Kinetic Models of Catalase Actlon
1919
TABLE I
.31(32) 35 (36) 0 , 1 6 7 26(2,5) 29(28) 0 25 19(39) 21(21) 0 50 14(15) 16(17) 1 00 11112) 13(14) 2 00 8 ( 8 ) lO(10) 4.00 4(4) 1(5) 8 00 lil) 2(1) 12.00 O(0) I(0) 24.00 O(0) O(0) 0 083
18(18) 25 (25) 31(31) 30(31) 26(26) 18(18) 8(9)
3(3) l(1) O(0)
3 (2) 4(4)* 4(3) 5(4) 8(7) 9(9) 15(15) 15(76) 28(27) 24(25) 46(45) 33(31) 61(63) 31(30) 71(70) 27(28) 73(73) 27(27)
Figures in parentht'sck are calrulated by the compukr program. The others are experinienlally determined. E and F were not resolved analytically in the samples taken a 1 these time intervals and the figures ralculated by the computer fnr 1C and for F were added t.ogether for comparison.
*
stants may be assigned upon the basis of the proposed mechanism of mefhyl glucoside formation.' In deriving the equations no allowance was made for variation of concentrations within the resin particles or transport of the reactant into and the products out of the resin. The primary objective was to determine whether satisfactory equations
could be derived. The next step in the investigation will be the determination of rate constants for the single-phase reaction, under identical conditions, using p-toluenesulfonic acid as a catalyst. A comparison of the two sets of rate constants should then show the direction and magnitude of the effect of the resin upon the reaction. Further experimental work designed t o test the validity of the proposed mechanism, as applied to D-galactose, is also in progress. Supplementary Material Available. Derivations for rate equations and the computer program will appear following these pages in the microfilm edition of this volume of the journal. Photocopies of the supplementary material from this paper only or microfiche (105 X 148 mm, 24X reduction, negatives) containing all of the supplementary material for the papers in this issue may be ohtained from the ,Journals Department, American Chemical Society, 1155 16th St., N.W., Washington, D. C. 20036. Remit check or money order for $3.00 for photocopy or $2.00 for microfiche, referring to code number JPC-74-1918.
References and Notes ( 1 ) R. H. Pater. R. A. Coelho and D.F. Mowery, Jr.. J. Org. Chem., 38, 3272 (1973). (2) See paragraph at end of text regarding supplementary material,
Stable Intermediates in Kinetic Models of Catalase Action M. L. Kremer' and S. Baer Department of Fhysical chemist^, The Hebrew University,Jerusalem. Israel (Received September 24, 1973: Revised Manuscript R e c e h d May 20, 1974) Publication costs assisted by the Central Research Fund of The Hebrew University
The peroxidatic model of catalase action (A) and one of its extensions (C) are discussed. A mathematical relation between the final concentration of the intermediate of model A and the initial peroxide concentration is derived. I t is shown that the steady-state concentration is the upper limit of the possible final concentrations of the intermediate complex. In case C, which is a consecutive two-complex system, the first complex disappears a t the end of the reaction while the second complex persists a t infinite time. The terminal concentration of this complex may vary, depending on initial conditions and on the rate constants, between zero and the total enzyme concentration. Some aspects of the models relative to catalase kinetics are discussed.
Introduction Both experimental and theoretical aspects of catalase kinetics drew considerable attention in the past. Theoretical treatments were, as a rule, confined to the steady state.' Many important results were, however, obtained under circumstances in which the system was removed from the steady state.2 I t is therefore of interest to investigate the mathematical properties of the kinetic equations of model systems under non-st,eady-state conditions. The analysis reveals many interesting features, all of which are, however, not yet accessibie to experimental verification. A complete integration of the rate equations is in general not possible. A discussion must thus rely largely on the
analysis of the differential rate equations themselves together with their numerical integration with the help of computers. In this connection the kinetics of the intermediates is of special interest as they exhibit, in some cases, characteristic behavior in that they do not disappear a t the end of the reaction. In one particular case the intermediate accumulates in a two stage process, the srcond stage occurring a t the end of the reaction when the concentrations of all other reacting species tend to zero.
The Peroxidatic Mechanism The simplest mechanism for catalatic. action has been proposed by Chance, et al.' The Journal 01 Physical Chemistry lid 75 N3 19 1974
M. L..
1920
E
Kremer and S. Baer
kS--!-+ES
r - 1 1
1
ES A S -!-+E P Y
+
P
('4)
E denotes catalase haematin and S hydrogen peroxide. ES is the enzyme substrate complex. Small type letters denote concentrations. P is the total concentration of catalase haematin. The relative rate of formation of the intermediate and of' peroxide (the latter is a negative quantity) is given by
Hy integrating this expression between t = 0 and t = m (corresponding to the limits xo,O and O,p,, respectively) we obtain
-E, a 8
Figure 1. pm as a function of &: e = 1 pM, R(A) = 0.1, p * ( A ) = 0.909 pcM NB) = 1, p*(B) = 0 500 p&f RC) = 10, p*(C) = 0 091 fi fu.
