Chapter 3. Quantum mechanical treatment of enzyme reactions

摘要

3 Quantum mechanical treatment of enzyme reactions By GA�BOR NA�RAY-SZABO� and DO�RA K. MENYHA�RD Department of Theoretical Chemistry Lora�nd Eo�tvo�s University H-1117 Budapest Pa�zma�ny Pe�ter st. 2 Hungary 1 Introduction In principle any molecular process can be described accurately using quantum mechanics. Precise solutions of the basic equations need however formidable computer capacity even within the framework of transition-state theory. At present only gasphase reactions involving small molecular species with up to ten non-hydrogen atoms can be treated appropriately at the pure ab initio level; in the case of larger systems simplified models and various numerical approximations must be used. Fortunately in spite of the extreme complexity and size of enzymatic systems we now have at hand an entire arsenal of molecular models and computational methods for their satisfactory treatment.The starting point for any quantitative approach to an enzyme mechanism is an adequate model. Almost all explicit calculations are based on the experimentally determined three-dimensional structure of the enzyme under study that can be obtained via protein crystallography1–4 or nuclear magnetic resonance techniques.2–5 This information should be completed with knowledge about the environment of the system such as structural and bulk water ionic strength pH and other factors potentially influencing enzyme activity. It is quite frequently however not appropriate to suppose that high-performance ab initio calculations with a large basis set and inclusion of electron correlation e§ects on a small model of a dozen active-site atoms deprived of their environment are necessarily more reliable than e.g.a classical electrostatic calculation on a large almost complete model that includes all protein atoms with appropriate protonation states surrounding water counter-ions and other essential components of the biophase. Accordingly the selected model and applied method should harmonise in the sense that their degree of sophistication should be similar or if this is not the case the results of the calculation should be discussed appropriately. To help the reader to avoid inconsistency between the selected model and the computational method applied we treat geometric models of enzymatic reactions in more detail in Section 2. In Section 3 mostly quantum mechanical computational methods as well as some other ones used to complement them to provide a better description on a larger model will be discussed.The basic philosophy of the computational treatment of enzyme structure and function is the pragmatic approach that has proved to be quite produc- 49 tive e.g. in case of the computational study of molecular structure vibration spectra or reaction mechanisms involving small systems with C N O and H atoms. The pragmatic approach does not insist on the solution of the basic equations of quantum mechanics containing only universal physical constants such as the velocity of light or charge and mass of the electron but does make use of the information deduced from a variety of physical and chemical studies. Such information is e.g. that electrons are distributed in more or less separated shells around atomic nuclei or the majority of molecules can be constructed from building blocks (chemical bonds) that are transferable to a certain extent from one molecule to another.If we do not rely on this preconception solution of the Schro� dinger equation becomes much more laborious even for the simplest systems experimental facts can be reproduced with much less accuracy at a higher cost and thus the cost/performance ratio of computational methods remains relatively low. One of the most successful pragmatic approaches is molecular mechanics,6 which has also been widely applied to the study of enzymes.7 The method is based on a relatively simple analytical representation of the potential energy surface and makes use of a set of empirical parameters.The relative mathematical simplicity of the potential energy functions makes them suitable for the description of enzymes where the number of atoms to be treated explicitly is very large often 10000 or more when including the solvent and other components of the environment. Empirical energy functions allow one to calculate forces on enzyme atoms and to do molecular dynamics calculations for the entire system including the protein and its environment using reasonable computer time and storage capacity. Calculations permit one to obtain specific information that is complementary with experimental data e.g. energetic aspects of ligand binding changes in side-chain conformation e§ects of point mutation or the mechanism of folding. However details of enzymatic reactions related to bond formation and breaking need a more sophisticated treatment such as the hybrid quantum mechanical molecular mechanical methods which will therefore also be surveyed.Another important approach to computer modelling of enzymatic systems is protein electrostatics. While it is based on quantum mechanics a classical treatment can be justified allowing a vast reduction in computational e§ort yet maintaining considerable accuracy of calculated quantities. This is because at large distances the interaction between charged or polar species such as amino-acid residues is determined by classical electrostatics that can be adequately modelled by relatively simple mathematical equations the precise analytical or numerical solution of which provides a good estimate of energy changes during enzymatic processes.Though the electrostatic approach does not account for quantum mechanical e§ects thus it might be beyond the scope of the present review we devote special attention to it in Section 3. As we will see below enzyme mechanisms cannot be fully understood without protein electrostatics therefore an adequate quantum mechanical treatment needs to be often completed by inclusion of an electrostatic contribution to the energy or the respective term in the Hamiltonian of the model system. Though organic chemistry has provided a variety of systems that can be considered as enzyme mimics8 there is now ample evidence indicating that enzymatic reactions show some specific features that have no analogues in the conventional treatment of classical organic mechanisms.This is why we devote special attention to these features 50 G. Na� ray-Szabo� and D. K. Menyha� rd in Section 4 since these are the most interesting details of enzyme action adequate treatment of which may lead to a much better understanding of these complicated events with a potential for precise modelling and the ability to predict molecular phenomena. We give an overview of recent important work on the quantum mechanical treatment of enzyme reactions in Section 5. To date quite a few enzymes have been discussed on a relatively sound theoretical basis and their number is steadily increasing. It seems to be justified to say that in the foreseeable future we will be able to treat enzymatic processes with an accuracy that is su¶cient for a reliable modelling even for some predictions to the level achieved for the majority of classical organic chemical reactions.2 Models Owing to their relative mathematical simplicity molecular mechanics and molecular dynamics use full models of enzyme systems with explicit consideration of all components protein cofactor (if one exists) substrate water and counter-ions. However such a model is far beyond the scope of any quantum mechanical method thus enzyme systems must be partitioned into various regions that are arranged like shells within an onion (Fig. 1). The active site or central machinery (C) is surrounded by a polarisable environment (P) which is embedded in a non-polarisable region of the protein with ionisable side-chains (N) and water including an atmosphere of charged counter-ions as solvent (W). Some regions may overlap and some may be dropped completely from the model depending