Modelling the kinetics of enzyme-catalysed reactions in frozen systems: the alkaline phosphatase catalysed hydrolysis of di-sodium-p-nitrophenyl phosphate

www.elsevier.com/locate/ifset

Innovative Food Science and Emerging Technologies 5 (2004) 335–344

Modelling the kinetics of enzyme-catalysed reactions in frozen

systems: the alkaline phosphatase catalysed hydrolysis

of di-sodium-p-nitrophenyl phosphate

Netsanet Shiferaw Terefe, Ann Van Loey, Marc Hendrickx*

Department of Food and Microbial Technology, Laboratory of Food Technology, Faculty of Agricultural and Applied Biological Sciences,

Katholieke Universiteit Leuven, Kasteelpark Arenberg 22, Leuven B-3001, Belgium

Received 24 February 2003; accepted 9 May 2004

Abstract

The applicability of the William, Landel and Ferry (WLF) equation with a modification to take into account the effect of melt-dilution

and an empirical log-logistic equation were evaluated to model the kinetics of diffusion-controlled reactions in frozen systems. Kinetic

data for the alkaline phosphatase catalysed hydrolysis of disodium-p-nitrophenyl phosphate (DNPP) in four model systems with different

glass transition temperatures at maximal freeze-concentration (TgV) comprising sucrose (TgV=� 34.2 jC), maltodextrin (TgV=� 14.4 jC),carboxymethylcellulose (CMC), (TgV=� 12.6 jC) and CMC–lactose (TgV=� 23.1 jC) in a temperature range of � 28 to 0 jC were used.

The modified WLF equation was used with a concentration-dependent glass transition temperature (Tg) as well as TgV as reference

temperatures. For both cases, the equation described well the reaction kinetics in all the systems studied. The log-logistic equation also

described the kinetics in all model systems except in the vicinity of the melting temperature of ice. The effect of melt-dilution on reactant

concentration was found to be significant only in the dilute model systems near the melting temperature of ice.

D 2004 Elsevier Ltd. All rights reserved.

Keywords: Alkaline phosphatase; Glass transition; Freeze-concentration; Melt-dilution; Frozen systems; WLF

Industrial relevance: Significant deviation from the Arrhenius equation has been observed for the kinetics enzyme-catalysed reactions in the subfreezing range.

This work offers a clear insight into the kinetics of such enzyme controlled reactions in frozen systems suggesting changes in the conformation of the enzyme

studied. A potentially useful empirical equation is proposed for describing kinetics in frozen foods.

1. Introduction equation has been observed for the activities of enzymes

The kinetics of enzyme-catalysed reactions in frozen

systems has been studied quite extensively due to the impact

of enzyme activity on the quality evolution of biological

materials during frozen storage. During freezing, enzyme-

catalysed reactions in cellular systems accelerate or decrease

less than expected, due to freeze-induced enzyme delocal-

ization. This is less common for enzyme-catalysed reactions

in non-cellular systems as the rate-enhancing effect of

freeze-concentration is mostly overbalanced by the rate-

depressing effect of a freeze-induced increase in salt con-

centration with the accompanying change in pH and ionic

strength (Fennema, 1975a). A distinct depression in activity

that is more pronounced than that predicted by Arrhenius

1466-8564/$ - see front matter D 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ifset.2004.05.004

* Corresponding author. Tel.: +32-16-32-15-85; fax: +32-16-32-19-60.

E-mail address: [email protected] (M. Hendrickx).

such as invertase, proteinase and lipase, trypsin, alkaline

phosphatase and peroxidase in the subfreezing temperature

range in non-cellular systems (Fennema, 1975a; Lund,

Fennema, & Powrie, 1969; Maier, Tappel, & Volman,

1954; Mullenax & Lopez, 1975; Sizer & Josephson,

1942). In general, significant deviation from the Arrhenius

equation is observed for the kinetics of enzyme-catalysed

reactions in the subfreezing temperature range, even when

freezing does not occur (Fennema, 1975a). Among the

explanations presented for this deviation are: reversible

enzyme inhibition by one or more constituents of the

solution (Kistiakowsky & Lumry, 1949), low temperature

reversible denaturation of the enzyme due to increased

formation of intramolecular and/or intermolecular hydrogen

bonds (Hultin, 1955; Kavanau, 1950; Maier et al., 1954;

Tappel, 1966), change in viscosity, pH and salt concentra-

tion due to low temperature and/or freeze-concentration and

N. Shiferaw Terefe et al. / Innovative Food Science and Emerging Technologies 5 (2004) 335–344336

eutectic formation (Fennema, 1975a; McWeeny, 1968; Siz-

er, 1943; Tappel, 1966).

