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An on-line method for the collection and analysis of quasi-steady-state kinetic data for enzyme-catalyzed reactions
ANALYTICAL
BIOCHEMISTRY
69, 590-606
(1975)
An On-Line Method for the Collection Analysis of Quasi-Steady-State Kinetic for
Enzyme-Catalyzed
and Data
Reactions’
CHARLES WALTER The University
of Houston, Department Houston, Texas
of Chemical 77004
Engineering,
HEIDI EBERSPAECHERAND J. PATRICK HUGHES The University of Texas System Cancer Center, M.D. Anderson Hospital Tumor Institute at Houston, Department of Biomathematics, Biochemistry and Biology, and Community Health Computing, Incorporated Houston, Texas 77025
and
Received May 13, 1974; accepted June 19, 1975 A special mixing device for initiating enzyme-catalyzed reactions is used to rapidly achieve an unperturbed quasi-steady state. An on-line computer is employed to sample the initial conditions, the mixing time, and concentrations that change as a function of time during this quasi-steady state phase. A statistical method for estimating initial, quasi-steady state rates from the time course of the enzymecatalyzed reaction is described. Practical considerations for using this parameter estimation system lead to the conclusion that for the enzyme-catalyzed reaction tested, the extent overall reaction should be above .2% for high initial substrate concentrations, and above 1% for initial substrate concentrations in the range of the Michaelis constant. Application of this method to a typical enzyme-catalyzed reaction suggests that objective estimates of initial rates from a given set of concentrations and corresponding times can be obtained with a standard error in the range of 2-3%, but that reproducibility is not better than about 10%. When this procedure was used to estimate initial rates for the glycerol dehydrogenasecatalyzed oxidation of glycerol by NAD, it was found that this enzyme did not behave according to the classical “Michaelis-Menten” mechanism of enzyme action.
Quasi-steady-state (1) enzyme kinetic data usually consists of a series of substrate or product concentrations, c,, obtained at discrete times, I*, or of a continuous record of c(t) plotted on a stripchart recorder. The usual procedure in enzyme studies is to use the kinetic data to obtain a single parameter called the “initial, quasi-steady-state rate” (v(O)). v(0) obtained in this way is not an experimental observable. It is a parameter that was estimated from observables. As in the case of other estimated ’ This work was supported in part by grants from the NSF (GB-7 110 and GB-20612) the Robert A. Welch Foundation (G517), and the NIH (CA-l 1430).
Copyright @ 1975 by Academic Press, Inc. All rights of reproduction in my form resewed.
ENZYME-CATALYZED
591
REACTIONS
quantities, it is necessary to describe explicitly the method one uses to obtain estimates for initial, quasi-steady-state rates. It is the purpose of this paper to describe an objective method for obtaining accurate estimates of initial quasi-steady-state rates and their variances. MATHEMATICAL MODEL FOR THE TIME-COURSE OF ENZYME-CATALYZED REACTIONS
We (2) have found that during the early, quasi-steady-state stages of many enzyme reactions, an approximate relationship between the product concentration, P and time, r is
[II
A + BP(f) + CP(r)’ = r where A, B and C are constant coefficients. If the substrate, experimental observable, P can be estimated from P(f) = S(0) - S(t)
S is the
PI
P(0) = 0,
provided insignificant quantities of S and P are bound to the enzyme at any given instant. Thus Eq. [2] is usually a valid approximation provided S(0) % E(0). Differentiation of Eq. [l] yields the quasi-steady-state rate: 6(t) = l/[B Substitution rate :
+ 2CP(t)].
131
of P(0) = 0 into Eq. [3] yields the initial,
quasi-steady-state
v(0) = P(O) = l/B.
[41
If correction of Eq. [4] due to a small off-set at t = 0 is necessary, it can be accomplished by solving Eq. [l] for P when t = 0, and inserting the result into Eq. [ 31. STATISTICAL
An estimation
METHOD
FOR
PARAMETER
ESTIMATIONS
of A, B, and C in Eq. [ I] can be obtained
from Eq.
[S-73:
A = (~1~170 - 7~~71+ GT~T.J&
[51
B = (ng, - n-2~0- 7~~~~)/8,
[61
c = (77370- 7r#g,- 7f47*)/&
[71
where ?T1= XP”XP4 - (ZP”)“, 7r* = XPZP4 - XP2ZP3, 7r3 = ZPXP3 - (XP”)‘,
592 = 7~~= 7~~= 70 = 71= 5-2= 6=
7~4
WALTER,
EBERSPAECHAR
NZP2 - (ZP)“, NZP4 - (ZP”)“, NZP3 - CPZP2, Et, X.tP, X.tP2, 2XPZP2ZP3 - N(ZP3)2 + NZP2ZP4
AND
HUGHES
- (ZP”)” - (ZP)2XP4,
and N is the total number of data points collected. B, and C are
The variances of A,
var(A) = 7r1$/6,
PI
var(B) = 7~+#&,
[91
var(C) = r4$/8,
[lOI
where +t2-Ar0-BT1-CT2 N-3
’
MATERIALS Glycerol
dehydrogenase used in these experiments was obtained from Aerobacter aerogenes. NAD, and Trizma Base were obtained from Sigma Chemical Company; spectroquality glycerol was obtained from Matheson, Coleman, and Bell Company; and the phosphates and carbonates for the buffers were obtained from J. T. Baker Company. Tris was passed through a DEAE-Sephadex A-50 (Pharmacia, Inc.) ion-exchange column. Water was prepared by distillation from a large Barnstead still followed by redistillation on a smaller, all glass Barnstead still. Hardware
Figure 1 is a system diagram describing the hardware used here. The components include: (a) detection system; (b) mixing device; (c) interface; (d) timing mechanism; (e) analog-to-digital converters (A/D converters); (f) small digital computer; (g) devices for analog and digital display; (h) a device for data storage; (i) a device for inputting scale factors, etc. and printing out data and estimated parameters. Our detection systems include two different spectrophotometers and a spectrofluorometer. The methods are not limited to these devices and could easily be extended to NMR machines, polarimeters, refractometers, ORD machines, etc. The mixing device we use is illustrated in Fig. 2. When in operation, this device is situated over a sample holder located in the light path of a spectrophotometer or spectrofluorometer. The sample holder contains a known volume of a solution containing all of the ingredients necessary
ENZYME-CATALYZED
593
REACTIONS
Lint
0. D.
Tapes
0 Q
1 Q
ov or 1.4v Second Spectrometer
ov or 1.4v
SYSTEM
DIAGRAM
FIG. 1. System diagram for collecting and analyzing enzyme kinetic data.
