Steven K. Burger

Quantum mechanics/molecular mechanics minimum free-energy path for accurate reaction energetics in solution and enzymes: Sequential sampling and optimization on the potential of mean force surface

To accurately determine the reaction path and its energetics for enzymatic and solution-phase reactions, we present a sequential sampling and optimization approach that greatly enhances the efficiency of the ab initio quantum mechanics/molecular mechanics minimum free-energy path (QM/MM-MFEP) method. In the QM/MM-MFEP method, the thermodynamics of a complex reaction system is described by the potential of mean force (PMF) surface of the quantum mechanical (QM) subsystem with a small number of degrees of freedom, somewhat like describing a reaction process in the gas phase. The main computational cost of the QM/MM-MFEP method comes from the statistical sampling of conformations of the molecular mechanical (MM) subsystem required for the calculation of the QM PMF and its gradient. In our new sequential sampling and optimization approach, we aim to reduce the amount of MM sampling while still retaining the accuracy of the results by first carrying out MM phase-space sampling and then optimizing the QM subsystem in the fixed-size ensemble of MM conformations. The resulting QM optimized structures are then used to obtain more accurate sampling of the MM subsystem. This process of sequential MM sampling and QM optimization is iterated until convergence. The use of a fixed-size, finite MM conformational ensemble enables the precise evaluation of the QM potential of mean force and its gradient within the ensemble, thus circumventing the challenges associated with statistical averaging and significantly speeding up the convergence of the optimization process. To further improve the accuracy of the QM/MM-MFEP method, the reaction path potential method developed by Lu and Yang [Z. Lu and W. Yang, J. Chem. Phys. 121, 89 (2004)] is employed to describe the QM/MM electrostatic interactions in an approximate yet accurate way with a computational cost that is comparable to classical MM simulations. The new method was successfully applied to two example reaction processes, the classical SN2 reaction of Cl-+CH3Cl in solution and the second proton transfer step of the reaction catalyzed by the enzyme 4-oxalocrotonate tautomerase. The activation free energies calculated with this new sequential sampling and optimization approach to the QM/MM-MFEP method agree well with results from other simulation approaches such as the umbrella sampling technique with direct QM/MM dynamics sampling, demonstrating the accuracy of the iterative QM/MM-MFEP method.

Quadratic string method for determining the minimum-energy path based on multiobjective optimization

Based on a multiobjective optimization framework, we develop a new quadratic string method for finding the minimum-energy path. In the method, each point on the minimum-energy path is minimized by integration in the descent direction perpendicular to path. Each local integration is done on a quadratic surface approximated by a damped Broyden-Fletcher-Goldfarb-Shanno updated Hessian, allowing the algorithm to take many steps between energy and gradient calls. The integration is performed with an adaptive step-size solver, which is restricted in length to the trust radius of the approximate Hessian. The full algorithm is shown to be capable of practical superlinear convergence, in contrast to the linear convergence of other methods. The method also eliminates the need for predetermining such parameters as step size and spring constants, and is applicable to reactions with multiple barriers. The effectiveness of this method is demonstrated for the Muller-Brown potential, a seven-atom Lennard-Jones cluster, and the enolation of acetaldehyde to vinyl alcohol.

Sequential quadratic programming method for determining the minimum energy path

A new method, referred to as the sequential quadratic programming method, is presented for determining minimum energy paths. The method is based on minimizing the points representing the path in the subspace perpendicular to the tangent of the path while using a penalty term to prevent kinks from forming. Rather than taking one full step, the minimization is divided into a number of sequential steps on an approximate quadratic surface. The resulting method can efficiently determine the reaction mechanism, from which transition state can be easily identified and refined with other methods. To improve the resolution of the path close to the transition state, points are clustered close to this region with a reparametrization scheme. The usefulness of the algorithm is demonstrated for the Muller-Brown potential, amide hydrolysis, and an 89 atom cluster taken from the active site of 4-oxalocrotonate tautomerase for the reaction which catalyzes 2-oxo-4-hexenedioate to the intermediate 2-hydroxy-2,4-hexadienedioate.