on the degree of sophistication of the treatment.In the following we discuss various regions in detail providing a guide for theire site For enzymes without prosthetic groups or cofactors the minimum-size active site model (C) is composed of one to three essential side-chains and a truncated substrate (e.g. formamide replacing a peptide bond). Thus the minimum number of nonhydrogen atoms within an active-site model ranges between 10 and 15 which is tractable even with highly sophisticated quantum mechanical (molecular orbital or density functional) methods with an appropriate account of electron correlation. Such reduced models are suitable for a gross description of bond fission and bond formation as well as substrate conformational changes or electronic excitations localised to the active site.In the case of single-point calculations we should use the active-site geometry which might be available from the Protein Data Bank,9 or any set of Cartesian co-ordinates obtained by some experimental technique or much less frequently by protein (e.g. homology) modelling.1,2 If performing geometry optimisation atoms at the periphery of the active site must be tethered at their experimentally determined positions to avoid artificial distortions and keep other atoms within the geometrical frame of the rest of the protein that has been dropped from the model. For a concrete model a boundary of the active site must be defined to provide a chemically realistic system that o§ers an appropriate target for the quantum mechanical calculation.Covalent bonds linking side-chains of the active site to the protein backbone have to be cut and in most cases the resultant dangling bonds are saturated 51 Quantum mechanical treatment of enzyme reactions Fig. 1 Onion-shell model of malate dehydrogenase. Active site (C black space-filling representation) reactive part of the cofactor the substrate and His-177; polarisable protein environment (P dark grey capped-stick representation) enzyme residues with at least one atom within 1 pm distance from the substrate; non-polarisable protein environment (N light grey tube) other enzyme residues and solvent W explicit water molecules (spherical representation). by hydrogen atoms. Since H–X bonds are more or less polar if XN O or S and this may cause ambiguous charge accumulation on peripheral atoms C–C single bonds preferably the CA–CB bond should be cut to provide an adequate active-site model.Some software packages use suitably defined pseudo atoms for saturation.10 A general shortcoming of most partitioning models is that the hydrogen atoms used to saturate dangling bonds in C may be in steric conflict with surrounding protein atoms at the boundary. As a result convergence problems may arise in the quantum mechanical calculation for C or owing to the clashing of atoms spurious contributions emerge in the molecular mechanics energy expression. There are some important cases (e.g. electron-transfer reactions in haem proteins) where the minimum-size active site should contain several dozen or even around one hundred atoms.For these only semiempirical or small basis-set ab initio molecular orbital methods can be applied which may bring much more reliable results than a higher level quantum mechanical calculation on a smaller inadequate model. An example is cytochrome C peroxidase where localisation of an unpaired electron in compound I a reaction intermediate is a crucial problem (see details in Section 5). For this study it proved to be very di¶cult to define the appropriate size of the active site 52 G. Na� ray-Szabo� and D. K. Menyha� rd Fig. 2 Active-site model of cytochrome C peroxidase applied for the study of electron transfer during catalysis. model to be able to determine the most probable location of the unpaired electron. Our experience with the calculation of spin distribution for various structures derived from that shown in Fig.2 by truncation has been that the results are extremely sensitive to the definition of the model adding or dropping a few atoms introduced considerable changes. Sometimes it is not initially clear what is the appropriate size for the model of the active site. In such cases we may do calculations on models of gradually increasing size and compare results. The appropriate size is that for which a certain ‘saturation’ of the calculated properties can be observed e.g. the charge distribution or the energy of a proton transfer does not change if switching from the smaller model to the larger one. Protein core A thorough understanding of enzymatic mechanisms cannot be achieved solely on the basis of active-site models.There are at least two important phenomena electrostatic catalysis and protein fluctuation that can be described only with larger models that also incorporate those amino-acid residues of the enzyme that do not participate 53 Quantum mechanical treatment of enzyme reactions directly in the catalytic process. Owing to the great number of atoms to be considered approximations should be introduced which however may provide fair results. It is known that proteins produce strong electrostatic fields around themselves that essentially influence their properties and activity (see e.g. ref. 11 for a recent review). It is possible to consider this e§ect in a quantum mechanical calculation on the activesite model if we incorporate the external potential in the Hamiltonian that emerges from monopoles representing each atom of the protein core surrounding it.These monopoles should be placed at atom sites obtained from an experimental determination of the macromolecular structure. Recent versions of sophisticated quantum mechanical software such as the GAUSSIAN package o§er this possibility. The user may consider up to a few hundred Coulomb terms in the input thus a fair treatment of electrostatic e§ects is possible.12 Increasing the number of monopoles considered leads to increased computer time and storage requirements thus it is advised that atoms within a sphere of 1.0 to 1.5nm radius around the active site be considered others can be dropped from the model. Net charges are obtained from various libraries consisting of transferable monopoles assigned to each amino-acid residue.The simplest way to determine these charges is by Mulliken population analysis following a molecular orbital calculation on the residue model. It is however more appropriate to use the so-called potentialderived monopoles that correctly reproduce the electrostatic potential beyond the van der Waals volume of the residue.13 To reduce the number of terms to be considered in the model Hamiltonian dipoles and higher multipoles can be used for the reproduction of the electrostatic field emerging from the protein. The dielectric response can be mimicked by using polarisable dipoles14 or within the frame of reaction field theories which will be treated in more detail in Section 3. At present protein fluctuations can be treated exclusively on the basis of molecular dynamics (see Section 3).Accordingly if an enzymatic mechanism is influenced by such fluctuations the combined quantum mechanical molecular mechanical method should be applied. This method allows one to incorporate all enzyme atoms even a great number of explicit water molecules in the model which may be quite important if e.g. hinge-like conformational changes are crucial in the catalytic mechanism.15 Water as solvent The main component of the biological phase where enzymes work is water a liquid with a large dielectric constant which thus exerts a quite strong influence on molecular transformations with the participation of polar species. Since most enzyme mechanisms involve such species the e§ect of water as solvent is crucial. It is not enough to consider the active site and protein environment in a precise model a quantitative description of the enzyme reaction is possible only if surrounding water is also treated adequately.Water can participate in enzyme processes through three ways. First one or more water molecules may act