Towards accurate ab initio QM/MM calculations of free-energy profiles of enzymatic reactions

Multiscale Free Energy Simulations: An Efficient Method for Connecting Classical MD Simulations to QM or QM/MM Free Energies Using Non-Boltzmann Bennett Reweighting Schemes

The reliability of free energy simulations (FES) is limited by two factors: (a) the need for correct sampling and (b) the accuracy of the computational method employed. Classical methods (e.g., force fields) are typically used for FES and present a myriad of challenges, with parametrization being a principle one. On the other hand, parameter-free quantum mechanical (QM) methods tend to be too computationally expensive for adequate sampling. One widely used approach is a combination of methods, where the free energy difference between the two end states is computed by, e.g., molecular mechanics (MM), and the end states are corrected by more accurate methods, such as QM or hybrid QM/MM techniques. Here we report two new approaches that significantly improve the aforementioned scheme; with a focus on how to compute corrections between, e.g., the MM and the more accurate QM calculations. First, a molecular dynamics trajectory that properly samples relevant conformational degrees of freedom is generated. Next, potential energies of each trajectory frame are generated with a QM or QM/MM Hamiltonian. Free energy differences are then calculated based on the QM or QM/MM energies using either a non-Boltzmann Bennett approach (QM-NBB) or non-Boltzmann free energy perturbation (NB-FEP). Both approaches are applied to calculate relative and absolute solvation free energies in explicit and implicit solvent environments. Solvation free energy differences (relative and absolute) between ethane and methanol in explicit solvent are used as the initial test case for QM-NBB. Next, implicit solvent methods are employed in conjunction with both QM-NBB and NB-FEP to compute absolute solvation free energies for 21 compounds. These compounds range from small molecules such as ethane and methanol to fairly large, flexible solutes, such as triacetyl glycerol. Several technical aspects were investigated. Ultimately some best practices are suggested for improving methods that seek to connect MM to QM (or QM/MM) levels of theory in FES.

QM/MM study of the C-C coupling reaction mechanism of CYP121, an essential cytochrome p450 of Mycobacterium tuberculosis

Among 20 p450s of Mycobacterium tuberculosis (Mt), CYP121 has received an outstanding interest, not only due to its essentiality for bacterial viability but also because it catalyzes an unusual carbon-carbon coupling reaction. Based on the structure of the substrate bound enzyme, several reaction mechanisms were proposed involving first Tyr radical formation, second Tyr radical formation, and C-C coupling. Key and unknown features, being the nature of the species that generate the first and second radicals, and the role played by the protein scaffold each step. In the present work we have used classical and quantum based computer simulation methods to study in detail its reaction mechanism. Our results show that substrate binding promotes formation of the initial oxy complex, Compound I is the responsible for first Tyr radical formation, and that the second Tyr radical is formed subsequently, through a PCET reaction, promoted by the presence of key residue Arg386. The final C-C coupling reaction possibly occurs in bulk solution, thus yielding the product in one oxygen reduction cycle. Our results thus contribute to a better comprehension of MtCYP121 reaction mechanism, with direct implications for inhibitor design, and also contribute to our general understanding of these type of enzymes.

Modeling catalytic promiscuity in the alkaline phosphatase superfamily

In recent years, it has become increasingly clear that promiscuity plays a key role in the evolution of new enzyme function. This finding has helped to elucidate fundamental aspects of molecular evolution. While there has been extensive experimental work on enzyme promiscuity, computational modeling of the chemical details of such promiscuity has traditionally fallen behind the advances in experimental studies, not least due to the nearly prohibitive computational cost involved in examining multiple substrates with multiple potential mechanisms and binding modes in atomic detail with a reasonable degree of accuracy. However, recent advances in both computational methodologies and power have allowed us to reach a stage in the field where we can start to overcome this problem, and molecular simulations can now provide accurate and efficient descriptions of complex biological systems with substantially less computational cost. This has led to significant advances in our understanding of enzyme function and evolution in a broader sense. Here, we will discuss currently available computational approaches that can allow us to probe the underlying molecular basis for enzyme specificity and selectivity, discussing the inherent strengths and weaknesses of each approach. As a case study, we will discuss recent computational work on different members of the alkaline phosphatase superfamily (AP) using a range of different approaches, showing the complementary insights they have provided. We have selected this particular superfamily, as it poses a number of significant challenges for theory, ranging from the complexity of the actual reaction mechanisms involved to the reliable modeling of the catalytic metal centers, as well as the very large system sizes. We will demonstrate that, through current advances in methodologies, computational tools can provide significant insight into the molecular basis for catalytic promiscuity, and, therefore, in turn, the mechanisms of protein functional evolution.

