Computational Molecular Dynamics: Challenges, Methods, Ideas

Steered molecular dynamics (SMD) induces unbinding of ligands and conformational changes in biomolecules on time scales accessible to molecular dynamics simulations. Time-dependent external forces are applied to a system, and the responses of the system are analyzed. SMD has already provided important qualitative insights into biologically relevant problems, as demonstrated here for applications ranging from identification of ligand binding pathways to explanation of elastic properties of proteins. First attempts to deduce potentials of mean force by discounting irreversible work performed on the system are summarized. The non-equilibrium statistical mechanics underlying analysis of SMD data is outlined.

The function of many important proteins comes from their dynamic properties, and their ability to undergo conformational transitions. These may be small loop movements that allow access to the protein’s active site, or large movements such as those of motor proteins that are implicated with muscular extension. Yet, in spite of the increasing number of three-dimensional crystal structures of proteins in different conformations, not much is known about the driving forces of these transitions. As an initial step towards exploring the conformational and energetic landscape of protein kinases by computational methods, intramolecular energies and hydration free energies were calculated for different conformations of the catalytic domain of cAMP-dependent protein kinase (cAPK) with a continuum (Poisson) model for the electrostatics. In this paper, we will put the previous results into context and discuss possible extensions into the dynamic regime.

Molecular dynamics (MD) simulations of proteins provide descriptions of atomic motions, which allow to relate observable properties of proteins to microscopic processes. Unfortunately, such MD simulations require an enormous amount of computer time and, therefore, are limited to time scales of nanoseconds. We describe first a fast multiple time step structure adapted multipole method (FAMUSAMM) to speed up the evaluation of the computationally most demanding Coulomb interactions in solvated protein models, secondly an application of this method aiming at a microscopic understanding of single molecule atomic force microscopy experiments, and, thirdly, a new method to predict slow conformational motions at microsecond time scales.

A model of the conformational transitions of the nucleic acid molecule during the water adsorption-desorption cycle is proposed. The nucleic acid-water system is considered as an open system. The model describes the transitions between three main conformations of wet nucleic acid samples: A-, B- and unordered forms. The analysis of kinetic equations shows the non-trivial bifurcation behaviour of the system which leads to the multistability. This fact allows one to explain the hysteresis phenomena observed experimentally in the nucleic acid-water system. The problem of self-organization in the nucleic acid-water system is of great importance for revealing physical mechanisms of the functioning of nucleic acids and for many specific practical fields.

Protein-ligand interactions control a majority of cellular processes and are the basis of many drug therapies. First, this paper summarizes experimental approaches used to characterize the interactions between proteins and small molecules: equilibrium measurement of binding constant and standard free energy of binding and the dynamic approach of ligand extraction via atomic force microscopy. Next, the paper reviews ideas about the origin of different component terms that contribute to the the stability of protein-ligand complexes. Then, theoretical approaches to studying protein-small molecule interactions are addressed, including forced extraction of ligand and perturbation methods for calculating potentials of mean force and free energies for molecular transformation. Last, these approaches are illustrated with several recent studies from our laboratory: (1) binding of water in cavities inside proteins, (2) calculation of binding free energy from “first principles” by a new application of molecular transformation, and (3) extraction of a small ligand (xenon) from a hydrophobic cavity in mutant T4-lysozyme L99A.

To estimate free energy differences from a single simulation of the initial state one may either, use a series expansion of the free energy around the initial state, make an assumption in regard to the functional form of the free energy or treat the mutation as a single step perturbation. Of these the perturbation approach holds the greatest promise. The perturbation approach is fast, easy to implement and does not depend on empirically derived parameters or assumptions. Given an appropriate reference state the perturbation approach can be used to rapidly estimate solvation or binding free energies of a wide range of related compounds for use in force field development or structure based drug design.

