ROLAND F. BEERS,JR., AND IRWIN W. SIZER
290
0.2 0.4 0.6 0.8 1.0 WT. OF GEL (G/100 CC. SOLN.). Fig. 6.-Effect of temperature on the adsorption of NH4+by calcined silica gel from 0.05 N NH40Ac solution.
can chemisorb bases as weak as quinoline quite strongly, as shown by Mills, B,oedeker arid Oblad. l7 (17) G. A. Mills, E. R. Boedeker and A. G . Oblad, J . A m . Chem.
sot., l a , 1.554 (1950).
Vol. 57
It is apparent from the wide variation in acid strength of the different acid centers that the A1 atoms are riot all disposed in the structure in the same way. The most acidic spots are undoubtedly those intimately associated with silica in such a way that the A1 atoms acquire a coordination number of 4 (cf. ref.4). However, it is not possible to say on the basis of the present data whether the acid centers showing low acid strength in the low teinperature adsorption experiments do or do not acquire sufficient acidity a t cracking temperatures to be important to the cracking reaction. Acknowledgment.-The author acknowledges gratefully the assistance of members of the Inspection Section of the Socony-Vacuum Laboratories, Research and Development Division, and of Mr. J. L. Hammond in obtaining much of the analytical data. He also wishes to express his appreciation to the management of the SoconyVacuum Laboratories for permission to publish this paper.
KINETICS AND THERMODYNAMICS OF THE STEADY STATE SYSTEM OF CATALASE WITH HYDROGEN PEROXIDE BY ROLAND F. BEERS,J R . , ~A-N~D IRWIN W. SIZER Department of Biology, Massachusetts Institute of Technology, Cambridge, Massachusetts Received April 4, 1968
Equations for calculating the velocity constants of the two consecutive reactions between catalase and hydrogen peroxide from values of the over-all velocity constant and the concentration of the intermediate complex have been derived from steady state theory. The thermodynamic constantfi AH, AF and AS, of the consecutive reactions have been found t o be the same and indicate that the rate determining steps are the same.
Introduction Models of the kinetics of catalytic systems based on mathematical and chemical equations are essential for interpreting any theory for the mechanism of the action of a catalyst and for explaining the effects of inhibitors and accelerators on the catalyst-substrate system. On the basis of the mechanism of the catalase-hydrogen peroxide reaction proposed by several investigator^,^-^ catalase presents the unique feature of having a single substrate species which acts as both the acceptor (reductant) and donor (oxidant) molecule. In view of the fact that the hemes of catalase show no evidence of heme-heme interaction,' the reactions between catalase and peroxide are written thus by Chance8 (1) This work was done under an American Cancer Society, Inc., Fellowship recommended by the Committee on Growth of, the National Research Council. (2) The steady state analysis of this paper is from a thesis submitted by R. F. Beers, Jr., t o the Massachusetts Institute of Technology in June 1951 in partial fulfillment of the requirements for the degree of Doctor of Philosophy. (3) Naval Medical Research Institute, National Naval Medical Center, Bethesda 14, Maryland. (4) B. Chance, Biochem. J., 46, 387 (1950). (5) J. B. Sumner, A. L. Dounce and V. L. Frampton, J . Bioi. Chem., 136, 343 (1940). (6) B. Chance, ibid., 180, 947 (1949). (7) B. Chance, ibid., 179, 1299 (1949). (8) A recent paper by Chance9 indicates t h a t the active form of the substrate molecule is the undissociated acid, HOOH, and of the eneyine is the hydrated form, FeOHr. Should this prove to be correct it does not alter the steady state theory discussed below. The re-
FeO1-I , FeOOH
bl + HzOz ++ FeOOH + HzO kZ
ka + H202-++ FeOH + HzO + k-
(a) 0 2
'(b)
I
where FeOH represents one heme of catalase. In the lowest range of hydrogen peroxide employed, the specific reaction rate (velocity constant, k,) of the over-all reaction does not change in magnitude or order with variations in concentration of hydrogen peroxide. Theory.-The rate expressions for the substrate concentration, s, and the primary complex concentration, p , based upon the consecutive reactions (a) and (b) are ds/dt = --k,(e
+ b2p - k4ps
(1)
- p ) s - L ( p ) - krps
(2)
- p)s
and dp/dt
=
kl(e
actions (a) and (b) become
FeOHz
.
