Quantifying Electronic Effects in QM and QM/MM Biomolecular Modeling with the Fukui Function

QI Helena W., KARELINA Maria, KULIK Heather J.,＊

1 Introduction

Enzymes catalyze nearly every reaction that occurs in the cell, and atomistic simulations of enzymes can provide valuable insight into the source of enzymatic rate enhancements. In order to provide a balance between accuracy and computational cost, a multilevel approach is typically employed for enzyme modeling1-9, wherein a reactive region is treated quantum mechanically (QM), while the surrounding enzyme is treated with an empirical molecular mechanics (MM) force field. QM methods describe essential aspects of the enzyme environment(i.e., charge transfer, polarization, and bond rearrangement) that standard point charge MM models cannot. Furthermore, many of the limitations present in MM point charge force fields cannot be addressed even through the use of fixed charge,polarizable force fields10-12. Typical QM regions are on the order of tens of atoms in order to maximize efficiency9,13,14.Recent work, made possible through advances in electronic structure efficiency, has enabled large-scale simulation of entire proteins15,16and revealed that critical properties of enzyme active sites only reach converged asymptotic limits in the range of 500-1000 atoms in radial convergence studies17-28. In light of these challenges for QM/MM modeling, it becomes essential to identify the role of each residue in the active site in order to determine which residues must be treated quantum mechanically to develop an accurate mechanistic picture at a reasonable computational cost.

Within the field of conceptual density functional theory29,the Fukui function30:

has been invoked as a measure of reactivity, as the first step of a chemical reaction often involves a change in the electron density. Two finite forms of the three-dimensional Fukui function exist, either with integer electron addition to mimic the initial portion of a nucleophilic reaction:

The most reactive portions of a molecule are expected to have the largest value of the Fukui function. The condensed form31of the Fukui function30is more practical to assess,instead computing the change in atomic partial charges with electron removal or addition. Condensed Fukui functions have been previously invoked in the study of biological systems32,and frontier states have been hypothesized as a potential tool to identify the location of active sites in enzymes33in conjunction with semi-empirical quantum chemistry methods.

We recently introduced27Fukui shift analysis (FSA) as a tool to identify critical residues for systematic construction of QM regions in QM/MM simulation. In this approach, we focused on extracting residue-specific influences on core active site properties. We first compute the condensed Fukui functions31on the minimal active site with all remaining protein residues treated at the MM level. In particular, we compute the by-residue (RES) sum of partial charges over a residue:

and monitor how it changes with electron addition or removal to obtain a condensed form of the Fukui function. We employ by-residue sums in conjunction with real-space partitioned Voronoi deformation density (VDD) partial charges34to mitigate basis set sensitivity. We also showed27range-separated hybrids to be essential to obtaining meaningful Fukui functions over standard global hybrids, particularly when computing Fukui functions of a residue at a substantial distance from the core active site residues.

We then compute this by-residue condensed Fukui function as an electron is added:

or removed:

Rigorously, the condensed Fukui function31refers to the single-atom, partial-charge difference with electron addition or removal. Here we also use this terminology to refer to the by-residue-summed condensed Fukui function. Then, we carry out a systematic scan through either every residue in the protein or through every residue within a distance cutoff (e.g., 10 ?, 1 ? = 0.1 nm) of any portion of the core active site. Each residue in the scan is added one at a time to the QM region, and condensed Fukui functions are recomputed over this modified QM region on the same relevant active site residues (ASRs, see schematic in Fig.1). A background, median reference value is determined over all of the cases in which the added residue is not interacting with the ASRs, i.e., values obtained for distant neutral residues. The root-sum-squared (RSS) difference in all by-residue condensed Fukui functions on the active site with respect to the median reference is then a metric for the relative influence of a residue on the active site:

Fig.1 Schematic of FSA calculation. The condensed Fukui functions are summed over individual components of the active site (blue spheres, top) in the protein in a QM/MM calculation.These condensed Fukui functions are compared to their values in a protein with a QM region in which one protein residue is added(bottom). This process is repeated with each residue added one at a time (e.g., Glu as shown at bottom in sticks) throughout the protein or within a distance cutoff of the active site, and the magnitude in shift of the Fukui function on the active site (green glow around blue spheres) is evaluated with respect to a baseline or null result.

