NMR for direct determination of Km and Vmax of enzyme reactions based on the Lambert W function-analysis of progress curves

Article abstract of DOI:10.1016/j.bbapap.2011.10.011

¹H NMR spectroscopy was used to follow the cleavage of sucrose by invertase. The parameters of the enzyme's kinetics, K_m and V _max, were directly determined from progress curves at only one concentration of the substrate. For comparison with the classical Michaelis-Menten analysis, the reaction progress was also monitored at various initial concentrations of 3.5 to 41.8 mM. Using the Lambert W function the parameters K_m and V_max were fitted to obtain the experimental progress curve and resulted in K_m = 28 mM and V _max = 13 μM/s. The result is almost identical to an initial rate analysis that, however, costs much more time and experimental effort. The effect of product inhibition was also investigated. Furthermore, we analyzed a much more complex reaction, the conversion of farnesyl diphosphate into (+)-germacrene D by the enzyme germacrene D synthase, yielding K_m = 379 μM and k_cat = 0.04 s^{- 1}. The reaction involves an amphiphilic substrate forming micelles and a water insoluble product; using proper controls, the conversion can well be analyzed by the progress curve approach using the Lambert W function.

Full text of DOI:10.1016/j.bbapap.2011.10.011

Biochimica et Biophysica Acta 1824 (2012) 443–449
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NMR for direct determination of Km and Vmax of enzyme reactions based on the
Lambert W function-analysis of progress curves
⁎
Franziska Exnowitz, Bernd Meyer, Thomas Hackl
Institute of Organic Chemistry, University of Hamburg, Martin-Luther-King-Platz 6, 20146 Hamburg, Germany
a r t i c l e i n f o
a b s t r a c t
1H NMR spectroscopy was used to follow the cleavage of sucrose by invertase. The parameters of the enzyme's
kinetics, Km and Vmax, were directly determined from progress curves at only one concentration of the substrate.
For comparison with the classical Michaelis–Menten analysis, the reaction progress was also monitored at various
initial concentrations of 3.5 to 41.8 mM. Using the Lambert W function the parameters Km and Vmax were fitted to
obtain the experimental progress curve and resulted in Km=28 mM and Vmax=13 μM/s. The result is almost
identical to an initial rate analysis that, however, costs much more time and experimental effort. The effect
of product inhibition was also investigated. Furthermore, we analyzed a much more complex reaction, the
conversion of farnesyl diphosphate into (+)-germacrene D by the enzyme germacrene D synthase, yielding
Km =379 μM and kcat =0.04 s−1. The reaction involves an amphiphilic substrate forming micelles and a
water insoluble product; using proper controls, the conversion can well be analyzed by the progress
curve approach using the Lambert W function.
Article history:
Received 22 August 2011
Received in revised form 12 October 2011
Accepted 17 October 2011
Available online 29 October 2011
Keywords:
1H NMR
Enzyme kinetics
Progress curve analysis
Lambert W function
Invertase
Germacrene D synthase
© 2011 Published by Elsevier B.V.
1. Introduction
assumes a quasi-steady state approximation where the concentration
of the enzyme substrate complex is constant over time and produces a
Enzyme kinetics is important to understand the mechanisms of
enzyme reactions. Many spectroscopic methods are applied frequently
in enzyme kinetics. The powerful technique, NMR spectroscopy, by
which substrates and products can directly be quantified, is, however,
rarely used. Textbooks dealing with enzyme kinetics usually mention
NMR spectroscopy only incidentally [1,2]. On the other hand NMR
spectroscopic investigations of binding, dynamical and structural
properties of enzymes have become quite popular in recent years
[3].
Kinetic analysis of enzyme reactions by NMR has some advan-
tages. Measurements are performed in homogenous solutions and
the reaction progress can be monitored directly without labeling
of the substrate(s). In contrast, relative large sample amounts are
necessary for NMR spectroscopy compared to other sensitive standard
assays e.g. using fluorescence or radioactive labeling. The require-
ments on sample quantity were, however, significantly reduced by
technical developments in high resolution NMR in the last decade
[4,5].
time-independent hyperbolic relation of the two. Since its introduction
in 1913, including the Briggs–Haldane modification in 1925, the
Michaelis–Menten model has been widely used to describe enzyme
processes [6,7]. It has been proven to be a simple yet powerful approach
in the determination of enzyme parameters Km and Vmax. However,
measuring initial velocities at different concentrations by NMR
spectroscopy or other analytical techniques is a time and material
intensive procedure.
