Hans Degn
institute of Biochemistry Odense Un~verslty Odense, Denmark
Oscillating Chemical Reactions in Homogeneous Phase
The first report on an oscillating chemical reaction appeared in print in 1828. A. Th. Fechner described an electrochemical cell which produced an oscillating current (1). Towards the turn of the century W. Ostwald found an oscillation in the rate of dissolution of chromium in acid (2). About the same time J. Liesegang discovered the periodic precipitation patterns named after him (3). The electrochemical oscillations and the Liesegang phenomenon were for a long time the only well-known examples of chemical oscillations, and because they both involved diffusion gradients i t became a dogma that such gradients are necessary for oscillations to occur in chemical systems. Oscillations in closed homogeneous systems were considered impossible. Although there now seems to be abundant experimental evidence for the existence of homogeneous oscillating reactions there are still theoreticians who resist the idea, and also a few experimentalists think that alleged homogeneous oscillations are caused by dust particles, although nobody has explained how. The disbelief among qualified chemists is still so widespread that the question whether a chemical reaction in a closed homogeneous system can oscillate cannot be said to be definitively settled. It is hard to think of any other question which already occupied chemists in the nineteenth century and still has not received a satisfactory answer. Theories on Oscillating Reactions
The first important publication on homogeneous chemical oscillations was a theoretical paper by A. Lotka (4),which appeared in 1910. Lotka formulated the following hypothetical reaction scheme
own model from 1920, are mere modifications of the 1910-model. Lotka's two reaction schemes contain only two substances, and the corresponding differential equations are manageable. Chemical reaction schemes involving three or more species are, like the celestial many-body problem, impossible to solve rigorously. However, short of solution it can often be decided in some way or other whether a certain model has periodic solutions. I n addition, models can be scanned for periodic solutions by the help of computers. I n such ways many workers have found periodic solutions in a wide variety of hypothetical, but kinetically plausible reaction schemes (6-18). Many of these are created by modifications of Lotka's reaction scheme from 1910 and others from scratch. From these explorations it is now clear that what is required in a chemical reaction scheme in order to obtain oscillations, not surprisingly, is feed-back. The set of individual reaction steps must constitute a closed loop through which information in the form of activating or inhibitory effects can flow. It should he noticed that the closed loops in question are not necessarily circular sequences of chemical conversions. The minimal requirement for a closed feedback loop in an isothermal system is a sequence of chemical reactions, and one substance in the sequence exerting a positive or a negative effect upon the rate of a reaction either upstream or downstream in the sequence. Four general types of chemical feed-back loops can exist. These are indicated below.
111
The substance, A, is introduced a t a constant rate, and it is converted by an autocatalytic reaction into X. The latter disappears by a first-order reaction. Lotka found that the steady state concentration of A will be k&, and this concentration may he approached through a damped oscillation. This work by Lotka did not arouse much immediate interest, and later it came into the shadow of another paper, published by Lotka 10 years later, where he demonstrated that a reaction scheme containing two autocatalytic reactions in a sequence can generate undamped oscillations (5). It is now realized that Lotka's 1910-paper signified the breakthrough because most oscillating chemical models which have been published later, including Lotka's 302
/
lournol of Chemical Education
IV
-
r
activation
a, F! az
r
a
1
a"-, F! a.
+
inhibition
-a,F?aaF?
