THE JOURNAL OP BIOLOGICAL CHEMISTRY Vol. 245, No. 7, Iesue of April 10, pp. 1637-1647, 1970

Printed in U.S.A.

A Method for the Kinetic Analysis of Progress Curves Using Horse Serum Cholinesterase As a Model Case

(Received for publication, September 8, 1969)

JEAN KANT BALCOM" AND WALTER M. FITCH$

From the Department of Physiological Chemistry, University of Wisconsin, Madison, Wisconsin 5S706

SUMMARY

A technique which utilizes the kinetic information present in the progress curve for a reaction has been developed to determine the values of some kinetic parameters and has been used to study the hydrolysis of an artificial substrate, acetyl ester of 3-hydroxyphenyltrimethylammonium bromide by horse serum cholinesterase. In this method, many data points (each defined by a substrate concentration, a product concentration, and a velocity) can be obtained from each prog- ress curve. Data from at least two such curves are then fit to the rate equation for the reaction to determine the val- ues of the kinetic parameters, Km, V,,,, KIS, and KII. Such values, determined by this progress curve assay method, proved to be identical with those found by the usual initial velocity assay method.

The mechanism of the hydrolysis of the acetyl ester of 3-hydroxyphenyltrimethylammonium bromide by horse serum cholinesterase is erdered Uni-Bi. The reaction was found to have a Km of 0.149 mu and a V,,, of 2.21 pmoles nonacetylated phenol from the acetyl ester of 3-hydroxy- phenoltrimethylammonium bromide per min per mg of enzyme. One of the two products of the reaction (acetate) was shown to have little or no effect upon the reaction (Ki - 180 mM). The other product was shown to be a linear non- competitive inhibitor of the reaction. The effect of tem- perature upon the reaction was studied by use of the progress curve assay method. An Ea of 6500 cal per mole and a AH* of 5900 cal per mole at 25” was determined for the reaction. Both initial velocity assays and progress curve assays were used to study the effect of pH upon the reaction. The apparent pK of an active group in the enzyme was found to be 5.8 at 25”, and a heat of ionization of 7000 cal per mole was calculated for it. These results are consistent with a histidine being important in the catalytic activity of the en- zyme. The effect of ionic strength upon the reaction pa- rameters was investigated with the use of both progress curve and initial velocity assays. Km and KIS were found to increase with increasing ionic strength, while V, was found to decrease and KU to remain relatively constant

* Predoctoral Trainee supported by Grant Z-Tl-GM302, National Institutes of Health. Part of this work. is taken from a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

1 This investigation was supported by United States Public Health Service National Institutes of Health Grant NB-04565-06.

with increasing ionic strength. The effects of choline and N, N-dimethylaminoethanol upon the reaction were deter- mined by use of initial velocity assays. Choline was found to be both an alternate product modifier and a dead end inhibitor of the reaction. N, N-Dimethylaminoethanol was found to be an uncompetitive inhibitor of the reaction.

In this paper is presented a method for utilizing data obtained from progress curves to determine the kinetic parameters of an enzyme reaction. This method, in selected cases, may con- siderably reduce the number of assays which must be performed to determine certain kinetic constants as compared to the initial velocity measurement technique commonly employed. As will be shown in this paper, this method is particularly useful in experiments which seek to determine the values of the kinetic constants under a variety of experimental conditions (such as pH, temperature, and ionic strength experiments).

In this method the data derived from progress curves are fitted to the actual rate equation for the reaction rat’her than its integrated form. In contrast to the work of Kalow, Genest, and Staron (1)) Schwert (2)) and Fitch (3)) progress curve assays are run under conditions where the effect of one of the generated products is significant. Thus, the rate equation must include terms to describe the effect of this product.

Both this method and the more common initial velocity method were used to study the hydrolysis of an artificial substrate by horse serum cholinesterase under a variety of experimental conditions. Both methods are shown to give essentially identical results.

The reaction chosen for study was the hydrolysis of the arti- ficial substrate acetyl ester of 3-hydroxyphenyltrimethylam- monium bromide by horse serum cholinesterase. The products of the reaction are acetate and the phenol portion of neostigmine. Fitch first used this substrate in his studies on a cholinesterase from Pseudomonas Jluorescens (4).

The simplest mechanism which may be written for the cho- linesterase-catalyzed hydrolysis of certain esters is

where A is the ester substrate, P is the alcohol product, & is

1637

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1638 Kinetic Analysis of Progress Curves Vol. 245, No. 7

the acid product, E is the enzyme, and kl, kz, etc. are rate con- stants for the individual steps.’ Experiments described below (see “Effect of Product”) show that acetate (Q) has little effect on the hydrolysis of AcPN2 and that PN (P) is a linear noncompetitive inhibitor of the reaction. Therefore the full rate equation for the reaction must be reducible to the form

v= K,(l -,jr,(1+&)’ C2) This equation describes the velocity (v) at any time as a function of the variable substrate (A) and product (P) concentrations and the kinetic constants Vm, Km, KIS, and KII. In the kinetic experiments to be described, data points defined by a velocity (v), a substrate concentration (A), and a product concentration (P) were fitted to Equation 2 by a computer program written by Cleland (6) to determine the values for the kinetic constants (V,, Km, KIS, and KII).

EXPERIMENTAL PROCEDURE

Mater&--Both AcPN and PN were donated by Hoffman-La Roche, Inc., Nutley, New Jersey. Horse serum cholinesterase was obtained from Sigma as a lyophilized, salt-free powder, and on the basis of its stated activity to hydrolyze 5.5 pmoles of acetylcholine per min per mg of enzyme at pH 8.0, at 37”, and the work of Heilbronn (7), the enzyme preparation would appear to have been purified about 25-fold. In our work the enzyme hydrolyzed approximately 2.0 pmoles of AcPN per min per mg of enzyme at pH 7.2 and 25”. Enzyme solutions in standard phosphate buffer (0.025 M, pH = 7.2) were stored frozen until needed.