Combined quantum mechanics/molecular mechanics (QM/MM) methods in computational enzymology

Computational enzymology is a rapidly maturing field that is increasingly integral to understanding mechanisms of enzyme-catalyzed reactions and their practical applications. Combined quantum mechanics/molecular mechanics (QM/MM) methods are important in this field. By treating the reacting species with a quantum mechanical method (i.e., a method that calculates the electronic structure of the active site) and including the enzyme environment with simpler molecular mechanical methods, enzyme reactions can be modeled. Here, we review QM/MM methods and their application to enzyme-catalyzed reactions to investigate fundamental and practical problems in enzymology. A range of QM/MM methods is available, from cheaper and more approximate methods, which can be used for molecular dynamics simulations, to highly accurate electronic structure methods. We discuss how modeling of reactions using such methods can provide detailed insight into enzyme mechanisms and illustrate this by reviewing some recent applications. We outline some practical considerations for such simulations. Further, we highlight applications that show how QM/MM methods can contribute to the practical development and application of enzymology, e.g., in the interpretation and prediction of the effects of mutagenesis and in drug and catalyst design.

Why nature really chose phosphate

Phosphoryl transfer plays key roles in signaling, energy transduction, protein synthesis, and maintaining the integrity of the genetic material. On the surface, it would appear to be a simple nucleophile displacement reaction. However, this simplicity is deceptive, as, even in aqueous solution, the low-lying d-orbitals on the phosphorus atom allow for eight distinct mechanistic possibilities, before even introducing the complexities of the enzyme catalyzed reactions. To further complicate matters, while powerful, traditional experimental techniques such as the use of linear free-energy relationships (LFER) or measuring isotope effects cannot make unique distinctions between different potential mechanisms. A quarter of a century has passed since Westheimer wrote his seminal review, 'Why Nature Chose Phosphate' (Science 235 (1987), 1173), and a lot has changed in the field since then. The present review revisits this biologically crucial issue, exploring both relevant enzymatic systems as well as the corresponding chemistry in aqueous solution, and demonstrating that the only way key questions in this field are likely to be resolved is through careful theoretical studies (which of course should be able to reproduce all relevant experimental data). Finally, we demonstrate that the reason that nature really chose phosphate is due to interplay between two counteracting effects: on the one hand, phosphates are negatively charged and the resulting charge-charge repulsion with the attacking nucleophile contributes to the very high barrier for hydrolysis, making phosphate esters among the most inert compounds known. However, biology is not only about reducing the barrier to unfavorable chemical reactions. That is, the same charge-charge repulsion that makes phosphate ester hydrolysis so unfavorable also makes it possible to regulate, by exploiting the electrostatics. This means that phosphate ester hydrolysis can not only be turned on, but also be turned off, by fine tuning the electrostatic environment and the present review demonstrates numerous examples where this is the case. Without this capacity for regulation, it would be impossible to have for instance a signaling or metabolic cascade, where the action of each participant is determined by the fine-tuned activity of the previous piece in the production line. This makes phosphate esters the ideal compounds to facilitate life as we know it.

Exploring, refining, and validating the paradynamics QM/MM sampling

The performance of the paradynamics (PD) reference potential approach in QM/MM calculations is examined. It is also clarified that, in contrast to some possible misunderstandings, this approach provides a rigorous strategy for QM/MM free energy calculations. In particular, the PD approach provides a gradual and controlled way of improving the evaluation of the free energy perturbation associated with moving from the EVB reference potential to the target QM/MM surface. This is achieved by moving from the linear response approximation to the full free energy perturbation approach in evaluating the free energy changes. We also present a systematic way of improving the reference potential by using Gaussian-based correction potentials along a reaction coordinate. In parallel, we review other recent adaptations of the reference potential approach, emphasizing and demonstrating the advantage of using the EVB potential as a reference potential, relative to semiempirical QM/MM molecular orbital potentials. We also compare the PD results to those obtained by direct calculations of the potentials of the mean force (PMF). Additionally, we propose a way of accelerating the PMF calculations by using Gaussian-based negative potentials along the reaction coordinate (which are also used in the PD refinement). Finally, we discuss performance of the PD and the metadynamics approaches in ab initio QM/MM calculations and emphasize the advantage of using the PD approach.

Performance comparison of computational methods for modeling alpha-helical structures

Geometry optimization results are reported for secondary structural elements of small proteins and polypeptides. Emphasis is placed on how well molecular mechanics as well as semiempirical, ab initio, and density functional methods describe α-helical and related structures in purely theoretical models (Gly10, Ile10) as well as in realistic models (an α-helical region of calmodulin, and the complete structure of a small protein). Many of the methods examined here were found to provide unsatisfactory descriptions of the hydrogen-bonding interactions within polypeptide-type structures, as the α-helical canonical secondary structure motif was not reproduced accurately. Ab initio and DFT methods provided reasonable results only when solvation models were included, although Hartree-Fock failed even with solvation in one of the test cases; among the semiempirical methods, one of the PM6 implementations performed very well.