The Conformational Free Energy Thermodynamic Integration (CFTI) method, a new multidimensional approach for conformational free energy simulations, is presented. The method is applied to two problems of biochemical interest: exploration of the free energy surfaces of helical alanine (Ala) and α-methylalanine (Aib) homopeptides in vacuum and the cost of pre-organization of the opioid peptide Tyr-D-Pen-Gly-Phe-D-Pen (DPDPE) peptide for disulfide bond formation. In the CFTI approach a single molecular dynamics simulation with all ø and ψ dihedrals kept fixed yields the complete conformational free energy gradient for the studied peptides. For regular structures of model peptides (Ala)_n and (Aib)_n where n=6,8,10 and Aib is a-methylalanine in vacuum, free energy maps in the helical region of ø- ψ space are calculated, and used to roughly locate stable states. The locations of the free energy minima are further refined by the novel procedure of free energy optimization by steepest descent down the gradient, leading to structures in excellent agreement with experimental data. The stability of the minima with respect to deformations is studied by analysis of second derivatives of the free energy surface. Analysis of free energy components and molecular structures uncovers the molecular mechanism for the propensity of Aib peptides for the 3₁₀-helix structure in the interplay between the quality and quantity of hydrogen bonds. For the linear form of the DPDPE peptide in solution, free energy differences are calculated between four conformers: Cyc, representing the structure adopted by the linear peptide prior to disulfide bond formation, β _c and β _E, two slightly different β-turns previously identified as representative, stable structures of the peptide, and Ext, an extended structure. The simulations indicate that β _E is the most stable of the studied conformers of linear DPDPE in aqueous solution, with β _c , Cyc and Ext having free energies higher by 2.3, 6.3, and 28.2 kcal/mol, respectively. The free energy differences of 4.0 kcal/mol between β _c and Cyc, and 6.3 kcal/mol between β _E and Cyc, reflect the cost of pre-organizing the linear peptide into a conformation conducive for disulfide bond formation. Such a conformational change is a pre-requisite for the chemical reaction of S-S bond formation to proceed.

This article provides an overview of an algorithm used for the prediction of ionization constants of titratable residues in proteins. The algorithm is based on an assumption that the difference in protonation behavior of a given group in an isolated state in solution, for which the ionization constant is assumed to be known, and the protonation behavior in the protein environment is purely electrostatic in origin. Calculations of the relevant electrostatic free energies are based on the Poisson-Boltzmann (PB) model of the protein-solvent system and the finitedifference solution to the corresponding PB equation. The resultant multiple site titration problem is treated by one of two methods. The first is a hybrid approach, based on collecting ionizable groups into clusters. The second method is a Monte Carlo approach based on the Metropolis algorithm for extracting a sufficient number of low-energy ionization states out of all possible states, to obtain a correct estimation of thermodynamic properties of the system.

As examples of applications, we present the overall accuracy of predicted ionization constants for about 50 groups in 4 proteins, changes in the average charge of bovine pancreatic trypsin inhibitor at pH 7 along a molecular dynamics trajectory, and finally, we discuss some preliminary results obtained for protein kinases and protein phosphatases.

Two incarnations of the canonical ensemble probability distribution based on the generalization of statistical mechanics of Tsallis are described. A generalization of the law of mass action is used to derive equilibrium constants. Reaction rate constants for barrier crossing are derived using the transition state theory approximation. Monte Carlo and Molecular Dynamics algorithms which can be used to sample Tsallis statistical distributions are defined. The results are used to demonstrate that MC and MD algorithms which sample the Tsallis statistical distributions can be expected to enhance the rate of phase space sampling in simulations of many body systems.

A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the ammo acid labels and of the distances between the C _α atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Ångström. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic programming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has local minima within 1.3-4.7Å of the PDB geometry, with one exception that has an error of 8.5Å.

A personal account of work on long-timestep integration of biomolecular dynamics is presented, emphasizing the limitations, as well as success, of various approaches. These approaches include implicit discretization, separation into harmonic and anharmonic motion, and force splitting; some of these techniques are combined with stochastic dynamics. A Langevin/force-splitting approach for biomolecular simulations termed LN (for its origin in a Langevin/normal-modes scheme) is also described, suitable for general thermodynamic and sampling questions. LN combines force linearization, stochastic dynamics, and force splitting via extrapolation so that the timestep for updating the slow forces can be increased beyond half the period of the fast motions (i.e., 5 fs). This combination of strategies alleviates the severe stability restriction due to resonance artifacts that apply to symplectic force-splitting methods and can yield significant speedup (with respect to small-timestep reference Langevin trajectories). Extensions to sampling problems are natural by this approach.

A stochastic path integral is used to obtain approximate long time trajectories with an almost arbitrary time step. A detailed description of the formalism is provided and an extension that enables the calculations of transition rates is discussed.