An alternative approach is based on progress curve analysis,
where the concentration of the products and/or substrates is fol-
lowed over time at only one concentration in a single automated
experiment [8–10]. This method requires the integrated form of
the Michaelis–Menten equation that is implicit in the substrate
concentration and the calculation requires numerical integration.
This is followed by an appropriate nonlinear optimization routine
for an iterative estimation of the kinetic parameters. Later, in
1997 Schnell and Mendoza derived a closed form solution of the in-
tegrated Michaelis-Menten equation [11]. Starting from the integrated
form they obtained an expression for the substrate concentration as
a function of time (Eq. (1)). The solution is based on the Lambert W
Function (also called Omega Function) which is defined as the in-
verse of x ex, that is, W(x)+ln{W(x)}=ln(x) [12]. It was shown by
Goudar et al. and others that the Lambert W Function allows an an-
alytical solution of enzyme kinetics from single progress curves
[13–15]. The function is implemented in mathematical computing
Michaelis–Menten kinetics correlates the initial velocity with the
initial substrate concentration of an enzyme reaction. This approach
Abbreviations: FDP, farnesyl diphosphate; (+)-GDS, (+)-germacrene D synthase
⁎
Corresponding author. Tel.: +49 40 42838 2804.
E-mail address: hackl@chemie.uni-hamburg.de (T. Hackl).
1570-9639/$ – see front matter © 2011 Published by Elsevier B.V.
doi:10.1016/j.bbapap.2011.10.011

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F. Exnowitz et al. / Biochimica et Biophysica Acta 1824 (2012) 443–449
software and can be solved by nonlinear optimization routines (e.g.
Maple®, MATLAB®, R).
section. The spectra were analyzed with TOPSPIN 2.1 (Bruker). FIDs
were treated with 0.5 Hz exponential line-broadening function and
were zero-filled once. Integration of signals was performed using the
intser function of TOPSPIN.
(
!)
½Sꢀ0
m
½Sꢀ0−V′ maxt
½Sꢀ ¼ K′mW
exp
ð1Þ
K′
K′
m
2.4. Preparation of invertase solutions and data acquisition
Progress curve analysis of enzyme reactions has the advantage
that the reaction progress is monitored over the full reaction time
and that not only initial reaction rates are monitored. Under these
conditions product inhibition can’t be neglected and if present influences
the values of Km and Vmax. For that reason the apparent values K′m
and V′max are obtained in Eq. (1). Another advantage of progress curve
analysis originates from the fact that product inhibition can be easily
detected by recording progress curves at two initial concentrations.
Many aspects of the analysis of enzyme progress curves using non-
NMR techniques were summarized by Duggleby [10]. NMR spectroscopy
offers the easiest sample handling as well as a high sensitivity. Also, often
the direct analysis of the stereochemistry of the initial reactions products
can be determined (cf. below). We have chosen the enzyme invertase
(Saccheromyces cerevisea) used by Michaelis and Menten in order
to derive their kinetic model. Invertase hydrolyzes sucrose into glucose
and fructose (invert sugar) [6].
A sucrose stock solution (3.6 M in 25 mM acetate buffer, 50 mM
NaCl, 2 mM NaN3, D2O, pH 5.0) was added to a solution of invertase
provided in an 2 mL Eppendorf tube (to yield 600 μL with 2.5 μg in-
vertase in 25 mM acetate buffer, 50 mM NaCl, 2 mM NaN3, D2O, pH
5.0) and thoroughly mixed using an Eppendorf pipette. Then the reac-
tion mixture was transferred to a 5 mm NMR tube. After the insertion
of the NMR tube to the magnet the sample was locked and the exper-
iment started. Eventually the shim was corrected. Spectra were
recorded every 2 min applying 8 scans (34.4 s) and using 64, 72 or
180 transients in a pseudo 2D pulse sequence. The acquisition time
(AQ) was 3.35 s and a relaxation delay (D1) of 1 s was applied. The
initial concentrations of sucrose were 3.5, 8.6, 12.4, 14.8, 15.1 17.8,
22.2, 27.4 and 41.8 mM. For the investigation of product inhibition
two samples were equally prepared as described above, except that
one sample additionally contained 9 μL of a 1:1 mixture of glucose/
fructose (1.8 M in 25 mM acetate buffer, 50 mM NaCl, 2 mM NaN3,
D2O, pH 5.0).
