MINIREVIEW Barrier passage and protein dynamics in enzymatically catalyzed reactions Dimitri Antoniou 1 , Stavros Caratzoulas 1, *, C. Kalyanaraman 1 , Joshua S. Mincer 1 and Steven D. Schwartz 1,2 1 Department of Biophysics, Albert Einstein College of Medicine, Bronx, NY, USA; 2 Department of Biochemistry, Albert Einstein College of Medicine, Bronx, NY, USA This review describes studies of particular enzymatically catalyzed reactions to investigate the possibility that catalysis is mediated by protein dynamics. That is, evolution has crafted the protein backbone of the enzyme to direct vibra- tions in such a fashion to speed reaction. The review presents the theoretical approach we have used to investigate this problem, but it is designed for the nonspecialist. The results show that in alcohol dehydrogenase, dynamic protein motion is in fact strongly coupled to chemical reaction in such a way as to promote catalysis. This result is in concert with both experimental data and interpretations for this and other enzyme systems studied in the laboratories of the two other investigators who have published reviews in this issue. Keywords: protein dynamics; enzyme catalysis; tunneling; promoting vibration; promoting mode. INTRODUCTION The transmission of an atom or group of atoms from the reactant region of a reaction to the product region under the control of an enzyme is central to biochemistry. The manner in which the enzyme speeds this transfer is in some cases still not clear. What is known is the end effect; enzymatic reactions occur at rates many orders of magnitude more rapid than the corresponding solution phase reactions. This review will describe work recently completed in our group that has focused on examining the possibility that protein dynamics may in some enzymes play a central role in helping to produce the catalytic effect. These types of motions, which we have termed Ôrate promoting vibrationsÕ,are motions of the protein matrix that change the geometry of the chemical barrier to reaction. By this we mean that both the height and width of the barrier are changed. This unique role for the protein matrix has significant implications for the dynamics of the chemical reaction; in particular, causing a barrier to narrow can significantly enhance a light particle’s ability to tunnel, while masking the normal kinetic indicators of such a phenomenon. It is this feature that we have proposed as a unifying principle for some experimental data relating to tunneling in enzymatic reactions. This paper will describe our studies of rate promoting vibrations in enzymatic reactions with particular attention to the physical origins of the phenomenon. The structure of this paper will be as follows: in the next section, we will briefly review a number of different potential mechanisms for enzyme catalytic action along with promoting vibra- tions. Following this, we will describe the mathematical foundation for our theories in some detail. This section will be written for nonexperts, but will contain the necessary formulae for the specialist as well. It will include the relationship between the current theories and a well-known approach to charged particle transfer in biological reactions, namely the Marcus theory. In this section we will also describe a simple nonbiological chemical system in which the physical features of promoting vibrations may be easily understood – proton transfer in organic acid crystals. We will then describe how we have used these concepts to fit seemingly anomalous kinetic data for enzymatic reactions. In the next section, we explore how one might rigorously identify the presence of such a promoting vibration in any enzymatic reaction, and illustrate the concepts with appli- cations to specific enzyme systems. The paper then con- cludes with discussions of future directions for this research. POTENTIAL MODES OF ENZYMATIC ACTION The exact physical mechanisms by which enzymes cause catalysis is still a topic for vigorous dialogue [1–3]. The research described in this paper will argue for a strong contribution from a nontraditional source, i.e. directed protein motions. In order to place this concept into a context, we will briefly review other potential mechanisms for enzymes to cause catalysis. We emphasize that none of these mechanisms are mutually exclusive, and are probably all involved in catalysis to a greater or lesser extent in each enzyme system. One of the earliest and still widely accepted ideas used to explain this catalytic efficiency is the transition state-binding concept of Pauling [4]. In this picture, as a chemical substance is being transformed from reactants to products, the species that binds most strongly to the enzyme is at some Correspondence to S. D. Schwartz, 1300 Morris Park Ave., Bronx, NY 10461, USA. Tel.: + 1 718 430 2139, E-mail: sschwartz@aecom.yu.edu Abbreviations: NAC, near attack conformations; HLADH, horse liver alcohol dehydrogenase; YADH, yeast alcohol dehydrogenase. Note: a website is available at http://www.aecom.yu.edu/home/sggd/ faculty/schwartz.htm *Present address: Department of Chemical Engineering, Princeton University, NJ, USA. (Received 8 March 2002, revised 31 May 2002, accepted 6 June 2002) Eur. J. Biochem. 269, 3103–3112 (2002) Ó FEBS 2002 doi:10.1046/j.1432-1033.2002.03021.x intermediate point thought to be at or near the top of the solution phase (i.e. uncatalyzed) barrier to reaction. This preferential binding releases energy that stabilizes the transition state and thus lowers the barrier to reaction. This is a standard picture for nonbiological catalysis, and it also has significant experimental support. A critical observation is found using kinetic isotope effect methods. In this way, one can probe the chemical structure of the transition state in the catalytic event. Stable molecules can be designed that share the electronic properties of the transition state (usually identified by the electrostatic potential at the van der Waals surface). Furthermore, these molecules make highly potent inhibitors [5,6]. When substrate-like molecules that cannot react to form products bind, often a far lower level of inhibition is found. This result is said to be indicative of the fact that the transition state is strongly bound. It has been