The numerical integration of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or Hamiltonian partial differential equations, is a challenging task. Various methods have been suggested to overcome the step-size restrictions of explicit methods such as the Verlet method. Among these are multiple-time-stepping, constrained dynamics, and implicit methods. In this paper, we investigate the suitability of time-reversible, semi-implicit methods. Here semi-implicit means that only the highly oscillatory part is integrated by an implicit method such as the midpoint method or an energy-conserving variant of it. The hope is that such methods will allow one to use a step-size k which is much larger than the period e of the fast oscillations.

Systems with multiple time scales, and with forces which can be subdivided into long and short range components are frequently encountered in computational chemistry. In recent years, new, powerful and efficient methods have been developed to reduce the computational overhead in treating these problems in molecular dynamics simulations. Numerical reversible integrators for dealing with these problems called r-RESPA (Reversible Reference System Propagator Algorithms) are reviewed in this article. r-RESPA leads to considerable speedups in generating molecular dynamics trajectories with no loss of accuracy. When combined with the Hybrid Monte Carlo (HMC) method and used in the Jump-Walking and the Smart-Walking algorithms, r-RESPA is very useful for the enhanced sampling of rough energy landscapes in biomolecules.

Simulation of the dynamics of biomolecules requires the use of a time step in the range 0.5-1 fs to obtain acceptable accuracy. Nevertheless, the bulk of the CPU time is spent computing interactions, such as those due to long-range electrostatics, which vary hardly at all from one time step to the next. This unnecessary computation is dramatically reduced with the use of multiple time stepping methods, such as the Verlet-I/r-RESPA method, which is based on approximating “slow ” forces as widely separated impulses. Indeed, numerical experiments show that time steps of 4 fs are possible for these slow forces but unfortunately also show that a long time step of 5 fs results in a dramatic energy drift. Moreover, this is less pronounced if one uses a yet larger long time step! The cause of the problem can be explained by exact analysis of a simple two degree-of-freedom linear problem, which predicts numerical instability if the time step is just less than half the period of the fastest normal mode. To overcome this, a modification of the impulsive Verlet-I/r-RESPA method is proposed, called the mollified impulse method. The idea is that one modifies the slow part of the potential energy so that it is evaluated at “time averaged ” values of the positions, and one uses the gradient of this modified potential for the slow part of the force. Various versions of the algorithm are implemented for water and numerical results are presented.

The design and analysis of an explicit Split Integration Symplectic Method (SISM) for molecular dynamics (MD) simulations is described. SISM uses an analytical treatment of high frequency motions within a second order generalized leapfrog scheme. SISM is up to an order of magnitude faster than the commonly used leapfrog-Verlet (LFV) algorithm which is of the same order and complexity as SISM. The main restriction on time step in the general MD simulations, which stems from the high-frequency motion is, to a large extent, overcome in this approach. The simulation results for selected examples show that SISM posses long term stability and the ability to use long time steps. This should significantly extend the scope of the presently used algorithms and thus contribute to the general applicability of MD algorithms.

Geometric integrators are numerical timestepping schemes which preserve invariant structures associated to physical dynamical systems. For example, a symplectic integrator is one which preserves a strong differential invariant of the flows of Hamiltonian systems (the 2-form dq ^ dp associated to canonical variables q,p). For constrained systems such as the rigid body, preservation of geometric phase-flow structure is complicated by the choice of coordinates and the need for efficiency. Nowhere are these issues more critical than in the simulation of rigid body systems. In recent work, several alternative geometric approaches to rigid body systems integrators have been proposed and applied in molecular simulation. In this article, these methods are introduced and compared with a simple model problem.

We present new methods for time-dependent quantum mechanical simulations of large polyatomic systems and their applications to photochemical processes in clusters. Two related approaches are discussed: The Classical Separable Potential (CSP) approach, and its extension towards Configuration Interaction (CICSP). The former scheme assumes separability of the vibrational modes of the system, and describes each mode as moving in a mean field due to the other modes. The basic idea, which allows for quantum simulations of hitherto unaccesibly large systems, is that the effective single-mode potentials are obtained from a classical MD simulation that precedes the quantum calculation. The second approach represents an improvement that corrects for correlations between different modes, resulting in a scheme of good accuracy. Applications of the methods are presented for dynamics following photodetachment in a small I−(Ar) ₂ cluster (where comparison with numerically exact calculation is possible) and for photoexcitation dynamics and spectroscopy of atomic and molecular impurities in large clusters, such as I _{2(Ar) 17} and I _{2(Ar) 47} Future directions of method development are suggested in the light of the algorithmic aspects and the applications.