2. Materials and methods
2.5. Preparation of germacrene D synthase solutions and data acquisition
2.1. Chemicals
10 μL of a farnesyl diphosphate stock solution (10 mM FDP in
50 mM tris-d11, 300 mM NaCl, 20 mM NaN3, D2O, pH 7.8) was directly
added to 190 μL of a solution of (+)-germacrene D synthase provided
in a 3 mm NMR tube to give a final volume of 200 μL containing
2.5 μM germacrene D synthase (corresponding to 0.16 mg/mL) in
deuterated TBS (50 mM tris-d11, 300 mM NaCl, 20 mM NaN3, D2O,
pH 7.8, 5% DMSO-d6 (v/v), 1 mM MgCl2). The reaction solution was
mixed by shaking the NMR tube thoroughly. After insertion of the
NMR tube to the magnet the sample was locked, eventually the
shim corrected and the experiment started. Spectra were recorded
using a pseudo 2D pulse sequence and applying 128 scans on 32
sequential experiments. This pulse sequence contained the excitation
sculpting sequence for water suppression. Each single experiment had
a total acquisition time of 21 min 55 s.
Invertase (EC 3.2.1.26, ß-fructofuranosidase, S. cerevisea) was
obtained from Sigma-Aldrich (Steinheim, Germany) with a specific
activity of 200–300 u/mg enzyme (pH 4.6, 298 K). Sucrose and far-
nesyl diphosphate were purchased from Sigma-Aldrich (Steinheim,
Germany). Glucose and fructose were purchased from Merck
(Darmstadt, Germany). D2O was obtained from Deutero (Kastellaun,
Germany), DMSO-d6 and Tris–HCl-d11 from Eurisotop (Saarbrücken,
Germany).
2.2. Purification of (+)-germacrene D synthase
(+)-Germacrene D synthase (EC4.2.3.22, sesquiterpene synthase,
Solidago canadensis) was obtained by heterologous expression in
E. coli host strain BL21(DE3)pLysS (containing a N-terminal 6-times
histidine-tag) as previously described except that after adding IPTG
incubation was performed over night at 16 °C [16]. The enzyme
was purified from the culture medium by affinity chromatography
Ni-NTA-agarose (Qiagen, Hilden, Germany) according to the manu-
facturers' recommendations. After purification the elution buffer
was exchanged immediately with a deuterated tris buffer (50 mM
Tris–HCl-d11, 300 mM NaCl, 20 mM NaN3, D2O, pH 7.8) using Vivaspin
centrifugal concentrators (MWCO 10 kDa, GE Health Care, Freiburg,
Germany).
2.6. Micelle formation of farnesyl diphosphate
A sample was prepared containing 500 μM FDP in deuterated tris
buffer (50 mM tris-d11, 300 mM NaCl, 20 mM NaN3, D2O, pH 7.8, 5%
DMSO-d6) and a 1H NMR spectrum was measured (Fig. S6a, Supple-
mentary data). 1 mM MgCl2 was added to the same sample and the
experiment was repeated (Fig. S6b, Supplementary data). STD NMR
spectra were recorded using a spectral width of 7000 Hz and 32 k
time domain data points. The on resonance pulse was set to 5250 Hz,
the off resonance pulse to 40 kHz. Saturation was achieved by a train
of 90° Gaussian-shaped pulses of 50 ms yielding a total saturation time
of 3 s with an attenuation of 45 dB. Water suppression was achieved
using the excitation sculpting sequence. O1 was set on resonant to
the water signal at 3285 Hz. The temperature during acquisition
was 300 K.
2.3. NMR spectroscopy
All NMR experiments were performed at 298 K (invertase) or 285 K
(germacrene D synthase) using a Bruker Avance 700 MHz NMR spec-
trometer with a 5 mm inverse triple resonance probe head. Spectra
were recorded with a spectral width of 9763 Hz (invertase) and 64k
data points or 7000 Hz (germacrene D synthase) and 28 k data points.