argued, however, that the electrostatic character of the active site during the catalytic event is largely determined by whatever charge stabilization is needed as the reaction progresses. If an inhibitor is designed with the complement- ary charges, it will bind strongly to the active site. However, this does not imply that the method by which the enzyme produced catalysis was transition state binding and con- comitant release of energy [1]. A second approach, which might be viewed as the converse of transition state stabilization, is ground state destabilization. In this picture [7], the role of the enzyme is to make the reactants less stable rather than making the transition state more stable. Thus the energetic hill that must be climbed with thermal activation is lowered. Energies are all relative and so the end effect of this and the first mechanism are the same; lowering the relative energy difference between reactants and transition state. But it is clear that this view presents a very different physical mechanism. Recent calculations [8] seem to show that this model may well be dominant for the most efficient enzyme known, orotidine monophosphate decarboxylase. A third concept that has been also suggested. In solution, reactants are strongly solvated by water, the dominant component of most living cells. When enzymes bind reactants, they often exclude water, and this lowered dielectric environment may be more conducive to reaction [9–11]. This approach to catalysis tends to treat the catalytic event much like an electron transfer reaction in solution. The dominant description of electron transfer in solution is Marcus’ theory [12], and this approach has also been used to describe atom transfer [13]. The concept here is that the main barrier to reaction is, in fact, reorganization of the solvent as charged particles move, rather than the intrinsic chemical barrier due to transformation of the substrate. It is certainly true that such energy reorganization may be a significant component in many cases, but probably does not account for all catalysis in biological systems. A fourth recent suggestion by Bruice [14,15] is that the dominant role of an enzyme is to position substrates in such a way that thermal fluctuations easily take them over the barrier to reaction. The set of positions the enzyme encourages the substrate to take are known as Ônear attack conformationsÕ (NACs). Here, while the enzyme might bind strongly to a transition state structure, this binding energy is not thought to be released specifically to speed the reaction. The enzyme moulds the substrate so that it is on the edge of reacting and forming products. Because the enzyme helps the reactants to form the NAC, this view is philosophically a bit closer to the ground state destabilization view. It is, however, not a statistical energetic argument, but rather a chemical structure argument. A fifth possibility for the mode of action of enzymes is the principle subject of this paper, that is, motions within the protein itself actually speed the rate of a chemical reaction. There is significant relation between this possibility and the last view of catalysis described above, i.e. the creation of the NAC. It must be stressed, however, that the current view is a dynamic one. For this concept to be true, actual motions of the protein must couple strongly to a reaction coordinate and cause an increase in reaction rate. This is not simply preparation of a reactive species, but rather dynamic coupling. It is important to note that this is an entirely different view of the method by which the enzyme accomplishes rate acceleration. In this view, evolution has created a protein structure that moves in such a way as to lower a barrier and make it less wide. It must be emphasized that this lowering of the barrier is not statistical lowering of a potential of mean force through the release of binding energy, but rather the use of highly directed energy (a vibration) in a specific direction. Furthermore, this is not simply the statistical preparation of reactive species as in the NAC concept. Here, protein dynamics directly affect the reaction coordinate potential. Although this effect can be quite apparent for a tunneling system (the probability to tunnel increases exponentially with a reduction of the width of the tunneling barrier), it is equally important for systems where the reaction proceeds through classical transfer, because as the barrier is made narrower, it is also lowered. In order to understand how directed protein motions may cause catalysis, we need a theory of chemical reactions in a condensed phase. Our group has developed theories over the past 10 years, and this work, initially developed for simple condensed phases, such as polar media, forms the basis for our analysis. We now describe these theories in some detail. AN ENZYME AS A CONDENSED PHASE: THEORETICAL FORMULATION FOR THE STUDY OF CHEMICAL REACTION There are two requirements to enable the study of a chemical reaction in any system, be it as simple as a gas phase collision, or as complex as that in an enzyme. First, a potential energy for the interaction of all the atoms in the system is needed. This includes the interactions of all atoms having their chemical bonds changed, and those that do not. The second requirement is for a method to solve the dynamics of the equations of motion that allow one to follow the progress of the reacting species in the presence of the rest of the system from reactants to products. In this work, we assume that we are able to obtain the first requirement (the potential). In order to study the dynamics on this potential, however, one needs to solve the Schro- dinger equation for the entire collection of atoms. It is a well-known fact that this is difficult for three or four atoms, and so essentially impossible for the thousands of atoms in a reaction catalyzed by an enzyme. Various groups have taken a number of possible approaches to solve this problem. One may assume that quantum effects are minor, and use a purely classical 3104 D. Antoniou et al. (Eur. J. Biochem. 