FeH20z
kl + Hz02 -++ FeHzOz + HzO kZ
.
+ HzOZC+ FeOHz + OZ + HzO kk4
1
(9) B. Chance, J . Bioi. Chem., 194, 471 (1952). (10) R. K. Bonnichsen, B. Chance and H. Theorell, Acta Chem. Scand., 1, 685 (1947). (11) R. F. Beers, Jr., and I. W. Sizer, J. Bioi. Chem., 196, 133 (1952).
KINETICS AND THERMODYNAMICS OF CATALASE WITH HYDROGEN PEROXIDE
Mar., 1953
where e is the total molar heme concentration. Adding (1) and (2) gives (cf. 12). dp/dt
+ $kips = -ds/dt
ds/dt = -k.es
(4)
If we substitute (4) into (3) then dp/dt
dependent variables (cf. 14). From (8) and (14) it can be shown that 1
(3)
The empirical rate formula for the substrate concentration after an induction period of a few micro-seconds as a function of the total heme concentration is
+ 2k4ps = kaes
(5)
But (4)is readily integrated to give (6)
= sOe-kaet
where so is the initial substrate concentration after the induction period. Substituting (8) into ( 5 ) and solving for p yields
where p o is the initial primary complex concentration after the induction period. k, has a value of approximately 10’ 1. mole-l set.-', while e is M , and so is M for most experimental studies.l‘ Substituting these values into (7) gives
- p/e i Zks/2k1
(15)
IC,, p and e are readily measured experimentally. Therefore, equations (8) and (15) can be used to calculate the values of kl and k4. The experimental activation energy (E,) of the over-all reaction can now be resolved by equation (14) into the separate contributions of kl and kd. Using the relation, k = f exp ( - E I R T ) , where E is the “experimental energy of activation” and f is the temperature-independent “frequency factor”16 equation (14) becomes when simplified
Experimentally, it has been shown11J3that d2(ln ks)/(.l/T)zE 0: This implies (as can be shown by requiring that in equation (18) the second derivativc of In ks with respect to 1/T vanish) that E1 E E4
(17)
Moreover, if one substitutesfs exp ( - E s / R T ) for ks in equation (16) and differentiates with respect to R T , it can be shown that E. E E1 E E4
We observe that in less than a second the exponential term drops off to an insignificant value, since the magnitude of ‘2k& is never less than unity. Equation (7), therefore, reduces to the approximation
291
(18)
The conclusion, (17), can be reached in another way. From (8) and (15) can be derived (e Rk
- p ) / p % ka/ki E Rk, O r Efdfi exp (E1- Ea)/RT
(19) (20)
Therefore, unless E4 LZ El, Rk (as measured by (e p ) / p ) will vary as a function of temperature.16 Confirmation of the validity of this model can be Since e is constant p must be constant. Therefore established by determining whether p / e is a condp/dt E 0 (9) stant. Additional supporting data can be used to If, now, (1) is subtracted from (2) we have check equation (15) since all the variables of that equation have been published.16b17 Chancela has ds/dt = -2kl(e - p ) s 2kzp (10) shown with his rapid flow technique that p / e reSubstituting the value for p from (8) into (10) mains constant over a wide range of substrate conyields centration. The constancy of this ratio has also been confirmed when the notatin-glucose system is dsldt E - [2kl (1 (11) used as a continuous source of hydrogen p e r ~ x i d e . ~ We have repeated Chance’s experiments with catBut comparing (11) with (4) we observe that alase coupled with a crude preparation of notatin plus glucose and have also found that p / e is constant. We have in addition established the fact that (e - p ) / p or Rk does not change within experimental error (5%) over the temperature range of Therefore, if (6) and (13) are to be consistent, k z / s 0-50O.18 must be small compared to kl (cf. 13). Equation Experimental (13) simplifies to the general equation for the overApparatus.-For optical density measurements a Beckall velocity constant13 man spectrophotometer with 1 cm. path length quartz p / e E k&?ka
(8)
+
&)
2]es
(14)
It should be pointed out that p / e and Jcs are (12) K. J. Laidler and J. M. Socquet, THIS JOURNAL, 54, 519
(1950)’.
