11. KINETICS AND MECHANISMS THE INFLUENCE OF FLUCTUATIONS IN PROTEIN CHARGE AND CHARGE CONFIGURATION ON THE RATES OF ENZYMATIC REACTIONS* BY JOHN G. KIRKWOOD Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut Received 6th May, 1955 The Kirkwood-Shumaker theory of fluctuation forces between protein molecules arising from fluctuations in protonic charge and charge configuration is applied to an analysis of the interaction of the protein moiety of an enzyme molecule and a substrate molecule locally bound to an active site. It is established that if there is an increase in dipole moment of the active site + substrate complex in its activation to the transition state, interaction with vicinal basic groups of the protein by the fluctuation mechanism can produce a substantial decrease in the free energy of activation.This interactron is maximal at a pH equal to the pK of the conjugate acids of the effective basic groups. The proposed mechanism accounts in a satisfactory manner for the dependence of the rates of many enzymatic reactions on pH without invoking the simultaneous participation of two or more ionizable groups. An enzyme is a protein possessing an active catalytic site or prosthetic group to which the substrate molecule first associates and subsequently reacts by a mechan- ism determined not only by its local environment but also by the protein moiety of the molecule. The essential role of the protein in enzymatic activity is clearly demonstrated by the catalytic impotence of isolated co-enzymes or prosthetic groups under conditions permitting their dissociation from the protein.For brevity in the following discussion, we shall refer to the protein moiety of the enzyme molecule as the protein and to the attached coenzyme or prosthetic group, whether dissociable or not, as the active site. Abstractly, WE may say that the protein affects the rate of an enzymatic reaction either by forces which groups vicinal to the active site exert cn the associated complex of site and substrate molecule or by serving as a reservoir of mobile ions, for example, hydrogen ions, which exert a local catalytic effect when transferred to the complex. The influence of the protein may manifest itself either in the free energy of binding of the sub- strate to the active site, resulting in a change of the Michaelis-Menten constant K , or in the free energy of activation of the complex, resulting in an increase in the intrinsic rate constant k3 of the enzymatic reaction, which we shall suppose to follow the typical rate law, (1) dF1 - k3LElLSl - _ - _ dt K , + IS]' where [El and [S] are the concentrations of enzyme and substrate, respectively.In principle, any cf the usual types of intermolecular force acting between vicinal parts of the protein and the active site + substrate complex, both in its activated and unactivated states, could produce changes in K , and k3. If the forces are primarily electrostatic, their effect should be sensitive to pH and ionic strength. One of the characteristic features of enzymatic reactions is the sensitivity of their * Contribution no.1299 from the Sterling Chemistry Laboratory, Yale University. 78J . G . KIRKWOOD 79 rates to pH. For most hydrolytic enzymes, though not all, the rate as a function of pH follows a bell-shaped curve, which descends on both sides of an optimum pH corresponding to maximum rate. This behaviour has been ascribed to the simultaneous participation of two types of basic groups, either in the complex or on the protein contiguous to the complex, in the reaction, the conjugate acids of one being uinionized and of the other ionized. We shall presently show that this hypothesis, although possibly correct, is not necessary to account for the effect of pH on the rate of enzymatic reactions. In an earlier article,l we have shown that the mobility of their charge distribu- tions gives rise to a supplementary attractive force between protein molecules which would not exist between static constellations of electric charge.Proteins, considered as ampholytes, contain a large number of neutral and negatively charged basic groups, such as -NH2 and COO-, to which protons are attached. Except in highly acid solutions, the number of basic sites generally exceeds the average number of protons bound to the molecule, so that there exist many possible configurations of the protons, differing little in free energy, among which fluctuations may occur as the result of thermal motion. Similar fluctuations may occur in the configurations of other ions bound to the proteins, when the number of binding sites exceeds the average number of such ions which are bound.Fluctuations in the number and configuration of the mobile ions impart fluctuating charges and fluctuating electric multipole moments to the molecule. The fluctu- ating electric field of each molecule of a protein pair induces fluctuations in the charge constellation of its neighbour in such a manner as to produce an attractive force between them. In a similar manner, an attractive force is established be- tween a protein and a small molecule, of fixed charge distribution. In this instance the field arising from the charge of the small molecule, if ionic, or from its dipole moment, if polar, induces fluctuations in the charge distribution