Enzymatic Catalysis: The Emerging Role of ... - ACS Publications

J. Phys. Chem. B 2009, 113, 13465–13475

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REVIEW ARTICLE Enzymatic Catalysis: The Emerging Role of Conceptual Density Functional Theory Goedele Roos,*,†,‡,§,| Paul Geerlings,|,¶ and Joris Messens†,‡,§ Department of Molecular and Cellular Interactions, VIB, Pleinlaan 2, 1050 Brussels, Belgium, Structural Biology Brussels, Vrije UniVersiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium, Brussels Center for Redox Biology, Pleinlaan 2, 1050 Brussels, Belgium, and General Chemistry, Vrije UniVersiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium ReceiVed: April 15, 2009; ReVised Manuscript ReceiVed: June 30, 2009

Experimentalists and quantum chemists are living in a different world. A wealth of theoretical enzymologyrelated publications is hardly known by experimentalists, and vice versa. Our aim is to bring both worlds together and to show the powerful possibilities of a multidisciplinary approach to study subtle details of complicated enzymatic processes to a broad readership. MD simulations and QM/MM approaches often focus on the calculation of reaction paths based on activation energies, which is a time-consuming task. A valuable alternative is the reactivity descriptors founded in conceptual DFT like softness, electrophilicity, and the Fukui function, which describe the kinetic aspects of a reaction in terms of the response to perturbations in N and/or υ(r), typical for a chemical reaction, of the reagents in the ground state. As such, the relative energies at the beginning of the reaction predict a sequence of activation energies only based on the properties of the reactants (Figure 5). In 2003, Geerlings et al. published a key review giving a detailed description of the principles and concepts of conceptual DFT and highlighting its success to study generalized acid/base reactions including addition, substitution, and elimination reactions. Since the time that this review appeared, conceptual DFT has proven its strength in literally hundreds of papers with application to organic and inorganic reactions. Its role in unravelling enzymatic reaction mechanisms, in handling experimentally difficult accessible biochemical problems, and in the interpretation of biochemical experimental observations is emerging and very promising. Introduction Although studied for decades, enzymatic catalysis remains one of the most intriguing biochemical phenomena. Traditionally, enzymatic catalysis is studied with X-ray crystallography, NMR, kinetic assays, site-directed mutagenesis, and isotope effects.1,2 On the other hand, computer modeling provides a powerful arsenal that can be applied to study subtle details of complicated enzymatic processes.3 Nowadays, computational chemistry has reached an accuracy by which it can compete with experimental methods for small and medium sized molecules, up to 20 atoms, in predicting structural, energetical, and spectroscopical properties and reactivities.4 For larger systems such as proteins, zeolites, fullerenes, and nanotubes, not only are trends often well reproduced but also sometimes amazing agreement between theory and experimental data is obtained.5 Common computational techniques to study the structure and reactivity of enzymes are molecular dynamics (MD) simulations and transition state calculations via combined quantum mechanics (QM)-molecular mechanics (MM) or via a full QM approach.3,6-12 These methods focus on finding * Corresponding author. E-mail: [email protected]. Phone: +32-2-629 19 92. Fax.: +32-2-629 19 63. † VIB. ‡ Structural Biology Brussels, Vrije Universiteit Brussel. § Brussels Center for Redox Biology (http://redox.vub.ac.be/). | General Chemistry, Vrije Universiteit Brussel. ¶ Member of the QCMM Alliance Group Ghent-Brussels.

reaction paths by the computation of activation energies. A computationally less demanding method uses the reactivity descriptors founded in conceptual density functional theory (DFT).13,14 This method describes the preferred reaction energetics and thus kinetics in terms of the properties of the reagents in the ground state and is a successful tool to gain insight into how enzymes work. Conceptual DFT looks at the perturbation of the energy13,14 upon changing the number of electrons or the external potential, i.e., the potential felt by the electrons due to the nuclei. Conceptual DFT has been successfully used to study generalized acid/base reactions, including most of the organic reactions (additions, substitutions, eliminations)14b and the inorganic complexation reactions, and recently also redox reactions15 and pericyclic reactions.16 In this review, we show that conceptual DFT can give an extra dimension to the understanding of enzymatic catalysis, where the same archetypes of reactions mentioned above are encountered, especially to solve experimentally unreachable problems and to interpret experimental observations. In the field of enzymology, there is still an anxious fear to fully integrate conceptual DFT. Therefore, a review article, in which conceptual DFT is presented to a broad audience as a tool to gain insight in the subtle details of enzymatic reaction mechanisms, is not ahead of time, although the literature on protein-related applications of conceptual DFT is still limited. By discussing the described examples,17-21 we would like to show to nonspecialists the beauty and added value

10.1021/jp9034584 CCC: $40.75  2009 American Chemical Society Published on Web 09/15/2009

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Goedele Roos (born in 1980) has been working since 2007 as a postdoctoral fellow at the Fund for Scientific Research (FWO) affiliated with the Free University of Brussels (VUB). She obtained her Ph.D. at the VUB in 2007, working on the border between theory and experiment in two groups: the Structural Biology Brussels laboratory involved in structural enzymology and the Quantum Chemistry group active in fundamental and applied aspects of density functional theory (DFT). She is the author or coauthor of more than 10 research publications integrating DFT, biochemistry, and structural biology to gain full insight into enzymatic reaction mechanisms. As a member of the Brussels Center for Redox Biology, her present work focuses on proteins that are involved in thiol-based catalytic mechanisms and oxidative protein folding.