for the enzyme-catalyzed reaction except one. A stripchart recorder or direct readout from the spectrophotometer is used to insure that the initial condition is constant. During this period a small, known volume of a solution containing the missing ingredient is placed in the plastic spoon connected to the mixing device. The door to the spoon chamber is closed, the door between the spoon chamber and the sample holder is opened, a switch on the mixing device is activated, and the computer samples the initial condition and rings a bell. If necessary, the computer calculates the diluted initial condition expected after addition of the small volume to the rest of the assay mixture. Next, the plunger on the mixing device is smoothly depressed so that the spoon dips into the assay solution. A switch informs the computer that the spoon has been
WALTER,
EBERSPAECHAR
AND HUGHES
MIXING DEVICE
1 Spectrophofometer - Sample Houring Cover
2 Stirring LIeviceAdapterwith PlungerDoor 3 MwingSpoonLoading-Door 4 Plunger Housmgwth Arresting Wingnut 5 Sfmr Motor - Activating Micro-Switch
FIG.
6 Computer-fnferfaced Microswtch 7 Plungerwith Relsaamg Springs 8 Teflon Mixmg Spoon 9 Plunger stop
2. Mixing device used to introduce solutions into cuvettes.
depressed, and the computer begins to count the number of milliseconds the spoon is down. Simultaneously, a small motor rotates the spoon while it is held at its lowest point in the assay solution. Within 1 set, mixing is complete, and the plunger is allowed to smoothly return to its original position. When the spoon leaves its lowest position, the motor stops and the spoon stops rotating. As soon as the spoon clears the light-
ENZYME-CATALYZED
595
REACTIONS
(B&L)
ICARY)
(A & Bl
X
k
2 - (ZkX-b) 1
+ b 7 -
FIG. 3. Triple scaling amplifier interface analog diagram.
path of the instrument, a switch informs the computer that mixing is over. The total number of milliseconds required for mixing is estimated by dividing the total time the spoon is down by two; this estimate is retained in the computer’s memory as T,. The signal that mixing is over is also the signal to the computer to begin sampling the analog signal from the instrument as a function of time. The sampling continues 60 times/set for 508 set, or until it is aborted manually with a switch on the mixing apparatus. The interface consists of the three sections indicated in the schematic diagram in Fig. 3. Each section connects a particular instrument to an A/D converter, and each is designed to scale its instrument’s output signal to the range of the A/D converter (+ 1 V). The two aspects of the instruments’ output signals that need to be conditioned are the gain and the offset. All three sections of the interface share the same offset control (knob zero in Fig. 3), but each has its own gain control (knobs A + B, CARY, B + L in Fig. 3). A ranging procedure uses the computer and a digital oscilloscope to adjust the interface knobs to their proper setting for the anticipated output range of the instrument in use. The timing mechanism uses the 60 Hz of the line voltage from Houston Lighting and Power Company. The frequency of this signal is
596
WALTER,
EBERSPAECHAR
AND
HUGHES
extremely accurate, but appropriate software must be used to filter spurious spikes in the voltage (especially at 5:00 pm). The A/D converters are the standard nine-bit converters available from the Digital Equipment Corporation. These converters are capable of sampling rates up to 104/sec. Eight A/D converters are used in the hardware configuration in Fig. 1; as few as three would be sufficient to collect data from a single instrument. The computer used here is Digital Equipment Corporation’s LINC-8. Our present configuration has 16 A/D converters, 8K of 12-bit memory, a digital oscilloscope, two LTNC-tape drives, an incremental plotter, and a standard teletype. The analog data are displayed on the stripchart recorder or x-y plotter indicated in Fig. 1. After the data has been converted to digital form, it can be displayed on the digital oscilloscope and/or the incremental plotter. The analog data are stored by means of a stripchart recording trace. The digitized data are stored on a LINC-tape. The teletype is used to communicate with the computer and for printing data and estimated parameters. The digital oscilloscope is used to scale instrument signals for the A/D converters, and to display data. Software Figure 4 is a flowchart for the collection and analysis of the data. The software components include: (a) monitor; (b) data setup and calibrator, data acquisition, and data storage programs for each instrument; and (c) data analysis programs for S(t) and for P(t). These programs are written in LAP-6-3L. The monitor is used to put the floating point routines into the correct memory banks, to determine if a data acquisition program is needed, and to place the data acquisition and analysis programs into the proper memory banks. The data setup and calibrator portion of the data setup, acquisition, and storage program is used to examine the state of the data tape, create a data identification word, and calibrate the output from the interface so that the full range of the A/D converters can be used to digitize the data. The data acquisition part of this program is used to sample and store the initial condition, count and store the mixing time, sample, average and store the data as a function of time, control the termination of data collection, and display the collected data on the digital oscilloscope. The data is collected 60 times/set, averaged to obtain the mean value during that second, and stored as the averaged data point for the midsecond during which the 60 points were collected. This averaging is done to keep the number of core-stored data points obtained during a single experiment (there is insufficient time between sampling to write the data
ENZYME-CATALYZED
597
REACTIONS
Count Startlng Time: Set T = T (0)
Set T , = T (01 + 5 + J
I
I
1
I
J < 507
Calculate
A. 6. and C and Their Variances
by F,tt,ng P, and T , to
T=A+BP+CP2
FIG.
4. Flowchart
for collecting
and analyzing
enzyme
kinetic
data.
598
WALTER,
EBERSPAECHAR
AND HUGHES
on magnetic tape) within the limitations imposed by a small computer, and it also has the advantage that it serves as a filter for occasional spurious data points. After data collection is over, the data storage portion of this program writes the 508 averaged data points from computer memory to the LINC-tape. The data analysis program determines whether the data is in computer memory (i.e., from a current experiment), or must be called into memory from the LINC-tape. The program uses the data identification word, the mixing time, the initial conditions, and information requested of the user via the teletype to prepare the data so that it can be fit to Eq. [ 11. Finally, the program uses Eq. [5-lo] to fit an integrally increasing number of data points to Eq. [ 11. The coefficients (Eqs. [5-7]), their variances (Eqs. [g-lo]), and the initial, quasi-steady-state rate (Eq. [4]) are printed on the teletype for the requested sequence of values of N. An example of the input and output for the data analysis programs is illustrated in Fig. 5. In the first line, the program recognizes that the data is not in memory, but is stored on LINC-tape. The second line requests the block number where the date is stored. “175” was entered by the user. The fourth line notes that the data was collected on the Cary Model 16 Spectrophotometer. The fifth line requests the optical density DATA
FROM
TYPE
TAPE
3-DIGIT
L75 DATA
15 FROM
TYPE .s
UNIT
OCTAL
F”LL
THE SCALE
1. EN
WHERE
DATA
BEGINS.