More recently, Slade and Levine (1991) proposed that the

kinetics of diffusion-limited processes including enzyme-

catalysed reactions in frozen systems is governed by the

physicochemical properties of the freeze-concentrated ma-

trix surrounding the ice crystals. According to these authors,

in the freeze-concentrated ‘rubbery’ matrix above the glass

transition temperature of the maximally freeze-concentrated

matrix TgV, the Arrhenius equation is not applicable. Rather

the kinetics is governed by the William, Landel and Ferry

(WLF) equation, which was originally developed to charac-

terise the influence of temperature on mechanical relaxation

processes in synthetic polymers (Williams, Landel, & Ferry,

1955). Simatos and Blond (1991) questioned this approach,

stressing the need to consider the effect of melt-dilution on

viscosity over and above the direct effect of temperature

according to the WLF equation as well as the opposing effect

of melt-dilution on reactant concentration. Contrary to the

suggestion of Slade and Levine (1991) that the reference

temperature in the WLF equation for frozen systems should

be TgV, Champion, Blond, and Simatos (1997) argued that the

continuous evolution of concentration as a function of

temperature in frozen systems makes the use of a constant

reference temperature like TgVinappropriate. Instead, they

proposed using the glass transition temperature of the un-

frozen matrix at its actual concentration as a reference

temperature. They used this approach to model the alkaline

phosphatase-catalysed hydrolysis of di-sodium-p-nitro-

phenyl phosphate (DNPP) in frozen sucrose solutions of

different concentrations (Champion, Blond et al., 1997).

Kerr and Reid (1994) studied the temperature dependence

of viscosity of frozen sugar and maltodextrin solutions and

found that both approaches yield good fit of the experimental

data. They noted that using TgVis more practical as it is easier

to measure and independent of concentration while using Tgcame close to forming a ‘‘universal’’ relationship for all the

solutions studied.

Peleg, Engel, Gonzalez-Martinez and Corradini (2002)

proposed an alternative log- logistic equation, which is

purely phenomenological, arguing that there are problems

with the use of both the Arrhenius and WLF equations.

According to these authors, this alternative equation is not

based on the assumption that there is a universal analogy

between totally unrelated systems and simple chemical

reactions, which is explicitly assumed when the Arrhenius

equation is used. It also has no special reference temper-

ature as in WLF equation, whose physical significance is

not always clear. The equation was used to model the

kinetics of enzymatic browning in dried foods and model

systems, the flowability of fructose and melted cheese and

the kinetics of microbial inactivation with satisfactory

results.

In this work, the use of the WLF equation (with both

TgVand temperature-dependent Tg as references) with a

modification to include the effect of melt-dilution on

reactant concentration and the equation of Peleg et al.

(2002) are evaluated in modelling the kinetics of the

alkaline phosphatase catalysed hydrolysis of DNPP in four

carbohydrate solutions. The alkaline phosphatase-catalysed

hydrolysis was demonstrated to be diffusion-controlled in

sucrose solution at 25 jC (Simpoulos & Jencks, 1994) and

has been analysed as such in a temperature range between

� 24 and 20 jC in concentrated sucrose solutions (Cham-

pion, Blond et al., 1997; Champion, Blond, Le Meste, &

Simatos, 2000).

2. Materials and methods

2.1. Experimental protocol for the kinetic study

The alkaline phosphatase assay used is based on the

FIL–IDF 82A (1987) procedure modified according to

Kerr, Lim, Reid, and Chen (1993). In this reaction, the

substrate DNPP is hydrolysed by alkaline phosphatase to

release p-nitrophenol (which is coloured) and phosphate

ions. The substrate solution consisted of 0.13 g/l DNPP

(Across Organics, New Jersey, USA) in a buffer com-

posed of 0.018 M NaHCO3 and 0.042 M Na2CO3 (pH

10.4). A 0.4 g/l enzyme solution was also prepared in the

same buffer solution. The alkaline phosphatase used was

of type I-S obtained from bovine intestinal mucosa

(Sigma, St. Louis, MO, USA) with an activity of 23 U/

mg. The enzymatic reaction was studied in four model

systems: 20% sucrose, 5% low viscosity carboxymethyl-

cellulose (CMC) (Sigma), 20% maltodextrin with dextrose

equivalent (DE) of 13–17 (Aldrich, Milwaukee, WI,

USA) and a 5% CMC–lactose mixture with 1:1 ratio.