269) Ó FEBS 2002 approach to solve the dynamics [16]. We are specifically interested in studies of enzyme systems where quantum mechanics plays a significant role, through tunneling of a light particle, in the chemical step of the enzyme, and so the classical approach will not be expected to yield valid results. Another approach is to use a mixed quantum-classical formulation in which a subset of the atoms is treated quantum mechanically and the rest of the system is treated purely classically. In recent years, this approach has become popular with the pioneering work of such investigators as Gao [8]. We have chosen a different approach, largely on stylistic grounds. Rather than treating the full collection of atoms as a mixture of quantum and classical objects (something that is difficult to define rigorously), we have developed approximate approaches to treat the entire collection of atoms as a quantum mechanical entity. As mentioned above, both approaches are approximate, but we prefer to make the approximation uniform for the entire system. We have called our approach the ÔQuantum KramersÕ methodology [17,18]. Our ideas were motivated by the following approximations developed for the study of the classical mechanics of large, complex systems. It is known that for a purely classical system [19,20], an accurate approximation of the dynamics of a tagged degree of freedom (for example a reaction coordinate) in a condensed phase can be obtained through the use of a generalized Langevin equation. The generalized Langevin equation is given by Newtonian dynamics plus the effects of the environment in the form of a memory friction and a random force [21]. Thus, all the complex microscopic dynamics of all degrees of freedom other than the reaction coordinate are included only in a statistical treatment, and the reaction coordinate plus environment is treated as a modified one- dimensional system. What allows a realistic simulation of complex systems is that the statistics of the environment can in fact be calculated from a formal prescription. This prescription is given by the Ôfluctuation-dissipation the- oremÕ, which yields the relationship between friction and random force. In particular, this theory enables us to calculate the memory friction from a relatively short-time classical simulation of the reaction coordinate. The Quan- tum Kramers approach, in turn, is dependent on an observation of Zwanzig [22,23]; if an interaction potential for a condensed phase system satisfies a fairly broad set of mathematical criteria, the dynamics of the reaction coordi- nate as described by the generalized Langevin equation can be rigorously equated to a microscopic Hamiltonian in which the reaction coordinate is coupled to an infinite set of Harmonic Oscillators via simple bilinear coupling: H ¼ P 2 s 2m s þ V o þ X k P 2 k 2m k þ 1 2 m k x 2 k q k À c k s m k x 2 k  2 ð1Þ The first two terms in this Hamiltonian represent the kinetic and potential energy of the reaction coordinate, and the last set of terms similarly represent the kinetic and potential energy for an environmental bath. Here, s represents some coordinate that measures progress of the reaction (for example, in alcohol dehydrogenase where the chemical step is transfer of a hydride, s might be chosen to represent the relative position of the hydride from the alcohol to the NAD cofactor.) c k is the strength of the coupling of the environmental mode to the reaction coordinate, and m k and x k give the effective mass and frequency, respectively, of the environmental bath mode. A discrete spectral density gives the distribution of bath modes in the harmonic environment: JðxÞ¼ p 2 X k c 2 k m k x k dðx À x k ÞÀdðx þ x k Þ ½ ð2Þ Here d(x ) x k )istheÔDirac deltaÕ function, so the spectral density is simply a collection of spikes, located at the frequency positions of the environmental modes, weighted by the strength of the coupling of these modes to the reaction coordinate. Note that this infinite collection of oscillators is purely fictitious; they are chosen to reproduce the overall physical properties of the system, but do not necessarily represent specific physical motions of the atoms in the system. It would seem that we have not made a huge amount of progress; we began with a many-dimensional system (classical) and found out that it could be accurately approximated by a one-dimensional system in a frictional environment (the generalized Langevin equation.) We have now recreated a many-dimensional system (the Zwanzig Hamiltonian). The reason we have done this is twofold. First, there is no true quantum mechanical analogue of friction, and so there really is no way to use the generalized Langevin approach for a quantum system, such as we would like to do for an enzyme. Second, the new quantum Hamiltonian given in Eqn (1) is much simpler than the Hamiltonian for the full enzymatic system. Harmonic oscillators are a problem that can easily be solved by quantum mechanics. Thus, the prescription is, given a potential for the enzymatic reaction, we model the exact problem using Zwanzig Hamiltonian, as in Eqn (1), with the distribution of harmonic modes given by the spectral density in Eqn (2), and found through a simple classical computation of the frictional force on the reaction coordi- nate. Then, using methods to compute quantum dynamics developed in our group [24–29], quantities such as rates or kinetic isotope effects may be computed. Thus, the quantum Kramers method, developed in our group, consists of the following ingredients. Given a potential for the enzymatic reaction, we model the exact problem using Zwanzig’s Hamiltonian, as in Eqn (1), with the distribution of harmonic modes given by the spectral density in Eqn (2). The spectral density is obtained through a Ômolecular dynamicsÕ simulation of the classical system. Then, using methods developed in our group to carry out the quantum dynamics, quantities such as rates or kinetic isotope effects may be computed. This approach enables us to model a variety of condensed phase chemical reactions with essentially experimental accuracy [30]. There are deeper connections between this approach and another popular method of dynamics com- putation in complex systems. We have shown [30] that this collection of bilinearly-coupled oscillators is in fact a microscopic version of the popular Marcus theory for charged particle transfer [12,13]. The bilinear coupling of the bath of oscillators is the simplest form of a class of couplings that may be termed antisymmetric because of the mathe- matical property of the functional form of the coupling on reflection about the origin. This property has deeper implications than the mathematical nature of the symmetry Ó FEBS 2002 Barrier passage and protein dynamics (Eur. J. Biochem. 