The kinetics of p-nitrophenyl-β-d-cellobioside hydrolysis and transglycosylation by Thermobifida fusca Cel5Acd

Article abstract of DOI:10.1016/j.carres.2010.09.011

The hydrolysis of p-nitrophenyl-β-1,4-cellobioside (pNP-G2) by the catalytic domain of the retaining-family 5-2 endocellulase Cel5A from Thermobifida fusca (Cel5Acd) was studied. The dominant reaction pathway involves hydrolysis of the aglyconic bond, producing cellobiose (G2) and a 'reporter' species p-nitrophenol (pNP), which was monitored spectrophotometrically to track the reaction. We also detected the production of cellotriose (G3) and p-nitrophenyl-glucoside (pNP-G1), confirming the presence of a competing transglycosylation pathway. We use a mechanistic model of hydrolysis and transglycosylation to derive an expression for the rate of pNP-formation as a function of enzyme concentration, substrate concentration, and several lumped kinetics parameters. The derivation assumes that the quasi-steady-state assumption (QSSA) applies for three intermediate species in the mechanism; we determine conditions under which this assumption is rigorously justified. We integrate the rate expression and compare its integral form to pNP-versus-time data collected for a range of enzyme and substrate concentrations. The integral comparison gives a stringent test of the mechanistic model, and it serves to quantify the lumped kinetics parameters with good statistical precision, particularly a previously unidentified parameter that determines the selectivity of hydrolysis versus transglycosylation. The integrated rate expression accounts well for pNP-versus-time data under all circumstances we have investigated.

Full text of DOI:10.1016/j.carres.2010.09.011

Carbohydrate Research 345 (2010) 2507–2515
Contents lists available at ScienceDirect
Carbohydrate Research
journal homepage: www.elsevier.com/locate/carres
The kinetics of p-nitrophenyl-b-D-cellobioside hydrolysis and transglycosylation
by Thermobifida fusca Cel5Acd
⇑
John W. Dingee, A. Brad Anton
School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14850, United States
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 9 June 2009
Received in revised form 7 September 2010
Accepted 9 September 2010
Available online 17 September 2010
The hydrolysis of p-nitrophenyl-b-1,4-cellobioside (pNP-G2) by the catalytic domain of the retaining-
family 5-2 endocellulase Cel5A from Thermobifida fusca (Cel5Acd) was studied. The dominant reaction
pathway involves hydrolysis of the aglyconic bond, producing cellobiose (G2) and a ‘reporter’ species
p-nitrophenol (pNP), which was monitored spectrophotometrically to track the reaction. We also
detected the production of cellotriose (G3) and p-nitrophenyl-glucoside (pNP-G1), confirming the pres-
ence of a competing transglycosylation pathway. We use a mechanistic model of hydrolysis and transgly-
cosylation to derive an expression for the rate of pNP-formation as a function of enzyme concentration,
substrate concentration, and several lumped kinetics parameters. The derivation assumes that the quasi-
steady-state assumption (QSSA) applies for three intermediate species in the mechanism; we determine
conditions under which this assumption is rigorously justified. We integrate the rate expression and
compare its integral form to pNP-versus-time data collected for a range of enzyme and substrate concen-
trations. The integral comparison gives a stringent test of the mechanistic model, and it serves to quantify
the lumped kinetics parameters with good statistical precision, particularly a previously unidentified
parameter that determines the selectivity of hydrolysis versus transglycosylation. The integrated rate
expression accounts well for pNP-versus-time data under all circumstances we have investigated.
Ó 2010 Elsevier Ltd. All rights reserved.
Keywords:
p-Nitrophenyl-b-1,4-cellobioside
Thermobifida fusca Cel5A
Retaining glycosyl hydrolases
Transglycosylation
Cellulases
Enzyme kinetics
1. Introduction
another substrate molecule) can result in transglycosylation.4–6
Consequently, when nitrophenyl glycosides are used to study the
Glycosides of mono- and dinitrophenols are convenient sub-
strates for studying the activity of glycosyl hydrolases because
their hydrolysis yields an easily detectable nitrophenolic prod-
uct.1–3 These substrates typically contain mono- or disaccharide
glycones specific to the glycosyl hydrolase being studied and a
nitrophenolic aglycone that mimics a sugar unit, insofar as it fits
in the active site and does not interfere with the underlying hydro-
lytic mechanism. The small size of the substrate ensures high if not
complete specificity for bond cleavage at the aglyconic bond where
the nitrophenolic ‘reporter’ group is attached. The enzymatic reac-
tions can be terminated in alkaline conditions, deprotonizing a cat-
alytic proton donor and increasing the absorbance of the reporter
molecule, which usually has a peak around 400 nm and can be
quantified with either an extinction coefficient or a standard curve.
Thus, one can obtain reporter concentration-versus-time data with
high-precision for use in mechanistic and kinetics studies.
activity of retaining glycosyl hydrolases, increasing the substrate
concentration can activate this transglycosylation pathway,
channeling the nitrophenol reporter group into a ‘hidden’ non-
hydrolyzable product and reducing the observed reporter yield.
When this happens, the reaction appears to be substrate-
inhibited.7,8 However, the observed reduction in yield is caused
by the transglycosylation pathway, which is fundamentally differ-
ent from substrate inhibition, where substrate is simply quenching
the activity of the enzyme.9,10 The presence of an alternate path-
way for transglycosylation complicates the mechanistic picture
and the mathematical description of reaction kinetics for retaining
glycosyl hydrolases and their mutants.