Aliquots of 0.035 ml of enzyme solution were added to

microtest tubes containing 0.35 ml of the substrate and 0.35

ml of the food model solutions. The tubes were immediately

capped, mixed for 30 s, and frozen by immersion in liquid

nitrogen. The sample solutions were stored in cryostats

(HAAKE F3 and HAAKE F6, HAAKE Mess-Technik,

Karlsruhe, Germany) at various temperatures ranging from

� 28 to 20 jC. Samples were removed immediately after

temperature equilibration to serve as zero-time references

and at specified time intervals after storage between zero and

several days, depending on the storage temperature. Upon

removal from the cryostats, 0.35 ml of 28 g/l trichloroacetic

acid (TCA) was immediately added to the frozen sample to

inactivate the enzyme as thawing begins and therefore

preventing further hydrolysis. After the samples have thawed

in boiling water for 40 s, they were cooled in ice water. To

develop the colour from p-nitrophenol, the pH was raised by

adding 0.35 ml of 1.2 M NaOH solution. Absorbance was

measured at a wavelength of 425 nm using a spectropho-

tometer (LKB BIOCHROM, ULTROSPEC K) and con-

verted to concentration of p-nitrophenol using a standard

curve obtained with the model solutions. For each time/

temperature condition, samples were taken at least in tripli-

N. Shiferaw Terefe et al. / Innovative Food Science and Emerging Technologies 5 (2004) 335–344 337

cates and the average concentration was calculated and

plotted against time. The initial rate, V0, was taken as the

initial slope of the concentration versus time curve. In all

cases, SAS Statistical Software (SAS release 8.1, SAS

Institute, Cary, NC, USA) was used for linear and non-linear

regression analysis of the data.

2.2. WLF equation

2.2.1. Theoretical analysis

For the enzyme-catalysed reaction studied, Michaelis–

Menten kinetics (Eq. (1)) was assumed:

Eþ SWk2

k2ðE� SÞ!k3 P ð1Þ

where E, S, E–S and P represent enzyme, substrate,

enzyme–substrate complex and product, respectively, and

k1, k2 and k3 are reaction rate constants corresponding to the

enzyme–substrate association, the enzyme–substrate disso-

ciation and the transformation of the enzyme–substrate

complex into product. The reaction can be divided into

two steps: the first step is the diffusion of the reactants

toward each other and the formation of the enzyme–

substrate complex, which is controlled by the affinity

between the two molecules; the second step is the transfor-

mation of the enzyme–substrate complex into product

(Champion et al., 2000). The first step is influenced by

the translational mobility of the reactants. In viscous sys-

tems like the case being considered, transport processes are

very inefficient, thus this step can be rate controlling

(Bailey, North, & Pethrick, 1981).

Based on the steady state assumption, the initial rate can

be expressed as Eq. (2).

V0 ¼ K½E0�½S0� ð2Þ

where [E0] and [S0] are the initial enzyme and substrate

concentrations, respectively, and K is the overall rate con-

stant which can be expressed as Eq. (3).

K ¼ k3k1

k3 þ k2ð3Þ

When translational diffusion is rate controlling, the rate

of product formation is much faster than the rate of

dissociation of the enzyme– substrate complex (i.e.,

k3Hk2). In this case, the global rate constant approximates

to k1, which depends on the diffusion of the reactants

toward each other to form the enzyme–substrate complex.

Atkins (1995) analysed diffusion-controlled reactions and

related k1 to the diffusion coefficient of the two reacting

species according to Eq. (4).

K ¼ k1 ¼ 4pR*DNA ð4Þ

where R* is the collision diameter, D is the sum of the

diffusion coefficients of the two reacting species and NA is

Avogadro’s constant. Assuming that Stokes–Einstein equa-

tion (Eq. (5)) is applicable:

DE ¼ kT

6pgRE

and DS ¼ kT

6pgRS

ð5Þ

where k is the Boltzmann constant; DE, DS, RE and RS are

the diffusion coefficients and the hydrodynamic radii of

the enzyme and the substrate, respectively; and D is the

viscosity of the medium.

