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Pii: s0166-1280(03)00307-5

Journal of Molecular Structure (Theochem) 632 (2003) 297–307 How important is the refinement of transition state structures ` ngels Gonza´lez-Lafont, Jose´ M. Lluch* Departament de Quı´mica, Universitat Auto`noma de Barcelona, Bellaterra, Barcelona 08193, Spain Received 30 October 2002; revised 18 December 2002; accepted 18 December 2002 In this paper the need to use a second derivatives direct algorithm to refine the location of transition state structures obtained in enzymatic systems has been analyzed. The 25 approximate QM/MM transition state structures previously found by means ofa reaction coordinate approach for the three mechanisms of racemization of mandelate and propargylglycolate by mandelateracemase enzyme have been refined using a modified micro-iterative optimization method developed in this work. Therefinement of transition state structures is especially useful to assure that a structure, found as the highest potential energy pointon a profile depicted by a particular reaction coordinate, lies in the correct quadratic region. This is more important in thosesteps of the enzymatic process where the selected reaction coordinate may not reflect quite accurately the geometrical changestaking place in the active site.
q 2003 Elsevier B.V. All rights reserved.
Keywords: Quantum mechanical/molecular mechanical transition state structures; Reaction coordinate approach; Micro-iterative method;Second derivatives direct optimization method; Mandelate racemase reaction mechanisms the enzyme – substrate complex for a fixed positionof the nuclei. That is, using this technique a small Enzymes speed up reactions by many orders of region at the active site of an enzyme is described magnitude using fundamental physical processes to quantum mechanically, whereas the surrounding increase chemical reactivity. Methods that permit to protein is included by a simpler MM representation.
calculate and to explore the Potential Energy Classical molecular dynamics simulations have to be Surface (PES) of an enzymatic reaction are needed carried out to sample extensively the configurationspace, looking for new regions of the PES around to understand theoretically the enormous catalytic minimum energy structures representing possible power of enzymes. A hybrid quantum mechanical/ reactant and product complexes. However, both energetic and entropic factors make it impossible in can be used to obtain the potential energy of practice the molecular dynamics generation ofreactive trajectories going from the reactant region * Corresponding author. Tel.: þ34-93-581-2138; fax: þ34-93- to the product region in a canonical ensemble at a E-mail address: [email protected] (J.M. Lluch).
given temperature. Then kinetics information can be 0166-1280/03/$ - see front matter q 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0166-1280(03)00307-5 X. Prat-Resina et al. / Journal of Molecular Structure (Theochem) 632 (2003) 297–307 obtained by means of the Transition State Theory, a PES to locate the set of stationary points that statistical approach to real dynamics, which is able connects with the reactant and the product by to provide canonical rate constants kðTÞ .
means of the real reaction pathway should be According to Variational Transition State Theory highly recommended prior to the free energy the canonical rate constant depends on the calculations. So, the set of dividing surfaces should generalized free energy barrier, that is, the maxi- be raised along that real reaction pathway.
