Kinetics of enzyme deactivation: a case study

catalysis today ELSEVIER

Catalysis Today 22 ( 1994) 489-5 10

Kinetics of enzyme deactivation: a case study Giuseppe Toscano, Domenico Pirozzi, Michele Maremonti, Liliana Gianfreda *, Guido Greco Jr. Dipartimento

dr Ingegneria Chimica, Dipartimento di Scienze Chimico-agrarie, Federrco iI, Nap&, Italy

Universit& degli Studi di Napolr

Abstract Acid phosphatase (EC 3.1.3.2, from potato) thermal deactivation was studied by kinetic methods with specific reference to the effect of medium properties. The time-course of activity decay was employed as an indirect probe for analysing the structural changes undergone by the protein. The experimental data show convex log(activity) vs. time curves. Usually, departure from simple firstorder kinetics is explained by postulating enzyme aggregation, or enzyme heterogeneity, or formation of partially deactivated forms of the enzyme. None of the above models was entirely satisfactory in interpreting the experimental data. The current models for enzyme thermal deactivation therefore appear to be purely correlative. The effects of medium changes were accounted for on the basis of a phenomenological approach, the ‘equivalent temperature’ yielding a monoparametric model.

1. Introduction Basic aim of this study is the characterisation of enzyme stability in aqueous solution by the quantitative evaluation of the kinetics of irreversible deactivation within the framework of mechanistic and/or phenomenological models. Acid phosphatase (EC 3.1.3.2, from potato) was used as a sample enzyme. Currently, no industrial processes supported by enzymes or micro-organisms with phosphatasic activity are carried out [ I]. Klibanov and co-workers, however, proposed to employ phosphatases in order to separate mixtures of cis and rrans isomers of alicyclic alcohols [ 21 and racemates of D, L-threonine [ 31 by stereoselective hydrolysis of their phosphoric esters (Klibanov, A.M. US Pat. 465947 1) . Phosphatases are able to carry out synthesis or transfer reactions, as well. The synthesis of glucose-6-phosphate and glucose- 1-phosphate from glucose and pnitrophenylphosphate can be performed by acid phosphatase contained in Escher* Correspondingauthor. 0920-5861/94/$07.00 0 1994Elsevier Science B.V. All rights reserved ~~~~0920-S861(94)00127-8

490

Giuseppe Toscano et al /Catalysis Today 22 (1994) 489-510

ichiafreundii immobilised a biphasic water~hlorofo~

whole cells [ 41. The synthesis of glycerylphosphate in system can be achieved by an alkaline phosphatase

151. The detailed knowledge of deactivation kinetics is a prerequisite for bioreactor design and for mechanistic interpretations of enzyme deactivation phenomena, as well. In the first case, the quantitative description of deactivation dependency upon environmental variables (temperature, pH, concentration of reagents, products, additives, etc.) allows the prediction of the bioreactor useful life. In the latter instance, once a correspondence between deactivation mechanism and observed kinetics has been established, the deactivation kinetics might be used as a diagnostic tool to identify the deactivation mechanism and to develop a rational stabilisation technique. Enzyme deactivation may be due either to the loss of the native confo~ation (without rupture or formation of covalent bonds) or to chemical modification of functional groups of the active site. In this paper, we shall only deal with the former mechanism of deactivation. In the following, it will be referred to as ‘thermal deactivation’. A rather common assumption is that thermal deactivation mechanisms involve a single active enzyme form, i.e., the native one. Furthermore, first-order kinetics with respect to the native enzyme concentration are usually assumed for the deactivation process. Consequently, an exponential activity decay is predicted. Thermal deactivation of acid phosphatase, however, departs from first-order kinetics, as shown by the convex log(activity) vs. time curves 161. Such a behaviour is not restricted to acid phosphatase (see for examples 17-I 11). These deactivation patterns suggested kinetic schemes involving more than a straightforward, firstorder, two-state transition. Furthermore, the occurrence of multiple active forms seems quite probable. Therefore, the results obtained in the course of the present study might be of fairly general interest. Investigations on the correlation between the level of residual activity and the extent of structural modification undergone by the enzyme have been usually disregarded, because of the all-or-none assumption about the activity. Incidentally, the irreversibly denatured state is rather difficult to characterise and its presence is hard to detect because of the loss of activity (catalytic or other, such as the ability of antibody binding) I Tsou and co-workers compared the enzyme inactivation rate to the rate of confo~ational changes monitored by UV absorbance, fluorescence and circular dichroism changes and exposure of buried sulphydryl groups. For the denaturation by exposure to high temperatures and to chaotropic agents like guanidinium chloride and urea of the enzymes glyceraldehyde-6-phosphate dehydrogenase [ lo,12 J and creatine kinase [ 131, they observed that inactivation generally occurred before any observable confo~ational change took place. Tsou [ 141 inferred that the active site structure is perturbed before major conformational changes (subunits dissociation, full unfolding of tertiary structure, aggregation between unfolded macromolecules) become observable. This is reasonable, if one

