A hybrid recursion method to robustly ensure convergence efficiencies in the simulated scaling based free energy simulations

Recently, we developed an efficient free energy simulation technique, the simulated scaling (SS) method [H. Li et al., J. Chem. Phys. 126, 024106 (2007)], in the framework of generalized ensemble simulations. In the SS simulations, random walks in the scaling parameter space are realized so that both phase space overlap sampling and conformational space sampling can be simultaneously enhanced. To flatten the distribution in the scaling parameter space, in the original SS implementation, the Wang-Landau recursion was employed due to its well-known recursion capability. In the Wang-Landau recursion based SS free energy simulation scheme, at the early stage, recursion efficiencies are high and free energy regions are quickly located, although at this stage, the errors of estimated free energy values are large; at the later stage, the errors of estimated free energy values become smaller, however, recursions become increasingly slow and free energy refinements require very long simulation time. In order to robustly resolve this efficiency problem during free energy refinements, a hybrid recursion strategy is presented in this paper. Specifically, we let the Wang-Landau update method take care of the early stage recursion: the location of target free energy regions, and let the adaptive reweighting method take care of the late stage recursion: the refinements of free energy values. As comparably studied in the model systems, among three possible recursion procedures, the adaptive reweighting recursion approach is the least favorable one because of its low recursion efficiency during free energy region locations; and compared to the original Wang-Landau recursion approach, the proposed hybrid recursion technique can be more robust to guarantee free energy simulation efficiencies.     

Simulated scaling method for localized enhanced sampling and simultaneous "alchemical" free energy simulations: a general method for molecular mechanical, quantum mechanical, and quantum mechanical/molecular mechanical simulations

A potential scaling version of simulated tempering is presented to efficiently sample configuration space in a localized region. The present "simulated scaling" method is developed with a Wang-Landau type of updating scheme in order to quickly flatten the distributions in the scaling parameter lambdam space. This proposal is meaningful for a broad range of biophysical problems, in which localized sampling is required. Besides its superior capability and robustness in localized conformational sampling, this simulated scaling method can also naturally lead to efficient "alchemical" free energy predictions when dual-topology alchemical hybrid potential is applied; thereby simultaneously, both of the chemically and conformationally distinct portions of two end point chemical states can be efficiently sampled. As demonstrated in this work, the present method is also feasible for the quantum mechanical and quantum mechanical/molecular mechanical simulations.     

Generalized simulated tempering realized on expanded ensembles of non-Boltzmann weights

A generalized version of the simulated tempering operated in the expanded ensembles of non-Boltzmann weights has been proposed to mitigate a quasiergodicity problem occurring in simulations of rough energy landscapes. In contrast to conventional simulated tempering employing the Boltzmann weight, our method utilizes a parametrized, generalized distribution as a workhorse for stochastic exchanges of configurations and subensembles transitions, which allows a considerable enhancement for the rate of convergence of Monte Carlo and molecular dynamics simulations using delocalized weights. A feature of our method is that the exploration of the parameter space encouraging subensembles transitions is greatly accelerated using the dynamic update scheme for the weight via the average guide specific to the energy distribution. The performance and characteristic feature of our method have been validated in the liquid-solid transition of Lennard-Jones clusters and the conformational sampling of alanine dipeptide by taking two types of Tsallis [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] expanded ensembles associated with different parametrization schemes.     

Replica exchanging self-guided Langevin dynamics for efficient and accurate conformational sampling

This work presents a replica exchanging self-guided Langevin dynamics (RXSGLD) simulation method for efficient conformational searching and sampling. Unlike temperature-based replica exchanging simulations, which use high temperatures to accelerate conformational motion, this method uses self-guided Langevin dynamics (SGLD) to enhance conformational searching without the need to elevate temperatures. A RXSGLD simulation includes a series of SGLD simulations, with simulation conditions differing in the guiding effect and∕or temperature. These simulation conditions are called stages and the base stage is one with no guiding effect. Replicas of a simulation system are simulated at the stages and are exchanged according to the replica exchanging probability derived from the SGLD partition function. Because SGLD causes less perturbation on conformational distribution than high temperatures, exchanges between SGLD stages have much higher probabilities than those between different temperatures. Therefore, RXSGLD simulations have higher conformational searching ability than temperature based replica exchange simulations. Through three example systems, we demonstrate that RXSGLD can generate target canonical ensemble distribution at the base stage and achieve accelerated conformational searching. Especially for large systems, RXSGLD has remarkable advantages in terms of replica exchange efficiency, conformational searching ability, and system size extensiveness.     

