or powerful) and the ease with which the model equation used to fit the data can be changed to forms other than the simple Michaelis-Menten case. OMNITAB operations are carried out on an imaginary worksheet with 49 columns and 101 rows using commands in English in much the same way as one would use to explain the problem to someone who was to do the calculations on a desk calculator. For example, to add two columns of numbers one can use the command, ADD COLUMN 12 TO COL. 15 AND STORE THE RESULTS IN COL. 4. OMNITAB actually “sees” only the words ADD, 12, 15, and 4, while the other words are comments there only for the convenience of the programmer. Using similar, easily understood commands, one reads in the initial velocities together with the substrate and inhibitor concentrations, computes the reciprocal velocities and relevant substrate or inhibitor expressions according to the model under consideration, storing each new variable in separate columns. The command used for fitting the data has the form “FIT YIN COL. 1, WEIGHTS IN 2, X IN 3,4,5, STORE THE COEFFICIENTS IN 6 AND THE RESIDUALS IN COL. 7,” where Y is the dependent variable (reciprocal velocities) and “X” is the set of independent variables previously computed (and stored in columns 3,4,5 in this example). The best least-squares coefficients of the independent variables together with their standard deviations and the goodness of fit are stored in column 6 and are also automatically printed out together with other information on the tit (t and F statistics, means, and other data for analysis of variance). This single FIT command produces two pages of output describing the results. One can, of course, use any of the several alternative linear forms for the model equation provided that the weights are properly treated. The weight, Wi, of the ith data point is given by (22):

wi =

l/U,

(1)

A 1 (l/cr,z) N +=I where (+* is the standard deviation of the dependent variable, and N is the number of points. The standard deviation should, strictly speaking, be measured for each data point but this is rarely done because of the rather large number of measurements this would require. In order to have some basis for assigning the weights other than intuition, we measured the standard deviation of the velocities at just two velocities (one on the high side, one on the low) by replicated measurements. The variance was then assumed to follow an expression of the form o%*)

= cq”,

(2)

HEXOKINASE

TYPE

III

589

where c and (x are constants determined from the two measured standard deviations. The weights take the form for the reciprocal expression W(l/u,)

Substitution

= N&+(l/uJ/+

l/Og(l/uJ

= N~u,‘h=(uJl~

ui’ld(u,)

of (2) into

(31 gives

(3)

finally

WWUJ= =& i

(4)

We obtained a value for a of 1.89, whereas for equal weights in the reciprocal plot one would use a = 4 and for equal weights in the direct plot, a = 0. We carried out all of the fits using each of these values for (I and also carried out the usual graphical analysis by hand in order to obtain some notion about the sensitivity of the kinetic parameters to the different weighting schemes. The values for each of the kinetic parameters and their standard deviations were calculated from the coefficients and their standard deviations obtained by the program, using the standard formulas (24) for the propagation of errors, assuming they are uncorrelated. Comish-Bowden and Eisenthal(25, 261 have correctly pointed out some of the assumptions that are necessary for the method of least squares to obtain optimal coefficients. We have already treated the problem of obtaining the correct weights. Another consideration is that the errors in the dependent variable should be normally distributed about their mean. To examine this problem we made 30 repeated measurements of the initial velocity under the same conditions. We found that the standard deviation of the velocity was 4.0% of the mean, and the normalized third and fourth moment about the mean, p, and & were 0.15 and 2.48, respectively. The values for a normal distribution are 0.0 and 3.0, respectively. The values found are therefore judged to be sufficiently close to the values for the normal distribution that the method of least squares is not seriously jeopardized by the failure of this assumption. The OMNITAB computerprogmm. The most general model for reversible inhibition in a Sequential Bi Bi Mechanism which is first order with respect to each substrate is

!=A [(1+4)+>(1+4,) U

+2(1+-L)+E/(1+LJ

In the OMNITAB used to fit data inhibitor, and

(5)

example shown here, this model is where ATP is the substrate, Z is an B (n-glucose) is held constant, so

-1 = c, + C,(Z) + c, U

(i)

+

Cd(f),

(5al

590

SIANO,

OMNITAB

SEQUENTIAL

:

H.J.

