Classical Electrostatics for Biomolecular Simulations † ‡ § ∥ G

Review pubs.acs.org/CR Classical Electrostatics for Biomolecular Simulations † ‡ § ∥ G. Andreś Cisneros, Mikko Karttunen, Pengyu Ren, and Celeste Sagui*, † Department of Chemistry, Wayne State University, Detroit, Michigan 48202, United States ‡ Department of Chemistry and Waterloo Institute for Nanotechnology, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 § Department of Biomedical Engineering, BME 5.202M, The University of Texas at Austin, 1 University Station, C0800, Austin, Texas 78712-1062, United States ∥ Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, United States Biographies 806 Acknowledgments 806 References 806 1. INTRODUCTION Classical atomistic simulations, also known as molecular mechanics simulations, use simple potential-energy functions to model molecular systems at the atomic level. In this representation, atoms or groups of atoms are represented as spherical particles that interact through relatively simple potential functions such as Hooke’s law and Lennard-Jones and Coulomb potentials. These representations are then used CONTENTS to sample the conformational phase space of the molecules via 1. Introduction 779 simulation techniques such as Monte Carlo, ligand docking, 1.1. Electrostatic Problem 780 and molecular dynamics (MD). In MD simulations, the 2. Methods for Computing the Long-Range Electro- particles obey classical equations of motion, generally Newton’s static Interactions 781 laws or Langevin dynamics, which allow for the characterization 2.1. Ewald Summations 782 of the time evolution of the molecular structures, their 2.2. Particle−Mesh Methods That Use Fourier fluctuations and interactions, and therefore the investigation Transforms 784 of the system’s kinetic and thermodynamical properties. 2.3. Particle−Mesh Methods in Real Space 784 Since their introduction to the physics community during the 2.4. Fast Multipole Methods 786 1950s,1 MD methods have grown in complexity with 2.5. Local Methods 786 refinements both of the accuracy of the energy functionals 2.6. Truncation Methods 787 and of the sophistication of the methods used for the sampling 2.6.1. Reaction Field Methods 789 of the relevant phase space. The force fields used in 2.7. Charge-Group Methods 789 biomolecular simulations include a set of potentials based on 2.8. Treatments and Artifacts of Boundary physical models, along with a set of associated parameters Conditions 790 which are obtained by fitting to experimental and/or quantum 3. Accurate Representation of the Electronic Charge 791 simulations. The potentials are mathematical functions of the 3.1. Charge Assignment Schemes 791 nuclear coordinates only, since the Born−Oppenheimer 3.2. Point Multipoles 792 approximation2 allows the separation of the electronic and 3.3. Continuous Representations of Molecular nuclear degrees of freedom: classical force fields consider Charge Density 793 explicitly the latter, while the electronic charge is approximated 3.4. Polarization 795 by distributed charges or multipoles. Bonded atoms are 3.5. Charge Transfer 797 represented by two-body, three-body, and four-body terms, 3.6. Polarizable Force fields 798 based on bond distances and bond and dihedral angles. 4. Electrostatics in Multiscale Modeling 800 Nonbonded interactions, commonly modeled by Lennard- 4.1. Long-Range Electrostatics in QM/MM 800 Jones and Coulomb potentials, are generally described by 4.2. Continuous Functions in QM/MM 801 pairwise interactions. 4.3. Electrostatics in Coarse-Grained Models 802 Long-range electrostatic interactions are crucial for the 5. Other Developments 804 stability of proteins, nucleic acids, glycomolecules, lipids, and 6. Summary and Perspective 805 other macromolecules, and their interactions with solvent, ions, Author Information 806 Corresponding Author 806 Received: November 20, 2012 Notes 806 Published: August 27, 2013 © 2013 American Chemical Society 779 dx.doi.org/10.1021/cr300461d | Chem. Rev. 2014, 114, 779−814 Chemical Reviews Review and other molecules. Electrostatic interactions in biomolecular atom finds itself. Second, one has to decide how the assigned simulations have typically been modeled using the atom- atomic density should be represented mathematically. centered “partial-charge” approximation in which the full charge The partitioning of the extended molecular charge density of the system is replaced by point, fractional charges distribution is a key issue for improving the accuracy of current assigned to every atom. For instance, the simplest models of force fields for large-scale biomolecular simulations. Tradition- water assign one partial charge to each atom of the molecule. If ally, electrostatic interactions have been modeled using a set of higher accuracy is required to