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considers that amino acids involved in the active site structure (always relatively near to one another within the native tertiary structure) are generally arranged at fairly distant locations along the primary sequence, so that small changes in the tertiary structure are able to modify substantially the active site geometry. In the course of deactivation, heterogeneous mixtures of protein molecules at different stages of denaturation may be obtained. Therefore, arty observable quantity is a statistical average of the unknown values pertaining to each single form, with unknown weights, since the mixture composition is not established. If an enzyme solution has lost, say, 50% of its initial activity, it is not possible to discriminate a priori between the case where 50% of the enzyme molecules is 100% inactivated and that where 100% of molecules is 50% inactivated (even though the first interpretation is the most frequently adopted). In order to overcome these difficulties, in the present experimental work the measurement of the residual catalytic activity will be used as a tool to characterise the partially denatured forms. Indeed, the deviation from first-order kinetics can be explained by postulating the appearance of denature, pa~ially inactivated enzyme structures. Obviously, it is necessary to take into account alternative explanations, founded on aggregation and on enzyme heterogeneity, as well. Critical importance in discriminating among different models will be attributed to their ability to provide a quantitative description of the activity decay under different experimental conditions. Indeed, the consistency with the basic model assumptions imposes some constraints on the variability of model parameters. The authors of the present paper are well aware that the lack of detailed stmc~ral information on the enzyme used circumscribes possible mechanistic interpretations. Nevertheless, on the basis of structural and functional features common to all enzymes and of a formal analysis of the dependence of kinetics on experimental variables, it is possible to conjecture about the kind of conformational changes involved in enzyme deactivation. The experimental results discussed in the present paper reveal some unexpected features of the irreversible thermal deactivation of enzymes.

2. MateriaIs and methods 2. I. Reagents A commercial (partially purified) preparation of acid phosphatase was used for all the experimental runs of this work. It was purchased from Boeh~nger M~nheim as a lyophilised powder and used without further purification, unless otherwise specified. p-Nitrophenyl phosphate (PNPP), disodium salt, was used as the substrate. According to the producer, the specific activity of the commercial lyophilisate for PNPP hydrolysis at 37”C, pH 5.6 is approximately 2 U/mg. Considerable variations in initial enzyme activity were observed, however, depending on the

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batch. Therefore, all activity data discussed in the following have been normalised with respect to the mean value of the initial activity (aN = 5.88 X 10s3 mmollmin mg under the assay conditions specified in Section 2.4). All the other reagents were analytical grade. 2.2. Purification The enzyme was purified from the commercial prep~ation by ion-exchange, gel filtration and affinity chromatography, according to the method described in [ 151. h-exchange

~hr~matogruphy

A DEAE-SephaceI@ column ( 1.6 X 11 cm) was employed. The column was equilibrated with 50 mM Tris-HCl buffer, pH 7.5. At this pH value, the enzyme is charged negatively and therefore it is entirely adsorbed by the ion-exchange column. Subsequent elution is carried out with a linear NaCl concentration gradient. Desorption occurs at a concentration of approximately 40 mM NaCl.

The column used was a Sephadex@’G-150 ( 1.6X 80 cm), equilibrated with 100 mM Tris-HCl buffer, pH 7.5. The molecular-weight calibration was performed according to the method described by [ 16 ] . Standard proteins were: aldolase ( MW 158~, from rabbit muscle), albumin (MW 67000, from bovine serum), ovalbumin (MW 43000, from chicken egg), chymotrypsinogen A (MW 25000, from bovine pancreas), ribonuclease A (MW 13700, from bovine pancreas). Void volume was evaluated with Blue Dextran 2000e3 (MW 2 X 106). Affinity chromatography with concanavalin A immobilised on agarose gel The column ( 1.6X 2.5 cm) was prepared with 5 ml of agarose gel. ~uilibration

was performed with 1 M NaCt, 1 mM Mg&, 1 mM MnC12, 1 mM CaC12in 100 mM Na-acetate buffer, pH 6. Acid phosphatase is completely bound by concanavalin A. Subsequent elution is obtained by displacement with a 5% wtlvol methylcY-rt-glucosideaqueous solution. 2.3. Deactivation runs In order to reduce the thermal transient, all deactivation runs were started by mixing a small volume of concentrated enzyme solution with a buffer solution already at the deactivation temperature, in order to bring the enzyme solution to the chosen temp~ra~re quickly. The enzyme was dissolved in Na citrate buffer 200 mM pH 5.6 at a final concentration of 10 mg of lyophylisate per ml, unless otherwise specified. Enzyme samples were withdrawn periodically and cooled down to 4”C, in order to stop deactivation. Storage at this temperature took place until the activity assay. No activity recovery was observed during 48 h storage at 4°C. The assay

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was carried out at 45”C, independent of deactivation temperature. At this temperature, no appreciable deactivation occurred in the time course of the kinetic test. Enzyme sampling and activity assay were carried out by a computer-driven liquid sampler Sample Changer 222 (Gilson) with thermostated tube racks. 2.4. Activity assay Specific activity, defined as mmol of substrate reacted per minute and per mg of enzyme tcommerci~ lyophylisate), was measured by an initial-rate method. The reaction used in the activity assays wasp-nitrophenyl phosphate (PNPP) hydrolysis in Na citrate buffer 200 mM, pH 5.6 at 4S-C. The reaction was started by injecting a given volume of enzyme solution (from 0.05 ml to 0.15 ml, depending on residual enzyme activity level) into a PNPP solution (preheated at the reaction temperature), final volume of 1 ml. In all activity assays, the final PNPP concentration in the reaction volume was 6 mM ( 3 K,, [ 61) . After incubation ( 10 to 30 min), the reaction was stopped by adding 1 ml of 1 M NaOH. The ensuing pH shift produced aquick, irreversible deactivation of theenzyme. Product (p-nitrophenate) concentration was detected at 405 nm (extinction coefficient E= 18 (mM *cm) - ’ ) . In all instances, conversion was kept within lo%, in order to maintain the reaction rate nearly constant throughout the assay. The hydrolysis of PNPP by acid catalysis during the assay was accounted for by subtracting from the total reading the absorbance of a blank, i.e. a substrate sample that had undergone the same thermal history. In order to account for enzyme absorbance, the same enzyme volume as that employed in the corresponding kinetic test was injected into the blank, upon alkalisation, The specific activity is calculated accordingly: CPNP