Because each calculation consists of at most one residue in addition to the ASRs, this calculation may be readily parallelized and rapidly carried out. We previously demonstrated on a 64 atom active site that this scan could be carried out on the entire protein in as little as 5 min when parallelized and carried out with graphical-processing-unit(GPU)-accelerated quantum chemistry35,36. Running each calculation in serial, by comparison, would take around 44 h.Using conventional codes, the benefit of this approach would be even more apparent in comparison to the charge shift analysis method we developed for similar purposes, wherein a large-scale simulation of 1000 atoms or more was required22,27.

那時(shí)，我還寫過(guò)一篇《昭君何以“請(qǐng)出塞”》，因?yàn)槲覐膩?lái)不認(rèn)為王昭君是為了民族大義而自請(qǐng)遠(yuǎn)嫁的，也從來(lái)不認(rèn)為她是為愛(ài)情而去的。當(dāng)時(shí)讀宋人的《鶴林玉露》，其中看到批駁王安石的“漢恩自淺胡自深，人生樂(lè)在相知心”，指其“悖理傷道甚矣”，又給我“壯了膽”，就寫了這篇雜談，先后發(fā)表在《鄭州晚報(bào)》和《蘇州日?qǐng)?bào)》上。后來(lái)，我有機(jī)會(huì)先后去了湖北興山的昭君故里和呼和浩特的青冢，這種歷史觀念與現(xiàn)場(chǎng)觀感相融會(huì)，又寫了8000余字的散文《歷史深處的昭君背影》，發(fā)表十幾年后還有刊物轉(zhuǎn)載這篇散文。

We now examine in more detail the quantitative insight that this approach provides by i) computing distance-dependent and residue-pair-dependent effects on the condensed Fukui function in model residue pairs and ii) measuring how summed,condensed Fukui functions change with single QM residue addition in the active site of enzymes that carry out SN2 methyl transfer reactions. The rest of this article is outlined as follows.In section 2, we provide the computational details of the calculations employed in this work. In section 3, we present Results and Discussion. Finally, in section 4, we provide our Conclusions.

2 Computational details

2.1 Protein structure preparation

The protocol for whole-protein simulation was as follows.The protein crystal structures were obtained from the protein databank (PDB) for catechol O-methyltransferase (COMT, PDB ID: 3BWM37), the study of which we have previously described22,27,38,39, and L-isoaspartyl methyltransferase40(IsoAsp, PDB ID: 1JG4). The charge state of the apoenzyme was assigned using the H++ webserver41-44assuming a pH of 7.0 with all other defaults applied. As H++ removes all nonstandard residues, residues in the active site adjacent to cofactors were manually assigned protonation states, as described in our previous work.22,27,38,39The output of H++ was used as the starting point for subsequent topology and coordinate preparation using the AMBER45tleap utility prior to classical molecular dynamics (MD) and QM/MM simulation with AMBER45.

Protein residues were described by the AMBER ff14SB46force field, which is derived from the ff99SB47force field with updates to backbone torsional parameters. For nonstandard residues in COMT and IsoAsp, we employ the generalized AMBER force field (GAFF)48with partial charges assigned from restrained electrostatic potential (RESP) charges49obtained with GAMESS-US50at the Hartree-Fock level using a 6-31G*51basis set, as implemented by the R.E.D.S. web server52-54. Resolved water molecules in the crystal structures were preserved, whereas any crystallizing agents were removed.The Mg2+force field parameters used in COMT simulation were obtained from Ref.55, as validated in some of our previous work22,27,38,39. Each protein was solvated in a periodic rectangular prism box with at least a 10 ? buffer of TIP3P56water and neutralized with either Na+or Cl-counterions. The full simulation contained 25893 atoms (3411 protein or cofactor atoms) for COMT and 33066 atoms (3616 protein or cofactor atoms) for IsoAsp.

2.2 MM equilibration

All structures were equilibrated with classical (MM)molecular dynamics in AMBER. COMT structures were extracted directly from previous molecular dynamics studies27,38,39. Minimizations were carried out for 1000 steps with the protein restrained followed by 2000 steps of unrestrained minimization. Following minimization, a 10-ps NVT heating step was carried out to raise the system temperature to 300 K using a Langevin thermostat with collision frequency of 1.0 ps-1and a random seed to avoid synchronization artifacts. This step was followed by a 1-ns NPT equilibration using the Berendsen barostat with a pressure relaxation time of 1 ps. Production dynamics were collected for 100 ns for each protein. The SHAKE algorithm57was applied to fix all bonds involving hydrogen, permitting a 2-fs timestep to be used for all MD. For the long-range electrostatics, the particle mesh Ewald method was used with a 10-? electrostatic cutoff.