Substituting Eq. (5) in Eq. (4), and assuming that

RS =RE = 1/2R*, we obtain

K ¼ 8RT

3gð6Þ

where R is the universal gas constant. The substrate and the

enzyme are generally not of similar size in the case of

enzyme-catalysed reactions, i.e., RS =RE does not always

hold true. According to Atkins (1995), since Eq. (5) is

approximate, little extra error is introduced by assuming that

RS =RE = 1/2R*. Thus, the rate constant is independent of

the identity of the reactants and depends only on the

temperature and viscosity of the medium (Atkins, 1995).

The WLF equation (Williams et al., 1955), which relates

viscoelastic relaxation processes like viscosity with temper-

ature in amorphous polymers and other glass forming

liquids above their glass transition temperature, can be

written as Eq. (7).

loggTsqs

gsTq

� �¼ C1ðT � TsÞ

C2 þ ðT � TsÞð7Þ

where Ts is a reference temperature greater than or equal to

Tg, qs, gs, q and g are density and viscosity at Ts and T,

respectively. Combining Eqs. (6) and (7), and since q is a

weak function of temperature (Parker & Ring, 1995), we

obtain Eq. (8).

logK

Ks

� �¼ C1ðT � TsÞ

C2 þ ðT � TsÞð8Þ

As stated earlier, there are two views with regard to the

reference temperature to use in the WLF equation for

kinetics in frozen systems; TgV or concentration-dependent

Tg. The other issue raised with regard to the applicability of

the WLF equation to kinetics in frozen systems is the effect

of melt-dilution on reactant concentrations (Simatos &

Blond, 1991). For the enzyme-catalysed reactions in the

concentrated unfrozen matrix of frozen systems, Eq. (2) can

be written as Eq. (9).

V0f ¼ K½E0f �½S0f � ð9Þ

where the subscript f denotes concentrations in the

frozen system. However, the reaction rate is normally

obtained by measuring the concentration of either the


product or one of the reactants after thawing. Pincok and

Kiovisky (1966) derived an equation that relates the

concentration of the unfrozen matrix and the initial

solute concentration with the reaction rate in frozen

systems (as measured in the thawed phase) for a

second-order bimolecular reaction assuming that all the

solutes are present in the unfrozen phase. Accordingly,

Eq. (9) can be rewritten as Eq. (10).

V0 ¼Cf

Ct

K½E0�½S0� ð10Þ

where V0 is the initial rate as measured in the thawed

phase, [E0] and [S0] are the initial concentrations of the

enzyme and the substrate as measured in the thawed

phase and Cf and Ct are the total volumetric concentra-

tion of solute (reactants + products + inert) in the freeze-

concentrated matrix and the thawed phase, respectively.

In this equation, both K and Cf are temperature-depen-

dent. Cf is obtained from the equilibrium phase diagram

of the system. For diffusion-controlled reaction, combin-

ing Eqs. (8) and (10) yields Eq. (11).

V0 ¼Cf

Ct

Ks10C1ðT�TsÞC2þðT�TsÞ ½E0�½S0� ð11Þ

If a concentration-dependent Tg is used as a reference

temperature in the WLF equation, Eq. (11) becomes

V0 ¼Cf

Ct

Kg10C1ðT�TgÞC2þðT�TgÞ ½E0�½S0� ð12Þ

where Tg is the glass transition temperature of the

system corresponding to concentration Cf and Kg is the

reaction rate constant at temperature Tg.

At an arbitrary reference temperature, Tref

V0ref ¼Cfref

Ct

Kref ½E0�½S0� ð13Þ

Again, substituting Eq. (8) in Eq. (13), we get

V0ref ¼Cfref

Ct

Kgref 10C1ðT�Tgref ÞC2þðT�Tgref Þ ½E0�½S0� ð14Þ

Cfref is the total solute concentration in the freeze-concen-

trated matrix at Tref, Tgref is the Tg corresponding to Cfref and

Kgref is the reaction rate constant at Tgref (and is equal to Kg

as the viscosity at the glass transition temperature is the

same and K is dependent on viscosity alone for diffusion-

limited reactions). Dividing Eq. (12) by Eq. (14) and taking

the logarithm yields Eq. (15).

logV0

V0ref

� �¼ C1ðT � TgÞ

C2 þ ðT � TgÞ� C1ðTref � Tgref Þ

C2 þ ðTref � Tgref Þ

þ logCf

Cfref

ð15Þ

The third term on the right-hand side of Eq. (15) accounts

for the influence of melt-dilution on reactant concentration.

