Conformational transitions of a dipeptide in water: Effects of imposed pathways using umbrella sampling techniques

The free energy difference between two states of a molecular system separated by an energy barrier can generally be computed using the technique of umbrella sampling along a chosen reaction coordinate or pathway. The effect of a particular choice of pathway upon the obtained free energy difference is investigated by molecular dynamics simulation of a model system consisting of a glycine dipeptide in aqueous solution. Two different reaction coordinates connecting the so-called C₅ and C₇ conformations, one involving intramolecular hydrogen bonds and the other involving the peptide ?, ψ angles, are considered. The Gibbs free energy differences ΔG(C₅ – C₇) are small in both cases, 1.5 ± 1 kJ mol^?1 and 2.2 ± 1 kJ mol ^?1, respectively. The two different reaction coordinates yield free energy differences that are identical to within their statistical error. It is found that the exchange of solute–solute, solute–water, and water–water hydrogen bonds involves free energy changes of less than k_BT, which points at the existence of a multitutde of low free energy pathways connecting the C₅ and C₇ dipeptide conformations. © 1994 John Wiley & Sons, Inc.     

Sparse sampling of water density fluctuations near liquid-vapor coexistence

Abstract

The free energetics of water density fluctuations in bulk water, at interfaces, and in hydrophobic confinement inform the hydration of hydrophobic solutes as well as their interactions and assembly. The characterisation of such free energetics is typically performed using enhanced sampling techniques such as umbrella sampling. In umbrella sampling, order parameter distributions obtained from adjacent biased simulations must overlap in order to estimate free energy differences between biased ensembles. Many biased simulations are typically required to ensure such overlap, which exacts a steep computational cost. We recently introduced a sparse sampling method, which circumvents the overlap requirement by using thermodynamic integration to estimate free energy differences between biased ensembles. Here we build upon and generalise sparse sampling for characterising the free energetics of water density fluctuations in systems near liquid-vapor coexistence. We also introduce sensible heuristics for choosing the biasing potential parameters and strategies for adaptively refining them, which facilitate the estimation of such free energetics accurately and efficiently. We illustrate the method by characterising the free energetics of cavitation in a large volume in bulk water. We also use sparse sampling to characterise the free energetics of capillary evaporation for water confined between two hydrophobic plates. In both cases, sparse sampling is nearly two orders of magnitude faster than umbrella sampling. Given its efficiency, the sparse sampling method is particularly well suited for characterising free energy landscapes for systems wherein umbrella sampling is prohibitively expensive.     

Computing free energies using nested sampling-based approaches

Abstract

Nested sampling (NS) has emerged as a powerful statistical mechanical sampling technique to compute the partition function of atomic and molecular systems. From the partition function all thermodynamic quantities can be calculated in absolute terms, including absolute free energies and entropies. In this article, we provide a brief overview of NS within a Bayesian context, as well as overviews of how NS is used to compute the partition functions and thermodynamic quantities in the canonical and isothermal-isobaric ensembles. Then we introduce a new scheme, Coupling Parameter Path Nested Sampling, to estimate the free energy difference between two systems with different potential energy functions. The method uses a NS simulation to traverse the same path through phase space as would be covered in traditional coupling parameter-based methods such as thermodynamic integration and perturbation approaches. We demonstrate the new method with two case studies and confirm its accuracy by comparison to conventional methods, including Widom test particle insertion and thermodynamic integration. The proposed method provides a powerful alternative to traditional coupling parameter-based free energy simulation methods.     

Molecular Dynamics Simulations of some Small Organic Molecules

Abstract

Free energy differences between different conformers of D-ribofuranose, L-malic acid and meso-tartaric acid in solution were calculated using Molecular Dynamics simulations. In case of ribose the α → β transition was studied. For the acids attention was focussed on the transitions between the three possible staggered conformers with respect to the central C-C bond. In all cases a thermodynamic integration method was employed to evaluate the free energy difference. The use of an alternative technique, umbrella sampling, for ribose did not give promising results.

