ROBERT A. ALBERTYAND GORDON G. HAMMES
154
the nature of the electrochemical reactions taking place during corrosion. This may be done by comparing the experimental value of (dE,/dpH) for a given metal-solution combination with the values in Tables I and 11. Although the literature is rich in experimental studies of the variation of corrosion potentials with pH, yet few of these studiess-26-28have been carried out under rigorous experimental conditions. Thus, Stern6 observed a value of (dEc/dpH) = -0.0559 v. for pure iron in air-free 4% NaCl solutions. A value of 0.059 v. was observed by D'Ans and BreckheirneP for carbonyl and electrolytic iron, also in air-free acid media. Similar results were observed by Boiihoeffer and Jena.27 Makrides, Komodromos and HackermanZ8found a value of 0.047 v. for iron and mild steel in air-free acid solutions. This value corresponds to the expression (dEc/dpH) = -4(2.303. R T ) / ( Z 3)F for Z = 2, as it is clear from Table I. This indicates that the anodic dissolution of iron is an activation-controlled process (cf. equation 12) leading to the formation of ferrous ions, and that the discharge of hydrogen is controlled by the electrochemical desorption (cf. equation 3). This conclusion is in agreement with the observations of other workers on the anodic dissolution of iron29and the cathodic evolution of hydrogen.30
+
(26) J. D'Ans and W. Breckheimer, 2.Elektrochem., 66, 585 (19521. (27) K. Bonhoeffer and W. Jena, ibid., 65, 151 (1951). (28) A. Makrides, N. Komodromos and N. Hackerman, J . Eleclroc h e n . Soe., 102, 363 (1955). (29) M. Stern, ibid., 102, 663 (1955). (30) B. Conway and J. O'M. Bockris, J . Chen. Phys., 26, 532 (1957).
Vol. 62
It is clear that the work of Makrides, Komodromos and Hackerman2*confirms the occurrence of a symmetrical energy barrier for the rate of anodic dissolution as given by equation 12. Values of (dEo/dpH) ranging from 0.0559 to 0.059 v.6,26,27 may indicate a diffusion-controlled hydrogen evolution as is clear from Table I. However, this could not be the case since the limiting current of diffusion of hydrogen is found to occur a t potentials more cathodic than the corrosion potentiaL8 Such values can still be explained on the basis of a rate-determining electrochemical desorption of hydrogen when the energy barrier for the anodic dissolution of iron is not symmetrical. Thus, if the symmetry factor is designated by p, one gets from (3) and (12) (dE,/dpH) = -4(2.303RT)/[(2PZ
+ 3)FJ
For Z = 2 and p = 0.25, a value of 0.059 v. is obtained from the above expression at 25". It is possible that the symmetry factors for the dissolution of iron and the evolution of hydrogen may depend on the nature of the electrode surface, e.g., roughness, and on the nature and concentration of the various constituents of the electrode, and this may account for the difference between the results of some authors.6,28 The authors wish to express their thanks to Professor A. R. Tourky for his interest in this work, and to Professor J. O'M. Bockris for helpful discussions on the mechanism of hydrogen evolution.
APPLICATION OF THE THEORY OF DIFFUSION-CONTROLLED REACTIONS TO ENZYME KINETICS' BY ROBERT A. ALBERTY AND GORDON G. HAMMES~ ContriDutionfrom the Department of Chemistry, University of Wisconsin,Madison, Wisconsin Received JuEu 10, 1967
Extrapolation of the minimum values of the second-order rate constantsofor the reaction of fumarate ions with fumarase The ma nitude of this rate constant may be to zero ionic strength yields a value of about 3 x 1010 sec.-l M-1 at 25 accounted for theoretically by considering the diffusion of substrate into a hemisphericafreaction site on a plane with a reaction radius of 5 provided that the effect of electrical charges in increasing the rate is included. It is therefore concluded that the bimolecular reaction of fumarase with its substrates is diffusion controlled. Equations derived by Debye for the rate of diffusion controlled reactions and for the effect of electrostatic attractions or repulsions between the reactants have been used. The effect of electrolyte concentration on the experimentally determined minimum values of the second-order rate constants indicates a charge of +2 to f 3 on the enzymatic site. The application of these theoretical considerations to enzymatic reactions in general is discussed for the case where the effect of protein charges outside the enzymatic site may be neglected.
.
w.,
Introduction Studies of the steady-state kinetics of the forward and reverse reactions catalyzed by fumarase over a range of pH values have permitted the calculation of minimum values for the rate constants of the bimolecular reactions of the fumarase molecule with the two substrates.3 The fact that high values (1) This research was supported by grants from the National Science Foundation and from t,he Research Committee of the Graduate Sohool of the University of Wisconsin from funds supplied b y the Wisconsin Alumni Research Foundation. (2) W. A. R. F. Fellow, 1956-1957. (3) R. A. Alberty and W. €1. Peirce, J. Am. Chcm. Soc., 79, 1526 (1957).
(-IO9 sec.-l M-l) are obtained raises the question as to how fast such reactions may be expected to occur with a protein of molecular weight 220,000 having 6 or less catalytic sites per m~lecule.~ The rates of the reactions catalyzed by saccharase and catalase,5 carbonic anhydrase,e cholinesterase,7 and trypsin8 have been compared earlier with predictions of ideal gas kinetic theory. However, (4) N. Shavit, R. G. Wolfe, Jr., and R. A. Alberty, in preparation. (5) J. B. S. Haldane, Proc. Roy. Soc. (London), 8108, 569 (1931). (6) F. J. W. Roughton, Eroeb. Enzymorsch., 8 , 295 (1834). (7) L. H. Easson and E. Stedrnan, Proc. Roy. SOC.(London), 8121. 142 (1936). (8) J. A. V. Butler, J . Am. Chem. Soc., 63, 2971 (1941).
Feb., 1958
APPLICATION OF
THE
THEORY OF DIFFUSION-CONTROLLED REACTIONS
155
it seems much more satisfactory to discuss the development of the theory has been made by maximum rate of reaction of enzyme with substrate Collins16 and Collins and Kimball.17J8 The frequency factor for a second-order reaction in terms of a diffusion process and t o base the calculations on the diffusion coefficients for these may be greater than the rate constant for the substances. The fundamental equation for the corresponding diffusion controlled reaction,lg but rate of a diffusion controlled reaction was derived the actual value for a second-order rate constant by Smoluchowskig in his considerations of the rates would not be expected to exceed that calculated for of coagulation of colloidal suspensions. The in- diffusion into a sink, provided that the effects of fluence of electrical charges on the rate was taken electrostatic interactions (to be discussed in a later into account by Debyelo and by Verwey and section) are also taken into account. Application to Enzymatic Reactions.-In general, 0verbeek.l' Spherically Symmetric Diffusion.-Fick's first enzymatic reactions involve a reaction of the sublaw for spherically symmetric diffusion may be strate with the enzymatic site followed by further bimolecular and unimolecular steps. The rate written constants for bimolecular steps, which are of dq = 4 d D dt interest here, have in some cases been measured in the transient phase of the reaction,*O and in others where dq is the quantity diffusing through a spher- have been from the coefficients in the ical surface of radius r in time dt, e is the concen- steady-state rate equation for both the forward and tration and D is the diffusion coefficient. reverse reactions. The bimolecular reaction of a The differential equation for spherically sym- spherical enzyme molecule with a spherical submetric diffusion with the diffusion coefficient inde- strate molecule would not be expected t o be reprependent of concentration is sented by spherically symmetric diffusion of substrate into a sink with a radius equal to the sum of that of protein and substrate as illustrated in Fig. la. Since enzymes in general have been found to Smoluchowskig showed that the quantity q which have only one or several catalytic sites per molecule, flows into a fixed spherical sink of radius R in the concentration of substrate would not be maintime t from an infinite volume of initially homo- tained a t zero all over the surface of the protein geneous solution of concentration eo may be ob- molecule, even if the activation energy for the tained by integration of equation 2 and is given by reaction were zero. The actual situation may be approximated more closely by considering diff uq = 4nDRco + da (3) sion into a hemispherical site on the surface of the protein as illustrated in Fig. lb. The rate of the This equation neglects transient effects of the diffusion controlled reaction of substrate with the order of the time of a molecular displacement. site may be calculated using equation 4 with RIZ For this calculation it is assumed that no convection equal to the sum of the radii of the enzymatic site arises from the inverted density gradient which is and of the substrate and D12equal t o the sum of the produced, either because the time is so short, or self diffusion coefficients of substrate and the whole the distance is so small. After a transient phase of protein molecule.22 If the enzymatic site was on a diffusion, the contribution of the term t'/P becomes plane, the steady-state diffusion controlled rate negligible, and the rate of flow into the spherical would be half that calculated using equation 4. sink becomes constant. This is in contrast with Depending upon the solid angle through which linear diffusion where a steady state is not set up by substrate molecules could diffuse into the site, some diffusion into a sink. I n order to allow for the other fraction of the rate calculated using equation Brownian motion of the central particle the sum 4 could be used. By reference t o equation 3 it is of the diffusion coefficients, D12= D1 D2, of the seen that if the diffusion is into a sink on the surface two types of particles is used.12 of the protein, the transient phase will be of shorter I n applying equation 3 to a chemical reaction R duration than for diffusion into the larger sink. is replaced by the sum Rlz of the radii of the re- I n the absence of experimental data the diffusion actants (the "reaction radius"). Thus the second- coefficients may be estimated by use of Stokes' order rate constant IC (in sec.-l M-l) for reaction law or Longsworth's empirical extension.23 I n between unlike molecules may be expressed as the case of ions the self diffusion coefficient is t o be used. (4) By use of D12the relative Brownian motion of enzyme and substrate molecules is taken into Equation 4 has been useful in accounting for the account, but the rotational Brownian motion of the rate of quenching of f l ~ o r e s c e n c e . ~ ~Further -~~
(g)
[t
e]
+
(9) M.V. Smolucbowski, 2. physik. Ghem., 92, 129 (1917). (10) P . Debye, Truns. Electrochem. Soc., 8 2 , 265 (1942). (11) E.J. Verwey and J. Th. G. Overbeek, "Theory of the Stability of Lyophobio Colloids," Elsevier Publ. Co., Inc., Amsterdam, 1948,p. 165. (12) S, Chandrasekhar, Rev. lifod. Phys., 16, 1 (1943). (13) J. Q. Umberger and V. K. LaMer, J . Am. Chem. Soc., 67, 1099 (1945). (14) B. Williamson and V. K. LaMer, ibid., 70, 717 (1948,. (15) K . C. Hodges and V. R. Lahler, ibid., 70, 722 (1948).
(16) F. C. Collins, J . Colloid Sei., 6 , 499 (1950). (17) F. C. Collins and G . E . Kimball, {bid., 4, 425 (1949). (18) F. C. Collins and G. E. Kimball, Ind. Eng. Chem., 4 1 , 2551 (1949). (19) R. Noyes, J . Ghem. Phya., 2 2 , 1349 (1954). (20) B. Chance, J . B i d . Chern., 161, 553 (1943). (21) H. Theorell, A. P. Nygaard and R. Bonnichson, Acta Chem. Scand., 9, 1148 (1955). (22) Personal communication, Dr. J . H. Wang. (23) L. G. Longsworth. J . A m . Chem. Soc., 74, 4155 (1952); 76, 5705 (1853).
ROBERT A. ALBERTYAND GORDON G. HAMMES
156
Vol. 62
calculated by inserting an additional term into equation 1 which is proportional to the electrical potential gradient. The differential equation which must be solved in this case isl0.l1 dq = 4?rr2
(.bc + CdV - -) P
(8) ' (b) Fig. 1.-Radial diffusion (a) into the whole enzyme molecule or ( b ) into the enzymatic site. The dashed line indicates the distance at which the concentration in the steady state is half that at infinite distance.
protein molecule is ignored. The question arises as to whether the rotational motion is slow enough to permit the establishment of a steady-state concentration gradient around the enzymatic site. Since the relaxation time for rotational Brownian motion of a sphere is proportional t o the third power of the radius or to the molecular weight, the importance of this motion may be quite different for different enzymatic reactions. Effect of Electrical Charges.-In the above discussion the effect of electrical charges has been ignored. For simple enzymatic reactions these effects would primarily lead to a lowering or raising of the Michaelis constant. If the enzyme and substrate carry electrical charges, the rate of the diffusion controlled reaction will be greater or smaller than that calculated in the preceding section, depending upon whether the charges of substrate and enzymatic site (or the enzyme molecule as a whole) are of the opposite or of the same sign. When the net charge of the protein molecule is small, we might as a first approximation consider only the net charge of the enzymatic site and ignore the charges of the remainder of the protein molecule. The effect of electrostatic interactions on the maximum velocity of enzymatic reactions has been discussed by K i r k ~ o o dwho ~ ~ has emphasized the importance of fluctuations in the charge configuration. If electrostatic attractions or repulsions are operative in a diffusion controlled reaction it is obvious that the result of increasing the salt concentration will be t o increase the shielding of the charges from each other and t o reduce the effect on the rate. The unimolecular steps of the reaction occur within the ion atmosphere of the protein and therefore might be expected to proceed a t rates nearly independent of the electrolyte concentration. Therefore, probably most. of the effect of electrolyte ions on enzymatic reactions occurs in the reaction of the substrate with the enzyme. Another effect of changing the electrolyte concentration results from the binding of ions by the protein. The effect of electrical charges on the rate of a diffusion controlled bimolecular reaction may be (24)
J. Kirkwood, Diec.
Faraday Soc., 20, 7 8 (1955).
dt
dr
(5)
where V is the electrical potential energy a t distance r, p is the frictional coefficient and the other symbols are the same as for equation 1. The frictional coefficient may be expressed in terms of the diffusion coefficient by use of p = kT/D. Here k is Boltzmann's constant and T is the absolute temperature. If the transient phase of the reaction is ignored, the steady-state rate may be calculated from equation 5 as described by Debyeto and Verwey and Overbeek" if Vis a function of r only. The secondorder rate constant for a diffusion controlled reaction is given by 4KD12
k = f " ,V,kT JRiz
*
x
N = *4
DuRlzf
(6)
1000
1000
T'
If V = 0 equation 6 reduces to equation 4. It is convenient t o represent the factor by which the second-order rate constant is altered through electrostatic effects by f so that
f=
1
(7)
The magnitude off may be calculated theoretically for infinite dilution of electrolyte and for electrolyte concentrations in the limiting law region. Eigen25 has determined the second-order rate constants of several reactions in water involving hydrogen or hydroxyl ions and has shown that these rate constants can be account%d for by equation 6 with a reaction radius of 5 A. After a H+ (or HaO+) ion comes within this distance of approximately 2 hydrogen bond lengths of a basic ion or molecule, the reaction proceeds rapidly by proton tunneling. I n the case of reactions not involving ions of the solvent, it would not be expected that the reaction within the ion pair which is formed first would be so fast. Electrostatic Effects at Infinite Dilution.-At infinite dilution of electrolyte the electrostatic energy of interaction is of the form
where z1 and 2 2 are the number of charges on ions 1 and 2 (with signs), e is the electronic charge and E is the dielectric constant of the intervening medium. Insertion of this relation into equation 7 and integration leads to L/Riz exp(L/R12) 1
-
where L = zlzzez/ckT
(10)
A plot of log f versus zlz2is given in Fig. 2 for E = 80, T = 298" and Rlz = 5 X 10-8 em. As is apparent (25) M. Eigen, Z. physik. Chsm., 1, 3/4, 176 (1954); M. Eigen and L. De Maeyer, Z. Electrochem., 59, 980 (1965).
APPLICATION OF THE THEORY OF DIFFUSION-CONTROLLED REACTIONS
Feb., 1958
157
from this figure the repulsion due to charges of the same sign on enzymatic site and substrate is more effective in slowing down the reaction than the attraction of charges of opposite sign is in speeding up the reaction. For an increase in rate of an order of magnitude, xlzZ must be greater than six in the illustration used. I n addition to the charges on the enzymatic site, there are of course charges distributed throughout the protein molecule. Jf the protein molecule is spherical, we may consider as an approximation that the net charge is situated at the center of the sphere. The additional potential at any point in the solution due to this charge may be calculated. However, the rate may not be calculated using equation 6 because spherical symmetry has been destrioyed by introducing these additional charges. Since the substrate never approaches these charges closely, their effect per charge will be much less than that of the electrical charges on the site. Thus we might expect that as a first approximation the -3 effect on the kinetics of changing the net charge of -8 -6 -4 -2 0 +2 +4 +6 the whole protein molecule might be neglected so z I z2. long as the net charge is small. However, when the net charge of the protein becomes large, the Fig. 2.-Plot of log f versus zlz2 at various ionic strength values for e = 80, T = 29S0, and R,z = 5 X 10-8 cm. effect of these charges may be expected to be considerable. As the ionic strength approaches zero, y approaches The duration of the transient state when there unity. are electrostatic interactions has been discussed by To illustrate the dependence of f on z1z2, log f Montrol126 who gives the following equation for is plotted versus zlz2 in Fig. 2 for p = 0, 0.001 and the second-order rate constant. 0.010 and aqueous solutions at 25" with R12 = 5 X lo-* em. As the ionic strength is increased, L/Riz RIz exp( -L / R d IC = 4nN -- DizRia [exp(L,Rlz) - 1 4- (nDt)% the factor by which the reaction is speeded up 1000 (if X I and xz are of opposite sign) or the factor by (11) The second term in the brackets is the transient which the reaction is slowed down (if x1 and x2 are term. For the fumarase reaction it contributes of the same sign) is decreased. Equation 12 is valid only for KR The Limiting Law for the Effect of Added Elec- At infinite dilution the electrostatic factor f is trolyte.-If electrolyte is present in the solution correct for all values of zlzz since the assuniption in addition to enzyme and substrate, the enzyme of a coulombic potential imposes no restrictions on and substrate ions will be surrounded by ion atmos- the magnitude of the charges involved. However, pheres which will shield the charges and reduce the as the ionic strength increases, the theoretically magnitude of the effects discussed in the preceding permissable values of zlzzchange markedly because section, We cannot hope to obtain a quantitative of the approximations made in the derivation of the theory of these effects except in the limiting law potential function in ionic media.27 The effect of region. For low ionic strength values Debyelo has these approximations can be readily seen in Fig. 2 where the curve for p = begins to go downshown that ward when zlxz becomes sufficiently negative. This downward curvature corresponds to a physically unreal decrease in rate as xlzzbecomes more where h: is the reciprocal "thickness" of the ion negative. Thus for u = the theory begins to atmosphere given by break down seriously at zlz2= - 6, while for u = calculations show that reasonable results are obn = Xniei2]'/' = 0.3295 X 108 dji (13) tained up to zlz2 = 14. If IYLIR121
Feb., 1958
THEINTERACTION OF ANIONIC DETERGENTS
quently undergoes an intermolecular reaction. The electrostatic factor in the rate of association has been calculated above on the assumption that the effect on the rate of charges other than those in the enzymatic site may be ignored. Since the isoelectric point of fumarase is pH 7.0 at’ 0.1 ionic strength, the net charge will be rather small in the neutral pH range. However, at high pH values the electrical charge may become quite large. Application to Other Enzymatic Reactions.-If the bimolecular reaction of a low molecular weight substrate with an enzymatic site is diffusion controlled, the rate constant would be expected to fall in the range of about lo8 to 10’0 see.-’ M-l. If the substrate is a small molecule the reaction radius would not be expected to be greatly different from 5 A. Since Dll2 depends primarily upon the diffusion coefficient of the substrate and since the diffusion coefficient for spherical molecules varies with M-‘/a, Dl2 will have nearly the same magnitude for many enzymatic reactions. However, the electrostatic factors may be considerably different for different reactions. Some substrates, like adenosinetriphosphate, have high charges and some enzymatic sites may be highly charged. I n cases where the bimolecular reaction is diffusion-controlled, considerable difficulty is involved in measuring the rate by flow techniques. If the enzyme concentration is low in comparison with the initial substrate concentration, (S)o, the half-life for the transient state of the enzymatic reaction would be 0.7 X sec. for (S)o = 10-8 A4, sec. for (S)O= M for k = 109 and 0.7 X set.-' M-l. The oxidation of ferrocytochrome-c by hydrogen peroxide as catalyzed by yeast cytochrome-c
159
peroxidase has been studied by Chance.a4 He obtained a rate constant of 1.2 X 108 sec.-1 M-1 for the rate of formation of a ternary complex between ferrocytochrome-c and the peroxidase-peroxide complex. He was unable to account for this large reaction rate using collision theory and assuming only iron-iron collisions are effective in producing a reaction. Using equation 4 with a solid angle of 2a and estimating the other parameters as D12 = em., then k = 10-6 set.-' and Rlz = 5 X 2 X lo8 sec.-l M-I. The assignment of this value of the reaction radius is rather arbitrary since although only the iron atom of the substrate is involved in this reaction, steric and electrostatic effects are probably present which cannot be evaluated. However, according to this calculation, the reaction appears to be diffusion controlled. Gibson and R ~ u g h t o nhave ~ ~ obtained a rate constant of 4 X 108 sec.-l M-’for the combination of hemoglobin and nitric oxide. This is the fastest directly measured reaction rate of an oxygen-carrying pigment with any ligand. Using equation 4 with solid angle 2n, DI2 = 10-6 ems2 sec.-l and Rlz = 5 X cm., the theoretical rate constant is calculated to be 2 X 109 set.-' 214-’, which is in reasonable agreement with the experimental value. I n conclusion, the proposed method of accounting for the rate of formation of enzyme-substrate complexes gives reasonable results for the three types of complexes considered, namely, those between enzyme and charged substrate, enzyme and protein substrate, and enzyme and neutral substrate. (34) B. Chance, ”Enzymes and Enzyme Systems,” J. T. Edsall, ed., Harvard University Press, Cambridge, 1951, 1.1. 93. (35) Q. H. Gibson and F. J. W. Itoughton, J . Physiology, 136
(1957).
1NTER.ACTION OF ANIONIC DETERGENTS AND CERTAIN POLAR ALIPHATIC COMPOUNDS I N FOAMS AND MICELLES BY W. M. SAWYER AND F. M. FOWKES Shell Development Company, Emeryville, Calvomia Received July 11, 1067
The addition of non-ionic surface active compounds to solutions of a variety of anionic detergents has been shown to enhance foam stability. The effectiveness of such compounds depends strongly on the detergent. It is found that detergents increase in “susceptibility to foam stabilization” in the order (1)branched alkylbenzene sulfonates, (2) n-alkylbenzene sulfonates, (3) secondary alkyl sulfates, (4)2-n-alkane sulfonates and (5) primary alkyl sulfates. This is also the order of increasing surface tension of the detergents (without additives) a t concentrations greater than the critical micelle concentration (CMC). The olar aliphatic additives with straight hydrocarbon chains of 8-14 carbon atoms were the more effective foam-stabilizers anzamong these effectiveness increased in the order (1) primary alcohols, (2) glycerol ethers, (3) sulfolanyl ethers, (4) amides and (5) N-polar substituted amides. This order is, in eneral, the order of increasing surface activity and CMC-depress!ng activity. Monolayer com ositions calculated from surface tension and CMC measurements show that in the mixed monolayer there is negligible specid interaction between detergent and additive molecules, and that increasing foam stability in a series of detergent-additive pairs corresponds to an increasing mole fraction of additive in the adsorbed monolayer. The most stable foams were found with detergent-additive pairs having SO-SO% of additive in the adsorbed monolayers. The requirements for preferential adsorption of additives are the same as found for foam stability: solutions of the detergent (without additive) should have a high surface tension, the additive should give water a low surface tension, and should depress the CMC of the detergent. The mechanism of CMC depression by additives is discussed, and the importance of the distribution of additive between sites on the surface of a micelle and elsewhere in a detergent solution is pointed out.
Introduction Of Of Studies Of the surface ‘Odium dodecyl have shown that the presence of dodecaiiol reduces the Surface tenSiO11,
increases the surface viscosity, increases the foam (1) A. P. Brady, THISJOUBNAL,63, 56 (1949); ”Monomolecular Layers,” Am. Assoo. Adv. Sci., Washington, D. C., 1954, p. 33. (2) G. D. Miles and L. Shedlovsky, THISJOURNAL, 48, 57 (1944).