FROMM IOWA STATE JANUARY

s

FOR UNIT FOR UNIT OTHERWISE

CHANGE RAISE RAISE MULTIPLY DIVIDE AVERAGE DIVIDE 8 $ 8 $ $ 8

COLS

$W,A,B,U

1, 2, 3, 5

WTS. IN RECIPROCAL PLOTS PUT ALPHA WTS. IN DIRECT PLOTS PUT ALPHA = 0 DETERMINE ALPHA EXPERIMENTALLY

rb S CALCULATE s ADD 1.0 TC 0.0 AND DIVIDE 1.0 BY COL DIVIDE 1.0 BY COL DIVIDE 1.0 BY COL MULTIPLY COL 12 MULTIPLY COL 12 MULTIPLY COL 13 MULTIPLY COL 14 CALCULATE

THE

INDEPENDENT

STORE IN COL 6 1 AND STORE IN COL 2 AND STORE IN COL 3 AND STORE IN COL BY COL 13 AND STORE BY COL 5 AND STORE BY COL 5 AND STORE BY COL 5 AND STORE THE

11 12 13 IN IN IN IN

$(COEF SWREXPN $(1/A) $(1/B) $GiA*B) WA) Iww $(YA*B)

COL 14 COL 35 COL 36 COL 37

FIT

VARIABLES

NOT

IN

MODEL

FROM

$(-ALPHA) $(V”4) $(V**(-ALPHA)) $(V**(CALPHAB $(lN**(I-ALPHA)) $GVEIGHTS) SCNORMALIZED

16 16

FIT

STATEMENT

COL 5, 1 IN COL 6, l/A IN COL 12, l/B IN COL 13, l/A*B 14, I/A IN COL 35, I/B IN COL 36, I/(A*B) IN COL 37 Y IN

11, WTS

1 IN

RATE

EQ)

WEIGHTS

THE SIGN OF THE ELEMENT IN COL 4 1 TO THE +4.0 POWER AND STORE IN COL 1 TO THE l 1,4* POWER AND STORE IN COL COL 15 BY COL 16 AND STORE IN COL 17 1.0 BY COL 17 AND STORE IN COL 18 OF COL 17 AND STORE IN COL 19 COL 17 BY COL 19 AND STORE IN COL 20

I IN COL

= 4

VARIABLES

COL COL

ELIMINATE

FROMM

UNIVERSITY 1975

READ THE FOLLOWING DATA INTO 0.2OE-66 0.64E-03 0.64E-03 0.66 0.25366 O.O5E-03 0.64E-03 0.00 0.25366 O.O7E-03 0.64E-03 0.66 0.33396 O.lOE-63 0.64E-03 0.00 0.333-06 0.20E-03 O.ME-03 0.69 O.O7E-96 O.O4E-03 O.O4E-03 2.OE-03 O.C@E-66 O.O5E-03 O.ME-03 2.OE-03 O.llE-66 O.O7E-03 0.64E-03 2.OE-03 O.llE-96 O.lOE-03 0.64E-03 2.OE-63 O.l8E-66 0.20E-03 O.O4E-03 2.OE-03 READ ALPHA INTO COL 4 1.89

:

AND

BI BI PROGRAM

8

: $ $

ZYSKIND,

IN

20, X IN 5,6,12,35,

COEF

IN

22, RES

: CALCULATE 96 DIFFERENCE I ADD COL 23 TO COL 11 AND STORE IN COL 25 RAISE COL 25 TO THE -1.0 POWER AND STORE IN COL SUBTRACT COL 26 FROM COL 1 AND STORE IN COL 27 DIVIDE COL 27 BY COL 1 AND STORE IN COL 28 MULTIPLY COL 28 BY 196.0 AND STORE IN COL 29 ABS COL 29 AND STORE IN COL 30 AVERAGE COL 30 AND STORE IN COL 31 I $ PRINT OUT THE RESULTS 8 HEAD COL 1N HEAD COL 2/A HEAD COL 3/B HEAD COL 5/I HEAD COL llIlN(EXP) HEAD COL BO/WEIGHTS HEAD COL 23/RESIDUALS

