Calculation Of Free Energy Profile Research Articles

Combined QM/MM, Machine Learning Path Integral Approach to Compute Free Energy Profiles and Kinetic Isotope Effects in RNA Cleavage Reactions.

We present a fast, accurate, and robust approach for determination of free energy profiles and kinetic isotope effects for RNA 2'-O-transphosphorylation reactions with inclusion of nuclear quantum effects. We apply a deep potential range correction (DPRc) for combined quantum mechanical/molecular mechanical (QM/MM) simulations of reactions in the condensed phase. The method uses the second-order density-functional tight-binding method (DFTB2) as a fast, approximate base QM model. The DPRc model modifies the DFTB2 QM interactions and applies short-range corrections to the QM/MM interactions to reproduce ab initio DFT (PBE0/6-31G*) QM/MM energies and forces. The DPRc thus enables both QM and QM/MM interactions to be tuned to high accuracy, and the QM/MM corrections are designed to smoothly vanish at a specified cutoff boundary (6 Å in the present work). The computational speed-up afforded by the QM/MM+DPRc model enables free energy profiles to be calculated that include rigorous long-range QM/MM interactions under periodic boundary conditions and nuclear quantum effects through a path integral approach using a new interface between the AMBER and i-PI software. The approach is demonstrated through the calculation of free energy profiles of a native RNA cleavage model reaction and reactions involving thio-substitutions, which are important experimental probes of the mechanism. The DFTB2+DPRc QM/MM free energy surfaces agree very closely with the PBE0/6-31G* QM/MM results, and it is vastly superior to the DFTB2 QM/MM surfaces with and without weighted thermodynamic perturbation corrections. 18O and 34S primary kinetic isotope effects are compared, and the influence of nuclear quantum effects on the free energy profiles is examined.

Effect of Protein Flexibility from Coarse-Grained Elastic Network Parameterizations on the Calculation of Free Energy Profiles of Ligand Binding.

The characterization of the affinity and binding mechanism of specific molecules to a protein active site is scientifically and industrially relevant for many applications. In principle, this information can be obtained using molecular dynamics (MD) simulations by calculating the free energy profile of the process. However, this is a computationally demanding calculation. Currently, coarse-grained (CG) force fields are very well implemented for MD simulations of biomolecular systems. These computationally efficient force fields are a major advantage to the study of large model systems and/or those requiring long simulation times. The Martini model is currently one of the most popular CG force fields for these systems. For the specific case of protein simulations, to correctly maintain the macromolecular three-dimensional structure, the Martini model needs to include an elastic network (EN). In this work, the effect of protein flexibility, as induced by three EN models compatible with the Martini force field, was tested on the calculation of free energy profiles for protein-ligand binding. The EN models used were ElNeDyn, GoMartini, and GEN. The binding of triolein (TOG) and triacetin (TAG) to a lipase protein (thermomyces lanuginosa lipase-TLL) was used as a case study. The results show that inclusion of greater flexibility in the CG parameterization of proteins is of high importance in the calculation of the free energy profiles of protein-ligand systems. However, care must be taken in order to avoid unjustified large protein deformations. In addition, due to molecular flexibility there may be no absolute need for the center of the ligand to reach the center of the protein-binding site. The calculation of the energy profile to a distance of about 0.5 nm from the active site center can be sufficient to differentiate the affinity of different ligands to a protein.

Lessons learned about steered molecular dynamics simulations and free energy calculations.

The calculation of free energy profiles is central in understanding differential enzymatic activity, for instance, involving chemical reactions that require QM-MM tools, ligand migration, and conformational rearrangements that can be modeled using classical potentials. The use of steered molecular dynamics (sMD) together with the Jarzynski equality is a popular approach in calculating free energy profiles. Here, we first briefly review the application of the Jarzynski equality to sMD simulations, then revisit the so-called stiff-spring approximation and the consequent expectation of Gaussian work distributions and, finally, reiterate the practical utility of the second-order cumulant expansion, as it coincides with the parametric maximum-likelihood estimator in this scenario. We illustrate this procedure using simulations of CO, both in aqueous solution and in a carbon nanotube as a model system for biologically relevant nanoheterogeneous environments. We conclude the use of the second-order cumulant expansion permits the use of faster pulling velocities in sMD simulations, without introducing bias due to large dispersion in the non-equilibrium work distribution.