as proton donor or acceptor directly in the catalysed reaction; these have to be included explicitly in the active-site model. Another type is structural water that binds relatively sy to the protein surface and is therefore immobilised to a great extent. These structural water molecules may be treated in similar fashion to the protein core by incorporating the respective Coulomb term in the Hamiltonian of the active-site model their location is obtained from protein crystallography. In the 54 G. Na� ray-Szabo� and D.K. Menyha� rd case of a molecular dynamics study structural water molecules should be explicitly included in the model. The third most complicated type of biological water is bulk either modelled explicitly by a set of point charges or a single point dipole and a van der Waals volume excluding close contacts or implicitly by a continuous dielectric medium. An implicit way of considering the e§ect of bulk water is to use various e§ective dielectric constants (a number between 1.4 and 4 for ionised side-chains between 10 and 40 or a distance-dependent parameter) to scale down the Coulomb term in the Hamiltonian.16 Bulk water plays an important role in enzyme energetics. Owing to its strong dielectric behaviour it compensates electrostatic forces acting among interacting partners.Thus electrostatic energy e§ects that seem to be extremely large if gas-phase models are used for the calculation become even by an order of magnitude smaller in an aqueous medium. Early models of enzyme reactions neglected water therefore energy changes were severely overestimated.17 The aqueous solvent of an enzyme contains dissolved ions (mostly Na` K` Ca2` H 3 O` OH~ or Cl~) that balance its net charge making the whole system neutral their concentration is characterised by the ionic strength. Though located near ionised surface side-chains these ions are not fixed rather they form a loose cloud of charges the e§ect of which can be correctly treated only by statistical methods such as the Poisson–Boltzmann approach discussed later in Section 3. In addition to dissolved ions the substrate concentration may also influence the enzyme mechanism as in the case of superoxide dismutase where the catalytic reaction is di§usion-controlled.Brownian dynamics simulation techniques developed to calculate the rate at which reactant molecules di§using in solution would collide with the appropriate orientation for a reaction are suitable for a quantitative treatment.18 3 Methods Depending on the selected model quantum mechanical methods of various sophistication can be applied for a calculation. If we restrict the study to a model of the active site which may sometimes bring useful results even if the environment is neglected the relatively small number of atoms allows one to apply sophisticated ab initio molecular orbital19,20 or density functional21 methods.In the case of very large active-site models (with up to 100 non-hydrogen atoms) semiempirical19,22 methods can be used often yielding useful information on activation barriers and reaction heats. In the following we will discuss those methods that go beyond the bare active-site models. If environmental e§ects are crucial but the molecular events remain restricted to the active site variants of the reaction-field theory can be applied. On the other hand if protein fluctuations should be considered to provide an adequate description of the enzyme reaction molecular dynamics methods have to be used. Parameterisation of the force field used in these methods is often based on quantum mechanical calculations on simple models but for larger distortions especially for bond fission and formation no simple yet adequate mathematical expression of the corresponding energy term is available.Combined quantum mechanical molecular mechanical methods developed in the last decade o§er a fair solution to the problem. Finally there are some enzyme reactions (e.g. side-chain protonation) that can be precisely 55 Quantum mechanical treatment of enzyme reactions treated by classical electrostatics which we survey at the end of this section. Reaction-field theories A method based on the active-site–environment partition is the self-consistent reaction- field theory.23–25 For an enzyme model of Fig. 1 the active site or central region C may be represented by an e§ective Hamiltonian H C H C 0H*/5 (1) where the first term stands for the unperturbed active site and the interaction operator has the following form H*/5 kCSF ETkCSgM CT (2) where kC is the dipole moment operator SF ET and SgM CT denote the average fields of the permanent and induced-point dipoles representing the environment (EPNW cf.Fig. 1). The term M C is the expectation value of the dipole moment and g is a response function. The first term in eqn. (1) represents the coupling of the active site with the electrostatic field of the dipoles in the environment while the second term stands for second-order e§ects active-site dipoles induce a field in the environment which in turn interacts with the former. Both permanent and induced electrostatic fields emerging from the environment can be approximated in terms of the wave function of the active site C SF ETg 0S(C D kCD(CT (3) SgM CTg*S(C D kCD(CT (4) If we combine eqns.(3) and (4) we obtain an e§ective non-linear equation for the wave function (C H C 0kC (g 0 g*)S(C D kC D(CT(E C (C (5) g 0 and g* are the orientation and induced contributions to the response tensor g which is usually parametrised to reproduce selected empirical data. The self-consistent reaction-field model properly describes polarisation of the active site by the environment which may be quite important in enzyme reactions. It does not account for charge transfer between the active site and the environment therefore this has to be defined carefully because charge-transfer e§ects though small may play a role for hydrogen-bonded systems.26 The generalised form of the theory has been formulated at the level of molecular dynamics allowing one to study e.g.the role of protein fluctuations in enzyme reactions.27 The reaction field approach can also be formulated at the level of the density functional theory.28 Here the electronic energy of a molecule can be written as E;Sti D h(1) D tiT1 2 Po505(r)G(r,r@)o505(r@) drdr@E9#o-(r) (6) where h(1) is the one-electron Hamiltonian ti is the i-th one-electron orbital o505(r) is the total charge density including both electronic and nuclear contributions. The last term in eqn. (6) stands for the exchange and correlation energies. The Coulomb interaction operator G(r ,r@) is equal to 1/ D rr@ D in the gas phase. Applying the variation principle to eqn. (6) we obtain the Kohn–Sham equations for the oneelectron orbitals 56 G. Na� ray-Szabo� and D. K. Menyha� rd Gh(1)PG(r,r@)o505(r@) dr@l9#o-(r)Hti(r)eiti(r) (7) with l9#o-(r)dE9#o-(r)/do-(r) (8) and o-(r);Dti(r) D 2 (9) Solvation and protein core e§ects may be considered in the e§ective electron –electron interaction term G(r,r@) +e(r)+G(r,r@)4nd(r,r@) (10) where the dielectric constant e(r) is a function of the spatial co-ordinates the d(r,r@) Dirac delta represents a unit charge located at r@.For the active site that we treat by quantum mechanics e(r)1 thus the G(r,r@) Green’s function can be partitioned into a pure Coulomb part and a reaction contribution G*(r,r@) representing the environment (protein and surrounding solvent). Thus we obtain the following energy expression E;Sti D h(1)D tiT1 2 Po505(r)o505(r@)/ D rr@ D drdr@E9#o-(r) (11) 12 Po505(r)G*(r,r@)o505(r@) drdr@ The dependence on the environment enters only in the last term and a reaction-field problem has to be solved by iteration until self-consistency is reached.A simplified but numerically more feasible form of reaction-field theories is the protein dipoles Langevin dipoles method of Warshel.14,29 Here the active site is represented by a set of point charges that can be obtained by quantum mechanical calculations on initial transition and final states of the reaction. Protein