Selecting Collective Variables and Free-Energy Methods for Peptide Translocation across Membranes.

Adiabatic free energy dynamics (AFED) was introduced by Rosso et al. [J. Chem. Phys. 2002, 116, 4389] for computing free energy profiles quickly and accurately using a dynamical adiabatic separation between a set of collective variables or reaction coordinates and the remaining degrees of freedom of a system. This approach has been shown to lead to a significant gain in efficiency versus traditional methods such as umbrella sampling, thermodynamic integration, and free energy perturbation for generating one-dimensional free energy profiles. More importantly, AFED is able to generate multidimensional free energy surfaces efficiently via full sweeps of the surface that rapidly map out the locations of the free energy minima. The most significant drawback to the AFED approach is the need to transform the coordinates into a generalized coordinate system that explicitly contains the collective variables of interest. Recently, Maragliano and Vanden-Eijnden built upon the AFED approach by introducing a set of extended phase-space variables, to which the adiabatic decoupling and high temperature are applied [Chem. Phys. Lett. 2006, 426, 168]. In this scheme, which the authors termed "temperature accelerated molecular dynamics" or TAMD, the need for explicit coordinate transformations is circumvented. The ability of AFED and TAMD to generate free energy surfaces efficiently depends on the thermostatting mechanism employed, since both approaches are inherently nonequilibrium due to the adiabatic decoupling. Indeed, Maragliano and Vanden-Eijnden did not report any direct generation of free energy surfaces within the overdamped Langevin dynamics employed by these authors. Here, we show that by formulating TAMD in a manner that is closer to the original AFED approach, including the generalized Gaussian moment thermostat (GGMT) and multiple time-scale integration, multidimensional free energy surfaces for complex systems can be generated directly from the probability distribution function of the extended phase-space variables. The new TAMD formulation, which we term driven AFED or d-AFED, is applied to compare the conformational preferences of small peptides both in gas phase and in solution for three force fields. The results show that d-AFED/TAMD accurately and efficiently generates free energy surfaces in two collective variables useful for characterizing the conformations, namely, the radius of gyration, R(G), and number of hydrogen bonds, N(H).

Umbrella sampling has been a workhorse for free energy calculations in molecular simulations for several decades. In conventional umbrella sampling, restraining bias potentials are strategically applied along one or several collective variables. Major drawbacks associated with this method are the requirement of a large number of bias windows and the poor sampling of the transverse coordinates. In this work, we propose an alternate formalism that departs from the traditional umbrella sampling to mitigate these issues, where we replace umbrella-type restraining bias potentials with bucket-type wall potentials. This modification permits one to formulate an efficient computational strategy leveraging wall potentials and metadynamics sampling. This new method, called "bucket sampling", can significantly reduce the computational cost of obtaining converged high-dimensional free energy surfaces. Extensions of the proposed method with temperature acceleration and replica exchange solute tempering are also demonstrated.

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.

Using Committor and its Distribution to Assess the Convergence of Free Energy Calculations

Changes in free energy provide valuable information for molecular recognition, including both ligand-receptor binding thermodynamics and kinetics. Umbrella sampling (US), a widely used free energy calculation method, has long been used to explore the dissociation process of ligand-receptor systems and compute binding free energy. In existing publications, the binding free energy computed from the potential of mean force (PMF) with US simulation mostly yielded "ball park" values with experimental data. However, the computed PMF values are highly influenced by factors such as initial conformations and/or trajectories provided, the reaction coordinate, and sampling of conformational space in each US window. These critical factors have rarely been carefully studied. Here we used US to study the guest aspirin and 1-butanol dissociation processes of β-cyclodextrin (β-CD) and an inhibitor SB2 dissociation from a p38α mitogen-activated protein kinase (MAPK) complex. For β-CD, we used three different β-CD conformations to generate the dissociation path with US windows. For p38α, we generated the dissociation pathway by using accelerated molecular dynamics followed by conformational relaxing with short conventional MD, steered MD, and manual pulling. We found that, even for small β-CD complexes, different β-CD conformations altered the height of the PMF, but the pattern of PMF was not affected if the MD sampling in each US window was well-converged. Because changing the macrocyclic ring conformation needs to rotate dihedral angles in the ring, a bound ligand largely restrains the motion of cyclodextrin. Therefore, once a guest is in the binding site, cyclodextrin cannot freely change its initial conformation, resulting in different absolute heights of the PMF, which cannot be overcome by running excessively long MD simulations for each US window. Moreover, if the US simulations were not converged, the important barrier and minimum were missed. For ligand-protein systems, our studies also suggest that the dissociation trajectories modeled by an enhanced sampling method must maintain a natural molecular movement to avoid biased PMF plots when using US simulations.