In this study we characterize the hydrolysis and transglycosyla-
tion of a glycoside of cellobiose (G2) and p-nitrophenol (pNP) by
the catalytic domain of Thermobifida fusca Cel5A (Cel5Acd), a
34.5 kDa, endo-specific, family 5-2 glycosyl hydrolase.11 Cel5A
and its catalytic domain have been the subject of several mechanis-
tic studies with homogeneous substrates. For example, Barr com-
pared the activity of Cel5A on various cello-oligosaccharides and
found that cellotetraose (G4) and cellopentaose (G5) were com-
pletely hydrolyzed within 15 min under conditions where only
20% of cellotriose (G3) was hydrolyzed in 18 h, revealing that G3
is a particularly poor substrate for Cel5A.12 Also Barr was the first
Nearly all glycosyl hydrolases follow either an inverting or
retaining hydrolytic mechanism.4 If the retaining mechanism is fol-
lowed, then the reaction pathway contains a glycosyl/enzyme
intermediate; nucleophilic attack of this by a carbohydrate (e.g.,
⇑
Corresponding author. Tel.: +1 607 255 3629.
E-mail address: aba6@cornell.edu (A.B. Anton).
0008-6215/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.carres.2010.09.011

2508
J. W. Dingee, A. B. Anton / Carbohydrate Research 345 (2010) 2507–2515
to report transglycosylation by Cel5Acd, showing that G4-hydroly-
sis produced non-complementary products G2 and G3.12 Irwin
et al. showed that Cel5Acd could not hydrolyze 4-methylumbellife-
ryl-glycoside (MU-G1) but could hydrolyze pNP-G2, MU-G2, and
MU-G3 with release of the reporter group, pNP or MU.13 However,
Irwin’s study and several others involving Cel5A14–18 do not ad-
dress the possibility of a competing transglycosylation pathway.
We present a two-pathway mechanism for pNP-G2 hydrolysis
by Cel5Acd that includes the formation of pNP-reporter via simple
hydrolysis and a non-hydrolyzable pNP-G1 product via transglyco-
sylation. The same reaction network for competing hydrolysis and
transglycosylation has been proposed by others, and the rate
expression that it implies has been used to fit to initial-rate data
for reporter formation.7,8,19 We advance the knowledge gained in
the previous investigations by deriving rigorous conditions that
delimit applicability of the rate expression and by integrating it
to predict the full time-dependence of the pNP-reporter concentra-
tion. The integral model predicts reporter concentration as a
function of initial enzyme concentration, initial substrate concen-
tration, and four lumped kinetics parameters that we quantify by
a single curve-fit to a large set of pNP-versus-time data, collected
with varying initial concentrations of enzyme and substrate. The
integral kinetics model accurately accounts for all features of the
data, verifying its accuracy and by inference the reaction mecha-
nism on which it is based.
tubes were covered and placed in a 50 °C water bath. Every 15 min
three tubes were selected at random and moved to a cool water
bath, where 1 mL of 2 M Na2CO3 was added to stop the reaction.
After the last time-point (195 min), 2 mL of deionized (DI) water
were added to each tube, and the optical density was measured
with a spectrophotometer at 400 nm (OD400). The OD400 readings
were adjusted using the average zero-time value as a blank, and
pNP concentrations were calculated by comparing to a standard
curve.
Hydrolysis reactions for thin layer chromatography (TLC) were
run in 100
l
L volumes as described above but without the addition
L aliquots were spotted
of Na2CO2 or DI water. At completion, 10
l
directly on to a TLC plate (Whatman LK5D, 150 Å silica gel plates),
dried, separated, and developed as described previously.12,21,22
2.3. Data analysis
pNP concentrations were measured at 13 equal time increments
of 15 min each for all combinations of two different enzyme con-
centrations (1.4 and 4.0 lM) and four different substrate concen-
trations (1.0, 1.5, 2.0, and 2.5 mM). Each measurement was
repeated three times to generate an average pNP concentration
with an associated (sample) standard deviation. All 104 data points
were fit simultaneously with
a four-parameter, model-based
analytic function, which is described in Section 4, below. Best-fit
values of the four adjustable parameters were determined by min-
imizing v
2, the weighted sum of squared errors, via a Monte Carlo
2. Experimental
search of parameter space.23 The standard deviation of each fit-
parameter was estimated by forcing the parameter to vary from
its best-fit value, re-minimizing v2 with respect to the other three
parameters, and using the results to estimate the curvature of the
v2 function.23 Because the other parameters were allowed to vary
in these calculation, the standard deviations of fit parameters
grossly include the effects of correlations among them, although
they are not explicitly calculated here.24 The minimization pro-
gram was written in-house using Visual Basic/Excel.
2.1. Cel5Acd production
Cel5Acd was purified from Escherichia coli strain D1430 as
described previously for the xyloglucanase-expressing strains
D1170 and D1212 with the following modifications.20 The phe-
nyl-sepharose column was eluted with successive high-salt to
low-salt washes, using approximately 2-column-volumes of 1.2,
0.6, 0.3, and 0.0 M ammonium sulfate with a continuous 5 mM
Kpi buffer (pH 6.0), with Cel5Acd eluting in the final wash. The
Q-sepharose column was eluted with a 6-column-volume gradient
from 0 to 0.5 M NaCl, with a continuous 10 mM BisTris buffer (pH
5.6). Final protein fractions were exchanged into sodium acetate
buffer (50 mM NaAc, 0.02% NaAz, pH 5.5) and concentrated
3. Results
3.1. pNP-G2 hydrolysis by Cel5Acd
to roughly 60
l
M using a stirred cell ultrafiltration chamber with
The pNP product curves for eight combinations of initial Ce-
l5Acd and pNP-G2 concentrations are presented in Figure 1. The
product curves rise monotonically from zero, and hydrolysis pro-
ceeds to completion within six hours. In every case, the final pNP
product was less than the initial pNP-G2 substrate. For example,
only about 78% of the substrate is converted to reporter for
ST = 2.5 mM, and about 87% of the substrate is converted to repor-
ter for ST = 1.0 mM. This suggests that pNP-G2 had reacted by an
alternate pathway where hydrolysis did not occur adjacent to the
reporter group.
a
polyethersulfone ultrafiltration membrane (10 kDa MWCO,
Millipore, Billerica, MA).
Strain D1430 contains an internal L175Q mutation (Brian Barr,
Personal Communication, Loyola University, Baltimore, MD). The
mutation does not appear to affect the protein fold or activity, as
it purified like wild-type and had comparable activity on homoge-
nous substrates.