IN

26

IN

23

WEIGHTS)

KINETIC OMNITAB

STUDIES

OF

BOVINE

HEXOKINASE

TYPE

591

III

Program-Continued HEAD COL 29/% DIFFERENCE PRINT COLS 1 I2 ,3 , 5 ,20,23,29,11 FORMAT A(//,3X, ‘AVG. % DIFFERENCE IS, F9.4) ABRIDGE A ROW 1 OF COL 31 3 $ DO A COMPLETE ANALYSXS OF THE RESIDUALS $ SYSTEMATIC ERROR IN THE DATA OR MISFIT $ MODEL. STATISTICAL ANALYSIS OF COL 23

TITLEX TITLEY PLOT TITLEX PLOT TITLEY TITLEX PLOT STOP

IN ORDER DUE TO AN

TO DETECT INAPPROPRIATE

ANY

: CREATE THE PLOTS. $ l/A IN COL 11 VS COL 12 l/B COL 11 VS COL 13 RESIDUALS V COL 23 VSCOLl

where

the C’s are the constants actually found in the subprogram. These are then used to calculate (by hand) the desired values of V, and the K’s, together with their standard deviations. The modification of the FIT statement by the elimination of one or more independent variables is the only one necessary to encompass all of the models commonly used in enzyme kinetics. This feature of the OMNITAB FIT statement is a good deal more convenient than the corresponding manipulation in most FORTRAN programs, where whole blocks of code must be modified. In addition to the t values of the coefficients and the measures, S* and <% dim, of the goodness of fit of the data to the proposed model, the program carries out a very extensive analysis of the residuals which can also be used to detect any systematic variation of the error with velocity and substrate concentrations. This information can be used to make more reliable and quantifiably rigorous judgments about the important question of the appropriateness of an hypothesized kinetic model to explain the data than is possible by any graphical procedure. FIT

RESULTS

Initial rate experiments. Initial rate studies on purified bovine liver Type III hexokinase are shown in Figs. 1 and 2. Substrate inhibition was observed at concentrations about 0.15 mM glucose, and the level of glucose was therefore kept below this value. The double reciprocal plots were drawn by the plotter of the computer. The computer, programmed in OMNITAB language, was used to plot and evaluate the kinetic parameters using the model of Eq. (5). Table I describes the kinetic pa-

rameters which were obtained from this and other experiments. The Michaelis constants for ATP and n-glucose calculated from the data of Figs. 1 and 2 were 0.26 and 17 PM, respectively, using a value of 1.89 for cr. Grossbard and Schimke (2) reported values of 0.98 and 7.0 PM for the Michealis constants of ATP and n-glucose, respectively. The results of Figs. 1 and 2 which indicate convergence of the double reciprocal plots show that the kinetic mechanisms of liver hexokinase is sequential rather than Ping-Pong.2 Table I also verifies this because the standard deviation of KAB is about 13% of the least squares estimate of KAB and so it is certainly significantly different from zero. These findings are consistent with the view that all mammalian hexokinases exhibit sequential kinetic mechanisms. Dead-end inhibition studies. One of the most convenient and least ambiguous methods for determining whether kinetic mechanisms are ordered or random involves the use of dead-end competitive inhibitors (14). This procedure, which was employed originally to determine whether the kinetic mechanism of yeast hexokinase was Ordered or Rapid Equilibrium Random Bi Bi (14) requires that a deadend competitive inhibitor be available for each substrate. It is possible from the inhi4 The descriptive