atoms in P and N are represented by their net charges and induced-dipoles ash their atomic polarisabilities while water molecules inWare modelled by Langevin dipoles. The total electrostatic energy of C is approximated as a sum of its self-energy and mixed terms standing for the interaction with P and W.A commercially available software package,MOLARIS which is based on the protein dipoles Langevin dipoles method and extended by semi-microscopic models and a free-energy perturbation method has been developed and has been used with success for a variety of enzyme reactions.30 A special variant of reaction-field theories is the fragment self-consistent field method,31,32 where the full enzyme wave function is expanded in terms of strictly localised molecular orbitals corresponding to classical chemical bonds one-centre lone pairs two-centre p-bonds and many-centre n-bonds. This expansion allows a natural partition of the system with no need for saturation of dangling bonds thus avoiding the problem of spurious e§ects mentioned when discussing active-site models in Section 2.The Hamiltonian of C includes interaction terms with P and N represented by polarisable non-mixing bond orbitals and fixed atomic monopoles respectively. No charge transfer is allowed between regions but its e§ect can be reduced by the appropriate extension of C. Another variant has been proposed by Garmer in a recent 57 Quantum mechanical treatment of enzyme reactions study,33 where he implemented the e§ective interaction model in which explicit electronic structure calculations are allowed at a sophisticated ab initio level for active-site species. The rest of the protein and surrounding water is described using semiclassical terms involving electrostatics steric repulsion polarisation and dispersion. Computer simulation techniques To reduce the formidable computational work associated with a (hypothetical) full quantum mechanical treatment of a realistic enzyme model computer simulations utilising simple representations of the potential energy surface can be done.The validity of such simulations is limited by the accuracy of the mathematical model used to describe the potential surface therefore simulation studies should be always compared with experimental results to assess their validity. To date simulations of systems with 10000 to 50000 atoms are feasible using sophisticated computer hardware and software. They are quite useful to analyse and interpret experimental results and may often provide the only way to determine molecular properties that cannot be measured directly in enzymatic systems. The simplest representation of a potential surface is o§ered by molecular mechanics.6,7 The minimal mathematical model includes terms describing bond stretch bond angle deformation hindered rotation around bonds and non-bonded interactions between atoms separated by three or more bonds V(r 1 ,r 2 . . .,rN);"0/$4 12 K"(RR 0 )2;!/'-4 12 K!(hh0 )2;$*)$3!-4K$1cos(n/c) (12) ;ijM4eij(pij/rij)12(pij/rij)6qiqj/erijN Here K" K! and K$ are force constants R 0 h0 and c are equilibrium structural parameters for bonds bond angles and torsional angles respectively. Bonded terms can be parametrised quite well most force fields reproduce experimental bond lengths bond angles and torsional angles to a high accuracy. Non-bonding especially Coulomb parameters in the last term of eqn. (12) are more problematic (see ref.11 for a review). Net atomic charges can be obtained from molecular orbital calculations on small models and are amended to give values that reproduce the electrostatic potential beyond the van der Waals volume as precisely as possible.34 Eqn. (12) may be completed by various cross terms and further ones modelling e.g. transition metal compounds and metallo-proteins.35 These parameters together with the non-bonded ones form a force field that di§erentiates between various atom and bond types and therefore their number may be very large. Several force fields have been developed; they are denoted by acronyms such as AMBER,36 CHARMM,37 OPLS38 and CVFF,39 some of them are available in commercial software products. Application of force fields to enzyme reactions is most problematic if processes involving bond formation or breaking are under study.Though there have been attempts to face this challenge,14,40,41 neither of them are perfect. It is therefore simpler to combine quantum mechanical calculations (for the active site) with a force field (for the environment) in high-level simulations of enzyme reactions. This is why in the last decade a variety of propositions have been published to combine the quantum mechanical and molecular mechanical approaches. The basis of this hybrid quantum/ classical scheme is the onion-like model of biological macromolecules depicted in 58 G. Na� ray-Szabo� and D. K. Menyha� rd Fig. 1 which was first applied by Warshel and Levitt.42 Subsequently Singh and Kollman incorporated the field of protein atomic charges in the Hamiltonian,43 then Karplus and co-workers combined the semiempirical AM1 parametrised molecular orbital method with molecular mechanics.44 In the 1990s several variants have been developed independently on the basis of semiempirical molecular orbital,45–52 valence bond53 or density functional theory.54,55 The basic assumption in hybrid quantum/classical models is that the e§ective Hamiltonian of an enzyme system partitioned as in Fig.1 is written as the sum of four terms:56 H H QM H MM H QM@MM H"06/$!3 (13) The first quantum mechanical term in eqn. (13) has a conventional form with kinetic and Coulomb interaction terms H QM 12 ;+2 i ;ij 1/r*j ;iaZa/ria ;abZaZb/Rab (14) where i and j and a and b stand for electronic and nuclear co-ordinates respectively r is the electron–electron or electron–nuclear distance R is the nuclear–nuclear distance and Z denotes nuclear charges.The molecular mechanics Hamiltonian does not depend on co-ordinates of the quantum region thus it is not an operator but a mere number denoting the energy determined by a given force field. The QM/MM (quantum/ classical) interaction Hamiltonian is written as follows H QM@MM ;iM qM/riM ;aMZaqM/RaM ;aM (AaM/RaM 12BaM/RaM 6) (15) where i and a correspond to the electrons and nuclei in the quantum region while M stands for atoms in the molecular mechanics region. It is only the first term that has to be considered in the solution of the quantum problem since it depends on electronic co-ordinates. The second and third expressions are additive terms to the energy and may be further approximated according to the method applied for the quantum region.The last boundary term in eqn. (13) is introduced to mimic the rest of the enzyme system that cannot be treated explicitly. As mentioned in Section 2 the quantum region is cut out of its covalently bound environment therefore dangling bonds must be saturated if performing a quantum mechanical calculation on it. This saturation may be done by using dummy centres31 or hydrogen atoms. Once we have the molecular mechanical or hybrid force field we may perform a molecular dynamics simulation by integrating Newton’s equation of motion eqn. (16) over time for the system under study. Fi(t)miai(t)miL2ri(t)/Lt2 (16) where Fi(t) ai(t) and ri(t) are the force on atom i its acceleration and position at time t respectively.The force is computed as the negative gradient of the potential energy function in eqn. (12) Fi LV(r 1 ,r 2 . . .,rN)/Lri (17) There are established numerical methods available for the integration of eqn. (16) and once we have the atomic velocities we can calculate the temperature of the system 59 Quantum mechanical treatment of enzyme reactions at any time. Furthermore it is possible to generate a trajectory for the simulated system the history of motion over a period of time and in principle all its equilibrium and dynamic properties can be calculated from this trajectory. While molecular dynamics is generally useful for studying processes that occur on the picosecond time scale (molecular