2.2. Hydrolysis of pNP-G2 by Cel5Acd
pNP-G2 (N5759, >98% purity) and pNP (1048, spectrophotomet-
ric grade) were obtained from Sigma–Aldrich (St. Louise, MO).
Solutions of pNP and pNP-G2 were prepared at 10 mM in sodium
acetate buffer (50 mM NaAc, 0.02% NaAz, pH 5.5) and stored at
ꢀ20 °C.
3.2. Products distribution
To identify the distribution of final products, 4 lM Cel5Acd and
5 mM pNP-G2 were reacted to exhaustion (4 h), and the final solu-
tion was run on a TLC plate. The result, shown in Figure 2, confirms
that the major products of pNP-G2 hydrolysis are pNP and G2, with
minor products pNP-G1, G3, and trace amounts of G1. Figure 2 also
reveals that no pNP-G2 was present in the final reaction, that is, the
substrate reacted completely. This was validated by boiling the ex-
hausted reaction under basic conditions (2 M Na2CO3), which cat-
alyzes the complete hydrolysis of any remaining pNP-G2 to G2
and pNP. This method (data not shown) confirmed that no pNP-
G2 remained in the reaction at the final times in Figure 1.
All reactions were prepared in sodium acetate buffer (50 mM
NaAc, 0.02% NaAz, pH 5.5) from the same initial enzyme and sub-
strate solutions. Activity assays of Cel5Acd suspended in sodium
acetate buffer for up to 6 days at 50 °C confirmed that the enzyme
remained stable for the entire duration of the reaction under the
conditions employed. For each product curve Cel5Acd, pNP-G2,
and buffer were combined in a 10 mL screw-cap plastic tube, sha-
ken, and transferred in 100 lL aliquots into separate test tubes. The

J. W. Dingee, A. B. Anton / Carbohydrate Research 345 (2010) 2507–2515
2509
Figure 1. Hydrolysis time-course curves. pNP-G2 hydrolysis product (pNP) measured in time for 1.4
lM Cel5Acd (left) and 4.0
l
M Cel5Acd (right). The initial substrate
concentrations in each are: 1.0 mM (diamond), 1.5 mM (square), 2.0 mM (circle), and 2.5 mM (triangle). The solid curves represent fits to the data. (See Section 4.)
Figure 3. Sugar inhibition. Final concentrations of pNP product for reactions
containing increasing concentrations of glucose (white) or cellobiose (grey). The
initial concentrations of pNP-G2 substrate (2 mM) with Cel5Acd enzyme (1.7 lM)
are the same in each. The reactions were conducted in triplicate and terminated
after 4 h. (See Section 2.)
4. Discussion
Figure 2. Product compositions. TLC of a 4 lM Cel5Acd and 5 mM pNP-G2 reaction
4.1. Mechanism for hydrolysis and transglycosylation
after 4 h of hydrolysis. By lane: (a) glucose (G1) through cellopentaose (G5)
standard, (b) the reaction, (c) pNP-G1, and (d) pNP-G2. Note that a faint G1 band in
lane (b) is often observed but becomes nearly invisible when TLC plates are digitally
scanned for photographic reproduction.
Cel5Acd belongs to glycosyl hydrolase family 5, which share a
common TIM barrel fold, hydrolyze b-glycosidic bonds via a dou-
ble-displacement retaining mechanism, and contain eight invari-
ant amino acids.25 Family 5 is divided into subfamilies based on
amino acid sequence similarities above 25%26; Cel5A is in subfam-
ily 2, hence it is a family 5-2 glycosyl hydrolase. The 3D molecular
structure of Cel5Acd, solved using X-ray crystallography to 1.60 Å
resolution for the wild-type and to 1.77 Å resolution for the inac-
tive E355Q mutant with G4 bound, has been published in the Pro-
tein Data Bank (PDB: 2CKS, 2CKR). Comparison of this structure to
those of other family 5-2s reveals that Cel5Acd’s active site is a
3.3. Inhibition assays
To test for glucose or cellobiose inhibition, Cel5Acd/pNP-G2
reactions were loaded with increasing concentrations of G1 and
G2 and allowed to react until an apparent final state was reached.
The pNP yields for 1.7 lM Cel5Acd hydrolyzing 2 mM pNP-G2
after 4 h with 0 mM to 3 mM G1 or G2 are presented in Figure 3.
At these concentrations G2 does not influence the reaction
yield or composition, whereas G1 shows unexpectedly strong
inhibition at concentrations above about 2.0 mM. Compositions
were determined with TLC (data not shown), revealing that a
large fraction of pNP-G2 remained unreacted in the high-glucose
assays.
shallow cleft at the C-terminus of the a/b barrel with at least five
pyranose ring-binding sites or ‘subsites’ within the active site from
ꢀ3 to +2.27–29 Subsites are numbered in the negative and positive
directions toward the non-reducing and reducing end of the sub-
strate, respectively, with hydrolysis always occurring between
the +1 and ꢀ1 subsites.30

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J. W. Dingee, A. B. Anton / Carbohydrate Research 345 (2010) 2507–2515
Next we postulate a molecular-level mechanism for the hydro-
bond. TLC data in Figure 2 confirm this specificity, since the prod-
uct ratio of pNP-G1 (minor) to G1 (trace) is very large. Similar spec-
ificity has been documented for a homologous enzyme–substrate
system.31
lysis and transglycosylation of pNP-G2 by Cel5Acd, involving the
five observed subsites and an additional putative sub-site in the
+3 location. The overall mechanism, shown in Scheme 1, can be
thought of as two pathways branching from a shared intermediate.