(27).

nomenclature

is that

of Cleland

I/v

x 10-5

0; 8.00

I 16.00

I 2u.00

1 uo.00

T 32.00 l/Glucose

1 -3 U8.00

FIG. 1. A plot of reciprocal of initial velocity(u) tion of glucose in the presence of 1.65 mM (O), 0.50 mM MgATP*(0). Glucose was varied in the range at 2 mM. The reaction mixture, 3.1 ml, contained details are described in Experimental Procedure.

l/V

I 56.00

1 6U.00

1 72.00

x IO

with respect to reciprocal of molar concentrarnre CO), 0.28 rnxr CO), 0.19 rnxr (a), and 0.15 0.017-0.099 mM. Free MgS+ was maintained 0.013 units of enzyme. Other experimental

x 10-5

x 5

8 s n 8 a

;;

x CA 0.00

1 8.00

1 16.00

1 2U.00

l/ATP

I 32.00 x 10-2

1 uo.00

I UB.00

I 56.00

I GU.00

FIG. 2. A plot of reciprocal of initial velocity (v) with respect to reciprocal of molar concentration of MgATP*in the presence of .099 rnru (O), .046 rnru CO), .031 mM (a), .022 mM (a), and .017 rnM (0) glucose. MgATP*was varied in the range 0.15-1.65 mhi. Free Mg*+ was maintained at 2 mre. The reaction mixture, 3.1 ml, contained 0.013 units of enzyme. Other experimental details are described in Experimental Procedure. 592

KINETIC

STUDIES

OF

BOVINE

HEXOKINASE

TABLE A COMPARISON Model

Inhibitor

Parameter

OF THE KINETIC

Value

5

None

V, K.4 KB

K ( 8 G-f) S 6a

NAG

K, ( %Kiiff) S

6

ATP’-

K* ( %K&f~ S

8

ADP

K, K,i (%KIYlfl) S

9

G6P

Ki ( %K&T) S

“The compared deviation

X X x x

10eB hi/min 1o-4 M 10-S M 10-g hfr

M

M M

Value

1.31 8.62

21 5.9

1.94 7.71

19 7.5 1.94% 3.40

2.78 6.72

21 12 1.97% 4.41

12 36

2.04 2.13

2.52 1.90

12 34

1.61 2.32

12 34 2.52% 9.03

Graphical estimate

% Error

2.72 2.50 1.56 2.81

11 7.4 11 13 1.65% 4.69

2.57 1.49 4.58

8.0 7.0

1.93 1.66

2.52% 9.06

4.03 7.11 1.81

260 12 6.3 5.96% .64

2.18 8.87 1.74

8.0 14 7.6 5.32% .96

3.30 14.8 1.65

12.0 19 15 6.08% 1.08

1.04 18.4 3.25

6.51 1.43

63 9.1

11.0 1.30

74 9.6 3.31% 2.63

15.4 1.33

84 9.9 3.37% 2.80

2.2 2.07

x 105 X lo-’ X lo-’

% Error 6.0 4.7 6.1 12 1.48% 3.47

2.55% 8.58

M

(I = 4.0

2.76 2.63 1.71 2.45

x 10’ M

a = 1.89 Value

2.19% 2.82

x 10-a M x 10-s M

III”

3.6 3.5 4.2 14 1.48% 2.46

x 104

X 1o-4 X 1o-5 X lo-’

% Error

FOR HEXOKINASE

2.80 2.71 1.80 2.12

x 10’ x lo-‘M x 10-S M

593

III

I

PARAMETERS

a=0

unite

TYPE

x 106

3.81% 1.99

kinetic parameters found by using the method of least squares with three weighting schemes is with the results obtained by the method of reciprocal plots. The “8 error” is the standard as a percent of the mean, and v = 21 for all the models except 9, which had 16.

