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8/6/2019 2007 A4 HPLC Assay Anal Biochem
1/7
ANALYTICAL
BIOCHEMISTRY
Analytical Biochemistry 363 (2007) 128134
www.elsevier.com/locate/yabio
0003-2697/$ - see front matter 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.ab.2006.12.044
Natural substrate assay for chitinases using high-performance liquidchromatography: A comparison with existing assays
Inger-Mari Krokeide, Bjrnar Synstad, Sigrid Gseidnes, Svein J. Horn,Vincent G.H. Eijsink, Morten Srlie
Department of Chemistry, Biotechnology and Food Science, Norwegian University of Life Sciences, P.O. Box 5003, N-1432 Aas, Norway
Received 7 November 2006
Available online 4 January 2007
Abstract
The determination of kinetic parameters of chitinases using natural substrates is diYcult due to low Km values, which require the use
of low substrate concentrations that are hard to measure. Using the natural substrate (GlcNAc)4, we have developed an assay for the
determination ofkcat and Kmvalues of chitinases. Product concentrations as low as 0.5 M were detected using normal-phase high-perfor-
mance liquid chromatography (HPLC) with an amide 80 column (0.20 25 cm) using spectrophotometric detection at 210nm. By means
of this assay, kcat and Kmvalues for chitinases A (ChiA) and B (ChiB) ofSerratia marcescens were found to be 331 s1 and 91 M and
282 s1 and 4 2M, respectively. For ChiB, these values were compared to those found with commonly used substrates where the
leaving group is a (nonnatural) chromophore, revealing considerable diVerences. For example, assays with 4-methylumbelliferyl-(Glc-
NAc)2 yielded a kcat value of 18 2 s1 and a Km value of 306 M. For two ChiB mutants containing a Trp ! Ala mutation in the +1
or +2 subsites, the natural substrate and the 4-methylumbelliferyl-(GlcNAc)2 assays yielded rather similar Km values (5-fold diVerence at
most) but showed dramatic diVerences in kcat values (up to 90-fold). These results illustrate the risk of using artiWcial substrates for char-
acterization of chitinases and, thus, show that the new HPLC-based assay is a valuable tool for future chitinase research. 2007 Elsevier Inc. All rights reserved.
Keywords: Enzymatic assay; Chitinase; Natural substrate; HPLC
Chitin, a -1,4-linked polymer of N-acetylglucosamine
(GlcNAc), is an abundant biopolymer in nature. It is the
most important nonplant structural biopolymer, occurring
in, e.g., the exoskeletons of invertebrates, the cell walls of
fungi, and the digestive tracts of insects. Chitin is easily
derived from waste products such as shrimp shells. So far,
chitin is primarily used as a source for chitosan, a partiallydeacetylated soluble form of chitin, and for glucosamine.
Fragments of chitin or chitosan (chitooligosaccharides)
may inhibit certain chitin-degrading enzymes, giving them
potential as fungicides [1], insecticides [24], and antimala-
rials [5,6]. Chitooligosaccharides are environmentally
friendly because of their fast degradation in nature and,
since chitin does not occur in humans, chitin metabolism is
an interesting target area for development of drugs and
pesticides.
Chitin does not accumulate in nature because the poly-
mer is eVectively degraded by diVerent chitinases belonging
to the glycoside hydrolase enzyme families 18 and 19 [7].
Serratia marcescens has an eYcient chitinolytic machinery
and, when grown on chitin, three family 18 chitinases areexpressed: chitinases A (ChiA), B (ChiB), and C (ChiC) [8].
ChiA and ChiB are processive chitinases that digest the
chitin polymer in opposite directions producing mainly
(GlcNAc)2, while ChiC is a nonprocessive endochitinase
that hydrolyzes the polymer randomly, yielding longer chi-
tooligosaccharides [9,10]. The S. marcescens chitinases have
been characterized in several studies [811], using a variety
of substrates.
Kinetic analysis of chitinases is usually conducted with
artiWcial substrates such as 4-methylumbelliferyl-(GlcNAc)2
* Corresponding author. Fax: +47 64965901.