Convergence of QM/MM free-energy perturbations based on molecular-mechanics or semiempirical simulations

Lately, there has been great interest in performing free-energy perturbation (FEP) at the combined quantum mechanics and molecular mechanics (QM/MM) level, e.g. for enzyme reactions. Such calculations require extensive sampling of phase space, which typically is prohibitive with density-functional theory or ab initio methods. Therefore, such calculations have mostly been performed with semiempirical QM (SQM) methods, or by using a thermodynamic cycle involving sampling at the MM level and perturbations between the MM and QM/MM levels of theory. However, the latter perturbations typically have convergence problems, unless the QM system is kept fixed during the simulations, because the MM and QM/MM descriptions of the internal degrees of freedom inside the QM system are too dissimilar. We have studied whether the convergence of the MM → QM/MM perturbation can be improved by using a thoroughly parameterised force field or by using SQM/MM methods. As a test case we use the first half-reaction of haloalkane dehalogenase and the QM calculations are performed with the PBE, B3LYP, and TPSSH density-functional methods. We show that the convergence can be improved with a tailored force field, but only locally around the parameterised state. Simulations based on SQM/MM methods using the MNDO, AM1, PM3, RM1, PDDG-MNDO, and PDDG-PM3 Hamiltonians have slightly better convergence properties, but very long simulations are still needed (~10 ns) and convergence is obtained only if electrostatic interactions between the QM system and the surroundings are ignored. This casts some doubts on the common practice to base QM/MM FEPs on semiempirical simulations without any reweighting of the trajectories.

A hybrid elastic band string algorithm for studies of enzymatic reactions

A common challenge in theoretical biophysics is the identification of a minimum energy path (MEP) for the rearrangement of a group of atoms from one stable configuration to another. The structure with maximum energy along the MEP approximates the transition state for the process and the energy profile itself permits estimation of the transition rates. In this work we describe a computationally efficient algorithm for the identification of minimum energy paths in complicated biosystems. The algorithm is a hybrid of the nudged elastic band (NEB) and string methods. It has been implemented in the pDynamo simulation program and tested by examining elementary steps in the reaction mechanisms of three enzymes: citrate synthase, RasGAP, and lactate dehydrogenase. Good agreement is found for the energies and geometries of the species along the reaction profiles calculated using the new algorithm and previous versions of the NEB and string techniques, and also those obtained by the common method of adiabatic exploration of the potential energy surface as a function of predefined reaction coordinates. Precisely refined structures of the saddle points along the paths may be subsequently obtained with the climbing image variant of the NEB algorithm. Directions in which the utility of the methods that we have implemented can be further improved are discussed.

A general theoretical model for electron transfer reactions in complex systems

In this paper we present a general theoretical-computational model for treating electron transfer reactions in complex atomic-molecular systems. The underlying idea of the approach, based on unbiased first-principles calculations at the atomistic level, utilizes the definition and the construction of the Diabatic Perturbed states of the involved reactive partners (i.e. the quantum centres in our perturbation approach) as provided by the interaction with their environment, including their mutual interaction. In this way we reconstruct the true Adiabatic states of the reactive partners characterizing the electron transfer process as the fluctuation of the electronic density due to the fluctuating perturbation. Results obtained by using a combination of Molecular Dynamics simulation and the Perturbed Matrix Method on a prototypical intramolecular electron transfer (from 2-(9,9'-dimethyl)fluorene to the 2-naphthalene group separated by a steroidal 5-α-androstane skeleton) well illustrate the accuracy of the method in reproducing both the thermodynamics and the kinetics of the process.

Prechemistry versus preorganization in DNA replication fidelity

The molecular origin of nucleotide insertion catalysis and fidelity of DNA polymerases is explored by means of computational simulations. Special attention is paid to the examination of the validity of proposals that invoke prechemistry effects, checkpoints concepts, and dynamical effects. The simulations reproduce the observed fidelity in Pol β, starting with the relevant observed X-ray structures of the complex with the right (R) and wrong (W) nucleotides. The generation of free energy surfaces for the R and W systems also allowed us to analyze different proposals about the origin of the fidelity and to reach several important conclusions. It is found that the potential of mean force (PMF) obtained by proper sampling does not support QM/MM-based proposals of a large barrier before the prechemistry state. Furthermore, examination of dynamical proposals by the renormalization approach indicates that the motions from open to close configurations do not contribute to catalysis or fidelity. Finally we discuss and analyze the induced fit concept and show that, despite its importance, it does not explain fidelity. That is, the fidelity is apparently due to the change in the preorganization of the chemical site, as a result of the relaxation of the binding site upon binding of the incorrect nucleotide. Finally and importantly, since the issue is the barrier associated with the enzyme-substrate (ES)/DNA complex at the chemical transition state and not the path to this complex formation (unless this path involves rate determining steps), it is also not useful to invoke checkpoints while discussing fidelity.