269) 3105 properties. Antisymmetric couplings, when coupled to a double-well-like potential energy profile, are able to instan- taneously change the level of well depths, but do nothing to the position of well minima. This modulation in the position of minima is exactly what the environment is envisaged to do within the Marcus theory paradigm. As we have shown [30], the minima of the total potential in Eqn (1) will occur, for a two-dimensional version of this potential, when the q degree of freedom is exactly equal and opposite in sign to cs mx 2 , and the minimum of the potential energy profile along the reaction coordinate is unaffected by this coupling. Within Marcus’ theory, which is a deep tunneling theory, transfer of the charged particle occurs at the value of the bath coordinates that cause the total potential to become symmetrized. Thus, if the bare reaction coordinate potential is symmetric, then the total potential is symmetrized at the position of the Ôbath plus couplingÕ minimum. When this configuration is achieved, the particle tunnels; the activation energy for the reaction is largely the energy to bring the bath into this favorable tunneling configuration. While Marcus’ theory and our microscopic quantum Kramers theory are highly successful in many cases, in other cases, it is not possible to reproduce experimental results using such an approach. The reason for this is that the antisymmetric coupling contained within the Zwanzig Hamiltonian does not physically represent all possible important motions in a complex reacting system. In fact, such a reality was pointed out some time ago in seminal work of the Hynes group [31]. In some of our earlier work on hydrogen transfer in enzymatic systems, we were able to show that one could reasonably fit experimental kinetic data in such enzymatic systems with phenomenological applica- tion of the Hynes theories [32]. We became interested in a microscopic study of such systems in the examination of nonbiological proton transfer reactions, i.e. organic acid crystals. The simplest example is a carboxylic acid dimer, showninFig.1.Suchsystemshadbeenstudiedformany years [33–37], and they presented what seemed to be a chemical physics conundrum. While quantum chemistry computations seemed to show that the intrinsic barrier to proton transfer in these systems was reasonably high, and low experimental activation energies seemed to indicate a significant involvement of quantum tunneling in the proton transfer mechanism, careful measurements of kinetic iso- tope effects showed kinetics indicative of classical transfer. In order to study such systems, a rigorous theory, which allowed inclusion of symmetrically coupled vibrations, in addition to an environmental bath of antisymmetrically coupled oscillators, was needed. Mathematically, the simp- lest transformation of the Hamiltonian in Eqn (1) is given by: H ¼ P 2 S 2m s þ V o þ X k P 2 k 2m k þ 1 2 m k x 2 k q k À c k s m k x 2 k  2 þ P 2 Q 2M þ 1 2 mX 2 Q À Cs 2 MX 2  ð3Þ Note that in this case, the oscillator that is symmetrically coupled, represented by the last term in Eqn (3), is in fact a physical oscillation of the environment. We were able to develop a theory [38] of reactions mathematically represented by the Hamiltonian in Eqn (3), and using this method and experimentally available param- eters for the benzoic acid proton transfer potential, we were able to reproduce experimental kinetics as long as we included a symmetrically coupled vibration [39]. The results are shown in Table 1 below. The two-dimensional activa- tion energies refer to a two-dimensional system comprised of the reaction coordinate and a symmetrically coupled vibration. The reaction coordinate is also coupled to an infinite environment as described above. In this case, the symmetric motion has a clear physical origin: the symmetric motion of the carbonyl and hydroxyl oxygen atoms toward each other. Kinetic isotope effects in this system are modest, even though the vast majority of the proton transfer occurs via quantum tunneling. The end result of this study is that symmetrically coupled vibrations can significantly enhance rates of light particle transfer, and also significantly mask kinetic isotope signatures of tunnel- ing. A physical origin for this masking of the kinetic isotope effect may be understood from a comparison of the two- dimensional problem comprised of a reaction coordinate coupled symmetrically and antisymmetrically to a vibration. As Fig. 2 shows, antisymmetric coupling causes the minima (the reactants and products) to lie on a line; the minimum energy path, which passes through the transition state. In contrast, symmetric coupling causes the reactants and products to be moved from the reaction coordinate axis in such a way that a straight line connection of reactant and products would pass no where near the transition state. This, in turn, results in the gas phase physical chemistry phenomenon known as corner cutting [40–42]. Physically, the quantity to be minimized along any path from reactant to products is the action. This is an integral of the energy, and so loosely speaking, it is a product of distance and depth under the barrier that must be minimized to find an approximation to the tunneling path. The action also includes the mass of the particle being transferred, and so in the symmetric coupling case, a proton will actually follow a very different physical path from reactants to products in a reaction than a deuteron. (Not just in the trivial sense that one tunnels more than another). It is this following of a different physical path, even when tunneling dominates, Fig. 1. A benzoic acid dimer. Thereactioncoordinateinthiscaseisthe symmetric transfer of the hydroxyl protons to the carbonyl oxygen. The promoting vibration is the symmetric motion of the oxygens toward each other. Table 1. Activation energies for H and D transfer in benzoic acid crystals at T ¼ 300 K. Three values are shown: the activation energies calculated using a one- and two-dimensional Kramers problem and the experimental values. The values of energies are in kcalÆmol )1 . E 1d E 2d Experiment H 3.39 1.51 1.44 kcalÆ mol )1 D 5.21 3.14 3.01 kcalÆ mol )1 3106 D. Antoniou et al. (Eur. J. Biochem. 