Biochemical Review
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Some recent advances in enzyme kinetics
CHRISTOPHER W. WHARTON
Department of Biochemistry, University of Birmingham,
P.O. Box 363, Birmingham B15 27T, U.K.
linear-plot method. Boeker’s (1982a) method is similar to one
proposed long ago by Alberty & Koerber (1957), in which
AIPI/t is plotted against time. The initial rate is again given by
the ordinate intercept. The relative efficacy of these two plots for
In this short review, I have chosen to make a selection from the
recent literature on enzyme kinetics of those aspects that I
consider to represent advances in the conceptual approach to
the subject and/or to be particularly useful practical techniques.
Estimation of initial rates
The vast majority of enzyme kinetic analyses presented in the
literature involve the measurement and analysis of initial rates;
this is so because the interpretation of properly estimated initial
rates is a relatively robust procedure. It is, however, not
infrequent to find that it is difficult to obtain good estimates of
initial rates owing to premature curvature of product-time
traces. This may result, for example, from product inhibition or
enzyme inactivation. It is well known that it is notoriously
difficult to estimate tangent slopes accurately, and it is
undoubtedly the case that many results that have been presented
in the literature are subject to this source of error (see, e.g.,
Atkins & Nimmo. 1980; Waley, 1981). Accordingly, considerable attention has been paid in recent times to the
possibility of improving estimation of initial-rate values in
circumstances in which the progress curve is significantly
non-linear near the start of the reaction.
All of the methods described below rely on analysis of
the integrated Michaelis equation, which, of course, relates
substrate depletion or product accumulation to the passage of
time. Several empirical methods based on the fitting of
polynomials have been proposed Isee, e.g., Cornish-Bowden
(1975) for a critical discussionl, but these are not described here,
since they involve either laborious computation or the availability of a computer program. Szawelski (1983) has recently
proposed a very simple polynomial procedure based on the
smoothing method of Savitzky & Golay (1964).
The earliest suggestions as to how the estimates of initial rates
might be improved by a simple procedure was provided by Lee
& Wilson (1971). They proposed that a chord be constructed
(say between 0% and 20% reaction) and that its slope be taken
as the reaction rate at a substrate concentration represented by
the mean of the values at the chord ends, i.e. 10% in this case.
This procedure, although quite accurate if the reaction is less
than 50% complete, does not provide an estimate of the initial
rate at the initial substrate concentration, and so the estimate
will be affected by any product inhibition that occurs.
Cornish-Bowden (1975) has presented an ingenious method
based on the integrated Michaelis equation, which uses a form of
the direct linear plot (see below) for the estimation of initial
velocities in non-ideal circumstances. This method, illustrated in
Fig. 1 and described in the legend, performs well in many
circumstances provided that the extent of reaction is restricted
to approx. 50% when the approximated form of the integrated
equation is used. The approximation obviates the need for
calculations involving logarithms to be made and is very
accurate at low concentrations of product.
Boeker (1982a) has suggested a simple accurate method
based on a plot of AlPl/t against A[PI,where A[Pl is the change
in product concentration during time t (i.e. AIPl = [PI - [PI,).
This type of plot, which gives the initial rate at zero time as the
ordinate intercept, is easy to construct and seems to be accurate
in most circumstances. The method relies on the same
approximation used by Cornish-Bowden (1975) in the directVOl. 1 1
4 -I%
-3
KnlbpO,
-2
-[PI,+
-1
0
lPV2 (mM)
Fig. 1. Direct linear plot for the determination of initial rates
from data that does not demonstrate a linear IPI-versus-t
relationship near the start of the reaction, constructed in
accordance with the method of Cornish-Bowden ( 1 975)
The integrated Michaelis equation may be plotted in parameter
space with the use of ordinate intercepts of I P l / t and abscissa
intercepts of -[Pl/ln ( [ P l J ( [ P l , - [ P l ) ) . Such a plotting procedure gives a series of lines that, for perfect data, intercept at a
common point, namely Vapp,,K , ,,pp.). The above procedure,
which is exact, may be simplified without significant loss of
accuracy (provided that 1 P l / [ P ] , < 0.5) by approximation of the
logarithmic term for the abscissa intercepts by using the series:
Truncation of the series to the first term and appropriate
substitution and rearrangement gives the expression
-[PI, + [Pl/2 for the abscissa intercepts. The axes are drawn at
approx. 20-30° in order to enhance resolution of the ordinate
axis and so to separate more clearly the v,, estimate from the
other [Pl/t intercepts. The data used to construct the plot are
taken from Wharton & Szawelski (1981), and represent the
papain-catalysed hydrolysis of methoxycarbonylglycine p-nitrophenyl ester in 0.3 M-phosphate buffer, pH 5.98, at 25OC. The
initial substrate concentration was 3 . 4 2 m ~and the plot was
constructed with data taken between 18% and 6 I% of complete
reaction. The initial rate determined from Fig. 1 is
1.70 x lO-’M.s-’ as compared with a tangent estimate at t = 0
of 1.6 1 x lo-’ M s-’. A tangent estimate at 6 1% reaction
indicated that the rate had decreased to approx. 50% of the
initial rate at this point. The broken line, which gives the v,
value, is drawn through the ‘middle’ of the group of intercepts
of the other lines. Strictly, the correct value of v , is given by the
median, on the ordinate, of the lines drawn through each
intersection, but the improvement in accuracy is insufficient to
justify this complication.