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

Molecular simulation has been extensively used to calculate free energy associated with any molecular phenomenon in order to understand the thermodynamics and kinetics of the process. When the energetic barrier between the initial and the final stage is large, biasing collective variables (cv) that are some combination of atomic coordinates, allows sampling of all representative configurations and accurate calculation of free energy profiles. In this work, using all atom molecular dynamics simulation with biasing schemes like well-tempered metadynamics, umbrella sampling and replica exchange umbrella sampling, we calculate the free energy profile for translocation of small molecules through a model 1,2-dioleoyl-sn-glycero-3-phosphocholinebilayer as a function of distance from the bilayer center. We find that only for neutral solutes, the free energy calculation is straightforward. Using the committor analysis, we find that non-converged profiles result due to insufficient sampling of the transition state. We also demonstrate the usage of acceptance probabilities in a replica exchange calculation as a way to identify sampling problems. Based on our results, we provide useful guidance to detect and overcome convergence failure to infer useful physical behavior of a system.

Targeted Molecular Dynamics Calculations of Free Energy Profiles Using a Nonequilibrium Friction Correction.

As standard unbiased molecular dynamics (MD) simulations become impractical for sampling rare events, "targeted MD" employs a moving distance constraint to enforce rare transitions along some reaction coordinate x. To calculate free energy profiles from these nonequilibrium simulations via Δ G( x) = W( x) - Wdiss( x), apart from the (readily obtained) work W( x) performed on the system, the dissipated work Wdiss( x) is also required. By employing a second-order cumulant expansion of Jarzynski's equality combined with an analysis within Langevin theory, the dissipated work can be expressed via a nonequilibrium friction coefficient ΓNEQ( x) that may be calculated on-the-fly from constraint force fluctuations. Adopting the ion dissociation of NaCl in water as a test system, this friction correction is shown to result in accurate free energy profiles, even for a modest number of simulations and at high constraint velocities. As a bonus, the analysis of ΓNEQ( x) may yield valuable insight into the microscopic mechanism of friction.

From Aβ Filament to Fibril: Molecular Mechanism of Surface-Activated Secondary Nucleation from All-Atom MD Simulations.

Secondary nucleation pathways in which existing amyloid fibrils catalyze the formation of new aggregates and neurotoxic oligomers are of immediate importance for the onset and progression of Alzheimer's disease. Here, we apply extensive all-atom molecular dynamics simulations in explicit water to study surface-activated secondary nucleation pathways at the extended lateral β-sheet surface of a preformed Aβ9-40 filament. Calculation of free-energy profiles allows us to determine binding free energies and conformational intermediates for nucleation complexes consisting of 1-4 Aβ peptides. In addition, we combine the free-energy profiles with position-dependent diffusion profiles to extract complementary kinetic information and macroscopic growth rates. Single monomers bind to the β-sheet surface in a disordered, hydrophobically collapsed conformation, whereas dimers and larger oligomers can retain a cross-β conformation resembling a more ordered fibril structure. The association processes during secondary nucleation follow a dock/lock mechanism consisting of a fast initial encounter phase (docking) and a slow structural rearrangement phase (locking). The major driving forces for surface-activated secondary nucleation are the release of a large number of hydration water molecules and the formation of hydrophobic interface contacts, the latter being in contrast to the elongation process at filament tips, which is dominated by the formation of stable and highly specific interface hydrogen bonds. The calculated binding free energies and the association rates for the attachment of Aβ monomers and oligomers to the extended lateral β-sheet surface of the filament seed are higher compared to those for elongation at the filament tips, indicating that secondary nucleation pathways can become important once a critical concentration of filaments has formed.

Calculation of the intrinsic solvation free energy profile of an ionic penetrant across a liquid-liquid interface with computer simulations.