When TgV is used as the reference temperature in the WLF

equation and Tref = TgV, the second term in Eq. (15) disap-

pears and we obtain

logV0

V0gV¼ C1ðT � TgVÞ

C2 þ ðT � TgVÞþ log

Cf

CgV

� �ð16Þ

2.3. The equation of Peleg et al. (2002)

The general form of the log-logistic model of Peleg et al.

(2002) can be written as:

Y ¼ lnf1þ exp½cðT � TcÞ�gm ð17Þ

where Y is the rate parameter in question (V0 in our case),

and c, Tc and m are constants.

3. Results

3.1. WLF equation

The applicability of Eq. (15) was evaluated using the

experimental data on the kinetics of the alkaline phospha-

tase catalysed hydrolysis of DNPP in sucrose model system

as all the pertinent data for sucrose–water system are

available in the literature. The concentration of the unfro-

zen matrix at the reaction temperature was obtained from

the equilibrium phase diagram of sucrose–water system

according to Blond, Simatos, Catte, Dussap, and Gros

(1997). The data was converted into volumetric concentra-

tion [m/v] using the density data from Handbook of

Chemistry and Physics (1982). The concentration-depen-

dent Tg was determined using Gordon–Taylor equation

(Eq. (18)).

Tg ¼xsTgs þ kxwTgw

xs þ kxwð18Þ

where xs and xw are the mass fractions of the solute and

water in the unfrozen matrix, Tgs and Tgw their glass

transition temperatures and k is a constant. According to

Blond et al. (1997), for sucrose–water system, Tgs = 338

K, Tgw = 135 K and k = 5.46. The kinetic data obtained in

the sucrose model system together with the predicted curve

using Eq. (15) are shown in Fig. 1. The reference temper-

ature that was used in this case was � 27 jC, which is the

lowest experimental temperature. As can be seen from the

figure, a very good fit of the experimental data was

obtained with C1 = 17.8 and C2 = 126.1, which are different

from the ‘‘universal’’ coefficients and the reported values

for the viscosity of sucrose solution by Kerr and Reid

Fig. 1. WLF plot with concentration-dependent Tg as a reference and

consideration of the effect of melt-dilution on reactant concentration for the

alkaline phosphatase-catalysed hydrolysis of DNPP in sucrose model

system.


(1994) and Champion, Blond et al. (1997). Eq. (16) was

also used to model the kinetic data in the sucrose model

system. The concentration of the unfrozen matrix was

obtained from the phase diagram as above. The TgVofsucrose was determined to be � 34.2 jC using differential

scanning calorimetry (DSC) as described in our previous

work (Terefe, Mokewena, Van Loey and Hendricks, 2002).

The mid-point of the higher transition temperature was

considered as TgV, since it is found to be related to the

mechanical collapse of the amorphous matrix in real time

and is of greater importance than the true glass transition

temperature (Shalaeve & Franks, 1995). The WLF curve

thus obtained for the kinetics in the sucrose model system

Fig. 2. WLF plot with TgVas reference and consideration of the affect of

melt-dilution on reactant concentration for the alkaline phosphatase-

catalysed hydrolysis of DNPP in sucrose model system.

is shown in Fig. 2. Again as can be seen in the figure,

reasonably good fit of the experimental data is obtained

with rather different values of C1 and C2 (C1 = 35.27,

C2 = 148.6). In this case, V0gVwas also estimated as a

parameter in the non-linear regression of the equation,

since the reaction rate was determined only up to � 27

jC, as it was highly reduced below this temperature. Since

sufficiently good fit of the experimental data was obtained

using Eq. (16) and since complete physical data are not

available for the other model systems, only Eq. (16) was

used to model the kinetics in these systems. The correction

factor for the effect of melt-dilution on reactant concen-

tration, log(Cf/CgV), for the sucrose–water system is well

described by Eq. (19).

logCf

Cfref

� �¼ AðT � Tref Þ

Bþ Tð19Þ

where A and B are constants.

The experimental data according to Blond et al. (1997)

and the fitted curve using Eq. (19) are presented in Fig. 3.