+
817
818
BIOCHEMICAL SOCIETY TRANSACTIONS
the estimation of initial rates depends on the accuracy of the
approximations that are used in each case to simplify the
integrated Michaelis equation. No comparison has, as yet, been
made between these methods.
Waley (1981) has improved the method of Lee & Wilson
(1971) in that a chord estimate is made but the concentration of
substrate to which the chord slope corresponds is given exactly
by the equation:
ISI, =
ISl,-lSl,
In(ISI,/ISlJ
where IS], and ISl, are the substrate concentrations at the chord
ends (e.g. 0% and 20% of reaction) and IS], is the ‘corresponding’ concentration of substrate that relates to the chord slope.
This method, although more complicated, has the advantage of
being exact when product inhibition and enzyme inactivation do
not occur. An extension of the method, which makes use of two
chord slope estimates, provides good estimates of initial rates
when product inhibits competitively. It is also recommended for
use in cases of enzyme inactivation and inhibition by product in
a manner that is other than competitive. A surprising aspect of
the two-chord method is that it apparently allows the detection
of competitive product inhibition in a single experiment (provided that K,>K,); this was previously thought not to be
possible.
In view of the fact that there are now several quite simple
methods available for the estimation of initial velocities in
non-ideal circumstances, it is to be hoped that these will be used
by enzyme kineticists and others with the result that the
alarmingly inaccurate procedure of visual tangent estimation
will slowly become extinct.
Plotting methods for the analysis of initial velocities
The primary and secondary plotting methods developed for
the analysis of complex enzyme mechanisms (see, e.g., Cleland,
1977) are now well known and need not be discussed in the
present context. Here I discuss some aspects of the nature
and fidelity of the analysis of primary plots.
Perhaps the most remarkable example of lateral thinking and
ingenuity to have appeared in this field in recent years is the
direct linear plot (Eisenthal & Cornish-Bowden, 1974). This
exceptionally simple but conceptually elusive transformation to
parameter space is by now fairly well known, but still surprisingly
little used. Here I am concerned with a more recent variant of
the direct linear plot, presented by Cornish-Bowden & Eisenthal
(1978), and some of the general principles involved.
An example of the variant of the direct linear plot described
above is shown in Fig. 2. It is immediately obvious that a little
calculation, namely of [Sl/u, and I/v, values, is required, but in
my opinion this is outweighed by the fact that this form of the
plot does not involve extrapolation as required in the original
direct linear plot. This has the great advantage that the axes of
the plot may be scaled in a simple fashion. Owing to the
extrapolation involved it is frequently necessary to perform
more than one plot when using the original method, since
many of the intercepts can be off-scale. The disadvantage
incurred by this last observation is, however, somewhat
attenuated in view of the need only to count the intercepts in
order to find the median that represents the best estimate of the
Michaelis parameters in both forms of plot. It is frequently
possible to estimate the median even if a few intercepts are
off-scale: it is. however. more satisfactory to be able to make all
the intercepts demonstrable visibly in order to establish an
impression of the experimental scatter.
Perhaps the most remarkable property of direct linear plots is
their amenability to non-parametric statistical analysis. This
both eliminates a large amount of calculation in the estimation
of the best-fit Michaelis parameters and also dispenses with
arguments concerning the validity of the least-squares assumption and the ‘normality’ of the error distribution of the data. The
Eooh
1000
600
I
0
110
tJV
20
30
40
K,IV
Fig. 2. Direct linear plot constructed in accordance with the
method of Cornish-Bowden & Eisenthal(l978)
The data are taken from Wharton (1969) and represent the
papain-catalysed hydrolysis of N-benzoyl-L-serine methyl ester.
Initial rates were measured at pH7.0, I 0.1, at 25OC in the
presence of 5 mw-cysteine, the reaction vessel of the titrimeter
being flushed with N,. The Michaelis parameters (determined
from eight points), determined by using the method of Wilkinson
(1961), were:k,,,,= 168+6min-’, Km=52?4mM.Themedian
estimates of the parameters taken from the Figure are: k,,,, =
166min-‘, K , = 53mM. Clearly both methods are in excellent
agreement.
direct-linear-plot method has been extensively compared with
other ‘respectable’ methods (e.g. that of Wilkinson, 1961) for the
determination of Michaelis parameters. It has been found to
perform as well as these methods, which involve non-linear
least-squares-regression analysis, when normally distributed
data are presented. It has a clear advantage over these other
methods in cases where ‘outliers’ are present; these have a very
serious effect on least-squares analysis (unless weighted out), but
do not unfavourably bias median estimation.
The direct-linear-plot method does have three disadvantages
that can sometimes militate in favour of the use of one of the
other forms of analysis. For purely graphical analysis (i.e. where
a computer is not used) the plots become rather congested if more
than about six initial-rate estimates are used. This is easily overcome by the use of a computer program. but of course the
method loses some of its essential simplicity if a computer has to
be involved. Also. it is not possible in any simple fashion to obtain standard errors for the Michaelis parameters from direct
linear plots. although it is important to remember that the areas
enclosed by the intercepts represent joint confidence limits for
the Michaelis parameters (Cornish-Bowden & Eisenthal, 1974).