We introduce the novel concept of an intrinsic free energy profile, allowing one to remove the artificial smearing caused by thermal capillary waves, which renders difficulties for the calculation of free energy profiles across fluid interfaces in computer simulations. We apply this concept to the problem of a chloride ion crossing the interface between water and 1,2-dichloroethane and show that the present approach is able to reveal several important features of the free energy profile which are not detected with the usual, nonintrinsic calculations. Thus, in contrast to the nonintrinsic profile, a free energy barrier is found at the aqueous side of the (intrinsic) interface, which is attributed to the formation of a water “finger” the ion pulls with itself upon approaching the organic phase. Further, by the presence of a nonsampled region, the intrinsic free energy profile clearly indicates the coextraction of the first hydration shell water molecules of the ion when entering the organic phase.

Revisiting the Exact Relation between Potential of Mean Force and Free-Energy Profile.

Constraints are convenient in the calculation of free energy profiles via molecular dynamics simulations, but they subtly alter the phase space distribution. In a recent letter of a related title in this journal, Wong and York [J. Chem. Theory Comput. 2012, 8, 3998-4003] suggest replacing Fixman's mass-metric correction with its inverse and adding a Jacobian contribution to the potential of mean constraint force. This letter refutes both suggestions, discusses the underlying interpretation problems, and provides supporting numerical simulations.

Exploring the energetics of water permeation in photosystem II by multiple steered molecular dynamics simulations

The Mn4Ca cluster of the oxygen-evolving complex (OEC) of photosynthesis catalyzes the light-driven splitting of water into molecular oxygen, protons, and electrons. The OEC is buried within photosystem II (PSII), a multisubunit integral membrane protein complex, and water must find its way to the Mn4Ca cluster by moving through protein. Molecular dynamics simulations were used to determine the energetic barriers for water permeation though PSII extrinsic proteins. Potentials of mean force (PMFs) for water were derived by using the technique of multiple steered molecular dynamics (MSMD). Calculation of free energy profiles for water permeation allowed us to characterize previously identified water channels, and discover new pathways for water movement toward the Mn4Ca cluster. Our results identify the main constriction sites in these pathways which may serve as selectivity filters that restrict both the access of solutes detrimental to the water oxidation reaction and loss of Ca2+ and Cl− from the active site.

A Collective Coordinate to Obtain Free Energy Profiles for Complex Reactions in Condensed Phases.

Exploration of chemical reactions in complex explicit environments has become an affordable task with the use of hybrid quantum mechanics/molecular mechanics potentials which allow calculating free energy profiles of chemical reactions under the influence of the surroundings. Tracing these free energy profiles requires the selection of a reaction coordinate, which can be cumbersome for those processes involving more than a single chemical event in a concerted step. We here propose a collective coordinate to be used in the calculation of free energy profiles for complex reactions in condensed phases. This coordinate is based in the definition of the advance along a path introduced by Branduardi et al. (J. Chem. Phys.2007, 126, 054103) but modified to use internal coordinates which are more adequate for the description of chemical reactions. The coordinate is tested with the analysis of the isochorismate transformation to pyruvate and salycilate in aqueous solution and in the active site of PchB, a reaction that involves a CO bond breaking simultaneously with a proton transfer between two carbon atoms. The coordinate introduced here allows obtaining smooth and meaningful free energy profiles of the reaction.

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.

An experimentally guided umbrella sampling protocol for biomolecules

We present a simple method for utilizing experimental data to improve the efficiency of numerical calculations of free energy profiles from molecular dynamics simulations. The method involves umbrella sampling simulations with restraining potentials based on a known approximate estimate of the free energy profile derived solely from experimental data. The use of the experimental data results in optimal restraining potentials, guides the simulation along relevant pathways, and decreases overall computational time. In demonstration of the method, two systems are showcased. First, guided, unguided (regular) umbrella sampling simulations and exhaustive sampling simulations are compared to each other in the calculation of the free energy profile for the distance between the ends of a pentapeptide. The guided simulation use restraints based on a simulated "experimental" potential of mean force of the end-to-end distance that would be measured by fluorescence resonance energy transfer (obtained from exhaustive sampling). Statistical analysis shows a dramatic improvement in efficiency for a 5 window guided umbrella sampling over 5 and 17 window unguided umbrella sampling simulations. Moreover, the form of the potential of mean force for the guided simulations evolves, as one approaches convergence, along the same milestones as the extensive simulations, but exponentially faster. Second, the method is further validated by replicating the forced unfolding pathway of the titin I27 domain using guiding umbrella sampling potentials determined from actual single molecule pulling data. Comparison with unguided umbrella sampling reveals that the use of guided sampling encourages unfolding simulations to converge faster to a forced unfolding pathway that agrees with previous results and produces a more accurate potential of mean force.