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Joris Messens (born in 1962) is staff scientist and head of the Redox Regulation research group within Flanders Institute for Biotechnology (VIBVrije Universiteit Brussel). He created the Brussels Center for Redox Biology as a platform to stimulate redox and oxidative protein folding research in Belgium. After several years in the biotech industry, he became an expert in protein purification. He obtained his Ph.D. from the John Moores University of Liverpool (U.K.) with structural and functional work on arsenate reductase from Staphyloccocus aureus. He is the author or coauthor of more than 30 peer-reviewed publications with a central theme of thiol/disulfide exchange mechanisms.

The introduction of the electron density combined with an orbital ansatz by Kohn and Sham in an ingenious computational scheme, resulting in a set of Hartree-Fock-like equations,30 leads to a better quality/cost ratio when evaluating molecular properties.4 Further, many traditional concepts such as electronegativity, hardness, and softness were sharply defined and, as such, numerically evaluated in a DFT context, offering a framework to quantitatively study chemical reactivity.14 This sub-branch of DFT has been called “conceptual DFT” by its founding father, R. G. Parr,13b and although of course the fundamental DFT (basics and computational aspects) evidently involve lots of (sometimes intricate) concepts, the name of the subfield has not changed. “Chemical DFT” might be a valuable alternative. In the following paragraph, we will discuss these DFT-based concepts. Paul Geerlings (born in 1949) is full Professor at the Free University of Brussels (Vrije Universiteit Brussel) where he obtained his Ph.D. and Habilitation, heading a research group involved in conceptual and computational DFT with applications in organic, inorganic, and biochemistry. He is the author or coauthor of about 340 publications in international journals or book chapters. In recent years, he organized several meetings around DFT: in 2003, he was the chair of the Xth International Congress on the Applications of DFT in Chemistry and Physics (Brussels, September 7-12, 2003), and in 2006, he chaired “Chemical Reactivity” also in Brussels.

DFT-Based Concepts

The Basic Principles of DFT

The energy E is a functional (E ) E[N, υ(r_)]) of the number of electrons N and of the external potential υ(r_), i.e., the potential felt by the electrons due to the nuclei. A functional is a relation that takes a function as input and returns a scalar, with a given υ(r_) function and N corresponds a particular E. The energy of a system can be perturbed by changing N and/or υ(r_), mathematically expressed as a Taylor expansion of the E ) E[N, υ(r_)] functional. This leads to the response functions (Figure 1), giving the response of the system on perturbations in N and/or υ(r_), occurring at the microscopic level during a chemical reaction (Figure 2). These response functions are

Quantum chemistry mainly concentrates on Schrodinger’s time independent equation (SE) HΨ ) EΨ (where Ψ is the wave function, E the energy, and H the Hamiltonian) of which the exact solution is only accessible for a single electron system.22,23 Ψ is an immensely complicated function of the space and spin coordinates of all electrons. For many electron systems, several methods are available to approximate a solution of this complicated SE equation.5,22-25 In recent years, the DFT approach has gained importance. On the basis of the Hohenberg and Kohn theorems,26 DFT considers the electron density function F(r_), a function of only three spatial coordinates, as the carrier of all information of the system it describes.13,27-29

Figure 1. Response functions that quantify the response of a system to the perturbations in the number of electrons N and/or the external potential υ(r_).

that conceptual DFT can bring to explain reaction mechanisms, which are beyond the reach of experimentalists.

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µ)-

IE + EA 2

(1)

in which IE and EA indicate the vertical ionization energy and electron affinity, respectively. Exact theories show that the E ) E(N) curve is a series of straight line segments introducing a discontinuity for the derivative33 at integer N values (for a perspective, see ref 34). The finite difference approximation for η and S is given by Figure 2. Formula box: Energy derivatives and response functions. δnE/∂Nmδυ(r_)m′ (n e 2).

η)

IE - EA 2

(2)

S)

1 IE - EA

(3)

and

Figure 3. E versus N plot indicating the relation between the ionization energy (IE), electron affinity (EA), and electrophilicity (ω). N0 represents the neutral reference system with N electrons, N - 1 represents the corresponding cation, and N + 1 the corresponding anion. The minimum of the curve represents Nideal, the reference system which has taken up an ideal amount of electrons.

isolated system properties and serve as reactivity descriptors. The simplest cases are the response functions when only the number of electrons changes (∂nE/∂Nn)υ(r_) at constant external potential, i.e., constant molecular geometry. The first derivative of the energy with respect to the number of electrons is the chemical potential µ (Figure 2). µ equals the negative of the electronegativity χ and measures the tendency of an electron to escape from the electronic cloud.13,31 This quantity bridges the fundamentals of DFT (and so computational DFT) with conceptual DFT, as it appears in the variational equation of DFTsthe road to the “best” density by minimizing the energysas the Lagrangian multiplier introduced by the demand that the electron density should at all times be normalized to the numbers of electrons. The second derivative of the energy with respect to the number of electrons is the global hardness η (Figure 2). η is the resistance of the system to changes in the number of electrons.13,32 For a long time, no methods for quantifying hardness were available. A breakthrough was reached by Parr and Pearson,32 identifying the chemical hardness as the second derivative of the energy with respect to the numbers of electrons, evaluated at a fixed molecular geometry. This derivative can be approximated as the difference between the vertical ionization energy (IE) and the electron affinity (EA) (Figure 3). Assuming a quadratic relationship between the energy E and the number of electrons N (Figure 3), the finite difference approximation to µ for a system can be written as13,14

