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Biochemical Education 27 (1999) 87—91

What students must know about the determination of

enzyme kinetic parameters

Francesco Ranaldi*, Paolo Vanni, Eugenio Giachetti?

Department of Biochemical Sciences, University of Florence, viale Morgagni 50, I-50134 Firenze, Italy

Abstract

After a brief overview of the limits of the graphical methods used to determine enzyme kinetic parameters, the paper shows the

results of their application to simulated velocity data, influenced by experimental errors of increasing magnitude. The comparison

indicatesthat the best method to evaluate »???and K?is nonlinearregression, even in the presence of constantrelative error; whereas

the double reciprocal plot should be avoided, unless used with a proper weighting factor. The paper also suggests a simple method to

obtain computer-simulated velocity data, with which the student may get direct practise and experience. ? 1999 IUBMB. Published

by Elsevier Science Ltd. All rights reserved.

1. Introduction

Most biochemistry textbooks assign a limited space to

enzyme kinetics, especially to the questions inherent in

the methods for determining kinetic parameters. More-

over, in some of the most popular books, it is common to

encounter vague or misleading sentences about this

topic. Neither applied biochemistry nor laboratory

courses seem to provide the students with the appropri-

ate fundamental information required for a correct

approach to kinetic calculations. We think this scanty

training may be the reason for the fact that in many

biochemical publications the constants characteristic of

enzyme catalysis are still determined by means of simple

linear regression applied to the double reciprocal plot,

also known as Lineweaver and Burk plot.

This is not a trivial question. We have observed that in

some practical instances, a Michaelis constant deter-

mined with this procedure may differ up to 150% from

that evaluated with nonlinear regression. We can also

show that the use of simple linear regression applied

to Lineweaver and Burk transformation may even lead to

wrong interpretationsabout the effect of an inhibitor [1].

Biochemistryis considered to be the most‘‘exact’’ of all

the sciences in the field of Biology, and we must do our

*Corresponding author.

?E-mail: giachetti@scibio.unifi.it.

best to consolidate this status. Some contribution in this

sense can come from an early training of the student,

the future researcher, by providing him or her with the

proper computational tools: this will also promote the

development of a critical mind for the analysis of experi-

mental data in general.

The primary aim of the present paper is to focus the

teacher’s attention on the limits of the methods com-

monly suggested to determine enzyme kinetic parame-

ters. We offer a critical overview of the various graphical

methods and their comparison in terms of accuracy and

precision. At the end, we suggest a simple procedure to

obtain computer-simulated velocity data, influenced by

suitable experimental errors, with which the student may

practise and verify the statements we make.

2. Accurate or precise?

The evaluation of the kinetic constants is a central

point in enzyme research. Apart from its content in terms

of rate constants and beyond its merely mathematical

meaning: substrate concentration yielding half-maximal

reaction velocity, the K?(or K?) can provide a lot of

biochemical and physiological information about an en-

zyme [2]. The K?establishes an approximate value for

the intracellular level of substrate; since K?is a constant

characteristic of each enzyme, its value may be used for

discriminating isoforms from different organisms, or tis-

sues, or developmental stages; if we know the value of

0307-4412/99/$20.00 ? 1999 IUBMB. Published by Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 7 - 44 1 2( 9 8 ) 0 0 30 1 -X

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Table 1

Common linear transformations of Henri—Michaelis—Menten equation

Plot ofPlotted variables Slope

y-axis intercept

x-axis intercept

Lineweaver—Burk

Hanes

Eadie—Hofstee

Eadie—Scatchard

1/v vs. 1/[S]

[S]/v vs. [S]

v vs. v/[S]

v/[S] vs. v

K?/»???

1/»???

!K?

!1/K?

1/»???

K?/»???

»???

»???/K?

!1/K?

!K?

»???/K?

»???

Note: The first three transformations were suggested by Woolf before 1932; they had never been published by the original author, but they are all

quoted in an enzymology textbook and in a subsequent paper by Haldane and Stern [3, 4]. Moreover, the Eadie—Scatchard plot [5] is practically the

same as the Eadie—Hofstee plot [6—8], the only difference being the axes swap. All the four linear transformations must be therefore considered

‘‘rediscovered’’ later.