bition patterns relative to each substrate to make a choice between ordered and random mechanisms. In the case of the Rapid Equilibrium Random Bi Bi mechanism a competitive dead-end inhibitor for A, I,, will react as follows:

E +I, =EZ,,Ki; EB + I, = EZ,B, Kii; EZ, + B = EZ,B,Kiii

When a dead-end competitive inhibitor for the rate expression described by Eq. (6) is modified so that the K,IA term is not altered by inhibitor; however, the KB/B term will be multiplied by the factor (1 + Zb/Kii), here Kit represents the dissociation constant for the reaction EA + Zb = EAZb and I, in Eq. (6) is replaced by lb, i.e.,

B is used in place of A,

j,1 = $ m CI+:+:

Equation (5) will be modified to account for the effect of inhibitor as shown in Eq. (6).

(6)

+KAB (A)(B)

(

I+!!

Kii >

(64

(1 +$!)I

For the Ordered Bi Bi mechanism, a competitive inhibitor of substrate A will cause a kinetic effect which is described by Eq. (6) but where K,i is replaced by Ki. On the

SIANO,

594

ZYSKIND,

other hand, a unique rate expression is obtained when a dead-end competitive inhibitor for B(16) is used. The only interaction of Ib with enzyme is, EA + Ib = EAIb, Ki. Equation (7) describes the rate equation for inhibition by lb. 1 1 -=Vi

[

Vm

+;c:+

x*

$>

+&,.

(7f

Equation (7) shows that a competitive inhibitor for B in an Ordered Bi Bi mechanism will act as an uncompetitive inhibitor with respect to substrate A. It is therefore possible to make a choice between the Ordered and Random Bi Bi mechanisms. In the experiments reported here, Nacetylglucosamine, a competive inhibitor for glucose, and ATP4-, a competitive inhibitor for the substrate MgATP2-, were used. Figures 3-6 illustrate the effects of the dead-end inhibitors on the kinetics of l/v

AND

FROMM

the liver hexokinase reaction. The data demonstrate that the competitive inhibitor for one substrate is a noncompetitive inhibitor for the other substrate. These findings are consistent with random substrate binding [Eqs. (6) and (6a)] and serve to exclude ordered mechanisms for liver hexokinase. It is of interest to note that all four mammalian hexokinase isozymes studied, brain (5, 6), muscle (a), glucokinase (91, and Type III liver hexokinase, exhibit very similar kinetic mechanisms. It is of some interest to see the effect of different weighting functions upon the goodness of fit and the values obtained for the kinetic parameters. Table I shows the effect of letting a! = 0 (equal weights in the direct plot), cy = 4 (equal weights in the reciprocal plot), and (Y= 1.89, which is our experimentally determined value. The graphical estimate is obtained by an “eyeball fit” of the data using the usual reciprocal plots. The values obtained for V,, KA, Ks, and KAB are seen to be rather insensi-

x 10-h

FIG. 3. A plot of reciprocal of initial velocity (u) with respect to reciprocal of molar concentration of glucose in the absence (0) and presence of 0.10 rnre CO), 0.19 mM (a), 0.32 mr.r (a), and 0.54 mM (0) N-acetylglucosamine. MgATPZconcentration was held constant at 0.5 rnbf and glucose was varied in the range 0.017-0.108 mru. Free Mg2+ was maintained at 2 mM. The reaction mixture, 3.1 ml, contained 0.0061 units of enzyme. Other experimental details are described in Experimental Procedure.

8 ti

I 0.00

I e.00

I 16.00

I 2U.00

1 X.00

l/ATP

I UO.00

I 118.00

I 56.00

FIG. 4. A plot of reciprocal of initial velocity (v) with respect to reciprocal tion of MgATP*in the absence (0) and presence of 0.10 mru (0),0.19 mM CO), 0.54 rnM (0) IV-acetylglucosamine. Glucose concentration was held constant MgATP*was varied in the range 0.X-1.7 mr.r. The reaction mixture, 3.1 ml, units of enzyme. Other experimental details are described in Experimental

8 d

I 0.00

I 1.00

I E.00

1 3.00 l/ATP

1 GU.00

x lo-*

I 11.00 x 1O-3

I 5.00

I 6.00

I 7.00

molar concentra0.32 mM C(B), and at 0.94 mru and contained 0.0061 Procedure.