Conformational memories and a simulated annealing program that learns: Application to LTB4

The use of computer simulations in all areas of chemistry is growing rapidly because of the powerful insights that they have provided into many interesting phenomena. As investigators continuously examine more sophisticated problems, they need increasingly more powerful tools. Hence, much effort has gone into the development of algorithms which might extend the scope and power of standard dynamic and Monte Carlo techniques. In the Monte Carlo regime, the most common area subject to improvement is the choice of a trial move. In the ordinary case, trial moves are generated uniformly at random. In the extended and hopefully improved case, trial moves are generated randomly but not uniformly. In this article we present a new and totally general method of biased sampling which is applicable to any flexible molecule. In our method, multiple simulated annealing runs are performed to reveal populated and unpopulated regions of the multidimensional conformation space. The second phase of the simulation is done at a fixed temperature with sampling only from populated regions found in the first phase. Because the simulated annealing runs quickly reveal unpopulated regions of the conformation space, the volume of conformation space that needs to be sampled in the second phase of the algorithm is reduced by many orders of magnitude. Additionally, because no energy minimization is used, these populations represent a canonical ensemble which may be used to estimate conformational free energies. © 1995 by John Wiley & Sons, Inc.     

Hierarchical parallelization of divide-and-conquer density functional tight-binding molecular dynamics and metadynamics simulations

Massively parallel divide-and-conquer density functional tight-binding (DC-DFTB) molecular dynamics and metadynamics simulations are efficient approaches for describing various chemical reactions and dynamic processes of large complex systems via quantum mechanics. In this study, DC-DFTB simulations were combined with multi-replica techniques. Specifically, multiple walkers metadynamics, replica exchange molecular dynamics, and parallel tempering metadynamics methods were implemented hierarchically into the in-house Dcdftbmd program. Test simulations in an aqueous phase of the internal rotation of formamide and conformational changes of dialanine showed that the newly developed extensions increase the sampling efficiency and the exploration capabilities in DC-DFTB configuration space.     

Essential energy space random walks to accelerate molecular dynamics simulations: convergence improvements via an adaptive-length self-healing strategy

Recently, accelerated molecular dynamics (AMD) technique was generalized to realize essential energy space random walks so that further sampling enhancement and effective localized enhanced sampling could be achieved. This method is especially meaningful when essential coordinates of the target events are not priori known; moreover, the energy space metadynamics method was also introduced so that biasing free energy functions can be robustly generated. Despite the promising features of this method, due to the nonequilibrium nature of the metadynamics recursion, it is challenging to rigorously use the data obtained at the recursion stage to perform equilibrium analysis, such as free energy surface mapping; therefore, a large amount of data ought to be wasted. To resolve such problem so as to further improve simulation convergence, as promised in our original paper, we are reporting an alternate approach: the adaptive-length self-healing (ALSH) strategy for AMD simulations; this development is based on a recent self-healing umbrella sampling method. Here, the unit simulation length for each self-healing recursion is increasingly updated based on the Wang-Landau flattening judgment. When the unit simulation length for each update is long enough, all the following unit simulations naturally run into the equilibrium regime. Thereafter, these unit simulations can serve for the dual purposes of recursion and equilibrium analysis. As demonstrated in our model studies, by applying ALSH, both fast recursion and short nonequilibrium data waste can be compromised. As a result, combining all the data obtained from all the unit simulations that are in the equilibrium regime via the weighted histogram analysis method, efficient convergence can be robustly ensured, especially for the purpose of free energy surface mapping.     