CARY. OD
THEN
ZERO
SCALE
OD:
0 THlS
DATA
THERE HOW
ARE MAN”
WAS
COLLECTED
+5.060000E+002 POINTS PER
ON
03103.
DATA CALCULATION?
POINTS
NOT
INC
THE
IC.
400 FACTOR 160.8 FIT A B C
TO
FOR = = =
A 8
C = RATE
‘/AR= VAR= VAR=
INTERVAL=
+1.179874E+OOl +1.607394E+OOl
-
+,.938380E-002= +6.221251E-002
LETS
WITH WlSH
VAR=
VARVAR-
TO
THAT ANALYZE
+ CPXP
3.344730E-001 6.338327E-002 Z.S01916E-003
T = A * BP
YOU
MICRDMOLAR:
T = A + BP -
QUADRATIC:
FINISHED DO Y OK.
TO
= +6.185119E-002 FOR
= =
DATA
QUADRATIC:
+1.147360E.001 +1,616784E+OOl +1.518,55E-002-
RATE FIT
CONVERT
+6,09OE-002
TO
‘6
283E-002
TO
+6.287E
+ CPXP
1.930922E-001 2.857745E-002
8,834315E-004 INTERVAL = +6.157E-002
-002
DATA. ANOTHER
SET
ON
UNIT
1)
GO.
FIG. 5. Example input with the aid of a Cary dehydrogenase from A. 0.033 M ammonium ion,
and output for the data analysis program. The data were collected Model 16 spectrophotometer. The enzyme used was glycerol aerogenes. The initial assay conditions are 0.05 M glycerol, 0.5 M carbonate buffer, pH 9.0, and 5.67 mM NAD.
ENZYME-CATALYZED
REACTIONS
599
scale used in the experiment. “5” and “0” were entered by the user. The eighth line notes that the data were collected on March 3, and the ninth line that there are 508 averaged date points. The tenth line requests the interval beginning at t = 0 for the parameter calculations. “400” was entered by the user. This number means that 400 averaged data points will be used for the first fit, and 509 (because 509 is less than twice 400) for the second fit. The twelfth line requests the factor needed to convert the optical densities to micromolar concentrations. Since the extinction coefficient at 340 nm for NADH is 6.22 X 10m3 optical density units per micromolar, “160.8” ~M/OD is entered by the user. In the next 12 lines the 400- and 509-point fits are given as follows: A in seconds is followed by its variance; B in seconds per micromolar is followed by its variance; C in seconds per micromolar squared is followed by its variance; the initial, quasi-steady-state rate in micromolar per second (calculated from Eq. [4]) is followed by its range (calculated from the variance for B). After the final calculation, the output informs the user that the program is finished, and inquires if the user wants to analyze another set of data. “Y” is typed by the user to indicate “Yes,” and the computer responds that it is ready to proceed. PRACTICAL CONSIDERATIONS FOR USING THE PARAMETER ESTIMATION SYSTEM
The experimental data reported in this section were collected during the glycerol dehydrogenase (EC 1.1.1.6)-catalyzed conversion of NAD and glycerol to NADH and dihydroxyacetone. A Cary Model 16 spectrophotometer was used to detect the light transmission as a function of time, and a Cary kinetic interface was used to convert the data to optical densities. The initial condition and 508 averaged optical densities were stored for each experiment. The optical density range of the Cary was scaled by the interface so that - 1 V corresponded to an optical density of zero, and +1 V to an optical density of one-half. Thus, the largest possible digitized number from the A/D converters corresponds to an optical density of 0.500, and the smallest digitized number to an optical density of 0.000. Since the LINC-8 A/D converters convert analog signals to 9-bit numbers, an optical density range of zero to one-half is converted with an accuracy better than lo-:’ optical density units per bit. In other words, if an optical density of zero is digitized to the octal number 000, an optical density of 0.00 1 would be 00 1, 0.449 would be 776, and 0.500 is 777. The extent overall reaction required to obtain consistent estimates of A, B, and C in Eq. [ 1] depends on the nature of the particular chemical reaction one wishes to study. In the case of the glycerol dehydrogenasecatalyzed oxidation of 0.05 M glycerol, we find that optical densities should be collected until the extent overall reaction is at least
FIG. 6. Relationship io Eq. [l] in the text.
a IL 0
2
4
400
600
between the overall reaction The experimental conditions
Time Interval
200
h--l, 6
i0
IC)-
I: ,-
0.
2oc
400 200 Time Interval
600
‘1
iI
.
400
obtained
Time Interval
200
and the coefficients
-6OJ0
- 20.
L 8=/6 /
B
interval during which data are collected are identical to those in Fig. 5.
B
2c I-
25
30
by fitting
600
the data
ENZYME-CATALYZED
REACTIONS
601
P(t)/P(m) > 0.002. In Fig. 6 are plots of estimates of the coefficients in Eq. [ 1] and their variances (represented by the vertical bars) versus the time interval during which data were collected for each estimate. In all cases the number of data points used for each estimation should have been sufficient to obtain a consistent fit to Eq. [ 1] provided the interval during which the data was collected was sufficiently long. Since data was collected at equal time intervals, the number of data points used for each estimate increases linearly with the time interval. Thus, the estimates obtained for the IOO-set time interval were calculated from 100 averaged data points, and the estimates obtained for the last time interval were calculated from 509 averaged points. The estimates in Fig. 6 were calculated from an experiment wherein the initial concentration of NAD was 5.67 mM; A(Fig. 6A) and C (Fig. 6C) are monotonically increasing functions and B(Fig. 6B) is a monotonically decreasing function of the time interval used. The estimates in Fig. 7 were calculated from an experiment wherein the initial NAD concentration was 2.72 mM; A(Fig. 7A), B (Fig. 7B), and C (Fig. 7C) approach asymptotic values after passing through an extremum at an intermediate time interval. The estimates in Fig. 8 were calculated from an experiment wherein the initial NAD concentration was 0.068 mM; A (Fig. 8A) and C (Fig. 8C) are monotonically decreasing functions and B (Fig. 8B) is a monotonically increasing function of the time interval used. In each of these experiments, the parameters in Eq. [ 1] become independent of the time interval used to collect the data only after that time interval exceeds 300-400 sec. For the experiment in Fig. 6, this corresponds to an overall reaction of about 0.002; for the experiment in Fig. 7 it corresponds to an extent reaction of about 0.003; in Fig. 8 it corresponds to P(t)/P(m) of about 0.01. The reason the value of P(t)/P(m) required for coefficient consistency changes is probably because the order of the enzyme-catalyzed reaction increases (and hence the curvature of P(t) increases) from nearly zero when NAD(0) = 5.67 mM (Fig. 6) to nearly one when NAD(0) = 0.068 mM (Fig. 8). Since these results are typical of many experiments we have conducted for this enzyme-catalyzed reaction, we conclude that P(t)/P(w) should be above 0.002 for experiments with high initial coenzyme concentrations, above 0.01 for experiments with low initial coenzyme concentrations, and above a lower limit between 0.002 and 0.01 for experiments with intermediate coenzyme concentrations. EXPERIMENTAL
APPLICATIONS
Table 1 contains the average values and standard deviations for A, B, and C in Eq. [ 11, average values for their standard error, and the average value for the initial, quasi-steady-state rate obtained from 26 different experiments when NAD(0) is 0.2 14 mM, and 16 separate experi-
C
i
4
6
200 400 Time Inlervol
600
B
51 0
0~
5-
10.