1 8.00

FIG. 5. A plot of reciprocal of initial velocity (u) with respect to reciprocal molar concentration of MgATPZin the absence CO), and presence of 1.8 mM (a,), 2.8 mM C(B), and 3.9 mM (0) ATP’-. Glucose was held constant at 0.04 rnr+r and MgATP*was varied in the range 0.15-1.7 mM. The reaction mixture, 3.1 ml, contained 0.0061 units of enzyme. Other experimental details are described in Experimental Procedure. 595

596

SIANO, ZYSKIND,

AND FROMM

FIG. 6. A plot of reciprocal of initial velocity (u) with respect to reciprocal molar coucentration of glucose in the absence (0) and presence of 0.40 miu (O), 1.3 mM (a,), 2.4 rnbd (al, and 3.6 mM (0) ATP’-. MgATP*- was held constant at 0.5 mM and glucose was varied in the range 0.017-0.108 mM. The reaction mixture, 3.1 ml, contained 0.0027 units of enzyme. Other experimental details are described in Experimental Procedure.

tive to CX,whereas, at the other extreme, the values for the inhibition constants are somewhat more variable. The standard deviations of Kt for ADP and G6P obtained are large so that the best value for the Kls is not significantly different from zero. The term involving it is retained in the model equation, however, for logical completeness (see below). The two measures for the goodness of the fit of the data used to obtain the kinetic parameters of the model assumed are the average absolute % difference between the observed and calculated values for l/u, <% dim, and the standard deviation of the fit, defined as

where Y = N - n - 1 is the number of degrees of freedom left after fitting the N data points to the n + 1 adjustable parame-

ters in the model. The values for s and <% diff, cited when an inhibitor is used are those for the data fitted by the model with no constraints on the other kinetic parameters. The values of the kinetic parameters are obtained in a manner analogous to that used in the traditional reciprocal plot method, i.e., the values for KA, KB, and KAB are obtained from the data taken in the absence of an inhibitor by using model 5 and then they are used to obtain the inhibition constants through the analysis of data taken in the presence of an inhibitor. Product Inhibition Experiments In order to gain additional insight into the kinetic mechanism of liver hexokinase Type III product inhibition experiments were undertaken. In the original proposal by Alberty (28), it was suggested that kinetic studies in the presence and absence of product should permit one to make an unambiguous choice from among certain bireactant mechanisms. It was subse-

KINETIC

STUDIES

OF

BOVINE

quently shown by Fromm and Nelson (29) that this kinetic approach is compromised by the formation of abortive complexes of enzyme, substrate, and product. Nevertheless, kinetic experiments in the presence of product do permit one to come to definitive conclusions on substrate and product interactions with enzymes if information is already available on the nature of the kinetic mechanism. In data not presented it was observed that ADP is a noncompetitive inhibitor of both substrates ATP and glucose. In order to rationalize these findings with the rapid equilibrium Random Bi Bi mechanism, it is necessary to assume formation of the abortive ternary complexes, E-glucoseADP, and E-ATP-ADP. The former complex appears to be a reasonable possibility when considered in the context of what the transition state might be expected to be for a concerted mechanism involving hexokinase substrates and products. The abortive complex involving ADP and ATP appears to be a viable possibility only if one assumes that ADP may in some way bind at the glucose portion of the active site. Rudolph and Fromm (30) have shown from computer simulation studies of the steady-state Random Bi Bi mechanism that product inhibition patterns similar to those described for ADP may be applicable. Furthermore, the simulations also showed that dead-end competitive inhibitors will give patterns similar to those presented in Figs. 3-6 for the steady-state Random Bi Bi mechanism. If it is assumed that the kinetic mechanism approaches the rapid equilibrium case at the limit, Equation (8) may be used to describe the results of ADP (I) inhibition. 1 vi

=

&

[l

+

z

-5%

+:

(1

(1

+i&,

+ (AM?)