E-mail address:[email protected] (M. Srlie).
mailto:%[email protected]:%[email protected]:%[email protected]8/6/2019 2007 A4 HPLC Assay Anal Biochem
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Natural substrate assay for chitinases using HPLC / I.-M. Krokeide et al. / Anal. Biochem. 363 (2007) 128134 129
((GlcNAc)2-4MU), (GlcNAc)3-4MU, and p-nitrofenyl-
(GlcNAc)2 ((GlcNAc)2-pNP). These substrates are not
optimal because they present the enzyme with nonnatural
leaving groups and because of substrate inhibition [12].
Because of the length and the nature of the leaving groups,
non-natural substrates are of very limited use when assess-
ing the eVects of mutations meant to aVect natural sub-strate degradation. In addition, the leaving groups 4-MU
and pNP are corresponding bases to weak acids with pKavalues of 7.8 [13] and 7.2 [14], respectively, while acid disso-
ciation constants for sugars areV 12 [15]. At high pH (Glc-
NAc)2-4MU and (GlcNAc)2-pNP yield negatively charged
leaving groups while this will not be the case for a sugar
leaving group. Kinetic characterization with natural sub-
strates is diYcult because (1) low Km values require detec-
tion of low concentrations of substrate and product, (2)
most longer substrates contain more than one hydrolysable
glycosidic bond, (3) degradation products longer than
dimers are also substrates, and (4) some chitinases degrade
longer substrates processively. With respect to the latter
two problems, (GlcNAc)4 is an exception since most chitin-
ases convert this compound exclusively to two dimers,
which are not degraded any further.
We present a chitinase assay based on the use of the natu-
ral substrate (GlcNAc)4 and the use of a sensitive HPLC
setup to monitor substrate and product concentrations. We
have compared the natural substrate assay with existing
assays using wild-type ChiA and ChiB from S. marcescens
and two engineered variants of ChiB. The engineered enzymes
are ChiB-W97A and ChiB-W220A. Trp97 is located in the +1
subsite of the active site of ChiB while Trp220 is located in the
+2 subsite [16,17]. Mutation of these residues will thus aVectthe enzymes interaction with the leaving group.
Materials and methods
Chemicals
Tetra-N-acetylchitotetraose, 4-methylumbelliferyl-di-N-
acetylchitobiose, 4-methylumbelliferyl-tri-N-acetylchitotrii-
ose, para-nitrophenyl-di-N-acetylchitobiose, and acetonitrile
were purchased from SigmaAldrich.
Production and puriWcation of chitinases
ChiA [18], ChiB [19], and mutants of ChiB were puriWed
from periplasmatic extracts of the producer strains by
hydrophobic interaction chromatography, as described
previously [12]. Enzyme purity was veriWed using SDS/
PAGE and was above 95% in all cases. Protein concentra-
tions were determined using the Bradford assay kit
provided by Bio-Rad (Hercules, CA, USA).
Chromatography of chitooligosaccharides
Mixtures of (GlcNAc)4 and (GlcNAc)2 were separated
by normal-phase HPLC using a Tosoh TSK Amide 80
column (0.2025 cm) with an amide 80 guard column. The
sample size was 50L and the chitooligosaccharides were
eluted isocratically at 0.25mL/min with 70% (v/v) acetoni-
trile at room temperature. The chitooligosaccharides were
monitored by measuring absorbance at 210nm and the
(GlcNAc)4 concentrations were quantiWed by measuring
peak areas and by comparing these to those of standardsamples with known concentrations of (GlcNAc)4. Using
these standard samples, it was established that there was a
linear correlation between the peak area and the analyzed
(GlcNAc)4 concentration within the concentration range
0.5300M used in this study.
(GlcNAc)4 assay
Reactions were started by adding 0.5nM ChiA or ChiB
or 1.0 nM ChiB-W220A or 2.0nM ChiB-W97A (Wnal con-
centrations) to 1-mL solutions containing 100, 80, 60, 40,
20, 10, 5, or 2M (for ChiA and ChiB), 500, 400, 300, 200,
100, 50, or 25 M (for ChiBW220A), and 1000, 600, 400,
200, 100, 50, or 25 M (for ChiB-W97A) (GlcNAc)4 in
20 mM NaAc buVer, pH 6.1, and 0.1 mg/mL bovine serum
albumin (Wnal concentrations). After enzyme addition, 50-
L aliquots were transferred to a HPLC vial containing
150L of acetonitrile at appropriate time points. Reaction
conditions and timing were such that the (GlcNAc)4 con-
centration in the sample would not go below 80% of the
starting concentration. An aliquot taken before enzyme
addition was used as the tD 0 sample. The slopes of the
plots of substrate concentrations vs time were taken as the
hydrolysis rate. Then the hydrolysis rates were plotted vs
the substrate concentration in a MichaelisMenten plotand the experimental data were Wtted to the Michaelis
Menten equation using nonlinear Wtting in Origin 7. Alter-
natively, substrate concentrations divided by the hydrolysis
rates were plotted vs substrate concentrations in a Hanes
plot using a standard spreadsheet program to obtain kcatand Km values. The derived values using the two diVerent
approaches were the same within experimental errors.