As such, the experimental and quantum chemical determination of the hardness was made possible. The inverse of the global hardness is the global softness S.35 S correlates with the system’s polarizability.36 χ, η, and S describe a quantity of the molecule as a whole, i.e., a global quantity of the ground state. Differentiation of the energy with respect to υ(r_) introduces a local character into the global reactivity descriptors, resulting in the local hardness and softness, and the Fukui function (Figure 2).37 The Fukui function f(r_) indicates the regions where the molecule preferentially reacts (regioselectivity). f(r_) is related to the frontier orbital (highest occupied and lowest unoccupied molecular orbital) electron densities and can be considered as a generalization of Fukui’s molecular orbital concept.38 In a finite difference approximation and for a system of N0 electrons, the Fukui functions indicating the reactivity toward an electrophilic and nucleophilic attack are given by eqs 4 and 5, respectively:13,14

f -(r_) ≈ FN0 - FN0-1

(4)

f +(r_) ≈ FN0+1 - FN0

(5)

Note that this result was shown to be exact in exact DFT theory.39 When integrating the Fukui function over atomic regions, one finds the condensed Fukui functions for the nucleophilic and electrophilic attack on atom A:

f A- ) qA(N0) - qA(NO - 1)

(6)

f A+ ) qA(N0 + 1) - qA(NO)

(7)

qA(N0), qA(N0 + 1), and qA(N0 - 1) are the electronic populations for atom A in the neutral molecule (N0 electrons) and the corresponding anion (N0 + 1) or cation (N0 - 1), all evaluated at the geometry of the neutral molecule or more generally at the geometry of the N0 electron system (cf. the demand for constant external potential). For practical applications, the common way to calculate the functional derivatives involving differentiations with respect to

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Figure 4. (a) Global form of the HSAB principle: the interaction between acid A and base B is optimal when SA ) SB. (b) Local form of the HSAB principle: the interaction between the acidic part k of molecule A and the basic part l of molecule B is optimal when sk ) sl.

the number of electrons (Figure 2) is by using a finite difference approximation. Today, methods are being developed to calculate these response functions by avoiding the differentiation with respect to the numbers of electrons.40 Although, for example, promising results are obtained for the calculation of the Fukui function from the change in Kohn-Sham orbital energies when the external potential is perturbed,40 these new techniques are computationally still very expensive and not applicable yet to large systems involved in enzyme catalysis. The local softness s(r_) can be obtained by the multiplication of the global softness and the Fukui function and indicates the reactivity of a molecular center toward a comparable site in another molecule.35 The local hardness is not the inverse of the local softness,13 and its nature is still a matter of debate.41 Among others, the electronic charge or molecular electrostatic potential are often used as approximate expressions of the local hardness.42,43 The electrophilicity concept ω44 (Figure 3) is given by a combination of the hardness and the chemical potential (ω ) µ2/8η). ω indicates the stabilization energy when a ligand takes up an ideal number of electrons coming from an idealized sea of electrons with zero hardness and zero chemical potential.45 ω measures the reactivity toward an attack by a negatively charged nucleophile. HSAB Principle: An Example of “To Compute Is Not to Understand” Within the context of conceptual DFT, Parr, Lee, and Chattaraj46 presented evidence for the hard and soft acids and bases (HSAB) principle, originally formulated by Pearson.47 The HSAB principle states that hard acids prefer to react with hard bases and soft acids prefer to interact with soft bases, all other factors being equal. As such, the preferred interaction between the reaction partners can be rationalized through the difference in softness of the interacting parts (atoms, functional groups), which should be minimal for optimal interaction (Figure 4). When molecule A with an acidic region k interacts with molecule B with a basic region l, the interaction between k and l is optimal if their local softnesses match as close as possible. This is known as the local version of the HSAB principle48-50 (Figure 4). Using the HSAB principle, the preferred interaction can be described in terms of the properties of the reagents in the ground state. Assuming that reaction paths of similar reactions will not

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Figure 5. Noncrossing of the reaction paths: relation between the softness matching result ∆s and activation energy (E#). The relationship between the initial slope of the reaction profile, activation energy, and softness matching in the case of similar soft-soft interactions is shown.