vibrations conformational transitions hydrogen exchaactions ligand binding or fluorescence depolarisation) many interesting events such as folding hinge bending of protein regions or di§usion appear only over much longer time scales.The di§usion encounter of a substrate and the enzyme in solution is a very interesting phenomenon for which the technique of Brownian dynamics may be used to generate trajectories in the nanosecond or microsecond region.57 Here the solvent region (W in Fig. 1 excluding the substrate itself) is represented by continuum models and potentials of mean force by making use of the Langevin equation L2ri(t)/Lt2Fi(t)Ri(t)/mi cidri(t)/dt (18) where c is a frictional coe¶cient and R is a random force with zero mean and has no correlation with the systematic force. A vast simplification of computations is possible via the reduction of molecular details in the Brownian dynamics simulation.Accordingly a variety of biomolecular problems have been addressed with this method including di§usional encounter of enzyme–ligand complexes58 and large-scale motions of protein regions.59 Protein electrostatics The majority of enzyme processes involve one or more protonation steps where no quantum mechanical treatment is necessary to estimate relative pK values of side chains near the active site. Shift of protonation equilibrium states inside proteins can be precisely predicted by classical electrostatics however a sophisticated model that includes all protein atoms surrounding water and counter-ions should be used. Presently the best method for the calculation of protein electrostatic potentials is based on the numerical solution of the Poisson–Boltzmann equation as proposed by Honig and co-workers.60–62 Below we will treat this approach in detail.The method is based on the Poisson equation relating the spatial variation of the protein electrostatic potential V to the charge density o and the dielectric constant e +e(r)+V(r)4no(r)0 (19) For uniform polarisability represented by a single dielectric constant and with point charges eqn. (19) reduces to Coulomb’s law. If the polarisability is not uniform e.g. each atom is assumed to have a di§erent polarisability e(r) varies in space. It is possible to account for ionic strength e§ects via representing the mobile ions in Wof Fig. 1 by a mean-field approximation. According to Boltzmann the concentration of an ion i at a point r with a charge of qi is as follows Ci(r)C"6-,,iexpqiV(r)/kT (20) where C"6-,,i is the bulk ion concentration.The net charge density for mobile ions is given by o.(r);aiCi(r) (21) 60 G. Na� ray-Szabo� and D. K. Menyha� rd Replacing the charge density in eqn. (19) with the sum of densities due to mobile ions and the protein o(r)o.(r)o1(r) we obtain the Poisson–Boltzmann equation +e(r)+V(r)4nMj(r);C"6-,,iexpqiV(r)/kTo1(r)N0 (22) where j(r)1 for regions accessible to ions and is 0 otherwise. Eqn. (22) can be applied to systems with arbitrary geometry and non-uniform dielectrics. If the protein is not highly charged the exponential term in eqn. (22) can be linearised since the exponent is much smaller than unity and we obtain the following form +e(r)+V(r)4n2IV(r)/kTo1(r)0 (23) where I1 2 qi 2C"6-,,i the ionic strength.After solving the Poisson–Boltzmann equation the electrostatic energy expression is obtained from the following integral expression E4 ;P 1 0 Vi(j)q1i Ei(j)k1i dj (24) where summation runs over all charges and dipoles and Vi and Ei are the electrostatic potential and its gradient the field at the centre of each charge and dipole respectively. If all response functions are linear we obtain the simplest form E4 1 2;(Viq1i Eik1i) (25) Numerical solution of the Poisson–Boltzmann equations is quite complicated since they are non-linear three-dimensional partial di§erential equations and thus finite di§erence methods should be used. The commercial software DelPhi developed for large-scale applications,63 makes use of this procedure.4 Special features of enzyme reactions Though studies on enzyme active sites provide essential information on mechanisms it is not possible to completely understand the latter without consideration of a full model consisting of all essential components of the biomolecular system. It is important to know which e§ects originate from the environment and which can be explained using a bare active-site model alone. In the following we discuss three aspects that play a more-or-less important role in enzyme catalysis and may help us in understanding its very essence. Electrostatic catalysis Current opinion on enzyme catalysis is that the origin of rate acceleration is the energetic stabilisation of the transition state.64 Though entropy e§ects may also play a role it is very di¶cult to experimentally determine their actual contribution to an enzyme reaction and theoretical calculations are also problematic.65 A qualitative examination of the entropy contribution to the catalytic rate acceleration of serine proteases has shown that the magnitude of such e§ects is not very large.14,66 There is another type of entropy e§ect proximity that is associated with bringing reactive groups within bonding distance.67 These contributions seem also to be smaller than 61 Quantum mechanical treatment of enzyme reactions Fig.3 Complementarity of the electrostatic potential patterns of the active site of trypsin (left) and its environment (right). Positive and negative regions are shown by light and dark grey shading respectively. previously thought since enzymes are rather flexible.General acid–base catalysis involving proton transfers mediated by acidic or basic residues of the protein may also play a role in rate acceleration,67 however its importance is smaller than often supposed.14,66 There is now ample evidence that for a large number of enzymes protein electrostatics plays a crucial role in catalytic rate acceleration.14,66 In these mechanisms the polarity of the transition state of the reacting species is larger than that of the ground state and the pre-oriented enzyme dipoles stabilise the former which leads to rate acceleration.68–70 This e§ect cannot be described using active-site models alone; as we discussed in Sections 2 and 3 electrostatic e§ects are provided by the protein and aqueous environment that have to be properly accounted for.Elegant support for this hypothesis has been brought by point mutation experiments on serine proteases.71,72 Replacement of the buried charged aspartate component of the catalytic triad in trypsin and subtilisin by neutral asparagine and alanine respectively led to a rate reduction by about four orders of magnitude that can be attributed to a pure electrostatic e§ect.73 A simple visualisation of the electrostatic stabilisation of the active-site transition state is displayed in Fig. 3 where electrostatic potential patterns of the transition state modelled by the active site and its environment are complementary. Another electrostatic e§ect that may be important for enzyme action is the control of the redox potential of the active site or some of its components (e.g.a haem).74,75 In ref. 66 we discussed a number of enzymes where the major source of rate acceleration is 62 G. Na� ray-Szabo� and D. K. Menyha� rd shown to be electrostatics and further support is given in Section 5 of the present review. However it cannot be said that electrostatic e§ects are always the exclusive factor for enzyme catalysis. Richards and co-workers argue that chorismate mutase catalysis may be rationalised in terms of a combination of substrate strain and transition-state stabilisation.76 Another interesting finding is that in the predicted mechanism for orotidine monophosphate decarboxylase a requirement for catalytic activity is the provision of an environment of low dielectric (by the protein) which cannot be observed in aqueous solution.77 We will discuss these works in Section 5 in more detail.Di§usion control In souch as acetylcholinesterase the catalytic rate is controlled by di§usion.78 This