The final disadvantage relates to the extent to which any plotting
method demonstrates that data follow a specific physical law
(e.g. the Michaelis-Menten equation) as against allowing
parameter estimation from data actually known to follow a particular law. This, in my opinion. the direct-linear-plot procedure
does not readily distinguish random error from failure to obey
the Michaelis-Menten equation; one of the linear transformations of the Michaelis-Menten equation (see below) is probably
better suited for this purpose. It is usually easier to assess
systematic deviation from a supposedly straight line than from
a point in space. It has been suggested that the direct-linearplot procedure is particularly useful for ‘plotting at the bench’,
I983
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BIOCHEMICAL REVIEW
and this is clearly so provided that the reservations outlined
above are borne in mind.
One of the most surprising features characteristic of enzyme
kinetic analyses published in the literature is the extraordinary
tenacity with which the Lineweaver-Burk (double-reciprocal)
plot is retained in use. On very many occasions in the literature
(see, e.g., Dowd & Riggs, 1965), the serious disadvantages
particularly in its use for the display and visual assessment of
data have been carefully explained, and yet have, to an alarming
extent, apparently been ignored. Of the three linear transformations of the Michaelis-Menten equation that are available, the half-reciprocal form, which makes use of a plot of
ISllu, against IS], is much the most satisfactory (Wharton &
Eisenthal, 1981). The weights that should be applied to the
ordinate for regression analysis are relatively invariant over the
usual range of IS1 and vo values used in kinetic analyses, and the
substrate concentration values are plotted on a linear scale. This
form of plot is steadily being used more widely, but there is still a
long way to go before it can be said to have gained the general
acceptance that it deserves.
A number of publications (see, e.g., Cleland, 1979; Nimmo &
Mabood. 1979) have appeared in recent years concerned with
selection of the most appropriate weighting strategies. This
aspect presents severe problems, since it is unclear in virtually
every instance what form the error structure takes. It is, for
example. not generally established whether the error in enzyme
kinetic experiments can be regarded as normally distributed;
such a situation is most unfortunate, since this is a requirement
for the proper application of least-squares-regression analysis.
The amount of work required to establish the nature of the error
distribution for any experimental series is prohibitive, and is, in
practice, never investigated in any detail. Further, it is almost
always unclear as to how the error is related to the magnitude of
the measured initial rates (Storer et al., 1975). A very common,
but probably incorrect, assumption is that the error is
proportional to the value of the measured initial velocity
(Wilkinson, 196 1). However, an equally plausible assumption
might be that the error is constant and independent of the value
of the initial rate. These two assumptions lead to quite different
values for the weighting factors (v: and vo2) for the regression
of the Lineweaver-Burk form of the Michaelis-Menten
equation. Although it might seem sensible to choose a
compromise between these extremes (e.g. v:), in view of the
general lack of information concerning the nature of the error,
this, in fact. has been stated to represent a combination of the
worst features of both of the extreme weighting strategies
(Cornish-Bowden, 1981).
Cornish-Bowden & Endrenyi (198 I) have proposed a method
for the fitting of enzyme kinetic data that does not require prior
knowledge of the weights. This method, which calculates the
best weights, seems to work well when applied to simulated data,
but it is too recent to have been applied to laboratory data. It is
expected that this method, based on a ‘biweight’ regression
procedure, will become widely used when its apparent merits are
more widely appreciated and it has been tested with experimentally obtained results.
There can, of course, be no better approach than to determine
the error structure, realistically only insofar as the weighting
strategy is concerned, for each experimentat series by means of
repeat estimations at some substrate concentration values. In
this way the variances of individual points can be determined and
be used in the form of their reciprocals as weights. The amount
of effort required in order to define the variances with any
reasonable degree of accuracy has precluded the application of
this type of approach in all but a very small number of cases
(see, e.g., Storer ef al., 1975; Askelof et al., 1976).
It is pertinent to comment on the need for very accurate
estimates of the Michaelis parameters of enzyme reactions.
Accurate relative estimates are frequently required as in pH- or
temperature-dependence studies. However, it is suspected that if
VOl. 11
a consistent sub-optimal method is used for all such series of
experiments then the relative values are likely to be related in a
consistent and interpretable fashion. This factor and lack of a
clear need for accurate absolute values somewhat allay my fears
of serious reprecussions that might arise from the application of
less-than-ideal methods of kinetic analysis. Nonetheless it can
hardly be regarded as harmful that the mechanics of parameter
estimation have been so thoroughly examined in terms of
robustness.
Use of progress curves in enzyme kinetic analysis
As mentioned above, the vast majority of enzyme kinetic
experiments reported in the literature consist in the measurement and analysis of initial rates. Progress-curve analysis, which
makes use of the whole time course of a reaction, has, in
practice, been almost entirely neglected. This seems to be due to
a widely held belief that progress-curve analysis is fraught with
problems, some of them being of such a nature that they are
difficult to eliminate. Although there is certainly some truth in
this, it has become apparent in recent years that most of these
problems can be eliminated quite easily, provided that they are
clearly identified. For example, the availability of spectrophotometers with high sensitivity, low drift and an excellent
signal-to-noise ratio has all but eliminated the very serious
problem of error in the end point (i.e. IPlm)of the progress curve
(Newman et al., 1974; Wharton & Szawetski, 1982a). All
methods but one, the differentiation method, which is described
below (Wharton & Szawelski, 1982b), require an accurate
estimate of the end point in order to function properly, and there
is no viable alternative but to wait at the very least seven half
times until this is reached. It has proved to be essential
(Wharton & Szawelski, 1982a) to achieve an end point
measurement for each reaction; unfortunately, it has not proved
satisfactory to use a single estimate and apply this to several
reactions. The small differences between one reaction run and
another (often approx. 1-2%) can cause a significant effect on
the parameter estimates if they differ by as much as 1%. Thus a
significant time penalty is incurred by having to wait for the
completion of each reaction. It is also necessary to run several
reactions in order to assess and take into account the effect, if
any, of product inhibition. This cannot in general be taken into
account when a single reaction is used (but see above; Waley,
1981).