cross according to Klopman’s rule,51 the relative slopes of two or more similar reaction paths estimated on the basis of the response functions at the beginning of the reaction correlate with the relative energies in the transition state (Figure 5). As a result, no transition states and activation energies should be calculated. Transition state calculations give the exact activation energy but not necessarily insight into the reaction mechanism, which can sometimes easily be obtained from the numerical value of the various response functions (Figure 2). This goes along with Parr’s dictum: “To compute is not to understand”.52 In several case studies, we will show how such an approach can be followed in enzymatic catalysis. We will concentrate on local examples to disentangle, e.g., regioselectivity problems. How to Apply on Enzymatic Catalysis Phosphoenolpyruvate, the Cofactor of Several Synthases. The oldest example of the application of the HSAB principle to understand enzymatic catalysis can be found in the paper of Li and Evans.53 In this contribution, the chemical reactivity of the C-C double bond (C2-C3 in Figure 6a) of phosphoenolpyruvate (PEP) could be explained in terms of the HSAB principle. PEP is a cofactor for several enzymes, i.e., 5-enolpyruvylshikimate 3-phosphate (EPSP) synthase, an enzyme of the shikimate pathway in plants, and 3-deoxy-D-manno-2-octulosonate-8-phosphate (KDO8P) synthase, an enzyme of the glycolysis.54-57 EPSP synthase has an inverse alpha/beta barrellike structure containing two globular domains composed of beta sheets and alpha helices.58 Ligand binding converts the enzyme from an open state to a tightly packed closed state (Figure 6b).58,59 EPSP synthase is involved in aromatic acid biosynthesis,60 transferring the enolypyruvyl group of PEP to shikimate3-phosphate to form 5-enolpyruvyl-3-shikimate phosphate and inorganic phosphate. During this event, the C3 atom becomes protonated54,55 (Figure 6a) and acts as a hard base. On the other hand, in the reaction catalyzed by KDO8P synthase (Figure 6c), this C3 carbon acts as a soft base. Here, C3 attacks the aldehyde carbon atom of D-arabinose-5-phosphate to form 3-deoxy-D-manno-2-octulosonate-8-phosphate (Figure 6d).57 This reaction plays a crucial role in the biosynthesis of an unusual eight-carbon sugarm 3-deoxy-D-manno-2-octulosonate, an important constituent of lipopolysaccharide of most Gram-negative bacteria.61,62 For a long time, the dual reactivity of C3sonce as hard and in another condition as soft baseswas puzzling and experimental studies were unable to nail down the cause of this phenomenon. Conceptual DFT identified the C3 atom as a hard or soft base depending on the ionization state of PEP and on the conforma-

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Figure 6. (a) Schematic representation of the reaction catalyzed by 5-enolpyruvylshikimate 3-phosphate (EPSP) synthase. S3P, shikimate-3phosphate; Pi, inorganic phosphate. (b) Ribbon diagram of the overall structure of the closed conformation of EPSP synthase (PDB: 1G6S) with bound shikimate-3-phosphate (red), and inhibitor, glyphosate (green). (c) Ribbon diagram of the overall structure of KDO8P synthase (PDB: 2nwr) with bound phosphoenolpyruvate (PEP) (red). (d) Schematic representation of the reaction catalysed by 3-deoxy-D-manno-2-octulosonate-8-phosphate (KDO8P) synthase. Ara5P: D-arabinose-5-phosphate.

tion of the dihedral angle between the carboxylate and the C2-C3 double bond. The charge and Fukui function of the C3 atom change when the conformation of the molecule varies. For mono- (PEP1-) and trianionic PEP (PEP3-) (Figure 7), the negative charge and the Fukui function at C3 is minimal when the dihedral angle is close to 0° (Table 1), indicating a hard site at C3. When the dihedral angle is around 90°, the charge and Fukui function at C3 are maximal (Table 1), indicating a soft site. For dianionic PEP (PEP2-), C3 is a hard site when the dihedral angle is around 30° and a soft site when the dihedral angle is about 90° (Table 1). As such, the HSAB theory predicts a favored reaction with the hard proton acid when the C3 atom

behaves like a hard base and with the soft aldehyde carbon when the C3 atom acts as a soft base. The Disulfide Cascade Mechanism in Arsenate Reductase. Here, conceptual DFT scrutinizes the experimental results of the enzymatic reaction mechanism of Staphylococcus aureus arsenate redcuctase (ArsC).18,20,21 ArsC is part of the arsenic defense mechanism of the cell and catalyzes the reduction of arsenate to arsenite in a multistep disulfide cascade reaction (Figure 8).63 ArsC was extensively studied by various experimental techniques including X-ray crystallography, NMR, and kinetic assays.63-66 Three cysteines (Cys10, Cys82, and Cys89) have been pinpointed as the redox-active cysteines,65 and the

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Figure 7. Identification of the C3 atom of phosphoenolpyruvate (PEP) as a hard or soft base depending on the ionization state of PEP and on the conformation of the dihedral angle between the carboxylate and the C2-C3 double bond. Monoanionic (PEP1-) and trianionic (PEP3-) with a dihedral angle between the carboxylate and the C2-C3 double bond of 0° (C3 is a hard center) and 90° (C3 is a soft center). Dianionic PEP (PEP2-) with a dihedral angle between the carboxylate and the C2-C3 double bond of 30° (C3 is a hard center) and 90° (C3 is a soft center). Color code: carbon, gray; hydrogen, white; phosphor, orange; oxygen, red.