2.3 QM/MM simulation and analysis

Snapshots from MD production runs were extracted for QM/MM simulation. The periodic box was post-processed using the center of mass utility in PyMOL58to generate the largest possible spherical droplet centered around each protein that was circumscribed by the original rectangular prism periodic box.The resulting system was again processed with tleap to generate a system with spherical cap boundary conditions that were enforced with a restraining potential of 1.5 kcal·mol-1·?-2(1 cal =4.1868 J). Reactant, transition state, and product geometries for all enzymes were extracted from previous work22,27,38,39. All QM/MM simulations were carried out using TeraChem35,36for the QM portion and AMBER 1445for the MM component. The QM region is modeled with density functional theory (DFT)using the range-separated exchange-correlation functional ωPBEh59(ω = 0.2 bohr-1) with the 6-31G*51basis set, as validated in our previous work27. The FSA schemes rely on evaluation of partial charges obtained from the Voronoi deformation density (VDD) method34. The VDD partial charges use a promolecule definition to partition the real space density and are therefore relatively basis-set insensitive, as described in previous work. Preparation, automation, and analysis were carried out using in-house python scripts. A tutorial example of the workflow and accompanying scripts for protein analysis are provided on our website (http://hjkgrp.mit.edu/csafsa).

2.4 Gas phase Fukui calculations

Reference comparisons were generated for two-residue models extracted from select proteins (PDB IDs: 2Y7Y, 2ZF3,3RQ9, 4JGG, and 4N30). All pairs were identified as containing one Asp residue and another residue with a shortest atom distance (＜ 1.6 ?) to that Asp residue. These structures were protonated with the tleap module of AMBER45. Capping hydrogen atoms were added along the protein backbone vector with a scaled bond length (1.09 ? for C―H, 1.01 ? for N―H).The structures were geometry optimized in TeraChem35,36using B3LYP60-62/6-31G*51with all heavy atoms and the capping hydrogen atoms frozen and the remaining hydrogen atoms allowed to relax. An in-house code was used to translate each residue in a residue pair by their respective Cαatoms to produce increasing inter-residue separations. The Multiwfn63code was used to obtain bond-critical points (BCP) between residue pairs and evaluate the potential energy at the bond critical point (VBCP),which provides a good proxy for the relative hydrogen bond or electrostatic interaction strength between two residues.Condensed Fukui functions were evaluated at the ωPBEh59/6-31G*51level of theory using Mulliken charges.

3 Results and discussion

3.1 Revealing Fukui length-scales with specific protein residue pairs

In previous work27, we employed shifts in summed,condensed Fukui functions as a tool for constructing QM regions in QM/MM simulation (see Fig.1). Now, we return to simple residue-residue gas phase pairs in order to identify the extent to which the presence of an additional close protein residue will alter the frontier states of one protein residue. For instance, an aspartate in isolation will have a strong f-of -1.0, centered on the oxygen anion of the carboxylate sidechain. The presence of additional residues at short distance, e.g. in hydrogen bonding configurations, is expected to shift this condensed Fukui function. We focus here on gas phase residue pairs due to suggestions by Merz and coworkers32that the measure of the Fukui in large-scale biological systems might require multiple electron addition or removal due to the larger system size. This observation also led us to neglect the inclusion of explicit water molecules in our simulation. If an implicit solvation model16was employed with a low dielectric constant around 4 typical of a protein environment, we would expect slightly reduced Fukui function perturbations from our gas phase values, whereas screening in an aqueous environment would have a stronger effect.