269) Ó FEBS 2002 that causes the kinetic isotope effects to be masked. It was this low level of primary kinetic isotope effect that suggested a similarity between the proton transfer mechanism in the organic acid crystal and that of enzymatic reactions. While coupled motions of nearby atoms in enzymatic reactions have been used to explain anomalous kinetic isotope effects [43], these were studies in a classical picture with semiclas- sical tunneling added (the Bell correction; [44]) and they could not be used to account for enzymatic reactions in a deep tunneling regime. Klinman and coworkers have helped pioneer the study of tunneling in enzymatic reactions. One focus of their work has been the alcohol dehydrogenase family of enzymes. Alcohol dehydrogenases are NAD + -dependent enzymes that oxidize a wide variety of alcohols to the corresponding aldehydes. After successive binding of the alcohol and cofactor, the first step is generally accepted to be complex- ation of the alcohol to one of the two bound Zinc ions [45]. This complexation lowers the pK a of the alcohol proton and causes the formation of the alcoholate. The chemical step is then transfer of a hydride from the alkoxide to the NAD + cofactor. They [46] have found a remarkable effect on the kinetics of yeast alcohol dehydrogenase (a mesophile) and a related enzyme from Bacillus stereothermophilus, a thermo- phile. A variety of kinetic studies from this group have found that the mesophile [47] and many related dehydro- genases [48–51] show signs of significant contributions of quantum tunneling in the rate-determining step of hydride transfer. Remarkably, their kinetic data seem to show that the thermophilic enzyme actually exhibits less signs of tunneling at lower temperatures. Recent data of Kohen & Klinman [52] also show, via isotope exchange experiments, that the thermophile is significantly less flexible at mesophi- lic temperatures, as in the results of Petsko et al. [53], who conducted studies of 3-isopropylmalate dehydrogenase from the thermophilic bacteria Thermus thermophilus.These data have been interpreted in terms of models similar to those we have described above, in which a specific type of protein motion strongly promotes quantum tunneling; thus, at lower temperatures, when the thermophile has this motion significantly reduced, the tunneling component of reaction is hypothesized to go down even though one would normally expect tunneling to go up as temperature goes down. Additionally, the Klinman group has investigated the catalytic properties of various mutants of horse liver alcohol dehydrogenase (HLADH). HLADH in the wild-type has a slightly less advantageous system to study than yeast alcohol dehydrogenase, because the chemistry is not the rate determining step in catalysis for this enzyme. Two specific mutations have been identified, Val203 fi Ala and Phe93 fi Trp, which significantly affect enzyme kinetics. Both residues are located at the active site; the valine impinges directly on the face of the NAD + cofactor distal to the substrate alcohol. Modification of this residue to the smaller alanine significantly lowers both the catalytic efficiency of the enzyme, as compared to the wild-type, and also significantly lowers indicators of hydrogen tunnel- ing [54]. Phe93 is a residue in the alcohol binding pocket. Replacement with the larger tryptophan makes it harder for the substrate to bind, but does not lower the indicators of tunneling [55]. Bruice’s recent molecular dynamics calcula- tions [56] produce results consistent with the concept that mutation of the valine changes protein dynamics, and it is this alteration, missing in the mutation at position 93, which in turn changes tunneling dynamics. (We note the recent experimental results from Klinman’s group [57] in which no decrease in tunneling is seen as the temperature is raised.) A final set of enzymes now thought to exhibit dynamic protein control of tunneling hydrogen transfer is that in the amine dehydrogenase family. Scrutton and coworkers have extensively studied these enzymes [58]. Though similarly named and having a similar end effect as the alcohol dehydrogenases, they employ radically different chemistry. These enzymes catalyze the oxidative deamination of primary amines to aldehydes and free ammonia. In this case, however, rather than a chemical step of hydride transfer, the rate determining chemical step is proton transfer; and in fact these enzymes catalyze a coupled electron proton transfer reaction. Electrons are coupled to some cofactor, for example, in the case of aromatic amine dehydrogenase, the cofactor is tryptophan-tryptophyl qui- none. Kinetic studies have shown that methylamine dehy- drogenase exhibits not only relatively large primary kinetic isotope effects (unlike the alcohol dehydrogenases), but also very strong temperature dependence in the measured activation energy. This experimental data has been inter- preted as showing that the enzyme works via a promoting vibration [59], as we have suggested for bovine serum amine oxidase [32], and for various forms of HLADH [60]. Here, the primary kinetic isotope effect is % 17, rather than 3 or 4. s q s 0 +s 0 A,S S A Fig. 2. This diagram shows the location of stable minima in two- dimensional systems. In one case a vibrational mode is symmetrically coupled to the reaction coordinate, and in the other, antisymmetrically coupled. The figure represents how antisymmetrically and symmetri- cally coupled vibrations effect position of stable minima – that is reactant and product wells – in modulating the one dimensional double well potential (before coupling along the x axis). The x axis, s,repre- sents the reaction coordinate, and q the coupled vibration. The points on the figure labeled S and A are the positions of the well minimal in the two dimensional system with symmetric and antisymmetric coup- ling, respectively. An antisymmetrically coupled vibration displaces those minima along a straight line, so that the shortest distance between the reactant and product wells passes through the transition state. In contradistinction, a symmetrically coupled vibration, allows for the possibility of Ôcorner cuttingÕ under the barrier. For example, a proton and a deuteron will follow different paths under the barrier. Ó FEBS 2002 Barrier passage and protein dynamics (Eur. J. Biochem. 