I? = k4/k1. p* = e/(l + H) is the steady-state concentration of ES. EquaticJn 2 relates the final concentration of ES (p,) to the initial concentration of peroxide (xg). It is seen that for x g > 0, p.. is positive, ie.,ES persists a t the end of the reaction. Equation 2 also shows that pm increases by increasing xo. The increase is, however, limited by the upper hound p*. The dependence of p-. on x o a t different R is shown in Figure I. Both the shape of the curves and the iupper limit depend on R. A t low H (relatively low rate constant of reactlon of ES with s),p * is close to e and the curve approaches a straight line with a slope of unity. I t corresponds to the quantitative formation of ES from its components. With increasing R, the peroxidatic step (reaction 4) hecomc?s rnore important. As a consequence, p* decreases and the line becomes curved (an increasing fraction o f t h e peroxide is decomposed into oxygen and water). In the ct)ur:je o f a single run p increases monotonically toward its f i n d limit. 'Phis behavior follows from the rate ctq uat ion
x[kie
db;/dt
- (kl
+ k4)p]
+
and from the condition p 5 e / ( 1 R ) . This time course of p is also known from analog and digital computer simulations of mechanism A.:$."
The Two-Complex Mechanism Mechanism A accounts for many aspects of catalase action." A t high x g , however, an extension of this mechanism is necessary."~"One of these extensions involves the resolution of the formaticm of the active intermediate into two consecutive sleps. tkcause of the interesting kinetic features of this system and because of its potential importance for the mechanism of the decomposition of HzOz by iron centered catalysts in general, this mechanism will be examined in the following in some detail.7-10 l!!
The Journal of
Pt7dSlCnl
1 +- S 7 ES,
C:17emistry Vol 78 No. 19 1974
Considering the kinetics of the intermediates we find that exhibits "normal" behavior in the sense that it rises to a maximum then declines to zero. The kinetics of ESll, on the other hand, is more complex. From the rate expressions for ESI and ESll we can derive the following relationship
p1
-
+
KM = (k2 + k : d / k l ; KM' = K M ( k : d k 4 ) .pz* is the concentration of ESII in the double steady state (both with regard to p1 and p2) and is given by
pz* = ( k , / k , ) e / ( x +-
ICM')
(4)
Outside the steady state p2* is a quantity defined by eq 4. It follows from eq 3 that as long as pz is less than p?*, p:! is increasing. (The expression in the bracket can be positive only for the combination dplldt > 0 and dpz/dt > 0, or for dplldt < 0 and dpz/dt > 0. The combination dpl/dt > 0 and dpzldt < 0 is excluded because of the sequence of the intermediates.) I t follows from the same argument that once p2 has reached pz*, it cannot fall below it again. In this case the final value of p z cannot be lower than ( k : J k4)e/K~'. Next we show that, under certain circumstances, p 2 can surpass pz*. We first assume that xo is sufficiently high that a t some stage of the reaction a steady state is established (pz = p2*). As the reaction progresses and the peroxide is consumed, the steady state will ultimately be disturbed. Because of the sequence o f the intermediates, first dplldt will become negative. Assuming that pz = p2* (p2 cannot then be lower than p2*), it follows from eq :3 that dpzldt must be positive. Thus, in the late p h a w of th.e reaction, when the steady state breaks down, p2 rises nboiie its extrapolated steady-state leLiel, which itself is increasing toward a n upper limit. p2- will attain some value between (h:&4)e/Kbf' and e. If x o is low, p z will tend, naturally, to lower limits. The entire range of' possible final values of pz extends thus from 0 to e. These conclusions are illustrated by some digital computer simulations shown in Figure 2. In the calculations e and xg were held constant. k2 was put equal to zero. k 1 and k4 were taken both as 1 X IO6 M-' set.-'. k:>was varied between 2 and 2 X lo6 sec-'. For k:! = 2 x IO6 sec-', p:! ap-
Kinetic Models o f Catalase Action
1921
Discussion
3,
01
02
03
04
05
06
m sec
The outstanding feature of both mechanisms A and C is the persistence of the active intermediate a t infinite time. A similar case has been encountered before by Benson, who found that in the autocatalytic scheme
AI-B A + B A C
B did not disappear a t the end of the reaction.14 Defining the degree of formation of ES, relative to its maximal value, f = p,/p*, we can write eq 2 in the form
5 = f(1 - R ) - 2R In e C
E
proaches closely the steady-state limit shown as a broken line (curve A). For hz = 0, this limit is the same for all values of h:j and is given by e k l / ( k l h4). For the given pap2- exceeds the rameters it is 5 X IO-’< M . By decreasing h:%, steady-state limit (curves I3 and C). A t hn = 2 sec-l, p z m becomes equal to 0.99c (curve C). This behavior can be understood in the following way. The factor causing p:! to exceed p2* is the negative value of dplldt in the declining phase of p1. If the rate constant of the irreversible decomposition of ESI (h:J is high, p1 remains low throughout the reaction (Idplldtl p2*, it decreases: p z always tends to p2*. Since under the chosen conditions x