We examine five residue pairs where Asp is held fixed as one residue in the pair and the other residue is varied by identity and nature and strength of the hydrogen-bonding donor. In all cases,we select unusually short non-bonding distances (＜ 1.6 ?)between Asp and the pairing residue (see Computational details).We then remove an electron from the overall residue pair and sum the partial charges over the Asp residue in both the N and N - 1 electron systems in order to obtain a revised summed f-in the presence of a second residue. Overall, we observe weak or no correlation between how the summed f-shifts from -1.0 and the strength of hydrogen bonding interaction, as assessed through the potential energy at the bond critical point (Fig.2). Instead,chemical identity plays a critical role. For instance, the presence of an Arg residue is unlikely to shift the Asp f-because its positively charged nature makes it difficult to ionize. Instead,Arg should have a large positive f+, but nearly any residue neighboring Asp will likely dominate the condensed f+. Indeed,the summed condensed Fukui function remains 90% of the isolated value, although we note that such a deviation would still be considered significant in QM region construction for QM/MM simulation (see Sec.3.2).

Fig.2 Condensed Fukui function on an Asp residue (left axis, red bars, as indicated by red arrow) as a function of the paired residue with potential energy at the bond critical point, VBCP/2 overlaid(right axis, blue points, as indicated by blue arrow) for five representative Asp-residue pairs (see structures in Fig.3).

Examining neutral residues reveals more varied trends that can be rationalized by the nature of the bonding partner (Fig.2).The N―H hydrogen bond donors of Asn (D-N) and His (D-H2)produce the largest deviations from the isolated Asp condensed Fukui function at around -0.5 and -0.7, respectively. A second His configuration (D-H1) is unusual in that a C―H acts as the hydrogen bond donor, slightly reducing the effect on the condensed Fukui function to around -0.8, despite comparable hydrogen bonding distances (Fig.3). This variation could be rationalized somewhat based on the lower Pauling electronegativity of carbon (χ = 2.55) versus nitrogen (χ =3.04). That is, the His pair (D-H2) is holding onto less electron density and therefore loses less when an electron is removed.However, such a rationale does not explain why Asp-Ser pairs with a strong O―H hydrogen bond donor have a relatively small deviation of the summed condensed Fukui function from the ideal case (see Figs.2 and 3). Thus, the shift of a condensed Fukui function summed over a central residue will depend on the orientation (e.g., H1 vs. H2 Asp-His cases) and identity of neighboring residues (e.g., Asp-Asn vs. Asp-Arg). In all cases,the presence of proximal residues in a short hydrogen-bonding configuration gives rise to some Fukui shift, however, for our purposes of QM region determination (see Sec.3.2).

Fig.3 Five representative Asp-residue pairs extracted from crystal structures and labeled by the associated PDB ID. The shortest distance between the two residues is indicated with a yellow rod and labeled accordingly. Each residue is indicated by the single letter code in a manner consistent with Fig.2.

Fig.4 Condensed Fukui function on an Asp residue as a function of inter-residue separation, d, in ? for Asn-Asp residue pair shown in inset. The x-axis is the shortest inter-residue distance, which is controlled by translating the residue pairs by their Cα atoms away from each other with respect to the X-ray crystal separation (1.43 ?, PDB ID: 3RQ9). The asymptotic, isolated limit is reached between 4.0 and 5.0 ? for the condensed Fukui function.

A second question beyond residue-residue pairing sensitivity of the condensed Fukui function sum shifts is distance dependence. In particular, we have motivated27the condensed Fukui function as a tool to evaluate QM region consistency and convergence in QM/MM simulation, which has been predominantly17-27assessed with radial convergence tests.Therefore, it is useful to determine the typical length scales at which a residue stops perturbing a partner residue’s summed condensed Fukui function. We compute this distance dependence for the Asp-Asn pair, which showed the strongest shift of the summed condensed Fukui function on Asp for the crystal structure, which had a shortest non-bonded distance(O-…H) of 1.43 ?. Here, we translate and optimize the Asn structure as it moves away from Asp and recompute the summed condensed Fukui function on Asp (Fig.4). We observe that the shift on the condensed Fukui function decays rapidly and smoothly with distance. Overall, the summed condensed Fukui function on Asp reaches its isolated limit when the two residues’closest distance is roughly 4-5 ?. This analysis is useful to put our earlier observations with FSA27in context. Although we could have computed interaction energies between these residues, this smooth decay i) both indicates that our functional choice is good and ii) delineates the length-scale over which FSA will work.