(1

+-L)

(8) +i,]

.

In Eq. (a), Ki, Kii, and K
HEXOKINASE

TYPE

597

III

E-ATP + ADP = E-ATP-ADP, Ktr; E-glucose + ADP = E-glucose-ADP, Kiii. Table I indicates the values obtained for the dissociation constants. One of the interesting characteristics of mammalian hexokinases is their potent inhibition by glucose-6-P. In the case of brain hexokinase glucose-6-P is a competitive inhibitor of ATP and a noncompetitive inhibitor of glucose (2, 31) whereas this product is a noncompetitive inhibitor of both substrates with the muscle (Type II) isozyme (8). It was found (data not presented) that glucose-6-P is a competitive inhibitor of ATP and either a noncompetitive or an uncompetitive inhibitor of glucose from experiments with Type III hexokinase. Inhibition with respect to glucose was statistically more significant when the model for noncompetitive inhibition was used. Equation (9) describes the rate expression for the rapid equilibrium Random Bi Bi mechanism assuming the formation of the abortive complex, enzyme-glucose-glucose-6-P. vi1

=

;

[l

+2

(1

+&,

(g)

In Eq. (9), Z is taken to be glucose-6-P and KI and Kii are described by the following reactions: E + glucose-6-P = E-glucosedP, Ki; E-glucose + glucose-6-P = E-glucose-glucose-6-P, Kii; E-glucose-6-P + glucose = E-glucose-glucose-6-P, Ki
598

SIANO,

ZYSKIND,

nase by reversing inhibition by glucose-gP (12, 32). This effect occurs in the physiological range of Pi only when glucose-6-P is present. On the other hand, there is some evidence that Pi in muscle tissue inhibits hexokinase activity independently of glucose-6-P (8). It was of interest to determine the effect of Pi on the activity of liver hexokinase. Figure 7 illustrates that Pi inhibits the liver enzyme at levels of Pi in the physiological concentration range. In experiments which are not presented, it was found that Pi is a linear competitive inhibitor of ATP and a linear noncompetitive inhibitor of glucose. Furthermore, Pi does not appear to reverse the effects of glucose6-P when it is present as a product inhibitor. These findings suggest that liver hexokinase III responds to Pi very much the way muscle hexokinase (Type II) does (8). DISCUSSION

The initial rate data of this report were treated with model or equation fitting computer programs written in the OMNITAB language. To the best of our knowledge all other model fitting programs which are currently in vogue that use weighted velocities, weight the ui’s to the 4th power. There is no reason to believe that weighting to the 4th power is valid. The data of Table I illustrate quite clearly the importance of doing replicate analysis in order to establish the factor (Y and the proper weighting of the kinetic data.

AND FROMM

aor

01 0

10

20

FIG. 7. A plot of initial velocity as a function of total concentration of phosphate. Glucose, MgATP2-, and Mg”+ concentrations were held constant at 66 PM, 0.167 mrq and 8 mM, respectively. Ionic strength was maintained at 0.11 by supplemental addition of NaCl. In addition to the above, each reaction mixture contained 20 mM HEPES buffer, pH 7.7, 5 mM &mercaptoethanol, 120 PM TPN+, and substantial excess of glucose-6-P dehydrogenase. In calculation of free magnesium concentration 100 M-l was used as the binding constant for PI.