Force-momentum-based self-guided Langevin dynamics: a rapid sampling method that approaches the canonical ensemble

The self-guided Langevin dynamics (SGLD) is a method to accelerate conformational searching. This method is unique in the way that it selectively enhances and suppresses molecular motions based on their frequency to accelerate conformational searching without modifying energy surfaces or raising temperatures. It has been applied to studies of many long time scale events, such as protein folding. Recent progress in the understanding of the conformational distribution in SGLD simulations makes SGLD also an accurate method for quantitative studies. The SGLD partition function provides a way to convert the SGLD conformational distribution to the canonical ensemble distribution and to calculate ensemble average properties through reweighting. Based on the SGLD partition function, this work presents a force-momentum-based self-guided Langevin dynamics (SGLDfp) simulation method to directly sample the canonical ensemble. This method includes interaction forces in its guiding force to compensate the perturbation caused by the momentum-based guiding force so that it can approximately sample the canonical ensemble. Using several example systems, we demonstrate that SGLDfp simulations can approximately maintain the canonical ensemble distribution and significantly accelerate conformational searching. With optimal parameters, SGLDfp and SGLD simulations can cross energy barriers of more than 15 kT and 20 kT, respectively, at similar rates for LD simulations to cross energy barriers of 10 kT. The SGLDfp method is size extensive and works well for large systems. For studies where preserving accessible conformational space is critical, such as free energy calculations and protein folding studies, SGLDfp is an efficient approach to search and sample the conformational space.     

Folding simulations with novel conformational search method

A novel scheme for fast conformational search has been developed by combining the replica exchange method (REM) with the generalized effective potential concept. The new method, referred to Q-REM [S. Jang et al. Phys. Rev. Lett. 91, 058305 (2003)], is expected to provide a useful alternative to the conventional REM for effective conformational sampling of complex systems. The authors have performed folding simulations of the Trp-cage miniprotein using Q-REM. All atom level simulations with generalized Born solvent access-area solvation model show that successful folding can be observed with much smaller number of replicas in Q-REM compared to the conventional REM. It can be concluded that the new method has potential to significantly improve sampling efficiency, allowing simulations of more challenging systems.     

Enhanced sampling method in molecular simulations using genetic algorithm for biomolecular systems

We propose a molecular simulation method using genetic algorithm (GA) for biomolecular systems to obtain ensemble averages efficiently. In this method, we incorporate the genetic crossover, which is one of the operations of GA, to any simulation method such as conventional molecular dynamics (MD), Monte Carlo, and other simulation methods. The genetic crossover proposes candidate conformations by exchanging parts of conformations of a target molecule between a pair of conformations during the simulation. If the candidate conformations are accepted, the simulation resumes from the accepted ones. While conventional simulations are based on local update of conformations, the genetic crossover introduces global update of conformations. As an example of the present approach, we incorporated genetic crossover to MD simulations. We tested the validity of the method by calculating ensemble averages and the sampling efficiency by using two kinds of peptides, ALA3 and (AAQAA)₃. The results show that for ALA3 system, the distribution probabilities of backbone dihedral angles are in good agreement with those of the conventional MD and replica-exchange MD simulations. In the case of (AAQAA)₃ system, our method showed lower structural correlation of α-helix structures than the other two methods and more flexibility in the backbone ψ angles than the conventional MD simulation. These results suggest that our method gives more efficient conformational sampling than conventional simulation methods based on local update of conformations. © 2018 Wiley Periodicals, Inc.     

Adaptive lambda square dynamics simulation: An efficient conformational sampling method for biomolecules

A novel, efficient sampling method for biomolecules is proposed. The partial multicanonical molecular dynamics (McMD) was recently developed as a method that improved generalized ensemble (GE) methods to focus sampling only on a part of a system (GEPS); however, it was not tested well. We found that partial McMD did not work well for polylysine decapeptide and gave significantly worse sampling efficiency than a conventional GE. Herein, we elucidate the fundamental reason for this and propose a novel GEPS, adaptive lambda square dynamics (ALSD), which can resolve the problem faced when using partial McMD. We demonstrate that ALSD greatly increases the sampling efficiency over a conventional GE. We believe that ALSD is an effective method and is applicable to the conformational sampling of larger and more complicated biomolecule systems. © 2013 Wiley Periodicals, Inc.     