mum value of the generalized free energies The point now is how precise the location of the associated with a set of dividing surfaces built up transition state structures has to be to produce a along a suitable reaction pathway taken as a reliable reaction path. Is the location of the transition reference. The generalized free energies can be state structure as the maximum energy point along an obtained, for instance, from molecular dynamics energy profile built up as a function of a conveniently simulations using the umbrella sampling technique chosen reaction coordinate sufficient or after that the with an adequate biasing potential or by means of transition state structures have to be refined? The statistical perturbation theory Once the reactant purpose of this paper is to use the different reaction and the product have been localized, a progress channels we have previously found for the coordinate connecting them and based on suitable racemization of mandelate and propargylglycolate by internal coordinates can be adopted to define the mandelate racemase enzyme to shed some light to that reaction pathway. As an alternative, knowing the valence bond structures of both reactant and This paper is structured as follows. In Section 2 we product, a mapping potential as a function of briefly review the methods used for locating transition the diagonal elements of an empirical valence state structures in enzymatic catalysis, in Section 3 we present the enzymatic system where we have tested define a reaction pathway as a collective reaction the different methods, and in Section 4 we compare coordinate analogous to the solvent coordinate used the results obtained with the different strategies. A in Marcus theory for electron transfer reactions. The EVB provides a very effective way of exploring thePES of a substrate – enzyme complex between anyset of possible intermediates. Thus it is essential todefine the problem in terms of feasible reactants, 2. Methods to locate stationary points in big products and intermediates. In all cases, nuclear quantum effects and corrections accounting for therecrossing of the dividing surface can be introduced Using QM/MM potentials the energy calculation is expensive enough to look for effective methods that reach convergence with as minimum number approaches become very fruitful, their practical of steps as possible. On the other hand, the high implementation is generally based on a reference number of degrees of freedom in enzymatic path constructed using the information extracted systems makes the usage of standard second order from the reactant and product. However, this methods, such as Newton – Raphson, too compu- procedure could lead to inaccurate results. Due to tational demanding due to the construction and the complexity of the enzymatic reactions, the real manipulation of a very big second derivatives reaction pathway can be very different from the matrix (Hessian). Different strategies have already apparent one at first glance. The enzymatic reaction been applied to locate transition state structures in can actually take place through several parallel and such big systems All of them try to find a kinetically competitive channels, each one consist- compromise between effectiveness and low compu- ing of multiple steps, involving several intermedi- tational cost. On the other hand, although the usage ates in going from the reactant to the product, then of suitable internal coordinates is an alternative leading to a priori unexpected reaction paths. As a way to achieve converged structures effectively in consequence, an exploration of the corresponding X. Prat-Resina et al. / Journal of Molecular Structure (Theochem) 632 (2003) 297–307 optimization with Cartesian coordinates will beconsidered.
The easiest procedure to look for transition state structures is to freeze one degree of freedom of oursystem, a coordinate (e.g. a distance, angle or dihedralor a combination of them) that is representative of thereaction we are studying, and to perform restrainedminimizations scanning this coordinate from reactantsvalue to products all along what it is supposed to bethe reaction path. In this case we obtain a discreteenergy profile and the highest point of this profile istaken as the transition state structure of the reaction.
The crucial aspect is to choose an adequate reactioncoordinate and to perform the scan with as manypoints as possible. It is not always so intuitive tochoose such a coordinate, and even when thecoordinate is the right one we will show that insome cases the proposed transition state is an incorrectstructure.
In our test cases we have used an harmonic potential with a constant of 10,000 kcal/mol/A restrain the chosen coordinate during the minimiz-ation that is performed with the LBFGS method Fig. 1. Scheme for the location of transition state structures in In this paper the distance between the acceptor atom enzymatic systems moving only a core while keeping the rest of and the hydrogen that is being transferred in each step environment atoms frozen (top). Micro-iterative scheme (bottom).
has been always used to define the reactioncoordinate.
A next step is the usage of methods that knowing small part of the system (core) is participating in the reactant and product structure make use of the the reaction, then keeping the rest of the degrees of energy and its first derivatives to find the transition state structure and an approximation to the minimum NR or RFO search is performed only for this core, energy path To our knowledge only Conjugate avoiding the environment to relax. The intuitive Peak Refinement method has been applied to next solution is permitting the environment to relax using an inexpensive optimization method (e.g.
Conjugate Gradient LBFGS ABNR while in the core a transition state structure searchis performed with the NR-like method. Both Another strategy consists in the direct location processes are carried out alternating one and the of the transition state structure with methods that other until self-consistency (This is the use the second derivatives of the energy (e.g.
and is the method that we will consider in our comparative study as the approach that gives the tational cost mentioned above the treatment of the most refined structure. This so-called micro-itera- whole set of degrees of freedom at the same level tive method can be achieved in several manners is not feasible. In a first approximation only a depending on the minimizer for the environment, X. Prat-Resina et al. / Journal of Molecular Structure (Theochem) 632 (2003) 297–307 the interaction between core and environment, andhow often the environment must be relaxed duringthe transition state structure search in the core. Allthese features will not be discussed here.