(13) Several authors have considered the second step, k4, t o be the ka and k4 are assumed to be equal.“ Part of this misinterpretation arose from,a failure to realize that k~ is determined as a function of e while k4 is determined as a function of p only. The same argument may be applied to kr, which is a function of (e p). ’ (14) B. Chance and D. Herbert, Biochem. J., 46, 402 (1950); B. Chance, J. Biol. Chem., 179, 1341 (1949). rate determining processa in which case
-
cuvettes and a hydrogen discharge lamp for a light source has been used. The cell compartment is temperature controlled with cooling coils.11 Because the primary complex of catalase is unstable a t high temperatures, it has been found necessary to construct a temperature controlled mixing apparatus which permits determinations of optical density immediately after mixing. Two I-ml. tuberculin syringes (one for catalase, the other for the notatin system), encased in a water jacket connected in series with the cool(15) J. A. Christiansen, J. Colloid Sci., 6 , 213 (1951). (16) A. C. Burton, J . Cell. Comp. P h ~ s i o l .9, , 1 (1936). (17) E. Chance, J. BioE. Chem., 179, 1311 (1949). (18) R. E’. Beers, Jr., and I. W. Sizer, Federation Proc., 11, 11 (1952).
292
ROLAND F. BEERS, JR.,AND IRWIN W. SIZER
in coils of the spectrophotometer, are clamped ip a horizontafposition over the phototube housing unit.'" Procedure.-Temperature equilibrium in the syringes is reached in less than a minute. During chis .time the operator adjusts the dark current and slit width of the spectrophotometer for use a t 4050 A. The plunger is pushed rapidly, emptying both syringes into the cuvette. A few bubbles of air are then forced from the air syringe through the solution in the cuvette and an optical density reading is taken. For active preparations of glucose oxidase the concentration of the primary complex of catalase remains constant for only a few seconds unless more air is bubbled through the solution. Each sample is checked three or four times by bubbling air into the system and taking density readings. The same solution is used for successive measurements of the absorption curve, until there is evidence that the optical density at 4050 A. does not return to the initial value after aeration.
'
Vol. 57
optical density of the Soret band of catalase drops rapidly during the first 30 minutes. This does not appear to be related in any way to the formation of the secondary complex between catalase and hydrogen peroxide reported by Chance4 and also observed by us.
