Puddle-jumping: a flexible sampling algorithm for rare event systems

Methods for Classical-Mechanical Molecular Simulation in Chemistry: Achievements, Limitations, Perspectives

More than a half century ago it became feasible to simulate, using classical-mechanical equations of motion, the dynamics of molecular systems on a computer. Since then classical-physical molecular simulation has become an integral part of chemical research. It is widely applied in a variety of branches of chemistry and has significantly contributed to the development of chemical knowledge. It offers understanding and interpretation of experimental results, semiquantitative predictions for measurable and nonmeasurable properties of substances, and allows the calculation of properties of molecular systems under conditions that are experimentally inaccessible. Yet, molecular simulation is built on a number of assumptions, approximations, and simplifications which limit its range of applicability and its accuracy. These concern the potential-energy function used, adequate sampling of the vast statistical-mechanical configurational space of a molecular system and the methods used to compute particular properties of chemical systems from statistical-mechanical ensembles. During the past half century various methodological ideas to improve the efficiency and accuracy of classical-physical molecular simulation have been proposed, investigated, evaluated, implemented in general simulation software or were abandoned. The latter because of fundamental flaws or, while being physically sound, computational inefficiency. Some of these methodological ideas are briefly reviewed and the most effective methods are highlighted. Limitations of classical-physical simulation are discussed and perspectives are sketched.

Global structure search for molecules on surfaces: Efficient sampling with curvilinear coordinates

Efficient structure search is a major challenge in computational materials science. We present a modification of the basin hopping global geometry optimization approach that uses a curvilinear coordinate system to describe global trial moves. This approach has recently been shown to be efficient in structure determination of clusters [C. Panosetti et al., Nano Lett. 15, 8044-8048 (2015)] and is here extended for its application to covalent, complex molecules and large adsorbates on surfaces. The employed automatically constructed delocalized internal coordinates are similar to molecular vibrations, which enhances the generation of chemically meaningful trial structures. By introducing flexible constraints and local translation and rotation of independent geometrical subunits, we enable the use of this method for molecules adsorbed on surfaces and interfaces. For two test systems, trans-β-ionylideneacetic acid adsorbed on a Au(111) surface and methane adsorbed on a Ag(111) surface, we obtain superior performance of the method compared to standard optimization moves based on Cartesian coordinates.

Towards fast, rigorous and efficient conformational sampling of biomolecules: Advances in accelerated molecular dynamics

BACKGROUND

Accelerated molecular dynamics (aMD) has been proven to be a powerful biasing method for enhanced sampling of biomolecular conformations on general-purpose computational platforms. Biologically important long timescale events that are beyond the reach of standard molecular dynamics can be accessed without losing the detailed atomistic description of the system in aMD. Over other biasing methods, aMD offers the advantages of tuning the level of acceleration to access the desired timescale without any advance knowledge of the reaction coordinate.

SCOPE OF REVIEW

Recent advances in the implementation of aMD and its applications to small peptides and biological macromolecules are reviewed here along with a brief account of all the aMD variants introduced in the last decade.

MAJOR CONCLUSIONS

In comparison to the original implementation of aMD, the recent variant in which all the rotatable dihedral angles are accelerated (RaMD) exhibits faster convergence rates and significant improvement in statistical accuracy of retrieved thermodynamic properties. RaMD in conjunction with accelerating diffusive degrees of freedom, i.e. dual boosting, has been rigorously tested for the most difficult conformational sampling problem, protein folding. It has been shown that RaMD with dual boosting is capable of efficiently sampling multiple folding and unfolding events in small fast folding proteins.

GENERAL SIGNIFICANCE

RaMD with the dual boost approach opens exciting possibilities for sampling multiple timescales in biomolecules. While equilibrium properties can be recovered satisfactorily from aMD-based methods, directly obtaining dynamics and kinetic rates for larger systems presents a future challenge. This article is part of a Special Issue entitled Recent developments of molecular dynamics.