Vr

a = 1OOOA tcEVE

(1)

where c,,Npis product (~-nitrophenol) concentration, cn (mglml) and V, (ml) are enzyme concentration and solution volume, respectively, At (min) is the incubation time and V, the overall reaction volume. Performing the activity assay off-line allowed us to adjust the analytical conditions to the activity level attained in any given enzyme sample. Indeed, as already stated, the amount of enzyme employed for the activity assay may be increased, as well as the incubation time, in order to obtain a reaction yield within the sensitivity limits of the analytical method. High enzyme concentration ( 10 mg/ml) was used in the deactivation batch and enzyme samples were assayed upon dilution, if necessary. 2.5. Statistical methods Model parameters were estimated by a weighed least square method. The weighed sum of squares of errors (SSE), i.e., the difference between the measured

Giuseppe Toscano et al. f Cat&is

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Today 22 (1994) 489-510

values a, and those calculated by the model ti,, was numerically

minimised:

13

SSE= Cwi(a,-a^i)’

(2)

The statistical weights wi = 1 laf were employed. This choice implicitly assumes that the percentage error is constant over the range of activity values explored, i.e., a’( a) a a*.This ~sumption can be physic~ly justified as follows. In order to get reliable results throughout the whole interval of activity values explored, the final product concentration at the end of each assay should always be of the same order of magnitude. As activity decays, if the other assay variables are not adjusted one has to use higher and higher enzyme amounts in order to obtain an adequate product concentration. Therefore, the relative error on activity remains approximately constant since the absolute error on product concentration does not change (see Eq. 1). Definite trends appear in residuals a - ri,, quite independently of the model chosen to correlate the data. The trends are a consequence of the fact that sets of consecutive samples with decreasing activity were assayed with the same enzyme mount. Therefore, product yield varied somewhat within each set, so that product concentration was not measured with uniform reliability throughout. Furthermore, although initial temperature transients were reduced to a minimum, nevertheless they may still affect the activity decay of the very first samples at the higher temperatures used. Goodness of fit was evaluated by the index of determination R2, defined as p’

SST - SSE

SST

1. N

where SST= CWiaf I

Parameter estimates have been reported as (parameter value) + (standard deviation). Approximate standard deviation of parameters was evaluated from the linearised models. In order to reduce parameter interactions (high covariances), the following transformation of parameters was used for the Arrhenius dependence: k=k,exp(

-E/RT)

=&iexp[

-(E/R)(l/T-m)]

(4)

where the overbar denotes mean value over the range explored.

3. Results and discussion 3.1. Enzyme specijcity and reaction kinetics Acid phosphatase catalyses the hydrolysis of phosphoric monoesters and phosphoproteins (with some specificity for those having an aromatic moiety), in a

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moderately acid environment (optimal pH 5.6). Phospho~c diesters are not hydroIysed. The irreversible thermal deactivation of acid phosphatase in aqueous solution under various experimental conditions (temperature, pH, presence of additive like sorbitol, urea, methanol, saccharose, etc. > was extensively studied [ 6,17-2 11. Acid phosphatase was chosen as a sample enzyme because of its relatively low stability. Indeed, deactivation runs can be carried out at relatively low temperatures (less than 7O”C), even at the optimal pH. Avoiding extreme conditions rules out covalent m~i~cations such as peptide bond hydrolysis, cystein oxidation, amino acid racemization etc. [22]. The activity assay involves the measurement of the initial rate of PNPP hydrolysis. The latter yields an estimate of kcnt,provided substrate concentration is much higher than K,, throughout the kinetic assay. In the present case, k,,, allows a simple interpretation at the molecular level. Indeed, all the proposed mech~isms for phosphors monoeste~ hydrolysis by acid phosphatase support the existence of a phospho~l-enzyme as a reaction intermediate [ 23-25 J . The minimal reaction scheme is as follows: ROH 3

E + ROPQ,H-

e

Es ROPO,H-

E+HPO; -

where E is the enzyme and R an aIky1 or aromatic group. Within the limits of the quasi-steady state approximation, the apparent kinetics cannot be distinguished from the classical Michaelis-M~nten [ 261: k,,lElotSl ll=

K,+[s]

where

k

k2k =-an(jI(,=kv~‘k2.--$g Cat

k,+ k3

(5) 1

At the optimal pH (about 5.6)) different phosphoric monoesters are hydrolysed with different k,,, [ 241. This implies that the inte~ediate hydrolysis is rapid ( k3 X- k2) and that alcohol (or phenol) release is controlling (cf. [ 261 p. 199200) _Therefore, at a molecular level, the k,,, = k2 value accounts for p-nitrophenol release rate. 3.2. Deactivation mechanisms Enzyme deactivation is often assumed to involve only two different enzyme forms: native (active) N and denatured (fully deactivated) D: N--+D

(ii)

Giuseppe Tascaraa et al. f Catalysis Today 22 (1994) 489-510

4%

The deactivation process is expected to be first order with respect to native enzyme concentration, if it is not due to chemical modifications:

Upon integration one gets: (7)

IN1 = lN10 exp( -klI

where [N] Ois the initial, native-enzyme concentration As a consequence, an exponential decay of the overall specific activity (i.e., referred to the total enzyme concentration) is predicted:

a,CNl WI,

a=----=a,

exp( -kt)

where aN is the specific activity pertaining to the native enzyme, However, enzyme thermal deactivation often yields convex log(activity) vs. time curves, as in the present case (Fig. 1) . This suggests that a more complex kinetic pattern is followed, other than a simple two-state transition. In the literature on protein denaturation (cf. [ 261, p. 415-429), departure from first-order kinetics is usually explained by the occurrence of one or more of the following instances: (1) the rate-controlling step in the denaturation process is po~ymo~ecular (aggregation) many monomolecular reactions proceed simultaneously with different rates (2) (enzyme heterogeneity) (3) the process occurs in serial steps with the appearance of partially denatured forms of the enzyme (biphasic scheme).

1o-7L--..__e._..0

5

10

I

I

15

20

J 25

time (hr) Fig. 1. Thermal deactivation of acid phosphatase from potato as a function of deactivation mental conditions: 10 mg ~yophylisate/ml in 200 mM Na citrate buffer, pH 5.6.

temperature.

Expen-

0

1

2 time (hr)

3

4

Fig. 2. Acid phosphatase deactivation at 60°C in Na citrate buffer 200 mM pH 5.6 with lyophylisate concentrations OF 1 and 10 mg/mI.

As we shall see, none of the above models is entirefy satisfaetoty to describe the kinetics of activity loss of acid phosphatase. 3.2.1. Aggregation According to Mozhaev and Mart&k [27] and Klibanov [ 281, aggregation among different enzyme molecules is a major cause of irreversible deactivation. The phenomenon is described as a two-stage process. The first stage may consist in more than one elementary step and it may involve only reversible confo~ational changes and oligomer association equilibria [ 29,301. The second stage is intrinsically polymole~ular and irreversible. Aggregates are stabilised by covalent bonds ~i~te~ole~ular disulphide bridges) [ 291 or by non-covalent intemctions (hydrophobic interactions, electrostatic forces, hydrogen bonds) [ 3 I-331. Aggregation is an intrinsically polymolecular phenomenon. As a consequence, if aggregation were the rate-controlling step, the overall rate of deactivation should increase by increasing enzyme concentration. Indeed, if we integrate the nth-order kinetics d[Nl -= dt

-k,z[N]”

(9)

with the initial condition [N] = [ivlo, we obtain:

where ffNis the specific activity of the native form. Eq. 10, with n=3/2, well fits the deactivation data at 6O”C,as reported in Fig. 2. As shown in Fig. 2, however, two thermal deactivation runs carried out with two widely different acid ~hos~ha~se concentration levels ( 1 and 10 mglml, respec-

tively ) yield coinciding time courses. This clearly rules out aggregation as the ratelimiting step in enzyme deactivation. Aggregation does actually take place since massive precipitate formation is apparent. It is, however, a later event that follows irreversible, enzyme-concentration-independent conformational changes leading to deactivation. The result depicted in Fig. 2 rules out other, intrinsically polymolecular mechanisms of chemical deactivation, as well, such as chemical reaction of the enzyme with non-identi~ed impu~ti~s [ 261 or hydrolysis of peptide bonds by proteolytic enzymes in crude preparations [ 341. By inspection of Fig. 2, it can be seen that the agreement between Eq. 10 and experimental data is excellent, even though the concentration dependence is not obeyed. We shall consider Eq. 10 later, for an alternative physical explanation. 3.2.2. Enzyme heterogeneity Heterogeneity of commercial enzyme preparations, i.e., occurrence of isoenzymes or micro heterogeneity due to post-translational modifications, has been conjectured as a possible cause of deviation from simple exponential decay [ 351. A model based on the simultaneous, irreversible, first-order deactivation of two isoenzymes, with different specific activity and stability, does predict a convex (double exponential) decay. In its simplest form, the deactivation path can be summarised as:

N1-+Q (iii)

&--+b where totally On zyme,

N, and N2 are two native isoenzymes and D,, Dz are the co~esponding, inactivated structures. assumption of first-order kinetics (unimolecular process) for either isoenone gets:

WI1 = INIloexp( -k,t) Nl

= [&IO exp( -&I

(11)

where [ N1 1o and [&I 0 are the initial concentrations of the two isoenzymes, kl and k2 are the co~esponding, first-order deactivation constants. For future reference, we suppose k, > k2, i.e. the isoenzyme I is the less stable one. The overall, apparent activity is given by

a=~dW ++[&I [N~ln+ [NJ, a2l?Jzlo +

IN,],+ [&lo

=

a~[N,l, [N,],+

[N2,0exp(-kit)

exp( - k2t) = LYexp( - k,t) + p exp( - k2t)

(12)

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Table 1 Parameterestimate for two-isoenzyme model

Tc,(“(3

PI(ff+P)

k, (l/h)

k, (l/h)

50 52.5 55.5 57.5 60

0.142 f 0.008 0.120 f 0.008 0.062 rt 0.006 0.017 kO.004 0.003 f 0.001

0.797 * 0.039 1.538f0.103 2.470f0.181 3.081 f0.344 7.528 f 0.557

0.080 f 0.146 f 0.278 k 0.360 + 0.603 f

R2 0.004 0.005 0.009 0.024 0.046

0.9961 0.9925 0.9846 0.9348 0.9340

where cr and /3 are the specific activities of isoenzyme 1 and 2, respectively, times the corresponding, initial weight fractions in the native mixture. In contrast to the model hypothesis, we have found, however, that (a) the purification of the commercial preparation by low-pressure liquid chromatography did not show any heterogeneity, within the limits of the method resolution; (b) Eq. 12 is unable to correlate the time course of deactivation at different temperatures, and to predict the observed effect of a sudden temperature variation. As regards point (a), in order to bring into evidence enzyme heterogeneity, we performed the purification of the raw enzyme preparation by low-pressure liquid chromatography, according to the experimental technique described in Materials and Methods. Kruzel and Morawiecka [ 151, under the same operational conditions, found the enzyme to be heterogeneous, since a fraction was not adsorbed during the sample-loading phase. In contrast, in the course of the ion-exchange chromatography run we performed on the partially purified, commercial preparation, all phosphatasic activity is adsorbed in the loading phase. Furthermore, we observed a single activity peak in the eluate under NaCl gradient. The partially purified commercial preparation used in the present work therefore coincides with the enzyme fraction extensively purified by Kruzel and Morawiecka [ 151, since both enzyme fractions are completely adsorbed on loading. During gel-filtration runs, no enzyme heterogeneity becomes apparent. Indeed, we observed a single activity peak, corresponding to a molecular weight of 114 kDa. This value is about twice as that obtained by Kruzel and Morawiecka [ 151 with thin-layer gel filtration at pH 4.5. It is substantially the same, however, as that reported by Kubicz [ 361, with polyacrylamide gel electrophoresis in non-denaturing conditions, pH 9.5. The disagreement could be due to a pH-dependent association equilibrium (monomer at acid pH, dimer at basic pH) . Affinity chromatography with concanavalin A shows that acid phosphatase from potato is a glycoprotein, according to literature reports [ 15,371 . Again, a single activity peak is observed. As far as the limits of the model predictions are concerned, it should be noted that all experimental curves should be correlated by the same, deactivation-temperature independent, values of (Yand p, since each parameter is related to the specific activity of a well defined enzyme form and all activity tests were performed

500

Giuseppe Toscano et al. /Catalysis Today 22 (1994) 489-510

at the same temperature of 45°C independent of deactivation temperature. Therefore, all specific activities should be temperature independent. Therefore, the activity fraction pl ( cr + p) of the more stable form should be temperature independent, as well. This is not the case, as shown by the data reported in Table 1. 3.2.3. ‘Biphasic’ model The apparent non-first-order kinetics of the time course of activity decay can be explained by a serial mechanism involving more than one intermediate, partially deactivated form: N+D,

+DZ+.

. . +D,

(iv)

The time course of the overall activity is given by: a=cl

exp( -h,t)

+c2 exp( -h2t)

+. . . +c, exp( -A,t)

(13)

where ci and Ai depend on the specific activity of each single form and on the kinetic constant of each deactivation step, as well. The temperature dependence of the parameters provides a test criterion for the model: the kinetic constants should depend on Td according to the Arrhenius law, whereas the specific activities of each form should be independent of Td, since the activity assays were performed at a constant temperature. The following mechanism was proposed for the thermal deactivation of acid phosphatase from potato by Gianfreda et al. [ 171: N-+X-D

(v)

where N is the native enzyme, X a partially deactivated, intermediate form and D a fully deactivated form. If both transitions are monomolecular and k, and k2 are the kinetic constant of the first and of the second deactivation step, respectively, one gets:

4Nl

-= dt

-k,[N]

4x1

-=k,[N] dt

(14) -k2[X]

Upon integration with the initial conditions

[N] = [N] 0. [X] = 0 we obtain:

WI = [Nl, exp( - k,t) [X]

=

[NlOkl

k,bp(

We eventually

a=%[Nl = Q

- k2t)- exp( -

(15) k,t) 1

get:

+e[Xl [N],

=(aN-$)exP(

exp( - k,t) + p exp( - k,t)

-kit)

+$exp(

-k2t)

(16)

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Table 2 Parameter estimate for the biphasic model

adaN

0.132~0.008 0.91oIfro.244 17Z~Oil5~0

% AC (callmol) k, =k&exp[ -(E,IR)(lIT--1/T)] &J Cl/h) 15,(callmol) k&&exp[ - (Eg’R)( lfl-m) J

2.332 f 0.237 47800 rt 5800

&Cl/h)

0.235 i 0.038 44000 f 2000

Et (calllmol)

where aN and a, are the specific activities of the forms N and X, respectively. The equation is the same as that obtained for the isoenzyme model, but for the different meaning of the pre-exponential factors. The monomolecular deactivation constants ki and kz show an Arrhenius dependence on Td. Again, however, the pre-exponential factors do not show the expected dependence on Td. Indeed, the specific activity of the intermediate form ax is not independent of Td as it should, since the activity assays have been performed at the constant temperature of 45°C. Mechanism (iv) is not able to explain the observed dependence of a, on Id, since it assumes the existence of a single, deactivation-temperature independents inte~ediate X. Therefore, Gianfreda et al. [ 171 proposed the following modification of the previous mechanism: via parallel reactions, the native form N gives rise to an equilibrium between a finite number of intermediate forms X1, Xz, .... x,: N-+(X,=X,=...SX,J-+D

(vi)

The equilibrium distribution of different forms X, changes with Td. If the intermediate forms are characterised by different specific activities, the apparent activity of a ‘lumped’ form (X) depends on Td, too. The simplest way to account for the sigmoidal form of ax versus Td is to postulate the existence of two different forms: X1, active, and Xz, inactive. Increasing deactivation temperature displaces the equilibrium towards the inactive form. By denoting the X1 activity with ax1 one gets: axi ax=i%=l++Oexp(

axI -AGIRT)

(17)

The temperature dependence of ax is well described by Eq. 17. All the model diameters are reported in Table 2. The equilib~um assumption is partially contradicted by the substantial irreversibility of deactivation. Indeed, as already discussed in Materials and Methods, we

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have observed no activity recovery upon keeping the p~ally deactivated enzyme solution at 4°C for up to 48 h, prior to the activity assays. This result could still be compatible with the equilibrium hypothesis, since it could be explained by a kinetic ‘freezing’ of the equilib~um X , +X, at 4°C. A set of experimental runs were performed by suddenly varying the deactivation temperature, in order to investigate in greater detail the equilibrium hypothesis. A first experimental run was performed at 6O”C, according to the usual procedure. The deactivation was protracted until the second rectilinear portion of the semilogarithmic plot was attained. According to the biphasic model, this implies that all the active enzyme is in the (X) form. At this stage, the deactivation temperature is increased suddenly from 60 to 65°C. A further double exponential decay is observed. This transient, preceding the final, exponential decay can be interpreted as the time-progressive displacement of the X1 +X2 equilibrium towards the inactive form X2, governed by a kinetic constant of the same order of magnitude as k, (Fig. 3). According to this hypothesis, however, a displacement in the equilibrium should occur on cooling, as well. Indeed, an increase in the X1 fraction concentration should take place, giving rise to a corresponding increase in overall activity. Obviously, the latter must be followed eventually by an activity decrease with a rate depending on the final deactivation tempem~re. In order to check this possibility, we investigated the activity decay ensuing a sudden tem~rature decrease 60°C --+55°C. As one can see from Fig. 3, no evidence of any activity recovery has been observed, under these experimental conditions. This indicates an intrinsic inconsistency of X t + XZ the model since a kinetic ‘freezing’ of the rea~gement of the equilib~um does not seem plausible. Indeed, at 55°C the kinetic constant of the equilib~um rea~angement should be such as to produce an observable, though temporary, increase in activity, unless the activation energy of the rearrangement is extremely high. 3.3. The ‘equivalent temperature’ On the basis of the preceding discussion, the current models for enzyme thermal deactivation appear to be simply correlative ones, even though they might be based on plausible, yet unfounded, physico-chemical assumptions. Therefore, we have attempted to characterise the effect of medium changes in a compact way, on the basis of the following empirical observations. Different sets of medium parameters (deactivation temperature, pll, ionic strength, additive concentration) can yield the same deactivation profile, even though the kinetics of deactivation are strongly affected by each of them. Indeed, the experimental data produced under widely different experimental conditions can be interpolated with deactivation profiles produced in a ‘standard’ medium, i.e., 200 mM Na citrate buffer, pH 5.60, no additives (Fig. 4). For each data set an ‘equivalent temperature’ Te,, can be thus identified [ 201.

Giuseppe Toscano et al. /Catalysis

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6

7

time (hr) Fig. 3. Deactivation run at 6O*Cwith sudden temperature changes at t= 3.5 h up to 65°C and down to 55°C.

In order to calculate Teq, it should be noted that the info~ation contained in Table 2 enables us to predict the activity versus time profile a( r,T,) produced at any given deactivation temperature Td_ For each set of experimental data, the corresponding value of Teq is calculated as that that minimises the sum of the weighed square residues between the experimental data and a (t,TJ for deactivation at T,, under ‘standard’ conditions. A trial and error procedure has been adopted in order to calculate Teq. Although a( t,TJ is based upon the serial model, the results obtained do not depend on the latter, since the role of the specific deactivation model adopted is purely instrumental. That au ‘equivalent temperature’ does actually exist, has been verified for enzyme deactivation in the presence of sorbitol, a stabilising agent, in the range 0 to 3 M at T= 60°C [ 191; for deactivation in the presence of urea, in the range 0 to 2.4 M at T= 45°C [ 211 and for a set of different experimental conditions [ 201. To some extent, the existence of an ‘equivalent temperature’ rules out any model based upon more than one independent reaction step. Indeed, that the same set of

w

’

I

/

0

1

2

I

timei

/

I

4

5

6

Fig. 4. identical activity decays under different experimental conditions (data replotted from 1201)

504

Giuseppe Toscano et al. /Catalysis Today 22 (1994) 489-510

Fig. 5. Qualitative dependence of k on the deactiv~on temperature T, and the additive (stabilizer) ~oncen~tjon c.