Discussion
The validity of the proposed kinetic model of the catalase-hydrogen peroxide system has been verified by the agreement between the predicted and observed constancy of p / e over a wide concentration range of substrate. Additional confirmation should be obtained from the published values of k., k1 and p/e. Table I1 quotes experimental values of k , kl and p / e for horse liver," beef liver" Results and bactenall3 catalase and calculated values of The absorption curve of the primary complex is kl, k4 and Rk based on the given values for k, and similar to that previously published by Chance.19 The extinction coefficient reported for horse liver Plea TABLE I1 catalase (E$$'-, = 340) has been used to calculate SPECIFIC REACTION RATES X extinction coefficients of the primary complex. Horse Horse Beef The deflection in the extinction coefficient a t 4050 eryth. Bacterial liver liver (4 hemes) (4 hemes) (3 hemes) (3 hemes) A. produced by complex formation has been found to be between 45 and 50, the same as that reported ks (exp.1 8.75 13.3 10.0 6.7 by Chance. We have also observed that under anaki (CXP.) 7.5 15.0 erobic conditions the primary complexrapidlyreverts kl (theory) 5.83 11.0 7.6 5.0 to the uncombined form. The addition of a few kr (theory) 17.5 16.5 15.0 10.0 bubbles of air makes possible the rapid restoration e -~ / e 0.25 0.40 0.33 0.33 of the primary complex. At low concentrations of Rk 3.0 1.5 2.0 2.0 notatin, however, the primary complex is stable Ref. 17 13 17 11 for periods up to 30 minutes, depending upon The large discrepancy between the theoretical the temperature. Since the rate of oxidation by glucose oxidase and experimental values of kl in Table 11,is conis zero order a t high concentrations of glucose the siderably in excess of any experimental error. steady state concentration of hydrogen peroxide Two compensatory errors have been made by under the experimental conditions specified may be previous workers in determining the value of kl assumed to be proportional to the concentration of from kinetic data.13 The maximum reduction of the glucose oxidase. A hundred-fold dilution of the extinction coefficient a t 4050 A. has been taken the notatin (glucose constant) produced no change as an indication of completion of the reaction bein the concentration of the primary complex ( p / e tween free catalase and hydrogen peroxide (treated essentially as an irreversible second-order process), is constant) in confirmation of Chance. l a Table I presents three typical sets of values of while it actually represents a saturation of only 25% Rk a t 2 and 45'. Since values a t intermediate for horse erythrocyte and 40% for bacterial cattemperatures show no deviation, it is apparent alase. Correcting this error gives a value of kl that within experimental error there is no detect- too small because the rate of change of p is also a able change in concentration of the primary complex function of k4. In calculating kl' the assumption had been made that a t sufficiently low concentraover this temperature range. tions the k4 reaction could be neglected. HowTABLE I ever, as indicated from equation (lo), k4 exceeds Control Sample kl for all catalase systems in which p / e is less than Temp., dendenNo. i%' 0.5 (Rk = 1). Since the maximum value reported Exp. f0.5O sity sity samp. dev. p e - P Rk so far is only 0.4, IGq cannot be neglected. ThereA 3 681 582 4 0.5 99 201 2.03 41 681 580 6 1.5 101 199 1.97 fore, the published values of kl are in error and their B 2 483 414 3 0.0 69 144 2.09 similarity to the theoretical values cannot be used 43 483 418 4 0.0 65 148 2.27 as an additional proof of the kinetic model. C 2 620 534 6 1.5 86 191 2.22 In order to test quantitatively the validity of this 45 620 535 2 0.0 85 192 2.25 kinetic model, either kl or k4 should be known from Average 2.14 ;t 5 % dev. experimental data. The simplest differential equa. However, the concentration of the primary tion which may be integrated to solve for ki is complex appears to fall with the age of the catalase obtained by eliminating k4 by subtracting equation preparation, although two separate shipments of (4) from (2) catalase from the supplier had the same value. ds/dt - dp/dt = -2kl(e - p ) s (21) At higher temperatures in the presence of glucose and in the absence of any glucose oxidase, the A similar equation has been discussed by Chance.18 However, a rigorous solution of (21) is not possible . (18a) R. F. Beers, Jr., and I. R. S i ~ e r Anal. , Chem., in press. a t the present time because adequate rate curves (19) B. Chance, Acta Chem. S.*ond., 1, 236 (1947).
