Efficient and Direct Generation of Multidimensional Free Energy Surfaces via Adiabatic Dynamics without Coordinate Transformations

The protein folding funnel paradigm suggests that folding and unfolding proceed as directed diffusion in a multidimensional free energy surface where a multitude of pathways can be traversed during the protein's sojourn from initial to final state. However, finding even a single pathway, with the detail chronicling of intermediates, is an arduous task. In this work we explore the free energy surface of unfolding pathway through umbrella sampling, for a small globular α-helical protein chicken-villin headpiece (HP-36) when the melting of secondary structures is induced by adding DMSO in aqueous solution. We find that the unfolding proceeds through the initial separation or melting of aggregated hydrophobic core that comprises of three phenylalanine residues (Phe7, Phe11, and Phe18). This separation is accompanied by simultaneous melting of the second helix. Unfolding is found to be a multistage process involving crossing of three consecutive minima and two barriers at the initial stage. At a molecular level, Phe18 is observed to reorient itself towards other hydrophobic grooves to stabilize the intermediate states. We identify the configuration of the intermediates and correlate the intermediates with those obtained in our previous works. We also give an estimate of the barriers for different transition states and observe the softening of the barriers with increasing DMSO concentration. We show that higher concentration of DMSO tunes the unfolding pathway by destabilizing the third minimum and stabilizing the second one, indicating the development of a solvent modified, less rugged pathway. The prime outcome of this work is the demonstration that mixed solvents can profoundly transform the nature of the energy landscape and induce unfolding via a modified route. A successful application of Kramer's rate equation correlating the free energy simulation results shows faster rate of unfolding with increasing DMSO concentration. This work perhaps presents the first systematic theoretical study of the effect of a chemical denaturant on the microscopic free energy surface and rates of unfolding of HP-36.

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.

We carried out ab initio molecular dynamic simulations in order to determine the free energy surfaces of two selected reactions including solvents, namely a rearrangement of a ruthenium oxoester in water and a carbon dioxide addition to a palladium complex in carbon dioxide. For the latter reaction we also investigated the gas phase reaction in order to take solvent effects into account. We used two techniques to reconstruct the free energy surfaces: thermodynamic integration and metadynamics. Furthermore, we gave a reasonable error estimation of the computed free energy surface. We calculated a reaction barrier of ΔF = 59.5 ± 8.5 kJ mol−1 for the rearrangement of a ruthenium oxoester in water from thermodynamic integration. For the carbon dioxide addition to the palladium complex in carbon dioxide we found a ΔF = 44.9 ± 3.3 kJ mol−1 from metadynamics simulations with one collective variable. The investigation of the same reactions in the gas phase resulted in ΔF = 24.9 ± 6.7 kJ mol−1 from thermodynamic integration, in ΔF = 26.7 ± 2.3 kJ mol−1 from metadynamics simulations with one collective variable, and in ΔF = 27.1 ± 5.9 kJ mol−1 from metadynamics simulations with two collective variables.

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.

Many biomolecules undergo conformational changes associated with allostery or ligand binding. Observing these changes in computer simulations is difficult if their timescales are long. These calculations can be accelerated by observing the transition on an auxiliary free energy surface with a simpler Hamiltonian and connecting this free energy surface to the target free energy surface with free energy calculations. Here, we show that the free energy legs of the cycle can be replaced with energy representation (ER) density functional approximations. We compute: (1) The conformational free energy changes for alanine dipeptide transitioning from the right-handed free energy basin to the left-handed basin and (2) the free energy difference between the open and closed conformations of β-cyclodextrin, a "host" molecule that serves as a model for molecular recognition in host-guest binding. β-cyclodextrin contains 147 atoms compared to 22 atoms for alanine dipeptide, making β-cyclodextrin a large molecule for which to compute solvation free energies by free energy perturbation or integration methods and the largest system for which the ER method has been compared to exact free energy methods. The ER method replaced the 28 simulations to compute each coupling free energy with two endpoint simulations, reducing the computational time for the alanine dipeptide calculation by about 70% and for the β-cyclodextrin by > 95%. The method works even when the distribution of conformations on the auxiliary free energy surface differs substantially from that on the target free energy surface, although some degree of overlap between the two surfaces is required. © 2016 Wiley Periodicals, Inc.