Sampling enhancement for the quantum mechanical potential based molecular dynamics simulations: a general algorithm and its extension for free energy calculation on rugged energy surface

An approach is developed in the replica exchange framework to enhance conformational sampling for the quantum mechanical (QM) potential based molecular dynamics simulations. Importantly, with our enhanced sampling treatment, a decent convergence for electronic structure self-consistent-field calculation is robustly guaranteed, which is made possible in our replica exchange design by avoiding direct structure exchanges between the QM-related replicas and the activated (scaled by low scaling parameters or treated with high "effective temperatures") molecular mechanical (MM) replicas. Although the present approach represents one of the early efforts in the enhanced sampling developments specifically for quantum mechanical potentials, the QM-based simulations treated with the present technique can possess the similar sampling efficiency to the MM based simulations treated with the Hamiltonian replica exchange method (HREM). In the present paper, by combining this sampling method with one of our recent developments (the dual-topology alchemical HREM approach), we also introduce a method for the sampling enhanced QM-based free energy calculations.     

Efficient sampling of protein conformational space using fast loop building and batch minimization on highly parallel computers

All‐atom sampling is a critical and compute‐intensive end stage to protein structural modeling. Because of the vast size and extreme ruggedness of conformational space, even close to the native structure, the high‐resolution sampling problem is almost as difficult as predicting the rough fold of a protein. Here, we present a combination of new algorithms that considerably speed up the exploration of very rugged conformational landscapes and are capable of finding heretofore hidden low‐energy states. The algorithm is based on a hierarchical workflow and can be parallelized on supercomputers with up to 128,000 compute cores with near perfect efficiency. Such scaling behavior is notable, as with Moore's law continuing only in the number of cores per chip, parallelizability is a critical property of new algorithms. Using the enhanced sampling power, we have uncovered previously invisible deficiencies in the Rosetta force field and created an extensive decoy training set for optimizing and testing force fields. © 2012 Wiley Periodicals, Inc.     

Multicanonical schemes for mapping out free-energy landscapes of single-component and multicomponent systems

Multicanonical (MUCA) sampling is a powerful approach for simulating large domains of thermodynamic macrostate space that relies on mapping out either the density of states or a free energy of the system as a function of a suitable "order parameter." The purpose of this study is to extend and apply to more complex systems the method introduced in a previous paper [M. K. Fenwick and F. A. Escobedo, J. Chem. Phys. 120, 3066 (2004)] that uses Bennett's acceptance ratio method for estimating MUCA free energies. Four types of MUCA schemes are considered according to what order parameter is adopted and how the macrostate space is traversed: a la grand canonical ensemble, a la semigrand canonical ensemble, a la semigrand isothermal-isobaric ensemble, and a la isothermal-isobaric ensemble. Two types of systems are studied, the first is a two-component Lennard-Jones mixture that exhibits a vapor-liquid transition, and the second is a hard-cuboid containing system that exhibits an isotropic-liquid crystalline transition. These systems are simulated with different MUCA schemes and the resulting free-energy profiles are used to determine phase-coexistence conditions. For the Lennard-Jones systems, it is also demonstrated that different types of MUCA simulations can be conveniently performed over different macrostate regions and the results can be subsequently pieced together into a continuous weighting function.     

Generalized essential energy space random walks to more effectively accelerate solute sampling in aqueous environment

Molecular dynamics sampling can be enhanced via the promoting of potential energy fluctuations, for instance, based on a Hamiltonian modified with the addition of a potential-energy-dependent biasing term. To overcome the diffusion sampling issue, which reveals the fact that enlargement of event-irrelevant energy fluctuations may abolish sampling efficiency, the essential energy space random walk (EESRW) approach was proposed earlier. To more effectively accelerate the sampling of solute conformations in aqueous environment, in the current work, we generalized the EESRW method to a two-dimension-EESRW (2D-EESRW) strategy. Specifically, the essential internal energy component of a focused region and the essential interaction energy component between the focused region and the environmental region are employed to define the two-dimensional essential energy space. This proposal is motivated by the general observation that in different conformational events, the two essential energy components have distinctive interplays. Model studies on the alanine dipeptide and the aspartate-arginine peptide demonstrate sampling improvement over the original one-dimension-EESRW strategy; with the same biasing level, the present generalization allows more effective acceleration of the sampling of conformational transitions in aqueous solution. The 2D-EESRW generalization is readily extended to higher dimension schemes and employed in more advanced enhanced-sampling schemes, such as the recent orthogonal space random walk method.     