It was shown that one needs a fairly accurate picture of the accessible conformational space in case of flexible molecules like the ones considered here before one can determine meaningful free energy differences. Large hysteresis effects between forward and reverse simulated transitions were observed, but contrary to the general belief they are no direct measure of the accuracy of the calculated ΔG values. In all cases the ΔG values resulting from the simulations and from NMR experiments agree within the, considerable, error limits and for the different forms of D-ribose, L-malic acid and L-tartaric acid the relative order of their populations is also correctly reproduced.     

Free Energy Calculations Predict Sequence Specificity in DNA-Drug Complexes

Abstract

Molecular dynamics simulations of DNA-netropsin complexes in water were performed using the thermodynamic cycle-perturbation method to calculate the free energy difference between complexes with an adenine-containing binding site and corresponding complexes where adenines are replaced by 2,6-diaminopurines (dap). The calculations predict a free energy difference of 3.7±0.9 kcal/mol (at 300K) in favour of netropsin binding to an (AATT)₂ DNA sequence compared to a (dapdapTT)₂ sequence.     

Free energy simulations for protein ligand binding and stability

Abstract

We summarize several computational techniques to determine relative free energies for condensed-phase systems. The focus is on practical considerations which are capable of making direct contact with experiments. Particular applications include the thermodynamic stability of apo- and holo-myoglobin, insulin dimerization free energy, ligand binding in lysozyme, and ligand diffusion in globular proteins. In addition to provide differential free energies between neighboring states, converged umbrella sampling simulations provide insight into migration barriers and ligand dissociation barriers and analysis of the trajectories yield additional insight into the structural dynamics of fundamental processes. Also, such simulations are useful tools to quantify relative stability changes for situations where experiments are difficult. This is illustrated for NO-bound myoglobin. For the dissociation of benzonitrile from lysozyme it is found that long umbrella sampling simulations are required to approximately converge the free energy profile. Then, however, the resulting differential free energy between the bound and unbound state is in good agreement with estimates from molecular mechanics with generalized Born surface area simulations. Furthermore, comparing the barrier height for ligand escape suggests that ligand dissociation contains a non-equilibrium component.     

Kinetics and thermodynamics of gas diffusion in a NiFe hydrogenase

We have investigated O₂ and H₂ transport across a NiFe hydrogenase at the atomic scale by means of computational methods. The Wild Type protein has been compared with the V74Q mutant. Two distinct methodologies have been applied to study the gas access to the active site. Temperature locally enhanced sampling simulations have emphasized the importance of protein dynamics on gas diffusion. The O₂ diffusion free energy profiles, obtained by umbrella sampling, are in agreement with the known kinetic data and show that in the V74Q mutant, the inhibition process is lowered from both a kinetic and a thermodynamic point of view. Proteins 2011. © 2012 Wiley Periodicals, Inc.     

A Comparison of Seven Fast but Approximate Methods to Compute the Free Energy of Deprotonation for Amino Acids in Aqueous Solution

Abstract

In recent years, a variety of methods based on statistical mechanics have been successfully applied to calculate free energy differences of chemical reactions from molecular simulation. The accuracy and computational efficiency vary strongly between these methods. Seven approximate but fast methods to calculate free energy differences are compared in terms of accuracy and efficiency with the accurate but expensive thermodynamic integration method as reference, using 28 protonation and deprotonation reactions of aspartic acid in aqueous solution as test cases. At least two simulations are required to obtain an accurate free energy difference between two states of the system. Both, the averaged one-step perturbation method and the linear response method yield the most accurate results, while the latter method shows the fastest convergence.     

Free energy surface for rotamers of cis-enol malonaldehyde in aqueous solution studied by molecular dynamics calculations

Two-dimensional free energy surfaces for four rotamers of cis-enol malonaldehyde in water have been investigated by umbrella sampling molecular dynamics (MD) calculations. Biasing potential used in the umbrella sampling calculation was adopted to be the minus of conformational free energy preliminary obtained by the thermodynamic integration MD calculations for the rigid malonaldehyde whose stretching and bending were all fixed. The calculated free energy surface shows that, in water, a rotamer that has an intramolecular hydrogen bond is most stable among the rotamers. This is the same as that in vacuum, while order of relative stability of the other three rotamers is different in water and in vacuum. Inclusion of intramolecular vibrations changed the free energy surface little, i.e. at most 2.6 kJ/mol, which is much smaller than the solvation free energy. Free energy barriers from the most stable intramolecular hydrogen bonded rotamer to the others are lowered by hydration but they are still very high, >50 kJ/mol, such that the malonaldehyde molecule spends most of its time in water taking this conformation. Thus, reaction coordinate for intramolecular proton transfer reaction in water may be constructed assuming this rotamer.     