kinetic and the~odynami~ constants ruling the phenomenon should be achieved under widely different experimental conditions sounds rather implausible. In contrast, this result could be explained under the hypothesis that the enzyme follows a constrained denaturation pathway and that a unique (though still unknown) parameter regulates the overall process. Indeed, let the deactivation rate be governed by a single parameter k that depends on deactivation temperature Td, additive concentration C, pH, ionic strength, etc.: u = a( t,k) . If k is a continuous function of all operational variables, the same range of k values can be explored for different sets of experimental conditions. With specific reference to the runs performed with urea and sorbitol, let us specifically consider two different experimental conditions: ( i) constant deactivation temperature Td = Td and variable additive ~oncen~ation C; (ii) constant additive concentration C= C* and variable deactivation temperature Te It is possible to identify a biunivocal correspondence between additive concentration in the range (i) and deactivation temperature in the range (ii). Reference should be made to Fig. 5, where a purely qualitative plot of k is reported, as a function of both Td and C. For any given additive concentration C, in range (i), one has to draw the perpendicular from point (Ti ,C) up to the corresponding value of k. Then, the contour line at constant k has to be followed, until its projection onto the (T;,(Z) plane crosses range (ii), thus determining the corresponding point with the same deactivation profile. Let us consider a deactivation model characterised by IZparameters, say k,, . ..k.,. Within the same framework, each parameter depends on deactivation temperature and additive concentration. If reference is made to the two ranges (i) and (ii), in order to achieve the same correspondence between points (T: ,C) and ( Td,C* ) , the contour lines of each model parameter must have the same projection onto the Td,C plane. If this is the case for any contour line, the n parameters are functionally dependent and ultimately can be reduced to a single one. Indeed, for any two parameters kl and kZ, the projections of the contour lines onto the Td,C plane are

Giuseppe Toscano et al. /Catalysis

Today 22 (1994) 489-$10

505

identified by equations kl (T&C) = k; and k2( Td,C) = k; , where I%;and kl are constant values in the range of k, and k,, respectively. If, by varying the contour line, one can set a correspondence between k; and k,* values such that the related projections are the same, then a continuous function k: = @(k; ) is defined. Therefore k2= @(k, ) and, by subsequent substitution, the number of model parameters can be reduced to one. 3.4. A monoparametric

phenomenoZogicaE model

If the order of reaction does not depend on expe~mental conditions, the model initially proposed for enzyme aggregation becomes monoparametric, even though its physico-chemical foundations do not seem to be experimentally validated. Indeed, by assuming n = 312 as suggested by the results of discussion in Section 3.2.1, upon rearrangement, Eq. 10 becomes:

where &$ is the specific activity of the native form and therefore does not depend on experimental conditions. Eq. 18 is able to fit deactivation data with reasonable accuracy (see Tables 3-5 and Figs. 6-8) and with the minimum number of adjustable parameters (one). Table 3 Parameter estimate for the phenomenolo~c~ model k’ fmmol-I”

7-aW)

(min mg)“‘h-‘)

RZ

4.0 f 0.2 50 8.0f0.3 52.5 25.14 0.7 55.5 71.2k2.9 57.5 219.9 f 1.7 60 k’=k&exp[ -(WR)(IIT-m)] k;, (mmoles-I’” (min mg)“2 h-‘) E (callmol)

0.9584 0.9560 0.9704 0.8912

0.9012 R2= 0.9747 24.20 It 0.04 90000 It 11000

‘Standard’ medium. Table 4 Parameter estimate for the phenomenol~~c~ model C (W

k’ (mmol-“2

0.9 1.8 2.1 3.6

62.8 f 6.0 34.7 rt 1 9 19.3* 1.1 12.2iO.8

?“= 60°C. additive: sorbitol.

(min mg)“‘h-‘)

R2

0.9146 0.9762 0.9836 0.9893

Table 5 Parameter estimate for the phenomenoXogical model c CM)

k’ (n~mol-“2 (minmg)“* h-l)

R2

2.2 2.6 3.0

20.7;1; 1.1 34.3 It: 1.9 58.9 13.7

0.9611 0.9704 0.9564

Ta= 45°C. additive: urea.

L-

i

/

I

f

I

10 15 time (hr) Fig. 6. Deactivation runs of Fig. 1 correlated by the phenomenological model

0

1

2 time f&f

3

4

Fig. 7. Deactivation runs in presence ofsorbitolat60”C corefated by the phenome:nofogicaI model (data mplotted from 1191).

Parameter k’ only depends on solvent properties (additive type and concentration, pH) and on Td according to the Arrhenius law. It is noteworthy that empirical models formally derived from rzth-order kinetics were often employed for non-first order enzyme deactivation since the beginning of this century ( [26], p. 41.5). Because of the reasons discussed in Section 32.1,

Giuseppe Toscano et al. /Catalysis Today 22 (1994) 489-510

1O-5

0

Fig. 8. Deactivation from [21]).