Methods for finding transition states on reduced potential energy surfaces

Three new algorithms are presented for determining transition state (TS) structures on the reduced potential energy surface, that is, for problems in which a few important degrees of freedom can be isolated. All three methods use constrained optimization to rapidly find the TS without an initial Hessian evaluation. The algorithms highlight how efficiently the TS can be located on a reduced surface, where the rest of the degrees of freedom are minimized. The first method uses a nonpositive definite quasi-Newton update for the reduced degrees of freedom. The second uses Shepard interpolation to fit the Hessian and starts from a set of points that bound the TS. The third directly uses a finite difference scheme to calculate the reduced degrees of freedom of the Hessian of the entire system, and searches for the TS on the full potential energy surface. All three methods are tested on an epoxide hydrolase cluster, and the ring formations of cyclohexane and cyclobutenone. The results indicate that all the methods are able to converge quite rapidly to the correct TS, but that the finite difference approach is the most efficient.

Quantum mechanics/molecular mechanics strategies for docking pose refinement: distinguishing between binders and decoys in cytochrome C peroxidase.

We investigate the effect of systematically applying molecular dynamics (MD) and quantum mechanics/molecular mechanics (QM/MM) to docked poses in an attempt to improve the correspondence between theoretical prediction and experimental observation. The proposed scheme involves running a short time scale MD simulation on a docked ligand pose (and any known structurally important crystal structure waters in the active site), followed by QM/MM minimization. Both of these steps are relatively fast for moderately sized ligands; longer time scale MD involving the protein is not found to improve the results. The final binding energy is given in terms of the QM/MM total energy, a van der Waals correction, and a term to account for desolvation effects. This methodology is first tested with a trypsin inhibitor, for which we establish the importance of running MD before reoptimizing with QM/MM. The method is then applied to cytochrome c peroxidase using a set of binders and decoys. In this example, the proposed methodology affords much better discrimination between binders and decoys than the traditional docking approach used. For both systems presented, application of this protocol results in a significantly better energetic ranking and a smaller root mean squared deviation from known crystallographic ligand poses. This work highlights the importance of including polarization effects through QM/MM and of sampling with MD to refine a set of initial docked poses.

Dual Grid Methods for Finding the Reaction Path on Reduced Potential Energy Surfaces.

Two new algorithms are presented for determining the minimum energy reaction path (MEP) on the reduced potential energy surface (RPES) starting with only the reactant. These approaches are based on concepts from the fast marching method (FMM), which expands points outward as a wavefront on a multidimensional grid from the reactant until the product is reached. The MEP is then traced backward to the reactant. Since the number of possible grid points that must be considered grows exponentially with increasing dimensionality of the RPES, interpolation is important for maintaining manageable computational costs. In this work, we use Shepard interpolation, which we have modified to resolve problems in overfitting. In contrast to FMM, which accurately locates the MEP, the new algorithms focus on locating the single rate-limiting transition state and provide only a rough estimate of the MEP. They do this by mapping out the RPES on a coarse grid and then refining a least action path on a finer grid. This is done so that the majority of the interpolation is done on the finer grid, which minimizes the amount of extrapolation inherent in an outward searching algorithm. The first method scans the entire PES before iteratively locating the transition state (TS) for the MEP on the lower bound estimate of the fine PES. The second method explores the coarse grid in a similar manner to FMM and then iteratively locates the rate-limiting TS in the same manner as the first method. Both methods are shown to be capable of rapidly obtaining (in less than 30 constrained optimization cycles) an approximation to the MEP and the rate limiting TS for three example systems: the 4-well potential, the molecule N-hydroxymethyl-methylnitrosaminee (HMMN), and a cluster model of DNA-uracil glycosylase.

Practical calculation of molecular acidity with the aid of a reference molecule.