The Lineweaver and Burk plot [9] (also referred as double reciprocal plot) has the advantage of containing the experimental error in only one

plotted variable (v); however it has the drawback that small errors on the determinationsof v are enlarged when reciprocals are taken. As a result, since

the errors on v are most likely to be significant at low values of v, these measurements affect the slope of the plot in a decisive manner.

Also the Hanes plot [10] contains a v term in the denominator; however, the [S]/v ratio probably introduces a self-correction preventing the error

propagationimpliedin thetransformation of velocity equation,as Wilkinson pointed out [11]. Moreover, in this plot, the pointscorrespondingto low

v values cluster on the left side, near the y-axis, so to affect the slope of the plot to only a small degree.

TheEadie—Hoftseeand Eadie—Scatchardplotsdo not involve reciprocalsof v,but herebothplotted variablesare subjectto experimentalerror. This

latter feature may imply the advantage of calling attention to points that deviate significantly from the theoretical pattern; but at the same time it

conflicts with the theoretical basis itself of least-squares fitting (see below).

K?, we can normalise enzyme assays so that »???pro-

vides an actual measure of total enzyme concentration.

Finally, by evaluating the effects of ligands, we can ident-

ify regulatory pathways or the kinetic mechanism of an

enzyme-catalysed reaction. With this in mind, it is easily

realised how important it is to have measures as exact

and precise as possible of the kinetic constants, »???and

K?. In particular, if the principal intent is to compare

parameters estimated from two similar experiments (e.g.

with and without inhibitor), the method yielding the

smallest variance (i.e. the more precise) should be chosen;

if the principal intent is to estimate the ‘‘true’’ values as

closely as possible (i.e. to discriminate isoenzymes) the

influence of bias must be considered (i.e. the more accu-

rate).

3. Graphical methods

Giventhe hyperbolicrelationshipbetween? and [S] in

Michaelis—Menten kinetics, any attempt to obtain

»???and K?from a plot of ? vs. [S] would undoubtedly

produce errors. To overcome this problem, transforma-

tions of the Henri—Michaelis—Menten equation were in-

troduced to transform the dependence between the two

variables into a linear relationship. In this way, the kin-

etic parameters may be determined by simply drawing

a straight line.

Table 1 shows some features of the four most common

linear transformations of

equation. Each of them has advantages and disadvan-

tages which are briefly summarised in the legend to the

table. A major concept the student should learn early is

that all of these are the consequence of a more general

Henri—Michaelis—Menten

feature, namely, whenever the form of an equation is modi-

fied,the relative weightingof the experimental observations

alters in a definite manner, as Lineweaver and Burk em-

phasised in their original paper [9]. These authors also

quantified the proper weighting to be applied in two

contemporaneous papers [12, 13] but this part of their

work appears to be widely neglected.

A graphical method deserving a separate discussion is

the so-called direct linear plot introduced by Eisenthal

and Cornish-Bowden [14]. The direct linear plot is

a parametric plot: axes, instead of variables,represent the

kinetic parameters, »???and K?, and each experimental

observation is plotted as a straight line rather than as

a point. Any point on the line has as co-ordinates a given

pair of parameter values that satisfy Henri—Michaelis—

Menten equation for that observation. In the absence of

experimental errors, n lines of this kind have a common

intersection point whose co-ordinates provide the only

parameter values satisfying Henri—Michaelis—Menten

equation for all n observations. In practice, one will

obtain a set of intersection points and the best estimates

of K?and »???can be taken as the medians of the

corresponding abscissa and ordinate values, respectively

(seeFig. 1). Themain advantageof the directlinearplot is

thatof being theoreticallyindependentfromthe nature of

experimental error.