Assays with nonnatural substrates
The kinetic parameters of ChiB variants for the (Glc-
NAc)2-4MU substrate at pH 6.1 were determined as thor-oughly described elsewhere [12,20] with an enzyme
concentration of 0.2nM and substrate concentrations in the
5 to 50-M range. The kinetic parameters of wild-type
ChiB toward (GlcNAc)2-pNP were determined in the same
way, except that the substrate concentration range was
adapted to the much higher Km for this substrate. ChiB
converts the (GlcNAc)3-4MU substrate exclusively to (Glc-
NAc)2 and (nondetectable) GlcNAc-4MU. The kinetic
parameters of wild-type ChiB toward the (GlcNAc)3-4MU
substrate were determined using a two-step assay based on
detection of substrate depletion, a ChiB concentration of
0.2 nM, and substrate concentrations in the 1 to 10-M
range [21]. In short, reactions were conducted as usual but,
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130 Natural substrate assay for chitinases using HPLC / I.-M. Krokeide et al. / Anal. Biochem. 363 (2007) 128134
before adding the 0.2M Na2CO3 stop solution, the reaction
mixtures were incubated for less than 1 min with a 300-fold
excess of ChiA to convert all remaining substrate to
(detectable) free 4MU. In all cases, the substrate concentra-
tions divided by the hydrolysis rates were plotted vs sub-
strate concentration in Hanes plots using a standard
spreadsheet program to obtain kcat and Km values.
Results
(GlcNAc)4 as substrate
(GlcNAc)4 productively binds in subsites 2 to +2 of
ChiA and ChiB of S. marcescens, yielding two (GlcNAc)2molecules as products that are not subject to further hydro-
lysis [8,12,21]. Longer substrates have several productive
binding modes and yield products that in turn are sub-
strates for the enzymes, complicating kinetic analysis of the
overall reaction. Productive binding of (GlcNAc)3 yields
only products that are not further hydrolyzed, (GlcNAc)2and GlcNAc. However, this substrate is less suitable than
(GlcNAc)4 because it occupies fewer subsites and because
there are two productive binding modes, at least in some
chitinases such as ChiA (binding in 2 to +1 and in 1
to +2) [21]).
HPLC analysis of chitooligosaccharides
Normal-phase HPLC-based quantitative analysis of sac-
charides with an amide 80 column has been used previously
[2124] but, so far, the use of this method for determination
of kinetic parameters (implying measurement of very lowconcentrations of sugar) has not been reported. In the pres-
ent study, the necessary increase in sensitivity was achieved
through a combination of adjustments. First, we used a col-
umn with a considerably reduced diameter (0.20cm vs
0.48 cm used in previous studies). A smaller column diame-
ter yields a smaller degree of sample dilution compared to a
larger column diameter and thus results in sharper peaks.
Second, injection volumes were 50 L. Since water elutes
the chitooligosaccharides, the samples were diluted to 75%
(v/v) acetonitrile before application to allow for retardation
on the column of the chitiooligosaccharides before elution
with 70% (v/v) acetonitrile. Together these adjustments per-mitted detection of (GlcNAc)4 concentrations down to
0.5M. Fig. 1 shows the HPLC chromatograms resulting
from enzymatic hydrolysis of (GlcNAc)4 by ChiB.