269) 3107 Another enzyme studied by this group is aromatic amine dehydrogenase. This enzyme is especially interesting because it is fairly nonspecific in the substrates it will bind and catalyze. In particular, in the series benzylamine, dopamine, and tryptamine, primary kinetic isotope effects range from a low of 4.8 in benzylamine to a high of 54.7 in tryptamine [58]. In addition, the three substrates demonstrate markedly differing temperature dependencies in their kinetic isotope effects. Scrutton and coworkers have described this enzyme as one that demonstrates both promoting vibrations and the overall importance of barrier shape rather than just barrier height in biochemistry. It seems then that there is a growing body of evidence that protein dynamics could well play a central role in enzymatic catalysis, well beyond standard pictures of loop motions that cause substrate binding and change electrostatic environments as substrates are transformed to products. In fact, in cases where tunneling seems to play a significant role, as indicated by kinetic isotope effect experiments, directed motion of the protein could well be responsible for a significant fraction of the catalytic mechanism. What is lacking in the ongoing analysis, is a tool that allows, through a knowledge of protein structure and an assump- tion of a potential function for the protein, the rigorous identification of the presence or absence of such a symmet- rically coupled/promoting vibration. Such a theoretical approach is especially important in light of the fact that there is currently no general experimental method to detect such a protein motion as it impacts catalysis. While spectroscopic methods can, with extraordinary sensitivity, interrogate localized motions in proteins, as we have described above, the defining nature of a promoting vibration is to be found in the nature of the coupling of that motion to the reaction coordinate. There is no experimental tool available to directly measure this coup- ling. The next section details our theoretical approach to the problem, and a recent application to alcohol dehydroge- nase. THE DETECTION OF PROMOTING VIBRATIONS IN PROTEINS The quantity that naturally describes the way in which an environment interacts with a reaction coordinate in a complex condensed phase is the spectral density. In Eqn (2), the spectral density could be seen to give a distribution of the frequencies of the bilinearly-coupled modes, convolved with the strength of their coupling to the reaction coordi- nate. The concept of the spectral density is, however, quite general, and the spectral density may be measured or computed for realistic systems in which the coupling of the modes may well not be bilinear [61]. We have also shown [18] that the spectral density can be evaluated along a reaction coordinate. One only obtains a constant value for the spectral density when the coupling between the reaction coordinate and the environment is in fact bilinear. We have shown that a promoting vibration is created as a result of a symmetric coupling of a vibrational mode to the reaction coordinate and, as described previously, this is quite a general feature of motions in complex systems. Analytic calculations demonstrated that such a mode should be manifest by a strong peak in the spectral density when it was evaluated at positions removed from the exact transition state position, in particular in the reactant or product wells. In cases where there is no promoting vibration, while the spectral density may well change shape as a function of reaction coordinate position, there will be no formation of such strong peaks. Numerical experiments completed in our group have shown a delta function at the frequency position of the promoting vibration as the analytic theory predicted when we study a model problem in which a vibration is coupled symmetrically. The results of such calculations are shown in Figs 3 and 4 [62]. These are spectral densities calculated for the proton in a potential for proton transfer between two carbon centers immersed in argon; shown in Fig. 3 at the transition state, and in Fig. 4 with the proton at a position near the reactant well. A more stringent test of the approach is to be found in a similar computation when, rather than explicitly including a symmetrically coupled vibration, we simply create a system in which proton transfer occurs between two vibrating atoms of a complex. There we expect to find a promoting vibration, but the identity of this vibration is not manifest in the model form, rather it is buried in the dynamics of the atomic motions. In fact, when we compute the spectral density for such a proton transfer system with the proton held in the reactant well and the effective mass of the vibrating system equal to 100 amu, we obtain the result shown in Fig. 5. Given the 0 50 100 150 200 250 300 0 5e-05 0.0001 0.00015 0.0002 0.00025 J(ω) Fig. 4. The spectral density for the same system as in Fig. 3, but now measured in the reactant well. 0 50 100 150 200 0 1e-05 2e-05 3e-05 4e-05 5e-05 J(ω) Fig. 3. A spectral density for proton transfer between two carbon centers with a symmetrically coupled vibration measured exactly at the transition state – the point of minimum coupling. 3108 D. Antoniou et al. (Eur. J. Biochem. 