The first pathway involves simple hydrolysis via the traditional
double-displacement mechanism of glycosyl hydrolases.4–6 Here,
enzyme (E) and pNP-G2 substrate (S) combine reversibly to form
a Michaelis complex (C1), followed by a nucleophilic attack, which
cleaves the aglyconic bond, releasing pNP reporter (R) and forming
The mechanism also assumes that pNP-G4 shifts by one subsite
in the non-reducing-end direction after formation. This shift can be
interpreted as a processive motion, where the molecule remains in
the active site and slides into its new state, or as a rapid release and
rebinding event, where the molecule transiently leaves the active
site. pNP-G4 is not observed in solution by TLC, but its release
would be difficult to detect because longer chain oligosaccharides
are typically hydrolyzed much faster than shorter ones.9,32–35
Although the movement of pNP-G4 within the active site must oc-
cur, specifying the exact mechanism (either processive or release/
rebinding) is not necessary, as it is irrelevant to our model for the
reaction kinetics.
an
a
-glycosyl/enzyme intermediate (C2). A water molecule then
-glycosyl/enzyme interme-
enters the active site and attacks the
a
diate, cleaving the glycosyl/enzyme bond and releasing cellobiose
product (P).
In the second transglycosylation pathway, the intermediate
complex binds another substrate molecule and forms a second
Michaelis complex (C3). The C4 hydroxyl group at the non-reducing
end of the second substrate acts as a nucleophile, breaking the
glycosyl/enzyme bond and forming a non-covalently bound pNP-
cellotetraose (pNP-G4) molecule. The pNP-G4 then shifts by one
subsite in the non-reducing-end direction before being hydrolyzed
via the double-displacement mechanism to secondary G3 product
(P*) and secondary pNP-G1 reporter (R*). The actions of the cata-
lytic nucleophile, Glu-355, and catalytic acid/base, Glu-263, are
highlighted here, although the entire process includes many more
active-site residues.
Cellobiose accumulates as
a non-hydrolyzable product in
solution and is known to be a strong competitive inhibitor of
other cellulases.36 Cellobiose could also act as an acceptor, com-
peting with the pNP-G2 in the transglycosylation reaction to
form G4 instead of pNP-G4. The G4 would be difficult to detect
if rapidly hydrolyzed to G2, but a more telling result would be
an increase in pNP formation, since replacing pNP-G2 with G2
in the transglycosylation pathway would ensure that more
pNP-G2 is converted to pNP by the hydrolysis pathway.9 Figure 3
shows that the pNP-yield is unaffected by G2 (up to 3 mM),
indicating that neither hydrolysis nor transglycosylation is inhib-
ited here.
The mechanism in Scheme 1 assumes the initial cleavage event
leading to the intermediate C2, occurs strictly at the aglyconic
Scheme 1. Proposed mechanism for hydrolysis and transglycosylation of pNP-G2 by Cel5Acd. The substrate and active site are presented in a simplified cartoon form and are
not to scale. The locations of the ꢀ3 and +3 subsites are implied on the left and right side of the active site. Sites ꢀ3 through +2 are observed directly in the 2CKR crystal
structure, whereas the existence of a +3 subsite is not yet proven. Glycosyl residues are shown as filled circles, and pNP residues are shown as empty circles.

J. W. Dingee, A. B. Anton / Carbohydrate Research 345 (2010) 2507–2515
2511
The kis are rate constants for the various elementary reactions in
Comparative assays with G1 (cf. Fig. 3) revealed that glucose
strongly inhibits the reaction at concentrations greater than about
2.0 mM, where cellobiose shows no inhibition. This is surprising,
since cellobiose is typically a stronger inhibitor than glucose. Pre-
vious studies of the hydrolysis of small oligosaccharides with
Cel5A12,14–18 and Cel5Acd12,13 have not specifically addressed the
possibility of glucose inhibition, presumably because it was
deemed too unlikely to be worthy of consideration. We do not in-
clude glucose inhibition in our mechanism, nor do we include the
effects of the minor products G3 and pNP-G1, because all three
species are present at very low concentrations in our measure-
ments. A focused investigation of glucose inhibition for this system
may prove interesting, but it is outside the scope of this study.
and KÃm
¼
are pseu-
kꢀ1þk2
kꢀ4þk5
Scheme 2, and the parameters Km
¼
k4
do-Michaelis–Menten (MM) constantsk1derived from them.
Eqs. 1a–e are to be solved with the initial conditions
C1(0) = C2(0) = C3(0) = R(0) = 0 and S(0) = ST, but they are nonlinear-
ly coupled and cannot be solved in closed form. One might be
tempted at this point to solve 1a–e numerically while varying
the seven rate constants ki to optimize the agreement between
the calculated RðtÞ and the data of Figure 1. To do this is perilous,
however, because the simple structure and statistical uncertainty
(error bars) of the data in Figure 1 cannot possibly justify a
curve-fit with seven degrees of freedom. It is likely one would find
multiple parameter sets with poor fit-quality and very low statis-
tical precision.23 One would like a rational way, guided by reliable
chemical insight, to simplify the equations and reduce the number
of fitting parameters.