Before the performance of kinetic measurements the NMR magnet
was shimmed using a protein sample in the same buffer that was
used later (1.5 μg/mL invertase in 25 mM acetate buffer, 50 mM NaCl,
2 mM NaN3, D2O, pH 5.0; 2.5 μM germacrene D synthase in 50 mM
tris-d11, 300 mM NaCl, 20 mM NaN3, D2O, pH 7.8). The spectrometer
was matched and tuned and the sample removed. The procedures for
sample preparation and data acquisition are described in the following
2.7. Analysis of invertase reaction
For linear regressions and calculation of Michaelis–Menten kinetics
the software OriginPro 8.5.0G SR1 (OriginLab Corporation, Northampton,
MA, USA) was used. Progress curves were fitted in MATLAB 7.10.0.499
(R2010a) (MathWorks, Inc., Germany). Initial velocities were plotted
against initial substrate concentrations and the hyperbolic curve was
fitted according to the one site binding model (pharmacokinetics,

F. Exnowitz et al. / Biochimica et Biophysica Acta 1824 (2012) 443–449
445
OriginPro) that is equivalent to the Michaelis–Menten equation
(Eq. (2)).
In order to start the experiment sucrose has to be added to the
invertase solution, the sample has to be inserted into the magnet
and the magnet eventually has to be shimmed. The spectrometer
was prepared prior to the experiment with an enzyme solution
(tuning, matching and shimming). Addition of sucrose stock solution
to a prepared invertase solution in an NMR tube did not yield a ho-
mogeneous mixture. Therefore both solutions were thoroughly mixed
before they were filled into the NMR tube.
d½Sꢀ V max½Sꢀ
−
¼
ð2Þ
dt
Km þ ½Sꢀ
For the analysis of progress curves a script was utilized applying
non linear regression on the Lambert W solution of the Michaelis–
Menten equation as it was published previously [15]. Progress curve
analysis requires an estimation of Km and Vmax. This can be achieved
by linearization of the integrated form of Michaelis–Menten equation,
e.g. according to Eqs. (3) and (4) [14,17]. Estimates of Km and Vmax are
calculated from the slope and intersection of y as shown in Fig. S4
(Supplementary data). Tables 1 and S2 summarize the results of the
Lambert W function-analysis of progress curves.
Fig. 1 shows all progress curves measured for the hydrolysis of
sucrose. Each kinetic experiment was analyzed with respect to the
concentrations of sucrose (H-1 of the glucose residue). In order to
obtain the integral I0 of sucrose at the beginning of each conversion
(t=0 s), I0 was calculated from the sum of the integrals I1 of sucrose
(H-1 of glucose residue) and I1 of the anomeric proton of α-glucose
in the first spectrum. After the calculation of the concentrations, linear
regression was applied to the first four data points of each progress
curve (Fig. S1, Supplementary data). The slope directly yields the initial
velocity for each experiment (Table S1, Supplementary data). The exact
concentrations of all substrates and products were calculated using
the peak of the acetate methyl group (buffer) as an internal reference
(Fig. S2, Supplementary data). In addition, the time for starting the
experiment was calculated from the linear equation by satisfying
the constraint that the intersection of y is the same as the initial concen-
tration [S]0. In order to account for saturation effects, proton spectra of a
mixture of sucrose, glucose and fructose were acquired applying the
same conditions that were used for kinetic measurements (buffer,
temperature, spectral parameters) but with increasing relaxation delays
between 1 and 45 s. Each signal that was analyzed was corrected by a
constant factor derived from the two proton spectra applying relaxation
delays of 1 and 45 s (Fig. S3, Supplementary data).