Since the phase diagrams of different carbohydrate–water

systems have been shown to have a similar shape to that of

sucrose (Lim & Reid, 1991), it was assumed that Eq. (19)

can be used to approximate log(Cf/Cfref) for other binary

model systems as suitable literature data was not available.

Thus, substituting Eq. (19) into Eq. (16), we obtain Eq. (20),

which is a direct function of temperature.

logV0

V0ref

� �¼ C1ðT � Tref Þ

C2 þ ðT � Tref Þþ AðT � Tref Þ

Bþ Tð20Þ

Eq. (20) was used to model the experimental data

obtained in the ‘rubbery’ matrices (above TgV) of maltodexin,

Fig. 3. log(Cf/CgV) versus temperature for sucrose–water system according

to the data of Blond et al. (1997) (Tref = TgV).

Table 1

The estimated values of the parameters in Eq. (20) for the alkaline

phosphatase catalysed hydrolysis of DNPP in different model systems

Model system C1 C2 A B

CMC 2.3F 0.2a 9.2F 1.2a 0.0406F 0.0242a � 1.27F 0.69a

CMC–lactose 6.9F 1.4 33.4F 7.8 0.0550F 0.0315 � 2.18F 1.09

Maltodextrin 4.2F 0.2 10.1F 0.7 0.0049F 0.0017 0.18F 0.12

a Standard error of regression.


CMC and CMC–lactose model systems. The TgVvalues forthese systems of � 14.4, � 12.6 and � 23.1 jC, respec-tively, as reported in Terefe et al. (2002), were used in the

analysis. The WLF plots for the kinetics in these systems are

presented in Fig. 4. The reaction kinetics in these model

systems was fitted well by Eq. (20).

The WLF coefficients and the parameters for the correc-

tion factor for the different systems are presented in Table 1.

As can be seen, the WLF coefficients estimated, when using

constant TgVas a reference in the equation, are different from

one another and the ‘‘universal’’ values. However, we should

not necessarily expect to get the same values as the ‘‘uni-

versal’’ ones. In the present case, the WLF equation was used

to take into account both the direct effect of temperature and

the indirect effect of temperature through melt-dilution on

viscosity. In addition as Ferry (1970) stated, the actual

variation from polymers to polymers is too large to allow

the use of such ‘‘universal’’ values except as a last resort in

the absence of experimental data. In addition, change in pH

and solute concentration may have an effect on the activity of

the enzyme. Kerr and Reid (1994) also observed greater

variation among the WLF coefficients and greater difference

with the ‘‘universal’’ values when TgVwas used as a referencein the WLF equation to model viscosity of frozen maltodex-

trin and sugar solutions.

The values of A and B estimated for the different model

systems were used to calculate the correction factor for

Fig. 4. WLF plots with TgVas reference and consideration of the effect of melt

hydrolysis of DNPP in different model systems.

melt-dilution in Eq. (20). The correction factor versus

temperature curves for the different model systems are

presented in Fig. 5. As can be seen from Figs. 3 and 5,

the effect of melt-dilution on reactant concentration seems

to be significant only near the melting temperatures of ice in

the respective model matrices. It cannot explain the reported

failure of the WLF equation (Simatos, Blond, & Le Meste,

1989; Kerr et al., 1993; Manzocco, Nicoli, Anese, Pitotti, &

Maltini, 1999; Terefe & Hendrickx, 2002) to describe

kinetics in frozen systems. Simatos and Blond (1991) made

similar conclusions based on their analysis of a reaction

half-life for a second-order reaction.

3.1.1. The equation of Peleg et al. (2002)

Eq. (17) was also used to model the kinetics of the

alkaline phosphatase catalysed hydrolysis of DNPP in the

four model systems. In this case, the kinetic data in the

glassy state of the maltodextrin, CMC and CMC–lactose

-dilution on reactant concentration for the alkaline phosphatase-catalysed

Table 2

The estimated parameter’s in the equation of Peleg et al. (2002) for the four

model systems

Model system Tc c m

CMC � 10.7F 0.4a 0.529F 0.0525a 11.7F 1.7a

CMC–lactose � 12.8F 1.0 0.447F 0.1170 11.5F 4.1

Maltodextrin � 7.1F 0.6 0.499F 0.0568 11.5F 2.4

Sucrose � 3.0F 0.3 0.378F 0.0072 25.1F 2.5

a Standard error of regression.