A widespread impression seems to have taken hold that the
integrated Michaelis equation, expressed in its general form:
Vapp,. t = (ISlo-lsl)+K,,a,,,,~In(lSIo/ISI)
or in terms of product:
is applicable only to simple enzyme reactions. In fact the
equation is applicable in a wide variety of circumstances,
including most forms of inhibition, reversibility and, provided
that one of the substrates is saturating, to some bisubstrate
systems (Alberty & Koerber, 1957; Schwert, 1969; Bates &
Frieden, 1973a,b; Laidler & Bunting, 1973; Cornish-Bowden,
1975; Duggleby & Morrison, 1977, 1978; Orsi & Tipton, 1979;
Wharton & Szawelski, 1982a).
A serious problem that can arise in attempts to apply
progress-curve analysis is that of enzyme inactivation during the
course of a reaction. This is tested for by using the method
described by Selwyn (1965). If such an effect is found, then
progress-curve analysis is inappropriate, and initial-rate studies
will have to be used; indeed, the initial rates may need to be
estimated from curved traces by using one of the methods given
above. Provided that the various problems that can arise in
progress-curve analysis can be identified and taken into account,
however, the potential savings in time, materials and effort are
very considerable and fully justify application of the technique.
8 20
A wide variety of methods have been developed for the
determination of kinetic parameters from progress curves. These
include linear least-squares regression, non-linear regression,
iterative procedures and polynomial fitting (for a review of these
procedures see Orsi & Tipton, 1979).
Over the last decade several attempts have been made to
simplify the analytical procedures available for treating
progress-curve data. For plotting of progress-curve data the
original method of Walker & Schmidt (1944). which makes use
of a plot of IPl/t against In{ISl,/(ISl,-IPl))
(or one of the
other linear transformations of the integrated Michaelis
equation), has been used for many years. This is rather a
cumbersome form of plot and requires considerable computation time, even if, for example, a pocket calculator is used.
Lee & Wilson (197 1) proposed a form of chord-slope analysis
of progress curves (see above). The slopes of chords drawn to
the curve are taken as initial rates at the mean substrate
concentration. Although they did not explicitly recommend
construction of linear (e.g. half-reciprocal) plots from single
progress curves, confining themselves to initial-rate estimates,
their method could clearly be adapted to such a purpose.
Yun & Suelter (1977) noted this application and thoroughly
examined the properties of such plots with regard to the nature
and extent of errors introduced by the approximation that the
chord-slope initial rate corresponds to the mean substrate
concentration. (Unfortunately they recommended use of the
double-reciprocal plot for plotting of the chord slopes; halfreciprocal plots would be more suitable.) They showed that the
introduction of error is insignificant if AISl/[Slo< 0.3, i.e. if the
chords traverse less than 30% of the conversion of substrate into
product. They further presented two methods for correcting for
error that is introduced if chords that traverse larger sections of
the progress curve than specified are used. The procedures
required to correct this form of error involve either the
construction of chords having constant A[Sl/lSl, and then
correcting the determined K, value or adjustment of the mean
substrate concentrations in order to give the correct ‘corresponding’ substrate concentrations.
Thus, despite the above criticisms, the method works well if
A[Sl/[Sl, < 0.3 and the half-reciprocal plot is used. The
accuracy vis-a-vis noise and slope estimation will be somewhat
restricted, since the chords must traverse less than 30% of the
progress curve.
More recently, Wharton & Szawelski ( 1 9 8 2 ~ )have applied
the half-time method (see. e.g., Hammes, 1978), which is widely
used in chemical kinetics, to the analysis of the integrated
Michaelis equation. Substitution of the half-time relations t = tt
and IS] = ISlo/2 leads to the expression:
BIOCHEMICAL SOCIETY TRANSACTIONS
(1975) proposed a direct linear plot of the integrated Michaelis
equation and provided an approximation to the logarithmic
term. This was described above as being useful for the
determination of initial rates and is shown in Fig. 1. The method
described above (Wharton & Szawelski, 1982a) can, of course,
also be used for this purpose.
The half-time method is not restricted to this particular
fraction of reaction, and any value from say ‘nine-tenths’ time
down to perhaps ‘one-fifth’ time can be used, but the extent of
reaction to be measured for each fractional time becomes rather
small below the latter value. The fraction of one-half was chosen
because it is so widely used in chemical kinetics and it does seem
to be a suitable compromise between the values that may
reasonably be used.
Wharton & Szawelski ( 1 9 8 2 ~ )provide a discussion of the
applicability of progress-curve analysis (see also Yun & Suelter,
1977; Orsi & Tipton, 1979) and have proposed some rules that,
if followed, allow reliable results to be obtained.
Wharton & Szawelski (19826) have described a completely
novel differentiation method of progress-curve analysis that does
not rely on or require determination of the end point of the
reaction. The method, shown in Fig. 4, is based on consideration
of the first three derivatives of the integrated Michaelis equation
with respect to time. These are:
and
200 -
150
h
v
,100
-
u
in which the logarithmic term is now a constant. A plot of
successive t i values against the ISl, values from which
the tt values are calculated yields a straight line of intercept
In 2.K,~,pp.,/Vapp,and slope 1/2Vapp..A half-time plot of a
chymotrypsin-catalysed reaction is shown in Fig. 3. The method
is conceptually simple and is exact. Half-time plots are easy to
construct and have given good results with several enzyme
systems, including the reversible fumarase-catalysed reaction.
The half-time equation has been recast in parameter space to
yield the following equation:
-
0
1
2
3
4
5
6
IO’IsI (M)
Fig. 3. Half-time plot constructed in accordance with the method
of Wharton & Szawelski ( 1 9 8 2 ~ )
The data are for the a-chymotrypsin-catalysed hydrolysis of
methoxycarbonylglycine p-nitrophenyl ester in 0.1 M-phosphate
buffer, pH 6.98. The reaction was monitored at 420 nm, and the
Islo
1112‘ Krn(app.)
a‘.,
= -+
initial substrate concentration was 57 PM. The data used in Fig.
21,
tt
3 were obtained from a kinetic run comparable with that used
for Fig. l(a) of Wharton & Szawelski (1982~).The Michaelis
A direct linear plot of the half-time equation is constructed by
parameters obtained by unweighted linear-regression analysis of
using values of ISl,,/2tt (which is the slope of a chord from [Sl,
the half-time equation were: k,,,, = 6.07 0.05 x 10-2s-1,
to [Sl,/2) on the ordinate and values of -[Sl,,/2 on the abscissa.
For perfect data the lines intersect at a point, namely VaPp,, K , = 7.64 & 0.25 PM, in very good agreement with the parameters quoted in the publication cited above.
In 2.K,,,pD,,, in the first quadrant of the plot. Cornish-Bowden
1983
BIOCHEMICAL REVIEW
Analogue or digital differentiation (twice) of an emerging
progress curve permits real-time estimation of the Michaelis
parameters. All other methods require an estimate of the end
point of the reaction before computation can be commenced. If
the third derivative is set equal to zero, we find that there is a
turning point (minimum) in the second derivative when
ISl,=K,,,pp,,/2: there is also an inflexion in the first derivative
at this point. The ordinate values of the first and second
derivatives are VnPp,/3 and 4 V,pp,2/27Km(app,,respectively at
this point. Several strategies present themselves for determination of the Michaelis parameters. For example, K , may be
determined by reference to the abscissa value (time) of the
minimum in the second derivative to the original progress curve
(see Fig. 4). Vapp,may then be determined from the value(s) of
either of the derivatives at this point. The method of choice for
real-time analysis makes use of the values of the derivatives at
the time of the second derivative minimum; no other information. apart from the molar absorption coefficient of the
substrate. is required. Indeed, computation may halt immediately after the minimum in the second derivative has been
passed, no reference being made to the original progress curve.
A major problem with all differentiation procedures is their
marked propensity for noise amplification. We have used the
smoothing/differentiation procedure of Savitzky & Golay
( 1 964) in order to achieve differentiation with an acceptable (if
greater than hoped for) degree of noise propagation. Lowfrequency modes (see Fig. 4) have proved particularly troublesome. Extensive studies on electronic analogue differentiation
systems have not, as yet, proved to be successful, largely as a
result of transients that occur on start-up of the system. Further
work on this is expected to produce circuits that will perform in
an acceptable fashion, but the time constants of the filters have
to be large in order to achieve efficient noise reduction. We
confidently expect this real-time method in either form to find
widespread applications. quite probably in relation to fastreaction experiments (e.g. stopped-flow), where time is of the
essence in the data-gathering process and in the analysis of the
kinetic parameters.
Some applicalions of isotope eflects in enzyme kinetics
The use of isotope methods in enzyme kinetics has assumed
an ever-increasing importance in recent years. Isotopes are used
for two fundamentally different purposes in enzyme kinetics.
Firstly, they are used in order to detect and measure very
specific processes by the use of radioactive tracer isotopes.
Secondly. they are used to measure kinetic isotope effects, which
arise from the mass changes that result from isotopic substitution: non-radioactive isotopes are frequently used for these
experiments. The former use of isotopes is exemplified by
equilibrium isotope exchange, in which a small quantity of
labelled substrate or product is added to the system under study
in order that the rate of transference across the equilibrium of
substrate or product may be measured. The isotope used in these
experiments is radioactive so that the rate of transference across
the equilibrium of the compound in question may be easily
measured. In this type of experiment the substrates and/or
products are labelled in such a way as to avoid a kinetic isotope
effect (i.e. at a site remote from the reaction centre). Alternatively, the kinetic isotope effect is minimized by using
heavy-atom isotopes, e.g. 32P.
Although equilibrium isotope exchange is now very widely
used, an alternative but potentially superior form of isotope
exchange known as flux-ratio analysis seems to have been very
little used (Britton, 1966; Britton & Clarke, 1972; Britton &
Dann. 1978: Gregoriou et al., 1981). The flux-ratio method
involves the measurement of rates or fluxes between particular
substrate and product pairs under steady-state conditions. If the
measured fluxes are ratioed, the expressions that describe the
dependence of the flux rates on the substrate or product
concentrations are greatly simplified. These relatively simple
VOl.