The method that we have implemented makes use of the RFO technique for the location of the transition Fig. 2. Racemization reaction of mandelate.
state structure in the core region The minimiz- norm of the whole system is converged with a ation of the environment is carried out with LBFGS and it is performed every time the RFO convergesinstead of at every step of this transition state structure The reaction coordinate method and the micro- search. The interaction between the core and the iterative procedure have been implemented in the environment during the minimization of the latter is always the full QM/MM energy at every step. That is, we do not calculate ESP charges and fix it all along the the transition state structures of the reaction mechan- minimization process like other authors have done isms of the enzymatic system that is presented in In our case this strategy was easier to implement, permits to obtain a real interaction between both The common procedure has been to take the regions and gives the possibility of including some structures obtained with the reaction coordinate quantum mechanical atoms in the environment zone.
method as the input for the micro-iterative method.
This method can easily find a transition state structure with low computational cost. If the initialstructure is appropriate it is faster than the reaction coordinate method mentioned above. When the coreis not selected adequately and some atoms of the 3.1. The reaction, the different substrates environment zone participate directly or indirectly in the reaction, then some coupling between the twozones might appear that avoids to reach the desired Mandelate racemase catalyzes the reversible iso- convergence. The solution to this problem is the merization of both enantiomers of mandelate selection of a bigger core zone. A bigger core zone In addition, propargylglycolate () has been implies the calculation and manipulation of a bigger found to be a moderately good substrate for Hessian matrix. The drawbacks can be solved with racemization. Both substrates can evolve from an approximated initial Hessian matrix and a partial reactants to products through three parallel mechan- diagonalization instead of a full diagonalization isms . The residues that belong to the active center (note that RFO only needs one eigenvector to calculate the displacement; this partial diagonaliza- tion can be carried out with the standard LAPACK change that occurs at each step of the mechanisms. We will consider the enantiomer (S) as the reactant and the The micro-iterative scheme we have used in this (R) as the product. For propargylglycolate both mechanisms I and II require a previous proton transfer,through the transition state (TS) structure 1, from the 1. Minimization of the selected environment with auxiliary proton donors Lys164 (mechanism I) or LBFGS keeping the core frozen until convergence.
2. Transition state structure search with RFO moving only the atoms of the core. This process is carriedout until convergence as well.
3. Check for the gradient norm of the total system: if it is not minimized, it returns to point 1. Theprocess will not be finished until the gradient Fig. 3. Racemization reaction of propargylglycolate.
X. Prat-Resina et al. / Journal of Molecular Structure (Theochem) 632 (2003) 297–307 transfer neither from Lys164 nor Glu317 beingrequired.
For mandelate mechanism I consists of the same six steps with the same transition state structures as inmechanism I for propargylglycolate. Mechanism IIconsist of only five steps because the change ofconfiguration of the Ca atom of the substrate takesplace in a concerted way with the proton donationfrom His297 at TS5 (TS4 does not exist in this case).
Mechanism III for mandelate involves formally twosteps, but the well between the two correspondingtransition state structures is so shallow that, for the Fig. 4. Mandelate racemase active center. The X symbol stands for purpose of this paper, it can be considered as a one- the phenyl or ethinyl groups when the substrates are mandelate or Glu317 (mechanism II). In both mechanisms the It has to be underlined that the description of the proton attached to the Ca atom (the one adjacent to different steps is extremely oversimplified. In chemi- the carboxylate group) of the substrate is being cal systems so complicated like enzymes, many atoms abstracted by Lys166 at TS2. His297 is migrating at belonging to a lot of residues move in a significant TS3, approaching to the substrate and moving away way in each step, accompanying the main change that from Glu247. The Ca atom of the substrate, the defines that step. For instance, many atoms have to stereogenic center, is changing its configuration from readapt smoothly their positions to the new charge (S) to (R), through a sp2 hybridisation, at TS4. A proton distribution along a proton tranfer, or, conversely, transfer from His297 to the Ca atom of the substrate is some residues have to migrate prior to a proton occurring at TS5. Finally, the proton transfer that happened in the first step is reversed through TS6. Incontrast, mechanism III for propargylglycolate is an 3.2. The theoretical model used: the potential energy asynchronic concerted mechanism which involves proton abstraction by Lys166, configuration changeof the Ca atom of the substrate and protonation from For the sake of comparison with the results His297 within a unique step, no previous proton already published, the enzymatic model used here Fig. 5. Mandelate racemase mechanisms (see text). Mechanisms I and II need the previous protonation of the carboxyl group of the substratebefore the isomerization. Mechanism III reaches products in one concerted step. Bold type atoms belong to the substrate.
