What students must know about the determination of enzyme kinetic parameters
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 indicates that the best method to evaluate Vmax and Km is nonlinear regression, even in the presence of constant relative 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 computersimulated velocity data, with which the student may get direct practise and experience.
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Article: Determination of Enzyme Kinetic Parameters on Sago Starch Hydrolysis by Linearized Graphical Methods
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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, I50134 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 computersimulated 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.
?Email: 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 computersimulated 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 halfmaximal
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
03074412/99/$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 variablesSlope
yaxis intercept
xaxis 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 selfcorrection 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 yaxis, 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 leastsquares 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
enzymecatalysed 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
lineartransformations 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 socalled direct linear plot introduced by Eisenthal
and CornishBowden [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 coordinates 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 coordinates 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
leastsquares method to his [S]/? vs. [S] plot [10], the
linear transformationsof
equation were originally conceived to obtain the kinetic
Henri—Michaelis—Menten
88
F. Ranaldi et al. / Biochemical Education 27 (1999) 87—91
Page 3
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
leastsquares 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 leastsquaresmethod 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.
by means 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
89
Page 4
Table 2
Means and variances of the K?values determined by different methods of calculation from simulated velocity data. ‘‘True’’ K?"0.1
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
$SD
0.101
0.004
0.100
0.004
0.101
0.004
0.100
0.003
0.101
0.003
0.100
0.004
0.099
0.006
0.097
0.008
0.098
0.007
0.096
0.006
0.099
0.006
0.098
0.008
0.099
0.007
0.094
0.014
0.098
0.009
0.094
0.008
0.103
0.009
0.096
0.010
Note: NLR, nonlinear regression; DLP, direct linear plot; SD, standard deviation (square root of variance).
Fig. 2. An example of the Microsoft Excel sheet, used to simulate
experimental data. Cells B4—B14 contain Michaelis—Menten equation
‘‘B1)A?/(D1#A?)’’; cells D4—D14 contain the formula ‘‘B?#(B?)C?),
where i stands for the number of the row.
deviation imposed. For each set, kinetic parameters have
been determined by means of nonlinear regression
(NLR), simple linear regression applied to the four trans
formations previously described, and direct linear plot
(DLP). The results are shown in Tables 2 (for K?) and
3 (»???values, respectively), in terms of mean$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
Page 5
Table 3
Means and variances of the »???values determined by different methods of calculation from simulated velocity data. ‘‘True’’ »???"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
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
[1] F. Ranaldi, C. Iacoviello, P. Vanni, C. R. Soc. Biol. 189 (1995)
657—665.
[2] I.H. Segel, Enzyme Kinetics, Wiley, New York, 1975.
[3] J.B.S. Haldane, K.G. Stern, in: Allegemeine Chemie der Enzyme,
Verlag von Steinkopff, 1932, p. 119.
[4] J.B.S. Haldane, Nature 179 (1957) 832.
[5] G. Scatchard, Ann. N. Y. Acad. Sci. 51 (1949) 660.
[6] G.S. Eadie, J. Biol. Chem. 146 (1942) 85—93.
[7] B.H.J. Hofstee, Science 116 (1952) 329—331.
[8] B.H.J. Hofstee, Nature 184 (1959) 1296—1298.
[9] H. Lineweaver, D. Burk, J. Am. Chem. Soc. 56 (1934) 658—666.
[10] C.S. Hanes, Biochem. J. 26 (1932) 1406—1421.
[11] G.N. Wilkinson, Biochem. J. 80 (1961) 324—332.
[12] H. Lineweaver, D. Burk, W.E. Deming, J. Am. Chem. Soc. 56
(1934) 225.
[13] D. Burk, (1934) in: F.F. Nord, R. Weidenhagen (Eds.), Ergebmisse
der Enzymforschung, vol. 3, pp. 23—56.
[14] R. Eisenthal, A. Cornish—Bowden, Biochem. J. 139 (1974)
715—720.
[15] P. Askelof, M. Korsfeldt, B. Mannervik, Eur. J. Biochem. 69
(1976) 61—67.
[16] D.B. Siano, J.W. Zyskind, H.J. Fromm, Arch. Biochem. Biophys.
170 (1975) 587—600.
[17] B. Mannervik, in: L. Endrenyi (ed.), Kinetic Data Analysis,
Plenum Press, New York, 1981, pp. 235—270.
[18] L. Endrenyi, F.H.F. Kwong, in: L. Endrenyi (ed.), Kinetic Data
Analysis, Plenum Press, New York, 1981, pp. 89—103.
[19] J.E. Dowd, D.S. Riggs, J. Biol. Chem. 240 (1975) 863—869.
[20] G.L. Atkins, I.A. Nimmo, Biochem. J. 149 (1975) 775—777.
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