The Empirical Valence Bond as an Effective Strategy for Computer-Aided Enzyme Design

Abstract

The ability of the empirical valence bond (EVB) to be used in screening active site residues in enzyme design is explored in a preliminary way. This validation is done by comparing the ability of this approach to evaluate the catalytic contributions of various residues in chorismate mutase. It is demonstrated that the EVB model can serve as an accurate tool in the final stages of computer aided enzyme design (CAED). The ability of the model to predict quantitatively the catalytic power of enzymes should augment the capacity of current approaches for enzyme design.

1. Introduction

Computer aided enzyme design (CAED) has become a subject of major attention and major activity in recent years (e.g [1–5]). Unfortunately the resulting constructs have been significantly less effective than the corresponding natural enzymes [1,6–8] and the reasons for this limited successes are not completely clear. One way to judge the effectiveness of different approaches for enzyme design is to use them in predicting the effect of different mutations on the catalytic rates of the corresponding enzymes.

Previous studies [9–18] have demonstrated that the empirical valence bond (EVB) method can give reliable prediction of enzyme activity and mutational effect. Although other quantum mechanics/molecular mechanics (QM/MM) approaches are also useful (e.g. [19,20]), the EVB is much less time demanding and thus allow for the expensive conformational search needed in enzyme design. Thus we focus here on the examination of the performance of the EVB method in reproducing observed mutational effects in different forms of chorismate mutase (CM). The results obtained in this preliminary study are quite promising, indicating that the EVB can be used in the final screening stage in computer aided enzyme design.

2. Methods

As stated above we explore here the performance of the EVB in computer aided enzyme design. The EVB has been described extensively in many works (e.g. [9,12,14]) and thus we provide here only key information about this method.

The EVB begins with the resonance states (or more precisely, diabatic states) corresponding to classical valence-bond structures. These basis states are mixed to describe the reacting system. The potential energies of the diabatic states (H₁₁ and H₂₂) and the mixing term (H₁₂) are represented by the Hamiltonian matrix elements,

$$H_{ii}={\epsilon }_i={\alpha }_{\mathit{gas}}^i+U_{\mathit{intra}}^i(\mathit{R},\mathit{Q})+U_{\mathit{inter}}^i(\mathit{R},\mathit{Q},\mathit{r},\mathit{q})+U_{\mathit{solvent}}^i(\mathit{r},\mathit{q})$$ (1)

$$H_{ij}=A\mathit{exp}(-a\mid \mathrm{\Delta }{R}^{\prime }\mid )$$

Here R and Q represent the atomic coordinates and charges, respectively, of the reactants or products (“solute”) in the diabatic states, and r and q are the coordinates and charges of the surrounding water or protein (“solvent”). ${\alpha }_{\mathit{gas}}^i$ is the energy of the i^th diabatic state in the gas-phase, where all the fragments are taken to be at infinity; $U_{\mathit{intra}}^i(\mathit{R},\mathit{Q})$ is the intramolecular potential of the solute system (relative to its minimum) in this state; $U_{\mathit{inter}}^i(\mathit{R},\mathit{Q},\mathit{r},\mathit{q})$ represents the interaction between the solute atoms and the surrounding solvent atoms; and $U_{\mathit{solvent}}^i(\mathit{r},\mathit{q})$ represents the potential energy of the solvent.

The adiabatic ground-state energy (E_g) and the corresponding eigenvector (C_g) are obtained by solving the secular equation

$${\mathit{H}}_{\mathit{EVB}}{\mathit{C}}_g=E_g{\mathit{C}}_g$$ (2)

The simplicity of the EVB formulation makes it relatively straightforward to obtain analytical derivatives of the potential surface by using the Hellmann-Feynman theorem for the first derivatives of Eg, and thus to sample the EVB energy surface by molecular-dynamics (MD) simulations. This is done by a combined free energy perturbation (FEP) umbrella sampling (US) procedure that provides the free energy function (Δg^‡(x)) that is needed to calculate the activation free energy (Δg^‡). The details of the FEP/US mapping procedure used to evaluate the EVB free energy surface are described elsewhere [12], and here we review only essential points for the simple case of two diabatic states. In such a case we use a mapping potential of the form,

$${\epsilon }_m=(1-{\theta }_m){\epsilon }_1+{\theta }_m{\epsilon }_2$$ (3)

where θ_m changes from 0 to 1 in n+1 fixed increments (θ_m = 0/n, 1/n, 2/n, …, n/n). The free energy ΔG_m associated with changing λ from 0 to m/n can be evaluated by a free-energy perturbation (FEP) procedure. The free energy functional that corresponds to the adiabatic ground state surface, E_g, is obtained by the FEP-umbrella sampling (FEP/US) method as described elsewhere ([9,12,14]). The EVB has been used extensively by our group and has been implemented (sometimes under different names) by leading research groups (see [21]).