In particular, this characteristic length scale of 4-5 ? suggests that shifts on the Fukui function persist for longer distances beyond the traditional distance cutoff for most strong non-covalent interactions such as hydrogen bonds (i.e., 3.0 ?).However, this analysis also highlights and confirms our earlier observation27that FSA in QM/MM simulation will, by definition, neglect many-body effects. That is, if a distant residue is a critical hydrogen bond partner of another residue that hydrogen bonds with the core active site but does not participate in direct hydrogen-bonding interactions with the substrate, it may not be detected with FSA. The continued motivation to focus on single-residue shifts on summed condensed Fukui functions, however, is in the simplicity of the interpretation of these effects, as we will now describe.

3.2 Analysis of Fukui function dependence on environment in QM/MM

We recently introduced27analysis of Fukui function shifts on core active site residues as a tool to determine quantitative QM region sizes for QM/MM simulation. In addition to being highly parallelizable, this technique also provides insight into the nature of interactions between residues and substrates. Here, we consider how the by-substrate summed condensed Fukui functions vary with addition of protein environment in a representative methyltransferase, catechol O-methyltransferase(COMT)64. In this enzyme, an S-Adenosyl-L-methionine (SAM)cofactor donates a methyl group to a catecholate anion substrate in a rate-limiting, SN2 methyl transfer reaction65,66. We can calculate summed condensed Fukui functions over SAM,catecholate, or the coordinating Mg2+ion that facilitates methyl transfer by doubly coordinating the catecholate structure. The SAM and Mg2+formal charges are positive, whereas the catecholate carries a net negative charge. For Mg2+and catecholate, only a single summed Fukui function is non-zero and exhibits any shifts: the nucleophilic for Mg2+and the electrophilic for catecholate. The catecholate acts as a nucleophile in this methyl transfer reaction, and the accumulation of charge on catecholate to form the anion is most closely related to its strength as a nucleophile. Therefore it is reasonable that the N - 1 to N electron Fukui function of catecholate (i.e., the electrophilic one) provides the greatest insight. SAM, on the other hand, carries two positively charged groups (an NH3+and an S-CH3+) and one negatively charged group (a carboxylate), leading to nonzero shifts in both nucleophilic and electrophilic summed condensed Fukui functions, depending on the geometry studied. In the FSA method, we compute an overall RMS change in the Fukui functions, but identifying geometric and residue sensitivity of the Fukui shifts, as we will now do, provides further chemical insight. It is expected that depending upon the portion of the reaction coordinate, each of the reactants will have differing Fukui functions, particularly as we move from reactants where catecholate is a reactive anion to products where SAM is no longer a cation.

A first focus on the catecholate substrate reveals strong shifts in the summed f-Fukui function for several residues,particularly for the reactant case where the bare O-is most available to interact with the surrounding protein environment(Fig.5). Reactant Fukui shifts are nearly their maximum value of-1 for key residues surrounding the catecholate in the active site including Asn41, Val42, Gly66, Ala67, Asp141, and Asn170.These residues generally surround the Mg2+-catecholate interface and are buried behind SAM as well (Fig.5). The pattern of Fukui shifts changes in the transition state (TS) structure as the oxygen becomes partially methylated. Here, a number of the same residues still exhibit a shift, but the intensity of the shift becomes smaller (around 0.2e) for most residues with large shifts in the reactant, except for Asp141, which no longer shifts at all. The intensity of some shifts increase or appear for the first time, including Met40, Tyr71, and Ser119. Finally, in the product geometry, the oxygen is fully methylated, and there is no shift in any of the Fukui functions that is over our predetermined threshold (0.05e, dotted lines in Fig.5). Overall,the majority of the closest residues to the substrate exhibit a Fukui shift but several further from the central active site do as well. The disappearance of any Fukui shift in the product state reveals that methylation dampens the ability of the substrate to lose electrons and also likely weakens its overall interaction with the protein.

Fig.5 (a) COMT reactant structure with all residues with significant condensed Fukui function shifts overall shown as sticks in light gray and substrate (SAM, catecholate, and Mg2+ shown in dark gray) also shown. The residues with significant shifts for the fcondensed Fukui function summed over catecholate in reactant,transition state, or product are shown in color (orange indicates one is above threshold, green indicates two, and purple, if applicable, indicates all three). (b) Shifts in the f- condensed Fukui function summed over catecholate substrate for 16 active site residues illustrated in (a). The reactant (top pane), transition state(middle pane), and product (bottom pane) shifts are shown relative to their respective baselines upon single residue addition to the QM region. Dotted lines indicate minimum thresholds of 0.05 or-0.05 and are used for assigning coloring in (a).