Using NMR a complete analysis of the substrate and the products of
sucrose cleavage is possible (see also Fig. S2, Supplementary data). This
is demonstrated in Fig. 2 for an initial concentration of sucrose of
27.4 mM. The anomeric protons of free α- and β-glucose (H-1), H-
1 of free β-fructofuranose and H-5 of β-fructopyranose were used
for integration. Signals of the minor product α-fructofuranose could
not be assigned unambiguously. The stereochemistry for the hydrolysis
of sucrose by invertase has been studied previously [19–21]. Invertase
is a ß-fructofuranosidase, hence the glycosidic bond is cleaved on the
fructose residue. In agreement with the ß-fructofuranoside activity
α-glucose is formed directly in the reaction, β-glucose only by mutaro-
tation as is evident from Fig. 2A. Fig. 2B also reveals that the hydrolysis
works under retention of the configuration of the β-D-fructofuranosyl
residue as it was reported previously [21,22]. The formation of β-D-
fructopyranose by mutarotation is evident from the time delay in its
t
1
V max
½Sꢀ0−½Sꢀ
KM
V max
ꢀ
ꢁ
ꢀ
ꢁ
¼
þ
ð3Þ
ð4Þ
ln ½Sꢀ0=½Sꢀ
ln ½Sꢀ0=½Sꢀ
ꢀ
ꢁ
ln ½Sꢀ0=½Sꢀ
t
KM
1
¼
þ
½Sꢀ0−½Sꢀ V max ½Sꢀ0−½Sꢀ
V max
2.8. Analysis of (+)-germacrene D synthase reaction
The Progress curve for the transformation of FDP was fitted in
MATLAB 7.10.0.499 (R2010a) (MathWorks, Inc., Germany). For the anal-
ysis a script was utilized applying non linear regression on the Lambert
W solution of the Michaelis-Menten equation [15]. The transformation
of FDP is not complete (cf. main text, Fig. 6). The amount of residual
unreacted FDP was calculated from the average of the last twelve data
points (starting at 27075 s) to yield [S]y0 =34.9 μM. Because Eq. (1)
supposes complete transformation of the substrate, [S]y0 was added
to Eq. (1) and subtracted from the initial concentration [S]0 to yield
Eq. (5).
ꢂ
ꢃ
8
<
9
2
0
1
3
½Sꢀ0−½Sꢀy0 −V0maxt
=
½Sꢀ0−½Sꢀy0
0
4
@
A
5
½Sꢀ ¼
K
MW
exp
þ ½Sꢀy0
ð5Þ
KM0
KM0
:
;
3. Results and discussion
The progress of the invertase reaction was monitored for different
initial concentrations of sucrose. Spectra were acquired every 2 min
over a period of several hours. Each spectrum was acquired with
eight scans and a total acquisition time of 35 s. From each progress
curve the initial rate was determined for a classical Michaelis–Menten
analysis. Additionally, each progress curve was analyzed according to
Eq. (1). The initial concentrations were chosen to be below, close to
and above the literature value for Km of 25 mM of sucrose [18].
0.05
3.5 mM
8.7 mM
12.3 mM
0.04
0.03
0.02
0.01
0.00
14.8 mM
15.1 mM
17.8 mM
22.3 mM
27.4 mM
42.2 mM
Table 1
Kinetic parameters derived by progress curve analysis at different initial sucrose
concentrations.
c0 (sucrose) [mM]
Km [mM]
vmax [μM/s]
3.50
8.70
12.4
14.8
15.1
17.8
22.3
27.4
41.8
15.5 2.1
26.3 1.5
26.9 1.3
28.2 0.7
30.0 2.3
26.8 0.7
26.7 0.6
29.2 0.3
38.8 2.0
8.69 1.1
13.1 0.6
12.9 0.5
13.2 0.3
13.4 0.8
13.0 0.2
12.0 0.2
12.1 0.1
13.5 0.4
0
5000
10000
15000
20000
t [s]
Fig. 1. Overview of progress curves recorded for the hydrolysis of sucrose by invertase.
Initial concentrations of sucrose are listed in the legend (every second data point is
shown).

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F. Exnowitz et al. / Biochimica et Biophysica Acta 1824 (2012) 443–449
A
A
8.0x10-6
0.030
0.025
0.020
0.015
0.010
0.005
0.000
6.0x10-6
4.0x10-6
2.0x10-6
R2
0.9929
OH
OH
Vmax 9.01 0.4 µM/s
KM 17.1 1.8 mM
O
HO
HO
O
HO
OH
HO
OH
OH
0.0
0.00
OH
0.01
0.02
0.03
0.04
c(sucrose) [M]
0
5000
10000
15000
20000
t [s]
B
0.03
RMS 2.99x10-5
B
0.025
0.02
0.015
0.01
0.005
0
0.030
0.025
0.020
0.015
0.010
0.005
0.000
Vmax
KM
12.0 0.1 µM/s
28.7 1.3 mM
OH
OH
OH
HO
OH
OH
O
O
OH
HO
OH
HO
0
0.5
1
1.5
2
x 104
t [s]
Fig. 3. Determination of Km and Vmax. A: Michaelis–Menten plot of invertase reaction derived
from initial velocities of each progress curve. The fit afforded Km=17 mM and Vmax =9 μM/s.