11
82 1
expressions for the various fluxes demonstrate a substrate- or
product-concentration-dependence that is highly characteristic of a particular kinetic mechanism. Although equilibrium
isotope exchange can be regarded as a highly sensitive method
of detecting minor pathways, this greatly complicates the
interpretation of the kinetic mechanism. The flux-ratio method
provides a very much more determinate method of establishing
the major pathway and for distinguishing between random and
ordered reaction pathways. Thus the equilibrium method does
not indicate the extent to which the various pathways that may
be detected contribute to the effective flux under steady-state
conditions. Since the flux-ratio method involves steady-state
measurements, the degree of partitioning between different
pathways is indicated. In addition, the existence of dead-end
inhibitor complexes, which complicates the interpretation of
equilibrium isotope effects, does not affect flux-ratio measurements, since there is no flux through these complexes in the
steady state.
It is surprising that this incisive method, which is not difficult
to effect in practical terms, has not been more widely applied.
This is perhaps due to the rather complicated fashion in which
the method was originally presented. More recently the method
has been described in a way that makes plain its theoretical
simplicity and practical utility (Gregoriou et al., 198 I ; CornishBowden & Gregoriou, I98 1).
Kinetic isotope eflects
A problem that frequently arises in the study of kinetic
isotope effects (usually deuterium) of enzyme-catalysed reactions is that of determining the intrinsic isotope effect, i.e. the
true value of the isotope effect on the rate constant of the step in
which it occurs (e.g. a proton-transfer step). This problem arises
because the kinetic expressions for the rates, equilibria and
overall rate constants usually consist of assemblies of rate
constants, only one of which may exhibit an intrinsic isotope
effect. This often has the effect of seriously attenuating the
observed isotope effect. The chemical catalytic step is sometimes not rate-limiting, as for example in the case of horse liver
alcohol dehydrogenase, where NADH loss is the rate-limiting
process (Wratten & Cleland, 1963, 1965). There is thus a need
for a procedure that enables the isolation of the intrinsic isotope
effect from the often complex assemblages of rate constants in
which the data are obtained as a result of rate measurements.
)
Northrop (1975, 1977; see also Cook & Cleland, 1 9 8 1 ~ has
provided such a technique, and a brief, simplified, outline of the
method is given here. It is applied to situations in which a
primary isotope effect is present at some stage of the
mechanism. The analysis is effected by using V , the apparent
maximum velocity in a given direction, and V/K as working
parameters, since these achieve the best possible separation of
variables. At least in simple mechanisms V contains only the
rate constants of non-equilibrium steps, whereas V / K is
composed of all rate constants up to and including the first
irreversible step. By contrast uo, the initial rate, and K, an
appropriate kinetic saturation constant, are complex expressions involving all the rate constants. If the values of V and V/K
obtained in the presence of hydrogen-containing substrate
and/or solvent are divided by those obtained in the presence of
deuterium, then the resulting ratios can be written in a form in
which the intrinsic isotope effect is confined to a single term. The
intrinsic isotope effect occurs in the numerator, and another
term, which is isotope-independent, occurs in numerator and
denominator. For the V ratio this isotope-independent term
comprises a ratio of rate constants that is known as the ‘ratio of
catalysis’ and defines for a simple mechanism a ratio of
irreversible steps. Similarly, for the V/K ratio, which has the
same functional form as the V ratio, the isotope-independent
portion, also a ratio of rate constants, is known as the
‘commitment to catalysis’, which expresses the ratio of reaction
to return to substrate from ES complex. The intrinsic isotope
BIOCHEMICAL SOCIETY TRANSACTIONS
effect may be separated from the 'commitment to catalysis' for
the V / K ratio by measurement of the tritium isotope effect and
use of the Swain-Schaad relationship, which relates deuterium
to tritium isotope effects. This procedure cannot be applied to
resolution of the V ratio, since IOO'X) substitution with tritium
would be required.
The method described in outline above is capable of
generalization to include many-stepped reactions and reversibility, and has been used, at least in part, on several occasions
;
(e.g. Damgaard. 1977, 1980: Cook & Cleland, 1 9 8 1 ~ Hermes
et al., 1982). Northrop ( 1977) has stated that errors arising from
failure of the Swain-Schaad relationship, in which the error is
likely to be approx. 10% (Albery & Knowles, 1977), are not
likely to exceed those that result from the experimental error
incurred in the measurement of apparent isotope effects.
However, it would be wise to bear in mind the possibility that
this relationship may not hold even to the above degree of
accuracy in all circumstances. particularly if steric restriction
effects are involved in a binding or catalytic event. Indeed,
Albery & Knowles (1977) have stated that the experimental
data must be characterized by less than 3'X) error if useful
deductions are to be made. Nonetheless the tritium isotope effect
has been measured and used to perform an analysis of this type
on yeast alcohol dehydrogenase with propan-2-01 as substrate
(Cook & Cleland, 198 Ib). The use of relatively poor unnatural
substrates is frequently a convenient way of ensuring that a
chemical step is rate-limiting.
Proton inventories
Most kinetic isotope effects are measured at 100%)isotopic
substitution. Thus most solvent deuterium kinetic isotope effects
are measured in pure 'H,O relative to H,O. It is usually possible
to obtain some information on any proton transfer that occurs
in a reaction provided that a well-defined single reaction step is
measured or if the intrinsic isotope effect is determined as
outlined above. The proton-inventory method, in which the
solvent kinetic isotope effect is measured at several atom
fractions (i.e. several concentrations) of deuterium relative to
hydrogen in the solvent. is capable to providing much more
detailed information about the reaction mechanism (Schowen,
1972, 1977: Albery. 1975: Szawelski & Wharton, 1981).