Fig. 5. The estimated correction factor for melt-dilution versus temperature

for CMC, CMC–lactose and maltodextrin model systems.


systems were also included. As shown in Fig. 6, the

equation described the kinetic data well except in the

vicinity of the melting point of ice. However, the fit is less

satisfactory in the glassy state, though it is not readily seen

Fig. 6. Experimental versus predicted initial rate of hydrolysis of the alkaline phosp

the equation of Peleg et al. (2002).

in the figure due to the order-of-magnitude difference

between the reaction rate in the glassy state and that in

the ‘rubbery’ state. On the other hand, the equation de-

scribed the experimental data equally well as the WLF

equation in the ‘rubbery’ state of all the model systems

except around the melting temperature of ice. The estimated

parameters for the different model systems are given in

Table 2.

4. Discussion

The applicability of the WLF equation is based on the

assumption that diffusivity is inversely proportional to the

hatase catalysed hydrolysis of DNPP in the four model systems according to


viscosity of the medium. The applicability of the Stokes–

Einstein equation was assumed in the derivation. However,

the applicability of the Stokes–Einstein equation is limited.

Phillies (1989) states that, the Stokes–Einstein equation is

applicable only for the diffusion of large dilute spheres in

low viscosity (g < 10 mPa) solvents. The equation becomes

inadequate, when the diffusing molecule is small compared

with the size of the molecules of the medium in which it

diffuses and when the viscosity is high (Parker & Ring,

1995). Champion, Hervet, Blond, Le Meste and Simatos

(1997) have demonstrated that the mobility of the DNPP

molecule in sucrose solutions can be described by this

equation except in the vicinity of Tg. However, no such

information is available with regard to the other model

systems. In addition, only translational diffusion is consid-

ered in the treatment. However, as an enzyme (which is a

macromolecule) is involved in the reaction, its active site

occupies only a very small part of the total molecular

volume. Thus, for the formation of the enzyme–substrate

complex to take place, the enzyme might have to undergo

some form of rotational diffusion into a specific conforma-

tion in which the active site comes into contact with the

substrate (Bailey et al., 1981). Besides, diffusivity may not

solely depend on the macroscopic viscosity of the medium.

Obstruction by the molecules in the medium might play a

role. Such an effect has been reported for the diffusivity of

molecules in different gels (Brown & Jhonson, 1980;

Hendrickx, Abeele, Engels, & Tobback, 1986; Slade,

Cremers, & Thomas, 1966). Moreover, the viscosity depen-

dence of molecular diffusion can be counteracted by the

increase in the hydrodynamic volume of the reacting mol-

ecules due to solvation and the effect of freeze concentration

on local concentration of the reactants (Manzocco, Nicoli,

Anese, Pitotti, & Maltini, 1999).

The validity of the approach that was used in the

derivation of the correction factor in the modified WLF

equation depends on the assumption that all the solutes in

the frozen system are in the unfrozen fraction, i.e., only pure

ice freezes out and that for the same initial composition, the

composition of the freeze-concentrated unfrozen fraction is

dependent only on temperature. There is a possibility for the

eutectic crystallisation of the buffer salts and the model

matrices. However, in most real systems, the above assump-

tion is valid (Fennema, 1975b; Franks, 1985). The freeze-

concentration effect might also result in changes in pH,

ionic strength, and dielectric constant, which might have an

effect on the activity of the enzyme (Fennema, 1975a;

Tappel, 1966). A reversible denaturation of the enzyme

due to freezing is also possible. These may make the

reaction activation-controlled rather than diffusion-con-

trolled. Some or all of these factors may account for the

discrepancy between the reaction rate and viscosity in the

sucrose model system.

The empirical relation for the correction factor of melt-

dilution in Eq. (20) estimated values of the same order of

magnitude as experimental data (see Fig. 5). It can be

potentially useful for estimating the effect of melt-dilution

(freeze-concentration), in the absence of experimental phase

equilibrium data. However, the result of this study as well as

others (Fennema, 1975b) indicate that the effect of melt-

dilution is important only in dilute systems and near the

melting temperature of ice. As Fennema (1975b) noted, in

many instances, foods have reasonably high initial concen-

tration of solutes and the expected rate enhancements during

freezing would be small and would extend over a rather

small temperature range. Certainly, this will not affect

stability at normal frozen storage temperature. Rather the

potential damage from freeze-induced rate enhancement is

greatest during freezing and specially thawing which is

inherently slower.