Assay System-The reaction was followed by the measurement of the product PN at 279 rnp by use of a Beckman DK-2 re- cording spectrophotometer. Standard assay conditions used 0.025 M potassium phosphate buffer, pH = 7.2, P/2 = 0.2. Assay volumes of 3 ml and light paths of 10 mm were used. The reaction was started by the addition of enzyme to a cuvette which already contained buffer, substrate, any desired affecter solution, and 0.2 M NaCl sufficient to bring the final volume to 3 ml. Unless otherwise noted the assay temperature was 25”.

Bu$ersStandard phosphate buffer (pH = 7.2, I’/2 = 0.2) was used in all experiments except those which studied the effects of ionic strength or pH upon the kinetic parameters. For the ionic strength experiments, the ionic strength was varied by the addition of NaCl to 0.025 M potassium phosphate buffer. The phosphate buffer was tested and shown to be capable of maintaining a constant pH during the reaction at all the ionic strengths used in the experiments.

1 For this reaction, the full rate equation may be written in the Cleland nomenclature (5) as

v= (1)

where V, = kaka/(ka + w; J&n = (kakl, + k&6)l(klk3 + k&-3);

ki, = kzlh; k, = kzlks; ki, = (k3 + k5)lk4; a& = (k2 + k3)/k4;

keq = klk3kS/k2k4766; ki* = ks/ka.

2 The abbreviations used are: AcPN, acetyl ester of 3-hydroxy- phenyltrimethylammonium bromide; PN, nonacetylated phenol from AcPN; DMAE, N,N-dimethylaminoethanol.

In the experiments which examined the effects of pH, a com- bination buffer of 0.025 M glycyl glycine (pK = 8.15)-0.025 M

potassium phosphate (pK = 6.9)-0.025 M sodium citrate (pK values = 5.9 and 4.46) was used. This combination buffer was found to buffer adequately over the pH range 4.5 to 8.5. Each buffer constituent was tested for possible ion effects upon the kinetic parameters. No significant effects were found.

Progress Curves-In progress curve assays the reaction was followed until the reaction was approximately 75% complete. The cuvette was placed aside and then returned to the spectro- photometer for a final optical density reading about 1 hour later when the reaction had gone to completion with no detectable substrate remaining. Suitable data were obtained from the progress curves as follows.

1. Evenly spaced lines (usually at l2-set intervals) were drawn perpendicular to the t’ime axis of the optical density versus time recording; the intersection of each line with the progress curve defined a point. The time interval selected determines the number of points obtained. If too few points are used, the fit to the rate equation will be poor.

2. For each point the velocity (v), the substrate concentration (A), and the product concentration (P) were determined. The optical density reading at each point, after corrections for the optical density caused by enzyme and the cuvette, is directly proportional to the amount of product PN (1 mu PN has an optical density of 1.82; the contribution of the substrate to the optical density at 279 rnp is negligible). Since product is generated only at the expense of substrate in a 1: 1 relationship, the substrate concentration at any point is calculated by sub- tracting the product concentration at the point in question from the initial substrate concentration (Ao). The velocity is defined as the slope of the chord joining the two points on either side of the point in question. The rate of uncatalyzed hydrolysis of AcPN, which occurs at high pH and which is first order with respect to AcPN, was determined at all pH values greater than 6.5. The observed velocities were corrected for the rate of uncatalyzed hydrolysis. These data were then fitted to Equa- tion 2 to determine values for Km, Vm, KIS, and KII. The fit to the rate equation requires at least two values of the quantity (A + P) to prevent the matrix of values from being indeter- minant. Therefore data from at least two progress curves with different initial substrate concentrations were combined for each determination of a set of parameters.

Initial Velocity-The same general procedure was used in performing initial velocity assays as for progress curve assays. However, (a) less enzyme was used in initial velocity assays in order to obtain a linear trace, (6) the assays were run for only a few minutes, and (c) final optical density readings were not required.

In the initial velocity method each assay yielded only one point. The substrate and product concentrations at each point were assumed to be equal to the initial concentrations of these components. The velocity of the assay was taken to be the slope of the tangent drawn to the initial portion of the curve.

Xtutistics-Data from both time course assays and initial velocity assays were routinely fitted to appropriate rate equations with the use of computer programs developed by Cleland (6). These programs give least square estimates of the values of the kinetic parameters and the standard error associated with each parameter. In making replots, each point was always weighted inversely to the square of the associated standard error.

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Issue of April 10, 1970 J. K. Balcom and W. M. Fitch 1639

The statistical computations assume an independent observa- tion for each data point as is true in the initial velocity assays. However, data from a single progress curve are not independent observations since the position of a point is constrained by the position of adjacent points on the smooth curve. Therefore, while the programs obtain satisfactory values for the kinetic parameters, the associated standard errors as well as the variance of the over-all fit will be underestimated when progress curve data are used as input to these programs.

In an effort to obtain more accurate estimates of the standard errors from progress curve data, the following analytical approach was tried. Upon substituting the quantity (A0 - A) for P in Equation 2 the following equation was obtained.

Dividing Equation 3 by the factor

( 1+$& )

gives

VA

(4)

where

v= VWZ

( 1 + ST - ss )

K= iq1+&)

( -40 Km

l+EY-KIS ) and

KI = KT1 Aofl Km ---

l+ KIId KIS >

Data from several progress curves (distinguished by their A0 values) were each fitted to Equation 4. This was followed by appropriate replots of K, V, and KI verSus Ao. These replots

were expected to yield values of K,, V,, KIX, and KII com- parable to those obtained by a total fit of the data from all the progress curves to Equation 2. More reliable estimates of the

standard errors were expected from this approach since a method was available to handle the problem of dependence in the data. However, although data fitted to Equation 4 yielded reasonable values for the parameters K/V and KI, the data were found to be ill conditioned to determine the K and V parameters sepa- rately. The difficulty appears to be caused by the A0 values of the data. That is, while data obtained from progress curves with A0 values equal or less than 1 mM are sufficient to deter- mine K/V and KI, progress curves with higher A0 values would be required to determine K and V separately. Thus the values of K and V which were calculated from the computer fit to Equation 4 were associated with standard errors of the order of 50% of the parameter value. Attempts to use these values

for K and V in the replots were unsuccessful. Specifically, the critical hyperbolic replot of K against A0 yielded a negative value for KIS which made it impossible to solve for the values of the other parameters. Therefore this approach was aban- doned.