Determination of kcat, Km, and eYciency constants (kcat/Km)
for wild-type ChiB with diVerent substrates
Enzymatic hydrolysis of (GlcNAc)4 by ChiB follows
MichaelisMenten kinetics as depicted by Fig. 2A. Kinetic
data for hydrolysis of (GlcNAc)4 and three diVerent artiW-
cial substrates are shown in Table 1. For (GlcNAc)4 hydro-
lysis, the nonlinear Wt of theoretical data to experimental
data yielded kcat and Km values of 282 s1 and 4 2M,
respectively. These values diVer considerably from values
obtained with the often-used substrate (GlcNAc)2-4MU,
which are 18 2 s1 and 316M, respectively [20]. Even
larger diVerences were observed with the (GlcNAc)2-pNP
substrate which yielded kcat and Km values of 1.40.5s-1
and 18135M, respectively. For (GlcNAc)3-4MU (which
is converted to (GlcNAc)2 and GlcNAc-4MU), the kcatvalue is 574 s1 and the Km value is 71M and both are
about two times larger than those for (GlcNAc)4.
The diVerences between the substrates are most apparent
when using eYciency constants (kcat/Km; Table 1). ChiB has
a 700 times greater eYciency toward (GlcNAc)4 compared
to (GlcNAc)2-pNP and a 12 times higher eYciency
compared to (GlcNAc)2-4MU. Results obtained with the
(GlcNAc)3-4MU substrate were similar to those obtained
with (GlcNAc)4.
Determination of kinetic parameters for ChiA with
(GlcNAc)4
The (GlcNAc)2-4MU susbtrate is often used because its
product is easy to detect at low concentrations and because
it tends to display seemingly natural Km values, i.e., in the
lower micromolar range. However, for some enzymes with
extended substrate binding clefts this short substrate may
be suboptimal because of the occurrence of multiple bind-
ing modes and cooperativity between these. This is for
example observed for ChiA [12]. With the (GlcNAc)4 sub-
strate, ChiA displayed normal MichaelisMenten kinetics
(Fig. 2B), yielding kcat and Km values of 331 s1 and
91M, respectively (Table 2).
Fig. 1. HPLC analysis of (GlcNAc)4 hydrolysis by ChiB. The starting con-
centration of (GlcNAc)4 was 40 M, and chromatograms obtained after
hydrolysis of 0% (bottom), approximately 10% (middle), and 20% (top) of
the substrate are shown. Both the tetramer and the dimer peak are split
into two peaks due to the anomer equilibrium of the chitooligosaccha-
rides. The anomers (approximately 60%) elute before the anomers
(approximately 40%; see, e.g., [21]). Due to the small diameter of the col-
umn and the relatively large sample volume, the anomeric forms are only
partly separated. The Wrst 8 min of the chromatograms were omitted for
clarity.
8 10 12 14 16 18 20
0
5
10
15
20
25
Absorbance/mV
Time /min
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Natural substrate assay for chitinases using HPLC / I.-M. Krokeide et al. / Anal. Biochem. 363 (2007) 128134 131
Determination of kcat and Km values for engineered chitinases
using (GlcNAc)4 and (GlcNAc)2-4MU as substrates
Activity assays based on natural substrates are particu-larly important for the analysis of the properties of engi-
neered enzymes carrying site-directed mutations near the
catalytic center. To analyze and illustrate this, we have char-
acterized two ChiB variants with mutations in subsite +1
(W97A) and +2 (W220A) (Fig. 3) with both (GlcNAc)4 and
(GlcNAc)2-4MU (Table 2). When (GlcNAc)4 was used, kcatand Km values of 126 2 s
1 and 807 2M were obtained
for ChiB-W97A, while the same values were 81 s1 and
17557M for the (GlcNAc)2-4MU substrate. For ChiB-
W220A, the kcat and Km values were 451 s1 and
71 1M and 0.50.1s1 and 824M when using (Glc-
NAc)4 and (GlcNAc)2-4MU, respectively. Interestingly,while the diVerence between the substrates is modest with
respect to Km values, it is as large as 90-fold with respect to
kcat values.
Discussion
Comparison of the various substrates
Up to now, (GlcNAc)2-4MU, which produces the
Xuorescing 4MU leaving group upon hydrolysis, has been
the most used substrate in chitinase assays [12,20,22,25].
Obvious advantages of the use of (GlcNAc)2-4MU are the
time window and simplicity of the assay and its sensitivity.
Product formation is readily and quickly detected using
Xuorescence spectroscopy. There are also several and seri-
ous disadvantages such as substrate inhibition, a nonnatu-
ral leaving group, and nonlinear kinetics, [12,22,26]. Analternative for (GlcNAc)2-4MU is (GlcNAc)2-pNP, which
yields the yellow chromophore pNP. This substrate has the
same disadvantages as (GlcNAc)2-4MU. The data ofTable 1
show that the kinetic parameters determined with
(GlcNAc)2-pNP are very diVerent from the presumably
more realistic parameters obtained with (GlcNAc)4.