Coarse-grained (multiscale) simulations in studies of biophysical and chemical systems

Recent years have witnessed an explosion in computational power, leading to attempts to model ever more complex systems. Nevertheless, there remain cases for which the use of brute-force computer simulations is clearly not the solution. In such cases, great benefit can be obtained from the use of physically sound simplifications. The introduction of such coarse graining can be traced back to the early usage of a simplified model in studies of proteins. Since then, the field has progressed tremendously. In this review, we cover both key developments in the field and potential future directions. Additionally, particular emphasis is given to two general approaches, namely the renormalization and reference potential approaches, which allow one to move back and forth between the coarse-grained (CG) and full models, as these approaches provide the foundation for CG modeling of complex systems.

Conical intersections in solution: formulation, algorithm, and implementation with combined quantum mechanics/molecular mechanics method

The significance of conical intersections in photophysics, photochemistry, and photodissociation of polyatomic molecules in gas phase has been demonstrated by numerous experimental and theoretical studies. Optimization of conical intersections of small- and medium-size molecules in gas phase has currently become a routine optimization process, as it has been implemented in many electronic structure packages. However, optimization of conical intersections of small- and medium-size molecules in solution or macromolecules remains inefficient, even poorly defined, due to large number of degrees of freedom and costly evaluations of gradient difference and nonadiabatic coupling vectors. In this work, based on the sequential quantum mechanics and molecular mechanics (QM/MM) and QM/MM-minimum free energy path methods, we have designed two conical intersection optimization methods for small- and medium-size molecules in solution or macromolecules. The first one is sequential QM conical intersection optimization and MM minimization for potential energy surfaces; the second one is sequential QM conical intersection optimization and MM sampling for potential of mean force surfaces, i.e., free energy surfaces. In such methods, the region where electronic structures change remarkably is placed into the QM subsystem, while the rest of the system is placed into the MM subsystem; thus, dimensionalities of gradient difference and nonadiabatic coupling vectors are decreased due to the relatively small QM subsystem. Furthermore, in comparison with the concurrent optimization scheme, sequential QM conical intersection optimization and MM minimization or sampling reduce the number of evaluations of gradient difference and nonadiabatic coupling vectors because these vectors need to be calculated only when the QM subsystem moves, independent of the MM minimization or sampling. Taken together, costly evaluations of gradient difference and nonadiabatic coupling vectors in solution or macromolecules can be reduced significantly. Test optimizations of conical intersections of cyclopropanone and acetaldehyde in aqueous solution have been carried out successfully.

Paradynamics: an effective and reliable model for ab initio QM/MM free-energy calculations and related tasks

Recent years have seen tremendous effort in the development of approaches with which to obtain quantum mechanics/molecular mechanics (QM/MM) free energies for reactions in the condensed phase. Nevertheless, there remain significant challenges to address, particularly, the high computational cost involved in performing proper configurational sampling and, in particular, in obtaining ab initio QM/MM (QM(ai)/MM) free-energy surfaces. One increasingly popular approach that seems to offer an ideal way to progress in this direction is the elegant metadynamics (MTD) approach. However, in the current work, we point out the subtle efficiency problems associated with this approach and illustrate that we have at hand what is arguably a more powerful approach. More specifically, we demonstrate the effectiveness of an updated version of our original idea of using a classical reference potential for QM(ai)/MM calculations [J. Phys. Chem. 1995, 99, 17516)], which we refer to as paradynamics (PD). This approach is based on the use of an empirical valence bond (EVB) reference potential, which is already similar to the real ab initio potential. The reference potential is fitted to the ab initio potential by an iterative and, to a great degree, automated refinement procedure. The corresponding free-energy profile is then constructed using the refined EVB potential, and the linear response approximation (LRA) is used to evaluate the QM(ai)/MM activation free-energy barrier. The automated refinement of the EVB surface (and thus the reduction of the difference between the reference and ab initio potentials) is a key factor in accelerating the convergence of the LRA approach. We apply our PD approach to a test reaction, namely, the S(N)2 reaction between a chloride ion and methyl chloride, and demonstrate that, at present, this approach is far more powerful and cost-effective than the metadynamics approach (at least in its current implementation). We also discuss the general features of the PD approach in terms of its ability to explore complex systems and clarify that it is not a specialized approach limited to only accelerating QM(ai)/MM calculations with proper sampling, but rather can be used in a wide variety of applications. In fact, we point out that the use of a reference (CG) potential coupled with its PD refinement, as well as our renormalization approach, provides very general and powerful strategies that can be used very effectively to explore any property that has been studied by the MTD approach.