The standard error estimates determined by the Cleland programs are reported in this paper as our best estimate of the goodness of fit. The extent to which they may be too good in the time course assays is unknown.

COMPARISON OF INITIAL VELOCITY AND PROGRESS

CURVE METHODS

The following experiments were performed under standard assay conditions to show that the kinetic parameters determined by both the time course method and the initial velocity method were equivalent.

Initial Velocity Experiments-At each of seven PN concen- trations, six samples containing different initial substrate con- centrations were assayed by the initial velocity techni’que. Reciprocal plots (l/v ver.sus l/A) are shown in Fig. 1. Replots of the intercepts and slopes of these reciprocal plots are shown in Fig. 2. The 42 data points were used to determine the values of the kinetic parameters by two analytical methods. In the “two-stage” analysis, data obtained at each PN concen- tration were fitted separately to the equation

Vd v=K,+

(5)

to determine apparent “V,” and “K,” values at each PTU’ concentration and thereby the values for the slopes “Km”/ “V,” and intercepts l/“TI,” of the reciprocal plots. The fit to the PN = 0 data determined the true values of V, and K,. KII and KIS were determined by weighted straight line fits to the intercept and slope replots, respectively. The values of Vm, Km/I/T,, KIS, and KII determined by this method are shown in Table I. The second method was a total fit of all 42

2.0

1.5

> 1

1.0

0.5

0 2 4 6 8 IO

I/A (mM)

J

FIG. 1. Effect of PN on reciprocal plots. Initial velocity assays performed at pH 7.2, r/2 = 0.2, temperature = 25” in the presence of 0.0 mu PN (A), 0.1 mu PN (B), 0.2 mM PN (C), 0.3 mM PN (D), 0.4 mu PN (E), 0.5 mM PN (F), and 0.6 mM PN (G). The data at each PN concentration were fit to Equation 5 by a program written by Cleland (6) to determine the slopes and inter- cepts of each straight line.

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1640 Kinetic Analysis of Progress Curves Vol. 245, No. 7

data points to Equation 2. The results of this fit are also listed in Table I.

Progress Curve Experiment-A series of nine progress curve assays with nine initial substrate concentrations ranging from 0.1 to 1.0 InM was run. The reactions were followed to comple- tion and data points calculated at 1Bsec intervals along each curve as described under “Progress Curves.” The resulting 812 data points were combined and fitted to Equation 2 to determine K,,JVn, V,, US, and KII. The results of this fit are pre- sented in Table I.

From the results shown in Table I, it was concluded that all three methods of analysis gave, within experimental error, the same values of K,,,, V,, KIS, and KII.

REFINEMENT OF PROGRESS CURVE METHOD

In order to reduce the amount of work required to determine V,, K,, KIS, and KII values from progress curves, a reanalysis was carried out to determine how many of the previous 812 data points were needed to get a satisfactory fit to the rate equation. This reanalysis led to the following conclusions. (a) Two progress curves with different initial substrate levels are both sufficient and necessary to provide data suitable for fitting Equation 2, provided the A0 values of the two reactions are not too similar. Two curves with initial substrate concen- trations of 0.15 and 0.5 mM were used routinely in subsequent work. (6) Points were routinely calculated at 12-set intervals along each curve although other intervals up t’o 96 set were also tried on the same data with the results shown in Table II. The

O-

f

I 1 I 1 I I I

0.2 0.4 0.6

[PN] (mM)

values of K, and V, obtained are not significantly different although the standard errors increase as the interval increases. V, shows a similar tendency. (c) The last half of each curve contributes little to the accurate determination of these pa- rameters and therefore may be eliminated as a source of data points. Therefore, all subsequent curves were only run until the optical density was approximately 75% of that expected for the complete hydrolysis of the initial AcPN concentration with a final optical density reading taken approximately 60 min af- ter each run.

EFFECT OF PRODUCT

In progress curve assays, product is continuously generated from substrate and therefore it is of prime importance to deter- mine the form of the rate equation which describes the effects of the products (acetate and PN) upon the reaction. The follow- ing experiments determined that acetate had no significant effect upon the reaction under the normal experimental condi- tions used and that PN was a linear noncompetitive inhibitor of the reaction.

Acetate-The effect of acetate upon the reaction was investi- gated by use of both the initial velocity assay and progress curve assay techniques. Six initial velocity assays (A0 = 0.1, 0.15, 0.2, 0.3,0.5, and 1.0 mM) were run in the presence and absence of 40 mM sodium acetate. This is 80 times the amount of acetate that would be liberated during a progress curve assay with A0 = 0.5 mM. The values of Km/V?, and V,, determined for each

TABLE II

Effect of time interval on calculated values of V,,, and K,

Two progress curves with initial AcPN concentration of 0.15 and 0.5 mM were run under the standard assay conditions and optical density values from the continuous curve read at intervals representing the time in seconds. The same two curves are used in each case but as the interval increases, the number of data points available necessarily decreases

Interval

SIG

12 24 36 48 60 72

FIG. 2. Replots of intercepts (A) and slopes (B) of the recip- rocal plots shown in Fig. 1 versus PN concentrations. The lines shown are weighted least squares fits.