B: Progress curve analysis of invertase reaction with initial sucrose concentration
of 27.4 mM. The data was fitted according to Eq. (1) and yielded Km =29 mM and
Vmax =12 μM/s.
0
5000
10000
15000
20000
t [s]
Fig. 2. NMR spectroscopic characterization of sucrose (27.4 mM) hydrolysis by invertase
(every second data point is shown). A: Analysis of glucose formation; the reaction directly
yields α-glucose, β-glucose is formed by mutarotation. The sum of total glucose and
sucrose concentration was plotted as a control for signal stability over time. B: Analysis
of fructose formation; cleavage of the glycosidic bond occurs under retention of
the configuration of the anomeric center. Thus, the reaction directly yields β-
fructofuranose, β-fructopyranose is also formed by mutarotation as indicated by
the lag phase at the beginning of the curve.
the fit is self-evident and the RMS is 3 10−5. For other initial concentra-
tions between 8.7 and 22.3 mM we determined Km=28.2 1.8 mM
and Vmax =12.7 0.7 μM/s (185 15 u/mg Invertase). The results
of all fits are summarized in Table 1. Noticeable Km differs at the
highest and lowest concentrations applied in the assay. For an initial
concentration of 3.5 mM sucrose values obtained for Km and Vmax are
15.5 mM and 8.68 μM/s (128 u/mg sucrose), respectively. This indi-
cates that substrate concentrations should not be too far away from
the Km as has been reported earlier for the analysis of progress curves
[10]. For an initial concentration of 42 mM sucrose an increase in Km
is observed but Vmax is still in the range of all other fits. This assumes
competitive inhibition of the enzyme reaction, which could be originated
from product inhibition. The errors quoted in table 1 reflect the quality of
the fit and do not include systematic errors. These would become evident
by running the time course repeatedly.
In order to investigate the effect of product inhibition, the reaction of
invertase was repeated with 27 mM sucrose in the presence of a 1:1
mixture of glucose and fructose (26 mM). Fitting the progress curve
according to Eq. (1) yielded Km=38 mM and Vmax=12 μM/s (Figs. 4
and S5 in the Supplementary data). The experiment demonstrates
that product inhibition can be clearly seen already from two progress
curves. Product inhibition has been known to be present in the action
of invertase [23]. Fructose has been reported to be a competitive inhib-
itor and should result as observed here in an increase of Km. Glucose is
reported to reduce Vmax as a (partial) non-competitive inhibitor.
formation. Thus, all stereochemical information about the enzymatic
reaction is obtained from a single experiment.
For comparison a Michaelis–Menten plot was derived from the
NMR data (Fig. 3A, Table S1 in the Supplementary data). Initial rates
were obtained by linear regression of the first four data points of each
reaction and plotted against initial substrate concentration. The data
was fitted directly to the Michaelis–Menten equation and afforded
a Km of 17 mM and a Vmax of 9 μM/s (133 u/mg invertase). The Km
obtained by this method is slightly below the literature value of
25 mM and the specific activity is slightly below the supplier's spec-
ification of 200–300 u/mg invertase.
Fig. 3B shows the experimental data for an initial concentration of
27.4 mM sucrose and the progress curve analysis applying Eq. (1).
Fitting of the data requires an estimation of Km and Vmax that can
be derived by a linearization of the integrated form of Michaelis-
Menten equation (see experimental section and Fig. S4). The fit afforded
a Km of 29 mM and a Vmax of 12 μM/s (179 u/mg invertase). The quality of

F. Exnowitz et al. / Biochimica et Biophysica Acta 1824 (2012) 443–449
447
0.030
0.025
0.020
0.015
0.010
0.005
0.000
recorded spectra of 128 scans at intervals of 22 min over a period of
12 h. The signals of H-2 [5.47 ppm (t, 1H, J=7.0 Hz)] of FDP and H-5
[5.76 ppm (d, 1H, J=15.9 Hz)] of (+)-germacrene D were used for
the kinetic analysis in the spectra of FDP transformation. Saturation ef-
fects were neglected (T1b1.5 s for both protons). In Fig. 5A the integrals
of both protons as well as their sum are plotted against reaction time.