4.2. Kinetics model—derivation
The kinetics problem can be simplified from a seven-parameter
fit to a four-parameter fit and solved analytically by applying the
quasi-steady state approximation (QSSA) to all three complex spe-
cies, that is, by setting dCi=dt ¼ 0 .37,38 Although the QSSA is assu-
redly valid for intermediate complexes in simple MM reactions, it
is not clearly valid for all three complexes in this case, since they
are formed in sequence.39 In the Appendix we derive inequality
relationships that delimit when the QSSA applies, and we enumer-
ate which assumptions are necessary for it to apply here. With the
QSSA simplification for the complexes, Eqs. 1a–e reduce to the fol-
lowing four-parameter, coupled-pair of ODEs:
In Scheme 2 we present a network of elementary chemical reac-
tions involving nine distinct chemical species, the minimum num-
ber necessary to account for the mechanism of Scheme 1. For
simplicity, we lump the entire transglycosylation event (both for-
mation and hydrolysis of pNP-G4) into a single, non-reversible
reaction. The reaction network of Scheme 2 is the same as has been
presented previously for several other enzyme–substrate systems
that transglycosylate.7,8,19
Since there are nine species, nine mass balances (ordinary dif-
ferential equations, ODEs) are required to describe the time-depen-
dent concentrations of all the species in a closed batch reactor. If
the reactor is charged initially with known concentrations of en-
zyme ET and substrate ST, there are four algebraic relationships that
account for conservation of the total amounts of enzyme, pNP, and
glycosyl molecular fragments in the reactor and for the stoichiom-
etric necessity that R* = P*. Consequently, only five of the nine
ODEs are independent. When substrate is present in large excess,
for example, ðET =ST Þ ffi 10ꢀ3 as in our experiments, some other
simplifications are possible, and the five independent mass bal-
ances reduce to the following:
2
~
~
1 dR
ET dt
k
catS þ kcat;2S
¼
¼
;
ð2aÞ
ð2bÞ
Km þ S þ ~ S2
1
~
Km;2
2
~
~
1 dS
ET dt
k
catS þ 2kcat;2S
:
Km þ S þ ~ S2
1
Km;2
~
The four lumped kinetics parameters in these are:
k3KÃ
k3KÃ þ k2KÃm þ k5Km
~
kcat
¼
¼
k2;
ð3aÞ
ð3bÞ
ð3cÞ
ð3dÞ
dC1
dt
dC2
dt
dC3
dt
m
m
¼ k1ðET ꢀ C1 ꢀ C2 ꢀ C3ÞS ꢀ k1KmC1;
¼ k2C1 ꢀ k3C2 ꢀ k4SC2 þ kꢀ4C3;
ð1aÞ
ð1bÞ
ð1cÞ
ð1dÞ
ð1eÞ
k3KÃ
k3KÃ þ k2KÃm þ k5Km
~
Km
Km;
m
m
k3KÃ þ k2KÃ þ k5Km
k2 þmk5
m
¼ k4SC2 ꢀ k4KÃ C3;
~
Km;2
¼
¼
;
m
dS
dt
dR
dt
k2k5
¼ ꢀk2C1 ꢀ k5C3;
¼ k2C1:
~
kcat;2
:
k3KÃ þ k2KÃ þ k5Km
m
m
The rate expression for reporter formation 2a and the lumped
parameters 3a–d have been identified before for similar systems,
and their combination has been called the ‘modified Michaelis–
Menten equation’ for reaction networks of this type.7,8,19 Eqs. 2a
and 2b can be integrated via a procedure described in the Appendix
to give SðtÞ implicitly and RðtÞ as a function of SðtÞ:
ꢀ
ꢁ
x
Km;2
S
ST
!#
~
~
k
catET t ¼
ðST ꢀ SÞ ꢀ Km ln
~
"
ꢀ
ꢁ
x
Km;2
x
x
þ S
~
þ
Km
ꢀ
x
1 ꢀ
ln
;
ð4aÞ
ð4bÞ
~
þ ST
ꢂ
ꢀ
ꢁꢃ
1
2
x
x
þ S
R ¼
ðST ꢀ SÞ ꢀ
x
ln
:
þ ST
Scheme 2. Reaction network. Network of elementary chemical reactions corre-
sponding to the mechanism of Scheme 1, with ki representing the various reaction
rate constants. Each species is defined as follows: free Cel5Acd = E, pNP-G2
substrate = S, product G2 = P, secondary product G3 = P*, reporter pNP = R, second-
A new lumped parameter appears naturally in these as a conse-
quence of the integration procedure:
ꢀ
ꢁ
kcat
k3KÃ
2k5
k3
2k4
k
ꢀ4
~
ary reporter pNP-G1 = R*, enzyme–substrate Michaelis complex = C1, covalent
glycosyl/enzyme intermediate = C2, and the secondary Michaelis complex between
substrate and -glycosyl/enzyme intermediate = C3.
a-
m
x
¼
¼
¼
1 þ
k5
:
ð5Þ
~
2kcat;2
a

2512
J. W. Dingee, A. B. Anton / Carbohydrate Research 345 (2010) 2507–2515
Table 1
4.3. Kinetics model—implications
~
~
~
Best-fit values of parameters kcat, Km, Km;2, and
x; their standard deviations; and their
relative precisions (= best-fit value/standard deviation  100%)a
For complete hydrolysis, the pNP-group in the substrate be-
comes either free pNP reporter (R) or the aglycone residue in a
non-hydrolyzable pNP-G1 secondary reporter (R*). The final distri-
bution of pNP in these two products, that is, the selectivity of the
reaction for hydrolysis versus transglycosylation, can be quantified
by Eq. 4a with S?0. One finds
Parameter
Value
Rel. precision (%)
kcat (minꢀ1
)
~
19.4 4.6
7.48 1.8
0.91 0.3
2.69 1.4
1.30 0.03
24
24
33
52
2.3
kcat;2 (mMꢀ1 minꢀ1
)
~
~
Km (mM)
~
Km;2 (mM)
ꢂ
ꢀ
ꢁꢃ
x
(mM)
R
ST
1
2
x
1 ꢀ ln
ST
x
þ ST
1
f ¼
¼
;
ð6Þ
a
~
Note: kcat;2 was calculated using Eq. 5. The solid lines in Figure 1 were generated
x
using these values in Eqs. 4a and 4b.
where f is the fraction of pNP in the initial substrate that is ulti-
mately converted to R, and 1 ꢀ f is the fraction of pNP in the sub-
strate converted to R*. When ST is small compared to
substrate appears as reporter, and the transglycosylation pathway
x, f?1, all
Scheme 2 could otherwise be targeted to affect the long-time prod-
uct distribution. For example, any mutation that increases k3 rela-
is effectively extinguished. Conversely, when ST is large compared
tive to k4 or k relative to k5, insofar as the ratio kꢀ4=k5 can be
x
, f?12, and R and R* are produced in equal amounts. This is
ꢀ4
to
made large relative to one, would directly favor hydrolysis over
transglycosylation. Note, however, that changes in the magnitudes
the maximum possible yield for the transglycosylation pathway, be-
cause one substrate molecule must act as a donor and hydrolyze in
the self-transfer reaction before another can act as an acceptor and
of the other elementary rate constants k1, kꢀ1, or k2 caused by such
~
mutations would only affect the lumped kinetics parameters kcat
,
transglycosylate. We conclude that the single kinetics parameter
x,
~
~
Km, and Km;2 according to Eqs. 3a–c, hence the overall rate of reac-
tion according to Eqs. 4a and 4b, but not the final distribution of
products according to Eq. 6.