3
%-,*a"-
The two last schemes do not contain feed-back unless the reaction steps in the sequence are reversible. Lotka's scheme from 1910 is the simplest kinetically plausible example of a reaction scheme with feed-back of type I. All four types of feed-back have been shown to produce sustained oscillations in hypothetical reaction schemes. No hypothetical reaction scheme with periodic solutions is known which does not contain a t least one of the four types of feed-back. We now know a multitude of hypothetical reaction schemes which can oscillate. However, they all as-
sume that reactants are in unlimited supply and, therefore, they can only be realized in open systems. It may be questioned whether the predictions of oscillatory kinetics in open systems apply to systems which are closed to mass transfer. Different authors have approached this problem by investigating the differential equations which describe general types of closed reaction systems. It was first shown that any closed isothermal system of first-order reactions will approach its equilibrium without oscillations (IS). Later this negative conclusion was extended to reaction schemes of any order or combination of orders in systems which are iwthermal or exchange heat with the surroundings by black body radiation only (14). This result depends on the application of the principle of detailed balancing, which prevents net transport around a closed loop in a chemical reaction system a t equilibrium. The general statement that the principle of detailed balancing excludes oscillations in closed homogeneous systems (15) is erroneous because the principle of detailed balancing does not apply far from equilibrium. Whereas oscillation around the equilibrium during the approach to equilibrium is excluded, oscillation around a transient which is converging on the equilibrium is not excluded. People who believe that oscillations cannot occur in a closed homogeneous system mean oscillations around the equilibrium as illustrated in Figure 1A. Those who maintain that oscillations do exist in closed systems mean oscillations around a quasi-steady state as indicated in Figure 1 B. Let us consider a reaction in an open system which is nourished by a constant input of reactant and produces a sustained oscillation, provided the rate of supply of reactant is within certain limits. If we can devise a chemical reaction which can form this reactant a t a constant rate within these limits we
Figure 1. Two types d orsillotionr imaginable in a closed homogeneous reaction system. Curve A shows an oscillation around the equilibrium during the approach to equilibrium. This type of oscillation is excluded by the principle of detailed boloncing. Curve B shows on orcillotlon around quori-steady states during the .pproach to equilibrium. This type is not excluded b y detailed balonring or m y other known law of nature.
can add that reaction to the system and dispense with the openness. The oscillation does not depend on whether the reactant is introduced by mass transport or it is formed by a chemical reaction. As long as the reaction is made available a t the proper rate the oscillation will go on in the closed system. Admittedly there are no means to secure an absolutely constant rate of supply of reactant in a closed system since the precursor of this reactant will be depleted causing a decreasing rate. However, a constant rate of formation of the reactant is no prerequisite for the occurrence of oscillations (9). As long as the rate stays within the limits where oscillations would occur in an open system with constant input rate, the system will oscillate whether the rate is constant or slowly decreasing. I n some of the cases of chemical oscillations described in the following, one of the reaction products is a gas which is allowed to escape from the liquid phase during the experiments. Nevertheless we consider the system closed, if there is no evidence that the gaseous product takes part in any reaction, including the reverse of its own formation, in the oscillating reaction system. Thus, when deciding whether or not a system is closed we only worry about substances which have a primary effect on the kinetics by entering into reactions. Secondary effects such as changes of ionic strength, pH, etc., caused by the unreactive products are disregarded. In all the investigations of oscillating models mentioned above it was assumed that the temperature is constant. This condition is impossible to achieve in a real system. There will always be a temperature difference between the reaction mixture and the beat sink, dependingon the rate of heat absorption or release in the reaction. This temperature difference is variable during the course of the reaction and it may not always be negligible in conventional reaction vessels such as a cuvet in a spectrophotometer. The occurrence of temperature oscillations in such a reaction system does not necessarily mean that the temperature is a variable on which the oscillation depends. In fact this is rarely true, although a few examples are known of so-called thermokietic oscillations where the heat of the reaction is involved in a feed-backloop causing oscillations. The conclusion of the theoretical works reviewed above is that oscillations can occur in models which are plausible in so far as their elements are all in accord with conventional principles of chemical kinetics. On the basis of the principle of detailed balancing, oscillations are theoretically excluded only at equilibrium and near equilibrium. Thus oscillations cannot take place around the equilibrium concentration during the approach to equilibrium. There are no known principles of chemistry which theoretically exclude oscillations in homogeneous systems far from equilibrium. Sustained oscillating reactions can exist in homogeneous phase. I n a closed homogeneous system the oscillations will sooner or later vanish as the reactants are used up. In an open homogeneous system, where reactants and products are transported in and out, sustained oscillations can be maintained indefinitely. Oscillating Reactions in lsolhermal Homogeneous Systems Closed to Mass Transfer Bmy's Oscillating Reaction
During investigations of the effect of iodine on the decomposition of hydrogen peroxide Bray (16) disVolume 49, Number 5, May 1972
/
303
covered in 1920 that a reaction mixture composed of hydrogen peroxide and potassium iodate in dilute sulfuric acid can produce an oscillation in the coucentration of free iodine and in the rate of oxygen evolution. I n this reaction system iodine is formed when hydrogen peroxide reduces iodate. 2HI0, + 5HsOn = IS + 502 6Hz0 (1) Iodine is consumed again in another reaction where it is oxidized back to iodate by hydrogen peroxide.