is again a phenomenon that cannot be understood by studying the active site alone; however Brownian dynamics simulations can be done to obtain more information on the fine details of the mechanism as done by Tan et al.79 for the di§usion of the substrate N-methylacridinium towards the enzyme. Their results indicate that electrostatic steering of the ligands makes a substantial contribution to the e¶ciency of the enzyme and the ionic strength of the medium is an important factor in determining the catalytic rate. It has to be mentioned however that the e§ects that influence the di§usion rate are very small relative to the huge electrostatic e§ect of the protein environment.After the protein changes the rate by more than ten orders of magnitude and the reaction becomes di§usion controlled the small e§ect of surface charges is only a modulation of the finer details of the process. This is supported by the work of Wlodek et al.80 who stated that a negatively charged glutamate side-chain located in the close vicinity of the enzyme active site contributes significantly to the catalytic e¶ciency of the enzyme in a dual manner. First it lowers the electrostatic potential and generates a favourable gradient at the bottom of the reaction gorge thus enhancing di§usion penetration of the positively charged substrate to the reactive site. Second it stabilises the transition state for the first step of catalysis via electrostatic interaction with the imidazole ring of the catalytic histidine protonated upon generation of the tetrahedral intermediate.Their simulations yielded a reasonable qualitative agreement between calculated and experimental values of the rate reduction upon mutation indicating the adequacy of the above conclusions. Another case where Brownian dynamics simulations are necessary is superoxide dismutase where the enzymatic rate is modulated by di§usion. Experimental studies indicated that the ionic strength has a significant e§ect on the rate as in the case of acetylcholinesterase,81 and the approach of the substrate to the active site is facilitated by the enzyme electrostatic field. Brownian dynamics calculations82 provided a rate and its variation with salt concentration that is in reasonable agreement with experimental results.Dynamic aspects The protein environment may also have an e§ect on enzymatic mechanisms via its fluctuations. An example is the serine protease reaction where experiments show that for amide substrates the rate-limiting step is acylation while for ester substrates it is deacylation. Based on their hybrid quantum/classical calculations Kollman and coworkers proposed that enzyme fluctuation e§ects may be responsible for this phenom- 63 Quantum mechanical treatment of enzyme reactions enon.83Afluctuation bringing the oxygen of the active serine side-chain within contact distance of the carbonyl carbon of the amide or ester substrate could decrease the activation energy to 18 and 9 kcal mol~1 respectively. Applying the same argument to the deacylation step the activation barrier becomes 12 kcal mol~1 just between the above values which explains the reversal of rate-limiting steps for the two substrates.Further cases where dynamic aspects may become crucial for the enzyme mechanism are those where large-scale motions of the protein are necessary to provide the active site where the actual reaction takes place. An example is phosphoglycerate kinase15 where two adjacent lobes of the enzyme approach during catalysis and provide the active-site cleft in the final stage of the reaction. Whether this hinge-bend type motion has an e§ect on the resulting rate or not can be decided only on the basis of molecular dynamics calculations on the full enzyme model. 5 Overview of recent work There are numerous papers available on the quantum mechanical treatment of enzyme mechanisms.Since earlier results have been summarised in detailed reviews,16,17,66 in the following we discuss only recent calculations published in the 1990s. The main target of most quantum mechanical studies on enzymes is the active site therefore groups of related enzymes can be formed according to the key component of their central machinery which may be a hydroxy group of a serine or threonine side-chain an aspartate or a histidine residue acting as a general acid or general base a structural water molecule located near the active site nicotinamide dinucleotide as a cofactor or a metal centre. In the following we discuss enzyme groups according to the above classification. Enzymes with an activated side-chain hydroxy group A number of otherwise unrelated enzymes make use of the same functional group for the initial attack at the substrate a hydroxy belonging to a serine or threonine side-chain.The most frequently and thoroughly studied enzymes are the serine proteases which consist of a catalytic triad of Asp His and Ser; the latter two residues form the core of the active site. For a recent review of the computational studies on the mechanism see ref. 66. The active serine (made su¶ciently nucleophilic by the other residues of the active site) attacks the carbonyl carbon of an amide or ester substrate then a proton transfer takes place from it to the catalytic imidazole side-chain and simultaneously a high-energy tetrahedral intermediate appears which breaks down to form an acyl enzyme (Scheme 1).84 This in turn hydrolyses via the reverse route to regenerate the active form.All serine proteases contain an oxyanion hole made up of two backbone amide hydrogen atoms. Several computational and experimental studies indicate that the major driving force of catalysis by serine proteases is electrostatics e.g. this assumption has led to the successful calculation of point mutation e§ects on the catalytic rate.73 A thorough computational study of the details of the mechanism was carried out by Kollman and co-workers.83,85 They performed molecular mechanics and molecular dynamics as well as semiempirical molecular orbital calculations and used acetate methylimidazole and methanol to represent Asp His and Ser of the active site respectively. 64 G. Na� ray-Szabo� and D. K. Menyha� rd Scheme 1 Mechanism of the hydrolytic reaction catalysed by serine proteases.The oxyanion hole was modelled by two water molecules amide and ester reactants by CH 3 NHCOCH 3 and CH 3 OCOCH 3 respectively. Studies based on the bare active site indicated that formation of the first tetrahedral intermediate is the rate-limiting step for both amide and ester substrates. However when protein fluctuations were also considered deacylation became rate-limiting as already mentioned above. The reaction was described by a slight approach of Ser towards the substrate followed by a concerted attack and proton transfer. In agreement with earlier studies the function of the buried aspartate and the oxyanion hole was shown to be the electrostatic stabilisation of the tetrahedral intermediate.65 Quantum mechanical treatment of enzyme reactions Scheme 2 Formation of the tetrahedral intermediate in acetylcholinesterase as proposed by Wlodek et al.80 Acetylcholinesterase also catalyses the hydrolysis of an ester bond that of acetylcholine to yield choline and acetate. Its catalytic triad which contains a reactive Ser is Asp · · · His · · · Ser,86 while in the Torpedo californica species87 it is Glu · · · His · · · Ser located at the bottom of a narrow 2 nm deep gorge. As already mentioned the reaction between acetylcholinesterase and its substrate is di§usion controlled. Brownian dynamics and quantum chemical calculations explain the experimentally observed 100-fold decrease of k#!5/K M upon Glu199Gln mutation.80 It has been suggested in agreement with other authors,88 that the role of Glu-199 is to provide the driving force leading the substrate to the active site.This proposition is based on ab initio molecular orbital calculations with the sd 3-21G basis set on simple models of the following amino acids Ser-200 His-440 Glu-237 (catalytic residues) amide groups of Gly-118 