t
L
Mar., 1953
KINETICSAND THERMODYNAMICS OF CATALASE WITH HYDROGEN PEROXIDE
293
Certain conclusions concerning the rate deterfor ds/dt and dp/dt for the initial pre-steady state period of the catalytic process .have not been pub- mining step may be made. The similarities of the lished. Solutions utilbing a differential analyzer heat, free energy and entropy of activation for kl have been discussed by Chancez0 in reference to and k4 and the fact that the same substrate species unpublished data. It is not quite clear whether is involved in both reactions indicate that the rate his theoretical values of kl calculated from equa- determining steps are the same in the two reactions. tion (15)20 are compared with published experi- Diffusion of the substrate is the common denomimental values.l3 For the present the most accurate nator of both reactions, but the thermodynamic values of kl and k4 can be determined only with constants are not typical of a diffusion process. The decrease in randomness of the system ensteady state data. The obsewation that Rk and the observed activa- countered with each reaction process is no doubt tion energy of k8 do not change with tempera- related to the structural characteristics of the ture10s*1.14 justifies the conclusion that k,, kl and kd protein which set special steric requirements which must be fulfilled in the formation of the complex. have the same activation energies. From the statistical theory of reaction rates we Non-linearity of the Arrhenius equation plot may estimate the differences in the values of the associated with many enzymatic processes has been two entropies and free energies of activation attributed by various authors to a temperature sensitive equilibrium between active and inactive k(,,a, = ( x T / ~ ) ~ - A F w I / R= T ( ~ T / h ) e A s ( i , k ~e-AH(l,r)/RT /R forms of the enzyme (see Kistiakowsky, et aL21). (22) A possible explanation based on the requirement of where K is Boltzmann’s constant, h Planck’s constant, two partial irreversible rate determining steps T the absolute temperature, R the gas constant, in an enzyme system has not been given serious and AH is E1,4 - RT. The transmission co- consideration heretofore. The authors have exefficient is assumed to be unity. tended Christiansen’s treatment of non-catalytic Table I11 presents calculations of the various systems to include the unique kinetics of catalase. thermodynamic constants of three sources of From a theoretical standpoint it is important to catalase based on data reported in the literature note the possibility of a deviation from the Arand on the assumption that all three catalases are rhenius equation on the basis of these two intercharacterized by the fact that El = E4. Despite the pretations for the special case of catalase. wide variations in experimental activation energies, Acknowledgment.-The authors wish to thank the entropies and free energies of activation are Professors David F. Waugh and Charles D. approximately the same. Coryell for their helpful discussions on several aspects of this paper, and Mrs. June Rosenberg TABLE I11 for her technical assistance. EXPERIMENTAL ENERGY, FREEENERGY AND ENTROPY OF ACTIVATION OF THE Two REACTIONS, kl AND k 4 , MOLARCONCENTRATION OF HEMEAT 25’
FOR A
Reac-
Eenp,
kcal.
AF, kcal.
e.u.
Horse erythrocyte (4hemes)
kl kc
1.7 1.7
7.54 6.94
-21.5 -19.5
Bacterial (4 hemes)
kl
1.4 1.4
7.2 6.96
-21.5 -20.7
Beef liver (3 hemes)
ki
0.6 0.6
7.60 7.23
-25.5 -24.2
Source of catalase
tion
k4
k4
A&
(20) B. Chance, “Modern Trends in Physiology and Biochemistry,” E. 9. G. Barron, Editor, Academic Press, Inc., New York, N. Y., 1951,
p. 25.
Note Added September 17, 1952.-We are pleased to note that Chance, et have reached independently some of the general results (eq. (8) and (15) above) previously obtained by one of US.^ Our present derivation, however, is somewhat more rigorous than our earlier, or than Chance’s. On the subject of rigor, we should like to remark that in the current paper b Chadce, et their attempt (pp. 308 ff.) to integrate dp/& analytically is based on two contradictory requirements, uiz., that 8 E so (p. 308) and that e so (p. 309). Also, in this integration, the appeal to “correction factors” from computer studies seems a further admission that the analysis has only descriptive value. (21) G. B. Kistiakowsky and R. Lumry, J . Am. Chem. SOC.,71, 2006 (1949). (22) B. Chance, D. S. Greenstein and F, J, W. Roughton, Arch. Eiochem. Biophya., 37, 301 (1952).