TABLE 1: Charge (q) and Fukui Function (f) of PEP at Different Ionization States and Dihedral Anglesa PEP-1

PEP-2

PEP-3

angle

q

f-

q

f-

q

f-

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0

-0.3313 -0.3356 -0.3432 -0.3520 -0.3492 -0.3531 -0.3864 -0.3918 -0.3948 -0.3955

0.4053 0.4143 0.4350 0.4575 0.4461 0.4617 0.4652 0.4765 0.4839 0.4910

-0.4459 -0.3180 -0.3976 -0.4037 -0.4109 -0.4183 -0.4250 -0.4779 -0.4804 -0.4809

0.4823 0.4325 0.4172 0.4132 0.4175 0.4277 0.4434 0.4984 0.4809 0.4927

-0.4337 -0.4341 -0.4356 -0.4488 -0.4555 -0.4620 -0.4666 -0.4695 -0.4707 -0.4697

0.4974 0.4962 0.4950 0.5235 0.5346 0.5469 0.5581 0.5678 0.5739 0.5744

a PEP-n stands for phosphoenolpyruvate in different ionization states (Figure 7). “angle” refers to the dihedral angle between the carboxylate and the C2-C3 double bond (Figure 7). Data reproduced from ref 53.

residues Thr11, Arg16, and Ser17 were found to be crucial for catalysis.20,66 However, the limitations of today’s biochemical approaches left a number of questions unsolved. First, the protonation state of the enzyme substrate complex is experimentally beyond reach due to the inherent instability of arsenate esters.67 Second, the experimental determination of the pKa of the catalytically important thiol groups (Cys10, Cys82, and Cys89) is not straightforward, because these redox-active cysteine residues are all involved in the successive steps of the reaction mechanism.63

Figure 8. (1) The reaction starts with the nucleophilic attack of Cys10 on arsenate leading to a covalent enzyme-arseno intermediate. (2) Arsenite is released after the nucleophilic attack of the thiol of Cys82. A Cys10-Cys82 intermediate is formed, and the redox helix partially unfolds. (3) At the end of the reduction cycle, Cys89 attacks Cys82, forming a Cys82-Cys89 disulfide. The redox helix is looped out and presents the disulfide bridge at the surface of the enzyme to thioredoxin. (4) Thioredoxin (Trx) regenerates the reduced form of arsenate reductase for a subsequent catalytic cycle. (Reproduced from ref 21 with permission. Copyright 2009 Wiley-VCH Verlag GmbH & Co. KGaA.)

TABLE 2: Reactivity between Arsenate and Thiolatea ∆s (au) -

CH3S (gas phase) CH3S- (ε ) 20.7)

H2AsO4-

HAsO42-

3.25 2.49

0.37 1.30

a Differences in local softness (∆s) between As and S calculated using ∆s(r) ) |s+(As) - s-(S)| in gas phase and solvent (ε ) 20.7), as a model for the enzymatic environment at the B3LYP/6-31+G** level. Data reproduced from ref 68.

Cys10 Preferentially Attacks Dianionic Arsenate. The first step in the multistep catalytic mechanism of ArsC consists of a nucleophilic displacement reaction carried out by Cys10 on arsenate (Figure 8, step 1).63 The protonation state of enzymebound substrate could not experimentally be obtained due to the instability of arsenate esters.67 Therefore, the HSAB principle is used to assess the protonation state of enzyme-bound arsenate. The difference in local softness between the attacking nucleophilic sulfur atom of Cys10 and the receiving electrophilic arsenic atom of arsenate is minimal in the case that dianionic arsenate was considered (Table 2).68 As such, the nucleophilic attack of Cys10 on dianionic arsenate is favored over the attack on monoanionic substrate. At first glance, the result seems to be rather strange, since there is an unfavorable Coulomb repulsion between the dianionic substrate and the proximal nucleophilic thiolate in the active site of ArsC. However, calculation of the natural population analysis charge showed the transfer of 0.3 negative charge units from arsenate to ArsC by the numerous enzyme-substrate interactions, reducing the electrostatic repulsion. In addition, calculation of the binding energy showed that the binding of dianionic arsenate in ArsC turned out to be 82 kcal/mol more favorable than that of monoanionic arsenate.18