84 96

93 49 29 21 16 13 10 9

SM

151 f 11 155 f 13 147 A 16 151 f 15 156 f 19 157 -I: 19 153 f 27 139 f 21

TABLE I

Cmnparison of kinetic parameters calculated by three methods of analysis

No. of data points Km

-

,

V9ti

CJOI PN/min)/(mg enzyme)

1.90 f 0.08

1.96 f 0.09 1.92 f 0.11 2.00 f 0.11 2.06 f 0.13 2.15 f 0.15 2.18 f 0.23 2.14 I 0.19

Method of analysis KmO/Vltt VW& KIS KII

(pwwles PN/min)/ (w ef=yme) m&l ft2.u

Two-stage analysis of initial velocity data. . . . . . . . . . . . . . . . 0.073 st 0.002” 2.14 f 0.02 0.099 f 0.010 1.28 zk 0.24 Total fit analysis of initial velocity data.. . . . . . . . . . . . . . . 0.068 -I: 0.007 1.99 f 0.06 0.082 & 0.010 1.67 f 0.34 Total fit analysis of progress curve data. . . . . . . . . . . . . . . . . . 0.068 f 0.002 2.21 f 0.03 0.091 rt 0.003 0.96 f 0.08

0 K,,, units are millimolar AcPN. Vm units are as in Column 2. b In this and subsequent tables the values of the kinetic parameters are given Z!Z the computer calculated least squares estimates

of the standard errors.

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TABLE III

Effect of acetate on kinetic parameters

K,,, units are millimolar AcPN. Vm units are as in Column 2.

Acetate

&Ml

0

40 0

26.7

7

Method of analysis I

Km/Vm

Initial velocity 0.095 f 0.008 Initial velocity 0.101 & 0.015 Progress curve 0.090 f 0.009 Progress cnrve 0.104 f 0.011

a N.D., not determined.

set of data (with or without acetate present) by fits to Equation terms to describe dead end inhibitions by PN, may be reduced to 5, are presented in Table III. There does not appear to be any significant effect of acetate.

Equation 2 by setting KZi,/Ki, equal to KIX and Ki, equal to KII.

To double check this conclusion, a pair of progress curve assays (with A0 values = 0.15 and 0.5 mM) were run under standard conditions and standard conditions plus 26.7 mu sodium acetate. The data from each pair of curves run under the same conditions were combined and fitted to Equation 2 to determine values for G/V,, V,, KIS, and KII. Since there are no terms in Equation 2 which describe the effect of acetate upon the reaction, effects caused by acetate should manifest themselves as apparent changes in Km/V,, V,, KIS, and KII. However, as can be seen in Table III, acetate did not significantly affect the values of these parameters.

The two rate equations which include terms to describe the effect of possible dead end inhibition by PN through combination with free enzyme and the enzyme-substrate complex are, respec- tively :

VVl?A V=

K m

1 + (K,K~~ + Kdup + K;,PZ

Km&&~ GJWL

where

In both the initial velocity and progress curve experiments, the value of K, in the presence of acetate was higher than in the absence of acetate. Although the difference in K, values was not statistically significant, the direction of the change in K, with acetate was that which would be expected if acetate exerted a small competitive effect upon the reaction. A rough calcula- tion of “Ki” for acetate based on the difference in the observed K, values resulted in values of 168 and 190 mM for the initial velocity and progress curve experiments, respectively. Since the acetate concentration in time course assays rarely exceeded 0.5 mM, the effect of acetate under the conditions used was insig- nificant in any case. Since added acetate (Q) had little or no effect upon the reaction, kg must be >> kg. Therefore, we may safely remove all terms in Equation 1 containing ka/lce in their denominators (i.e. K,, and Ki,) to give

kl K, = 2 with E + P --- A EP ks

and

where

ks Ki = Klo/K9 with EA + P w EAP

k IO

Because of the presence of the P2 term, Equation 7 predicts that the replot of the slopes against PN concentration will be parabolic whereas Equation 8 predicts that the replot of the intercepts will be parabolic. Since, the replots of the slopes and intercepts were both observed to be linear and computer fits of the experimental data to Equations 7 and 8 gave meaningless results, it was concluded that, if PN dead end inhibition is occurring at all, the contribution of the P2 terms in Equations 7 and 8 must be negligible. Setting the P2 terms in Equations 7 and 8 to zero reduces them to the form of Equation 2. Since these equations differ only by constants included in the KIS and KII terms of Equation 2, they are indistinguishable from Equation 6. The data, in any case, are described by Equation 2 and therefore the 42 data points obtained in this experiment were fitted to Equation 2 to determine the values of the kinetic parameters shown in Table I.

PN-PN, unlike acetate, was found to be an inhibitor of the reaction. Although it was not possible to determine whether PN participated as a dead end inhibitor as well as a normal product inhibitor, the following experiments indicated that the inhibition by PN was linear noncompetitive and therefore could be described by Equation 2.

At each of seven PN concentrations, six samples containing different substrate levels (0.1,0.15, 0.2, 0.3, 0.6 and 1.5 mM) were assayed by the initial velocity method. As can be seen from the reciprocal plots in Fig. 1 and the replots shown in Fig. 2 the inhibition by PN was linear noncompetitive, i.e. both the slopes and the intercepts of the reciprocal plots varied linearly with PN. Equation 2 is the general form for such a linear noncompetitive inhibition. Equation 6, which does not include

V??l

&mles PN/min)/ (w e*.w+e)

2.01 & 0.09 2.33 f 0.21 1.59 f 0.07 1.56 f 0.07

-

-

KIS KII

N;.a N.D.

0.086 f 0.008 0.087 f 0.007

N.D. 0.51 f 0.12 1.76 f 1.25

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EFFECT OF TEMPERATURE

By use of the progress curve method, values of K,/Vm, V,, KIX, and KII were determined at various temperatures for horse serum cholinesterase. From the V, values were obtained values of E,, AH*, and &I,,.