The importance of measuring isotope effects on well-defined
reaction steps in order to compile a proton inventory cannot be
over-emphasized: the results will be largely uninterpretable if
this is not so. Since the proteolytic enzymes have a rather broad
specificity. it has proved possible to use synthetic substrates in
order to measure solvent isotope effects that are well-defined in
this respect. The catalysis of the hydrolysis of reactive esters
(e.g. p-nitrophenyl esters) is usually cleanly limited by deacylation (Szawelski & Wharton. 1981; Wharton & Eisenthal,
1981). Accordingly, a proton inventory can be constructed on
deacylation with the use of reactive esters. Amide substrates
(e.g. p-nitroanilides) generally show rate-limiting acylation,
and so this type of substrate can be used to measure isotope
effects on this step of the reaction. This type of procedure can be
extended to measure reverse steps, and this allows a relatively
complete analysis of the reaction pathway to be achieved.
Solvent isotope effects measured for the purposes of protoninventory analysis are analysed by using the Gross-Butler
equation:
kn = k 0
n(lLn+nlT)
,
n( I - n + 4:)
in which k , is the rate constant in H,O, k , is the rate constant at
atom fraction n of ,H,O. v is the number of protons involved in
the reaction, $7 is the fractionation factor (see below) of the ith
exchangeable site of the ground state and 6 is the corresponding factor in the transition state. The fractionation
factors are equilibrium constants for the exchange of deuterium
with hydrogen at a particular site. They may also be seen as the
extent of deuteration of a particular site in a I : 1 mixture of H,O
and ,H,O. A simple single-proton isotope effect with ground
state # of unity is given by the reciprocal of the fractionation
factor. All sites that have # values of unity are not included,
since the Gross-Butler term also equals unity. All sites that do
not change their fractionation factors on going from ground
state to transition state also cancel, which leaves us with a
residue of 'active protons'.
A simple analysis shows that a plot of k , / k , against n is linear
if a single proton is 'in flight' in the transition state and the
ground state 9 is unity. Similarly, if two equivalent protons are
'in flight' in the transition state and the ground state 4 values are
unity, a plot of (k,/ko); against n is linear. 4" can be calculated
from the slopes of these plots. In more complicated systems
where, in addition to there being more than one proton 'in flight'.
not all the ground-state # values are unity. non-linear leastsquares-regression analysis is used in order to attempt to
determine the correct form of the Gross-Butler equation (see.
e.g., Szawelski & Wharton. 1081). It is important to make use of
structural information in order to assess as far as possible likely
0" values and to estimate the number of protons that are likely
to be involved on or associated with the reaction co-ordinate.
The possibility that secondary isotope effects and/or medium
effects (which result from many protons) are responsible for the
observed isotope effect may also be assessed. The method is. in
principle, extremely powerful, and when applied wisely supplies
abundant high-quality information concerning the reaction
co-ordinate being studied. The results. as is generally the case.
are best interpreted in conjunction with independent evidence.
This, in relatively well-defined cases, allows analysis of the
reacting system in terms of proton transfer vis-i~visheavy-atom
motions, an approach to a potential-energy map of the
reaction. As with all methods. there are drawbacks; the most
serious is the deduction of an over-simplified model owing to
fortuitous cancellation of terms in the numerator and
denominator that do not refer to the same atoms (Kresge. 1064).
Fig. 4. Example of rhe application of [he sequenrial differenrialion tnerhod of Wharton &
Szawelski (19826)
The reaction shown is the trypsin-catalysed hydrolysis of benzyloxycarbonyl-L-arginine
p-nitroanilide in 0. I M-Tris buffer. pH 7.8, at 25°C. ( a ) The result of differentiation
without smoothing. ( h ) The result of application of the smoothing/differentiation
method of Savitzky & Golay (1964). The ordinate scale applies only to the product
concentration. the derivatives having been scaled to fit the span of the plots. This
accounts for the difference in the appearance of the plots ( a ) and (h). since noise
excursions contribute to the scaling procedure. The Michaelis parameters are
V = 8.0 x 1 0 - 4 m ~ . s - -K,,
' . = 0 . 2 6 m ~ in
. very good agreement with parameters obtained
by means of the method of Atkins & Nimmo (1973). The reaction was recorded on
magnetic tape with the use of a 10-bit A/D converter, and subsequently input to an ICL
1906 computer, which was used for all computation.
1983
823
04
/
03
Oi
01
h
z
E
v
Y
I
I
1000
I
I
I
I
2000
3000
2000
3000
2
O3
1
02
01
1
0
1000
Time (s)
VOl.
11
I
4000
824
BIOCHEMICAL SOCIETY TRANSACTIONS
Albery (1975) has presented a method of analysis of be performed at low temperatures, where processes that would
proton-inventory experiments that relies only on measurements occur extremely rapidly at room temperature can be observed
at zero, half and unity atom fractions of deuterium. We have on a convenient time scale. The viscosity is, in many
recently used this method to provide an almost completely circumstances, sufficiently low to allow the use of a stopped-flow
automatic system of model selection. This relies on measure- technique, which permits the observation of quite rapid events
ment of isotope effects with several different substrates at low even at low temperature. By means of judicious variation of
concentration. If the substrates are appropriately chosen to have temperature and the nature of the substrates used in these
a common rate-limiting step that is adjacent to the binding step, experiments. a comprehensive description of papain-catalysed
then the ground-state fractionation factor(s) of the enzyme may reactions has been achieved (Angelides & Fink. 1978. 1979a.b).