‘5.
NO
200 400 Time lntervol
L
I 6
600
B=/8 3 C
- SC
- 6C
- 4c
- 2c
C
ZC
C
460 Time Interval
260
/--
660
C= 018
FIG. 7. Relationship between the overall reaction interval during which data are collected and the coefficients obtained by fitting the data to Eq. [l]. The experimental conditions are identical to those in Fig. 5 except the initial concentration of NAD is 2.72 mM.
A
8
IO
12
ENZYME-CATALYZED
REACTIONS
603
604
WALTER,
EBERSPAECHAR
AND
HUGHES
TABLE 1 AVERAGE VALUES OF THE COEFFICIENTS AND
THEIR
STANDARD
ERROR
(EQ.
[I])
(SE)
Experiment 1: 26 Determinations when NAD(0) is 0.214 mM A = -4.3 r 4.5 set SE(A) = 1.2 set B = 44.6 5 4.5 sec/wM SE(B) = .9 sec/yM C = 0.75 k 0.52 sec/& SE(C) = .l sec/& v(O) = 0.0227 t 0.0022 ptvilsec Experiment 2: 16 Determinations when NAD(0) is 0.696 mM A = -3.9 t 3.4 set SE(A) = 1.1 set B = 25.0 k 2.6 set/q SE(B) = .6 sec/pM C = 0.09 k 0.16 sec/j.d SE(C) = .14 sec/pM* v(0) = 0.0405 -t 0.0044 pM/SeC
ments when NAD(0) is 0.696 mM. The values of the coefficients used to calculate the averages in Table 1 were consistent during the time interval beginning when the reaction was started and beyond 400 set (See Figs. 6-8). If the corresponding values of A and C are used to correct the extrapolated initial slope for the small zero-time offset, v, = l/(BZ - 4AC)“’
[Ill
v, is not different from v(0) - l/B by more than 1%. The standard deviation calculated for any one parameter from the set of 26 or 16 individual values is usually larger than the standard error obtained for the same parameter. For example, A has a standard deviation in the range off. lOO%, but the standard error of A is about l/3 smaller. When the initial value of the transmitted light is matched exactly by electronically zeroing the initial optical density against a reference solution, the estimate of IAl is small (a few seconds), but it is not exactly zero. C also has a standard deviation in the range of k 100%, and the standard error for this coefficient is l/5 smaller in Experiment 1 but only slightly smaller in Experiment 2. These results illustrate why it is necessary to include the constant and the square terms in Eq. [l] (to obtain an acceptable fit), even though the estimated values of A and C are not necessarily reproducable from experiment to experiment. Furthermore, B = Iv(O) has a standard deviation in the range of k lo%, but the standard error of B is about l/5 smaller. This illustrates that the methods outlined here can be used to obtain objective estimates of initial rates from a given set of P and t with a standard error in the range of 2-3%, but even using these methods, reproducibility is not better than about 10%. We have reported elsewhere (2) the comparison between rates estimated by manual means and those obtained from the same data via these objective procedures. Since we are using extremely accurate
ENZYME-CATALYZED
FIG. 9. l/B = v(0) versus ferent initial concentrations those in Fig. 5.
v(O)/NAD(O) of NAD.
The
REACTIONS
605
plot for initial rate estimates obtained at 12 difother experimental conditions are identical to
equipment to measure mass, volume, light transmission, mixing time, and reaction times, it is unlikely that other estimates of initial, quasisteady-state rates obtained for the same reaction, especially by less objective procedures would result in better reproducibility. Figure 9 is a typical plot of v(0) = l/B versus v(O)/NAD(O) for a series of initial, quasi-steady-state rates obtained at 12 different NAD(0). This type of plot is linear if the set of v(0) and NAD(0) fit the rectangular hyperbola ‘(‘)
V,NAD(O) = NAD(0) + K,'
[I21
and it is the best of the three possible linear forms to use to detect deviations from Eq. [ 121 (3). As before, the values of v(0) used to construct Fig. 9 were consistent during the time interval between zero and beyond 400 sec. If the corresponding values of A and C are used to correct the extrapolated initial slope for the small zero-time offset, 0.99 < B/(B* - 4ACP2 < 1.01. When the 12 values of v(O) and NAD(0) in Fig. 9 were fit by nonlinear regression to Eq. [ 121, V, was 0.0639 pM/sec, and K, was 0.177 mM. The straight line corresponding to these values and Eq. [ 121 is drawn in Fig. 9. When error bars of length corresponding to 2 10% of each v(0) are drawn by connecting each point with a hypothetical straight line through the origin, the two
606
WALTER,
EBERSPAECHAR
AND
HUGHES
lowest points, and their error bars fall to the right of the straight line; the next four lowest points, and their error bars fall to the left of the line. This result suggests that the points in Fig. 9 are curved concave up, and that they do not fit Eq. [ 121 (3). ACKNOWLEDGMENT The authors are indebted to Mr. Cho-Yan Yeung who collected the experimental data used to construct Fig. 9.
REFERENCES 1. Walter, C. (1966) J. Theor. Bid. 11, 18 l-206. 2. Barrett, M. J., and Walter, C. (1970) Enzymobgia 3. Walter, C. (1974) J. Biol. Chem. 249, 699-703.