of the protein leading to an average polarization of the macromolecuIe and an attractive force in excess of that due to the interaction of the average charge distribution with the small molecule.Such an attractive force will exist between an enzyme and its substrate, if the substrate is ionic or polar and there is a suitable constellation of basic groups vicinal to the active site to which the substrate molecule is locally bound. As we shall see, this interaction is maximal at a pH equal to the pK of the conjugate acid of the effective basic group. Before proceeding to an analysis of the fluctuation force between the protein and the active site + substrate compIex of an enzyme, we shall briefly describe the origin of this force. According to the general theory of statistical mechanics, the potential of average force W(R) between two molecules separated by a distance R, one or both of which possess several internal configurations, is given by fl = l/kT, where V is the potential energy of the pair of molecules in fixed internal con- figurations and the averages are to be taken over all configurations with the un- correlated canonical distribution functions, appropriate to the isolated molecules separated by an infinite distance.* The function W(R) is the work required to bring the pair of molecules from infinite separation to the separation R.The value of W at contact is related to the standard free energy AFO of formation of * We wish to call attention to an error in eqn. (2) of the article by Kirkwood and Shumaker.1 The second term in the second line should read - (p/2)[< V2)av - ( V)3v] instead of - (p/2)( V2)av.This error has been corrected in eqn. (2) of the present article. Eqn. (5) of ref. (1) should also be corrected by omitting the second term W(W(R) of the first line.80 RATES OF ENZYMATIC REACTIONS a complex, consisting of the molecular pair, from the isolated molecules in the following manner AFOIRT = W/kT, log K = - W/kT, (3) where K is the equilibrium constant for the formation of the complex. If the complex undergoes subsequent chemical reaction through a transition state with free energy of activation AF*, we have AF*/RT = A W/kT AW= W * - W. (4) where W* is the local free energy of the molecular pair in the activated state. We shall now investigate the effect of fluctuations in the charge distribution of the protein on the binding of a dipolar substrate of low molecular weight to an enzyme and on the rate of the decomposition of the substrate.We suppose that the substrate molecule is specifiacally bound to the active site of the enzyme by local forces with a potential WO, in which the protein does not participate, and that in the activated state the local free energy is Wo*. Under the influence of the protein, we suppose the local free energies of the unactivated and activated states to be W' and W*. If KnL and Krno and k3 and k3' are the Michaelis- Menten constants and intrinsic rates of the reaction, with and without protein participation, they are related in the following manner, where for simplicity in the first of eqn. ( 5 ) we have assumed that k3 is small relative to k2, the rate of dissociation of ES the complex into enzyme and intact substrate, a restriction which can easily be removed.We now suppose that the protein possesses v basic groups, the conjugate acids of which have dissociation constants Ki and that the substrate possesses a dipole moment p, equal to p' in the uiiactivated state and p* in the transition state leading to its decomposition. If e is the protonic charge and zie is the permanent charge of basic group i, the potential energy Y arising from the electrostatic interaction of the basic groups, their conjugate acids and the dipole of the substrate molecule, for fixed configuration of the protons is given by the expression, where yf is the angle between the dipole moment of the substrate molecule and the radius vector of length Ri from basic group i, De an effective dielectric constant depending upon geometry and the interior dielectric constant cf the protein DO, and the exterior dielectric constant D of the solvent.The occupation variables Xi specify the proton configuration, X i being equal to unity if a proton is situated on basic group i and zero otherwise. With the use of the methods of Kirkwood and Shumaker 1 for calculating the mean values (Xi), and ( X i X i ) a v , eqn. (2) leads to the following expression for the free energy of interaction of the protein with the substrate, in excess of WO due to its local binding to the active site, [H+ J ] e p cos V i i= 1 [H'] + Ki D,Ri*J . G . KIRKWOOD 81 where [H+] is the hydrogen ion activity, The first term of eqn. (7) arises from the average charges of the basic groups and the second term arises from fluctuations in charge configuration.In the calculation of the mean values over charge configurations, electrostatic interaction between the protons has been neglected, but may be taken into account by numerical calculation if desired. Although the contribution from the average charge distribution may in certain instances play a dominant role, we remark that it will be very sensitive to the geometrical arrangement of the charges, since cos yj can assume both positive and negative values, and appropriately situated