It was revealed that the QCMD model is of canonical Hamiltonian form with symplectic structure, which implies the conservation of energy. An efficient and reliable integrator for transfering these properties to the discrete solution is the symplectic and explicit PICKABACK algorithm. The only drawback of this kind of integrator is the small stepsize in time induced by the splitting techniques used to discretize the quantum evolution operator. Recent investigations concerning Krylov iteration techniques result in alternative approaches which overcome this difficulty for a wide range of problems. By using iterative methods in the evaluation of the quantum time propagator, these techniques allow for the stepsize to adapt to the classical motion and the coupling between the classical and the quantum mechanical subsystem. This yields a drastic reduction of the numerical effort. The pros and cons of both approaches as well as the suitable applications are discussed in the last part.

The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes — analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK.

This paper presents results from quantum molecular dynamics simulations applied to catalytic reactions, focusing on ethylene polymerization by metallocene catalysts. The entire reaction path could be monitored, showing the full molecular dynamics of the reaction. Detailed information on, e.g., the importance of the so-called agostic interaction could be obtained. Also presented are results of static simulations of the Car-Parrinello type, applied to orthorhombic crystalline polyethylene. These simulations for the first time led to a first principles value for the ultimate Young’s modulus of a synthetic polymer with demonstrated basis set convergence, taking into account the full three-dimensional structure of the crystal.

The nonlocal dynamical coherent potential approximation (NDCPA) is formulated to calculate a single-electron(exciton) Green’s function of polaron due to the interaction of an electron(exciton) with phonons with dispersion. This approximation is an extension of the dynamical CPA. The NDCPA provides an efficient means calculating of an approximate Green’s function for a dynamical model of electrons (excitons) strongly coupled to optical or acoustical phonons, in the entire ranges of the electron(exciton)-phonon coupling strengths and electron (exciton) transfer. The electron(exciton)-phonon coupling in the Hamiltonian may involve terms of any order with respect to the phonon operators. A set of recurrent equations is derived in the case of a system at zero temperatures, from which the coherent potential can be obtained as a function of energy E and momentum k. A simple algorithm for the polaron spectra calculations is obtained for a linear electron(exciton)-phonon coupling in the antiadiabatic limit. The algorithm is applied to calculate absorption spectra of excitons linear and locally coupled to phonon without dispersion.

Many realistic biomolecular simulations require use of periodic boundary conditions to create a surface-free environment for the molecule of interest and associated solvent molecules to interact. Electrostatic interactions are the principal computational cost of such simulations. We have implemented two codes: a parallel variant of an Ewald summation method which computes the effect of infinite periodic boundary conditions, and a parallel variant of a multipole algorithm which explicitly computes the interactions within a large but finite periodic system. Each has a regime of applicability, with Ewald favoring smaller systems and fewer processors, and the multipole methods favoring larger systems and more processors. Simulations can now include a full treatment of periodic electrostatics to three or four significant figures of accuracy for a computational cost equivalent to that of a 12Å cutoff simulation.

Parallel molecular dynamics programs employing shared memory or replicated data architectures encounter problems scaling to large numbers of processors. Spatial decomposition schemes offer better performance in theory, but often suffer from complexity of implementation and difficulty in load balancing. In the program NAMD 2, we have addressed these issues with a hybrid decomposition scheme in which atoms are distributed among processors in regularly sized patches while the work involved in computing interactions between patches is decomposed into independently assignable compute objects. When needed, patches are represented on remote processors by proxies. The execution of compute objects takes place in a prioritized message-driven manner, allowing maximum overlap of work and communication without significant programmer effort. In order to avoid obfuscation of the simulation algorithm by the parallel framework, the algorithm associated with a patch is encapsulated by a single function executing in a separate thread. Output and calculations requiring globally reduced quantities are similarly isolated in a single thread executing on the master node. This combination of features allows us to make efficient use of large parallel machines and clusters of multiprocessor workstations while presenting minimal barriers to method development and implementation.

Research interests in molecular dynamics (MD) and its applications have increased significantly over the past few decades. This is due partly to the advances in software and hardware components of computer technology. The main computational goal of recent research work in molecular dynamics has been to reduce the computational cost of the force calculations which evidently accounts for approximately ninety percent of the total CPU time for most MD simulations. This paper describes parallel algorithms for force calculations using the force decomposition approach. These parallel algorithms have been tested and found to be highly portable and scalable. Numerical experiments on IBM SP/2 indicate that these algorithms have improved speedups and efficiencies.