6
C. Her et al. / Carbohydrate Research 455 (2018) 5e9
[
(
15,16]. This approach allows the conversion of sucrose to glucose
-D-glucose) and fructose via invertase to be monitored in real-
d½Pꢂ
d½Sꢂ
V
max½Sꢂ
a
n
¼
¼ ꢃ
¼ K
[1]
dt
dt
þ ½Sꢂ
time, and the subsequent mutarotation process that converts
D-glucose to -D-glucose. In this manuscript, we extend this pro-
a
-
M
b
Where [S] is the substrate concentration, Vmax is the maximal
rate of enzymatic turnover (sucrose to glucose and fructose), and
represents the Michaelis-Menten half-saturation constant. The
first-order differential equation (Equation [1]) can be integrated to
obtain the integral form of the MichaeliseMenten equation
cedure to demonstrate the role of sucralose in altering the
Michaelis-Menten kinetics of sucrose hydrolysis via invertase. The
combination of real-time NMR measurements and progress curve
analysis is used to determine the Michaelis-Menten constant (K
and maximum velocity (Vmax). The results from the progress curve
analysis indicate that sucralose has the characteristic of
K
M
M
)
[
20e22] as shown in Equation [2]:
a
ꢀ½
ꢁ
competitive inhibitor to invertase, which significantly reduces the
catalytic efficiency of the enzyme.
Sꢂ
0
KM ln
þ ½Sꢂ ꢃ ½Sꢂ ¼ Vmax
t
[2]
0
½
Sꢂ
The Lambert-W function is a mathematical function, in the form
of an exponential function, and has several applications in com-
puter science, mathematics, and physical sciences [23,24]. Mathe-
matically, the exponential function and the natural logarithmic
function, ln(x), are exponentially related. Similarly, W(x) is defined
2
. Materials and method
Invertase (EC 3.2.1.26,
purchased from Sigma-Aldrich with a specific activity of >300 u/mg
of solid (pH 4.6, 303 K). Sucrose, D-(þ)-Glucose, D O (99.9 atom %
D) and 3-(Trimethylsilyl) propionic-2, 2, 3, 3-d acid sodium salt
TSP) were also purchased from Sigma-Aldrich. Sucralose was
purchased from Alfa Aesar.
b-fructofuranosidase, S. cerevisiae) was
y
2
as the inverse of the function satisfying ye ¼ x and its solution
4
expressed by the Lambert-W(x) function like y ¼ W(x).
(
By substituting y ¼ ½Sꢂ=K in Equation [2] and rearranging, we
M
get Equation [3]:
ꢀ½
ꢀ
ꢁꢁ
Sꢂ ꢃ Vmax
t
½Sꢂ
y
0
0
ye ¼ xðtÞ ¼ exp
þ ln
tꢁ
2
.1. Sample preparation
K
M
K
M
ꢀ
[3]
½
Sꢂ
½Sꢂ ꢃ Vmax
0
0
For all the NMR kinetics experiments, pH 5.0 was used. The
sucrose (80 mM) and invertase (5 g/mL) concentrations were kept
¼
exp
K
K
M
m
M
constant, while the sucralose concentrations varied. The sucralose
concentrations used were 2 mM, 10 mM, 20 mM, 40 mM, 80 mM,
The left-hand side of Equation [3] is analogous to the Lambert-
W function in the Corless et al., 1996 article [24]. Thus, using the
definition of the Lambert-W function (y ¼ W(x)), an expression for
y can be obtained as that expressed in Equation [4]:
120 mM, and 160 mM. A timer was set at the beginning of the
addition of the invertase solution to account for the delay time (the
time before the collection of the NMR spectra).
ꢂ½
ꢃ
Sꢂ
À
Á
0
y ¼ W
exp ½Sꢂ ꢃ Vmaxt=KM
[4]
0
K
M
2.2. Real-time NMR measurements of the hydrolysis of sucrose via
invertase
Further substituting y ¼ ½Sꢂ=K
M
back in Equation [4], we get
Equation [5]:
A Varian-Agilent 400 MHz NMR spectrometer was used to
collect the NMR data. Each 1D-proton NMR spectrum was collected
with a spectral width of 14.88 ppm over 32768 number of points
ꢂ½
ꢀ
tꢁꢃ
Sꢂ
½Sꢂ ꢃ Vmax
o
o
½
Sꢂ ¼ K W
exp
[5]
M
K
K
M
M
(
with Ernst angle [17]). Each spectrum was signal averages over 24
Equation [5] relates the substrate concentration at any time ([S])
to its initial concentration ([S] ), the MichaeliseMenten kinetic
parameters Vmax and K . Equation [5] is used to fit the real-time
experimental data obtained for enzyme kinetics using the anal-
ysis code written in the R-Statistical environment [25].
transients and 1s relaxation delay between the transients leading
to a total time of 90.0 s per experiment (24 ꢁ 3.75 s per transient).
The 1D-proton NMR spectra were collected in an array fashion one
after another and were collected as necessary to follow the reaction
to completion. The NMR data were processed using the program
MNova NMR™. The area under the curve of the sucrose resonance
at 5.41 ppm), alpha-D-glucose and beta-D-glucose resonance (at
.22 ppm and 4.64 ppm), and sucralose resonance (at 5.49 ppm)
were used to calculate their concentration using TSP (at 0 ppm) as
the standard. Progress curve analysis was done following the pro-
cedure in the following reference [16].
0
M
3
. Results
(
5
3
.1. NMR spectral differentiation between sucrose, glucose, and
sucralose
Glucose, sucrose, and sucralose all have at least one proton with
a distinct resonance, granting the ability to perform real-time
analysis of the enzyme kinetics via NMR spectroscopy. Fig. 1
shows the NMR spectra of sucrose (Fig. 1a), sucralose (Fig. 1b)
and glucose (Fig.1c). The C1 proton for sucrose shows a double peak
(doublet) close to the resonance frequency of 5.41 ppm, while the
same proton in the sucralose is shifted downfield to 5.49 ppm. Also,
the C5 and C4 protons of sucralose resonate at 4.42 ppm and
4.54 ppm, respectively (Fig. 1b). As shown in Fig. 1c, the NMR
spectrum for glucose depicts a doublet at 5.22 ppm (C1 proton) [26]
2
.3. Progress curve analysis of the real-time NMR data
The catalyzed breakdown, or hydrolysis, of sucrose, can be
shown by the Michaelis-Menten equation. Schnell and Mendoza
22] presented the integrated form of the Michaelis-Menten
[
equation using the Lambert-W function with an application
developed by Goudar and co-workers [18,19]. A brief description of
enzyme kinetics is given here following our earlier work [16]:
The MichaeliseMenten equation in the differential form can be
used to describe the dynamics of substrate depletion as Equation
for
-D-glucose. The relative intensities of the
glucose peaks (Fig. 1c) are dependent on the thermodynamic
a-D-glucose and the second doublet at 4.64 ppm (C1 proton) for
b
a
-D-glucose and -D-
b
[1]: