Effect of subunit dissociation, denaturation ... - Wiley Online Library

Effect of Subunit Dissociation, Denaturation, Aggregation, Coagulation, and Decomposition on Enzyme Inactivation Kinetics: II. Biphasic and Grace Period Behavior Robert W. Lencki* Department of Chemical Engineering, Universite Laval, Sainte-Foy, Quebec, Canada GlK 7P4 Joseph Arul Department of Food Science and Technology, Universitb L aval, Sainte-Foy, Qubbec, Canada G1K 7P4 Ronald J. Neufeld Department of Chemical Engineering, McGill University, Montreal, Qubbec, Canada H3A 2A7 Received June 8, 19921Accepted July 27, 1992 A model previously developed to characterize enzymatic in-

activation behavior was used to explain the non-first-order biphasic and grace period phenomena that are often observed with oligomeric enzymes. Luciferase and urease were used as model enzymes for this study. It was hypothesized that with a dimeric enzyme such as luciferase, the oligomer initially dissociates reversibly into two native monomer species. These native monomers can then reversibly denature and irreversibly aggregate and coagulate. With the hexamer, urease, two trimers are formed that can subsequently aggregate to form an inactive hexamer. The dissociated monomer species of luciferase do not possess catalytic activity, so the inactivation mechanism is biphasic; the first slope of a first-order kinetic plot is influenced by the reversible oligomer/monomer/denatured-monomer transition whereas the second slope is associated with either irreversible aggregation or coagulation. In contrast, the trimer of urease has the same activity as the hexamer; therefore, during the initial hexamer-trimer transition, little activity loss occurs. However, as the trimer concentration increases, activity decreases a s a result of trimer aggregation. As a result, grace period inactivation behavior is observed. 0 1992 John Wiley & Sons, Inc. Key words: enzyme inactivation biphasic grace period INTRODUCTION

Most enzymes display first-order inactivation kinetic^.'^ However, the exact phenomenological meaning of the first-order rate constant is not always apparent. It has been previously shown that dissociation,20 denaturation,” decomposition,’ a g g r e g a t i ~ n ,and ~ ~ coagulation” reactions can all occur during the inactivation process. In addition, dissociation and denaturation generally lead * To whom all correspondence should be addressed at present address: University of Guelph, Dept. of Food Science, Guelph, Ont., Canada, N1G 2W1. Biotechnology and Bioengineering, Vol. 40, Pp. 1427-1434 (1992) Sons, Inc.

0 1992 John Wiley &

to reversible inactivation whereas decomposition, aggregation, and coagulation will irreversibly inactivate the enzyme.24Nevertheless, even with such a wide range of phenomena occurring simultaneously, if certain reaction rates are significantly higher than others, the overall inactivation process can still display apparent first-order kinetics .24 When the rates of two or more inactivation reactions are of the same order of magnitude, complex non-firstorder kinetics can potentially occur. Not surprisingly, two types of complex non-first-order kinetics have been observed. The loss of activity of Cypridina luciferase as a function of time and temperature’ is a classic example of the first type of complex inactivation behavior (Fig. 1). At low temperatures (e.g., 40°C), the inactivation curve is biphasic: Rapid inactivation is followed by a decelerated decay which eventually levels off, approaching an activity plateau. As the temperature increases from 40 to 5OoC, the initial inactivation stage becomes more and the second less prominent, until, at 52S°C, the first stage almost totally dominates. This biphasic behavior is a relatively common phenomenon in enzymatic inactivation The second type of non-first-order inactivation behavior involves a grace period: An initial stable phase where little activity loss is observed is followed by periods of accelerating and decelerating inactivation. Grace periods have been previously observed with enzymes such as galactosyl transferase3 and ~ -~ -fru c t o fu ra n o s i d a s e . ~~ Non-first-order inactivation behavior was originally explained by a parallel model that assumed that the native enzyme exists in a variety of forms with each form possessing its own activity and rate of inactivation.’ This explanation could be considered appropriate if

CCC 0006-3592/92/01101427-08

1.0

I

L,

0.8

0.6 a?

20.4 0.2

0.0

I

0

1

,

2

I

3

TIME (h)

+ , 4

I

5

6

Figure 1. Inactivation of luciferase as a function of temperature:' ( 0 )40°C; (0)45°C; (A) 47.5"C; (0) 50°C; (+) 52.5"C.

isoenzymes are present in solution or if the enzyme is cross-linked or immobilized. Nevertheless, this inactivation mechanism is limited; it can explain biphasic decay but is incapable mathematically of demonstrating a grace period. Furthermore, an analysis of inactivation behavior of enzymes in solution exposed to a rapid step change in temperature has indicated that at least some enzymes displaying biphasic behavior do not follow a parallel rnechani~rn.'~ More recently, it has been hypothesized that biphasic behavior results from the presence of an intermediate in the native-to-denatured tran~ition.'~ This model can generally be described by the following progression: P e Z e D (1) where P is the native enzyme, Z is an intermediate, and D is the denatured structure. The kinetic equation resulting from this inactivation mechanism has the advantage that it can demonstrate certain types of grace period beha~i0r.l~ A further modification of this mechanism has been proposed that involves the existence of several intermediates which are in equilibrium with each other.13 In addition, a general model composed of both series and parallel steps has also been suggested" and has been used to analyze the inactivation of a wide variety of enzymes." Unfortunately, the exact nature of the intermediate species in these models has never been explicitly defined. Furthermore, it is often assumed in these models-without any theoretical justificationthat one or more steps in the inactivation pathway are irreversible. Other analyses of biphasic inactivation have been strictly and do not incorporate any understanding of the underlying phenomena. The goal of this study was to use a mechanism previously developed to analyze first-order inactivation to better understand the enzyme species and underlying phenomena that can produce non-first-order behavior. Two model enzymes were chosen for this analysis which

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display non-first-order biphasic (luciferase) and grace period (urease) behavior. Luciferase was chosen because appropriate inactivation versus temperature results already exist for this enzyme.' We conducted experiments with urease since no suitable grace period inactivation versus temperature results could be found in the literature. For each of these two enzymes, the general inactivation equations were simplified using assumptions relevant to the catalyst in question. A kinetic analysis of the resulting mechanism was then performed, and the solutions obtained for each enzyme were compared to the literature or experimental results. EXPERIMENTAL

Jack bean urease (81 IU mg-') was obtained from Cooper Biomedical (Mississauga, Ontario). Reagent grade urea, HCI, and dithiothreitol were purchased from Fisher Scientific (Montreal, Quebec). All solutions were prepared using boiled-out distilled and deionized water and all glassware was cleaned with chromic acid and rinsed thoroughly before use. Following the advice of Blakeley et a1.: 2.0 X 10-6Mof dithiothreitol was also added to all urease solutions to inhibit enzyme polymer formation. Inactivation experiments were conducted by adding 25 mL of a 0.75 g L-' solution of urease into a sealed test tube and placing the test tube into a water bath at the desired temperature. At appropriate time intervals, 2 mL of the solution was removed. The urease activity of this sample was then immediately determined at the same temperature as the inactivation experiment. The pH-stat method4 was used to determine urease activity at a urea concentration of 0.625M. The apparatus consisted of a Radiometer model 80 autotitration system (Copenhagen, Denmark) fixed with a custom 0.050-L jacketed glass reactor connected to a Haake refrigerating water bath set at the appropriate temperature. A 0.10M HCI solution (calibrated by titration versus 0.1000M NaOH) was used to maintain the pH at 7.00 t 0.01. The system was interfaced to an IBM XT computer using custom hardware and software to accurately calculate the acid addition rate as a function of time. The pH-stat system was calibrated by comparing the rate of acid addition to the rate of ammonia production. Ammonia concentrations were determined using the phenol-hypochlorite reaction.33 MECHANISMS OF INACTIVATION

Globular monomeric proteins can inactivate via the following denaturation reaction: where PI and P: are the native and denatured structures of the monomeric protein. Since most enzymes are

BIOTECHNOLOGY AND BIOENGINEERING, VOL. 40, NO. 11, DECEMBER 20, 1992

oligomeric," dissociation also plays an important role in the inactivation process. For an enzyme with two subunits such as luciferase," it has been hypothesized that this process would involve two steps21:

5y

k2

k3 2P1 (3) p2 where the first stage is an association-dependent structural change and the second is an association/dissociation interaction. Enzymes with four or more monomers (urease has six l o ) can potentially have several oligomer-to-monomer pathways. Urease can form a stable trimer'; therefore, the first dissociation step would most likely be k4

k5 2P3 (4) e p 3 p 3 %y It has been previously assumed that the trimer then separates directly into three monomers by the following mechanismz4:

P3

BlPHASlC INACTIVATION BEHAVIOR: LUCIFERASE

The rate of change of all the enzymatic species of luciferase as a function of time ( t ) are given by Eqs. (9) to (14): dt

dm dt 2k,[PlP,] =

+ k-2[PlPl]

=

-k2[P2]

-

2k-3[Pi]2- kl[P1]

(9)

+

PI Pi PI -k.l 3 5

(5) It will be apparent in the subsequent analysis that the pathway of urease trimer dissociation will not significantly alter the model behavior under the conditions examined in this study so that the uncertain validity of this step will not have a serious impact on the present analysis. The dissociation and denaturation processes outlined above are in most cases completely reversible." Consequently, adjunct processes must be responsible for the irreversibility of enzyme inactivation. Ahern and Klibanov' have suggested that exposure to very high temperatures (i.e., 100°C) can lead to irreversible decomposition reactions, primarily the deamidation of asparagine. However, it is questionable whether decomposition is a significant reaction with the model enzymes used in this study because neither luciferase16nor urease" contain asparaginyl residues. Other potentially irreversible mechanisms generally involve intermolecular interaction^.^^^^' As previously defined,24under conditions where oligomer aggregation is inhibited, aggregation normally involves native monomers whereas coagulation occurs among denatured species. Thus, aggregation [Eq. (6)] and coagulation [Eq. (7)] can occur as follows: PI + P: P1* + P:

k

P:+l p:+i

(6)

(7) where Pl' and P: are aggregated and coagulated species, respectively. Consequently, the luciferase inactivation mechanism will involve the combined reactions of Eqs. ( 2 ) , (3), (6), and ( 7 ) . With urease, Contaxis and Reithel' have shown that two active trimeric species can also specifically aggregate to form an inactive hexamer P6' : 2P3 P6' (8) Thus, urease inactivation can be characterized by Eqs. ( 2 ) and (4)-(8).

d[P?l - dt

k , [ P l ] - k - l [ P ? ] - 2kg[P1*]2

d[P:] - - - iks[P1*][P:-I] - ikg[P:] [Pi*] dt

(i

I2)

(14)

A mass balance on all species yields m

m

[PIT = 2[P2] + 2[PlPl] + [Pl] + %[PI] + C i [ P i * ] i=2

i=l

(15)

where [PIT is the total concentration of enzyme monomers in solution. If it is assumed that the denatured, aggregated, and coagulated species are inactive and at t = 0 only the P2 species is present, then enzymatic activity relative to the activity at t = 0 is given by

where k+,k++,and k are the proportionality constants relating the concentrations of [P2],[ P I P l ] ,and [ P l ] to their enzymatic activities, respectively. Equations (9)(16) can be theoretically solved simultaneously to predict the inactivation behavior of luciferase. This complex model can, however, be simplified by making suppositions with regard to the relative magnitudes of the various kinetic constants. As stated p r e v i ~ u s l ythe , ~ ~native-to-denatured transition is generally rapid in comparison to the dissociation steps; thus, it can be supposed that the PI and P: species are approximately at equilibrium during the inactivation transition. At moderate temperatures and at low [PIT,the irreversible aggregation [Eq. (6)] and coagulation [Eq. (7)] steps can also be presumed to be slow

LENCKI, ARUL, AND NEUFELD: BlPHASlC AND GRACE PERIOD ENZYME INACTIVATION KINETICS

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versus the steps of Eqs. ( 2 ) and ( 3 ) . At this point in the analysis, we will assume that the reaction rates of Eqs. (6) and (7) are equal to zero. Further simplifications can be made concerning the activity of inactivation intermediates. In most cases, isolated monomer species do not possess residual catalytic activity.l2If this is true, then k = 0. Moreover, the concentration of the dissociation intermediate [PIP1]will most likely be low since this species should dissociate quickly. Hence, the steady state approximation that d [ P I P l ] / d t= 0 as well as the assumption that k 3 % k - z can be made. The above simplifications can now be applied to the inactivation model. Differentiating Eq. (16) and assuming that k = 0 and d [ P I P l ] / d t= 0 gives the following: d(A/Ao) dt

2

v

2

I 1 .o

0.5

[PIT dt %

k - z , Eq. (10)

An expression for [ P I ]can be arrived at by rearranging Eq. (15), assuming that [ P I P l ] [P:] , (i > l ) , and [PI*] (i > 1 ) are equal to zero and K1 = k l / k - , (i.e., the [ P ; ] and [ P I ]species are at equilibrium): [PIT - 2pz1 ( 1 + KI)

By substituting Eq. (20) into Eq. (19), rearranging, and , following supposing that (&Ao) = 2 [ P 2 ] / [ P I Tthe equation for the rate of change of @ / A o )as a function of time can be obtained:

Q

1.5

Xt

(17)

Substituting Eq. (18) into Eq. (9) and the resulting expression into Eq. (17) produces the following equation:

[Pl] =

0.0

0.0

1

Once again, if d[PIPl]/dt= 0 and k3 can be rearranged to give

\

Figure 2. Inactivation of a dimeric enzyme as a function of the constant Xr for varying Y / X values: (--) Y / X = 0.1; (- -) 1.0; (- -) Y / X = 3.0; (- - -) Y / X = 10.0. Y/X

ure 2 are approximately equal to - k 2 . The value of k Z will increase as the temperature is raised, and this would explain why, with luciferase, the initial slope of a first-order plot becomes more negative with increasing temperature (Fig. 3). All the curves of Figure 2 eventually reach a plateau value which is determined by the value of Y / X . An increase in temperature will also increase Y/X, as a result, the plateau will occur at lower A/Ao values. Eventually, when Y / X is very high, a simple first-order plot will be observed with a slope exactly equal to - k z [cf. Eq. (21)].It is interesting to note that the constant X is also a function of the total enzyme This is not monomer concentration in solution ([PIT). surprising since higher enzyme concentrations will favor the second-order reassociation reaction in Eq. (3). When [PIT is increased, X will increase, Y / X will de-

where

Figure 2 provides a graphical representation of a general solution for Eq. (21). This equation was normalized by plotting A/Aoversus Xt for various values of Y / X . Since it was assumed in the development of Eq. (21) that the Pl and P: species are essentially at equilibrium and k s and k 9 are zero, the curves of Figure 2 represent the transition from when only the P2 species is present in solution at t = 0 to when all the Pz, PIPl,PI, and PT species are at equilibrium at t = m. It has been shown previously24that the initial slopes of the curves of Fig-

1430

0.1

0

I

2

3

TIME (h)

4

5

Figure 3. Semilog plot of the inactivation of luciferase as a function of temperat~re:~ (0) 40°C;(0) 45°C;(A) 475°C;(0) 50°C; (+) 52.5"C.


crease, and consequently, the value of the A/% plateau will increase. Thus, the enzyme will appear to be more stable at higher concentrations [that is, if Eqs. ( 6 ) and ( 7 ) are not also accelerated by this same concentration increase]. Gianfreda et aI.l4 have in fact observed this high concentration stabilization phenomenon during the biphasic inactivation of acid phosphatase. With luciferase (Fig. l ) , it is evident that for all temperatures, a stable plateau is never reached and there is a slow decrease in activity after the initial rapid decay. Chase7 observed that if a dilute luciferase solution was first heated to 48°C for a period of 15 min, 55% of the original activity was lost. However, when this solution was immediately cooled to room temperature, a slow reactivation was noted; the enzyme regained 85% of its original value after 24 h. These results support our supposition that the initial inactivation reactions are reversible. The slowness of this reactivation also affirms that a second-order reassociation reaction is most likely present [Eq. ( 3 ) ] .The fact that after 24 h only 15% of the luciferase was inactive suggests that the final step is irreversible and slow with respect to the initial reversible steps. It can also be inferred from these results that if an enzyme solution is dilute and the reactivation experiment is not conducted for a long enough period of time, the enzyme may appear to be irreversibly inactivated while in fact the reactivation process is limited by slow kinetics. The slow decay observed in the plateau region of Figure 1 is most likely due to irreversible aggregation and coagulation reactions. It has been previously shown that if aggregation [Eq. ( 6 ) ] dominates and Eqs. ( 2 ) and ( 3 ) are approximately at equilibrium, inactivation kinetics will take on the following formz4:

4

+

I

-5

+ +

3- +

a

u

+

0

"

A

1

2

3

TIME (h)

4

5

Figure 4. Plot of (A0/A)"' versus time for the inactivation of luciferase as a function of temperature:' (0) 40°C; (0) 45°C; (A) 47.5"C; (0) 50°C; (+) 52.5"C.

GRACE PERIOD INACTIVATION BEHAVIOR: UREASE

As a first step, the rate of change of all the enzymatic species in Eqs. ( 2 ) and (4)-(8) as a function of time ( t ) can be determined:

Therefore, a plot of (Ao/A)"' versus t should be linear with a slope of ks(8KzK3[P]T)1'2. Likewise, if coagulation [Eq. ( 7 ) ] is the only mechanism present and it is assumed that coagulated species P? with i > 2 are initially at low concentrations, then d[P:]/dt = -2k9[PTI2 and the same plot of (AO/A)1'2 versus t would have a slope of k 9 K 1 ( 8 K 2 K 3 [ P ] T )A 1 ' plot 2 . of (AO/A)I"versus t for luciferase is approximately linear at extended reaction times for all temperatures (Fig. 4 ) . If aggregation or coagulation is the dominant mechanism, no simple direct correlation would exist between the second biphasic slope of a semi-log plot and the model kinetic constants. Nevertheless, the slopes of Figure 4 are not a strong function of temperature. This is expected because aggregation and coagulation in unstirred dilute enzyme solutions is a process that would most likely be diffusion limited. Since protein diffusion is not a strong function of temperature, this would give credence to the argument that either Eq. (6) or ( 7 ) is the secondary reaction of this biphasic inactivation. LENCKI, ARUL, AND NEUFELD: BlPHASlC AND GRACE PERIOD ENZYME INACTIVATION KINETICS

1431

Next, a mass balance on all species yields

6[P3P3] + 6[P6f]+ 3[P3]

[PIT = 6[P.5]

m

m

+ 3[PlP,Pl] + [PI] + i=2 cli[P,']+ E i [ P : ] r=l

(33)

If it is assumed that only the hexamer and trimer are active and, at t = 0, only the P6 species is present, enzymatic activity ( A ) relative to the activity at time zero ( A o )is then given by A 6[P6] =Ao [PIT

6k**[P3P3] 3ko[P3] 3koo[Z'1PlP1] +-~ * [ P ] T ~ * [ P ] T k*[P]r (34) where k*, k**, ko, and koo are the proportionality constants relating the concentrations of [PSI,[P3P,],[&], and [ P I P I P Ito ] their enzymatic activities, respectively. The inactivation behavior of urease can now be predicted by simultaneously solving Eqs. (24)-(34). Figure 5 illustrates the inactivation behavior of urease at 45, 50, and 52.5"C. At temperatures below 45"C, inactivation was a slow process, whereas at 55"C, almost total inactivation was observed within the first half hour of exposure. A distinct grace period was observed and the length of this phase appeared to decrease with increasing temperature. At 45 and 50°C, only an accelerated rate phase following the grace period was observed. But at 52.5"C, the inactivation curve was more sigmoidal, with a decrease in slope at extended reaction times. In the case of urease, the aggregation of the trimer to form an inactive hexamer [Eq. ( S ) ] is known to be very rapid and the dominant irreversible reaction'; as a result, the other aggregation and coagulation reactions can be presumed to make a negligible contribution to the overall inactivation scheme (i.e., k s and k9 are approximately zero). The assumption that at moderate temperatures the trimer aggregation reaction [Eq. ( S ) ] +

+

should also dominate over trimer dissociation [Eq. (5)] and trimer reassociation [Eq. (4)] leads to the supposition that k;( is much greater than k6 or k - 5 . Furthermore, if k5 and k;(are of the same order of magnitude, it can be deduced that k5 S k - 5 . Simplifications can also be obtained from presumptions regarding the relative concentrations of some of the intermediate species. For example, the dissociation intermediate P3P3 can be assumed to be present at low concentrations because this species should almost instantaneously dissociate. Based on this postulate, the steady state assumptions that d[P,P,]/dt = 0 as well as the simplification k5 k-4 can then be made. Consequently, Eq. (25) can be equated to zero and rearranged to give

Substituting Eq. (35) into Eq. (24) and assuming that k5 is much greater than k - 4 and k-5 then gives

If it is assumed that the P3 produced by dissociation is channelled toward P ; production so insignificant amounts of PIPIPl,PI, and P? are present in solution, the mass balance of Eq. (33) can be simplified and rearranged to produce (37) If k;(is much greater than k 6 and k - 5 , then Eq. (26) can also be reduced to

Experimental results have shown that the catalytic activities per subunit of the P6 and P3 species are essentially equivalent" (i.e., k* = k o). In accordance with this, the expression for overall activity [Eq. (34)]can be redefined as

Equations ( 3 6 ) - ( 3 9 ) can now be solved simultaneously to obtain an expression for A/Ao as a function of time and the kinetic constants k 4 and k:. To accomplish this, Eq. (36) can first be directly integrated to give

0.0

/

[P6]= -exp( -k,t) 6

0

I

I

I

I

1

2

3

4

TIME (h)

5

Inserting Eq. (40) into Eq. (38) gives the expression

Figure 5. Inactivation of urease as a function of temperature: (0) 45°C; ( 50°C; ! I (A) ) 52.5"C.

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Following the solution to a similar equation arrived at by Chien,* Eq. (41) can be solved to give:

K = (2ks"[PIT)/ where P = (iJ1(2i&))/(Hj1)(2i& ( 3 k 4 ) , T = exp(-kd), and i = -1. The terms J o , HA1),J , , and Hi1)are Bessel functions of the first ( J ) and third ( H ) kind of order 0 and 1, respectively. Equations (40) and (42) can now be introduced into Eq. (39) to provide an expression for the relative activity as a function of time. Figure 6 provides a graphical representation of this solution to Eq. (39). The graph was normalized so that k4t is on the x axis and each curve is for a different value of K . From a mathematical point of view, to display sigmoidal inactivation behavior, one of the inactivation reactions should have an order higher than 1. This is the case with our model since the trimer aggregation step [Eq. (S)] is second order. Initially, the transformation from the Pbto P3 species will not result in a significant loss of activity. But since Eq. (8) is second order, the P3 concentration must build up before the second step of the inactivation process can occur. This intermediate buildup is not required in the simple series-type mechanism so only a parabolic curve and not a true grace period (the initial phase of sigmoidal behavior) will be observed. Similarly, at long reaction times the concentration of P6 and consequently P3 will eventually decrease. The second-order nature of the reaction given by Eq. (8) will then exhibit a decreasing rate of inactivation, and this will produce the second stage of sigmoidal behavior. The normalized curves in Figure 6 all show a distinct grace period and, at high values of K , sigmoidal behavior. The value of K will generally increase as the temperature rises; therefore, the three curves of Figure 5 closely mirror the curves of Figure 6 corresponding to increasing values of K ,

$1

1 .o

0.8

0.6

With luciferase, as with most enzymes, the dissociated subunits (e.g., Pl) do not possess residual activity. Urease appears to be an exception, possessing an active P3 trimer. It is also interesting to note that other enzymes such as galactosyltransfera~e,~ P-gluc~sidase,~ and P-D-fructofuranosidase,3°which demonstrate grace period behavior, also have active dissociation products; galactosyltransferase is a tetramer with active monom e r ~P-glucosidase ,~~ is a dimer with active rnonorner~,'~ and P-D-fructofuranosidase is a tetramer with active dimers." Nevertheless, further work will be required before it can be stated that dissociation product activity is a general prerequisite for grace period behavior. DISCUSSION AND CONCLUSIONS

It is evident from the two examples given in this publication that the inactivation model previously developed to explain first-order inactivation phenomenaz4can be modified to explain complex non-first-order behavior. Using the two model enzymes, luciferase and urease, and appropriate assumptions with regard to the relative rates of the various reactions involved, inactivation curves demonstrating both biphasic and grace period phenomena could be obtained. The model also was able to explain the effect of temperature on both these inact ivation behaviors. Like the simpler series models that are currently found in the literature [e.g., Eq. (l)], the proposed inactivation model can properly fit non-first-order experimental results. However, the strength of our model is that when used to analyze the inactivation of a particular enzyme, a much better understanding of the underlying inactivation reactions (whether they be reversible or irreversible) as well as the nature of the reaction intermediates is obtained. But perhaps more importantly, our model takes into account interactions among the inactivation reactions. In fact, these interactions are the basis for non-first-order behavior. Nevertheless, in this study, only the two enzymes luciferase and urease were examined. Obviously, further work with many other enzymes will be required to confirm the universality of our model. This work will, however, eventually lead to a better understanding of enzyme inactivation phenomena. References

d

20.4 0.2

0.0

k,t

Figure 6. Inactivation of a hexameric enzyme as a function of the constant k 4 t ; K values: (--) 0.5; (- -) 1.0; ( - - ) 2.0; (- - -) 4.0; (-- - - - -) 8.0; (. '. ' ' .) 16.0.

1. Agarwal, P. K. 1985. Heterogeneous denaturation of enzymes: A distributed activation energy model with nonuniform activities. Biotechnol. Bioeng. 27: 1554-1563. 2. Ahern, T. J., Klibanov, A . M . 1985. The mechanism of irreversible enzyme inactivation at 100°C. Science 228: 1280-1284. 3. Belon, P., Louisot, P. 1974. Glycoprotein biosynthesis in the aortic wall. 111. Study of soluble galactosyl transferase in intima1 cells. Int. J. Biochem. 5: 409-415. 4. Blakeley, R . L., Webb, E. C., Zerner, B. 1969. Jack bean urease (EC 3.5.1.5). A new purification and reliable rate assay. Biochemistry 8: 1984-1990. 5. Caminal, G., Lafuente, J., Lopez-Santin, J., Poch, M., Sola, C. 1987. Application of extended Kalman filter to identification of enzyme deactivation. Biotechnol. Bioeng. 29: 366-378.

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