38,
140-146.
BIOCHEMISTRY
69, 590-606
(1975)
An On-Line Method for the Collection Analysis of Quasi-Steady-State Kinetic for
Enzyme-Catalyzed
and Data
Reactions’
CHARLES WALTER The University
of Houston, Department Houston, Texas
of Chemical 77004
Engineering,
HEIDI EBERSPAECHERAND J. PATRICK HUGHES The University of Texas System Cancer Center, M.D. Anderson Hospital Tumor Institute at Houston, Department of Biomathematics, Biochemistry and Biology, and Community Health Computing, Incorporated Houston, Texas 77025
and
Received May 13, 1974; accepted June 19, 1975 A special mixing device for initiating enzyme-catalyzed reactions is used to rapidly achieve an unperturbed quasi-steady state. An on-line computer is employed to sample the initial conditions, the mixing time, and concentrations that change as a function of time during this quasi-steady state phase. A statistical method for estimating initial, quasi-steady state rates from the time course of the enzymecatalyzed reaction is described. Practical considerations for using this parameter estimation system lead to the conclusion that for the enzyme-catalyzed reaction tested, the extent overall reaction should be above .2% for high initial substrate concentrations, and above 1% for initial substrate concentrations in the range of the Michaelis constant. Application of this method to a typical enzyme-catalyzed reaction suggests that objective estimates of initial rates from a given set of concentrations and corresponding times can be obtained with a standard error in the range of 2-3%, but that reproducibility is not better than about 10%. When this procedure was used to estimate initial rates for the glycerol dehydrogenasecatalyzed oxidation of glycerol by NAD, it was found that this enzyme did not behave according to the classical “Michaelis-Menten” mechanism of enzyme action.
Quasi-steady-state (1) enzyme kinetic data usually consists of a series of substrate or product concentrations, c,, obtained at discrete times, I*, or of a continuous record of c(t) plotted on a stripchart recorder. The usual procedure in enzyme studies is to use the kinetic data to obtain a single parameter called the “initial, quasi-steady-state rate” (v(O)). v(0) obtained in this way is not an experimental observable. It is a parameter that was estimated from observables. As in the case of other estimated ’ This work was supported in part by grants from the NSF (GB-7 110 and GB-20612) the Robert A. Welch Foundation (G517), and the NIH (CA-l 1430).
Copyright @ 1975 by Academic Press, Inc. All rights of reproduction in my form resewed.
ENZYME-CATALYZED
591
REACTIONS
quantities, it is necessary to describe explicitly the method one uses to obtain estimates for initial, quasi-steady-state rates. It is the purpose of this paper to describe an objective method for obtaining accurate estimates of initial quasi-steady-state rates and their variances. MATHEMATICAL MODEL FOR THE TIME-COURSE OF ENZYME-CATALYZED REACTIONS
We (2) have found that during the early, quasi-steady-state stages of many enzyme reactions, an approximate relationship between the product concentration, P and time, r is
[II
A + BP(f) + CP(r)’ = r where A, B and C are constant coefficients. If the substrate, experimental observable, P can be estimated from P(f) = S(0) - S(t)
S is the
PI
P(0) = 0,
provided insignificant quantities of S and P are bound to the enzyme at any given instant. Thus Eq. [2] is usually a valid approximation provided S(0) % E(0). Differentiation of Eq. [l] yields the quasi-steady-state rate: 6(t) = l/[B Substitution rate :
+ 2CP(t)].
131
of P(0) = 0 into Eq. [3] yields the initial,
quasi-steady-state
v(0) = P(O) = l/B.
[41
If correction of Eq. [4] due to a small off-set at t = 0 is necessary, it can be accomplished by solving Eq. [l] for P when t = 0, and inserting the result into Eq. [ 31. STATISTICAL
An estimation
METHOD
FOR
PARAMETER
ESTIMATIONS
of A, B, and C in Eq. [ I] can be obtained
from Eq.
[S-73:
A = (~1~170 - 7~~71+ GT~T.J&
[51
B = (ng, - n-2~0- 7~~~~)/8,
[61
c = (77370- 7r#g,- 7f47*)/&
[71
where ?T1= XP”XP4 - (ZP”)“, 7r* = XPZP4 - XP2ZP3, 7r3 = ZPXP3 - (XP”)‘,
592 = 7~~= 7~~= 70 = 71= 5-2= 6=
7~4
WALTER,
EBERSPAECHAR
NZP2 - (ZP)“, NZP4 - (ZP”)“, NZP3 - CPZP2, Et, X.tP, X.tP2, 2XPZP2ZP3 - N(ZP3)2 + NZP2ZP4
AND
HUGHES
- (ZP”)” - (ZP)2XP4,
and N is the total number of data points collected. B, and C are
The variances of A,
var(A) = 7r1$/6,
PI
var(B) = 7~+#&,
[91
var(C) = r4$/8,
[lOI
where +t2-Ar0-BT1-CT2 N-3
’
MATERIALS Glycerol
dehydrogenase used in these experiments was obtained from Aerobacter aerogenes. NAD, and Trizma Base were obtained from Sigma Chemical Company; spectroquality glycerol was obtained from Matheson, Coleman, and Bell Company; and the phosphates and carbonates for the buffers were obtained from J. T. Baker Company. Tris was passed through a DEAE-Sephadex A-50 (Pharmacia, Inc.) ion-exchange column. Water was prepared by distillation from a large Barnstead still followed by redistillation on a smaller, all glass Barnstead still. Hardware
Figure 1 is a system diagram describing the hardware used here. The components include: (a) detection system; (b) mixing device; (c) interface; (d) timing mechanism; (e) analog-to-digital converters (A/D converters); (f) small digital computer; (g) devices for analog and digital display; (h) a device for data storage; (i) a device for inputting scale factors, etc. and printing out data and estimated parameters. Our detection systems include two different spectrophotometers and a spectrofluorometer. The methods are not limited to these devices and could easily be extended to NMR machines, polarimeters, refractometers, ORD machines, etc. The mixing device we use is illustrated in Fig. 2. When in operation, this device is situated over a sample holder located in the light path of a spectrophotometer or spectrofluorometer. The sample holder contains a known volume of a solution containing all of the ingredients necessary
ENZYME-CATALYZED
593
REACTIONS
Lint
0. D.
Tapes
0 Q
1 Q
ov or 1.4v Second Spectrometer
ov or 1.4v
SYSTEM
DIAGRAM
FIG. 1. System diagram for collecting and analyzing enzyme kinetic data.