269) Ó FEBS 2002 success of the methodology to detect the presence of a promoting vibration in test calculations, the next goal is to apply the methodology to a real enzyme system. The choice we made was from the alcohol dehydrogenase family. Our previous studies of alcohol dehydrogenase enzymes involved yeast alcohol dehydrogenase (YADH) and a mutant of alcohol dehydrogenase from Bacillus stereothermophilus. YADH is advantageous for the study of kinetic isotope effects and enzyme dynamics, because the chemical step is rate determining and commitment factors need not be found. We began our studies of promoting vibrations in enzymes with HLADH [63] for two reasons: first, there is as yet no crystal structure for YADH, and such a structure is needed as a starting point for any dynamics study of a protein. Second, there are a number of mutants of HLADH, which allow detailed study of the influence of protein composition on protein dynamics, and how dynamics relates to kinetics of catalysis. Our analysis began with the 2.1-A ˚ crystal structure of Plapp and coworkers [64]. This crystal structure contains both NAD + and 2,3,4,5,6-pentafluorobenzyl alcohol com- plexed with the native HLADH (metal ions and both the substrate and cofactor.) The fluorinated alcohol does not react and go onto products because of the strong electron withdrawing tendencies of the flourines on the phenyl ring, and so it is hypothesized that the crystal structure corresponds to a stable approximation of the Michaelis complex. We then replaced the fluorinated alcohol with the unfluorinated compound to obtain the reactive species as in Luo et al. [56]. This structure was used as input into the CHARMM program [65]. Both crystallographic waters [64] (there are 12 buried waters in each subunit) and environ- mental waters were included via the TIP3P potential [66]. The substrates were created from the MSI/ charmm param- eters. The NAD cofactor was modeled using the force field of Mackerell et al. [67]. The lengths of all bonds to hydrogen atoms were held fixed using the SHAKE algorithm. A time step of 1 fs was employed. The initial structure was minimized using a steepest descent algorithm for 1000 steps followed by an adapted basis Newton–Raphson minimiza- tion of 8000 steps. The dynamics protocol was heating for 5 ps followed by equilibration for 8 ps, followed finally by data collection for the next 50 ps. Using CHARMM ,we computed the force autocorrelation function on the reacting particle. The force is calculated in CHARMM as a derivative of the velocity. This is a numerical procedure that can, of course, introduce error. We have recently found that spectral densities may also be calculated from the velocity autocorrelation function directly, and these spectral densi- ties exhibit exactly the same diagnostics for the presence of a promoting vibration, as do those calculated from the force. In addition, the Fourier transform of the force autocorre- lation function can be shown to be related to the Fourier transform of the velocity autocorrelation function times a square of the frequency. This square of the frequency tends to accentuate high frequencies. In a simple liquid, this is not a problem because there are essentially no high frequency modes. In a bonded system, such as an enzyme, many high frequency modes remain manifest in autocorrelation func- tions, and it is advantageous to employ spectral densities calculated from Fourier transforms of the velocity function. We will not have an ÔexactÕ reaction coordinate at our disposal, but this does not affect the calculation. The diagnostic of the promoting vibration is simply the presence of a strong variation in the spectral density as the reacting particle (in this case the hydride) is moved from the reactant well to the product well. As long as it is moved on a vector that contains some component of the reaction coordinate, a sharp spike will appear in the spectral density at a frequency corresponding to the promoting vibration, possibly shifted by a small amount [63]. Thus, appearance of a strong peak in the spectral density along the line connecting the alcoholate and the NAD + should be found close to the actual frequency of the promoting vibration. We then calculated the force or velocity autocorrelation function on the transferring particle, i.e. the hydride. A search through position space in the vicinity of the transition state will yield spectral densities in which a peak moves to ever-smaller frequencies. The result with the smallest frequency should be very close to the bare frequency of the promoting vibration, and incidentally would locate the transition state in the enzymatic environment. If the hypothesized promo- ting vibration is present, we can immediately check the frequency to ascertain if the predicted frequency is similar to the frequency of motion of an expected residue, which is in the putative protein motion. For example, a position correlation function on Val203 should yield an oscillatory function with period of oscillation close to that found in the spectral density calculation, if in fact the hypothesis that this residue is involved in a correlated motion that creates a promoting vibration is correct. As a final first test of our approach to the study of promoting vibrations in enzymes, we subjected the Val203 fi Ala mutant to the same computational procedure. Recall that this mutant has a smaller side chain impinging on the face of the NAD + distal to the alcohol. It is motion of this 203 residue into the cofactor, pushing the cofactor ring system closer to the alcohol, which is hypothesized to result in the creation of the promoting vibration. With a smaller sidechain, we expect Ôless of a pushÕ, which will be made manifest in a weaker coupling of the promoting vibration to the reaction coordinate. In turn this weaker coupling would appear in 0 100 200 300 400 500 ω (cm -1 ) 0 0.1 0.2 0.3 0.4 0.5 J(ω) Fig. 5. The spectral density in the reactant well for a similar model, but with no manifestly symmetrically coupled promoting vibration. In this case the carbon centers move toward each other, and their motion creates a promoting vibration similar to the benzoic acid system shown in Fig. 1. Ó FEBS 2002 Barrier passage and protein dynamics (Eur. J. Biochem. 