Exploring free-energy profiles through ion channels: Comparison on a test case

The calculation of free-energy profiles in proteins, and, more specifically, in ion channels, is a challenge for modern numerical simulations due both to the convergence problems associated with the electrostatics of the environment and to the difficulties in modeling the fields acting on the permeating ions. The present study is aims at comparing three different simulation techniques available in the literature on a nanometric channel protein chosen as a test case, with the purpose of establishing their real predictivity and limits.

Through the channel and around the channel: Validating and comparing microscopic approaches for the evaluation of free energy profiles for ion penetration through ion channels.

Microscopic calculations of free energy profiles for ion transport through biological ion channels present a very serious challenge to modern simulation approaches. The main problem is due to the major convergence problems associated with the heterogeneous landscape of the electrostatic environment in ion channels and with the need to evaluate the profile associated with the transfer of the ion from bulk water to the channel environment. This problem is compounded by the lack of reliable and relevant benchmarks that can discriminate between alternative approaches. The present study is aimed at reducing the above problems by defining benchmarks that are directly relevant to ion channels and can also give converging results. This is done by constructing a series of models of a truncated gramicidin channel with different numbers of water molecules and by comparing the profiles for going around the channel and through the channel. These discriminating models are then used to validate and compare the adiabatic charging free energy perturbation (FEP) approach combined with an umbrella sampling approach (Warshel, A. J. Phys. Chem. 1982, 86, 2218) and the potential of mean force (PMF) approach used frequently in studies of ion channels. It is found that both approaches work quite well until one moves to the case of the fully solvated channel. In this limit, the PMF approach may give different results for the overall work of going through the channel and around the channel, while the FEP approach gives physically consistent results. The present benchmark also indicates that the weighted histogram analysis method (WHAM) approach does not offer a significant advantage over earlier approaches at least as much as studies of ion channels are concerned. Finally, it is concluded that the FEP approach may be more useful in evaluating the overall barrier for moving ions from water to ion channels and that in some cases it might be beneficial to use the FEP approach for selective points along the channel and then to connect these points by PMF calculations.

Calculation of Free Energy Profiles for Elementary Bimolecular Reactions by ab Initio Molecular Dynamics: Sampling Methods and Thermostat Considerations

A study designed to refine the procedure for performing ab initio molecular dynamics calculations (AIMD) on chemical reactions is presented. Of key interest is the calculation of changes in free energy along the entire reaction path. Several simple elementary reactions are studied with the Car−Parrinello projector augmented-wave (CP−PAW) density-functional theory (DFT) methodology. The illustrative gas-phase bimolecular addition reactions are (i) a σ complexation of BH3 + H2O → H2O·BH3, (ii) the Diels−Alder reactions of butadiene with ethene, C4H6 + C2H4 → cyclohexene, 1,3-cyclopentadiene (CP) and ethene, CP + C2H4→ norbornene, and the stereoselective reaction of 5-amino-CP with ethene, amino-CP + C2H4 → amino-norbornene, (iii) the carbene cyclopropanation Cl2C + C2H4→ Cl2C3H4, and (iv) the dimerization of ketene. These reactions were used to test both the slow-growth and point-wise thermodynamic integration (STI and PTI) methods of phase-space sampling as well as the Nose−Hoover and Andersen thermostats....