which has units of concentration and emerges only in the integral
form of the rate expression, uniquely determines the tendency of
any enzyme–substrate system that follows the reaction network
in Scheme 2 to branch toward hydrolysis or transglycosylation
When ST is small enough compared to
x, to render hydrolysis
the dominant outcome, measured RðtÞ curves will evidence simple
MM kinetics. Inspection of the rate Eqs. 2a and 2b might lead one
to anticipate this, since simply dropping the S2-terms in those
equations reduces them to the familiar MM form.40 This is an
incorrect way to proceed, however, as it fails to account for subtle
relationships among the parameters that conspire to determine the
relative magnitudes of the S- and S2-terms. The correct way to pro-
ceed is to evaluate the asymptotic behavior of the integrated Eqs.
products. In a relative sense, enzyme–substrate pairs with large-
x
are hydrolyzers, and those with small- are transglycosylators.
x
Figure 4 presents Eq. 6 as a dimensionless plot of f ¼ R =ST versus
1
log10ðST =
x
Þ, and can be used to estimate
x from a single long-time
measurement of reporter yield for any enzyme–substrate system
that follows Scheme 2.
Scheme 1 provides a rational guide for mutating active sites to
target specific steps in the reaction pathway. Each mutation would
likely change the magnitude of several rate constants in Scheme 2,
and these changes would be evidenced as changes in the magni-
tudes of the lumped kinetics parameters that control the shape
of the pNP-reporter versus time curves [cf. Eqs. 3a–d and Table 1].
Consider, for example, the +3 subsite in Scheme 1. The inability
of G2 to inhibit transglycosylation (Fig. 3) suggests that the +3 sub-
site plays a pivotal role in the association of pNP-G2 in the glyco-
syl/enzyme intermediate prior to transglycosylation (Scheme 1). A
mutation at the +3 subsite that reduces the propensity of pNP-G2
substrate to bind to C2 in Scheme 2 would reduce k4, and a reduc-
4a and 4b for ST ; S ꢁ
x. Expanding the logarithmic terms to lead-
ing order in S=
x
and ST =
x
gives:
ꢀ
ꢁ
S
ꢀ
ꢀ
k
catET t ¼ ðST ꢀ SÞ ꢀ Km ln
;
ð7aÞ
ð7bÞ
ST
R ¼ ST ꢀ S;
where
kcat
Km
k3KÃ
m
~
ꢀ
ꢄ
ꢄ
ꢅ
ꢅ
kcat
¼
¼
¼
k2;
ð8aÞ
ð8bÞ
ðk3KÃ þ k2KÃ ꢀ k5KmÞ
~
m
m
1 ꢀ
x
tion in k4 relative to all other rate constants would decrease
Eq. 5), thereby reducing the enzyme’s ability to transglycosylate.
This particular mutation targeted k4, but Eq. 5 for can be used
x (cf.
Km
k3KÃ
~
m
ꢀ
Km
¼
Km:
ðk3KÃ þ k2KÃ ꢀ k5KmÞ
~
Km
m
m
x
1 ꢀ
x
more generally to identify which reaction rate constants in
Eq. 7a duplicates exactly the functional form of the integrated MM
equation40 but with effective constants kcat and Km. The effective
MM constants in this circumstance depend on the rate constants
for the entire reaction mechanism, even though products of the
transglycosylation pathway may not have been detected. This is be-
cause all intermediates in the mechanism are sampled regardless of
which pathway dominates the product distribution.
ꢀ
ꢀ
The preceding analysis reveals a general danger of investigating
the hydrolysis activity of enzymes by performing experiments for a
limited range of initial substrate concentrations. If one does not
realize that transglycosylation can occur and accidentally selects
values of ST that are small compared to
x for the system of interest,
one would mistakenly conclude that only hydrolysis can occur, and
that it occurs via a simple MM mechanism with a single interme-
ꢀ
ꢀ
diate complex, in which case kcat and Km would be functions of only
three elementary rate constants. In fact the mechanism may in-
ꢀ
ꢀ
volve three intermediate complexes, in which case kcat and Km
are the more complicated functions of elementary rate constants
given by Eqs. 8a and 8b.
Figure 4. Hydrolysis selectivity. Plot of Eq. 6, final fractional yield of hydrolysis
products f ¼ R =ST versus
x=ST .
1

J. W. Dingee, A. B. Anton / Carbohydrate Research 345 (2010) 2507–2515
2513
4.4. Kinetics model—comparison to pNP-reporter data
5. Conclusions
Data were collected in a region of parameter space where
, that is, where the transglycosylation yield varied consid-
Nitrophenyl glycosides are commonly used substrates for
studying the reactivity of retaining glycosyl hydrolases, yet the
deceptively simple kinetics of nitrophenol formation can be com-
plicated by concurrent transglycosylation pathways. Herein we
propose a two-pathway mechanism for the hydrolysis and trans-
glycosylation of pNP-G2 by Cel5Acd, and from it we develop a rate
expression for reporter formation. We provide quantitative justifi-
cation for using the QSSA for three intermediate complexes when
integrating the rate expression, and we fit the integral form of
the rate to pNP-versus-time data to extract values of several
lumped kinetics parameters. One parameter, which emerges from
the integration and is quantified very precisely by the curve-fits
we present, uniquely determines the variation of hydrolysis versus
transglycosylation selectivity as a function of initial substrate con-
centration. We demonstrate that when the substrate composition
is sufficiently low, the two-pathway mechanism predicts simple
MM kinetics for hydrolysis, even though three intermediate com-
plexes are involved in the reaction. Our kinetics model and mech-
anistic explanation should be useful for interpreting results for
other retaining glycosyl hydrolase systems where competing
hydrolysis and transglycosylation are evident.