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1642 Kinetic Analysis of Progress Curves Vol. 245, No. 7

TABLE IV was determined to be 6523 cal per mole. AH* at 25” was cal- E$ect of temperature on kinetic parameters culated to be 5931 cal per mole, and a value of 1.45 for &IO was

At each temperature, data from two progress curves with initial determined. substrate concentrations of 0.15 and 0.5 mM were combined and Shukuya has reported an E, of 7,700 and a &IO of 1.5 for the fitted to Equation 2 to determine the values for K,/Vm, V,, KZS, human serum choline&erase-catalyzed hydrolysis of acetyl- and KZZ. Assays were run in standard phosphate buffer (pH = choline (8). A &I0 of 1.4 has been reported for the horse serum 7.2, r/2 = 0.2). cholinesterase-catalyzed hydrolysis of acetylcholine (9). In

1939, Glick determined a &, of 1.36 and an E, of 5,100 for the hydrolysis of acetylcholine by crude horse serum (10). He also noted a sharp temperature optimum for the enzyme at 40”. Fitch, with ACPN as substrate, found an E, of 11,900 cal per mole below 31” and an E, of 7,400 cal per mole above 31” for a cholinesterase from P. JEuorescens (3).

Tt?Illper- ature vm

o”M PN/ m~~)/(+w alms)

11.05O1.4 f 0.1 16.05 1.6 f 0.1 20.40 2.2 f 0.1 25.40 2.2 zk 0.1 30.45 2.8 f 0.1 36.20 3.4 f 0.1 40.50 2.9 f 0.1

Km”/ Vm KIS

0.18 f 0.03 0.042 f 0.005 0.12 f 0.01 0.069 f 0.006 0.16 f 0.01 0.174 Z!Z 0.014 0.10 f 0.01 0.168 f 0.013

0.050 f 0.0050.147 f 0.014 0.074 zt 0.0090.371 -I: 0.036 0.076 f 0.0060.350 f 0.032

-

KII

0.77 f 0.25 0.32 f 0.06 0.35 zk 0.05 0.56 f 0.12 0.52 f 0.10 0.88 f 0.25 2.4 f 1.5

EFFECT OF PH

Effect of pH on V,, K,, KIS, and KII-A pair of progress curves with initial substrate levels of 0.15 and 0.5 llzM were run at each of 11 different pH values ranging from 4.57 to 8.45 and were fitted to Equation 2 to determine the values for Km/V,,

5 K,,, units are millimolar AcPN. V,,, units are as m borumn 1. V,, KIX, and KII shown in Table V. Initial velocity assays,

.6-

3.1 3.2 3.3 3.4 3.5

I/OK X IO3

FIG. 3. Effect of temperature on log V,. The point at 40.5” was excluded from the least squares line fit to the data. The slope of the plot is equal to - 1424. Since the slope of an Arrhenius plot = E,/-2.303 R, E, is equal to 6520 cal per mole. Also since E, = AH* + RT, AH* = 5930calpermoleat25°.

At each of seven temperatures ranging from 11.05-40.5”, two progress curves with initial substrate levels of 0.15 and 0.5 mM were obtained. Data from each pair of curves run at the same temperature were combined and fitted to Equation 2 to determine values for Km/V,, V,, KIS, and KII. Table IV presents the results of this experiment. An Arrhenius plot of the V, data is shown in Fig. 3. The point shown at 40.5’ was excluded from the straight line fit to the data since subsequent experiments showed that reversible denaturation occurs at approximately 40". Reproducible results were not obtained at temperatures lower than 11.05’. From the slope of the Arrhenius plot, E,

with initial substrate concentrations of 0.1, 0.15, 0.2, 0.3, and 0.6 InM were also performed at nine of the same pH values ranging from 4.57 to 8.45. The values for K,/V, and V, determined from these assays are also shown in Table V. Preliminary experiments determined that the enzyme was not appreciably denatured in an irreversible manner over the pH range 4.5 to 8.5.

Determination of pK of Active GrouyFig. 4 is a plot of log V, versus pH at 25” with data obtained from both progress curve and initial velocity assays. The data were fit by eye to a tracing of a theoretical curve representing the amount of a (conjugate) base present as a function of pH in the vicinity of its pK. This theoretical curve is the solid line shown in Fig. 4. The curve is asymptotic to two straight lines with slopes of +l and 0 whose intersection (5.83 in Fig. 4) is the apparent pK of a group whose base form is presumably necessary for enzyme activity (11). Plots of log V,/K, and log KIS versus pH both gave apparent pK values of 6.0 although the fit of these data to the theoretical curve was not as good as that shown in Fig. 4. Since histidine is the only group present in proteins with a pK in this region, it can be inferred that a histidine group is im- portant in the combination of free enzyme with substrate as well as in the catalytic event.

Determination of Heat of Ionization of Active Group-In order to confirm the identity of histidine as the active group, the heat of ionization (AHi) of the active group was determined. The preceding experiment on the effect of pH was repeated at 15, 20, 30, and 35” with the use of progress curve assays. A plot of log Vm veraus pH similar to Fig. 4 was made for each temperature and the value of the pK determined. Fig. 5 is a plot of pK uetsus “C. Since AHi = 2.303 X RT2 (dpK/dt), the value for AHd was calculated from the slope of Fig. 5 and found to be 6,950 cal per mole. Greenstein and Winitz (12) have tabulated the apparent heats of ionization for various amino acids. They are generally below 2,000 or above 10,000 cal per mole except for the tyrosine hydroxyl, the cysteine sulfhydryl, and the histidine imidazole for which the AHi are 6,200, 6,500, and 6,900 cal per mole, respectively. This result, in conjunction with the above noted pK around 6.0, leads to the conclusion that histidine is again the most likely active group.

Laidler (13), with use of the data of Hase (14), calculated essential pK values of 7.7 and 6.2 for the horse serum cholin-

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Issue of April 10, 1970 J. K. Balcom and W. M. Fitch 1643

TABLE V

E$ect of pH on kinetic parameters at 66’

Values for the kinetic parameters were determined at each pH by fit of data from two progress curves with initial substrate con- centrations of 0.15 and 0.5 mM to Equation 2 (A) or by fit of data from five initial velocity assays to Equation 5 (B). All assays were run in combinat,ion buffer (0.025 M glycyl glycine-0.025 M potassium phosphate-O.025 M sodium citrate) of appropriate pH. Veloc- ity data taken from assays run at pH values >6.5 were corrected for uncatalyzed hydrolysis of AcPN.