Essential energy space random walk via energy space metadynamics method to accelerate molecular dynamics simulations

To overcome the possible pseudoergodicity problem, molecular dynamic simulation can be accelerated via the realization of an energy space random walk. To achieve this, a biased free energy function (BFEF) needs to be priori obtained. Although the quality of BFEF is essential for sampling efficiency, its generation is usually tedious and nontrivial. In this work, we present an energy space metadynamics algorithm to efficiently and robustly obtain BFEFs. Moreover, in order to deal with the associated diffusion sampling problem caused by the random walk in the total energy space, the idea in the original umbrella sampling method is generalized to be the random walk in the essential energy space, which only includes the energy terms determining the conformation of a region of interest. This essential energy space generalization allows the realization of efficient localized enhanced sampling and also offers the possibility of further sampling efficiency improvement when high frequency energy terms irrelevant to the target events are free of activation. The energy space metadynamics method and its generalization in the essential energy space for the molecular dynamics acceleration are demonstrated in the simulation of a pentanelike system, the blocked alanine dipeptide model, and the leucine model.     

Overcoming entropic barrier with coupled sampling at dual resolutions

An enhanced sampling method is proposed for ab initio protein folding simulations. The new method couples a high-resolution model for accuracy and a low-resolution model for efficiency. It aims to overcome the entropic barrier found in the exponentially large protein conformational space when a high-resolution model, such as an all-atom molecular mechanics force field, is used. The proposed method is designed to satisfy the detailed balance condition so that the Boltzmann distribution can be generated in all sampling trajectories in both high and low resolutions. The method was tested on model analytical energy functions and ab initio folding simulations of a beta-hairpin peptide. It was found to be more efficient than replica-exchange method that is used as its building block. Analysis with the analytical energy functions shows that the number of energy calculations required to find global minima and to converge mean potential energies is much fewer with the new method. Ergodic measure shows that the new method explores the conformational space more rapidly. We also studied imperfect low-resolution energy models and found that the introduction of errors in low-resolution models does decrease its sampling efficiency. However, a reasonable increase in efficiency is still observed when the global minima of the low-resolution models are in the vicinity of the global minimum basin of the high-resolution model. Finally, our ab initio folding simulation of the tested peptide shows that the new method is able to fold the peptide in a very short simulation time. The structural distribution generated by the new method at the equilibrium portion of the trajectory resembles that in the equilibrium simulation starting from the crystal structure.     

Application of paramagnetic NMR-validated molecular dynamics simulation to the analysis of a conformational ensemble of a branched oligosaccharide

Oligosaccharides of biological importance often exhibit branched covalent structures and dynamic conformational multiplicities. Here we report the application of a method that we developed, which combined molecular dynamics (MD) simulations and lanthanide-assisted paramagnetic NMR spectroscopy, to evaluate the dynamic conformational ensemble of a branched oligosaccharide. A lanthanide-chelating tag was attached to the reducing end of the branched tetrasaccharide of GM2 ganglioside to observe pseudocontact shifts as the source of long distance information for validating the conformational ensemble derived from MD simulations. By inspecting the results, the conformational space of the GM2 tetrasaccharide was compared with that of its nonbranched derivative, the GM3 trisaccharide.     

Synergistic approach to improve "alchemical" free energy calculation in rugged energy surface

The authors present an integrated approach to "alchemical" free energy simulation, which permits efficient calculation of the free energy difference on rugged energy surface. The method is designed to obtain efficient canonical sampling for rapid free energy convergence. The proposal is motivated by the insight that both the exchange efficiency in the presently designed dual-topology alchemical Hamiltonian replica exchange method (HREM), and the confidence of the free energy determination using the overlap histogramming method, depend on the same criterion, viz., the overlaps of the energy difference histograms between all pairs of neighboring states. Hence, integrating these two techniques can produce a joint solution to the problems of the free energy convergence and conformational sampling in the free energy simulations, in which lambda parameter plays two roles to simultaneously facilitate the conformational sampling and improve the phase space overlap for the free energy determination. Specifically, in contrast with other alchemical HREM based free energy simulation methods, the dual-topology approach can ensure robust conformational sampling. Due to these features (a synergistic solution to the free energy convergence and canonical sampling, and the improvement of the sampling efficiency with the dual-topology treatment), the present approach, as demonstrated in the model studies of the authors, is highly efficient in obtaining accurate free energy differences, especially for the systems with rough energy landscapes.