As a result of these studies Fink has proposed that papain
be uniquely determined. Such knowledge of the ground-state
fractionation factor(s) greatly enhances the ease with which the undergoes at least one structural transition during the catalytic
various transition-state fraction factors may be assigned cycle. These structural events are fast compared with the
chemical steps of the catalysis. and could not easily have been
(Szawelski, 1983).
The proton-inventory method has been applied to a number detected by conventional kinetic analysis. The accumulation of
of enzyme-catalysed reactions, mainly by Schowen and his the tetrahedral intermediate in the acylation of papain at pH 9.3
colleagues. For example. Venkatasubban & Silverman (1980) by a p-nitroanilide substrate has also been observed (Angelides
have performed proton-inventory analysis on the carbonic & Fink, 1979b). Recently the method has been applied to the
anhydrase-catalysed hydration of CO,. The observed isotope study of leucine aminopeptidase (Lin & Van Wart, 1982). This
effect requires two or more protons to be 'in flight' in the is clearly a powerful methodology, but it seems certain that
transition state, probably as components of water bridges. outstanding vigilance is required with respect to the elimination
Quinn et al. (1980) have studied amidohydrolases and have of observations due to artifacts. Anyone who has been involved
shown that the fully specific substrates (i.e. asparaginase/ in stopped-flow experimentation will be well aware that it is only
asparagine: glutaminase/glutamine) show dual concerted proton too easy to be, at least temporarily, misled by artifactual effects
transfer, whereas asparaginase acting on glutamine shows (bubble formation and dispersion are the most trivial of these).
one-proton catalysis. It is thus suggested that the fully specific In more viscous systems at low temperature the likelihood of the
substrate is required in order to organize the active-centre occurrence of such effects is greatly magnified. Provided that
sufficient vigilance is maintained in this respect, the technique
groups to an extent sufficientto provide concerted catalysis.
A similar finding has been reported for the serine proteases by has a bright future.
Elrod et al. (1980). Some oligopeptide substrates that are
apparently subject to dual-concerted-proton-transfer catalysis Potpourri
are proposed to achieve a compression in the hydrogen-bonding
In this section two interesting methods that do not fit clearly
chain of the reaction co-ordinate. This results from interaction into any of the sections presented above are very briefly
with the oligopeptide remote from the active site. Non-specific described. Lin et a / . (1982) have proposed an ingenious thermal
substrates such as p-nitrophenyl acetate show single-proton variation method for separating the rate constants of the
catalysis, since they are not capable of remote interaction. Michaelis-Menten (Briggs-Haldane) mechanism. Analysis of
Substrates of intermediate specificity show behaviour between the temperature-dependence of K , and V and application of the
these extremes. Hunkapiller er a / . (1976) have also shown that constraint implied by the Arrhenius equation allows, in
the u-lytic proteinase-catalysed hydrolysis of oligopeptide principle, the separation of the three constituent rate constants.
p-nitroanilides is characterized by a dual-concerted-proton- Although theoretically delightful, this method may prove
transfer mechanism.
difficult to apply in practice. owing to ( a ) the limited
We can confidently expect that this powerful technique will be temperature range that can be used in enzyme kinetic studies
widely applied when its merits have become more widely and (b) to the inherent error in the computed Michaelis
appreciated.
parameters. It is nonetheless a method that one would very
Heavy-atom isotope effects (O'Leary, 1977) on C. N and 0 much like to see applied, and perhaps extended to other systems.
can now be measured with great accuracy by using the
Boeker (19826) has presented a general method for the
isotope-ratio mass spectrometer or by means of the equilibrium
integration of enzyme kinetic rate equations. It was previously
perturbation method (Schimerlik et al., 1975). This type of generally supposed that such integrated forms could only be
isotope effect, which is not discussed here in any detail, provides obtained on a mechanism-by-mechanism basis (see. e.g.,
very useful independent evidence that, when combined with Schwert, 1969). The method is ingenious and should prove most
solvent-kinetic-isotope results, can allow a more complete useful in future studies, both simulated and experimental, of the
interpretation of the system under study. We have used such progress curves of more complex mechanisms. It was not
nitrogen heavy-atom isotope results (O'Leary eta/., 1974) in the included in the section on progress-curve analysis since it has
interpretation of our experiments on papain acylation
not yet been applied to this area.
(Szawelski, 1983). A complication that arises in the interpretation of nitrogen isotope effects is the influence of the state Finale
of protonation of the nitrogen atom (O'Leary, 1977).
Any reviewer, other perhaps than those who are allowed
extensive
space, can be criticized for their personal prejudice in
Cryoenzymology
terms of choice of topics. I have purposely restricted the content
It came as something of a surprise to me, and I suspect to of this review to topics in enzyme kinetics that 1 consider to be
many other enzymologists. to find that it has been reported that of recent special interest and promise. Only time will tell how
many proteins retain a largely native structure when cooled to wise that choice has been. but it is clear that the Renaissance in
cryogenic temperatures in high concentrations of organic enzyme kinetics currently in full swing will continue. albeit
solvents (e.g. 60% dimethyl sulphoxide), (Donzou et a/., 1970; maybe along other lines.
Hui Bon Hoa & Douzou. 1973; Fink, 1973; Douzou, 1977:
Fink & Geeves. 1979).
enzymes, for example chymotrypsin and papain, have
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Angelides & Fink, 1978). This fact has allowed kinetic studies to
1983
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