The modified WLF equation, with both concentration-

dependent Tg and TgV as references, described well the

kinetics in the sucrose model system. However, the later

seems to be more practical as TgV is a measurable

quantity. The equation proposed by Peleg et al. (2002)

also described the kinetics of the alkaline phosphatase

catalysed hydrolysis of DNPP in the ‘rubbery’ matrices of

the four model systems equally well as the modified

WLF equation except in the vicinity of the melting

temperature of ice. However, as noted earlier, this

becomes an issue only in dilute systems, where the effect

of melt-dilution is significant. Hence, the equation is

potentially useful for modelling kinetics in frozen sys-

tems, as the process under consideration need not to be

diffusion controlled. The drawback is that the parameters

in the equation have no physical meaning. They also do

not have the significance attributed to them in the case of

microbial or enzyme inactivation in the context of kinet-

ics in frozen systems.

5. Conclusion

The results of this study indicate that the effect of melt-

dilution on reactant concentration is significant only in

dilute systems and near the melting temperature of ice.

Thus, the reported failure of the WLF equation to describe

kinetics in frozen systems might have to do with the

relationship between reactant mobility and the macroscopic

viscosity of the medium and/or the complex situation that

may arise due to low temperature as such, freezing and the

accompanying freeze-concentration with the consequent

changes in pH, ionic strength and dielectric constant,

which might cause changes in the conformation of the

enzyme and render the reaction activation-controlled rather

than diffusion-controlled. Further investigation in this

direction will be useful. Nevertheless, quantifying the

effect of melt-dilution (freeze-concentration) based on

phase equilibrium data is useful in the selection of suitable

cryostabilisers for formulated foods that undergo signifi-

cant deterioration during freezing or thawing. The results

of this study also indicate that the empirical equation of


Peleg et al. (2002) is potentially useful for describing

kinetics in frozen systems, as the process under consider-

ation needs not be diffusion controlled and since it can be

used to describe kinetics in the whole subfreezing temper-

ature range.

6. List of symbols

A, B Parameters in the empirical equation to describe

the correction factor for melt-dilution.

CMC Carboxymethylecellulose

Ct Total solute concentration in the thawed system in

% [m/v]

Cf Total solute concentration in the unfrozen matrix of

the frozen system in % [m/v]

Cfref Total solute concentration in the unfrozen matrix of

the frozen system at a reference temperature in %

[m/v]

CgV Total solute concentration in the unfrozen matrix of

the frozen system at TgV in % [m/v]

xs Mass fraction of the solute in the unfrozen matrix

of frozen system

xw Mass fraction of water in the unfrozen matrix of

frozen systems

C1, C2 WLF coefficients at a reference temperature

DE, DS The diffusion coefficients of the enzyme and the

substrate, respectively

D The sum of the diffusion coefficients of the

enzyme and the substrate

DNPP Disodium-p-nitrophenylphosphate

DE Dextrose equivalent

DSC Differential scanning calorimetry

[E0], [E0f] Concentrations of the enzyme as measured in the

thawed and frozen systems, respectively

k A constant in the Gordon–Taylor equation

k1, k2, k3 Reaction rate constants corresponding to the

enzyme–substrate association, the enzyme–sub-

strate dissociation and the transformation of the

enzyme–substrate complex into product

K The overall reaction rate constant

Kref The overall reaction rate constant at the reference

temperature

[S0], [S0f] Concentrations of the substrate as measured in

the thawed and frozen systems, respectively

T Temperature (in jC or K)

TCA Trichloroacetic acid

TgV Glass transition temperature of a maximally freeze

concentrated system

Tref, Ts Reference temperature (in jC or K)

Tgs Glass transition temperature of the dry solute in

(K)

Tgw Glass transition temperature of pure water in (K)

V0 Initial rate of hydrolysis in (Ag p-nitrophenol/ml h)

V0ref Initial rate of hydrolysis at the glass transition

temperature (Tref)

V0gV Initial rate of hydrolysis at the glass transition

temperature (TgV)WLF Williams, Landel and Ferry

Acknowledgements

This research has been supported by the Interfaculty

Council for Development Co-operation of the Katholieke

Universiteit Leuven.

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Modelling the kinetics of enzyme-catalysed reactions in frozen systems: the alkaline phosphatase catalysed hydrolysis of di-sodium-p-nitrophenyl phosphate

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