Unconstrained enhanced sampling for free energy calculations of biomolecules: a review

Abstract

Free energy calculations are central to understanding the structure, dynamics and function of biomolecules. Yet insufficient sampling of biomolecular configurations is often regarded as one of the main sources of error. Many enhanced sampling techniques have been developed to address this issue. Notably, enhanced sampling methods based on biasing collective variables (CVs), including the widely used umbrella sampling, adaptive biasing force and metadynamics, have been discussed in a recent excellent review (Abrams and Bussi, Entropy, 2014). Here, we aim to review enhanced sampling methods that do not require predefined system-dependent CVs for biomolecular simulations and as such do not suffer from the hidden energy barrier problem as encountered in the CV-biasing methods. These methods include, but are not limited to, replica exchange/parallel tempering, self-guided molecular/Langevin dynamics, essential energy space random walk and accelerated molecular dynamics. While it is overwhelming to describe all details of each method, we provide a summary of the methods along with the applications and offer our perspectives. We conclude with challenges and prospects of the unconstrained enhanced sampling methods for accurate biomolecular free energy calculations.     

First principles investigation of pressure dependent stability,phonon, Debye temperature,physical, mechanical and thermodynamic properties of Rh3Al intermetallic compound

ABSTRACT

Pressure dependence of stability, phonon, Debye temperature, physical, mechanical and thermodynamic properties of Rh₃Al intermetallic compound were investigated by first-principles The calculated cohesive energy (E_c), formation enthalpy (ΔH) show that Rh₃Al is a thermodynamically stable compound. Properties related to the phonons of Rh₃Al were also obtained. In addition, the transverse sound velocity (ν_s), longitudinal sound velocity (ν_l), average sound velocity (ν_m) and Debye temperature (Θ_D) of Rh₃Al were calculated by using the VRH method along with pressure range from 0 to 60?GPa. The values of lattice parameters, bulk modulus and its first-order pressure derivative are consistent well with other works. The band structure indicates that Rh₃Al compound exhibits a metallic character. Moreover, the total density of states, partial density of states, Mulliken charges and electron density difference have been analysed to explain the physical properties. Based on the stress–strain approach and the Born stability criteria, the mechanical properties were evaluated by elastic constants (C_ij), other modulus (B, E, G), (B/G) ratio, Poisson’s ratio (ν), the anisotropic index (A), hardness (H) and compressibility (K) for this intermetallic compound. Finally, the thermodynamic properties, including enthalpy, free energy, entropy and heat capacity are discussed range from 0 to 1000?K.     

Tryptophan-47 rotational isomerization in variant-3 scorpion neurotoxin. A combination thermodynamic perturbation and umbrella sampling study.

A combination thermodynamic perturbation and umbrella sampling study predicts two free energy wells for the rotational isomerization of the variant-3 scorpion neurotoxin tryptophan-47 indole side chain. One well has the indole side chain in the crystallographic orientation; the other has the indole rotated approximately 220 degrees to form a new conformation with a relative free energy of 3 +/- 2 kcal/mol. The activation barrier is 8.5 kcal/mol from the crystallographic well, from which transition state theory predicts a rate of escape of 2 x 10(5) s-1. Correlations in the displacements of side chains neighboring tryptophan-47 and the isomerization reaction coordinate last up to 20 ps. Favorable conditions of experimental verification are discussed.     