Gly-119 Ala-201 (oxyanion hole) Glu-199 and the substrate CH 3 COOCH 3 . The model geometry was optimised keeping Ca and peptide N atoms constrained to their crystallographic positions while the mutant was obtained by replacing Glu-199 by Gln-199. Calculations reproduced the dramatic activity decrease upon mutation and revealed the dual action of Glu-199. One e§ect is the enhancement of the di§usion of the substrate the other is the stabilising role of Glu-199 on the transition state formed during the first catalytic step through electrostatic interaction with the protonated catalytic His-440 residue (Scheme 2).The same calculations support a charge relay mechanism for the reaction catalysed by the Glu199Gln mutant. Aspartyl glucose aminidase is a lysosomal enzyme that catalyses the cleavage of N-linked oligosaccharide chains from Asn.89 The enzyme belongs to the family of the N-terminal nucleophile aminohydrolases90 for which a serine-protease-like catalytic mechanism was supposed.91,92 The reactive amino acid in the case of this enzyme is not a Ser rather it is the terminal Thr-183 which in itself provides the nucleophile that attacks the carbonyl carbon of the substrate and the base abstracting a proton from the nucleophile. Pera� kyla� and Kollman93 used quantum mechanical and molecular dynamics calculations to gain insight into the reaction mechanism making use of earlier results of Pera� kyla� and Rouvinen.94 Molecular orbital calculations (optimisation at the HF/6-31G* level energy computation at the MP2/6-31G*/HF/6-31G* level) on a model of the active site (Nd-methylasparagine as substrate 2-aminoethanol in place of Thr-183 and the oxyanion hole represented by 3-hydroxypropionic amide) 66 G.Na� ray-Szabo� and D. K. Menyha� rd were followed by dynamic simulations keeping the geometry of the active-site region constrained. Finally molecular orbital calculations including protein electrostatic and solvation e§ects were carried out. Both the acylation and deacylation processes were simulated. During acylation the attack of the nucleophilic oxygen on the carbonyl of the substrate was found to be the rate-limiting step while in deacylation it is the formation of the reaction product from the tetrahedral intermediate.Inclusion of the protein environment considerably enhanced both reactions by reducing their activation energies. Especially Asp-47 Lys-207 and Arg-211 were found to contribute favourably to the stabilisation of the transition-state complexes formed during the rate-limiting steps. It was found that in the deacylation reaction water molecules enter the active site and stabilise the ionic species through numerous H bonds. Enzymes with catalytic aspartate and histidine residues Aspartate or glutamate side-chains may act as acidic groups participating in reactions catalysed by various enzymes.An important family of such enzymes is the aspartic proteases. Their active site is comprised of two Asp-Thr-Gly repeats which face each other forming a dyad. Based on pH/hydrolysis profiles two mechanistically implicated deprotonation events can be associated with the Asp dyad95 and it was shown that the active site carries a formal negative charge.96 The exact mechanism of action is still under debate since the position of a water molecule tightly bound between the two catalytic aspartate side-chains has not been unequivocally determined. A possible mechanism based on crystallographic results on the enzyme complexed with transition- state analogues is given in Scheme 3.97 Earlier theoretical studies using both ab initio SCF calculations98 and the MNDO/H method99,100 indicated the existence of a prototropic equilibrium between the two aspartate side-chains.These works concentrated on examining the protonation scheme of Scheme 3 while another ab initio study101 explored all possibilities. Quantum mechanical and electrostatic calculations were carried out using several di§erent basis set and di§erent models of the system. The models were as follows (i) a formic acid/formate anion moiety and a water molecule; (ii) model (i) extended by two methanol and two formamide molecules at the position of Ser-35 Thr-218 Gly-34 and Gly-217; (iii) model (ii) replacing the listed amino acids by partial atomic charges. The geometry of the models was optimised for all four protonation arrangements of the active site of endothiapepsin and pepsin. The most important conclusion of the calculations was that in the native enzyme water prefers to form a bifurcated H-bond with the inner oxygen atoms of the active site.In both considered enzymes Asp-32 is predicted to be ionised. The theoretically derived water positions are in much better agreement with experimentally determined values than in the case of the linear structures supposed earlier. The above results also have implications for substrate binding. Previous calculations98 proposed that during the reaction displacement of the water molecule by the substrate must take place however the results of Beveridge and Heywood101 imply that the water remains trapped between the bound substrate and the active carboxyl groups. This point is in agreement with crystallographic studies;102,103 however for complete consistency a reversal between the ionisation of active-site aspartates must 67 Quantum mechanical treatment of enzyme reactions Scheme 3 Catalytic mechanism of action of aspartyl proteases as proposed by Veerapandian et al.102 The initial structure is depicted after Beveridge and Heywood.101 occur during substrate binding since in the proposed mechanisms Asp-215 should be deprotonated. Oldziej and Ciarkowski have also reached similar conclusions.104 Applying both semiempirical molecular orbital (AM1 and PM3) and density functional methods to supramolecular models with 160 to 190 atoms of the free enzyme and the Michaelis complex they concluded that the dissociation within the Asp dyad is ligand sensitive. It was found that though in the free enzyme and in the intermediates Asp-32 prefers to be ionised while Asp-215 is protonated at its outer oxygen in the Michaelis complex and in the product state the situation is reversed Asp-215 becomes ionised just as foreseen by Beveridge and Heywood.101 Electronic e§ects were found to be of utmost importance in determining the energy change along the catalytic path.Goldblum et al.105 have carried out calculations on a model based on the crystal structure of the enzyme with bound difluoroketone inhibitors102,103 that are assumed to be transition-state analogues of the tetrahedral intermediate formed as the initial product of the water attack on the substrate peptide bond. A full conformational search of proton positions in the active site was carried out by the MNDO/H method,99 using the following residues to model the enzyme environment Asp-32 Thr-33 Gly-34 Ser-35 Asp-215 Thr-216 Gly-217 and Thr-218.The inhibitor was also introduced into the full native structure of HIV-1 protease and its energy was minimised by molecular mechanics. Results of both quantum chemical and force field calculations indicate that Asp-32 is ionised while Asp-215 is protonated at its external oxygen atom. The interaction between the water molecule and the 68 G. Na� ray-Szabo� and D. K. Menyha� rd Scheme 4 Putative mechanisms for the isomerisation of dihydroxyacetone phosphate into D-glyceraldehyde-3-phosphate by triosephosphate isomerase. (Upper) After Bash et. al.,110 (lower) after Pera� kyla� and Pakkanen.111 69 Quantum mechanical treatment of enzyme reactions substrate was also modelled and it was found that the attack of water becomes energetically favourable if its oxygen and the carbonyl carbon of the peptide are at a distance of about 150 pm.Several possible routes of the two proton transfers were also tested but all resulted in very low activation energies. Based on this finding the authors proposed that bringing the substrate to close proximity of the water molecule should constitute the greater nergy of the first tetrahedral-intermediate formation. Another