Another alternative is (GlcNAc)3-4MU, which may be
converted either to (GlcNAc)2 and GlcNAc-4MU (as does
ChiB) or to (GlcNAc)3 and 4MU (as does ChiA) [12]. In the
latter case, product formation can readily be determined,
but the method has most of the disadvantages described
above for (GlcNAc)2-4MU. For ChiB, we have been able to
develop a method for measuring the conversion of (Glc-
NAc)3-4MU to (GlcNAc)2 and GlcNAc-4MU under con-
ditions that permit determination of kinetic parameters
[21]. The method is based on measuring product disappear-
ance, which is achieved by converting remaining product
quantitatively to 4MU through a short incubation with a
large excess of ChiA. This assay has the advantage that the
+1 subsite is occupied by a natural leaving group (i.e., a
sugar unit). Not unexpectedly, the results obtained with
(GlcNAc)3-4MU for wild-type ChiB resembled those
obtained with (GlcNAc)4. A serious disadvantage of this
assay is the need to purify two enzymes, especially large
amounts of ChiA, which is time consuming, and it works
Fig. 2. MichaelisMenten plots from HPLC analyses of (GlcNAc)4 hydrolysis with ChiB (A), ChiA (B), and the engineered enzymes ChiB-W220A (C) and
ChiB-W97A (D) at pH 6.1, 37 C. Solid symbols are experimentally determined data for the rate of substrate disappearance vs substrate concentrations
and the solid lines are the best nonlinear Wt using the MichaelisMenten equation.
0 20 40 60 80 100
0,000
0,005
0,010
0,015
0,020
R
ate/(M/s)
(GlcNAc)4/M
0 100 200 300 400 500
0,00
0,01
0,02
0,03
0,04
(GlcNAc)4/M
R
ate/(M/s)
0 20 40 60 80 100
0,000
0,005
0,010
0,015
0,020
Rate/(M/s)
(GlcNAc)4/M
0 200 400 600 800 1000 1200
0.00
0.05
0.10
0.15
Rate/(M/s)
(GlcNAc)4/M
A C
DB
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132 Natural substrate assay for chitinases using HPLC / I.-M. Krokeide et al. / Anal. Biochem. 363 (2007) 128134
only for enzymes that exclusively convert (GlcNAc)3-4MU
to (GlcNAc)2 and GlcNAc-4MU.
Although the rate-limiting step in the reaction mecha-
nism of family 18 chitinases is not known [17,20,27], it is
clear that a change in chemical properties of the leaving
group may aVect catalytic eYciency with regard to both the
pKa and the ability to interact with the enzyme. At pH 6.1,
about 2% of 4-methylumbelliferol and about 7% ofpara-
nitrophenol will be negatively charged while virtually all of
the (GlcNAc)2 and GlcNAc-4MU leaving groups will be
protonated and free of charge. It is interesting to note that
the substrate yielding the by far lowest kcat value ((Glc-
NAc)2-pNP) has the most acidic and the smallest leaving
group. ChiB is more eVective toward (GlcNAc)2-4MU
whose leaving group is slightly less acidic and considerably
larger. Several studies have shown that enzymesubstrate
interactions in subsites +1 and +2 are important for
catalytic eYciency [17,22,28,29]. The present data do not
permit discrimination between the roles of the acidity and
Ta le 1
kcat and Km of ChiB from Serratia marcescens for substrates with diVerent leaving groups, at 37 C, pH 6.1
a Ref[21].b Ref[20].
Leaving group (RH) kcat (s1) Km (M) kcat/Km (s
1 M1)
Di-N-acetylchitobiose
28 2 4 2 7
4-Methylumbelliferyl-N-acetylglucosamine
57 4a 7 1a 8
4-Methylumbelliferol
18 2b 31 6b 0.6
ara-Nitrophenol
1.4 0.5 181 35 0.01
O
ROH
OH
NH
O
OOH O
OH
OH
NH
O
O
OHOH
OH
NH
O
OOH O
OH
OH
NH
O
+ RHH2O
O
OH
OH
NH
O
OO
OH
OH
NH
O
OHOH
O
OO
OH
OH
NH
O
OH
O
O OOH
OH
NO
O
Ta le 2
Kinetic parameters for engineered and natural chitinases from Serratia
marcescens at 37C, pH 6.1, using (GlcNAc)4 and 4MU-(GlcNAc)2 as
substrates
a (s1).b (M).c (s1M1).d The combination of sigmoidal behavior and substrate inhibition
precludes determination of kinetic parameters [12].