Review Article The Cys82 and Cys89 Thiolates Are Stabilized in ArsC. The pKa’s of Cys82 (second reaction step, Figure 8) and Cys89 (third reaction step, Figure 8) could not be experimentally obtained because all redox-active residues are involved in the successive steps of the reaction mechanism.63 On the basis of the correlation between the natural population analysis (NPA) charge on the sulfur of the thiolate compound and the pKa,20,69 we were able to estimate the pKa of Cys82 and Cys89. The correlation between the pKa and the NPA charge can be explained in terms of the HSAB principle. The pKa indicates the reactivity of a residue toward protonation, which implies a hard proton, preferring a hard interaction partner. Since hard-hard interactions are mostly electrostatic in nature, the atomic charge as an approximation of the local hardness41-43 can act as a reactivity descriptor in a reaction toward a proton. Not only the pKa’s of Cys82 and Cys89 could be calculated, but also the structural elements influencing the pKa of the thiolates were identified.20 The pKa of Cys82 is calculated to be 5.0, which is 3.3 units lower than the pKa of free cysteine. Cys82 is stabilized as a thiolate by an eight-residue-long R-helix N-terminal flanked by Cys82 and C-terminal by Cys89 (called the redox helix) and the Cys82-Thr11 hydrogen bond.20 Further, the presence of arsenate in the active site of ArsC brings Arg16 within hydrogen bonding distance to Cys82, leading to an additional pKa decrease of 0.95 pKa units.70 Thus, the substrate itself contributes to the nucleophilic character of Cys82. The pKa of Cys89 located at the C-terminal of the redox helix is 2.7 units higher than the pKa of free cysteine. After the formation of the Cys10-Cys82 disulfide bridge in the second reaction step (Figure 8), the redox helix partially unfolds. The resulting absence of the helix macrodipole lowers the pKa of Cys89 with 3.4 units compared to the pKa of free cysteine, generating the nucleophilic thiolate form.21 The Mixed Disulfide Dissociation Mechanism of Thioredoxin. The mechanism behind the dissociation of the mixed disulfide complexes between thioredoxin (Trx) and its substrates is unknown and has already been debated for 20 years.71-78 Thioredoxins are ubiquitous reductases that control several essential functions of life, including promotion of cell growth, inhibition of apoptosis, and modulation of inflammation. All thioredoxins have a similar three-dimensional fold comprising a central core of four β-strands surrounded by three R-helices.79 All feature a conserved active-site loop containing two redoxactive cysteine residues in the sequence Cys1-Gly-Pro-Cys2 (Figure 9a). To break a disulfide bond in a substrate, first, a disulfide bond between the substrate and Cys1 of thioredoxin is formed. Second, this intermediate mixed disulfide complex is dissociated by Cys2. The mechanism by which Cys2 is activated to dissociate the mixed disulfide was studied using the first structural data available of a thioredoxin-protein complex between Trx from B. subtilis (with active-site numbering: Cys29-Gly30-Pro31-Cys32) and its endogenous substrate, arsenate reductase (ArsC) (PDB code: 2IPA).80 In this mixed disulfide structure, a Cys29Trx-Cys89ArsC intermediate disulfide (Figure 9b) is formed by the nucleophilic attack of Cys29Trx on the exposed Cys82ArsC-Cys89ArsC in oxidized ArsC. It has been suggested that Asp23Trx and Cys82ArsC were the residues responsible for deprotonation of Cys32Trx.72-78 This assumption was ruled out by experimental tests via complex formation studies where wild type Trx and Trx D23A were combined with triple cysteine mutants of ArsC.22 Molecular dynamics (MD) simulations have shown that Cys32Trx is stabilized and activated as a nucleophile by hydrogen bonds, while Cys32Trx is approaching Cys29Trx within contact distance via conformational

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Figure 9. (a) A ribbon diagram of the overall structure of thioredoxin (Trx) (PDB: 1XOB) with the active site Cys1-Gly-Pro-Cys2 shown in atom type. (b) Part of the structure of the mixed disulfide complex between thioredoxin and arsenate reductase (PDB: 2IPA).

TABLE 3: Reactivity Analysis of the Bs_Trx-ArsC Complex Dissociation: (A) Global Softness (S), Fukui Function (f + or f -) and Local Softness (s+ or s-), Obtained as sk( ) f k(S, of the Sulfur Atoms of the Nucleophilic Cysteines in the Bs_Trx-ArsC Complex; (B) Reactivity of Cys32Trx and Cys82ArsC toward the Cys29Trx-Cys89ArsC Mixed Disulfide as Measured by the Difference in Local Softness (∆s)a (A) Bs_Trx-ArsC complex (2IPA)

cysteine residue

S

f +/f -

s+/s-

Cys32Trx Cys29Trx Cys89ArsC Cys82ArsC

6.47 5.97 5.97 8.59

0.865 0.522 0.122 0.856

5.59 3.12 0.73 7.33

(B) ∆s

Cys32Trx/ Cys29Trx

Cys32Trx/ Cys89ArsC

Cys82ArsC/ Cys29Trx

Cys82ArsC/ Cys89ArsC

2.48

4.86

4.21

6.60

a

All quantities are obtained at the B3LYP/6-31+G** level of theory. Data reproduced from ref 21.

changes at the interface of the Cys29Trx-Cys89ArsC mixed disulfide. MD or experimental studies alone could not scrutinize the mechanism behind the experimentally observed regioselectivity. Information regarding the selectivity of the nucleophilic attack was obtained from a DFT reactivity analysis. In the Trx-ArsC complex, four possible reactions between the attacking nucleophilic cysteines (Cys32Trx and Cys82Trx) and the accepting electrophilic disulfide (Cys29Trx-Cys89ArsC) can be considered. The minimal local softness difference favors the nucleophilic attack of Cys32Trx on Cys29Trx (Table 3).21 The Cys29Trx-Cys89ArsC disulfide is less soft than Cys32Trx and Cys82ArsC, and Cys32Trx is less soft than Cys82ArsC. As such, the high reactivity of Cys32Trx toward the Cys29Trx-Cys89ArsC disulfide is consistent with the lower softness of Cys32Trx compared to Cys82Trx. On the basis of the Fukui function, it was found that Cys29Trx was more susceptible to nucleophilic attack than Cys89ArsC.

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Roos et al. TABLE 4: Local Softness sk( and Group Softness sG( of the Reactive Centers in Nicotinamide and Lumiflavinea molecule

site (k)

s( k

nicotinamide

C-2 C-3 C-4 C-5 C-6 Ht C-4 + C-5 + C-6 Ht + C-5 + C-4 C-2 + C-3 + C-4 Ht + C-4 + C-3 C-3 + C-4 + C-5 C-3 + Ht + C-5 N-1 C-4a N-5 C-5a C-5a + N-5 + N-1 C-4a + N-5 + C-5a C-4a + N-5 + N-1

0.226 1.575 0.013 1.117 0.256 0.256 sG( ) 1.452 sG( ) 1.452 sG( ) 1.880 sG( ) 1.910 sG( ) 2.771 sG( ) 2.948 0.212 1.159 2.048 0.000 sG( ) 2.260 sG( ) 3.207 sG( ) 3.409

lumiflavine

a The group softness including more than one atomic center k is ( ( obtained via sG( ) ∑k∈G s( k with sk ) fk S at the B3LYP/6-31G level of theory. See Figure 10 for the identification of the atomic centers. Data reproduced from ref 17.