Understanding how different classes of molecules move across biological membranes is a prerequisite to predicting a solute's permeation rate, which is a critical factor in the fields of drug design and pharmacology. We use biased molecular dynamics computer simulations to calculate and compare the free energy profiles of translocation of several small molecules across 1,2-dioleoyl- sn-glycero-3-phosphocholine (DOPC) lipid bilayers as a first step toward determining the most efficient method for free energy calculations. We study the translocation of arginine, a sodium ion, alanine, and a single water molecule using the metadynamics, umbrella sampling, and replica exchange umbrella sampling techniques. Within the fixed lengths of our simulations, we find that all methods produce similar results for charge-neutral permeants, but not for polar or positively charged molecules. We identify the long relaxation time scale of electrostatic interactions between lipid headgroups and the solute to be the principal cause of this difference and show that this slow process can lead to an erroneous dependence of computed free energy profiles on the initial system configuration. We demonstrate the use of committor analysis to validate the proper sampling of the presumed transition state, which in our simulations is achieved only in replica exchange calculations. On the basis of these results we provide some useful guidance to perform and evaluate free energy calculations of membrane permeation.

The transition of an α-helix to a β-sheet in proteins is among the most complex conformational changes seen in biomolecular systems. Due to long time scales involved in the transition, it is challenging to study such protein conformational changes using direct molecular dynamics simulations. This limitation is typically overcome using an indirect approach wherein one computes the free energy landscape associated with the transition. Computation of free energy landscapes, however, requires a suitable set of collective variables that describe the transition. In this work, we demonstrate the use of path collective variables [D. Branduardi, F. L. Gervasio and M. Parrinello, J. Chem. Phys., 2007, 126, 054103] and combine it with the finite temperature string (FTS) method [E. Weinan, W. Ren and E. Vanden-Eijnden, J. Phys. Chem. B, 2005, 109, 6688-6693] to determine the molecular mechanisms involved during the structural transition of the mini G-protein from an α-helix to a β-hairpin. The transition from the α-helix proceeds via unfolding of the terminal residues, giving rise to a β-turn unfolded intermediate to eventually form the β-hairpin. Our proposed algorithm uses umbrella sampling simulations to simulate images along the string and the weighted histogram analysis to compute the free energy along the computed transition path. This work demonstrates that the string method in combination with path collective variables can be exploited to study complex protein conformational changes such as a complete change in the secondary structure.

Understanding the permeation of molecules through lipid membranes is fundamental for predicting the cellular uptake of solutes and drug delivery mechanisms. In molecular simulations, the usual approach is to compute the free energy (FE) profile of a molecule across a model lipid bilayer, which can then be used to estimate the permeability of the molecule. Umbrella Sampling (US), which involves carrying out a series of biased simulations along a defined reaction coordinate (usually the bilayer normal direction), is a popular method for the computation of such FE profiles. However, US can be challenging to implement because the results are dependent on the strength of the biasing potential and the spacing of windows along the reaction coordinate, which, in practice, are usually optimized by an inefficient trial and error approach. The Steered Molecular Dynamics implementation of the Jarzynski Equality (JE-SMD) has been identified as an alternative to equilibrium sampling methods for measuring the FE change across a reaction coordinate. In the JE-SMD approach, equilibrium FE values are evaluated from the average of rapid non-equilibrium trajectories, thus avoiding the practical issues that come with US. Here, we use three different corrections of the JE-SMD method to calculate the FE change for the translocation of two aromatic substrates, phenylalanine and toluene, across a lipid bilayer and compare the accuracy and computational efficiency of these approaches to the results obtained using US. We show evidence that when computing the free energy profile, the JE-SMD approach suffers from insufficient sampling convergence of the bilayer environment and is dependent on the characteristic of the aromatic substrate itself. We deduce that, despite its drawbacks, US remains the more viable approach of the two for computing the FE profile.