X. Prat-Resina et al. / Journal of Molecular Structure (Theochem) 632 (2003) 297–307 is the same that we used in our previous studies of Potential energies (kcal/mol) for the racemization of mandelate vinylglycolate substrates A more detailed description of the method can be found there.
Here we will only mention a few aspects. Thestructure with the Protein Data Bank code 1MNS has been used to build up our model. The zone chosen to be represented quantum mechanically with a semiempirical Hamiltonian (PM3 in all constituted by the substrate, the magnesium atom, a water coordinated to it, and the lateral chains of the residues that participate actively in the isomeriza- The corresponding values obtained from the micro-iterative tion reaction (). This means 80 or 88 atoms, hydrogen link atoms to cap the seven QM/MM practically removed by performing a scanning with frontiers included, for propargylglycolate or more intermediate points along the reaction coordi- mandelate complexes, respectively. The rest of theenzyme and solvation waters are treated with the nate. On the contrary, in the case of propargylgly- colate important differences exist among some that a total of 3963 atoms will constitute the whole energy barriers obtained with the two methods.
These differences are mainly centered in step 4, The region corresponding to the moving atoms when stereogenic center Ca changes its configur- includes all the residues that fall into a sphere of 15 A ation. Another significant discrepancy can be seen centered at the magnesium atom. This implies 1299 at the TS1 of the mechanism I. A geometry moving atoms for the mandelate case. In what refers analysis of the corresponding transition state to the micro-iterative scheme the core zone has been structures will shed some light to this point.
chosen to include the important atoms at every step.
The full QM/MM interaction between core and environment used in this paper enables us to excludefrom the core region the QM atoms which are not The more relevant distances for the transition state relevant in a particular mechanism step. This fact structures corresponding to the racemization of permits us to work with a smaller Hessian for the core.
mandelate and propargylglycolate according to mech- In any case, the selection of the two regions has to be anisms I, II and III (located using the reaction Table 2Potential energies (kcal/mol) for the racemization of propargylgly- 4. Comparison of the results obtained with colate using the reaction coordinate method The potential energy barriers obtained by using the reaction coordinate and the micro-iterative methods for the racemization of mandelate and respectively. No significant differences are observed among the two sets of energy barriers in the case The corresponding values obtained from the micro-iterative of mandelate, no other than those that could be X. Prat-Resina et al. / Journal of Molecular Structure (Theochem) 632 (2003) 297–307 ˚ ) for the transition state structures corresponding to the racemization of mandelate and propargylglycolate according to mechanism I using the reaction coordinate method The same distances obtained using the micro-iterative method are given in brackets.
coordinate method and the micro-iterative method) It can be seen that there are no significant are presented in respectively. Since many divergences between both sets of transition state atoms move in each step, we have also compared the structures in most of the steps of the mechanisms. On positions of the main residues of the active center at the other hand, the discrepancies in general have no the transition state structures located using the two important consequence on the potential energy methods. To this aim we have calculated the root barriers: See, for instance, that a deviation of even mean square (RMS) of the difference between the coordinates of the atoms at the transition state mechanism II for mandelate, along with some RMS structures obtained employing the reaction coordinate method and the ones located by means of the micro- in the active center, produces a change of only iterative method. These RMS values are shown in 2 0.31 kcal/mol in the corresponding energy barrier for mechanisms I, II and III, respectively.