269) 3109 our computations as a smaller peak in the spectral density at positions remote from the transition state. The results from these calculations are shown in Figs 6, 7, and 8, with the spectral density now indicated by G(x), showing that we compute this spectral density from a velocity autocorrelation function rather than a force auto- correlation function. We now employ a velocity autocorre- lation function for a purely technical reason. In a simple fluid, relatively slow frequency motions dominate all envi- ronmental modes. Note for example, the first peak in Fig. 3 occurs at about 50 cm )1 , and the spectral density is essentially zero by 200 cm )1 . In a protein, there are vibrational modes extending up to CH stretches in the thousands of wavenumbers. It can be shown that the relationship between the force and the velocity autocorre- lation functions is simply multiplication by the square of the frequency. Thus in the force autocorrelation function, even very weakly coupled modes, can be dominant when their frequency is very high. When the methodology is applied to the enzyme, we find exactly the expected results. First, Fig. 6 shows the spectral density for the hydride when held at the reactant well, the product well and the transition state. We find strong peaks in the spectral density for the reactant and product configuration, with the spectral density for the transition state configuration appearing flat. We note that the results of the transition state computation are not zero; they simply are so much lower in magnitude, than the results at the reactants or products that they appear to be zero. In fact the spectral density at the transition state is exactly of the shape one would expect for the spectral density for a protein, and this result is shown alone in Fig. 7. This is what was found in Figs 3, 4 and 5. In addition, the spectral density for the hydride held at the transition state in the Val203 fi Ala mutant is exactly as was expected, i.e. similar frequency peaks at lower intensity. It is interesting to note that in locating the transition state location, defined in this instance as the position of minimum coupling of the promoting vibration to the reaction coordinate, we found that this location differs slightly between the wild-type and mutant enzymes. This is a further indication that protein dynamics play a central role in the catalytic effect in these systems. CONCLUSIONS In this review, we have described work pursued in our group over the past 5 years demonstrating the potential for protein involvement in catalysis, and theoretical methods that confirm the importance of such motions. These results have relied heavily on experimental results from the laboratories of the two other groups contributing reviews to this volume. Our initial involvement in this area came as a result of trying to understand why, in both biological and nonbiological systems, there seemed to be cases of significant involvement of quantum tunneling without the expected high primary kinetic isotope effect. 0 500 1000 1500 2000 2500 3000 ω (cm 1 ) 0.0 10.0 20.0 30.0 G S (ω) (MC) Fig. 7. The spectral density computed at the point of minimal coupling in Fig. 6, shown alone. Note that the spectral density is an order of magnitude smaller at the point of minimal coupling than in the reac- tant or product wells. This result is similar in this respect to the result obtainedinFigs3and4. 0 500 1000 1500 2000 2500 3000 ω (cm 1 ) 50.0 50.0 150.0 250.0 350.0 450.0 550.0 G S (ω) (R) (P) (MC) Fig. 6. The spectral densities for wildtype horse liver alcohol dehy- drogenase computed with the hydride held in the reactant well (r), the product well (p), and at the point of minimal coupling (mc). 0 100 200 300 400 500 ω (cm -1 ) 0 100 200 300 G S (ω) Val 203 → Ala mutant wild-type Fig. 8. A comparison of the spectral densities at the points of minimal coupling for the wildtype HLADH and for the mutant Val203 fi Ala. The smaller residue in the 203 position in the mutant is less strongly coupled to the reaction coordinate, hence the lower peaks. Note that the point of minimal coupling occurs at slightly different locations in the two proteins. 3110 D. Antoniou et al. (Eur. J. Biochem. 269) Ó FEBS 2002 Having understood this puzzle, it is important to mention that new problems have arisen. The first and foremost is the large timescale separation between the promoting vibration and the chemical turnover of the enzyme systems involved. The dominant peaks in the spectral densities indicate motions on the 150-cm )1 frequency scale. This corresponds to sub-picosecond vibrations. Clearly, many cycles of the promoting vibration must occur before it is effective in helping to cause chemical turnover. This is, of course, not without precedence; motions such as loop closures in proteins often happen many times before catalysis occurs. The generally accepted explanation is that such ineffective motions are the result of incorrectly placed groups or substrate in the enzyme active site. In many ways, this issue corresponds to finding the actual Ôreaction coordinateÕ in any condensed phase problem. For example, in a proton transfer in a polar solvent, reaction is not actually limited by movement of the proton, but actually by rearrangement of the solvent around the moving charged particle. Thus, what specific motions and placements of atoms within the enzyme and substrates are needed for catalysis will be a subject of significant concern for theoretical research. A second question of almost philosophical import is the extent to which evolution has utilized protein dynamics in concert with quantum tunneling to craft enzymes. It should certainly come as no surprise that tunneling is used in catalysis. Evolution knows nothing about which equation, Newton’s or Schrodinger’s, is needed to understand dynamics. All that is needed is the creation of a path from reactants to products. It is interesting to consider that the coupling of tunneling with protein dynamics in the form of promoting vibrations may have been used to create exquisite sensors of chemical substrates. Because tunneling is exponentially dependent on tunneling distance, small changes in distance in a promoting vibration can distinguish between different substrates. 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MINIREVIEW Barrier passage and protein dynamics in enzymatically catalyzed reactions Dimitri Antoniou 1 , Stavros Caratzoulas 1, *, C. Kalyanaraman 1 , Joshua S. Mincer 1 and Steven. amine dehydrogenase. This enzyme is especially interesting because it is fairly nonspecific in the substrates it will bind and catalyze. In particular, in the series benzylamine, dopamine, and tryptamine,. Cha, Y. , Jonsson, T., Grant, K.L. & Klinman, J.P. (1992) Role of internal thermodynamics in determining hydrogen tunneling in enzyme -catalyzed hydrogen transfer reactions. Bio- chemistry 31,