A set of linear free energy models are presented for determining the pK(a) values of amines, alcohols, and carboxylic acids. Models are determined from a series of pK(a) predictors, taken both from traditional natural atomic orbital analysis (NAO) and from a novel approach introduced here of using a reference molecule: an ammonium ion for amines and a hydrogen sulfide molecule for alcohols and carboxylic acids. Using these reference molecules, we calculate the barrier to proton transfer and show that a number of properties associated with the transition state are correlated with the pK(a). By considering 38 predictors, we obtain a four-variable model for amines and a three-variable model for oxygen-containing compounds. The model for amines is based on 145 compounds and has a root mean squared error (RMSE) of 0.45 and R(2) = 0.98. The oxygen set has 48 molecules: RMSE = 0.26, and R(2) = 0.993. Similar, linear, and multilinear models are constructed after separating the sets into chemically similar categories: alcohols, carboxylic acids, and primary, secondary, tertiary, and aromatic amines. This separation gives simpler models with relatively low RMSE values, where the most important predictor of the pK(a) is the difference in energy between transferring the proton from the reference molecular base to the conjugate acid from the data set.

Moving least-squares enhanced Shepard interpolation for the fast marching and string methods.

The number of the potential energy calculations required by the quadratic string method (QSM), and the fast marching method (FMM) is significantly reduced by using Shepard interpolation, with a moving least squares to fit the higher-order derivatives of the potential. The derivatives of the potential are fitted up to fifth order. With an error estimate for the interpolated values, this moving least squares enhanced Shepard interpolation scheme drastically reduces the number of potential energy calculations in FMM, often by up 80%. Fitting up through the highest order tested here (fifth order) gave the best results for all grid spacings. For QSM, using enhanced Shepard interpolation gave slightly better results than using the usual second order approximate, damped Broyden-Fletcher-Goldfarb-Shanno updated Hessian to approximate the surface. To test these methods we examined two analytic potentials, the rotational dihedral potential of alanine dipeptide and the S(N)2 reaction of methyl chloride with fluoride.

Efficient parameterization of torsional terms for force fields.

A novel method is presented for fitting force‐field dihedral angles using an ensemble of structures generated from an ab initio Monte Carlo simulation. Importance sampling is used to achieve an efficient algorithm using a low level of theory to minimize the system at each step with the dihedral angles constrained, followed by dihedral fitting using the single point energies at a higher level of theory. The resulting method is an order of magnitude more efficient than the traditional method of doing a constrained scan over each dihedral independently. Also as the sampling is more uniformly distributed, the full surface is approximated to a greater accuracy. The dihedral fitting is done with a nonlinear optimization method to vary the phase as well as the force constant. The utility of the method is demonstrated by fitting dihedrals of methyl L‐lactate, diisopropyl fluorophosphate, isopentenyl phosphate, a leucine dipeptide, and two inhibitors of Signal Transducer and Activator of Transcription 5. The results show that the Monte Carlo scheme is more efficient than constrained scans and is particularly effective at approximating the underlying potential energy surface when the dihedral degrees are coupled.

A combined explicit-implicit method for high accuracy reaction path integration.

We present the use of an optimal combined explicit-implicit method for following the reaction path to high accuracy. This is in contrast to most purely implicit reaction path integration algorithms, which are only efficient on stiff ordinary differential equations. The defining equation for the reaction path is considered to be stiff, however, we show here that the reaction path is not uniformly stiff and instead is only stiff near stationary points. The optimal algorithm developed in this work is a combination of explicit and implicit methods with a simple criterion to switch between the two. Using three different chemical reactions, we combine and compare three different integration methods: the implicit trapezoidal method, an explicit stabilized third order algorithm implemented in the code DUMKA3 and the traditional explicit fourth order Runge-Kutta method written in the code RKSUITE. The results for high accuracy show that when the implicit trapezoidal method is combined with either explicit method the number of energy and gradient calculations can potentially be reduced by almost a half compared with integrating either method alone. Finally, to explain the improvements of the combined method we expand on the concepts of stability and stiffness and relate them to the efficiency of integration methods.