Ab Initio QM/MM Simulation with Proper Sampling: “First Principle” Calculations of the Free Energy of the Autodissociation of Water in Aqueous Solution

Quantum mechanical calculations of activation free energies of chemical reactions in condensed phases present a major challenge for computational chemistry. On one hand, it is important to use high-level ab initio methods to obtain reliable results. On the other hand, it is essential to perform sufficient configurational sampling to obtain meaningful free energies. Although the advance of quantum mechanical/molecular mechanics (QM/MM) approaches has made this problem tractable, it still requires an enormous amount of computer time. The present work advances several strategies that allow one to perform practical ab initio QM/MM calculations of free energy profiles in solutions and proteins. The basic idea is the use of a simple reference potential for the ab initio calculations (e.g., Bentzien; et al. J. Phys. Chem. B 1998, 102, 2293). One version of this approach evaluates the free energy of transfer from the reference potential to the ab initio potential by a single step free energy perturbation (FEP) approach. A new version evaluates this free energy by the linear response approximation (LRA), which involves running trajectories on both the reference and the ab initio potentials. The performance of both approaches is examined by calculating the potential of mean force for the autodissociation reaction of water in solution. It is found that the LRA approach allows one to obtain reasonable results even in cases where the ab initio and reference potentials are significantly different. The present work also explores options for increasing the size of the quantum mechanical region. Here it is shown that the constrained DFT (CDFT) method provides a promising strategy. Finally, the general issue of modeling the autodissociation reaction by quantum mechanical approaches is briefly considered. It is pointed out that the use of the empirical valence bond (EVB) approach in the sampling process should provide a way for evaluating the elusive nonequilibrium solvation effect.

On the use of the adiabatic molecular dynamics technique in the calculation of free energy profiles

A new molecular dynamics method for calculating free energy profiles for rare events is presented. The new method is based on the creation of an adiabatic separation between a reaction coordinate subspace and the remaining degrees of freedom within a molecular dynamics run. This is achieved by associating with the reaction coordinate(s) a high temperature and large mass, thereby allowing the activated process to occur while permitting the remaining degrees of freedom to respond adiabatically. In this limit, by applying a formal multiple time scale Liouville operator factorization, it can be rigorously shown that the free energy profile is obtained directly from the probability distribution of the reaction coordinate subspace and, therefore, no unbiasing of the configuration space or postprocessing of the output data is required. The new method is applied to a variety of model problems including a two-dimensional free energy surface and its performance tested against free energy calculations using the “blue moon ensemble” approach. The comparison shows that free energy profiles can be calculated with greater ease and efficiency using the new method.

Free Energy of Ionic Hydration

The hydration free energies of ions exhibit an approximately quadratic dependence on the ionic charge, as predicted by the Born model. We analyze this behavior using second-order perturbation theory. The average and the fluctuation of the electrostatic potential at charge sites appear as the first coefficients in a Taylor expansion of the free energy of charging. Combining the data from different charge states (e.g., charged and uncharged) allows calculation of free-energy profiles as a function of the ionic charge. The first two Taylor coefficients of the free-energy profiles can be computed accurately from equilibrium simulations, but they are affected by a strong system-size dependence. We apply corrections for these finite-size effects by using Ewald lattice summation and adding the self-interactions consistently. An analogous procedure is used for the reaction-field electrostatics. Results are presented for a model ion with methane-like Lennard-Jones parameters in simple point charge water. We find t...

Calculations of free energy profiles for the staphylococcal nuclease catalyzed reaction

Calculations of the free energy profile for the first two (rate-limiting) steps of the staphylococcal nuclease catalyzed reaction are reported. The calculations are based on the empirical valence bond method in combination with free energy perturbation molecular dynamics simulations. The calculated activation free energy is in good agreement with experimental kinetic data, and the catalytic effect of the enzyme is reproduced without any arbitrary adjustment of parameters. The enormous reduction of the activation barrier (relative to the reference reaction in water) appears to be largely associated with the strong electrostatic effect of the Ca2+ ion and the two arginine residues in the active site. This favorable electrostatic environment reduces the cost of the general-base catalysis step by almost 15 kcal/mol (by stabilizing the OH- nucleophile) and then stabilizes the developing negative charge on the 5'-phosphate group in the second step of the reaction by about 19 kcal/mol. The basic features of the originally postulated enzyme mechanism (Cotton et al., 1979) are found to be compatible with the observed activation free energy. However, the proposed modification of the mechanism (Sepersu et al., 1987), in which Arg 87 interacts only with the pentacoordinated transition state, is supported by the simulations. Further calculations on the D21E mutant also give results in good agreement with kinetic data.