enzyme where one of the basic catalytic groups is carboxylate is triosephosphate isomerase which catalyses the conversion of dihydroxyacetone phosphate into D-glyceraldehyde-3-phosphate (see the mechanism in Scheme 4).106 Glu-165 has been identified as the catalytic base,107 but another key residue is thought to be His-95108 which also takes part in the proton transfer.Asn-10 and Lys-12 by stabilising the reaction intermediates also play an important role in catalysis.109 Glu-165 abstracts the proton on C1 of the substrate and an intermediate enediol is produced that undergoes protonation and deprotonation by the adjacent basic and acidic catalytic groups and the product is formed. Bash et al.110 aimed to study this enzyme mechanism with the hybrid quantum mechanical (AM1)/molecular mechanical method. The active site was made up of the substrate and side chains of Glu-165 and His-95 initially singly or doubly protonated. The molecular mechanics region comprised all residues within a 1.6nm radius around the side chain OE2 of Glu-165 extended by surrounding water molecules using the X-ray structure of the enzyme inhibited by phosphoglycolohydroxamate (an intermediate analogue) as initial co-ordinate set.The first step of the reaction (Glu~substrate GluHenediol) was considered in detail. Calculations showed that while the interaction of negatively charged substrate with Glu~ is quite unfavourable the interaction energy is small for GluH and enediolate. The role of the enzyme was shown to be the stabilisation of a Glu~/substrate pair and lowering of the barrier height by stabilising the transition state. A specific contribution of several amino acids of the protein environment was detected especially that of Lys-12 and it was stressed that the surrounding aqueous solvent also plays a role in catalysis. Calculations suggested a novel possibility for the second step of the reaction.It was shown that neutral His can serve as a general acid in the reaction. The initially protonated positive His-95 would reorient from its position in the crystal structure leading to a deep energy well corresponding to the state after proton transfer. However there is no experimental evidence for such a structural rearrangement or formation of such a stable intermediate that would hinder further reaction. On the other hand the proton transfer step from His-95 to enediolate with neutral His provides an energy barrier of the order of the experimental estimate. Pera� kyla� and Pakkanen111 reached contradictory conclusions to those of Bash et al.110 when using the model assembly method together with the Poisson–Boltzmann electrostatic description of the e§ect of the enzyme for the characterisation of the enzyme mechanism.They applied di§erent levels of ab initio theory a 3-21G* basis set was used for geometry optimisation and energy values were calculated at MP2/3-21G and 6-31G level for a smaller and a quite large model of the active site. Models were built using the crystal structure of triosephosphate isomerase with bound glycerol-3- phosphate.112 In the smaller model acetamide was used in place of Asn-10 a methylammonium cation for Lys-12 imidazole/imidazolate for His-95 and butanoate/bu- 70 G. Na� ray-Szabo� and D. K. Menyha� rd Scheme 5 A proposed mechanism for xylose–xylulose conversion by D-xylose isomerase.114,119 tanoic acid for Glu-165. The larger model contained mimics of four additional closelying amino acids.In the first step of the reaction proton abstraction by Glu-165 the glutamate side-chain rotated to give an enediolate (see Scheme 4) and spontaneously released a proton from its oxygen resulting in the enediol intermediate. Of all intermediates this was found to be the energetically most stable. The authors found the deprotonation of the active site His-95 unfavourable in addition they could not locate a stable intermediate with protonated Glu-165. Therefore in contrast to the mechanism in Scheme 4 the reaction was proposed to proceed from the enzyme–substrate complex via an enediol with protonated His-95 and deprotonated Glu-165 to the product state. The discrepancy between the two conclusions may be owing to the di§erent treatment of environmental e§ects and further studies are needed to clarify the fine details.The imidazole side-chain of histidine residues may act both as a general acid or base if donating or receiving a proton respectively in the catalytic process. A case where histidine acts as a general base is the catalytic reaction by xylose isomerase; for a detailed review see ref. 113. The enzyme catalyses the reversible conversion of D-xylose into D-xylulose and is also capable of converting other sugars from aldose into ketose. 71 Quantum mechanical treatment of enzyme reactions It requires Mg2` Co2` or Mn2` for activation while Zn2` Ba2` Cu2` and Ca2` inhibit catalysis. On the basis of protein crystallography a detailed stereochemical mechanism was proposed (Scheme 5)114 that implies binding of the a-D-pyranose form of the substrate followed by ring opening proton shuttle and a hydride shift was supposed to be the rate-limiting step.115 Ab initio and semiempirical molecular orbital calculations on smaller and extended models support the mechanism outlined in Scheme 5 and predict the hydride shift to be rate limiting.116–118 These studies revealed two interesting features of the reaction.One is electrostatic stabilisation of a () charge distribution (similar to that in serine proteases) formed in the transition state of the ring-opening step the other is the role of charge transfer from the hydrogen-bonded protein side-chains and substrate to the metal ions as a key factor influencing the hydride shift. It has also been shown that an important factor for the catalytic rate decrease observed in the D254E/D256E double mutant is the altered electrostatic environment that is less capable of stabilising the transition-state complex.119 Hu et al.120 set out to study why and when the catalytic metal ion moves during the reaction. The proton shuttle and hydride shift reaction steps (called isomerisation in a shorthand notation) were broken down into five elementary processes each studied separately. The active-site region was treated with the semiempirical PM3 method once in vacuum and once in the surrounding protein using the hybrid quantum mechanical molecular mechanical approach. Calculations were carried out in two further groups with the catalytic metal ion at site A (initial site) and at a site called B where the ion moves during catalysis. Both molecular mechanics and molecular dynamics were used to relax the structures obtained for reactants products and the four intermediate states.A characteristic di§erence between the reaction in vacuo and in the protein was found to be that while the corresponding structures were quite similar the vacuum calculations failed to reproduce the movement of the catalytic metal as during the entire reaction pathway structures with the catalytic metal at site A were of lower energy. The proposed details of isomerisation are the following. First the catalytic metal moves to site B facilitating proton transfer from a water molecule coordinated to Mg2` to Asp-257. Then the generated OH~ ion abstracts a proton fromOHof D-xylose and a hydride shift from C2 to C1 of D-xylose takes place. Finally a proton is transferred from NH 3 ` of Lys-183 to O1 of the substrate and the NH 2 of Lys-183 receives a proton supposedly from Asp-257 through Asp-255.Enzymes with activated water molecules In several enzymes a water molecule activated by an adjacent metal or the protein environment alone plays the key role in catalysis. An example is carbonic anhydrase a zinc metalloenzyme which catalyses the reversible hydration of CO 2 according to the following reaction scheme:121 ZnOH~CO 2HZnHCO 3 ~(H 2 O)ZnOH 2 HCO 3 ~HZnOH~H`HCO 3 ~ The reaction is pH dependent at low values dehydration of bicarbonate is favoured while above pH7 hydration of CO 2 is preferred. Zinc bound ligands are alsond t