pairs of basic groups can mutually nullify their contributions. On the other hand, the fluctuation term contains the factors cos Zvi, which are always positive and permit the effects of a large number of basic groups with randomly oriented radius vectors to accumulate. Since we are particularly interested here in investigating the effect of the fluctuation term, we shall consider onIy instances in which the symmetry of the arrangement of the basic groups relative to the active site causes the contribution from their average charges to vanish.Furthermore, since the fluctuation contribution from each basic group diminishes as the inverse fourth power of its distance from the active site, we may expect only vicinal groups to be effective. If we assume that there are v, vicinal basic groups of a single type a, with acidic dissociation con- stants K,, which make the effective contribution, eqn. (7) reduce to the following expression, The effective distance r, of each basic group from the active site would be equal to the actual distance, if the v, groups were distributed at uniform intervals of arc on the circumference of a circle to the plane of which the substrate dipole moment is parallel. For an actual geometrical arrangement approximating this regular array, r, will be approximately equal to the actual distance.The effecive dielectric constant is difficult to estimate reliably, since it is quite sensitive to geometry 2 near the interface between the protein and the exterior aqueous solvent. If the surface of the protein is regarded as plane in the vicinity of the active site and a pair of charges are situated at a depth b below the interface in the protein, a simple electrostatic calculation for a charge situated at a distance r from the substrate dipole yields where DO is the interior dielectric constant of the protein and D the exterior di- electric constant of the solvent.With an interior dielectric constant cf about 3 and an exterior dielectric constant of 80, D, is estimated with eqn. (9) to be of the order of magnitude 10 for reasonable ratios blr. We are now prepared to estimste the effect of fluctuations in charge configura- tion on the Michaelis-Menten constant K, and the intrinsic rate constant k3 of the enzymatic reaction. Since the effect on K , is of secondary interest, we shall present explicit expressions only for the rate constant k3. By means of eqn. (5) and (8), we obtain at once,82 RATES OF ENZYMATIC REACTIONS where A$ is the increment of the square of the dipole moment of the active site + substrate complex in activation to the transition state leading to decom- position of the substrate.We remark that k3 possesses a maximum when the pH of the solution is equal to the pK, of the conjugate acids of the effective vicinal basic groups. If (10) in the form, we denote by k, the maximum value of k3, we may write eqn. log (k3/km) = - CT tanh2 1 1 , el4 - e - U tanh u = - . eU+ edU According to eqn. (11), log (k3/k,J is represented as a function of pH by a sym- metrical bell-shaped curve with a maximum at p&, of the type commonly observed experimentally. A degree of dissymmetry would be introduced by taking into account electrostatic interactions between the protons. In instances where two or more types of vicinal basic groups were effective in contributing to the fluctu- ation free energy, the actual curve could be broadened by superposing two or more terms of the form of the right-hand side of eqn.(11). In order to make a rough comparison of the theory with experiment, we shall consider briefly the observations of Bergmann and Fruton 3 on the effect of pH on the rate of hydrolysis of carbobenzoxy-L-glutamyl-L-tyrosine by pepsin. The rate constant k3 possesses a maximum at a pH of four. This clearly indicates within the framework of the present theory that the effective basic groups are COO-. The constant k3 diminishes to about half its maximum value in an interval of one unit of pH on either side of the maximum, which requires CT to be ap- proximately unity. At 300" K, with an effective dielectric constant of 10, a value of A p of activation of 3 0 , we estimate cr to be of the order of magnitude unity for about 10 carboxyl groups situated at a distance of 5A from the active site, Although this distance is rather smaller than one might desire, we may reasonably conclude that the predictions of the theory are in semi-quantitative agreement with experiment. In conclusion, we remark that the present theory provides a reasonable mechan- ism for the participation of the protein moiety of an enzyme molecule in the enzymatic reaction. In particular, it provides an unusually simple explanation for the effect of pH on the rate of the reaction. No assertion is made that the interaction of the bound substrate molecule with vicinal basic groups through the fluctuation force is the only mechanism by which the protein may participate in the reaction. The planning of more critical experiments to test the present theory and more detailed applications to the analysis of existing experimental data will be deferred until a later time. 1 Kirkwood and Shumaker, Proc. Nat. Acad. Sci., 1952,38,863. 2 Kirkwood and Westheimer, J. Chem. Physics, 1938, 6, 506. 3 Bergrnann and Fruton, J. Bid. Chem., 1939, 127,627.