The EVB calculations were evaluated by using the MOLARIS simulation program [22] using the ENZYMIX force field. The EVB activation barriers were calculated at the configurations selected by using the same free energy perturbation umbrella sampling (FEP/US) approach used in all of our EVB studies. The simulation systems were solvated by the surface constrained all atom solvent (SCAAS) model [22] using a water sphere of 18 Å radius centered on the center of mass of the substrate, while long-range electrostatic effects were treated by the local reaction field (LRF) method [22]. The EVB region consisted of the entire substrate, a total of 24 atoms. The FEP mapping was evaluated by 21 frames of 20 ps each for moving along the reaction coordinate with our all atom surface constrained spherical model. All the simulations were done at 300 K with a time step of 1 fs. In order to obtain reliable results we repeated the simulations 5 times with different initial conditions (obtained from arbitrary points in the relaxation trajectory). Along the EVB simulation there was no positional constraint on the protein atoms and the EVB parameters used are the same to those used in the previous EVB study on CM [23]. The charges of the EVB states (reactant state and product state) were obtained by using the Gaussian03 package [24]. This was done by optimizing the structures of the reactant and product in the gas phase using B3LYP [25,26] with the 6–31+g(d) basis set [27,28] and solvating the optimized structures in the PCM solvent model [29,30].

To exploit the power of the EVB method in quantitative studies of enzymatic reactions, it is essential to fit the EVB free energy profile of the reference reaction in water, adjusting the parameters to reproduce the observed barrier for the reaction in solution. This was done in our previous studies [23,31]. The parameters optimized in the above procedure were used in the evaluation of the activation free energies in the different CM systems without any further adjustment (just using the EVB parameters of the reaction in aqueous solution).

3. Results

In this work we chose as a benchmark the ability to reproduce the catalytic activity of different mutants of different forms of enzyme chorismate mutase (CM) that catalyzes rearrangement of chorismate to prephenate (e.g see Gajewski JJ, Jurayj J, Kimbrough DR, Gande ME, Ganem B & Carpenter BK (1987) J. Am. Chem. Soc. 109: 1170–1186). The studied systems are the trimer Bacillus subtilis chorismate mutase (BsCM), [32] the homodimeric chorismate mutase from Escherichia coli (EcCM) [33] and the monomeric chorismate mutase mMjCM obtained by Hilvert and coworkers [34] by topological redesign of the thermostable EcCM homologue from M. jannaschii (MjCM). The coordinates for BsCM, EcCM (X-ray) and mMjCM (NMR) structures were obtained from the Protein Data Bank, Brookhaven National Laboratory, with PDB access codes 1COM, 1ECM and 2GTV, respectively.

In this work we considered; a) the native EcCM and the V35I, V35A mutants that have been used recently by Mayo and coworkers [35], b) the monomer mMjCM that has been designed by Hilvert and coworkers [36,37], and c) the native BsCM and the R90G-BsCM and R90Cit-BsCM mutations of refs [38,39]. In all of these cases the observed values of k_cat/K_m and k_cat are known. The position of the mutated residues is shown in Fig 1. Here we explore the performance of the EVB screening approach.

Figure 1.

Scheme of the Chorismate Mutase active site. The figure shows the structure of the transition state and its interactions with the neighboring residues, (a) for the dimer and monomer and (b) for the trimer.

Our main strategy for calculations of activation energies, is the EVB method used by our group and others in quantitative studies of enzyme catalysis of many enzymes (e.g. [9]). The EVB has been described in great details elsewhere [12,13] and was used recently in a study of CM [23,31].

This approach can evaluate Δg^‡_cat (that corresponds to k_cat) rather than the Δg^‡_p considered in the previous section. Here we explored the catalytic power of the trimer (BsCM), dimer (EcCM) and monomer (mMjCM) by calculating the EVB surfaces for different mutants and native proteins considered in Table 1. In each of the systems considered we started by generating the given sequence from the native protein proceeding to 100 ps MD simulations in order to relax the given structures. Five structures were saved during each relaxation process and then used to generate the EVB surface and obtain the activation free energies and reorganization energies.

Table 1.

Calculated and observed Δg^‡_cat (ΔΔg^‡_cat).^a

	Δg‡cat (ΔΔg‡cat)calc	Δg‡cat (ΔΔg‡cat)obs
EcCM	15.3 ± 1.5 (0.0)	15.3(0.0)
V35I-EcCM	13.3 ± 0.6 (−2.0)	15.0(−0.3)
V35A-EcCM	15.2 ± 0.9 (−0.1)	15.7(0.4)
mMjCM	16.2 ± 1.7 (0.9)	16.8(1.5)
BsCM	16.6 ± 1.6 (1.3)	15.3(0.0)
R90Cit-BsCM	23.7 ± 2.5 (8.4)	21.1(5.8)
R90G-BsCM	23.8 ± 2.2 (8.5)	22.5(7.2)

^aEnergies in kcal/mol.

