Constructing a multidimensional free energy surface like a spider weaving a web

Complete free energy surface in the collective variable space provides important information of the reaction mechanisms of the molecules. But, sufficient sampling in the collective variable space is not easy. The space expands quickly with the number of the collective variables. To solve the problem, many methods utilize artificial biasing potentials to flatten out the original free energy surface of the molecule in the simulation. Their performances are sensitive to the definitions of the biasing potentials. Fast‐growing biasing potential accelerates the sampling speed but decreases the accuracy of the free energy result. Slow‐growing biasing potential gives an optimized result but needs more simulation time. In this article, we propose an alternative method. It adds the biasing potential to a representative point of the molecule in the collective variable space to improve the conformational sampling. And the free energy surface is calculated from the free energy gradient in the constrained simulation, not given by the negative of the biasing potential as previous methods. So the presented method does not require the biasing potential to remove all the barriers and basins on the free energy surface exactly. Practical applications show that the method in this work is able to produce the accurate free energy surfaces for different molecules in a short time period. The free energy errors are small in the cases of various biasing potentials. © 2017 Wiley Periodicals, Inc.     

Walking freely in the energy and temperature space by the modified replica exchange molecular dynamics method

Replica Exchange Molecular Dynamics (REMD) method is a powerful sampling tool in molecular simulations. Recently, we made a modification to the standard REMD method. It places some inactive replicas at different temperatures as well as the active replicas. The method completely decouples the number of the active replicas and the number of the temperature levels. In this article, we make a further modification to our previous method. It uses the inactive replicas in a different way. The inactive replicas first sample in their own knowledge‐based energy databases and then participate in the replica exchange operations in the REMD simulation. In fact, this method is a hybrid between the standard REMD method and the simulated tempering method. Using different active replicas, one can freely control the calculation quantity and the convergence speed of the simulation. To illustrate the performance of the method, we apply it to some small models. The distribution functions of the replicas in the energy space and temperature space show that the modified REMD method in this work can let the replicas walk freely in both of the two spaces. With the same number of the active replicas, the free energy surface in the simulation converges faster than the standard REMD. © 2016 Wiley Periodicals, Inc.     

Determination of free energy profiles by repository based adaptive umbrella sampling: bridging nonequilibrium and quasiequilibrium simulations

We propose a new adaptive sampling approach to determine free energy profiles with molecular dynamics simulations, which is called as "repository based adaptive umbrella sampling" (RBAUS). Its main idea is that a sampling repository is continuously updated based on the latest simulation data, and the accumulated knowledge and sampling history are then employed to determine whether and how to update the biasing umbrella potential for subsequent simulations. In comparison with other adaptive methods, a unique and attractive feature of the RBAUS approach is that the frequency for updating the biasing potential depends on the sampling history and is adaptively determined on the fly, which makes it possible to smoothly bridge nonequilibrium and quasiequilibrium simulations. The RBAUS method is first tested by simulations on two simple systems: a double well model system with a variety of barriers and the dissociation of a NaCl molecule in water. Its efficiency and applicability are further illustrated in ab initio quantum mechanics/molecular mechanics molecular dynamics simulations of a methyl-transfer reaction in aqueous solution.     

Free Energy Calculations using a Swarm‐Enhanced Sampling Molecular Dynamics Approach

Free energy simulations are an established computational tool in modelling chemical change in the condensed phase. However, sampling of kinetically distinct substates remains a challenge to these approaches. As a route to addressing this, we link the methods of thermodynamic integration (TI) and swarm‐enhanced sampling molecular dynamics (sesMD), where simulation replicas interact cooperatively to aid transitions over energy barriers. We illustrate the approach by using alchemical alkane transformations in solution, comparing them with the multiple independent trajectory TI (IT‐TI) method. Free energy changes for transitions computed by using IT‐TI grew increasingly inaccurate as the intramolecular barrier was heightened. By contrast, swarm‐enhanced sampling TI (sesTI) calculations showed clear improvements in sampling efficiency, leading to more accurate computed free energy differences, even in the case of the highest barrier height. The sesTI approach, therefore, has potential in addressing chemical change in systems where conformations exist in slow exchange.     