Use of a QM/MM-based FEP method to evaluate the anomalous hydration behavior of simple alkyl amines and amides: application to the design of FBPase inhibitors for the treatment of type-2 diabetes

Standard molecular mechanics (MM) force fields predict a nearly linear decrease in hydration free energy with each successive addition of a methyl group to ammonia or acetamide, whereas a nonadditive relationship is observed experimentally. In contrast, the non-additive hydration behavior is reproduced directly using a quantum mechanics (QM)/MM-based free-energy perturbation (FEP) method wherein the solute partial atomic charges are updated at every window. Decomposing the free energies into electrostatic and van der Waals contributions and comparing the results with the corresponding free energies obtained using a conventional FEP method and a QM/MM method wherein the charges are not updated suggests that inaccuracies in the electrostatic free energies are the primary reason for the inability of the conventional FEP method to predict the experimental findings. The QM/MM-based FEP method was subsequently used to evaluate inhibitors of the diabetes drug target fructose-1,6-bisphosphatase adenosine 5'-monophosphate and 6-methylamino purine riboside 5'-monophosphate. The predicted relative binding free energy was consistent with the experimental findings, whereas the relative binding free energy predicted using the conventional FEP method differed from the experimental finding by an amount consistent with the overestimated relative solvation free energies calculated for alkylamines. Accordingly, the QM/MM-based FEP method offers potential advantages over conventional FEP methods, including greater accuracy and reduced user input. Moreover, since drug candidates often contain either functionality that is inadequately treated by MM (e.g., simple alkylamines and alkylamides) or new molecular scaffolds that require time-consuming development of MM parameters, these advantages could enable future automation of FEP calculations as well as greatly increase the use and impact of FEP calculations in drug discovery.

Multiscale modeling of biological functions

Recent years have witnessed a tremendous explosion in computational power, which in turn has resulted in great progress in the complexity of the biological and chemical problems that can be addressed by means of all-atom simulations. Despite this, however, our computational time is not infinite, and in fact many of the key problems of the field were resolved long before the existence of the current levels of computational power. This review will start by presenting a brief historical overview of the use of multiscale simulations in biology, and then present some key developments in the field, highlighting several cases where the use of a physically sound simplification is clearly superior to a brute-force approach. Finally, some potential future directions will be discussed.

Minimum MD simulation length required to achieve reliable results in free energy perturbation calculations: case study of relative binding free energies of fructose-1,6-bisphosphatase inhibitors

In an attempt to establish the criteria for the length of simulation to achieve the desired convergence of free energy calculations, two studies were carried out on chosen complexes of FBPase-AMP mimics. Calculations were performed for varied length of simulations and for different starting configurations using both conventional- and QM/MM-FEP methods. The results demonstrate that for small perturbations, 1248 ps simulation time could be regarded a reasonable yardstick to achieve convergence of the results. As the simulation time is extended, the errors associated with free energy calculations also gradually tapers off. Moreover, when starting the simulation from different initial configurations of the systems, the results are not changed significantly, when performed for 1248 ps. This study carried on FBPase-AMP mimics corroborates well with our previous successful demonstration of requirement of simulation time for solvation studies, both by conventional and ab initio FEP. The establishment of aforementioned criteria of simulation length serves a useful benchmark in drug design efforts using FEP methodologies, to draw a meaningful and unequivocal conclusion.

Challenges and advances in validating enzyme design proposals: the case of kemp eliminase catalysis

One of the fundamental challenges in biotechnology and biochemistry is the ability to design effective enzymes. Despite recent progress, most of the advances on this front have been made by placing the reacting fragments in the proper places, rather than by optimizing the preorganization of the environment, which is the key factor in enzyme catalysis. Thus, rational improvement of the preorganization would require approaches capable of evaluating reliably the actual catalytic effect. This work considers the catalytic effects in different Kemp eliminases as a benchmark for a computer-aided enzyme design. It is shown that the empirical valence bond provides a powerful screening tool, with significant advantages over current alternative strategies. The insights provided by the empirical valence bond calculations are discussed with an emphasis on the ability to analyze the difference between the linear free energy relationships obtained in solution and those found in the enzymes. We also point out the trade-off between the reliability and speed of the calculations and try to determine what it takes to realize reliable computer-aided screening.