Fig.6 (a) COMT reactant structure with all residues with significant condensed Fukui function shifts shown as sticks in light gray and substrate (SAM, catecholate, and Mg2+ shown in dark gray). The residues with significant shifts for the f + condensed Fukui function summed over Mg2+ in reactant, transition state, or product are shown in color (orange indicates one is above threshold, green indicates two, and purple, if applicable, indicates all three). (b) Shifts in the f + condensed Fukui function summed over Mg2+ for 16 active site residues illustrated in (a). The reactant(top pane), transition state (middle pane), and product (bottom pane) shifts are shown relative to their respective baselines upon single residue addition to the QM region. Dotted lines indicate minimum thresholds of 0.05 or -0.05 and are used for assigning coloring in (a).

Comparison of the nucleophilic condensed Fukui function on Mg2+, which is the change in partial charge on Mg2+as an electron is added or removed to the system, reveals localized shifts with residue addition from the COMT enzyme environment (Fig.6). Here, the reactant state exhibits no shifts on the core active site. Presumably this result is due to the fact that the catecholate substrate tightly binds Mg2+in the reactants and therefore Mg2+is less likely to be sensitive to the remaining environment. In the transition state and products, however,significant shifts arise, with the largest being observed in the products. For the transition state, modest shifts in Asn41, Gly66,Asp141, and Asn170 are largely in agreement with the three large shifts for the product at Asp141, Asp169, and Asn170 (see structure in Fig.6). These latter three residues that are detected most in the product are in fact the critical residues that describe the Mg2+coordination sphere. The longer range interaction with Asn41 arises in a part of the catalytic cycle where Asn41 is slightly closer to Mg2+. Thus, we may conclude that Fukui shifts on the Mg2+ion are confined to first-coordination-sphere effects unlike substrate interactions.

The SAM substrate reveals richer and more distant interactions with the enzyme environment, likely owing to its numerous charged functional groups and larger size than Mg2+or catecholate (Figs.7 and 8). Focusing first on the shifts in the nucleophilic condensed Fukui function sum on SAM, we observe both a change in sign of shifts, particularly in the TS and product (P), as well as the first occurrence of a residue that exhibits an above-threshold shift in all three structures (Asp141,see Fig.7 structures). Overall, residues above and behind the substrate, such as Tyr68 and Tyr71, have significant Fukui shifts in the reactant that are less substantial in the TS and product.Instead, in the TS and product, larger shifts are observed for Mg2+coordination sphere residues, including Asp169 and Asn170. Those residues are also quite proximal to the demethylated SAM product. Evaluation of the electrophilic condensed Fukui function shifts reveals some of the largest shifts, especially in the TS and P (Fig.8). In particular, several residues (e.g., Glu90, Ser119, Asp141, Asn170) are all at or above threshold in all three structures. Glu90 and Ser119 are notable because they hydrogen bond to SAM hydroxyl and amino groups, respectively, and therefore modulate electron loss favorability in all three structures. The smallest electrophilic Fukui function shifts can be rationalized to occur in the reactants because here the SAM is still effectively carrying a formal positive charge and is more resistant to electron loss or shifts in electron loss. In later TS and P structures, the SAM becomes effectively neutral, making it more responsive to electron removal.

Fig.7 COMT reactant structure with all residues with significant condensed Fukui function shifts shown as sticks in light gray and substrate (SAM, catecholate, and Mg2+ shown in dark gray).The residues with significant shifts for the f + condensed Fukui function summed over SAM in reactant, transition state, or product are shown in color (orange indicates one is above threshold, green indicates two, and purple, if applicable, indicates all three). (b) Shifts in the f + condensed Fukui function summed over SAM for 16 active site residues illustrated in (a). The reactant(top pane), transition state (middle pane), and product (bottom pane)shifts are shown relative to their respective baselines upon single residue addition to the QM region. Dotted lines indicate minimum thresholds of 0.05 or -0.05 and are used for assigning coloring in (a).