PH / K?lZ”/VWL

A I

8.45 0.079 & 0.003 7.92 0.104 f 0.005 7.47 0.063 f 0.001 7.01 0.098 f 0.004 6.53 0.090 f 0.004 6.06 0.142 f 0.005 5.56 0.288 f 0.011 5.30 0.557 f 0.015 5.08 0.692 f 0.020 4.84 1.51 f 0.040 4.57 2.39 f 0.072

B

0.132 f 0.018 0.096 f 0.016 0.096 f 0.007 0.108 f 0.011 0.153 f 0.004 0.192 + 0.013 0.357 f 0.014

0.805 f 0.045

3.96 f 0.21 -

V7fl

A I

B

(rrmoles PN/min)/(mg enzyme)

1.90 f 0.07 2.40 f 0.32 2.43 f 0.13 1.93 f 0.21 1.74 f 0.06 1.81 f 0.08 2.46 f 0.09 1.84 f 0.12 1.43 f 0.05 1.74 f 0.04 1.14 f 0.04 1.11 f 0.05

0.552 f 0.018 0.577 zk 0.016 0.436 f 0.014 0.270 f 0.007 0.298 f 0.013 0.225 f 0.008 0.132 f 0.005 0.177 f 0.016

a K, units are millimolar AcPN. V, units are as in Column 2.

5.0 6.0 7.0 6.0

PH

FIG. 4. Log li, versus pH at 25”. V, was determined at each pH by the fit of data from two progress curves with A0 = 0.15 and 0.5 mM to Equation 2 (0) or by fit of data from five initial velocity assays to Equation 5 (0). The log V, data shown were fit by eye to a tracing of a theoretical curve (---) representing the amount of a (conjugate) base present as a function of pH in the vicinity of its pK. The curve is asymptotic to two straight lines (- - -) whose intersection (5.83) is the apparent pK of a group whose base form is presumably necessary for enzyme activity (11).

esterase-catalyzed hydrolysis of acetylcholine. Gutfreund (15) determined the pK for trypsin-catalyzed hydrolysis of benzoyl- L-arginine ethyl ester at two temperatures (25 and 35”), cal- culated a AHi of 7000 cal per mole, and concluded that a histidine was present at the active site of the enzyme. Shukuya and Shinoda (16) reported a pK of 5.85 and a AHi of 6500 cal per mole for the human serum cholinesterase-catalyzed hydrolysis of acetylcholine. Fitch, working with an atypical cholinesterase from P. Jluorescens, determined a pK, of 4.6 for the hydrolysis of AcPN by this enzyme (3).

EFFECT OF IONIC STRENGTH

E$ect of Ionic Strength on K,, V,, KIS, and KII-A pair of progress curves with initial substrate levels of 0.15 and 0.5 mM

KIS

A

n2.w

0.112 f 0.008 0.142 f 0.012 0.054 f 0.005 0.138 f 0.011 0.118 f 0.009 0.212 f 0.019 0.448 f 0.055

1.22 f 0.203 0.582 f 0.062

12 f 15 0.935 f 0.159

KII

A

T&w

1.20 f 0.45 0.68 f 0.19 9.23 zk 27.60 0.51 f 0.11 2.06 f 1.17 1.27 f 0.45 2.70 f 1.66 100. f 1980. , 2.70 f 1.22

v. large * v. large 1.39 f 0.62

Y a.

FIG. 5. pK versus “C. The pK at each temperature was deter- mined as in Fig. 4. The point at 35” was excluded in the straight line least squares fit to the data since it was anomalous and derived from the poorest fitting log Vm versus pH plot.

were run at each of nine different ionic strengths (0.126, 0.200, 0.356, 0.508, 0.658, 0.809, 0.997, 1.185, and 1.560). Data points were taken at 12-set intervals along each curve. Data from each pair of curves run at the same ionic strength were combined and fitted to Equation 2 to determine values for the kinetic parameters. KII did not change significantly with increasing ionic strength. The changes in K,, V,, and KIS with increas- ing ionic strength are shown in Fig. 6. Also plotted are the results of a repeat experiment with progress curves and the values for K, and V, determined by initial velocity experiments (five initial velocity assays with A 0 = 0.1, 0.15, 0.2, 0.3, and 0.6 mM

AcPN were performed at each ionic strength). From Fig. 6 it can be seen that K, and KIS increase linearly

with increasing ionic strength while V, remains unchanged or perhaps decreases slightly with increasing ionic strength. The increase in Km and KIS with increasing ionic strength can be attributed to the decreased dissociations of the charged substrate

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1644 Kinetic Analysis of Progress Curves Vol. 245, No. 7

and product from the enzyme in the presence of a more charged environment.

Other workers have made less complete studies on the effects of salts upon some kinetic parameters. Myers reported an increased K, and decreased V, for the human serum cholin- esterase-catalyzed hydrolysis of acetylcholine upon the addition of 0.5 in NaCl to the assay mixture (17). Gregoire, Limozin, and Gregoire (18) reported that raising the KC1 concentration in five steps from 0.033 to 0.264 M resulted in an increased K,,,

I

.6 t A

3.0 B 1

I I I I I c .

.

0 .2 .4 .6 .B 1.0 1.2 1.4

P

FIG. 6. Effect of ionic strength 6) on K,,, (A), V, (B), and KIS (C). Values were determined by progress curve assays ( l ), repeat progress curve assays (A), and initial velocity assays (M). The straight lines were calculated by weighted least squares fits (points weighted inversely to the square of their standard errors). The slopes and their associated standard errors for plots A, B, and C are 0.211 f 0.048, -0.248 f 0.183, and 0.131 f 0.019, respec- tively.

and decreased V, for the horse serum cholinesterase-catalyzed hydrolysis of acetylcholine. Myers (19) studied the effect of ionic strength on the “Ki” for eserine or prostigmine inhibition of the human serum cholinesterase-catalyzed hydrolysis of benzoyl choline and found that increasing the salt concentration of the assay mixture resulted in an increased “Ki”.