Improved Statistical Sampling and Accuracy with Accelerated Molecular Dynamics on Rotatable Torsions

In enhanced sampling techniques, the precision of the reweighted ensemble properties is often decreased due to large variation in statistical weights and reduction in the effective sampling size. To abate this reweighting problem, here, we propose a general accelerated molecular dynamics (aMD) approach in which only the rotatable dihedrals are subjected to aMD (RaMD), unlike the typical implementation wherein all dihedrals are boosted (all-aMD). Nonrotatable and improper dihedrals are marginally important to conformational changes or the different rotameric states. Not accelerating them avoids the sharp increases in the potential energies due to small deviations from their minimum energy conformations and leads to improvement in the precision of RaMD. We present benchmark studies on two model dipeptides, Ace-Ala-Nme and Ace-Trp-Nme, simulated with normal MD, all-aMD, and RaMD. We carry out a systematic comparison between the performances of both forms of aMD using a theory that allows quantitative estimation of the effective number of sampled points and the associated uncertainty. Our results indicate that, for the same level of acceleration and simulation length, as used in all-aMD, RaMD results in significantly less loss in the effective sample size and, hence, increased accuracy in the sampling of φ-ψ space. RaMD yields an accuracy comparable to that of all-aMD, from simulation lengths 5 to 1000 times shorter, depending on the peptide and the acceleration level. Such improvement in speed and accuracy over all-aMD is highly remarkable, suggesting RaMD as a promising method for sampling larger biomolecules.

Enhanced and effective conformational sampling of protein molecular systems for their free energy landscapes

Protein folding and protein-ligand docking have long persisted as important subjects in biophysics. Using multicanonical molecular dynamics (McMD) simulations with realistic expressions, i.e., all-atom protein models and an explicit solvent, free-energy landscapes have been computed for several systems, such as the folding of peptides/proteins composed of a few amino acids up to nearly 60 amino-acid residues, protein-ligand interactions, and coupled folding and binding of intrinsically disordered proteins. Recent progress in conformational sampling and its applications to biophysical systems are reviewed in this report, including descriptions of several outstanding studies. In addition, an algorithm and detailed procedures used for multicanonical sampling are presented along with the methodology of adaptive umbrella sampling. Both methods control the simulation so that low-probability regions along a reaction coordinate are sampled frequently. The reaction coordinate is the potential energy for multicanonical sampling and is a structural identifier for adaptive umbrella sampling. One might imagine that this probability control invariably enhances conformational transitions among distinct stable states, but this study examines the enhanced conformational sampling of a simple system and shows that reasonably well-controlled sampling slows the transitions. This slowing is induced by a rapid change of entropy along the reaction coordinate. We then provide a recipe to speed up the sampling by loosening the rapid change of entropy. Finally, we report all-atom McMD simulation results of various biophysical systems in an explicit solvent.

Conformational optimization with natural degrees of freedom: a novel stochastic chain closure algorithm

The present article introduces a set of novel methods that facilitate the use of "natural moves" or arbitrary degrees of freedom that can give rise to collective rearrangements in the structure of biological macromolecules. While such "natural moves" may spoil the stereochemistry and even break the bonded chain at multiple locations, our new method restores the correct chain geometry by adjusting bond and torsion angles in an arbitrary defined molten zone. This is done by successive stages of partial closure that propagate the location of the chain break backwards along the chain. At the end of these stages, the size of the chain break is generally reduced so much that it can be repaired by adjusting the position of a single atom. Our chain closure method is efficient with a computational complexity of O(N(d)), where N(d) is the number of degrees of freedom used to repair the chain break. The new method facilitates the use of arbitrary degrees of freedom including the "natural" degrees of freedom inferred from analyzing experimental (X-ray crystallography and nuclear magnetic resonance [NMR]) structures of nucleic acids and proteins. In terms of its ability to generate large conformational moves and its effectiveness in locating low energy states, the new method is robust and computationally efficient.