TABLE 5: Local Electrophilicity of the Reactive Centers in Lumiflavinea

Figure 10. (a) Electron transfer in flavoproteins. Substrate is reduced by electrons coming from flavin, leading to oxidized flavin which on its turn is kept reduced by NAD(P)H. (b) Hydride transfer reaction between lumiflavine (part of flavin) and nicotinamide (part of NAD(P)H) in flavoenzymes.

Hydride Transfer in Flavoenzymes. Flavoenzymes are a widespread and versatile family of redox enzymes, using flavins (riboflavin, flavin mononucleotide (FMN), flavin adenine dinucleotide (FAD)) as cofactors. They are involved in a large number of biological functions, among which are dehydrogenation of amino acids, mediation of one- and two-electron transfer to heme groups, activation of molecular oxygen leading to a flavohydroperoxide, DNA repair, and several photochemical processes.81-84 The same enzyme is able to catalyze reactions which vary widely from a mechanistic point of view.81 This feature of flavoenzymes discriminates them from most other enzymes using cofactors, which, in general, each catalyze a single type of chemical reaction.81 In the majority of the flavoproteins, a direct transfer of electrons takes place between the substrate and the flavin. In a next step, the flavin is rereduced or reoxidized by molecular oxygen or NAD(P)+/NAD(P)H.85 Here, we consider the reductive and oxidative half reactions in the case that substrate is reduced by flavin, which on its turn is rereduced by NAD(P)H (Figure 10a). The reactivity of flavins is restricted to the lumiflavine part and the reactivity of NAD(P)H to the nicotinamide ring (Figure 10b). A direct electron transfer in the form of a hydride ion takes place between nicotinamide and lumiflavine. Although the stereochemistry of the reaction was known for years and supported by crystallographic data,86-88 different mechanisms have been proposed to account for the hydride transfer:89-91 (1) transfer of a hydride ion in a single step, (2) two step mechanism via an electron transfer followed by a hydrogen atom transfer or vice versa, and (3) transfer of two

ω+ k

N-1

C-4a

N-5

lumiflavine lumiflavine protonated at N-1 lumiflavine protonated at N-5

0.087 / /

0.475 0.820 4.464

0.839 2.983 3.531

a + The local electrophilicity of atom k is obtained via ω+ k ) fkω at the B3LYP/6-31G level of theory. See Figure 10 for the identification of the atomic centers. Data reproduced from ref 17. /: not determined.

electrons and a proton in three separate steps. In addition, the regioselectivity of the hydride transfer was not known. On the basis of the similar group softness,92 conceptual DFT studies have led to the suggestion of a direct hydride transfer between the reactive regions in nicotinamide and lumiflavine. The group softness was calculated as the sum of the local softnesses of the involved atomic centers. Different atomic centers of lumiflavine and nicotinamide were tested, but the smallest difference in group softness was found between the C-3, Ht, and C-5 atoms of nicotinamide and the C-4a, N-5, and N-5 atoms of lumiflavine, supporting the hydride transfer between these regions (Figure 10b and Table 4).17 As such, the mechanism of the hydride transfer process can be understood in the context of the HSAB principle as an interaction between soft species. The regioselectivity of the reaction could be assessed by the electrophilicity. In the lumiflavine molecule, the local electrophilicity of the N-5 atom is higher than the local electrophilicity of the N-1 and C-4a atoms. As such, the N-5 atom will most likely receive the hydride ion. When N-1 is protonated, the local electrophilicity of N-5 increases almost 4-fold, while, when N-5 is protonated, the electrophilicity of C-4a increases 10 times (Table 5). As such, protonation of N-1 leads to hydride transfer to C-4a via N-5. QSAR Combined with DFT Concepts in Histone Deacetylase Inhibitors. Histone deacetylase (HDAC) inhibitors are very promising targets as anticancer drugs93,94 and play a key role in the treatment of fibrotic, autoimmune, inflammatory, and polyglutamin diseases.95-99 For the selection of new

Review Article

Figure 11. Ribbon diagram of the active site of human histone deacetylase (HDAC8) (PDB: 1W22) with a bound hydroxamic acid inhibitor.