One of the very attractive features of the OMNITAB program presented in this report is that it may be used with a large number of different models by simply changing a single command statement. The data of this report strongly suggest that the kinetic mechanism of liver hexokinase, Type III isozyme is of the Random Bi Bi type as depicted in Scheme I. This conclusion was reached from the results in Figs. 1 and 2 which show that the mechaE- ADP

E-ATP

SCHEME

30

qa Id

1

KINETIC

STUDIES

OF

BOVINE

nism is sequential and from the findings of Figs. 3-6 which indicate that the substrates add randomly to the enzyme. The product inhibition data also appear to support this contention; however, it is not clear at this time whether the random mechanism is of the steady-state or rapid equilibrium type. It is well established that liver hexokinase Type III exhibits substrate inhibition at elevated levels of glucose (2). This fact is also in keeping with either of the Random Bi Bi type mechanisms. In the case of the rapid equilibrium pathway two or more molecules of glucose would be required to add to hexokinase, one at the active site and one at an inhibitory site, in order to explain substrate inhibition. On the other hand, computer simulation studies by Rudolph and Fromm (30) of the steady-state Random Bi Bi Mechanism indicate that substrate inhibition is largely a consequence of certain relationships among various rate constants. A choice between these two possible mechanisms can certainly be made from isotope exchange studies, provided technical problems involving kinetic irreversibility of the system do not arise (11). Although it would appear that bovine liver hexokinase Type III is not functional at the levels of glucose present in blood because of inhibition by glucose above 0.1 mM, this may not be physiologically signiticant in Go. In ruminant animals such as the bovine, glucokinase activity is either very low or absent in liver tissue (321, and it appears that the liver does not serve to remove blood glucose in these animals. In ruminants, ingested glucose is thought to be converted to fatty acids in the gut, whereas glucose is produced via gluconeogenesis in liver (32). It is not at all clear whether Pi is involved in the regulation of liver hexokinase activity. The results of this report suggest that like muscle hexokinase (8), liver hexokinase is inhibited by Pi in the millimolar concentration range. We proposed what we believe is a rational explanation as to why PI serves to activate glucose-6-P inhibited brain hexokinase, and why it tends to reinforce glucose-6-P inhibition in muscle (8). The physiological impli-

HEXOKINASE

TYPE

599

III

cations of Pi inhibition of hexokinase in an aerobic tissue such as liver are not obvious. Part of the explanation may be that when glucosyl units are being provided from glycogen, i.e., when Pi and glucoseSP are elevated, it would be advantageous to inhibit that small amount of hexokinase activity which does normally exist in bovine liver. ACKNOWLEDGMENT The authors acknowledge assistance of W. R. Ellison.

the excellent

technical

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136,95-98. 21. “OMNITAB, A Computer Program for Statistical and Numerical Analysis,” National Bureau of Standards Handbook (19661, p. 101. 22. BEVINGTON, P. R. (1969) Data Reduction and Error Analysis for the Physical Sciences, p. 188, McGraw-Hill, N.Y. 23. ALBERTY, R. A. (1953) J. Amer. Chem. Sot. 75, 1928-1932. 24. BEVINGTON, P. R. (1969) Data Reduction and Error Analysis for the Physical Sciences, Chap. 4, McGraw-Hill, N.Y. 25. EIBENTHAL, R., AND CORNISH-BOWDEN, A. (19’74) Biochem. J 139, 7X-720.

AND FROMM A., AND EISENTHAL, R. (1974) Biochem. J 139, 721-730. CLELAND, W. W. (1963) Biochim. Biophys. Actn 67, 104-137. ALBERTY, R. A. (1958) J. Amer. Chem. Sot. 80, 1777-1782. FROMM, H. J., AND NELSON, D. R. (1962) J. Biol. Chem. 237, 215-220. RUDOLPH, F. B., AND FROMM, J. H. (1971) J. Biol. Chem. 246, 6611-6619. FROMM, H. J., AND ZEWE, V. (1962) J. Biol. Chem. 237, 1661-1667. BALLARD, F. J. (1965) Comp. Biochem. Physiol. 14,437-443.

26. CORNISH-BOWDEN, 27.

28. 29. 30. 31. 32.