Free Energy Perturbations in Ribonuclease T1 Substrate Binding. A Study of the Influence of Simulation Length,Internal Degrees of Freedom and Structure in Free Energy Perturbations

Abstract

We have studied the reliability of free energy perturbation calculations with respect to simulation protocol and simulation length in a real biological system, the binding of two different ligands to wildtype Ribonuclease T ₁ (RNT1) and to a mutant of RNT1 with Glu-46 replaced by Gln (RNT1-Gln46). The binding of the natural substrate 3′ GMP has been compared with the binding of a fluorescent probe, 2-aminopurine 3′ mono phosphate (2AP3′MP). These simulations predict that the mutant binds 2AP3′MP better than 3′GMP. Four complete free energy perturbations were performed that form a closed loop of four free energy differences, which should sum up to zero. This could be used as a tool for searching for systematic errors that are not detected by standard forward ? backward perturbations. The perturbation between 2AP3′MP and 3′GMP is quite straightforward and similar to what has been done by other groups. The perturbation between Glu46 and Gln46 is much more complex, involving as many as twelve atoms and a change of charge. This perturbation needs much longer simulation time, 500-600 ps, than used in free energy perturbations before. The increased simulation time is needed both to reach an equilibrium and to include several phases of fluctuations of the observed parameters in the production run. The extremely long simulation time is not such a severe problem as much of the work might be done on several different machines in parallel and cheap workstations are excellent for these calculations. Problems may also occur with values of the coupling parameter Λ close to 0 or 1, due to the high mobility of atoms as well as insertion/deletion in a previously unoccupied space involved in the perturbation.     

The Influence of Starting Coordinates in Free Energy Simulations of Ligand Binding to Dihydrofolate Reductase

Abstract

Molecular dynamics (MD) simulation combined with free energy perturbation (FEP) methods have been used to study the key structural differences and relative free energies for the binding of 6-methyl-N5-deazapterin (N8 protonated) and the 8-substituted compound, 6,8-dimethyl-N5-deazapterin (N3 protonated), to dihydrofolate reductase (DHFR). The free energy changes have been calculated using a variety of initial X-ray coordinates derived from bacterial and vertebrate (including human) DHFRs, and both with and without the reduced cofactor nicotinamide adenine dinucleotide (NADPH) bound. Given a sufficiently long simulation time for the FEP calculations (ca. 200 ps), all structures obtained after mutating 6,8-methyl-N5-deazapterin to 6-methyl-N5-deazapterin exhibited hydrogen bond formation between a backbone carbonyl group of DHFR and H(N8) of 6-methyl-N5-deazapterin, analogous to that found in the X-ray crystal structure of N5-deazafolate(N8 protonated) bound to human DHFR. However, both simulation and experiment suggest this additional H-bonding does not greatly enhance thermodynamic stability, with experiment indicating at most a factor of 2 difference in the relative affinities of the two ligand cations for vertebrate DHFR. Moreover, a binding differential of 10 in favour of the protonated 8-substituted compound is found experimentally for bacterial DHFR. The MD/FEP calculations suggest that the relative cost of ligand desolvation may largely cancel the lowering of free energy obtained in the active site, resulting in predicted binding differences within the range indicated by the vertebrate and bacterial DHFR experiments. However, the theoretical free energy changes could not be obtained with the accuracy required for the rationalization of the observed species dependence. While sampling difficulties are known to be inherent in MD simulation methodologies, these studies with several initial coordinate sets have demonstrated the contribution of coordinate choice to this problem. The results indicate that for demanding protein-ligand binding problems such as this one, the accuracy of the method may be no better than ± 2 kcal/mol.     

Peptide Free Energy Landscapes Calibrated by Molecular Orbital Calculations

Free energy landscapes of peptide conformations werecalibrated by ab initiomolecular orbital calculations, after enhancedconformational sampling using the multicanonical molecular dynamicssimulations. Three different potentials of mean force for an isolateddipeptide were individually obtained using the conventional force fields,AMBER parm94, AMBER parm96, and CHARMm22. Each potential ofmean force was calibrated based on the umbrella sampling algorithm fromthe adiabatic energy map that was calculated separately by the abinitiomolecular orbital method. All the calibrated potentials of mean forcecoincided well. The calibration was applied to a peptide in explicit water,and the calibrated free energy landscapes did not depend on the force fieldused in conformational sampling, as far as the conformational space waswell sampled.     