A bias-exchange approach to protein folding

By suitably extending a recent approach [Bussi, G.; et al. J. Am. Chem. Soc. 2006, 128, 13435] we introduce a powerful methodology that allows the parallel reconstruction of the free energy of a system in a virtually unlimited number of variables. Multiple metadynamics simulations of the same system at the same temperature are performed, biasing each replica with a time-dependent potential constructed in a different set of collective variables. Exchanges between the bias potentials in the different variables are periodically allowed according to a replica exchange scheme. Due to the efficaciously multidimensional nature of the bias the method allows exploring complex free energy landscapes with high efficiency. The usefulness of the method is demonstrated by performing an atomistic simulation in explicit solvent of the folding of a Triptophane cage miniprotein. It is shown that the folding free energy landscape can be fully characterized starting from an extended conformation with use of only 40 ns of simulation on 8 replicas.     

Coulomb replica‐exchange method: Handling electrostatic attractive and repulsive forces for biomolecules

We propose a new type of the Hamiltonian replica‐exchange method (REM) for molecular dynamics (MD) and Monte Carlo simulations, which we refer to as the Coulomb REM (CREM). In this method, electrostatic charge parameters in the Coulomb interactions are exchanged among replicas while temperatures are exchanged in the usual REM. By varying the atom charges, the CREM overcomes free‐energy barriers and realizes more efficient sampling in the conformational space than the REM. Furthermore, this method requires only a smaller number of replicas because only the atom charges of solute molecules are used as exchanged parameters. We performed Coulomb replica‐exchange MD simulations of an alanine dipeptide in explicit water solvent and compared the results with those of the conventional canonical, replica exchange, and van der Waals REMs. Two force fields of AMBER parm99 and AMBER parm99SB were used. As a result, the CREM sampled all local‐minimum free‐energy states more frequently than the other methods for both force fields. Moreover, the Coulomb, van der Waals, and usual REMs were applied to a fragment of an amyloid‐β peptide (Aβ) in explicit water solvent to compare the sampling efficiency of these methods for a larger system. The CREM sampled structures of the Aβ fragment more efficiently than the other methods. We obtained β‐helix, α‐helix, 3₁₀‐helix, β‐hairpin, and β‐sheet structures as stable structures and deduced pathways of conformational transitions among these structures from a free‐energy landscape. © 2012 Wiley Periodicals, Inc.     

A convective replica‐exchange method for sampling new energy basins

Replica‐exchange is a powerful simulation method for sampling the basins of a rugged energy landscape. The replica‐exchange method's sampling is efficient because it allows replicas to perform round trips in temperature space, thereby visiting both low and high temperatures in the same simulation. However, replicas have a diffusive walk in temperature space, and the round trip rate decreases significantly with the system size. These drawbacks make convergence of the simulation even more difficult than it already is when bigger systems are tackled. Here, we present a simple modification of the exchange method. In this method, one of the replicas steadily raises or lowers its temperature. We tested the convective replica‐exchange method on three systems of varying complexity: the alanine dipeptide in implicit solvent, the GB1 β‐hairpin in explicit solvent and the Aβ_25–35 homotrimer in a coarse grained representation. For the highly frustrated Aβ_25–35 homotrimer, the proposed “convective” replica‐exchange method is twice as fast as the standard method. It discovered 24 out of 27 free‐energy basins in less than 500 ns. It also prevented the formation of groups of replicas that usually form on either side of an exchange bottleneck, leading to a more efficient sampling of new energy basins than in the standard method. © 2012 Wiley Periodicals, Inc.     