Double-lock ratchet mechanism revealing the role of alphaSER-344 in FoF1 ATP synthase

In a majority of living organisms, FoF1 ATP synthase performs the fundamental process of ATP synthesis. Despite the simple net reaction formula, ADP+Pi→ATP+H2O, the detailed step-by-step mechanism of the reaction yet remains to be resolved owing to the complexity of this multisubunit enzyme. Based on quantum mechanical computations using recent high resolution X-ray structures, we propose that during ATP synthesis the enzyme first prepares the inorganic phosphate for the γP-OADP bond-forming step via a double-proton transfer. At this step, the highly conserved αS344 side chain plays a catalytic role. The reaction thereafter progresses through another transition state (TS) having a planar ion configuration to finally form ATP. These two TSs are concluded crucial for ATP synthesis. Using stepwise scans and several models of the nucleotide-bound active site, some of the most important conformational changes were traced toward direction of synthesis. Interestingly, as the active site geometry progresses toward the ATP-favoring tight binding site, at both of these TSs, a dramatic increase in barrier heights is observed for the reverse direction, i.e., hydrolysis of ATP. This change could indicate a "ratchet" mechanism for the enzyme to ensure efficacy of ATP synthesis by shifting residue conformation and thus locking access to the crucial TSs.

Computational modeling of carbohydrate-recognition process in E-selectin complex: structural mapping of sialyl Lewis X onto ab initio QM/MM free energy surface

To advance our knowledge of carbohydrate recognition by lectins, we propose a systematic computational modeling strategy to identify complex sugar-chain conformations on the reduced free energy surface (FES). We selected the complex of E-selectin with sialyl Lewis X (denoted E-selectin/SLe(x) complex) as a first target molecule. First, we introduced the reduced 2D-FES that characterizes conformational changes in carbohydrate structure as well as the degree of solvation stability of the carbohydrate ligand, and evaluated the overall free energy profile by classical molecular dynamics simulation combined with ab initio QM/MM energy corrections. Second, we mapped flexible carbohydrate structures onto the reduced QM/MM 2D-FES, and identified the details of molecular interactions between each monosaccharide component and the amino acid residues at the carbohydrate-recognition domain. Finally, we confirmed the validity of our modeling strategy by evaluating the chemical shielding tensor by ab initio QM/MM-GIAO computations for several QM/MM-refined geometries sampled from the minimum free energy region in the 2D-FES, and compared this theoretical averaging data with the experimental 1D-NMR profile. The model clearly shows that the binding geometries of the E-selectin/SLe(x) complex are determined not by one single, rigid carbohydrate structure but rather by the sum of averaged conformations fluctuating around the minimum free energy region. For the E-selectin/SLe(x) complex, the major molecular interactions are hydrogen bonds between Fuc and the Ca(2+) binding site in the carbohydrate-recognition domain, and Gal is important in determining the ligand specificity.

At the dawn of the 21st century: Is dynamics the missing link for understanding enzyme catalysis?

Enzymes play a key role in almost all biological processes, accelerating a variety of metabolic reactions as well as controlling energy transduction, the transcription, and translation of genetic information, and signaling. They possess the remarkable capacity to accelerate reactions by many orders of magnitude compared to their uncatalyzed counterparts, making feasible crucial processes that would otherwise not occur on biologically relevant timescales. Thus, there is broad interest in understanding the catalytic power of enzymes on a molecular level. Several proposals have been put forward to try to explain this phenomenon, and one that has rapidly gained momentum in recent years is the idea that enzyme dynamics somehow contributes to catalysis. This review examines the dynamical proposal in a critical way, considering basically all reasonable definitions, including (but not limited to) such proposed effects as "coupling between conformational and chemical motions," "landscape searches" and "entropy funnels." It is shown that none of these proposed effects have been experimentally demonstrated to contribute to catalysis, nor are they supported by consistent theoretical studies. On the other hand, it is clarified that careful simulation studies have excluded most (if not all) dynamical proposals. This review places significant emphasis on clarifying the role of logical definitions of different catalytic proposals, and on the need for a clear formulation in terms of the assumed potential surface and reaction coordinate. Finally, it is pointed out that electrostatic preorganization actually accounts for the observed catalytic effects of enzymes, through the corresponding changes in the activation free energies.

Multiscale simulations of protein landscapes: using coarse-grained models as reference potentials to full explicit models

Evaluating the free-energy landscape of proteins and the corresponding functional aspects presents a major challenge for computer simulation approaches. This challenge is due to the complexity of the landscape and the enormous computer time needed for converging simulations. The use of simplified coarse-grained (CG) folding models offers an effective way of sampling the landscape but such a treatment, however, may not give the correct description of the effect of the actual protein residues. A general way around this problem that has been put forward in our early work (Fan et al., Theor Chem Acc 1999;103:77-80) uses the CG model as a reference potential for free-energy calculations of different properties of the explicit model. This method is refined and extended here, focusing on improving the electrostatic treatment and on demonstrating key applications. These applications include: evaluation of changes of folding energy upon mutations, calculations of transition-states binding free energies (which are crucial for rational enzyme design), evaluations of catalytic landscape, and evaluations of the time-dependent responses to pH changes. Furthermore, the general potential of our approach in overcoming major challenges in studies of structure function correlation in proteins is discussed.