We recently introduced a method called conveyor belt (CB) thermodynamic integration (TI) for the calculation of alchemical free-energy differences based on molecular dynamics simulations. In the present work, the CBTI approach is generalized to conformational free-energy changes, i.e., to the determination of the potential of mean force (PMF) along a conformational coordinate ξ of interest. The proposed conveyor belt umbrella sampling (CBUS) scheme relies on the parallel simulation of K replicas k = 0,1, ..., K - 1 of the system, with K even. For each replica k, the instantaneous value of ξ is restrained to an anchor value λk. The latter anchor points are equally spaced along a forward-turn-backward-turn path (i.e., a CB) between two extreme values defining the ξ-range of interest. The rotation of the CB is controlled by a variable Λ (range from 0 to 2π) which evolves dynamically along the simulation. The evolution of Λ results from the forces exerted by the restraining potentials on the anchor points, taken equal and opposite to those they exert on the replicas. Because these forces tend to cancel out along the CB, the dynamics of Λ is essentially diffusive, and the continuous distribution of ξ-values sampled by the replica system is automatically close to homogeneous. The latter feature represents an advantage over direct counting (DCNT) and traditional umbrella sampling (TRUS), shared to some extent with replica-exchange umbrella sampling (REUS). In this work, the CBUS scheme is introduced and compared to the three latter schemes in the calculation of 45 standard absolute binding free energies. These correspond to the binding of five alkali cations to three crown ethers in three solvents. Different free-energy estimators are considered for the PMF calculation, and the calculated values are also compared to those of a previous study relying on an alchemical path, as well as to experimental data.

Energetics of Storage and Diffusion of Water and Cyclo-Octasulfur for a Nonpolar Cavity of RHCC Tetrabrachion by Molecular Dynamics Simulations

Molecular dynamics (MD) simulations are widely applied to estimate absolute binding free energies of protein-ligand and protein-protein complexes. A routinely used method for binding free energy calculations with MD is umbrella sampling (US), which calculates the potential of mean force (PMF) along a single reaction coordinate. Surprisingly, in spite of its widespread use, few validation studies have focused on the convergence of the free energy computed along a single path for specific cases, not addressing the reproducibility of such calculations in general. In this work, we therefore investigate the reproducibility and convergence of US along a standard distance-based reaction coordinate for various protein-protein and protein-ligand complexes, following commonly used guidelines for the setup. We show that repeating the complete US workflow can lead to differences of 2-20 kcal/mol in computed binding free energies. We attribute those discrepancies to small differences in the binding pathways. While these differences are unavoidable in the established US protocol, the popularity of the latter could hint at a lack of awareness of such reproducibility problems. To test if the convergence of PMF profiles can be improved if multiple pathways are sampled simultaneously, we performed additional simulations with an adaptive-biasing method, here the accelerated weight histogram (AWH) approach. Indeed, the PMFs obtained from AHW simulations are consistent and reproducible for the systems tested. To the best of our knowledge, our work is the first to attempt a systematic assessment of the pitfalls in one the most widely used protocols for computing binding affinities. We anticipate therefore that our results will provide an incentive for a critical reassessment of the validity of PMFs computed with US, and make a strong case to further benchmark the performance of adaptive-biasing methods for computing binding affinities.

Curling and Local Shape Changes of Red Blood Cell Membranes Driven by Cytoskeletal Reorganization

The recently introduced hills method (Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 12562) is a powerful tool to compute the multidimensional free energy surface of intrinsically concerted reactions. We have extended this method by focusing our attention on localizing the lowest free energy path that connects the stable reactant and product states. This path represents the most probable reaction mechanism, similar to the zero temperature intrinsic reaction coordinate, but also includes finite temperature effects. The transformation of the multidimensional problem to a one-dimensional reaction coordinate allows for accurate convergence of the free energy profile along the lowest free energy path using standard free energy methods. Here we apply the hills method, our lowest free energy path search algorithm, and umbrella sampling to the prototype S(N)2 reaction. The hills method replaces the in many cases difficult problem of finding a good reaction coordinate with choosing relatively simple collective variables, such as the bond lengths of the broken and formed chemical bonds. The second part of the paper presents a guide to using the hills method, in which we test and fine-tune the method for optimal accuracy and efficiency using the umbrella sampling results as a reference.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTEffects of solvent on the conformation and the collective motions of a protein. 3. Free energy analysis by the extended RISM theoryAkio Kitao, Fumio Hirata, and Nobuhiro GoCite this: J. Phys. Chem. 1993, 97, 39, 10231–10235Publication Date (Print):September 1, 1993Publication History Published online1 May 2002Published inissue 1 September 1993https://pubs.acs.org/doi/10.1021/j100141a053https://doi.org/10.1021/j100141a053research-articleACS PublicationsRequest reuse permissionsArticle Views152Altmetric-Citations33LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InRedditEmail Other access options Get e-Alerts

Enhanced sampling and free energy calculations for protein simulations.