While free energy surfaces are the crux of our understandingofmany chemical and biological processes, their accuracy is generallyunknown. Moreover, many developments to improve their accuracy areoften complicated, limiting their general use. Luckily, several toolsand guidelines are already in place to identify these shortcomings,but they are typically lacking in flexibility or fail to systematicallydetermine how to improve the accuracy of the free energy calculation.To overcome these limitations, this work introduces OGRe, a Pythonpackage for optimal grid refinement in an arbitrary number of dimensions.OGRe is based on three metrics that gauge the confinement, consistency,and overlap of each simulation in a series of umbrella sampling (US)simulations, an enhanced sampling technique ubiquitously adopted toconstruct free energy surfaces for hindered processes. As these threemetrics are fundamentally linked to the accuracy of the weighted histogramanalysis method adopted to generate free energy surfaces from US simulations,they facilitate the systematic construction of accurate free energyprofiles, where each metric is driven by a specific umbrella parameter.This allows for the derivation of a consistent and optimal collectionof umbrellas for each simulation, largely independent of the initialvalues, thereby dramatically increasing the ease-of-use toward accuratefree energy surfaces. As such, OGRe is particularly suited to determinecomplex free energy surfaces with large activation barriers and shallowminima, which underpin many physical and chemical transformationsand hence to further our fundamental understanding of these processes.

A methodology is proposed for the calculation of multidimensional free-energy landscapes of molecular systems, based on analysis of multiple molecular dynamics trajectories wherein adaptive biases have been applied to enhance the sampling of different collective variables. In this approach, which we refer to as the Force-Correction Analysis Method (FCAM), local averages of the total and biasing forces are evaluated post hoc, and the latter are subtracted from the former to obtain unbiased estimates of the mean force across collective-variable space. Multidimensional free-energy surfaces and minimum free-energy pathways are then derived by integrating the mean-force landscape with a kinetic Monte Carlo algorithm. To evaluate the proposed method, a series of numerical tests and comparisons with existing approaches were carried out for small molecules, peptides, and proteins, based on all-atom trajectories generated with standard, concurrent, and replica-exchange metadynamics in collective-variable spaces ranging from one to six dimensional. The tests confirm the correctness of the FCAM formulation and demonstrate that calculated mean forces and free energies converge rapidly and accurately, outperforming other methods used to unbias this kind of simulation data.

Is Protein Single Molecule Dynamics under Force Described by Two or More States?

The authors present a method based on a linear response theory that allows one to optimize the geometries of quantum mechanical/molecular mechanical (QM/MM) systems on the free energy surfaces. Two different forms of linear response free energy functionals are introduced, and electronic wave functions of the QM region, as well as the responses of electrostatic and Lennard-Jones potentials between QM and MM regions, are self-consistently determined. The covariant matrix relating the QM charge distribution to the MM response is evaluated by molecular dynamics (MD) simulation of the MM system. The free energy gradients with respect to the QM atomic coordinates are also calculated using the MD trajectory results. They apply the present method to calculate the free energy profiles of Menshutkin-type reaction of NH3 with CH3Cl and Claisen rearrangement of allyl vinyl ether in aqueous solution. For the Menshutkin reaction, the free energy profile calculated with the modified linear response free energy functional is in good agreement with that by the free energy perturbation calculations. They examine the nonequilibrium solvation effect on the transmission coefficient and the kinetic isotope effect for the Claisen rearrangement.

Free energy simulations play a pivotal role in diversebiologicalapplications, including enzyme design, drug discovery, and biomolecularengineering. The characterization of high-dimensional free energysurfaces underlying complex enzymatic mechanisms necessitates extensivesampling through umbrella sampling or string method simulations. Accurateranking of target-binding free energies across large ligand librariesrelies on comprehensive alchemical free energy calculations organizedinto thermodynamic networks. The predictive accuracy of these methodshinges on robust, scalable tools for networkwide data analysis andextraction of physical properties from heterogeneous simulation data.Here, we introduce FE-ToolKit, a versatilesoftware suite for the automated analysis of free energy surfaces,minimum free energy paths, and alchemical free energy networks (thermodynamicgraphs).