+
12
+ 5Hx01 = 2HIOs + 4Hn0
(2)
It would be expected that a quasi-steady state of iodine concentration would result from the two opposed reactions while the hydrogen peroxide is being used up by conversion to molecular oxygen and water. Usually this is also the case. However, there is a comparatively narrow range of concentrations of the reactants where an oscillation in the iodine concentration and in the rate of oxygen evolution occurs (Fig. 2). Because of
Figure 2. Orcillation in light obrorptian due to dissolved iodine I460 mpl in solution of hydrogen peroxide and potasium iodote in dilute rvlfuric acid. The reaction was initioted by the addition of hydrogen peroxide to on acid solution of iodate a t time zero. The ternperdure wor 6 5 %
the traditional disbelief in oscillations in homogeneous phase several workers have been searching for heterogeneous processes in this oscillating reaction system. Obviously the evolution of oxygen bubbles is a heterogeneous part of the reaction, but Bray himself forestalled objections on this point as he showed that a t proper concentrations and temperature he could make the reaction proceed so slowly that the oxygen left the solution by diiusion from the surface instead of by forming bubbles. The oscillation was unaffected under these conditions. Nevertheless some workers who reinvestigated the reaction have included physical effects in connection with the formation of oxygen bubbles in their explanation of the oscillation (17-18). It has also been suggested that subtle heterogeneities in the form of dust particles may influence the reaction system and cause the oscillation (19). , , However. no exnlanation of how dust particles can cause an oscillation is to be found in the literature. Assuming that the requirement for heterogeneity in oscillating reactions is theoretically unfounded I have investigated Bray's oscillating reaction with a special aim a t revealing the feed-back mechanism which theory requires (20). One clue to this problem is given by the fact that the rate of oxygen evolution has its maximum in connection with the phase of decreasing iodine concentration. This is unexpected because oxygen ap304
/
Journal o f Chemical Education
pears in the first and not the second of the stoichiometric equations (1) and (2). Evidently the stoichiometry of the iodine consumption is not correctly described by eqn. (2). It appears that this reactionisaccompanied by a dismutation of H201to oxygen and water so that the overall reaction is I,
+ (5 + 2n)HsOe = 2HIOa + (4 + 2n)Hn0+ noz
(3)
The magnitude of n has never been measured and there is no way to predict it on a stoichiometric basis. IIo\vever, it mwr br of 3 considernble ~ i z eiinw rnwt of ~ l l eoxygen i3 evolved in burits during r l w phnjc of decreasing iodine concentration. Another remarkable property of the iodine consuming reaction is its ability to be switched on and off suddenly as revealed by the very sharp inflection points on the curve in Figure 2. The indefinable stoichiometry indicates that the reaction may be a branched free radical chain reaction. It is also a characteristic feature of branched chain reactions that critical limits of reactant concentrations determine whether or not the reaction can proceed. When the reactant concentration is slightly below such a critical limit the reaction rate is infinitesimally small. Slightly above the limit the reaction accelerates in an explosive way. In the actual case it seems that the iodine consuming reaction depends on a critical concentration of iodine below which it cannot initiate. However, when it has initiated it will bring the iodine concentration down below the limit for initiation before it stops again. This behavior, which accounts for the oscillation, can be explained by a reaction scheme which has branching of a higher order than one, i.e., i t is an autocatalytic reaction where the rate of the autocatalytic steps depends on the concentration of the autocatalytic substance a t a higher power than one. Computer calculations have revealed that a model containing a quadratically branched chain reaction can produce concentration oscillation of a waveform, very similar to that of the iodine concentration oscillation in Bray's oscillatingreaction (8). Belousov's Oscillufing Reaction
The second case of oscillations in a closed homogeneous system was discovered in 1958 by Belousov (21). He mixed potassium bromate, ceric sulfate, and citric acid in dilute sulfuric acid and found that the ratio of concentrations of the ceric and cerous ions was oscillating. Zhabotinskii (22) has studied this reaction intensively. He found that oscillations were still obtained if citric acid was substituted by any of a number of carboxylic acids having the common configuration R I
CHB
I
COOH
The substitution of cerium by manganese could also be done without causing the oscillation to disappear. Most studies on this reaction have been done with cerium and malonic acid (Fig. 3). A solution of ceric ions is strongly yellow whereas a solution of cerous ions is colorless. At proper concentrations of ceric ions, malonic acid, and bromate, easily observable color changes may occur at a frequency of about 1per rnin at
Figure 4. Schematic representotion of reaction mechanism in the cerium system. The back activation (outocatalyrirl is m=umed to b e due to on intermediate, X, containing bromine. Figure 3. Orcillation in light obrorbtion due to ceric ion ( 3 1 7 ma) in solution of molonic add, potmtium bromote, ond rericsulfote in dilute rulfunc odd. The reaction war initiated b y the addition of molonic acid about 1 minute after mixing of the other components. The temperature war
60°C.
room temperature. The reaction is, therefore, excellently suited for classroom demonstrations and makes an interesting laboratory experiment.' Although it has been known for a much shorter time than Bray's oscillating reaction, Belousov's oscillating reaction has been subject to much more study (22-27). The reaction system consists of two parts. One part is the reduction of ceric ions by malonic acid, and the other part is the oxidation of cerous ions by hromate. Again it would be expected that the two opposed reactions would result in a quasi-steady state while the reactants, bromate and malonic acid, are slowly being consumed. The occurrence of oscillations indicates feed-back in the mechanism, and the kinetic investigations have revealed more than one feed-back loop. Firstly, the oxidation of cerous ions by brornate is strongly autocatalytic. Secondly, the bromine liberated in this oxidation process reacts with malonic acid to form dibromo-malonic acid which is a potent inhibitor of the autocatalytic reaction as well as it is highly labile. Bromide ions have also been wggested as feed-back inhibitors. However, their lifetime in the strongly oxidizing solution seems to be too short to be comparable to the period of the oscillation. A flow diagram of the reaction with feed-back loops of type I and type I11 is shown in Figure 4. Small temperature oscillations in this reaction have been reported (27). However, there is no evidence that the heat of the reaction is involved in the mechanism of the oscillation. Oscillations in Glycolytic Enzyme Sysfem
Biochemistry has furnished the third example of an oscillating chemical reaction in homogeneous phase. It was first observed by Ghosh (28) that suspensions of yeast cells could, under certain conditions, produce a train of damped oscillations in the content of reduced nicotinamide adenine dinucleotide (NADH). Later it was found that similar oscillations may occur in cell free extracts of yeast (29), and under suitable conditions this oscillation was found to be undamped (SO). The discovery of this phenomenon caused a boom in the interest in oscillating reactions, and it was intensively 'The two Companion papers directly following this article illustmte the use of the Belousov reaction in the classroom. The article by Field (45)gives several methods of presenting the reaction as s. lecture demonstration and discusses its mechanistic interpretation. Lefelhocz (44) presents a student experiment based on the kinetics of the oscillating reaction.
studied. It soon became clear that the oscillation was a property of the glycolytic enzyme system which causes the breakdown of glucose to pymvate in most living cells. However, it turns out that there is a bewildering number of inhibitory and activating effects exerted by diierent intermediates of the glycolytic system on its different enzymes. The problem is not to prove the existence of a feed-back loop but rather to choose out of a handsome selection the one(s) which is (are) important for the oscillation. It seems that most workers believe that the important target enzyme for the feed-back is phosphofmctokinase which is (1) inhibited by adenosine triphosphate (ATP, a substrate of the enzyme), (2) activated by fructose-1,6-diphosphate (a product), (3) activated by ADP (a product) etc. (51). Much scrutiny of the glycolytic oscillation was inspired by the idea that biological rhythms and time keeping must depend on chemical oscillators in the organisms. This is undoubtedly true but i t does not seem likely that a clock function is assigned to an enzyme system which represents the main thoroughfare of cellular metabolism. Another viewpoint is that the many feedback loops in glycolysis are meant to regulate the supplies of the glycolytic intermediates and the end product. According to this view the oscillations are an accidental, probably quite harmless consequence of a strong feed-hack regulation. Thermal Decomposition of Difhionife
Finally it was discovered by Lynn (32) that dithionite in aqueous solution decomposes in an oscillatory way. Although the phenomenon (32-33) seems we11 established there has not been done sufficient kinetic work on this reaction to identify the feed-back mechanism. Oscillating Reactions in Isothermal Homogeneous Systems Open to Mass Transfer Gos Phase Oxidofion of Phosphorus