Fig.8 COMT reactant structure with all residues with significant condensed Fukui function shifts shown as sticks in light gray and substrate (SAM, catecholate, and Mg2+ shown in dark gray).The residues with significant shifts for the f- condensed Fukui function summed over SAM in reactant, transition state, or product are shown in color (orange indicates one is above threshold, green indicates two,and purple, if applicable, indicates all three). (b) Shifts in the fcondensed Fukui function summed over SAM for 16 active site residues illustrated in (a). The reactant (top pane), transition state (middle pane), and product (bottom pane) shifts are shown relative to their respective baselines upon single residue addition to the QM region.Dotted lines indicate minimum thresholds of 0.05 or -0.05 and are used for assigning coloring in (a).

An overview of the residues highlighted through this analysis reveals both expected and surprising critical residues in the active site electronic environment. The coordination sphere residues around Mg2+(i.e., Asp141, Asp169, and Asn170) are readily anticipated to be critical to accurate Mg2+electronic structure, as are hydrogen-bonding partners to SAM (e.g.,Glu90). More surprising residues include nonpolar residues such as Val42, which exhibited a prominent Fukui shift in SAM and catecholate electrophilic Fukui function sums. Therefore, this method provides promise for systematic QM region determination27as well as identification of unexpected residue-substrate interactions that would not have been anticipated through chemical intuition alone. Indeed, activation and reaction energies in COMT using FSA-constructed QM regions are within ～1 kcal·mol-1accuracy of results with very large, converged (ca. 600-1000 atoms) QM regions in QM/MM simulation (i.e., 15.6 kcal·mol-1activation energy for FSA vs.15.9 kcal·mol-1for large QM)27.

Finally, an additional manner in which the condensed Fukui function can be used is to interpret catalytic differences among members of enzyme families (here, methyltransferases). In order to guide this analysis, we return to COMT and model the active site as the negatively charged catecholate substrate and positively charged SAM cofactor. For comparison, we also now consider the enzyme L-isoaspartyl methyltransferase (IsoAsp,Fig.9). This enzyme facilitates peptide repair of damaged aspartate residues through methylation.40The substrate co-crystallized with IsoAsp is a short, six-residue peptide with a modified aspartate in the middle. We treat the positively charged SAM cofactor and negatively charged isoaspartate as our quantum region. These two quantum regions have nearly the same number of atoms (63 for COMT vs. 64 for IsoAsp) and relatively similar nucleophiles (i.e., carboxylate oxygen anion versus catecholate anion) in the methyl transfer step. Therefore,analysis of the reactant, transition state, and product in methyl transfer to IsoAsp (see Fig.9) can provide new insight into sensitivity of condensed Fukui functions to the nature of the enzyme.

Fig.9 (left) L-isoaspartyl methyltransferase (IsoAsp) protein (PDB ID: 1JG4) shown in cartoon representation with SAM and L-isoaspartate substrates shown as sticks colored by element(blue nitrogen, red oxygen, yellow sulfur, and gray carbon). The remainder of the peptide substrate is shown as light gray lines.(right) Close up of active site structure for reactant (R), transition state(TS), and product (P) geometries with same coloring as at left.

In order to assess enzyme-dependent effects, we sum the partial charges over SAM or over the relevant substrate (i.e.,catecholate for COMT and isoaspartate for IsoAsp) and compute the summed condensed Fukui functions for each enzyme’s respective reactant, transition state, or product geometries. In all cases, the total condensed Fukui function must add to -1.0 but it can be distributed more over SAM or over the substrate,depending upon the point along the reaction coordinate or on the enzyme. For the reactants, both COMT and IsoAsp have nearly identical summed f+values with the majority (＞ 90%) of the added electron accumulating on SAM (Fig.10). This result is expected as SAM carries a net positive charge and the substrate already carries a net negative charge. The higher localization of negative charge in isoaspartate also explains why it carries a lower summed f+than catecholate. Comparison of the electrophilic f-instead reveals electron loss from the negatively charged substrates, again with slightly more (around 0.1e) loss from the catecholate substrate than from isoaspartate but otherwise comparable behavior. Note that due to the charge of the two substrates, electrophilic condensed Fukui functions monitor features of the substrate, despite the fact that the substrate acts as a nucleophile.