Using Selectively Applied Accelerated Molecular Dynamics to Enhance Free Energy Calculations

Accelerated molecular dynamics (aMD) has been shown to enhance conformational space sampling relative to classical molecular dynamics; however, the exponential reweighting of aMD trajectories, which is necessary for the calculation of free energies relating to the classical system, is oftentimes problematic, especially for systems larger than small poly peptides. Here, we propose a method of accelerating only the degrees of freedom most pertinent to sampling, thereby reducing the total acceleration added to the system and improving the convergence of calculated ensemble averages, which we term selective aMD. Its application is highlighted in two biomolecular cases. First, the model system alanine dipeptide is simulated with classical MD, all-dihedral aMD, and selective aMD, and these results are compared to the infinite sampling limit as calculated with metadynamics. We show that both forms of aMD enhance the convergence of the underlying free energy landscape by 5-fold relative to classical MD; however, selective aMD can produce improved statistics over all-dihedral aMD due to the improved reweighting. Then we focus on the pharmaceutically relevant case of computing the free energy of the decoupling of oseltamivir in the active site of neuraminidase. Results show that selective aMD greatly reduces the cost of this alchemical free energy transformation, whereas all-dihedral aMD produces unreliable free energy estimates.

Modelling proteins: conformational sampling and reconstruction of folding kinetics

In the last decades biomolecular simulation has made tremendous inroads to help elucidate biomolecular processes in-silico. Despite enormous advances in molecular dynamics techniques and the available computational power, many problems involve long time scales and large-scale molecular rearrangements that are still difficult to sample adequately. In this review we therefore summarise recent efforts to fundamentally improve this situation by decoupling the sampling of the energy landscape from the description of the kinetics of the process. Recent years have seen the emergence of many advanced sampling techniques, which permit efficient characterisation of the relevant family of molecular conformations by dispensing with the details of the short-term kinetics of the process. Because these methods generate thermodynamic information at best, they must be complemented by techniques to reconstruct the kinetics of the process using the ensemble of relevant conformations. Here we review recent advances for both types of methods and discuss their perspectives to permit efficient and accurate modelling of large-scale conformational changes in biomolecules. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.

Principles of flexible protein-protein docking

Treating flexibility in molecular docking is a major challenge in cell biology research. Here we describe the background and the principles of existing flexible protein-protein docking methods, focusing on the algorithms and their rational. We describe how protein flexibility is treated in different stages of the docking process: in the preprocessing stage, rigid and flexible parts are identified and their possible conformations are modeled. This preprocessing provides information for the subsequent docking and refinement stages. In the docking stage, an ensemble of pre-generated conformations or the identified rigid domains may be docked separately. In the refinement stage, small-scale movements of the backbone and side-chains are modeled and the binding orientation is improved by rigid-body adjustments. For clarity of presentation, we divide the different methods into categories. This should allow the reader to focus on the most suitable method for a particular docking problem.

Estimating kinetic rates from accelerated molecular dynamics simulations: alanine dipeptide in explicit solvent as a case study

Molecular dynamics (MD) simulation is the standard computational technique used to obtain information on the time evolution of the conformations of proteins and many other molecular systems. However, for most biological systems of interest, the time scale for slow conformational transitions is still inaccessible to standard MD simulations. Several sampling methods have been proposed to address this issue, including the accelerated molecular dynamics method. In this work, we study the extent of sampling of the phi/psi space of alanine dipeptide in explicit water using accelerated molecular dynamics and present a framework to recover the correct kinetic rate constant for the helix to beta-strand transition. We show that the accelerated MD can drastically enhance the sampling of the phi/psi conformational phase space when compared to normal MD. In addition, the free energy density plots of the phi/psi space show that all minima regions are accurately sampled and the canonical distribution is recovered. Moreover, the kinetic rate constant for the helix to beta-strand transition is accurately estimated from these simulations by relating the diffusion coefficient to the local energetic roughness of the energy landscape. Surprisingly, even for such a low barrier transition, it is difficult to obtain enough transitions to accurately estimate the rate constant when one uses normal MD.