CHOLINE-LIKE MODIFIERS

Choline, which is a product of the serum cholinesterase- catalyzed hydrolysis of many choline esters (20-24), and several choline-like substances were tested to determine whether they were capable of affecting the horse serum cholinesterase-catalyzed hydrolysis of AcPN. Two progress curves with initial substrate levels of 0.15 and 0.5 IRM were run under each of the conditions listed in Table V. Data from each pair of curves run under the same conditions were combined and fitted to Equation 2 to determine values for “Km”/“Vm”, “V,“, “KIX”, and “KII.” Since Equation 2 does not include terms to describe the effect of any of these substances, any effects of the tested substances should result in apparent changes in Km/V,, Vm, KIS, and KII. The values of the kinetic parameters determined in the presence of various potential modifiers are listed in Table VI. The most potent substances seem to be choline and N,N-dimethylamino- ethanol. Therefore these two compounds, which differ only by one N-methyl group, were selected for further study.

Choline-Choline in an AcPN-PN system may be classified as an alternate product (25). In order to test the effect of choline upon the hydrolysis of AcPN by horse serum cholinesterase, six initial velocity assays with initial substrate concentrations of 0.1, 0.15, 0.2, 0.3, 0.6, and 2.0 mM were run at each of the 13 following choline chloride concentrations: 0, 1, 2, 4, 6, 8, 10, 13.33, 16.67,20.0,26.67,35.0, and 45.0 IRM. Five initial velocity assays with initial substrate concentrations of 0.15, 0.2, 0.3, 0.6, and 2.0 IRM were run at 61, 81, and 101 IRM choline chloride. The data from each set of five or six assays were then fitted to Equation 5 to determine “K,,,” and “V,” at each choline con- centration. Reciprocal plots (1 /v versus l/A), each having a slope of “K,“/“‘V,” and an intercept of l/“V,,” were drawn for each choline concentration. Four representative plots are shown in Fig. 7. Replots of the slopes “K,“/“Vm” and intercepts 1 /“Vm” of these reciprocal plots against [choline] are shown in Fig. 8. The mechanism shown in Fig. 9 is proposed to account for the observed slope effect in addition to the expected hyperbolic intercept effect by the alternate product choline. The equation

TABLE VI

Effect of choline-like substances on kinetic parameters

K,,, units are millimolar AcPN. V, units are as in Column 2.

Additions to standard reaction solutiona “fGn”‘PV7n” “ Vfll” “KIS” “KII”

(,mdes PN/min)/ m enzyme) ?nM ?nM

None...................................... 0.089 3t 0.009 1.59 zlz 0.07 0.086 f 0.008 0.51 f 0.12 Choline chloride........................... 0.410 f 0.068 2.54 f 0.27 0.311 f 0.023 1.07 -f 1.14 N,N-Dimethylaminoethanol. . . . 0.200 f 0.016 0.85 zk 0.03 0.230 zk 0.019 0.99 & 0.27 N-Methylaminoethanol. . . . . . . . . 0.121 f 0.013 1.54 & 0.08 0.118 f 0.012 0.94 z!c 0.40 Eetaine................................... 0.142 xk 0.016 1.49 * 0.09 0.160 f 0.016 0.41 * 0.10 N,N-Dimethylglycine. 0.100 Z!Z 0.010 1.55 f 0.07 0.089 zk 0.007 0.88 f 0.31 Sarcosine.................................. 0.080 f 0.009 1.28 f 0.06 0.063 f 0.008 0.55 z!z 0.13

0 The concentration of the addition was 0.0267 M throughout.

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Issue of April 10, 1970 J. K. Balcom and W. M. Fitch 1645

I/A (mM)

FIG. 7. Effect of choline on reciprocal plots. Initial velocity assays were run in standard phosphate buffer @H = 7.2, I’/2 = 0.2) at 25” in the presence of 0.0 mM choline (O), 13.4 mM choline (O), 35.0mM choline (A), and 61.6mM choline (m).

which describes the effect of choline on the reaction as diagrammed in Fig. 9 may be written

where I = [choline], and V,, K,, IUS, KIN, and KID are kinetic constants where

KIN = oc3 + kS)(kl6 + kn) UC8 + kll)h

and

Cleland has written a computer program (6) to fit data consisting of points defined by a velocity, a substrate concentration, and an “I” (choline) concentration to Equation 9. When the initial velocity data (93 points) from the experiment described above were used, the following values were obtained: K, = 0.158 f 0.012 (mM); V, = 2.00 f 0.06 (pmoles PN per min per mg of enzyme); KIX = 7.54 i 0.71 (MM); KIN = 10.38 f 3.73 (mM); KID = 5.57 f 1.66 (mM). The values for these parameters were the basis of the curve shown in Fig. 8B.

It is of interest that choline actually increases the V, for the reaction. This implies that the alternative reaction sequence [EQ + choline + E acetylcholine + E + acetylcholine] is faster than the normal (EQ + E + Q) step. Thus a hyperbolic activation by choline on V, is observed. In this particular case, choline competitively inhibits the reaction by combining with free enzyme (E) as a dead end inhibitor but activates by increas- ing 8,. From the reciprocal plots shown in Fig. 7 it can be seen that the result of this competition is over-all inhibition below approximately 1 mM AcPN and over-all activation above 1 mM

substrate. N, N-Dimethylaminoethanol-At each of five DMAE con-

centrations, five initial velocity assays with initial substrate levels of 0.1, 0.15, 0.2, 0.3, and 0.6 mM were run. Each set of five assays performed at a single DMAE concentration were fitted to Equation 5 to determine “Km” and “Vm” at each DMAE

0 20 40 60 60 100

[CHOLINE] (mM)

FIG. 8. The slopes (A) and intercepts (B) of 16 reciprocal plots (four such reciprocal plots are shown in Fig. 7) versus the choline concentration at which the initial velocity assays were run. Five of six assays were run at each of 16 different choline concentra- tions. The slope and intercept of each reciprocal plot was deter- mined by a computer fit to Equation 5 (6). The smooth line shown in B was calculated by use of Equation 9.