4. Error structure in enzyme kinetic measurements

Although already in 1932, Hanes had applied the

least-squares method to his [S]/? vs. [S] plot [10], the

lineartransformations of

equation were originally conceived to obtain the kinetic

Henri—Michaelis—Menten

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F. Ranaldi et al. / Biochemical Education 27 (1999) 87—91

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Fig. 1. An example of the direct linear plot. Each experimental obser-

vation is reported as a line across !s?and ??, and each intersection

provides an estimate of K?(abscissa) and »???(ordinate). The best

estimates of the parameters are taken as the medians of the two sets of

estimates.

parameters by drawing a straight line manually. How-

ever, since fitting by eye a line to the experimental points

is in any case a subjective procedure, linear regression

was applied early to graphical methods. Another point

the student should clearly understand is that in every

experimental approach, the results of data analysis are

highly sensitive to appropriate assumptions about the na-

ture and the behaviour of experimental errors. As an

example, the basic assumptions for the validity of the

least-squares method are a normal distribution of errors,

absence of systematic errors and negligible error asso-

ciated with the independent variable. Once these require-

mentsare satisfied,the least-squaresmethod is validboth

in the case of equally accurate measurements (constant

variance) and of measurements with different accuracy.

Inthe latter case, a weightingfactorshouldbe introduced

in regression analysis.

There is substantial agreement that for the measure-

ment of enzyme kinetic data carried out by a careful

operator, the experimental error has a normal distribu-

tionand severalauthorsprovideevidencesupportingthis

[15—17]. As regards the error behaviour (for a complete

reference see Endrenyi and Kwong [18]), some studies

have shown that error magnitude is approximately pro-

portional to reaction velocity. In other studies, the ex-

perimental error has been found to be made up of two

components of variable size, one independent and the

other proportional to reaction velocity. In short, the

conditions of constant absolute and constant relative

errors may be considered the two borderline cases of the

topic. Thus, because there are no data allowing final

conclusions in this sense, error dimensions should be

evaluated experimentally each time from empirical vari-

ance. Several effective shortcuts to establish the error

behaviour have been reported [18].

5. Experimental

To assess which of the methods for determining en-

zyme kinetic parameters is to be preferred, we compared

the results obtained with each of the most common

procedures (nonlinear regression, the four linear trans-

formations described above, and the direct linear plot)

applied to computer simulated velocity data. Linear and

nonlinear regression analyses were performed using the

programme SYSTAT v 5.01.

Data simulation gives a wide range of chances in terms

of ‘‘quality’’ of the data by suitably varying structure and

magnitude of the experimental errors. In generating the

velocity data for our comparison, we assumed a normal

error distribution — as justified in the previous section.

Moreover, for didactic simplicity, we restricted our anal-

ysis to the case of constant relative error (see below).

Note that in this way we chose the most unfavourable

condition for the nonlinear regression without weighting

factor, and that in the case of constant absolute error,

simple nonlinear regression would be the optimal analy-

sis tool by definition. The latter statement could be

checked by the student by means of practical exercises.

6. Data simulation

Velocity data can be easily simulated using the func-

tion ‘‘Random Number Generation’’ of Microsoft Excel.

We have used version 5.0a, run on a Performa 6500

Power Macintosh.

The procedure is quite simple and may be summarised

as follows:

(1) Fix the ‘‘true’’ values of the kinetic parameters: i.e.,

»???"1 and K?"0.1.

(2) Select a given number of substrate concentrations

(e.g. 11), distributed over a range of 0.25—5 times the

K?value (see Table 2).

(3) For each substrate concentration calculate the

‘‘exact’’velocity(»????)

Michaelis—Menten equation.

(4) Generate a set of random numbers with normal dis-

tribution around the mean (zero) and with suitable

standard deviation (e.g. 0.05; 0.1; 0.15).

(5) Insert in the proper cells the formulas corresponding

to »????#(»????) random number) to obtain the ‘‘ex-

perimental’’ velocities (»???), containing constant rela-

tive errors.

An example of a simulation (with SD"0.05) is shown

in Fig. 2.

bymeans ofHenri—

7. Results of the comparison

Our comparison has been carried out using 150 sets of

simulated velocity data: 50 for each level of standard

F. Ranaldi et al. / Biochemical Education 27 (1999) 87—91

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Table 2

Means and variances of the K?values determined by different methods of calculation from simulated velocity data. ‘‘True’’ K?"0.1

NLR 1/v vs. 1/SS/v vs. Sv vs. v/Sv/S vs. v

DLP

n"50, DS 0.05

Small error

Mean

$SD

n"50, DS 0.10

Mean

$SD

n"50, DS 0.15

Large error

Mean

$SD

0.101

0.004

0.100

0.004

0.101

0.004

0.100

0.003

0.101

0.003

0.100

0.004

0.099

0.006

0.097

0.008

0.098

0.007

0.096

0.006

0.099

0.006

0.098

0.008

0.099

0.007

0.094

0.014

0.098

0.009

0.094

0.008

0.103

0.009

0.096

0.010

Note: NLR, nonlinear regression; DLP, direct linear plot; SD, standard deviation (square root of variance).