Convergence of Free Energy Profile of Coumarin in Lipid Bilayer

Atomistic molecular dynamics (MD) simulations of druglike molecules embedded in lipid bilayers are of considerable interest as models for drug penetration and positioning in biological membranes. Here we analyze partitioning of coumarin in dioleoylphosphatidylcholine (DOPC) bilayer, based on both multiple, unbiased 3 μs MD simulations (total length) and free energy profiles along the bilayer normal calculated by biased MD simulations (～7 μs in total). The convergences in time of free energy profiles calculated by both umbrella sampling and z-constraint techniques are thoroughly analyzed. Two sets of starting structures are also considered, one from unbiased MD simulation and the other from "pulling" coumarin along the bilayer normal. The structures obtained by pulling simulation contain water defects on the lipid bilayer surface, while those acquired from unbiased simulation have no membrane defects. The free energy profiles converge more rapidly when starting frames from unbiased simulations are used. In addition, z-constraint simulation leads to more rapid convergence than umbrella sampling, due to quicker relaxation of membrane defects. Furthermore, we show that the choice of RESP, PRODRG, or Mulliken charges considerably affects the resulting free energy profile of our model drug along the bilayer normal. We recommend using z-constraint biased MD simulations based on starting geometries acquired from unbiased MD simulations for efficient calculation of convergent free energy profiles of druglike molecules along bilayer normals. The calculation of free energy profile should start with an unbiased simulation, though the polar molecules might need a slow pulling afterward. Results obtained with the recommended simulation protocol agree well with available experimental data for two coumarin derivatives.     

Adaptively biased molecular dynamics for free energy calculations

We present an adaptively biased molecular dynamics (ABMD) method for the computation of the free energy surface of a reaction coordinate using nonequilibrium 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.     

Integrating diffusion maps with umbrella sampling: application to alanine dipeptide

Nonlinear dimensionality reduction techniques can be applied to molecular simulation trajectories to systematically extract a small number of variables with which to parametrize the important dynamical motions of the system. For molecular systems exhibiting free energy barriers exceeding a few k(B)T, inadequate sampling of the barrier regions between stable or metastable basins can lead to a poor global characterization of the free energy landscape. We present an adaptation of a nonlinear dimensionality reduction technique known as the diffusion map that extends its applicability to biased umbrella sampling simulation trajectories in which restraining potentials are employed to drive the system into high free energy regions and improve sampling of phase space. We then propose a bootstrapped approach to iteratively discover good low-dimensional parametrizations by interleaving successive rounds of umbrella sampling and diffusion mapping, and we illustrate the technique through a study of alanine dipeptide in explicit solvent.     

Scalable free energy calculation of proteins via multiscale essential sampling

A multiscale simulation method, "multiscale essential sampling (MSES)," is proposed for calculating free energy surface of proteins in a sizable dimensional space with good scalability. In MSES, the configurational sampling of a full-dimensional model is enhanced by coupling with the accelerated dynamics of the essential degrees of freedom. Applying the Hamiltonian exchange method to MSES can remove the biasing potential from the coupling term, deriving the free energy surface of the essential degrees of freedom. The form of the coupling term ensures good scalability in the Hamiltonian exchange. As a test application, the free energy surface of the folding process of a miniprotein, chignolin, was calculated in the continuum solvent model. Results agreed with the free energy surface derived from the multicanonical simulation. Significantly improved scalability with the MSES method was clearly shown in the free energy calculation of chignolin in explicit solvent, which was achieved without increasing the number of replicas in the Hamiltonian exchange.     

Using the local elevation method to construct optimized umbrella sampling potentials: Calculation of the relative free energies and interconversion barriers of glucopyranose ring conformers in water

A method is proposed to combine the local elevation (LE) conformational searching and the umbrella sampling (US) conformational sampling approaches into a single local elevation umbrella sampling (LEUS) scheme for (explicit‐solvent) molecular dynamics (MD) simulations. In this approach, an initial (relatively short) LE build‐up (searching) phase is used to construct an optimized biasing potential within a subspace of conformationally relevant degrees of freedom, that is then used in a (comparatively longer) US sampling phase. This scheme dramatically enhances (in comparison with plain MD) the sampling power of MD simulations, taking advantage of the fact that the preoptimized biasing potential represents a reasonable approximation to the negative of the free energy surface in the considered conformational subspace. The method is applied to the calculation of the relative free energies of β‐D ‐glucopyranose ring conformers in water (within the GROMOS 45A4 force field). Different schemes to assign sampled conformational regions to distinct states are also compared. This approach, which bears some analogies with adaptive umbrella sampling and metadynamics (but within a very distinct implementation), is shown to be: (i) efficient (nearly all the computational effort is invested in the actual sampling phase rather than in searching and equilibration); (ii) robust (the method is only weakly sensitive to the details of the build‐up protocol, even for relatively short build‐up times); (iii) versatile (a LEUS biasing potential database could easily be preoptimized for small molecules and assembled on a fragment basis for larger ones). © 2009 Wiley Periodicals, Inc. J Comput Chem 2010     