The first data point was set to 13 min. This takes into account 2 min
after addition of FDP to (+)-GDS for the submission of the sample to
the magnet, locking, shimming and starting the experiment. Additionally,
the spectrum represents the time averaged signal over a period of 22 min.
Therefore the time for each data point is set to the middle of this period.
The curve of FDP does not show typical exponential behavior and is
reduced compared to formation of (+)-germacrene D. The reduction
of signal intensities is attributed to micelle formation. For FDP in
aqueous solution a CMC of 12 mM is reported but we have observed
that micelle formation of FDP is promoted by magnesium ions even
at micromolar concentrations of Mg2+ [24]. Due to micelle formation sig-
nal intensities of FDP are reduced even in presence of DMSO. Under the
reaction conditions, 500 μM FDP and 1 mM MgCl2, the signal intensities
are reduced by 45% as a result of line broadening (Fig. S6, Supplementary
26.6 mM sucrose + 25.8 mM glucose/fructose
28.4 mM sucrose
0
2000
4000
6000
8000
t [s]
Fig. 4. Analysis of product inhibition. Comparison of progress curves of samples with
26 mM of fructose and glucose (1:1) added (○) and without addition of the products
(▲) to the reaction of invertase. Addition of glucose and fructose clearly reduces the rate
of the cleavage of sucrose. Fitting the progress curve of the reaction with added products
according to Eq. (1) yielded expectedly a higher value for Km=38 mM but the same
Vmax=12 μM/s. The increase in Km is attributed to competitive product binding. Fitting the
reaction progress without additional inhibition afforded Km=28 mM and Vmax=11 μM/s.
A
2.0
The inhibition constant Ki for glucose (Ki ≈0.4 M) is much larger
than that for fructose (Ki≈0.2 M) and thus the effect on Vmax is not ob-
servable at the concentrations used here. The effect of the added fruc-
tose is however clearly discernable. In agreement with the observed
product inhibition, progress curve analysis for an initial concentra-
tion of 42 mM sucrose led to an increase of Km to 38.8 mM. In fact,
in this experiment we have the product concentration at comparable
levels already after about 60% of reaction progress.
1.5
FDP
(+)-germacren D
FDP + (+)-germacren D
1.0
0.5
0.0
In a further investigation we applied 1H NMR spectroscopy to
study the reaction of (+)-germacrene D synthase, (+)-GDS, from
S. canadensis. The enzyme catalyzes the transformation of farnesyl
diphosphate (FDP) to (+)-germacrene D, a monocyclic sesquiterpene
that originates in the early steps of sesquiterpene biosynthesis. The
enzyme was characterized previously by Prosser et al. [16]. They
studied the Michaelis–Menten kinetics by a radioactive assay and
obtained a Km of 3 μM and a kcat of 0.02 s−1. The transformation of
FDP starts with the cleavage of the diphosphate group under formation
of a highly reactive carbenium ion. After cyclization, two hydride shifts
and proton abstraction (+)-germacrene D is formed. The enzyme
requires magnesium and/or manganese ions in the millimolar range
to support the binding of FDP and the departure of the diphosphate
group. The reaction catalyzed by (+)-germacrene D synthase is a single
substrate reaction.
We were interested in the interaction of FDP with the enzyme as
this type of interaction is also important in the posttranslational
farnesylation of several regulative proteins by protein farnesyltransfer-
ase. FDP binding to a protein is driven by ionic as well as hydrophobic
interactions due its amphiphilic nature. These types of interactions are
very difficult to investigate and they normally require a very careful
analysis of the state that the amphiphilic substrate is in. Often one
finds heterogeneous solutions that have a drastic effect on the binding
events. These consist of specific and possible unspecific hydrophobic
binding to the protein as well as micelle formation of the substrate.
We were able to characterize the model system (+)-germacrene D
synthase in detail, making possible the characterization of FDP binding
to the enzyme.