Beside the above-mentioned reactions, which can exhibit oscillations in a closed homogeneous system during the depletions of the initial supplies of reactants, a nurnber of oscillating reactions are known which occur in homogeneous systems open to some or all of the reactants. The first oscillating reaction of this type was discovered in 1832 by Munck af Rosenschold (34). He observed that a stoppered bottle, where he kept yellow phosphorus under u~ater,emitted light flashe.~ periodically. The oscillation depended on a small leak in the stopper allowing atmospheric oxygen to enter. This oscillation in the chemiluminescent gas phase oxidation of phosphorus in an open system has been conVolume 49, Number 5, Moy 1972
/
305
firmed by many workers. Unfortunately no modern studies of the mechanism of the oscillation have appeared. In his classical investigations of the kinetics of the gas phase oxidation of phosphorus Semjouov (35) does not mention the oscillation. However, he found that the reaction is a branched chain reaction. Thus the system has positive feed-back, and it resembles Lotka's model from 1910 except that there are two reactants entering instead of one. The transport processes of phosphorus from the water phase and oxygen from the outside might be expected to cause heterogeneity in the gas volume in the bottle where the reaction takes place. However, these transport processes are slow, and the period of the oscillation is long. Therefore, it seems reasonable to assume that diffusion and convect,ion within the gas phase are keeping i t practically homogeneous. When I observed the oscillation occurring spontaneously in a bottle with phosphorus which had been left undisturbed for several years I noticed that the intensity of the light flash seemed uniform in the whole gas volume, which would hardly be the case if the gas phase were heterogeneous. Oscillations in Peroxidare Catalyzed
Oxidation Reactions
Yamazaki et al. (36) have discovered that the peroxidase-catalysed oxidation of NADH by 0% in a solution open to 0% from a gas phase can oscillate. The oscillation vanishes as the NADH is used up. If a regeneration of NADH is provided by the addition of a suitable second enzyme and its cosubstrate in large excess, undamped oscillations occur (37). Oscillations also occur when NADH is substituted by dihydroxyfumaric acid or indole-3-acetic acid (38). A proposed reaction scheme for the peroxidase catalyzed oxidation of NADH is shown in Figure 5 (1%). It isseen that the reaction of one mole-
duced by such a chemical reaction or it is provided from outside by diffusion across a boundary between two homogeneous phases is mathematically indistinguishable, and it does not have any influence on the behavior of the system with respect to oscillations. Thermo-Kinetic Oscillations
I n addition to the reactions mentioned above a few oscillating reactions are known in homogeneous systems where significant temperature oscillations occur, abd the transport of heat out of the system is an essential part of the mechanism of the oscillation. These are the cool flame combustion of hydrocarbons (33) and hydrogen sulfide (40) which both occurs in systems closed to mass transfer, and the gas phase chlorination of methylchloride (41) which occurs in a system open to mass transfer. I n the case of the cool flame combustion processes there are two important feed-back loops. One is provided by a branched chain reaction and the other depends on a negative temperature coefficient of this chain reaction. The reaction causes the temperature to rise, and the rising temperature exerts a feed-back inhibition on the reaction. I n the gas phase chlorination of methylchloride the heat liberated in the process is also involved in a feed-back loop. However, in this case the temperature coefficient is positive, and, therefore, the feed-back is positive. The heat can be considered analogous to an autocatalytic substance. The system resembles the Lotka model from 1910 with the variation that the autocatalytic reaction is substituted by a reaction with thermic acceleration, and the reaction which removes the autocatalytic substance is substituted by the transport of heat out of the system. Conclusion
Oscillation in a homogeneous chemical system is not contradicted by theory, and there is ample experimental evidence for its existence. All known examples of homogeneous oscillating reactions were found by chance. One important task which remains to be done is the "synthesis" of a new oscillating reaction from reaction elements chosen purposely to create the right feed-back properties for oscillations to occur. Literature Cited
Figure 5. Schemotis presentation of reoction mechanism in oxidation of NADH H i b y 0%catalyzed b y peroxidare. The reoction is a free radical branched chain reaction. Three NAD' rodicols ore created every time one is consumed.