1

2 time

3

4

507

I 5

(hr)

runs in presence of urea at 45°C correlated by the phenomenological

model (data

replotted

these models were discarded in favour of the series-parallel schemes of reaction. A recent example is the study of horseradish peroxidase deactivation carried out by Weng et al. [ 111. These authors found that a nth-order kinetics with IZ= 1.7 is able to fit the time course of deactivation. However, they preferred a two-isoenzyme model on the basis of statistical considerations. From E!q. 18, the following feature of deactivation profiles is apparent: if we define a characteristic time T= u-~.~/c’ - ’ and plot the activity values against the dimensionless time t/T, all the experimental points fall upon the same curve. A similar property has been observed by Simpson and Kauzmann [ 381 for non-first order denaturation of ovalbumin monitored by polarimetry. Specific optical rotation [ a] measurements were performed at the wave length of sodium D-line (589 nm) . By plotting log ( [ a] - [ a] =) vs. the ratio between elapsed time and denaturation half-time, they observed that the shape of the resulting curves was nearly the same in various experimental conditions (different temperatures and urea concentrations, presence of various additives). The authors did not attempt to explain this property and their discussion of results proceeded along the lines of Sections 3.2.1-3. 3.5. Final remarks on the phenomenological

model

The constraint on the number of independent parameters in the analytical expression can be explained by supposing that the enzyme must follow the same, fixed denaturation pathway for different environmental conditions. The overall process cannot be resolved into separate steps, but it should be described as a continuous conformational change of the enzyme macromolecule in which the catalytic activity is gradually lost. During the process the number of catalytic sites is not reduced, but the activity of each site is decreased because of the time-progressive loss of the original conformation.

Gruseppe Tuscano et al. /Cufafysis

508

Today 22 (1994) 489-510

We must suppose that the active site structure can be perturbed, without complete loss of activity, before gross confo~atioual changes, diss~iation of subunits or aggregation takes place. This may be reasonabIe if one takes into account that the amino acid residues of the active site are positioned nearby in the folded structure, but are generally apart in the primary sequence, so that small conformational changes can substantially modify active site geometry. At this stage, the physical inte~retation of the monop~ame~ic model cannot go into further details, because of the lack of structural information on acid phosphatase from potato. From an entirely phenomenological point of view, however, we can make further remarks on the mathematical model. Under the hypothesis that each macromolecule follows identically the same denaturation pathway, we can identify each configuration attained by the macromolecule with a reaction coordinate. At all times, the rate of change of configuration (dxldt) is a function of the configuration attained at that time (x(t)) and of the (unknowns parameter only:

where F does not depend on environmental conditions (temperature and other medium properties). In order to obtain a more specific form, we have to take into account the observed cong~ency of the activity vs. dimensionless time curves. If x is defined as n=a(t)a,

.xE ]O,l]

(20)

it follows that X(f) =f(kQ

(21)

By differentiation one obtains: dx ~=kf(kt).

The inverse kt=f - ‘(x) exists sincefis strictly monotone. By substitution one gets dx

,,=kjy(n)] By settingg(x) = -fi-‘(~)]

one obtains

dx z-- --kg(x)

The ~nctional form of g(x) relies on st~cture-activity not explicitly known, at the moment.

relationships which are

~~us~ppe Toscano et al. /Catalyses Today 22 11994/ 489-510

509

In order to set the form of g(x) fully, we choose, in all respects empirically: g(x) =2.X?*

(25)

By integration one obtains 1 X=(l+kt)* If one assumes k = ai’k’,

lZq.

18 follows.

4. Conclusions The new model hypotheses for irreversible thermal deactivation of acid phosphatase from potato can be summarised as follows: (1) during denaturation the number of catalytic sites is not reduced, but the activity of each site is decreased because of the time-progressive loss of the original ~onfo~ation, probably due to slight, though irreversible, changes in structure; (2) the enzyme follows a constrained denaturation pathway and a unique (though still unknown) kinetic parameter k( time- ’ ), which depends on experimental conditions, regulates the overall process; (3) different activitydecay profiles obtained under different experimental conditions coincide, provided the activity is plotted against dimensionless time kt;

at any time, the rate of conformation change only depends on the conformation attained at that time and on parameter k, according to Eq. 24. Obviously, inferences about ~onfo~ation~ changes have to be validated by structural characterisation of the denatured form(s). In any case, the empirical models formally derived from nth-order kinetics deserve interest as a tool to describe the dependency of non-first-order kinetics on medium variables, with acceptable accuracy and with a minimum number of parameters. (4)

References [ 1J J.E. Bailey and D.F. O&s, Biochemical Engineering Fundamentals McGraw-Hill, New York, 1986, p. 176. 121 A.M. Klibanov, A. Chendrasekhar and B.N. Alberti. Enzyme Microb. Technol., 5 ( 1983) 265-268. 131 M.P. Scollar, G. Sigal and A.M. Klibanov, Biotechnol. Bioeng., 27 (1985) 247-252. 141 I. Chibata, T. Tosa and T. Sato, J. Mol. Catal., 37 (1986) l-24. 151 G. Carrea, Trends Biotechnol., 2(4) (1984) 102-106. [61 L. Gianfreda, G. Marrucci, N. Grizzuti and G. Greco, Jr., Biotechnol. Bioeng., 26 (1984) 518-527. [7] A.M. Chase, J. Gen. Physiol., 33 (1950) 535-546. 181 J. Fischer, R. Ulbrich, R. Ziemann, S. Flatau, P. Wolna, M. Schleiff, V. Phtschke and A. Schellenberger, J. Soiid Phase Biochem., 5 ( 1980) 79-96. 191 J.P. Henley and A. Sadana, Biotechnol. Bioeng.. 26 (1984) 959-969.

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