On the origin of the catalytic power of carboxypeptidase A and other metalloenzymes

Zinc metalloenzymes play a major role in key biological processes and carboxypeptidase-A (CPA) is a major prototype of such enzymes. The present work quantifies the energetics of the catalytic reaction of CPA and its mutants using the empirical valence bond (EVB) approach. The simulations allow us to quantify the origin of the catalytic power of this enzyme and to examine different mechanistic alternatives. The first step of the analysis used experimental information to determine the activation energy of each assumed mechanism of the reference reaction without the enzyme. The next step of the analysis involved EVB simulations of the reference reaction and then a calibration of the simulations by forcing them to reproduce the energetics of the reference reaction, in each assumed mechanism. The calibrated EVB was then used in systematic simulations of the catalytic reaction in the protein environment, without changing any parameter. The simulations reproduced the observed rate enhancement in two feasible general acid-general base mechanisms (GAGB-1 and GAGB-2), although the calculations with the GAGB-2 mechanism underestimated the catalytic effect in some treatments. We also reproduced the catalytic effect in the R127A mutant. The mutation calculations indicate that the GAGB-2 mechanism is significantly less likely than the GAGB-1 mechanism. It is also found, that the enzyme loses all its catalytic effect without the metal. This and earlier studies show that the catalytic effect of the metal is not some constant electrostatic effect, that can be assessed from gas phase studies, but a reflection of the dielectric effect of the specific environment.

The EVB as a quantitative tool for formulating simulations and analyzing biological and chemical reactions

Recent years have seen dramatic improvements in computer power, allowing ever more challenging problems to be approached. In light of this, it is imperative to have a quantitative model for examining chemical reactivity, both in the condensed phase and in solution, as well as to accurately quantify physical organic chemistry (particularly as experimental approaches can often be inconclusive). Similarly, computational approaches allow for great progress in studying enzyme catalysis, as they allow for the separation of the relevant energy contributions to catalysis. Due to the complexity of the problems that need addressing, there is a need for an approach that can combine reliability with an ability to capture complex systems in order to resolve long-standing controversies in a unique way. Herein, we will demonstrate that the empirical valence bond (EVB) approach provides a powerful way to connect the classical concepts of physical organic chemistry to the actual energies of enzymatic reactions by means of computation. Additionally, we will discuss the proliferation of this approach, as well as attempts to capture its basic chemistry and repackage it under different names. We believe that the EVB approach is the most powerful tool that is currently available for studies of chemical processes in the condensed phase in general and enzymes in particular, particularly when trying to explore the different proposals about the origin of the catalytic power of enzymes.

On unjustifiably misrepresenting the EVB approach while simultaneously adopting it

In recent years, the EVB has become a widely used tool in the QM/MM modeling of reactions in condensed phases and in biological systems, with ever increasing popularity. However, despite the fact that its power and validity have been repeatedly established since 1980, a recent work (Valero, R.; et al. J. Chem. Theory Comput. 2009, 5, 1) has strongly criticized this approach, while not discussing the fact that one of the authors is effectively using it himself for both gas-phase and solution studies. Here, we have responded to the most serious unjustified assertions of that paper, covering both the more problematic aspects of that work and the more complex scientific aspects. Additionally, we have demonstrated that the poor EVB results shown in Valero et al. which where presented as verification of the unreliability of the EVB model were in fact obtained by the use of incorrect parameters, without comparing to the correct surface obtained by our program.

The energy gap as a universal reaction coordinate for the simulation of chemical reactions

The selection of a proper reaction coordinate is a major bottleneck in simulations of chemical reactions in complex systems. Increasing the number of variables that are used to bias the reaction largely affects the convergence and leads to an unbearable increase in computational price. This problem can be overcome by employing a complex reaction coordinate that depends on many geometrical variables of the system, such as the energy gap (EGAP) in the empirical valence bond (EVB) method. EGAP depends on all of the coordinates of the system, and its robustness has been demonstrated for a variety of enzymatic reactions. In this work, we demonstrate that EGAP, derived from a classical representation, can be used as a reaction coordinate in systems described with any quantum chemistry Hamiltonian. Benefits of using EGAP as a reaction coordinate as compared to a traditional geometrical variable are illustrated in the case of a symmetric nucleophilic substitution reaction in water solution. EGAP is shown to provide a significantly more efficient sampling and allows a better localization of the transition state as compared to a geometrical reaction coordinate.