The rest of the residues of the active center give lower The step 4 of mechanism I and II of propargyl- values of RMS and are not included in the tables.
glycolate is a especial case in which the transition ˚ ) for the transition state structures corresponding to the racemization of mandelate and propargylglycolate according to mechanism II using the reaction coordinate method The same distances obtained using the micro-iterative method are given in brackets.
X. Prat-Resina et al. / Journal of Molecular Structure (Theochem) 632 (2003) 297–307 propargylglycolate points to different directions, the conformation of the amino group in Lys166 is corresponding to the racemization of mandelate and propargylgly- slightly different and the Ca atom presents a colate according to mechanism III using the reaction coordinate different degree of configurational change. The Asp195 residue is depicted to realize the different conformation of Lys166. Then, we can see in thiscase that the differences are not only in the distances associated to the transferring hydrogens ( but in the immediate surrounding. Note that this step 4 consists basically of the configuration change of the Ca atom, the reaction coordinate chosen in this case (the distance between the transferring hydrogen and the acceptor heavy atom) not being perhaps the most adequate.
It is important to remark that the initial structure used for each step in the micro-iterative method has The same distances obtained using the micro-iterative method been the one obtained in the reaction coordinate method, and in all cases but in the step 4 (mechanismsI and II) for propargylglycolate we have found a state structures obtained with the two methods differ negative eigenvalue from the beginning and a corresponding eigenvector that describes the step. In energy difference already seen in this step as well.
step 4 the initial structure (that is, the transition state The differences appear in residue Lys166 and in the structure obtained from the reaction coordinate substrate. In we can do a visual inspection of method) had no transition vector. This means that what is happening. That is, the hydroxyl group of the proposed structure according to the reaction ˚ ) for the transition state structures corresponding to the racemization of mandelate and propargylglycolate according to mechanism I X. Prat-Resina et al. / Journal of Molecular Structure (Theochem) 632 (2003) 297–307 ˚ ) for the transition state structures corresponding to the racemization of mandelate and propargylglycolate according to mechanism II coordinate method was away from the quadratic of the refinement of the coordinates and the potential region with the suitable curvature corresponding to energy of the located structure but with the problem of the actual transition state structure of this step. So in a transition state structure found by the reaction this case we are not just dealing with the convenience coordinate method that may not fulfill the adequate ˚ ) for the transition state structure corresponding to the racemization of mandelate and propargylglycolate according tomechanism III Fig. 6. Transition state structures located with the reaction coordinate method (upper structure) and the micro-iterative method (lower structure) for the step 4 in mechanism I of the substrate propargylglycolate.
X. Prat-Resina et al. / Journal of Molecular Structure (Theochem) 632 (2003) 297–307 mathematical conditions to be considered at least, as the application of a second derivatives direct method, an approximation to the real transition state structure like the one presented here, is always recommended, rather to warrant the real nature of transition state of the located structure (in other words, that the located propargylglycolate, the main divergence between structure lies on the quadratic region corresponding to both transition state structures comes from the two the actual transition state structure), than to refine the distances associated with the transferring hydrogen concrete values of the potential energy barriers and oxygen atom of the substrate: 1.451 and 1.056 A Indeed this work concerns just to a particular for the Lys166-N· · ·H and the H· · ·OOC distances, enzymatic reaction. However, we have tested the respectively, according to the reaction coordinate racemization of two substrates, that takes place through three different mechanisms, involving many arising from the micro-iterative method. These distinct steps. In all, 25 transition states have been differences, which could be avoided using a located, which provide a critical mass of information, denser grid along the reaction coordinate method, probably enough to think that the conclusions of this leads to the corresponding potential energy barrier paper can be quite general for the enzymatic reactions In this paper we have built up a new second We are grateful for financial support from the derivatives direct method to locate transition state Spanish ‘Ministerio de Ciencia y Tecnologı´a’ and the structures in enzymatic reactions that differs in some ‘Fondo Europeo de Desarrollo Regional’ through points from previous related algorithms. This method, project No. BQU2002-00301, and the use of the here called micro-iterative method, is based on a RFO computational facilities of the CESCA.
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