Rapid free energy calculation of peptide self-assembly by REMD umbrella sampling

We extend umbrella sampling with replica exchange steps to calculate free energies that are important in the self-assembly of peptides. This leads to a more than 10-fold speed up over conventional umbrella sampling, thereby providing a practical method to calculate these free-energy differences. This approach can also observe first-order phase transitions and pinpoint the location of the concomitant boundary. When conformational changes are involved, this method can handle peptides up to approximately 7 residues, providing a rapid and accurate assessment of the thermodynamic properties of model systems, and can thus be used to answer fundamental questions about peptide self-assembly. When no major conformational changes are involved, we expect the size limit to be equal to that of standard molecular dynamics.     

Self-healing umbrella sampling: a non-equilibrium approach for quantitative free energy calculations

We propose a new approach for the umbrella sampling method in molecular dynamics simulations of complex systems. An accelerated sampling of the slow degrees of freedom is achieved by generating a single self-adaptive trajectory that tends to span uniformly the reaction coordinate using a time dependent bias potential derived from the preceding history of the system. To show the convergent behavior and the efficiency of the method, we present the free energy surface of alanine dipeptide in water as a function of the backbone dihedral angles.     

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

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

The free energy landscape of small peptides as obtained from metadynamics with umbrella sampling corrections

There is considerable interest in developing methodologies for the accurate evaluation of free energies, especially in the context of biomolecular simulations. Here, we report on a reexamination of the recently developed metadynamics method, which is explicitly designed to probe "rare events" and areas of phase space that are typically difficult to access with a molecular dynamics simulation. Specifically, we show that the accuracy of the free energy landscape calculated with the metadynamics method may be considerably improved when combined with umbrella sampling techniques. As test cases, we have studied the folding free energy landscape of two prototypical peptides: Ace-(Gly)(2)-Pro-(Gly)(3)-Nme in vacuo and trialanine solvated by both implicit and explicit water. The method has been implemented in the classical biomolecular code AMBER and is to be distributed in the next scheduled release of the code.     

Biomolecular free energy profiles by a shooting/umbrella sampling protocol, "BOLAS"

We develop an efficient technique for computing free energies corresponding to conformational transitions in complex systems by combining a Monte Carlo ensemble of trajectories generated by the shooting algorithm with umbrella sampling. Motivated by the transition path sampling method, our scheme "BOLAS" (named after a cowboy's lasso) preserves microscopic reversibility and leads to the correct equilibrium distribution. This makes possible computation of free energy profiles along complex reaction coordinates for biomolecular systems with a lower systematic error compared to traditional, force-biased umbrella sampling protocols. We demonstrate the validity of BOLAS for a bistable potential, and illustrate the method's scope with an application to the sugar repuckering transition in a solvated deoxyadenosine molecule.     

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

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

Efficient calculation of relative binding free energies by umbrella sampling perturbation

An important task of biomolecular simulation is the calculation of relative binding free energies upon chemical modification of partner molecules in a biomolecular complex. The potential of mean force (PMF) along a reaction coordinate for association or dissociation of the complex can be used to estimate binding affinities. A free energy perturbation approach, termed umbrella sampling (US) perturbation, has been designed that allows an efficient calculation of the change of the PMF upon modification of a binding partner based on the trajectories obtained for the wild type reference complex. The approach was tested on the interaction of modified water molecules in aqueous solution and applied to in silico alanine scanning of a peptide‐protein complex. For the water interaction test case, excellent agreement with an explicit PMF calculation for each modification was obtained as long as no long range electrostatic perturbations were considered. For the alanine scanning, the experimentally determined ranking and binding affinity changes upon alanine substitutions could be reproduced within 0.1–2.0 kcal/mol. In addition, good agreement with explicitly calculated PMFs was obtained mostly within the sampling uncertainty. The combined US and perturbation approach yields, under the condition of sufficiently small system modifications, rigorously derived changes in free energy and is applicable to any PMF calculation. © 2014 Wiley Periodicals, Inc.     

Determination of conformational free energies of peptides by multidimensional adaptive umbrella sampling

We improve the multidimensional adaptive umbrella sampling method for the computation of conformational free energies of biomolecules. The conformational transition between the alpha-helical and beta-hairpin conformational states of an alanine decapeptide is used as an example. Convergence properties of the weighted-histogram-analysis-based adaptive umbrella sampling can be improved by using multiple replicas in each adaptive iteration and by using adaptive updating of the bounds of the umbrella potential. Using positional root-mean-square deviations from structures of the alpha-helical and beta-hairpin reference states as reaction coordinates, we obtained well-converged free energy surfaces of both the in-vacuum and in-solution decapeptide systems. From the free energy surfaces well-converged relative free energies between the two conformational states can be derived. Advantages and disadvantages of different methods for obtaining conformational free energies as well as implications of our results in studying conformational transitions of proteins and in improving force field are discussed.     