Artificial reaction coordinate "tunneling" in free-energy calculations: the catalytic reaction of RNase H

We describe a method for the systematic improvement of reaction coordinates in quantum mechanical/molecular mechanical (QM/MM) calculations of reaction free-energy profiles. In umbrella-sampling free-energy calculations, a biasing potential acting on a chosen reaction coordinate is used to sample the system in reactant, product, and transition states. Sharp, nearly discontinuous changes along the resulting reaction path are used to identify coordinates that are relevant for the reaction but not properly sampled. These degrees of freedom are then included in an extended reaction coordinate. The general formalism is illustrated for the catalytic cleavage of the RNA backbone of an RNA/DNA hybrid duplex by the RNase H enzyme of Bacillus halodurans. We find that in the initial attack of the phosphate diester by water, the oxygen-phosphorus distances alone are not sufficient as reaction coordinates, resulting in substantial hysteresis in the proton degrees of freedom and a barrier that is too low (approximately 10 kcal/mol). If the proton degrees of freedom are included in an extended reaction coordinate, we obtain a barrier of 21.6 kcal/mol consistent with the experimental rates. As the barrier is approached, the attacking water molecule transfers one of its protons to the O1P oxygen of the phosphate group. At the barrier top, the resulting hydroxide ion forms a penta-coordinated phosphate intermediate. The method used to identify important degrees of freedom, and the procedure to optimize the reaction coordinate are general and should be useful both in classical and in QM/MM free-energy calculations.

Linear energy relationships for the octahedral preference of Mg, Ca and transition metal ions

The geometry, atomic charges, force constants, and relative energies of the symmetric and distorted M(2+)(H(2)O)(4)(F(-))(2), M(3+)(H(2)O)(4)(F(-))(2), M(2+)(H(2)O)(3)(F(-))(2), and M(3+)(H(2)O)(3)(F(-))(2) metal complexes, M = Mg, Ca, Co, Cu, Fe, Mn, Ni, Zn, Cr, V, were calculated by using the B3LYP/TZVP density functional method in both gas phase and aqueous solution, modeled using the polarized continuum model. The deformation energy associated with moving one water ligand 12 degrees from the initial "octahedral" arrangement, in which all O-M-O, O-M-F, and F-M-F angles are either 90 degrees or 180 degrees, was calculated to examine the angular ligand flexibility. For all M(2+)(H(2)O)(4)(F(-))(2) complexes, this distortion increased the energy of the complex in proportion to the electrostatic potential-derived (ESP) charge of the metal, and in proportion to D(-10), where D is the distance from the distorted ligand to its closest neighbor. The octahedral stability was further examined by calculating the energies for the removal of a water ligand from the octahedral complex to form a square-pyramidal or trigonal-bipyramidal complex. The octahedral preference, defined as the negative of the corresponding binding energy of the ligand, was found to linearly correlate with the ESP charge of the metal in both the gas phase and aqueous solution. The obtained results indicate that quantum-mechanical covalent effects are of secondary importance for both the flexibility and the octahedral preference of M(2+)(H(2)O)(4)(F(-))(2) and M(3+)(H(2)O)(4)(F(-))(2) complexes. This conclusion and supporting data are important for the development of consistent molecular mechanical force fields of the studied metal ions.

QM/MM methods for biomolecular systems

Combined quantum-mechanics/molecular-mechanics (QM/MM) approaches have become the method of choice for modeling reactions in biomolecular systems. Quantum-mechanical (QM) methods are required for describing chemical reactions and other electronic processes, such as charge transfer or electronic excitation. However, QM methods are restricted to systems of up to a few hundred atoms. However, the size and conformational complexity of biopolymers calls for methods capable of treating up to several 100,000 atoms and allowing for simulations over time scales of tens of nanoseconds. This is achieved by highly efficient, force-field-based molecular mechanics (MM) methods. Thus to model large biomolecules the logical approach is to combine the two techniques and to use a QM method for the chemically active region (e.g., substrates and co-factors in an enzymatic reaction) and an MM treatment for the surroundings (e.g., protein and solvent). The resulting schemes are commonly referred to as combined or hybrid QM/MM methods. They enable the modeling of reactive biomolecular systems at a reasonable computational effort while providing the necessary accuracy.

Development and application of ab initio QM/MM methods for mechanistic simulation of reactions in solution and in enzymes

Determining the free energies and mechanisms of chemical reactions in solution and enzymes is a major challenge. For such complex reaction processes, combined quantum mechanics/molecular mechanics (QM/MM) method is the most effective simulation method to provide an accurate and efficient theoretical description of the molecular system. The computational costs of ab initio QM methods, however, have limited the application of ab initio QM/MM methods. Recent advances in ab initio QM/MM methods allowed the accurate simulation of the free energies for reactions in solution and in enzymes and thus paved the way for broader application of the ab initio QM/MM methods. We review here the theoretical developments and applications of the ab initio QM/MM methods, focusing on the determination of reaction path and the free energies of the reaction processes in solution and enzymes.