ST
ffi
x
erably with ST. All 104 data points for reporter concentration ver-
sus time in Figure 1 (with their sample standard deviations)
were fit simultaneously with Eqs. 4a and 4b using a non-linear,
weighted least-squares approach, as described in the Section 2.
The best-fit values, standard deviations, and relative precisions
~
~
~
for the fit-parameters kcat, Km, Km;2, and
x are presented in Table 1.
~
We also calculate and present the dependent parameter kcat;2
cat=2x for comparison, since this parameter appears naturally in
¼
~
k
the differential form of the mass balance for RðtÞ, Eq. 2a, used by
others for initial-rate analysis in similar systems.7,8,19 The solid
lines in Figure 1 represent the kinetics-model solution with these
parameters. The fit to the data for ET = 1.4
lent, as evidenced by the fact that the fit-curves pass within the
error bars ðÆ1 Þ for all but three of the 52 data points. The same
fit appears less good but still convincing when compared to the
data for ET = 4.0 M in Figure 1b. The fit-curve for ST = 1.0 mM
lM in Figure 1a is excel-
r
l
lies below the data for t P 40 min, and the fit-curve for
ST = 2.5 mM passes above the data points for all times. These devi-
ations may be an evidence that the measurements in Figure 1b
include systematic errors that increasing ET to 4.0
into a range of parameter-space where the QSSA for complex C3
is beginning to fail (cf. Appendix, 3 >0 in Eq. A4c), or that other
lM moves one
Acknowledgments
e
mechanistic complications, unaccounted for in our model, enter
the picture at high ET and ST. (Transglycosylation of G3, e.g., see
Section 4.1.) We have no information to discriminate among these
possibilities.
This project was supported by the Initiative for Future Agricul-
ture and Food Systems Grant no. 2001-52104-11484 from the
USDA Cooperative State Research, Education, and Extension Ser-
vice. The authors thank Dr. David Wilson, his staff, and his students
at Cornell University for their assistance and advice throughout the
experimental portion of this study.
Irwin et al. investigated the hydrolysis of pNP-G2 by Cel5Acd
as part of a larger, comparative study of cellulase enzymes.13
They report an activity of 14.4
sured at t = 30 min with ST = 2.5 mM. For comparison, the data
for ET = 1.4
M in Figure 1a give an activity of 12:0 Æ 0:4 at
t = 30 min, and the data for ET = 4.0 in Figure 1b give
15:7 Æ 1:4. A small deviation from Irwin’s result is not surpris-
ing, since t = 30 min penetrates the portion of the curves where
the hydrolysis yield varies nonlinearly with ET and ST (vide
infra).
lmol G2/min/lmol protein, mea-
Appendix
l
l
M
The QSSA is a versatile method for finding approximate solu-
tions to the mass balances for some kinetics problems.37,38 The
QSSA assumes that in a specific region of parameter space and
time, the rates of formation and destruction of a particular chem-
ical species are effectively equal, such that the time derivative of
its concentration is nearly zero. This simplifies that mathematical
problem at hand by reducing an ODE to an algebraic equation,
but at the expense of completeness, because when a concentra-
tion-derivative is lost, the solution of the remaining equations can-
not meet the initial condition for the species in question.
It is often assumed that the QSSA is valid for any intermediate,
that is, any species that is produced after a reaction starts and dis-
appears before the reaction finishes. Indeed, several analyses of
transglycosylation kinetics that precede this one present rate func-
tions that were derived with multiple QSSAs without pausing to
consider their validity.7,8,19 The QSSA is valid for intermediate com-
plexes in many but certainly not all enzymatic biochemical reac-
Eq. 4a reveals that the long-time amplitude of the RðtÞ curves
is a function of the single parameter
x. The copious long-time
data in the right panel of Figure 1 ensure that
x can be specified
to very high precision ( 2.3%, cf. Table 1) by curve-fits to the data
~
~
~
~
of Figure 1. Conversely, the parameters kcat, kcat;2, Km, and Km;2
cannot be determined with rewardingly high precision ( 20–
50%, cf. Table 1). These parameters only affect the shape of RðtÞ
where it rises from zero and bends toward its long-time, stea-
dy-state value, and there are fewer data points in this regime.
Extracting more parameters from fewer data ensures lower statis-
tical precision.
The most important quantitative result of the integral curve-
fits is the value of the selectivity-determining parameter
x
¼
tions, so one should be wary. Here we outline
a rigorous
1:30 Æ 0:03 mM, since it can be used in Eq. 6 to predict the final
product distribution as a function of initial substrate concentra-
tion. For example, reducing the substrate concentration to
ST = 0.10 mM would give f ¼ 0:98, that is, almost pure hydrolysis
procedure for re-scaling Eqs. 1a–e to delimit the validity of the
QSSA for the complexes in the reaction mechanism we propose,
and we use the QSSA to derive the mathematical model for hydro-
lysis kinetics we presented earlier as Eqs. 4a and 4b. We demon-
strate the scaling method not only to support our own work, but
with a hope that others will use the QSSA for biochemical kinetics
cautiously and pause to investigate whether it is valid.
Bowne, Acrivos, and Oppenheim were the first to recognize the
analogy between the QSSA of chemical kinetics and the ‘singular
perturbation theory’ of differential equations.41 This analogy pro-
vides a rigorous prescription for identifying when a QSSA is valid.42
One must first re-scale the problem carefully to a dimensionless
products, and since ST =
tions, the pNP-versus-time curves would evidence simple MM
x
¼ 0:10=1:30 ffi 0:08 under these condi-
kinetics according to Eqs. 7a and 7b. The apparent catalytic rate
constant and MM constant would be kcat ¼ 65:5 minꢀ1 and
ꢀ
ꢀ
Km ¼ 3:08 mM according to Eqs. 8a and 8b. Conversely, very high
substrate concentrations are required to achieve a near-maximum
yield of transglycosylation products for this system; for example,
ST ffi 50 mM is required to achieve f ffi 0:55.