Hamiltonian replica‐exchange simulations with adaptive biasing of peptide backbone and side chain dihedral angles

A Hamiltonian Replica‐Exchange Molecular Dynamics (REMD) simulation method has been developed that employs a two‐dimensional backbone and one‐dimensional side chain biasing potential specifically to promote conformational transitions in peptides. To exploit the replica framework optimally, the level of the biasing potential in each replica was appropriately adapted during the simulations. This resulted in both high exchange rates between neighboring replicas and improved occupancy/flow of all conformers in each replica. The performance of the approach was tested on several peptide and protein systems and compared with regular MD simulations and previous REMD studies. Improved sampling of relevant conformational states was observed for unrestrained protein and peptide folding simulations as well as for refinement of a loop structure with restricted mobility of loop flanking protein regions. © 2013 Wiley Periodicals, Inc.     

Determination of the chemical potential using energy-biased sampling

An energy-biased method to evaluate ensemble averages requiring test-particle insertion is presented. The method is based on biasing the sampling within the subdomains of the test-particle configurational space with energies smaller than a given value freely assigned. These energy wells are located via unbiased random insertion over the whole configurational space and are sampled using the so-called Hit-and-Run algorithm, which uniformly samples compact regions of any shape immersed in a space of arbitrary dimensions. Because the bias is defined in terms of the energy landscape it can be exactly corrected to obtain the unbiased distribution. The test-particle energy distribution is then combined with the Bennett relation for the evaluation of the chemical potential. We apply this protocol to a system with relatively small probability of low-energy test-particle insertion, liquid argon at high density and low temperature, and show that the energy-biased Bennett method is around five times more efficient than the standard Bennett method. A similar performance gain is observed in the reconstruction of the energy distribution.     

Protecting High Energy Barriers: A New Equation to Regulate Boost Energy in Accelerated Molecular Dynamics Simulations

Molecular dynamics (MD) is one of the most common tools in computational chemistry. Recently, our group has employed accelerated molecular dynamics (aMD) to improve the conformational sampling over conventional molecular dynamics techniques. In the original aMD implementation, sampling is greatly improved by raising energy wells below a predefined energy level. Recently, our group presented an alternative aMD implementation where simulations are accelerated by lowering energy barriers of the potential energy surface. When coupled with thermodynamic integration simulations, this implementation showed very promising results. However, when applied to large systems, such as proteins, the simulation tends to be biased to high energy regions of the potential landscape. The reason for this behavior lies in the boost equation used since the highest energy barriers are dramatically more affected than the lower ones. To address this issue, in this work, we present a new boost equation that prevents oversampling of unfavorable high energy conformational states. The new boost potential provides not only better recovery of statistics throughout the simulation but also enhanced sampling of statistically relevant regions in explicit solvent MD simulations.     

Sampling free energy surfaces as slices by combining umbrella sampling and metadynamics

Metadynamics (MTD) is a very powerful technique to sample high‐dimensional free energy landscapes, and due to its self‐guiding property, the method has been successful in studying complex reactions and conformational changes. MTD sampling is based on filling the free energy basins by biasing potentials and thus for cases with flat, broad, and unbound free energy wells, the computational time to sample them becomes very large. To alleviate this problem, we combine the standard Umbrella Sampling (US) technique with MTD to sample orthogonal collective variables (CVs) in a simultaneous way. Within this scheme, we construct the equilibrium distribution of CVs from biased distributions obtained from independent MTD simulations with umbrella potentials. Reweighting is carried out by a procedure that combines US reweighting and Tiwary–Parrinello MTD reweighting within the Weighted Histogram Analysis Method (WHAM). The approach is ideal for a controlled sampling of a CV in a MTD simulation, making it computationally efficient in sampling flat, broad, and unbound free energy surfaces. This technique also allows for a distributed sampling of a high‐dimensional free energy surface, further increasing the computational efficiency in sampling. We demonstrate the application of this technique in sampling high‐dimensional surface for various chemical reactions using ab initio and QM/MM hybrid molecular dynamics simulations. Further, to carry out MTD bias reweighting for computing forward reaction barriers in ab initio or QM/MM simulations, we propose a computationally affordable approach that does not require recrossing trajectories. © 2016 Wiley Periodicals, Inc.     