synthetic HDAC inhibitors, quantitative structure activity relationship (QSAR) models provide guidelines. Most of the QSAR analyses are qualitative in nature with modest interpretive power and predictive potential.100 To increase the interpretive character and to broaden the applicability of the QSAR models, the electrophilicity as a descriptor is inserted into QSAR models.100 In HDAC, the chemical hardness was shown to be a second useful descriptor in QSAR.20 HDCAs remove acetyl groups of N-acetyl-lysine on histones101 and play as such a role in gene expression. Histones can interact tightly with the negatively charged phosphate groups of DNA via the positively charged amine groups present on the lysine and arginine residues of their tails. Acetylation of the NH3 groups in Lys and Arg neutralizes this positive charge, by which the histone becomes unable to interact with DNA. HDACs remove this acetyl groups and thereby encourage the histone-DNA interaction by which the condensed and transcriptionally silenced chromatin is formed.101,102 The active site of HDACs comprises a coordinated Zn2+ ion. HDAC inhibitors bind in the active site and as such coordinate the zinc ion (Figure 11). The hardness of the zinc binding groups of HDAC inhibitors correlates well with the calculated relative interaction energies of these inhibitors.19 This relative interaction energy is calculated as the difference in interaction energy between a hydroxide and an inhibitor bound in the active site. With the expansion of QSAR with the electrophilicity of the reacting agents, inhibitors can be more efficiently screened. Outlook Quantum chemistry, DFT in particular, offers the chemist and biochemist a variety of techniques to accurately calculate activation energies. In order to trace back both the theoretical and experimental results to the properties of the reactants, DFT offers a complete toolbox with response functions indicating how a molecule responds to changes in the number of electrons or the external potential, both quantities being essential variables in a chemical reaction.14 In this way, the time-consuming explicit computation of the activation energies can be avoided. Moreover, chemical insight is provided in the reaction mechanism. In this review, different examples show the vitality of conceptual DFT in enzymatic catalysis. In these examples, the focus lies on the local HSAB principle to answer experimentally difficult to handle questions. In addition, results obtained from computational studies can lead to the design of new experiments, such as site directed mutagenesis, kinetic, X-ray, and NMR experiments (see journal cover), by which full insight into the enzymatic reaction is gained. In view of the complexity of biochemical systems, a general methodology is presented starting from a model. In

J. Phys. Chem. B, Vol. 113, No. 41, 2009 13473 this model, compromises are made between computational cost and accuracy. Often, only the active site is taken into account. Both X-ray and NMR structures are used as the starting point for the DFT calculations and as such no link between structure and dynamics is made. To implement the entire enzyme and structural dynamics, conceptual DFT studies should be combined with techniques as quantum mechanics/molecular mechanics (QM/MM) and molecular dynamics (MD) simulations in the future. This will enable the accurate exploration of conformational changes and the description of electrostatic forces on the catalytic site due to the entire protein environment. QM/MM and MD are finetuning biomolecular processes and needed for high level modeling of biomolecules. In DFT calculations, solvent effects are usually included via continuum models with a dielectric constant representing the bulk solvent.103-105 Improvements can be made by explicitly including water molecules, but the use of an explicit solvent model in a conceptual DFT context needs further elaboration (for initial and more recent work, see refs 106 and 15b). In enzymatic catalysis, conceptual DFT plays an emerging role. In the years ahead, we are looking forward to a wide range of biological applications of computational DFT, given the availability of several quantum chemical or density functional theory programs107-110 and the ever increasing computing performance. It is not the intention of the authors to “invert” Parr’s dictum52 and to question the validity and intrinsic value of these calculations. However, just as sometimes massive numbers of experimental data need unifying concepts to order them and to predict which chemical reactions are the most likely to occur, this is also the case with large series of computational results. Conceptual DFT can play an essential role in guiding theoreticians and experimentalists to get a simple, yet quantum mechanically based, picture of biochemical phenomena by rationalizing experimental and/or computational data. Acknowledgment. G.R. thanks the Fund for Scientific Research (FWO) for a postdoctoral fellowship. P.G. thanks the VUB and the FWO for continuous support to his group. PG thanks all present and previous members of his group for many years of exciting collaborations in Conceptual DFT. References and Notes (1) Herschlag, D.; Jencks, W. P. J. Am. Chem. Soc. 1989, 111, 7587– 7596. (2) Hengge, A. C.; Cleland, W. W. J. Am. Chem. Soc. 1990, 112, 7421–7422. (3) Na´ray-Szabo´, G. THEOCHEM 2000, 500, 157–167. (4) Koch, C. W.; Holthausen, M. C. A Chemist’s Guide to Density Functional Theory, 2nd ed.; Wiley VCH: Weinheim, Germany, 2001. (5) Jensen, F. Introduction to computational chemistry, 2nd ed.; Wiley: New York, 2007. (6) Bruice, T. C.; Kahn, K. Curr. Opin. Chem. Biol. 2000, 4, 540– 544. (7) Garcia-Viloca, M.; Gao, J.; Karplus, M.; Truhlar, D. G. Science 2004, 303, 186–195. (8) Friesner, R. A.; Guallar, V. Annu. ReV. Phys. Chem. 2005, 56, 389–427. (9) Senn, H. M.; Thiel, W. Curr. Opin. Chem. Biol. 2007, 11, 182– 187. (10) Moro, G.; Bonati, L.; Bruschi, M.; Cosentino, U.; De Gioia, L.; Fantucci, P. C.; Pandini, A.; Papaleo, E.; Piteae, D.; Saracino, G. A. A.; Zampella, G. Theor. Chem. Acc. 2007, 117, 723–741. (11) Leopoldini, M.; Marino, T.; Michelini, M. D.; Rivalta, I.; Russo, N.; Sicilia, E.; Toscano, M. Theor. Chem. Acc. 2007, 117, 765–779. (12) Karplus, M.; Kuriyan, J. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 6679–6685.

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