Probing protein fold space with a simplified model

We probe the stability and near-native energy landscape of protein fold space using powerful conformational sampling methods together with simple reduced models and statistical potentials. Fold space is represented by a set of 280 protein domains spanning all topological classes and having a wide range of lengths (33-300 residues) amino acid composition and number of secondary structural elements. The degrees of freedom are taken as the loop torsion angles. This choice preserves the native secondary structure but allows the tertiary structure to change. The proteins are represented by three-point per residue, three-dimensional models with statistical potentials derived from a knowledge-based study of known protein structures. When this space is sampled by a combination of parallel tempering and equi-energy Monte Carlo, we find that the three-point model captures the known stability of protein native structures with stable energy basins that are near-native (all alpha: 4.77 A, all beta: 2.93 A, alpha/beta: 3.09 A, alpha+beta: 4.89 A on average and within 6 A for 71.41%, 92.85%, 94.29% and 64.28% for all-alpha, all-beta, alpha/beta and alpha+beta, classes, respectively). Denatured structures also occur and these have interesting structural properties that shed light on the different landscape characteristics of alpha and beta folds. We find that alpha/beta proteins with alternating alpha and beta segments (such as the beta-barrel) are more stable than proteins in other fold classes.

A skewed-momenta method to efficiently generate conformational-transition trajectories

We present a novel computational method, the skewed-momenta method (Skew'M), which applies a bias to the Maxwell distribution of initial momenta used to generate ensembles of trajectories. As a result, conformational transitions are accentuated and kinetic properties are calculated more effectively. The connection to the related puddle jumping method is discussed. A reweighting scheme permits the exact calculation of kinetic properties. Applications are presented for the rapid calculation of rate constants for molecular isomerization, and for the efficient reconstruction of free-energy profiles using a straightforward modification of the Jarzynski identity.

On searching in, sampling of, and dynamically moving through conformational space of biomolecular systems: A review

Methods to search for low-energy conformations, to generate a Boltzmann-weighted ensemble of configurations, or to generate classical-dynamical trajectories for molecular systems in the condensed liquid phase are briefly reviewed with an eye to application to biomolecular systems. After having chosen the degrees of freedom and method to generate molecular configurations, the efficiency of the search or sampling can be enhanced in various ways: (i) efficient calculation of the energy function and forces, (ii) application of a plethora of search enhancement techniques, (iii) use of a biasing potential energy term, and (iv) guiding the sampling using a reaction or transition pathway. The overview of the available methods should help the reader to choose the combination that is most suitable for the biomolecular system, degrees of freedom, interaction function, and molecular or thermodynamic properties of interest.

Army ants algorithm for rare event sampling of delocalized nonadiabatic transitions by trajectory surface hopping and the estimation of sampling errors by the bootstrap method

The most widely used algorithm for Monte Carlo sampling of electronic transitions in trajectory surface hopping (TSH) calculations is the so-called anteater algorithm, which is inefficient for sampling low-probability nonadiabatic events. We present a new sampling scheme (called the army ants algorithm) for carrying out TSH calculations that is applicable to systems with any strength of coupling. The army ants algorithm is a form of rare event sampling whose efficiency is controlled by an input parameter. By choosing a suitable value of the input parameter the army ants algorithm can be reduced to the anteater algorithm (which is efficient for strongly coupled cases), and by optimizing the parameter the army ants algorithm may be efficiently applied to systems with low-probability events. To demonstrate the efficiency of the army ants algorithm, we performed atom-diatom scattering calculations on a model system involving weakly coupled electronic states. Fully converged quantum mechanical calculations were performed, and the probabilities for nonadiabatic reaction and nonreactive deexcitation (quenching) were found to be on the order of 10(-8). For such low-probability events the anteater sampling scheme requires a large number of trajectories ( approximately 10(10)) to obtain good statistics and converged semiclassical results. In contrast by using the new army ants algorithm converged results were obtained by running 10(5) trajectories. Furthermore, the results were found to be in excellent agreement with the quantum mechanical results. Sampling errors were estimated using the bootstrap method, which is validated for use with the army ants algorithm.

Conformational sampling for the impatient

Several new methods for sampling conformations of biomolecules have appeared recently. A brief review thereof is presented, with particular emphasis on applications that have been published, and suitability for different kinds of systems. Four methods (namely: RESPA, replica-exchange molecular dynamics, CONCOORD and Gaussian network method) are readily applicable for biomolecular systems.