for the enzyme-catalyzed reaction except one. A stripchart recorder or direct readout from the spectrophotometer is used to insure that the initial condition is constant. During this period a small, known volume of a solution containing the missing ingredient is placed in the plastic spoon connected to the mixing device. The door to the spoon chamber is closed, the door between the spoon chamber and the sample holder is opened, a switch on the mixing device is activated, and the computer samples the initial condition and rings a bell. If necessary, the computer calculates the diluted initial condition expected after addition of the small volume to the rest of the assay mixture. Next, the plunger on the mixing device is smoothly depressed so that the spoon dips into the assay solution. A switch informs the computer that the spoon has been
WALTER,
EBERSPAECHAR
AND HUGHES
MIXING DEVICE
1 Spectrophofometer - Sample Houring Cover
2 Stirring LIeviceAdapterwith PlungerDoor 3 MwingSpoonLoading-Door 4 Plunger Housmgwth Arresting Wingnut 5 Sfmr Motor - Activating Micro-Switch
FIG.
6 Computer-fnferfaced Microswtch 7 Plungerwith Relsaamg Springs 8 Teflon Mixmg Spoon 9 Plunger stop
2. Mixing device used to introduce solutions into cuvettes.
depressed, and the computer begins to count the number of milliseconds the spoon is down. Simultaneously, a small motor rotates the spoon while it is held at its lowest point in the assay solution. Within 1 set, mixing is complete, and the plunger is allowed to smoothly return to its original position. When the spoon leaves its lowest position, the motor stops and the spoon stops rotating. As soon as the spoon clears the light-
ENZYME-CATALYZED
595
REACTIONS
(B&L)
ICARY)
(A & Bl
X
k
2 - (ZkX-b) 1
+ b 7 -
FIG. 3. Triple scaling amplifier interface analog diagram.
path of the instrument, a switch informs the computer that mixing is over. The total number of milliseconds required for mixing is estimated by dividing the total time the spoon is down by two; this estimate is retained in the computer’s memory as T,. The signal that mixing is over is also the signal to the computer to begin sampling the analog signal from the instrument as a function of time. The sampling continues 60 times/set for 508 set, or until it is aborted manually with a switch on the mixing apparatus. The interface consists of the three sections indicated in the schematic diagram in Fig. 3. Each section connects a particular instrument to an A/D converter, and each is designed to scale its instrument’s output signal to the range of the A/D converter (+ 1 V). The two aspects of the instruments’ output signals that need to be conditioned are the gain and the offset. All three sections of the interface share the same offset control (knob zero in Fig. 3), but each has its own gain control (knobs A + B, CARY, B + L in Fig. 3). A ranging procedure uses the computer and a digital oscilloscope to adjust the interface knobs to their proper setting for the anticipated output range of the instrument in use. The timing mechanism uses the 60 Hz of the line voltage from Houston Lighting and Power Company. The frequency of this signal is
596
WALTER,
EBERSPAECHAR
AND
HUGHES
extremely accurate, but appropriate software must be used to filter spurious spikes in the voltage (especially at 5:00 pm). The A/D converters are the standard nine-bit converters available from the Digital Equipment Corporation. These converters are capable of sampling rates up to 104/sec. Eight A/D converters are used in the hardware configuration in Fig. 1; as few as three would be sufficient to collect data from a single instrument. The computer used here is Digital Equipment Corporation’s LINC-8. Our present configuration has 16 A/D converters, 8K of 12-bit memory, a digital oscilloscope, two LTNC-tape drives, an incremental plotter, and a standard teletype. The analog data are displayed on the stripchart recorder or x-y plotter indicated in Fig. 1. After the data has been converted to digital form, it can be displayed on the digital oscilloscope and/or the incremental plotter. The analog data are stored by means of a stripchart recording trace. The digitized data are stored on a LINC-tape. The teletype is used to communicate with the computer and for printing data and estimated parameters. The digital oscilloscope is used to scale instrument signals for the A/D converters, and to display data. Software Figure 4 is a flowchart for the collection and analysis of the data. The software components include: (a) monitor; (b) data setup and calibrator, data acquisition, and data storage programs for each instrument; and (c) data analysis programs for S(t) and for P(t). These programs are written in LAP-6-3L. The monitor is used to put the floating point routines into the correct memory banks, to determine if a data acquisition program is needed, and to place the data acquisition and analysis programs into the proper memory banks. The data setup and calibrator portion of the data setup, acquisition, and storage program is used to examine the state of the data tape, create a data identification word, and calibrate the output from the interface so that the full range of the A/D converters can be used to digitize the data. The data acquisition part of this program is used to sample and store the initial condition, count and store the mixing time, sample, average and store the data as a function of time, control the termination of data collection, and display the collected data on the digital oscilloscope. The data is collected 60 times/set, averaged to obtain the mean value during that second, and stored as the averaged data point for the midsecond during which the 60 points were collected. This averaging is done to keep the number of core-stored data points obtained during a single experiment (there is insufficient time between sampling to write the data
ENZYME-CATALYZED
597
REACTIONS
Count Startlng Time: Set T = T (0)
Set T , = T (01 + 5 + J
I
I
1
I
J < 507
Calculate
A. 6. and C and Their Variances
by F,tt,ng P, and T , to
T=A+BP+CP2
FIG.
4. Flowchart
for collecting
and analyzing
enzyme
kinetic
data.
598
WALTER,
EBERSPAECHAR
AND HUGHES
on magnetic tape) within the limitations imposed by a small computer, and it also has the advantage that it serves as a filter for occasional spurious data points. After data collection is over, the data storage portion of this program writes the 508 averaged data points from computer memory to the LINC-tape. The data analysis program determines whether the data is in computer memory (i.e., from a current experiment), or must be called into memory from the LINC-tape. The program uses the data identification word, the mixing time, the initial conditions, and information requested of the user via the teletype to prepare the data so that it can be fit to Eq. [ 11. Finally, the program uses Eq. [5-lo] to fit an integrally increasing number of data points to Eq. [ 11. The coefficients (Eqs. [5-7]), their variances (Eqs. [g-lo]), and the initial, quasi-steady-state rate (Eq. [4]) are printed on the teletype for the requested sequence of values of N. An example of the input and output for the data analysis programs is illustrated in Fig. 5. In the first line, the program recognizes that the data is not in memory, but is stored on LINC-tape. The second line requests the block number where the date is stored. “175” was entered by the user. The fourth line notes that the data was collected on the Cary Model 16 Spectrophotometer. The fifth line requests the optical density DATA
FROM
TYPE
TAPE
3-DIGIT
L75 DATA
15 FROM
TYPE .s
UNIT
OCTAL
F”LL
THE SCALE
1. EN
WHERE
DATA
BEGINS.