In a series of two papers, we report the detailed mechanism of cyclobutane pyrimidine dimer repair in aqueous solvent using ab initio molecular dynamics simulations (AIMD). Umbrella sampling is used to determine the free energy surface for dimer splitting. The two-dimensional free energy surface for splitting of the C5-C5' and C6-C6' bonds on the anion surface is reported. The splitting of the C5-C5' and C6-C6' bonds occurs on a picosecond time scale. The transition state along the splitting coordinate in the anion state coincides with a maximum in the free energy along the same coordinate on the neutral surface. The implication is that back electron transfer occurring before the anion reaches the transition state leads to reformation of the cyclobutane dimer, while back electron transfer after transit through the transition state, leads to successful repair. On the basis of our calculations for CPD splitting in water, we propose a framework for understanding how various factors, such as solvent polarity, can control repair efficiency. This framework explains why back electron transfer leads predominantly to unsuccessful repair in some situations, and successful repair in others. A key observation is that the same free energy surfaces that control dimer splitting also govern how the back electron transfer rate changes during the splitting process. Configurational changes of the dimer along the splitting coordinate are also documented.

A molecule in a bulk of material is pulled from all the sides whereas a molecule at the surface experiences no pull from the top and is pulled only from the sides and below. Bonding between molecules takes place with lowering of free energy. The bond energies between particles are negative because energy is evolved on bond formation. Therefore, atoms, molecules, or ions on the surface of a liquid or solid possess higher energies than those located in the bulk of the substance. The energy of the surface increases with the increase of the surface area. Hence, the surface free energy can be defined as the energy required to create 1 m2 of new surface. This energy can be expressed in Joule per square meter (or ergs per square cm). As the molecules in solids are rigid, no such effect is realized. Surface free energy is a characteristic property of the liquid phase making a free surface. Since reduction in the surface area reduces the free energy, the surface of the liquid will tend to contract. Liquids tend to form spherical drops because spheres have the least surface area to volume ratio among all geometrical shapes. The tensional force causing the contraction of the free liquid surface is known as surface tension. The surface tension is defined as the force in newtons acting at right angles to a line 1 m long on the surface. The work done in producing 1 m2 of new surface is equivalent to the work done in stretching the surface by 1 m along a 1 m line against the surface tension and this is the surface free energy, σ. Thus, the surface tension σ, newton per meter (Nm−1), is numerically equal to the surface free energy Jm−2. Surface tension is generally used with reference to liquid, whereas surface free energy and interfacial energy are used with reference to liquids and solids, respectively. Solids definitely have surface free energy, but due to their rigid surfaces, they are neither smooth nor free to allow the solid to fill the shape of a container, and therefore, the concept of surface tension is not applicable in solids. Table 6.1 lists the values of surface tension of a few liquids.

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

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.

The selective permeability of cellular membranes is a crucial property for controlled transport into and out of cells. Molecules that can bypass the cellular machinery and spontaneously translocate across membranes could be used as therapeutics or drug carriers. Peptides are a prominent class of such molecules, which include natural and man-developed antimicrobial and cell-penetrating peptides. However, the necessary peptide properties for translocation remain elusive. Computer simulations could uncover these properties once we have a good collective variable (CV) that accurately describes the translocation process. Here, we developed a new CV, which includes a description of peptide insertion, local membrane deformation, and peptide internal degrees of freedom related to its charged groups. By comparison of CVs, we demonstrated that all these components are necessary for an accurate description of peptide translocation. Moreover, the advantages and disadvantages of three common methods for free-energy calculations with our CV were evaluated using the MARTINI coarse-grained model: umbrella sampling, umbrella sampling with replica exchange, and metadynamics. The developed CV leads to the reliable and effective calculation of the free energy of peptide translocation, and thus, it could be useful in the design of spontaneously translocating peptides.