reproduce a specific property of fixed atom-centered point charges or “partial charges”. In fact, water, then extra charges (representing, for instance, the lone popular classical MD codes for biomolecular simulations assign pairs) and/or multipoles and/or polarization are added to the partial charges to virtually every atom in the system. The most water model. Even in the “simple”, purely partial-charge model popular methods for extracting charges from molecular wave of biomolecules, the long-range Coulomb interactions quickly functions are based on a fitting of the atomic charges to the become the computational bottleneck. Their treatment molecular electrostatic potential (MEP) computed with ab demands carefully constructed algorithms in order to avoid initio or semiempirical methods5 outside the van der Waals artifacts3 and to take advantage of the existing and emerging surface. These nonbonded potentials are then expressed as a computer architectures. sum of spherically symmetric atom−atom potentials. Such a Historically, simple models of biological molecules have been description is known to represent an important source of errors used mainly out of necessity due to restraints imposed by for current force fields,6 primarily because the monopoles or absence of experimental data (or data at low resolution) and partial charges can vary enormously with conformational algorithmic and computational limitations. In many cases, very changes.7 In fact, a realistic physical molecular representation simple models have proved quite successful, such as the coarse- generally requires dipole moments (e.g., to model the lone grained and lattice models used to identify the main features of pairs), and quadrupole moments (e.g., to model the π-bonds). protein folding. However, biological systems in general are very Alternatively, more charge sites can be added to the molecule.8 complex, and experimental data is often at too low resolution to In principle, such a “distributed multipole” description can infer details of, for instance, molecular recognition processes. exactly describe the potential due to the true charge density, at Since one of the main reasons to use molecular modeling is to points distant from the expansion centers where “penetration” predict molecular properties that are difficult to observe effects are negligible.6 However, even with these improvements, experimentally, molecular modeling often finds itself in a the fit to the MEP remains poor in regions near the atomic paradoxical situation where its predictions cannot be validated nuclei where the charge densities overlap. As a consequence, experimentally. What has become clear, though, is that the the electrostatic interaction energy must be corrected for the 9 monopole electrostatic approximation is inadequate, and that “penetration” effects at close range. Finally, even if a the importance of an accurate representation of electrostatics distributed multipole representation gives excellent agreement cannot be overemphasized for challenging situations such as with the MEP outside the van der Waals surface, and even if molecular recognition and computer-aided drug design.4 continuous functions (such as Gaussians) are used to represent In this review, we primarily treat electrostatic methods for the electrostatic potential inside the van der Waals surface, a classical biomolecular simulations in explicit solvent, with totally reliable and stable representation of the electronic special emphasis on the accuracy of the electrostatic density requires the inclusion of polarization to account for the representation. The first part of the article is dedicated to variation of the electronic density caused by intra- and reviewing computational methods for dealing with the long- intermolecular conformational changes, as well as general range nature of the electrostatic interactions, including changes in the atomic environment. Additional complications traditional methods as well as recent extensions and develop- arise when there are changes in ionization states, as discussed in ments. The second part of the article is dedicated to reviewing section 5. current methods for representing the molecular electronic To begin, consider the simplest representation of electro- charge cloud, that go well beyond the point-charge statics given by the monopolar approximation. The long-range representation. In addition, we discuss various multiscale effects are most pronounced for this potential, which decays approaches at the quantum mechanics/molecular mechanics with distance r as 1/r. The charge-assignment scheme based on level and at the molecular mechanics/coarse-grain level. The fitting to the MEP is mathematically an under-determined general trends in both

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