Comparison of the remaining transition state and product structures between COMT and IsoAsp reveals greater differences in summed condensed Fukui functions (see Fig.10).When computing the summed condensed Fukui functions for these transition-state and product structures, we either halve the partial charges of the methyl group being transferred (for the TS)or adjust to sum it in the substrate (for P). However, we note that the partial charge of the methyl group does not change dramatically for either N-1-electron or N+1-electron references,giving it a small (～0.03e) effect on the overall calculation of the summed condensed Fukui functions. For both COMT TS and P,the nucleophilic summed f+remains predominantly on the SAM.Conversely, for IsoAsp, a transition occurs starting at the TS toward accumulation of charge on the substrate with electron addition (around 0.2e), which then shifts to nearly complete electron addition to isoaspartate in the product. The summed condensed f-Fukui functions are in greater agreement between COMT and IsoAsp for transition state and product. Here, COMT again favors slightly more electron loss in the transition state from SAM, as it did in the reactant, and the overall electron loss from SAM is increased in the transition state for both COMT and IsoAsp. In the product, summed condensed f-Fukui functions are comparable for both COMT and IsoAsp with electron depletion nearly exclusively occurring from the methylated substrate. Thus, overall, a difference in substrate identity reveals different baseline Fukui function sums in differing geometries. The greater involvement of the substrate in IsoAsp suggests a more reactive substrate, possibly because the additional oxygen that is not methylated is bare and uncoordinated, unlike the hydroxyl group in catechol in the COMT reaction. These comparisons of Fukui functions have the potential to reveal subtle differences in reactivity in enzyme active sites that arise from differences in environment or substrate identity.

Fig.10 Comparison of summed condensed Fukui functions for COMT and IsoAsp methyltransferases in reactants (top), transition state(middle), and products (bottom) for a QM region that consists only of SAM and the substrate molecule (L-isoaspartate for IsoAsp or catecholate for COMT). The SAM molecule is shown in blue in all cases, and the substrate molecule for each enzyme is shown in salmon as a stacked bar graph. The total f + or f- summed condensed Fukui function is indicated as labeled. For the transition state, the charges corresponding to the methyl group are split in half between SAM and substrate. In the product, the methyl group charges are assigned solely to the product.

4 Conclusions

Multi-scale quantum-mechanical/molecular-mechanical (QM/MM) and large-scale QM simulation provide valuable insight into enzyme mechanism and structure-property relationships.Electron density analysis can provide insight into how the enzyme environment modulates reactivity at the enzyme active site. We have motivated the use of changes of summed-byresidue condensed Fukui functions in the presence of additional protein environment in the Fukui shift analysis (FSA) method in order to interpret residue-residue interactions in biomolecular simulation. By identifying how frontier states of an active site are altered through the presence of an additional QM residue, the FSA method enables the user to identify when QM treatment of a residue is essential. This method provides new insight into the chemical mechanism and origin of stabilization by the greater enzyme environment in addition to providing a systematic route to quantitatively converged QM/MM energetics. In these cases,large FSA values suggest the residue quantum-mechanically affects reactivity at the active site.

We first carried out quantitative tests of how distance and residue identity altered the magnitude of Fukui function shifts in representative residue pairs. We identified that strong hydrogen bonds between an aspartate and asparagine sidechain caused the electrophilic condensed Fukui function on the aspartate to deviate by more than 0.5e from the value (i.e., -1) of aspartate in isolation. This deviation decayed rapidly from the 1.43 ? inter-residue distance in the crystal structure to reach the isolated limit around 4-5 ?, revealing the spatial extent to which protein residues may likely alter each other’s respective electronic environments. Comparison of numerous aspartate-residue pairs(i.e., with serine, asparagine, histidine, or arginine) revealed a hydrogen-bond-donor-specific deviation in the electrophilic condensed Fukui function.

We also evaluated how interpretation of the Fukui function as a measure of relative nucleophilicity provides insight on representative enzymes that carry out SN2 methyl transfer. We identified enzyme-specific and reaction-specific deviations in the Fukui functions of the co-substrate SAM and reacting nucleophile. Overall, the FSA method represents a promising approach to invoke conceptual density functional theory for both the systematic identification of quantum mechanical effects in enzymes and for mechanistic interpretation of environmental effects on reactivity.

Acknowledgment: The authors acknowledge Adam H.Steeves for a critical reading of the manuscript.

Supporting Information: A PyMOL session file of all of the relevant structures in this paper is available free of charge via the internet at http://www.whxb.pku.edu.cn.

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