II 4 E, bEAI/“‘-,b’ t,E+Q

k2 k4 T-

FIG. 9. Proposed mechanism for choline effects. Choline com- bines with EQ as alternate product and with E as dead end inhibi- tor. E, free enzyme; A, AcPN; P, PN; Q, acetate; C, choline; QC, acetylcholine.

concentration. Reciprocal plots with slopes of “K,“/“Vm” and intercepts of l/“Vm” are shown in Fig. 10. Replots of the slopes and intercepts versus [DMAE] (shown in Fig. 11) determined that the inhibition by DMAE was linear uncompetitive (slopes unaffected; intercepts vary linearly with DMAE). Therefore the data were fitted to the equation

V?SA v= Km + A (10)

which describes an uncompetitive inhibition by I (DMAE) (6). KII is equal to lci&ir, where

E& + DMAE + EQDMAE

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1646 Kinetic Analysis of Progress Curves Vol. 245, No. 7

I I

I I I I I I

0 2 4 6 8 IO

I/A hM1

FIG. 10. Effect of DMAE on reciprocal plots. Initial velocity assays were run under standard assay conditions in the presence of 0.0 mM DMAE (o ), 6.67 mM DMAE (0 j, 13.35 mM DMAE (m), 20.02 rnM DMAE (A), and 26.67 mM DMAE (X). Each set of datawas fitted to Equation 5 to determine the slope and intercept values of each reciprocal plot.

0 4 8 IO 12 14 16

DMAEI (mM)

FIG. 11. Replots of slopes (A) and intercepts (B) versus DMAE. See Fig. 10 for reciprocal plots. The lines shown are weighted least squares fits (points weighted inversely to square of associated standard errors). The slope of Line A is 0.00086 f 0.00043; the slope of Line B is 0.0159 f 0.0024.

The computer program to fit Equation 10 was obtained from Cleland. The values of K,, V,, and KII determined by this fit were 0.224 f 0.016 (mM), 2.09 f 0.07 (pmoles PN per min per mg of enzyme), and 24.3 f 1.8 (mM), respectively.

DISCUSSION

Comparison of Initial Velocity Technique with Progress Curve Method-Reciprocal plots and replots of initial velocity data are invaluable aids in determining the proper form of the rate equa- tion for a reaction. Thus, the initial velocity method of analysis was used to determine the effects of acetate, PN, choline, and N ,N-dimethylaminoethanol on the hydrolysis of AcPN when the rate equation for the reaction in the presence of the above modifiers was not known. The main disadvantage in the use of the initial velocity method of kinetic analysis is that a fairly large number (approximately 16) of initial velocity assays are required in order to make each reciprocal plot and replot to at least four points. Init’ial velocity data may also be fit to the full rate equation for the reaction if the equation is already known or if

the number of possible rate equations is very small. A smaller number of initial velocity assays are required to obtain sufficient data for a total fit if the proper rate equation is already known.

One advantage of the progress curve assay method is the large number of data points that can be derived from each assay. Data from a single assay possess high internal consistency. Two assays with a total running time of approximately 15 min are sufficient to provide data for fitting the rate equation. In contrast to the initial velocity assay method, substrate and product concentrations are experimentally determined. This eliminates errors caused by incorrect pipetting or incorrect reagent preparation with respect to substrate and product concentrations. The determination of the velocity at any point during a progress curve assay is more exact than the determina- tion of the initial velocity of an assay by extrapolation to zero time; it is also less prone to subjective error. Another advantage of the progress curve assay is the comparatively small amount of substrate or substrates and product or products required for the determination of the kinetic constants. When substrates or products are difficult to prepare, this aspect might be of considerable importance.

The major drawback to this method is the unavoidable presence of product or products during the assay. This makes it difficult to study modifiers of the reaction when the full rate equation for the reaction, including terms to describe the effects of these substances, is not known. Once the complete rate equation for the reaction in the presence of a specific modifier is known, however, then data from progress curves may be used to determine the kinetic constants by a fit of the data to the com- plete rate equation. Of course, data from each progress curve is of necessity obtained at a single modifier concentration since only one reaction mixture is prepared. Thus, one might expect poor determinations of modifier constants unless several progress curves were run.

The total time required to get the kinetic constants in the progress curve method is somewhat greater than that required in the initial velocity method because more data points are handled in the progress curve method.

Comparison of Rate Equation and Integrated Rate Equations- A method for fitting progress data to rate equations has been presented and used successfully to determine kinetic parameters for the hydrolysis of an artificial substrate by horse serum cholinesterase. Thus it is clear that progress curve data need not be fit to the integrated form of the rate equation in order to obtain values for the kinetic parameters. Moreover, these progress curve data were fitted to the rate equation for the reaction under conditions unlike those of Kalow et al. (1) or Schwert (2), in that the effect of generated product was signifi- cant. Considering (a) the difficulty of integrating rate equations, (6) the complexity of the resultant equations and, on the other hand, (c) the availability, for many mechanisms, of computer programs (6) to fit the differential forms of the equations, it is preferable to fit progress curve data to the rate equations them- selves rather than their integrated forms. Nevertheless progress curve data obtained by the described method could be fit to integrated rate equations. In either case the full equation which describes the mechanism must be known.

Aclcnowledgment-We wish to thank Dr. W. W. Cleland for many helpful discussions and numerous computer programs.

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Issue of April 10, 1970 J. K. Balcom and W. ill. Fitch 1647

REFERENCES

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13. LAIDLER, K. J., Faraday Sot. Trans., 61, 550 (1954). 14. HASE, E., J. Biochem. (Tokyo), 39, 259 (1952). 15. GUTFREUND, H., Faraday Sot. Trans., 61, 441 (1954). 16. SHUKUYA, R., AND SHINODA, M., J. Biochem., (Tokyo) 43, 315

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Jean Kant Balcom and Walter M. FitchCholinesterase As a Model Case

A Method for the Kinetic Analysis of Progress Curves Using Horse Serum

1970, 245:1637-1647.J. Biol. Chem. 

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