Current computer simulations of biomolecules typically make use of classical molecular dynamics methods, as a very large number (tens to hundreds of thousands) of atoms are involved over timescales of many nanoseconds. The methodology for treating short-range bonded and van der Waals interactions has matured. However, long-range electrostatic interactions still represent a bottleneck in simulations. In this article, we introduce the basic issues for an accurate representation of the relevant electrostatic interactions. In spite of the huge computational time demanded by most biomolecular systems, it is no longer necessary to resort to uncontrolled approximations such as the use of cutoffs. In particular, we discuss the Ewald summation methods, the fast particle mesh methods, and the fast multipole methods. We also review recent efforts to understand the role of boundary conditions in systems with long-range interactions, and conclude with a short perspective on future trends.

We have implemented classical Ewald and particle-mesh Ewald (PME) based treatments of fixed and induced point dipoles into the sander molecular dynamics (MD) module of AMBER 6. During MD the induced dipoles can be propagated along with the atomic positions either by iteration to self-consistency at each time step, or by a Car–Parrinello (CP) technique using an extended Lagrangian formalism. In this paper we present the derivation of the new algorithms and compare the various options with respect to accuracy, efficiency, and effect on calculated properties of a polarizable water model. The use of PME for electrostatics of fixed charges and induced dipoles together with a CP treatment of dipole propagation in MD simulations leads to a cost overhead of only 33% above that of MD simulations using standard PME with fixed charges, allowing the study of polarizability in large macromolecular systems.

CONTENTS 1. Introduction 779 1.1. Electrostatic Problem 780 2. Methods for Computing the Long-Range Electrostatic Interactions 781 2.1. Ewald Summations 782 2.2. Particle−Mesh Methods That Use Fourier Transforms 784 2.3. Particle−Mesh Methods in Real Space 784 2.4. Fast Multipole Methods 786 2.5. Local Methods 786 2.6. Truncation Methods 787 2.6.1. Reaction Field Methods 789 2.7. Charge-Group Methods 789 2.8. Treatments and Artifacts of Boundary Conditions 790 3. Accurate Representation of the Electronic Charge 791 3.1. Charge Assignment Schemes 791 3.2. Point Multipoles 792 3.3. Continuous Representations of Molecular Charge Density 793 3.4. Polarization 795 3.5. Charge Transfer 797 3.6. Polarizable Force fields 798 4. Electrostatics in Multiscale Modeling 800 4.1. Long-Range Electrostatics in QM/MM 800 4.2. Continuous Functions in QM/MM 801 4.3. Electrostatics in Coarse-Grained Models 802 5. Other Developments 804 6. Summary and Perspective 805 Author Information 806 Corresponding Author 806 Notes 806 Biographies 806 Acknowledgments 806 References 806

The accurate simulation of biologically active macromolecules faces serious limitations that originate in the treatment of electrostatics in the empirical force fields. The current use of "partial charges" is a significant source of errors, since these vary widely with different conformations. By contrast, the molecular electrostatic potential (MEP) obtained through the use of a distributed multipole moment description, has been shown to converge to the quantum MEP outside the van der Waals surface, when higher order multipoles are used. However, in spite of the considerable improvement to the representation of the electronic cloud, higher order multipoles are not part of current classical biomolecular force fields due to the excessive computational cost. In this paper we present an efficient formalism for the treatment of higher order multipoles in Cartesian tensor formalism. The Ewald "direct sum" is evaluated through a McMurchie-Davidson formalism [L. McMurchie and E. Davidson, J. Comput. Phys. 26, 218 (1978)]. The "reciprocal sum" has been implemented in three different ways: using an Ewald scheme, a particle mesh Ewald (PME) method, and a multigrid-based approach. We find that even though the use of the McMurchie-Davidson formalism considerably reduces the cost of the calculation with respect to the standard matrix implementation of multipole interactions, the calculation in direct space remains expensive. When most of the calculation is moved to reciprocal space via the PME method, the cost of a calculation where all multipolar interactions (up to hexadecapole-hexadecapole) are included is only about 8.5 times more expensive than a regular AMBER 7 [D. A. Pearlman et al., Comput. Phys. Commun. 91, 1 (1995)] implementation with only charge-charge interactions. The multigrid implementation is slower but shows very promising results for parallelization. It provides a natural way to interface with continuous, Gaussian-based electrostatics in the future. It is hoped that this new formalism will facilitate the systematic implementation of higher order multipoles in classical biomolecular force fields.

We present an Adaptively Biased Molecular Dynamics (ABMD) method for the computation of the free energy surface of a reaction coordinate using non-equilibrium dynamics. The ABMD method belongs to the general category of umbrella sampling methods with an evolving biasing potential, and is inspired by the metadynamics method. The ABMD method has several useful features, including a small number of control parameters, and an O(t) numerical cost with molecular dynamics time t. The ABMD method naturally allows for extensions based on multiple walkers and replica exchange, where different replicas can have different temperatures and/or collective variables. This is beneficial not only in terms of the speed and accuracy of a calculation, but also in terms of the amount of useful information that may be obtained from a given simulation. The workings of the ABMD method are illustrated via a study of the folding of the Ace-GGPGGG-Nme peptide in a gaseous and solvated environment.