Taboo-based Monte Carlo Search as a Method to Improve Sampling Efficiency

Abstract

Taboo-based Monte Carlo search which restricts the sampling of the region near an old configuration, is developed. In this procedure, Monte Carlo simulation and random search method are combined to improve the sampling efficiency. The feasibility of this method is tested on global optimization of a continuous model function, melting of the 256 Lennard-Jones particles at T? = 0.680 and ρ? = 0.850 and polypeptides (alanine dipeptide and Metenkephalin). From the comparison of results for the model function between our method and other methods, we find the increase of convergence rate and the high possibility of escaping from the local energy minima. The results of the Lennard-Jones solids and polypeptides show that the convergence property to reach the equilibrium state is better than that of others. It is also found that no significant bias in ensemble distribution is detected, though taboo-based Monte Carlo search does not sample the correct ensemble distribution owing to the restriction of the sampling of the region near an old configuration.     

The Chemical Potential of Dense Liquids by the Coupled Test Particle Method

Abstract

A new method for chemical potential estimation is proposed which is based on the coupled particle approach. The coupled particle method defines an attractive solution to the weighting function problem in umbrella sampling, bridging the gap between f and g distributions at high density. A way of eliminating the origin singularity is suggested, which is similar in spirit to the restricted umbrella sampling of Shing and Gubbins, but which is based on geometric rather than energetic criteria.

The method is illustrated on the Lennard-Jones system up to a reduced density ρ? = 1.1 along the isotherm T? = 1.2 and results are compared with the test particle insertion method and empirical equations of state. The new method is particularly useful at high liquid densities where it is superior to the other methods relying on the degree of overlap of f and g distributions. It gives reliable estimates of the chemical potential in the whole range of liquid densities.     

Molecular-Level Thermodynamic Switch Controls Chemical Equilibrium in Sequence-Specific Hydrophobic Interaction of 35 Dipeptide Pairs

Applying the Planck-Benzinger methodology, the sequence-specific hydrophobic interactions of 35 dipeptide pairs were examined over a temperature range of 273–333 K, based on data reported by Nemethy and Scheraga in 1962. The hydrophobic interaction in these sequence-specific dipeptide pairs is highly similar in its thermodynamic behavior to that of other biological systems. The results imply that the negative Gibbs free energy change minimum at a well-defined stable temperature, T_s, where the bound unavailable energy, TΔS^o = 0, has its origin in the sequence-specific hydrophobic interactions, are highly dependent on details of molecular structure. Each case confirms the existence of a thermodynamic molecular switch wherein a change of sign in ΔCp^o(T)_reaction (change in specific heat capacity of reaction at constant pressure) leads to true negative minimum in the Gibbs free energy change of reaction, ΔG^o(T)_reaction, and hence a maximum in the related equilibrium constant, K_eq. Indeed, all interacting biological systems examined to date by Chun using the Planck-Benzinger methodology have shown such a thermodynamic switch at the molecular level, suggesting its existence may be universal.     

Role of a reaction coordinate in free energy calculations

Abstract

The free energy calculation method emerges as a viable technique for ‘in-silico’ calorimetry. Efficient sampling techniques and the good choice of a reaction path connecting the reactant and the product state enable accurate computations of the free energy differences. We argue that in many cases the thermodynamic integration technique has the lowest variance when the transformation between the reactant and the product state proceeds along the natural path of the studied chemical reaction. We provide examples of free energy calculations for the fragmentation of the charged clusters and the swapping reaction of oligomer formation in proteins that follow a tentative reaction mechanism.     

Simulating the free energy of passive membrane permeation for small molecules

Abstract

Computer simulations of passive membrane permeation provide important microscopic insights into the molecular mechanism of this important biological process that are complementary to experimental data. Our review focuses on the main approaches for calculating the free energy, or potential of mean force, for permeation of small molecules through lipid bilayers. The theoretical background for most currently used methods for potential of mean force calculation is described, including particle insertion, thermodynamic integration, umbrella sampling, metadynamics, adaptive biasing force and milestoning. A brief comparison of strengths and weaknesses of the competing approaches is presented. This is followed by a survey of results obtained by the different methods, with special attention to describing the mechanistic insights generated by modelling and illustrating capabilities of the different techniques. We conclude with a discussion of recent advances and future directions in modelling membrane permeation, including latest methodological enhancements, consideration of multiple slow variables and memory effects.