+
cule of the free radical NAD' with oxygen eventually results in the formation of 3 new molecules of this free radical. So i t turns out that a branched free radical chain reaction is also involved in this oscillating reaction 'system, giving rise to feed-back of type I. The transport of oxygen across the phase boundary is an integral part of the oscillating mechanism. The rate of transport is proportional to the difference in oxygen tension in the two phases. This gives rise to a rate equation for the transport process, which could, in principle, result from a chemical reaction, namely a reversible first-order reaction producing Oz from a precursor in large excess. Whether the reactant is pro306
/
Journal of Chemical Education
F m a ~ n n A. . Tn.. Sehueigo., 53, 141 (1828). O s ~ w ~ W., m , Phys. Zaitech.. 8 , 87 (1899). L m s m m a , J., Nolurw. Wochschr., 11, 353 (1896). Lola*, A. J., J . Phys. Chcm., 14, 271 (1910). LOTS*, A. J., J . A m w . Chcm. Soe., 42, 1595 (1920). Moon% M. J.. T ~ o n sFarodoy . Soo., 45, 1098 (1949). (7) SPmaLsn, R.A,. AND SNELL, F. M.. Natum. 191. 457 (1961). (8) L m o s r ~ nP., , AND DEGN,H . . Acto Chem. Scond., 21,791 (1967). (1) (2) (3) (4) (5) (6)
(9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)
HroorNa, J.. Ind. & Enp. Chem.. 59, (5). 19 (1967). MORALES,M . . AND MoXby, D.. Biophys. J . 7, 621 (1967). Se~Hov.E. E.. Eur. J . Bioeham., 4, 79 (1968). DEGN, H.. A N D MATER,D., Biochim. Biophys. Aolo, 180.29 (1969). Hm~nor,J. Z., Bull. Moth. Biophys., 15, 121 (1953). GnAr. B. F., T?ons. Farodoy Soc., 66, 363 (1970). S w ~ n ~C. z ,J., J. Cnelr. Eono.,46. 309 (1969). BRAY.W. C.,J . Amm. Chem. Soc.. 43, 1262 (1921). P a a ~ nM , . , AND CULLIS,C . , Trans. Fwaday Soc., 47, 616 (1951). S a ~ wD. , H., AND P ~ r ~ c m n H. n , 0.. J . Phya. Cham., 72,2692 (1968). RICE, F.. AND REIFF.O., J . Phys. Chem., 31, 1552 (1927). D ~ a l rH., , Acto Chem., Scond., 21, 1057 (1967). Bmmnsov, B. P.. Sb. ref, rodiots. mad. so. 1958. Medgi., Moscow (1959).
ZH*BOTINS=~. A. M..Biofizikc. 9, 306 (1964). Z ~ A B O T I N ~ KA.I ~M, . , Dokl. Akod. Nauk. S S S R , 157,392 (1964). D e o ~ H. . , Notura. 213, 589 (1967). Vnvmw. V. A,, Z n ~ s o ~ ~ ~A. s rN.. n , AND Zarnw. A. N . , Russ. J . Phya. Chcm., 42, 1649 (1968). G. J.. AND BBOICE, T . C . , Inovg. Chem., 10,382 (1971). (26) KABPEREH, (27) BDBBE, H . G., Natum Phys. Sci., 233, 137 (1971). , A N D CHANCE, B., Biooharn. Biophys. Rcs. Comm., 16, 174 (28) G ~ o s n A,, (22) (23) (24) (25)
(1964).
(29) Cnmo%B.. Hms, B., AND BETS.A_. Biochem. Biophw. Re8. Comm.. 16. 182 (1964). (30) PTE.K..A N D CHANCE. B., PTOC. N&.Acad. S C ~ 55.888 ., (1966). (31) Prr;, E. K.. Con. J . Bof.. 47, 271 (1969). (32) LYNN,5.. Ph.D.thesis. California Institute of Technology (1954). (33) RINHER,R. G., LYNN.8.. M * 8 0 ~ .D. I 4 . s A m CoRCOR*N. W. H.. Ind. & Eng. Chem., 4 (3). 282 (1965). (34) M n l r c ~bp P. pow. A,,". phys. them., 32, 216 (1834). N. N., "Some Problema in Chemical Kimtios;, princeton (35) SEMJONOY, University Prasa. Princeton, N. J.. 1959.
s..
I.. Y O X O TK., ~ , AND NAXAJIWA, R.. Biochm. Biophyb.RCII. (36) YAX&Z*KI, Comm.. 2 1 , 582 (1965). r r Nolure, , 222, 794 (37) N*~*uan*, S., Yoxor*, K., AND Y ~ v ~ z ~ I., (1969). (38) D ~ o NH,..Bioehdm. Biophus. A d o . 180. 271 (1969). (39) Day, R. A,. A N D PEASE, R.N.. J . Amer. Chcm. Soc., 62,2234 (1840).
iE; ;;~,"e;;,,%,;:.;;y:::5~;;;,~91(:;;;;~, (42) F m n , R. J., J. Cnex. Eonc., 49, 308 (1972). (43) Lesmnocs, Jonw F..J. C ~ YEooc., . 49, 312 (1972).
Volume 49, Number 5, May 1972
/
307