The calculated activation barriers of the different mutants are compared to the corresponding observed values in Table 1 and Fig. 2. As seen from the table and the figure the agreement between the calculated and observed results is excellent and can be considered to be a quantitative agreement. It should be noted in this respect that the performance of our calculations may not be fully appreciated by those who note the small deviations between the calculated an observed values in Fig. 2 while overlooking the fact that we actually reproduced quantitatively the absolute values of the activation free energies without any parameterization on the reaction in the enzyme. Obtaining such a quantitative prediction is very encouraging.

Figure 2.

Correlation between the calculated and observed activation free energies.

Thus the EVB approach can be used in the final screening stage of CAED approaches and perhaps can be defined as the “gold standard” of our approaches.

In order to assess the effectiveness of a given CAED approach it is important to have a clear idea on the computational resources needed to obtain the given result. Here we report the computer time needed for each of the approaches considered above, or in other words the price per performance ratio. The relevant estimates are summarized in Table 2. As seen from the table after appropriate relaxation it is possible to screen 9 mutations on 100 nodes each time considering that each node has two processors and then we could screen around 60 mutations overnight. While this may sound an investment of a major resource it might provide at present the most practical way of getting accurate screening.

Table 2.

The performance times of the Δg^‡_cat calculation. ^a

	Number of nodes	Number of processors	Computation time
MD Relaxation	1	1	3 hours
Δg‡cat using EVB	11	22	1,5 hours

^aThe calculations are done on the USC HPCC (High Performance Computing and Communication) Linux computer, using the Dual Intel P4 3.0 GHz 2GB Memory nodes. The time involves an average over the three forms of CM and also for calculating Δg^‡_cat is an average over five runs.

The present study focused on the EVB approach and left for a subsequent studies the exploration of the performance of faster but less quantitative approaches, including the LRA, LIE and the PDLD/S-LRA methods [40] that have been found to provide qualitative but physically reasonable trend (e.g [41]), and even ground-state simulations may be useful in the screening process [42]. It is also important to point out that the case of CM is somewhat simpler than cases where there are several transition states with similar energy [43]. However, after identifying the highest energy TS one can use a small number of EVB calculations (usually three) to identify the free energy of the rate limiting TS. Obviously if we find that direct calculations of the TS binding free energy are sufficiently accurate we can use just a single step of such calculations focusing on the highest TS.

4. Concluding remarks

Despite recent impressive advances in enzyme design (e.g. [7,8]), the enzymes generated by current design efforts are still far less efficient than naturally evolved enzymes. Overall, it seems to us that a part of the problem in previous CAED has been the limited focus on modeling of the actual chemical step in the actual enzyme active site. Probably, the main problem in designing enzymes with native activity is related to the ability to predict the proper transition state (TS) stabilization. This difficulty is due in a large part to the difficulty of capturing the preorganization effect. That is, as shown in a large series of studies (see ref. [9] for a review) the major part of the catalytic effect ( in any enzyme that was examined quantitatively) is due to the fact that the dipoles of the active site polar groups are already partially oriented toward the TS charges and thus do not have to pay for this polarization process. On the other hand, in the uncatalyzed reaction in water about half of the energy of interaction between the water dipoles and the TS is paid in the reorganization of the water molecules. Thus, the calculations of the TS activation free energy are very different than the calculations of the interaction between the charged TS and the active site. This is probably demonstrated in the best way in the recent study of the catalytic effect of ketosteroid isomerase [44]. Obviously, attempts to evaluate the catalytic effect by using gas phase models or by looking at the electrostatic interaction between different residues and the TS are unlikely to reproduce the correct catalytic effect since it is impossible to assess the preorganization effect without including the protein and its reorganization in the simulations.

One clear but non trivial way to progress in enzyme design is to calculate the activation free energies in the proposed active sites, and this work explored the feasibility of such a strategy by using the EVB approach. Thus our work is not so much about effective early screening but mainly about the requirements of the final stage in the screening process. In this respect we believe that the EVB provides a very effective way for performing the final stage in the screening process. We are also developing an examining more qualitative approaches for early screening and these approaches will be reported in subsequent studies.

Abbreviations

EVB

empirical valence bond

CAED

computer aided enzyme design

FEP/US

free energy perturbation/umbrella sampling

CM

chorismate mutase