Fig. 2. An example of the Microsoft Excel sheet, used to simulate

experimental data. Cells B4—B14 contain Michaelis—Menten equation

‘‘B1)A?/(D1#A?)’’; cells D4—D14 contain the formula ‘‘B?#(B?)C?),

where i stands for the number of the row.

deviation imposed. For each set, kinetic parameters have

been determined by means of nonlinear regression

(NLR), simple linear regression applied to the four trans-

formations previously described, and direct linear plot

(DLP). The results are shown in Tables 2 (for K?) and

3 (»???values, respectively), in terms of mean$standard

deviation. For both parameters, when the error is small

(SD"0.05), all the methods are extensively accurate (i.e.

close to the ‘‘true’’ value) and precise (i.e. small standard

deviations, small variances). However, as the magnitude

of the error increases, the limitations of the graphical

methods are highlighted. Whilst nonlinear regression

proves to be the most accurate and precise method of

calculation,the double reciprocal plot is clearly the worst

procedurebothin terms ofaccuracyand precision.Ofthe

other methods, the Hanes plot (S/? vs. S) appears to be

more accurate than Eadie—Hofstee and Eadie—Scatchard

plots, but is less precise than them. These results are in

good agreement with the conclusions of the excellent

paper by Dowd and Riggs [19], who performed a similar

comparison using a much larger set of simulated data

(500), but with only five ‘‘experimental’’ points per set,

instead of the 11 in the present study. An unsatisfactory

judgement must be given on the direct linear plot which,

although decidedly preferable to Lineweaver and Burk

transformation, appears to have no advantage with re-

spect to the other three graphical methods. This finding

does not fully agree with the report by Atkins and

Nimmo [20].

In conclusion, even with a constant relative error —

a condition that would require the application of

a weighting factor for the correct analysis — the un-

weighted nonlinear regression is the most reliable

method for determining enzyme kinetic parameters. On

the other hand, when used without a weighting factor,

double reciprocal plot is the less satisfactory procedure.

Though this is (or at least it should be) known since long,

this plot continues to be widely used.

8. Final remarks

Once it is revealed that procedures based on linear

transformations have poor reliability, the student should

be advised that graphical methods must not be com-

pletely abandoned in enzyme kinetics. In fact, the in-

formation they provide is of great value both to monitor

the pattern of the experimental data during their

measurement and, in particular, as a powerful diagnostic

tool in inhibition studies.

Thanks to the wide spread use of personal computers,

combined with their increasing power, nonlinear regres-

sion analysis is nowadays within everyone’s reach.

90

F. Ranaldi et al. / Biochemical Education 27 (1999) 87—91

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Table 3

Means and variances of the »???values determined by different methods of calculation from simulated velocity data. ‘‘True’’ »???"1

NLR1/v vs. 1/SS/v vs. Sv vs. v/Sv/S vs. v

DLP

n"50, DS 0.05

Small error

Mean

$SD

n"50, DS 0.10

Mean

$SD

n"50, DS 0.15

Large error

Mean

$SD

1.004

0.018

1.000

0.026

1.006

0.019

1.001

0.018

1.006

0.017

1.002

0.022

0.990

0.038

0.988

0.052

0.990

0.039

0.979

0.037

1.000

0.037

0.994

0.035

0.993

0.048

0.992

0.075

0.991

0.047

0.971

0.053

1.010

0.059

0.975

0.065

Note: NLR, nonlinear regression; DLP, direct linear plot; SD, standard deviation (square root of variance).

Teachers should direct the students to exploit this option

for calculating reliable kinetic parameters, and at the

same time he or she should be responsible enough to

warn them of the problems arising from the (mis)use of

double reciprocal plot.

References

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