0
10000
20000
30000
40000
t [s]
B
0.0005
0.0004
0.0003
0.0002
0.0001
0.0000
FDP
(+)-germacren D
0
10000
20000
30000
40000
t [s]
Fig. 5. Progress curves for the reaction of farnesyl diphosphate (FDP) with germacrene
D synthase to yield (+)-germacrene D. A: While the product formation shows typical
exponential kinetics, this behavior is not observed for FDP. The sum of both curves is
not constant over time. This observation is explained by micelle formation of FDP in
presence of MgCl2. B: In order to obtain the progress curve for the transformation of
FDP, the integrals in Fig. 5A were corrected as described. Calculating the concentrations
and plotting the concentration time course for FDP and (+)-germacrene D now shows
two curves that intersect at half of the concentration for both compounds.
The analysis of the progress curve was only possible under specific
conditions. 5% DMSO-d6 was added to the NMR buffer (deuterated TBS)
to keep the product (+)-germacrene D of the reaction in solution and to
reduce micelle formation of FDP. To keep the protein native over the
course of the reaction a reduced temperature of 285 K was used. We

448
F. Exnowitz et al. / Biochimica et Biophysica Acta 1824 (2012) 443–449
data). The effect of signal reduction is concentration dependent as can be
seen in Fig. 5A. When the concentration of FDP and hence of the micelles
is reduced the signal reduction decreases. This can be observed from the
sum of integrals that increases with the progress of the reaction. In order
to account for this effect, all integrals of FDP are corrected, assuming that
the sum of integrals for FDP and (+)-germacrene D from minute 13 to
211 have a constant value. In Fig. 5B the progress curves for FDP and
germacrene D are plotted after applying the correction and calculating
the concentrations.
Fig. 6 shows the progress curve derived under the conditions of the
NMR assay. The kinetic parameters obtained are Km=379 μM and
kcat=0.04 s−1. kcat is in the same range of the value obtained by Prosser
et al., but there is a 125-fold increase in Km. To some extend this increase
is explained by the change of reaction conditions. The radioactive assay
was conducted in a MOPSO buffer that contained 5 mM MgCl2 and was
overlaid with hexane to capture (+)-germacrene D from the aqueous
phase and the reaction temperature was 303 K [16]. In addition to
that we observe product inhibition by (+)-germacrene D. The enzy-
matic transformation yielded a clear reaction solution and gave no
indication for the formation of an emulsion. A strong association ten-
dency of (+)-germacrene D to the hydrophobic active site in this
aqueous environment is reasonable. This conclusion is supported by
the fact that approximately 6% of the substrate is not transformed even
after 12 h. Due to the energetics of the reaction the backward reaction
can be excluded and the incomplete transformation is thus not attributed
to an equilibrium being present. Taking these effects into account,
the increase of Km is reasonable.
planed carefully and it is very important to choose the adequate tech-
nique. NMR spectroscopy, however, can easily detect and quantify in
situ all reaction products and substrates. Also, stereochemistry of the
products formed is directly obtained from the spectra which other
methods cannot normally achieve. Obviously NMR is not the method
to characterize very fast enzyme reactions but many transformations
are eligible when the reaction conditions are adapted carefully. The
detection of product inhibition is possible by comparing the time
courses in only two experiments. The protocol presented here is also
a simple and direct approach for the measurement of enzyme kinetics
in the presence of synthetic inhibitors. The closed shell solution of the
Michaelis–Menten equation can be expanded to further kinetic models
including various types of inhibition models [14]. In routine applica-
tions it is not necessary to run experiments for as much time as docu-
mented here. The experiments can also be run with as little as ten
data points. Progress curve analysis of the change in substrate concen-
trations as a function of time requires only a very straightforward and
simple experimental setup to determine enzyme parameters with a
minimal number of experiments.
Acknowledgements
We thank Dr. Iris Altug, Prof. Dr. H.J. Bouwmeester and Prof. Dr. W.
A. König (†19.11.2004) for the vectors containing the germacrene D
synthase genes and Dr. Altug for her helpful advice. We acknowledge
an equipment grant from the DFG ME1830/1-1 for the 700 MHz NMR
spectrometer. Part of this work was supported by the VW Stiftung
and by DFG through SFB470 and Graduate College GRK464.
4. Conclusions
Appendix A. Supplementary data
We demonstrated here that NMR spectroscopy is a powerful tool
to analyze the kinetics of enzyme reactions using progress curves.
NMR spectroscopy is one of many methods that can be applied in sub-
strate or product detection. Experiments in enzyme kinetics have to be
Supplementary data to this article can be found online at doi:10.
1016/j.bbapap.2011.10.011.
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