A comparison of methods to compute the potential of mean force.

Most processes occurring in a system are determined by the relative free energy between two or more states because the free energy is a measure of the probability of finding the system in a given state. When the two states of interest are connected by a pathway, usually called reaction coordinate, along which the free-energy profile is determined, this profile or potential of mean force (PMF) will also yield the relative free energy of the two states. Twelve different methods to compute a PMF are reviewed and compared, with regard to their precision, for a system consisting of a pair of methane molecules in aqueous solution. We analyze all combinations of the type of sampling (unbiased, umbrella-biased or constraint-biased), how to compute free energies (from density of states or force averaging) and the type of coordinate system (internal or Cartesian) used for the PMF degree of freedom. The method of choice is constraint-bias simulation combined with force averaging for either an internal or a Cartesian PMF degree of freedom.     

Non-Boltzmann sampling and Bennett's acceptance ratio method: how to profit from bending the rules

The exact computation of free energy differences requires adequate sampling of all relevant low energy conformations. Especially in systems with rugged energy surfaces, adequate sampling can only be achieved by biasing the exploration process, thus yielding non-Boltzmann probability distributions. To obtain correct free energy differences from such simulations, it is necessary to account for the effects of the bias in the postproduction analysis. We demonstrate that this can be accomplished quite simply with a slight modification of Bennett's Acceptance Ratio method, referring to this technique as Non-Boltzmann Bennett. We illustrate the method by several examples and show how a creative choice of the biased state(s) used during sampling can also improve the efficiency of free energy simulations.     

BAR-based optimum adaptive steered MD for configurational sampling

The equilibrium and nonequilibrium adaptive alchemical free energy simulation methods optimum Bennett's acceptance ratio and optimum crooks' equation (OCE), based on the statistically optimal bidirectional reweighting estimator named Bennett's Acceptance Ratio or Crooks' equation, perform initial sampling in the staging alchemical transformation and then determine the importance rank of different states via the time-derivative of the variance. The method is proven to give speedups compared with the equal time rule. In the current work, we extend the time derivative of variance guided adaptive sampling method to the configurational space, falling in the term of steered MD (SMD). The SMD approach biasing physically meaningful collective variable (CV) such as one dihedral or one distance to pulling the system from one conformational state to another. By minimizing the variance of the free energy differences along the pathway in an optimized way, a new type of adaptive SMD (ASMD) is introduced. As exhibits in the alchemical case, this adaptive sampling method outperforms the traditional equal-time SMD in nonequilibrium stratification. Also, the method gives much more efficient calculation of potential of mean force than the selection criterion-based ASMD scheme, which is proven to be more efficient than traditional SMD. The OCE workflow is periodicity-of-CV dependent while ASMD is not. The performance is demonstrated in a dihedral flipping case and two distance pulling cases, accounting for periodic and nonperiodic CVs, respectively. © 2019 Wiley Periodicals, Inc.     

Different Enzyme Conformations Induce Different Mechanistic Traits in HIV-1 Protease

The influence of the dynamical flexibility of enzymes on reaction mechanisms is a cornerstone in biological sciences. In this study, we aim to 1) study the convergence of the activation free energy by using the first step of the reaction catalysed by HIV-1 protease as a case study, and 2) provide further evidence for a mechanistic divergence in this enzyme, as two different reaction pathways were seen to contribute to this step. We used quantum mechanics/molecular mechanics molecular dynamics simulations, on four different initial conformations that led to different barriers in a previous study. Despite the sampling, the four activation free energies still spanned a range of 5.0 kcal ⋅ mol⁻¹. Furthermore, the new simulations did confirm the occurrence of an unusual mechanistic divergence, with two different mechanistic pathways displaying equivalent barriers. An active-site water molecule is proposed to influence the mechanistic pathway.