Chitinase (GlcNAc)4 (GlcNAc)2-4MU
kcata Km
b kcat/Kmc kcat
a Kmb kcat/Km
c
ChiB 28 2 4 2 7 18 2 31 6 0.6
ChiA 33 1 9 1 4 n.d.d n.d.d n.d.d
ChiB-W97A 126 4 807 40 0.2 8 1 175 57 0.03
ChiB-W220A 45 2 71 3 0.6 0.5 0.1 82 4 0.01
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Natural substrate assay for chitinases using HPLC / I.-M. Krokeide et al. / Anal. Biochem. 363 (2007) 128134 133
the interaction potential of the leaving group. The data in
Table 1 do suggest though that both the occupancy of the +1
and +2 subsites and a high pKa of the leaving group are
beneWcial for catalytic eYciency because the longer, sugar-
like substrates yielded lower Km and higher kcat values.
The present data show that the use of (GlcNAc)4 to cir-
cumvent the problems connected to other substrates is fully
feasible. Both ChiA and ChiB showed straightforward
MichaelisMenten kinetics without substrate inhibition
and the range of accessible substrate concentrations was
such that kinetic parameters could be determined with rea-sonable accuracy. The results show that ChiA and ChiB
have similar activities toward soluble substrates, in line
with the rather similar architectures of their active sites and
substrate-binding grooves [11,16,17,30]).
Analysis of mutants
The W97A and W220A mutants displayed large
increases in Km with both substrates, as might be expected
upon mutating important interaction partners in subsites
that are known to make a dominant positive contribution
to ligand binding. The increase in Km was larger for the nat-ural substrate (GlcNAc)4, presumably because the wild-
type subsites are optimized for sugar binding rather than
binding of the 4MU group. Not unexpectedly, mutation of
Trp220, which primarily aVects subsite +2, had only a mod-
est eVect on the Km for (GlcNAc)2-4MU.
Mutational eVects on kcat showed a dramatic dependence
on the substrate used (Table 2). With the natural substrate,
both mutations lead to an increase in kcat (4.5 and 1.6 times
higher for ChiB-W97A and W220A, respectively) whereas
the kcat is reduced from 18 to 8s1 for ChiB-W97A and to
0.5s1 for ChiB-W220A with (GlcNAc)2-4MU. For the
ChiB-W220A mutant, the two substrates yield a 90-fold
diVerence in kcat and a 60-fold diVerence in the eYciency
constant kcat/Km. Apart from illustrating the diVerences
between the substrates, these observations also lead to ques-
tions concerning the mechanism and rate-limiting step of
catalysis. For example, the dramatic eVect of the ChiB-
W220A mutation on the kcat obtained with (GlcNAc)2-
4MU must mean that Trp220 is involved in catalysis, even in
the case of short substrates. It is not known how the 4MUgroup binds to the enzyme, but it is clear that the group is
large enough (i.e., larger than a single sugar) to interact with
parts of the +2 subsite, including Trp220. Another puzzling
issue concerns the fact that mutation of the two tryptophans
increased the kcat for the natural substrate. Interestingly,
Watanabe et al. [31] found that mutation of analogous try-
ptophans in chitinase A1 from Bacillus circulans led to
increased speciWc activity toward chitopentaose. Further
kinetic and mutational studies are necessary to Wnd possible
explanations for these observations [32].
In conclusion, the results obtained with wild-type ChiB
and, particularly, with the two mutants clearly show that
the use of easy-to-handle artiWcial substrates for detailed
characterization of family 18 chitinases may lead to wrong
conclusions. Thus, such characterization should be based
on a natural substrate assay, such as the one described here.
Acknowledgments
We are grateful for the help of Dr. Gustav Vaaje-
Kolstad for the making of Fig. 3. Part of this work was
funded by The Norwegian Research Council, Grants
140497 and 140440.
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