2514
J. W. Dingee, A. B. Anton / Carbohydrate Research 345 (2010) 2507–2515
form where all variables are O(1) [O(x) = ‘order of x’ in an asymp-
totic sense]. When this is accomplished, the dimensionless coeffi-
cient in front of each term quantifies its relative magnitude; the
QSSA is valid when the coefficient of a properly scaled time deriv-
ative is small compared to one.39
First we scale the dependent variables such that their new,
dimensionless representations are O(1) for all time. In this case,
the correct scale for R and S is clearly ST. To determine the scaling
factor bi for each complex Ci, assume that the enzyme achieves
equilibrium between all three complex states before an apprecia-
quantify the magnitudes of the derivative-terms on the left-
hand-side of Eqs. A3a–c and therefore determine which if any
QSSAs are valid.
T
By inspection of A4a and A4b, e1
;
e2 6 ES
,
and since
T
ffi Oð10ꢀ3Þ in our experiments, the QSSA is assuredly valid for
ET
ST
C1 and C2 in this analysis. However, Eq. A4c reveals that e3 ꢁ 1,
that is, the QSSA is valid for C3, is not ensured by excess substrate
K
alone. Note that e3 is the product of three terms: þK, which is at
ST
ꢀ3
ET
ET
k2
most Oð1Þ;
, which is at most OðS Þ ffi Oð10 Þ; and
,
ST þKm
k5þk
T
which is unbounded. Because the second term is small, one onꢀl4y
ble amount of product is formed or substrate is consumed. Return-
needs the third term to be Oð1Þ or less for the QSSA in C3 to be as-
dCi
ing to Eqs. 1a–c, let Ci = bi,
¼ 0, and S = ST, and then solve the
suredly valid; for instance, this is achieved if k2 6 k5 or k2 6 k .
ꢀ4
coupled algebraic equations dtto find:
If e3 is not small like e1 and e2 certainly are, how would this be
evident in the pNP-reporter data? One can show by manipulating
KST
b1 ¼
b2 ¼
ET ;
ET ;
ðA1aÞ
ðA1bÞ
the scaled mass balances A3a–e with e1
¼ e2 ¼ 0 (or can infer from
ðST þ KmÞðST þ KÞ
the reaction mechanism, Scheme 2) that increasing e3 from zero
delays the formation of complex C3 and increases the initial rate
of pNP formation by hydrolysis at the expense of pNP-G1 forma-
ST KÃ
m
ðST þ KÃ ÞðST þ KÞ
m
ST2
tion by transglycosylation. After a (dimensionless) time of Oðe3
Þ
b3 ¼
ET :
ðA1cÞ
ðST þ KÃ ÞðST þ KÞ
passes, complex C3 achieves a quasi-steady-state, and this effect
decays away. This would be evident as a temporary ‘‘bulge” in
the pNP-reporter curves of Figure 1 at short times that disappears
at longer times. Seeing no such feature in the data, only a monaton-
ic rise of the pNP-reporter curves in every case, we conclude that
e3 ¼ 0 is a reliable approximation here.
m
ꢄ
ꢅꢄ
ꢅ
Ã
ST þKm
k3
k2
k5
k2
In these equations K ¼
Km
þ
ST . Introduce dimension-
ST þKÃm
less representations of the dependent variables in 1a–e by substi-
Ci
tuting Xi ¼ b , Z ¼ S , and Y ¼ R , and then rearrange the constants
so all the terims onSTthe right hand side of each equation are dimen-
sionless and O(1). The re-scaled representation of Eq. 1e for reporter
formation is of most interest to us, since R(t) is the species we track
in our measurements. That equation becomes
ST
We invoke the QSSA for all three complexes in the dimension-
less equations by assuming e1
¼
e2
¼ e3 ¼ 0, which converts Eqs.
A3a–c from ODEs to algebraic equations. Solving A3a–c for X1
and X3 as functions of Z and substituting into A3d and A3e gives
the following:
ꢀ
ꢁ
ST
dY
¼ X1:
ðA2Þ
k2b1 dt
dZ
dT
c1Z þ 2c2Z2
The coefficient on the left-hand side has dimensions of time and
represents the timescale for reporter formation. Define a scaled
¼ ꢀ
¼
;
ðA5aÞ
ðA5bÞ
c3 þ c4Z þ c5Z2
c1Z þ c2Z2
ST
2b1
t
dY
dT
time variable T ¼ with tR
¼
, and recast all five ODEs in terms
tR
k
:
c3 þ c4Z þ c5Z2
of T. One finds:
ꢀ
ꢀ
ꢀ
ꢁ
ꢁ
ꢁ
dX1
dT
dX2
dT
dX3
dT
The scaled versions of the initial conditions for these equations are
e1
e2
e3
¼ Z ꢀ d1X1 ꢀ c1X1Z ꢀ c2X2Z ꢀ c3X3Z;
ðA3aÞ
ðA3bÞ
ðA3cÞ
Sðt¼0Þ
Rð0Þ
ZðT ¼ 0Þ ¼
¼ 1 and Yð0Þ ¼
¼ 0, and the new constants are
ST
ST
ꢄ
ꢅ
ꢀ
ꢁ
k
3b2
2b1
k
5b3
2b1
k3
K
mb2
b2
k5
Kmb3
k3 þ k4ST Z
k3 þ k4ST
c1
¼
¼
,
c2
¼
,
c3
¼
,
c4
¼
þ k2 þ k3
,
and
k
k
k2ST ET
k2ET
b2ST
¼ d2X1 ꢀ
X2 þ d3X3;
ꢄ
ꢅ
k5b3
1
k2
1
c5
þ k . Eq. A5a can be integrated to give T(Z), and dividing
ET
5
¼ X2Z ꢀ X3;
A5b by A5a to eliminate dT gives an equation that can be integrated
to find Y(Z). Performing these operations and returning the results
to dimensioned variables give Eqs. 4a and 4b, presented earlier in
this paper.
dZ
¼ ꢀX1 ꢀ d4X3;
¼ X1:
ðA3dÞ
ðA3eÞ
dT
dY
dT
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