CARY. OD
THEN
ZERO
SCALE
OD:
0 THlS
DATA
THERE HOW
ARE MAN”
WAS
COLLECTED
+5.060000E+002 POINTS PER
ON
03103.
DATA CALCULATION?
POINTS
NOT
INC
THE
IC.
400 FACTOR 160.8 FIT A B C
TO
FOR = = =
A 8
C = RATE
‘/AR= VAR= VAR=
INTERVAL=
+1.179874E+OOl +1.607394E+OOl
-
+,.938380E-002= +6.221251E-002
LETS
WITH WlSH
VAR=
VARVAR-
TO
THAT ANALYZE
+ CPXP
3.344730E-001 6.338327E-002 Z.S01916E-003
T = A * BP
YOU
MICRDMOLAR:
T = A + BP -
QUADRATIC:
FINISHED DO Y OK.
TO
= +6.185119E-002 FOR
= =
DATA
QUADRATIC:
+1.147360E.001 +1,616784E+OOl +1.518,55E-002-
RATE FIT
CONVERT
+6,09OE-002
TO
‘6
283E-002
TO
+6.287E
+ CPXP
1.930922E-001 2.857745E-002
8,834315E-004 INTERVAL = +6.157E-002
-002
DATA. ANOTHER
SET
ON
UNIT
1)
GO.
FIG. 5. Example input with the aid of a Cary dehydrogenase from A. 0.033 M ammonium ion,
and output for the data analysis program. The data were collected Model 16 spectrophotometer. The enzyme used was glycerol aerogenes. The initial assay conditions are 0.05 M glycerol, 0.5 M carbonate buffer, pH 9.0, and 5.67 mM NAD.
ENZYME-CATALYZED
REACTIONS
599
scale used in the experiment. “5” and “0” were entered by the user. The eighth line notes that the data were collected on March 3, and the ninth line that there are 508 averaged date points. The tenth line requests the interval beginning at t = 0 for the parameter calculations. “400” was entered by the user. This number means that 400 averaged data points will be used for the first fit, and 509 (because 509 is less than twice 400) for the second fit. The twelfth line requests the factor needed to convert the optical densities to micromolar concentrations. Since the extinction coefficient at 340 nm for NADH is 6.22 X 10m3 optical density units per micromolar, “160.8” ~M/OD is entered by the user. In the next 12 lines the 400- and 509-point fits are given as follows: A in seconds is followed by its variance; B in seconds per micromolar is followed by its variance; C in seconds per micromolar squared is followed by its variance; the initial, quasi-steady-state rate in micromolar per second (calculated from Eq. [4]) is followed by its range (calculated from the variance for B). After the final calculation, the output informs the user that the program is finished, and inquires if the user wants to analyze another set of data. “Y” is typed by the user to indicate “Yes,” and the computer responds that it is ready to proceed. PRACTICAL CONSIDERATIONS FOR USING THE PARAMETER ESTIMATION SYSTEM
The experimental data reported in this section were collected during the glycerol dehydrogenase (EC 1.1.1.6)-catalyzed conversion of NAD and glycerol to NADH and dihydroxyacetone. A Cary Model 16 spectrophotometer was used to detect the light transmission as a function of time, and a Cary kinetic interface was used to convert the data to optical densities. The initial condition and 508 averaged optical densities were stored for each experiment. The optical density range of the Cary was scaled by the interface so that - 1 V corresponded to an optical density of zero, and +1 V to an optical density of one-half. Thus, the largest possible digitized number from the A/D converters corresponds to an optical density of 0.500, and the smallest digitized number to an optical density of 0.000. Since the LINC-8 A/D converters convert analog signals to 9-bit numbers, an optical density range of zero to one-half is converted with an accuracy better than lo-:’ optical density units per bit. In other words, if an optical density of zero is digitized to the octal number 000, an optical density of 0.00 1 would be 00 1, 0.449 would be 776, and 0.500 is 777. The extent overall reaction required to obtain consistent estimates of A, B, and C in Eq. [ 1] depends on the nature of the particular chemical reaction one wishes to study. In the case of the glycerol dehydrogenasecatalyzed oxidation of 0.05 M glycerol, we find that optical densities should be collected until the extent overall reaction is at least
FIG. 6. Relationship io Eq. [l] in the text.
a IL 0
2
4
400
600
between the overall reaction The experimental conditions
Time Interval
200
h--l, 6
i0
IC)-
I: ,-
0.
2oc
400 200 Time Interval
600
‘1
iI
.
400
obtained
Time Interval
200
and the coefficients
-6OJ0
- 20.
L 8=/6 /
B
interval during which data are collected are identical to those in Fig. 5.
B
2c I-
25
30
by fitting
600
the data
ENZYME-CATALYZED
REACTIONS
601
P(t)/P(m) > 0.002. In Fig. 6 are plots of estimates of the coefficients in Eq. [ 1] and their variances (represented by the vertical bars) versus the time interval during which data were collected for each estimate. In all cases the number of data points used for each estimation should have been sufficient to obtain a consistent fit to Eq. [ 1] provided the interval during which the data was collected was sufficiently long. Since data was collected at equal time intervals, the number of data points used for each estimate increases linearly with the time interval. Thus, the estimates obtained for the IOO-set time interval were calculated from 100 averaged data points, and the estimates obtained for the last time interval were calculated from 509 averaged points. The estimates in Fig. 6 were calculated from an experiment wherein the initial concentration of NAD was 5.67 mM; A(Fig. 6A) and C (Fig. 6C) are monotonically increasing functions and B(Fig. 6B) is a monotonically decreasing function of the time interval used. The estimates in Fig. 7 were calculated from an experiment wherein the initial NAD concentration was 2.72 mM; A(Fig. 7A), B (Fig. 7B), and C (Fig. 7C) approach asymptotic values after passing through an extremum at an intermediate time interval. The estimates in Fig. 8 were calculated from an experiment wherein the initial NAD concentration was 0.068 mM; A (Fig. 8A) and C (Fig. 8C) are monotonically decreasing functions and B (Fig. 8B) is a monotonically increasing function of the time interval used. In each of these experiments, the parameters in Eq. [ 1] become independent of the time interval used to collect the data only after that time interval exceeds 300-400 sec. For the experiment in Fig. 6, this corresponds to an overall reaction of about 0.002; for the experiment in Fig. 7 it corresponds to an extent reaction of about 0.003; in Fig. 8 it corresponds to P(t)/P(m) of about 0.01. The reason the value of P(t)/P(m) required for coefficient consistency changes is probably because the order of the enzyme-catalyzed reaction increases (and hence the curvature of P(t) increases) from nearly zero when NAD(0) = 5.67 mM (Fig. 6) to nearly one when NAD(0) = 0.068 mM (Fig. 8). Since these results are typical of many experiments we have conducted for this enzyme-catalyzed reaction, we conclude that P(t)/P(w) should be above 0.002 for experiments with high initial coenzyme concentrations, above 0.01 for experiments with low initial coenzyme concentrations, and above a lower limit between 0.002 and 0.01 for experiments with intermediate coenzyme concentrations. EXPERIMENTAL
APPLICATIONS
Table 1 contains the average values and standard deviations for A, B, and C in Eq. [ 11, average values for their standard error, and the average value for the initial, quasi-steady-state rate obtained from 26 different experiments when NAD(0) is 0.2 14 mM, and 16 separate experi-
C
i
4
6
200 400 Time Inlervol
600
B
51 0
0~
5-
10.
‘5.
NO
200 400 Time lntervol
L
I 6
600
B=/8 3 C
- SC
- 6C
- 4c
- 2c
C
ZC
C
460 Time Interval
260
/--
660
C= 018
FIG. 7. Relationship between the overall reaction interval during which data are collected and the coefficients obtained by fitting the data to Eq. [l]. The experimental conditions are identical to those in Fig. 5 except the initial concentration of NAD is 2.72 mM.
A
8
IO
12
ENZYME-CATALYZED
REACTIONS
603
604
WALTER,
EBERSPAECHAR
AND
HUGHES
TABLE 1 AVERAGE VALUES OF THE COEFFICIENTS AND
THEIR
STANDARD
ERROR
(EQ.
[I])
(SE)
Experiment 1: 26 Determinations when NAD(0) is 0.214 mM A = -4.3 r 4.5 set SE(A) = 1.2 set B = 44.6 5 4.5 sec/wM SE(B) = .9 sec/yM C = 0.75 k 0.52 sec/& SE(C) = .l sec/& v(O) = 0.0227 t 0.0022 ptvilsec Experiment 2: 16 Determinations when NAD(0) is 0.696 mM A = -3.9 t 3.4 set SE(A) = 1.1 set B = 25.0 k 2.6 set/q SE(B) = .6 sec/pM C = 0.09 k 0.16 sec/j.d SE(C) = .14 sec/pM* v(0) = 0.0405 -t 0.0044 pM/SeC
ments when NAD(0) is 0.696 mM. The values of the coefficients used to calculate the averages in Table 1 were consistent during the time interval beginning when the reaction was started and beyond 400 set (See Figs. 6-8). If the corresponding values of A and C are used to correct the extrapolated initial slope for the small zero-time offset, v, = l/(BZ - 4AC)“’
[Ill
v, is not different from v(0) - l/B by more than 1%. The standard deviation calculated for any one parameter from the set of 26 or 16 individual values is usually larger than the standard error obtained for the same parameter. For example, A has a standard deviation in the range off. lOO%, but the standard error of A is about l/3 smaller. When the initial value of the transmitted light is matched exactly by electronically zeroing the initial optical density against a reference solution, the estimate of IAl is small (a few seconds), but it is not exactly zero. C also has a standard deviation in the range of k 100%, and the standard error for this coefficient is l/5 smaller in Experiment 1 but only slightly smaller in Experiment 2. These results illustrate why it is necessary to include the constant and the square terms in Eq. [l] (to obtain an acceptable fit), even though the estimated values of A and C are not necessarily reproducable from experiment to experiment. Furthermore, B = Iv(O) has a standard deviation in the range of k lo%, but the standard error of B is about l/5 smaller. This illustrates that the methods outlined here can be used to obtain objective estimates of initial rates from a given set of P and t with a standard error in the range of 2-3%, but even using these methods, reproducibility is not better than about 10%. We have reported elsewhere (2) the comparison between rates estimated by manual means and those obtained from the same data via these objective procedures. Since we are using extremely accurate
ENZYME-CATALYZED
FIG. 9. l/B = v(0) versus ferent initial concentrations those in Fig. 5.
v(O)/NAD(O) of NAD.
The
REACTIONS
605
plot for initial rate estimates obtained at 12 difother experimental conditions are identical to
equipment to measure mass, volume, light transmission, mixing time, and reaction times, it is unlikely that other estimates of initial, quasisteady-state rates obtained for the same reaction, especially by less objective procedures would result in better reproducibility. Figure 9 is a typical plot of v(0) = l/B versus v(O)/NAD(O) for a series of initial, quasi-steady-state rates obtained at 12 different NAD(0). This type of plot is linear if the set of v(0) and NAD(0) fit the rectangular hyperbola ‘(‘)
V,NAD(O) = NAD(0) + K,'
[I21
and it is the best of the three possible linear forms to use to detect deviations from Eq. [ 121 (3). As before, the values of v(0) used to construct Fig. 9 were consistent during the time interval between zero and beyond 400 sec. If the corresponding values of A and C are used to correct the extrapolated initial slope for the small zero-time offset, 0.99 < B/(B* - 4ACP2 < 1.01. When the 12 values of v(O) and NAD(0) in Fig. 9 were fit by nonlinear regression to Eq. [ 121, V, was 0.0639 pM/sec, and K, was 0.177 mM. The straight line corresponding to these values and Eq. [ 121 is drawn in Fig. 9. When error bars of length corresponding to 2 10% of each v(0) are drawn by connecting each point with a hypothetical straight line through the origin, the two
606
WALTER,
EBERSPAECHAR
AND
HUGHES
lowest points, and their error bars fall to the right of the straight line; the next four lowest points, and their error bars fall to the left of the line. This result suggests that the points in Fig. 9 are curved concave up, and that they do not fit Eq. [ 121 (3). ACKNOWLEDGMENT The authors are indebted to Mr. Cho-Yan Yeung who collected the experimental data used to construct Fig. 9.
REFERENCES 1. Walter, C. (1966) J. Theor. Bid. 11, 18 l-206. 2. Barrett, M. J., and Walter, C. (1970) Enzymobgia 3. Walter, C. (1974) J. Biol. Chem. 249, 699-703.
38,
140-146.