Adaptive umbrella sampling of the potential energy: modified updating procedure of the umbrella potential and application to peptide folding

Adaptive umbrella sampling of the potential energy is used as a search method to determine the structures and thermodynamics of peptides in solution. It leads to uniform sampling of the potential energy, so as to combine sampling of low-energy conformations that dominate the properties of the system at room temperature with sampling of high-energy conformations that are important for transitions between different minima. A modification of the procedure for updating the umbrella potential is introduced to increase the number of transitions between folded and unfolded conformations. The method does not depend on assumptions about the geometry of the native state. Two peptides with 12 and 13 residues, respectively, are studied using the CHARMM polar-hydrogen energy function and the analytical continuum solvent potential for treatment of solvation. In the original adaptive umbrella sampling simulations of the two peptides, two and six transitions occur between folded and unfolded conformations, respectively, over a simulation time of 10 ns. The modification increases the number of transitions to 6 and 12, respectively, in the same simulation time. The precision of estimates of the average effective energy of the system as a function of temperature and of the contributions to the average effective energy of folded conformations obtained with the adaptive methods is discussed. Received: 11 July 1998 / Accepted: 22 September 1998 / Published online: 17 December 1998     

Analysis of the statistical error in umbrella sampling simulations by umbrella integration

Umbrella sampling simulations, or biased molecular dynamics, can be used to calculate the free-energy change of a chemical reaction. We investigate the sources of different sampling errors and derive approximate expressions for the statistical errors when using harmonic restraints and umbrella integration analysis. This leads to generally applicable rules for the choice of the bias potential and the sampling parameters. Numerical results for simulations on an analytical model potential are presented for validation. While the derivations are based on umbrella integration analysis, the final error estimate is evaluated from the raw simulation data, and it may therefore be generally applicable as indicated by tests using the weighted histogram analysis method.     

Efficient calculation of low energy statistical rates for gas phase dissociation using umbrella sampling

Monte Carlo (MC) simulations can be used to compute microcanonical statistical rates of gas phase dissociation reactions. Unfortunately, the MC approach may suffer from a slow convergence and large statistical errors for energies just above the dissociation threshold. In this work, umbrella sampling is proposed as a device to reduce the statistical error of MC rate constants. The method is tested by computing the classical dissociation rate for the reaction [H5O2+]* --> H2O + H3O(+) over the range of internal energy 38 < E < or = 100 kcal/mol. Comparing with other literature methods, it is found that umbrella sampling reduces the computational effort by up to two orders of magnitude when used in conjunction with a careful choice of sampling distributions. The comparison between MC rate constants and classical Rice-Ramsperberg-Kassel harmonic theory shows that anharmonicity plays an important role in the dissociation process of the Zundel cation (H5O2+) at all energies.     

Scalable free energy calculation of proteins via multiscale essential sampling

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

Application of the multiensemble sampling to the hydration free energy

We demonstrate the feasibility of using multiensemble sampling method (MESM) to determine the free energy difference between two far states for which the configurational distributions do not overlap at all. The MESM is a recently developed non‐Boltzmann sampling technique. The free energy of charging a sodium ion in water is accurately calculated in a single simulation, introducing nine intermediate ionic states. This is due to the ability of the method to explore the relevant parts of configuration space equally for every state, and this ability comes from the universality of weighting function W and the simplicity in adjusting its parameters. Detailed procedures of adjusting the parameters are presented. The comparison with a free energy perturbation method (FEPM) shows that the MESM is more reliable and efficient. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1004–1009, 2001     

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

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

Calculation of quantum mechanical free energy at low temperature

The path integral method is used to calculate the quantum mechanical free energy at low temperature. Based on the variational harmonic reference system and implemented by the partial averaging technique, the path integral can be cast into the form of a classical configurational integral with the original potential replaced by an effective one. We compared this approach with other related methods and found that it gave better results than the others considered in this paper. Furthermore, the multidimensional implementation of this method is discussed. Received: 15 September 1997 / Accepted: 1 October 1997     

Non-equilibrium umbrella sampling applied to force spectroscopy of soft matter

Physical systems often respond on a timescale which is longer than that of the measurement. This is particularly true in soft matter where direct experimental measurement, for example in force spectroscopy, drives the soft system out of equilibrium and provides a non-equilibrium measure. Here we demonstrate experimentally for the first time that equilibrium physical quantities (such as the mean square displacement) can be obtained from non-equilibrium measurements via umbrella sampling. Our model experimental system is a bead fluctuating in a time-varying optical trap. We also show this for simulated force spectroscopy on a complex soft molecule--a piston-rotaxane.     

Calculation of free energy differences for water from computer simulations

Free energy differences for water at different temperatures have been calculated from Monte Carlo computer simulations using both ratio overlap and umbrella sampling methods. The problems of calculating precise values from these methods are discussed. Three models for water (ST2, TIPS2 and PE) have been used in these calculations which have been compared with experimental estimates for the configurational free energy. The importance of being able to predict accurate mean potential energies as well as accurate energy distributions is emphasised.     

Calculation of relative solvation free energy differences by thermodynamic perturbation method: Dependence of free energy results on simulation length

Molecular dynamics (MD) simulations in conjunction with the thermodynamic cycle perturbation approach has been used to calculate relative solvation free energies for acetone to acetaldehyde, acetone to pyruvic acid, acetone to 1,1,1-trifluoroacetone, acetone to 1,1,1-trichloroacetone, acetone to 2,3-butanedione, acetone to cyclopropanone, and formaldehyde hydrate to formaldehyde. To evaluate the dependence of relative solvation free energy convergence on MD simulation length and starting configuration two studies were performed. In the first study, each simulation started from the same well-equilibrated configuration and the length was varied from 153 to 1530 ps. In the second study, the relative solvation free energy differences were calculated starting from three different configurations and using 510 ps of MD simulation for each mutation. These results clearly indicate that, even for molecules with limited conformational flexibility, a simulation length of 510 ps or greater is required to obtain satisfactory convergence and, for the mutations of large structural changes between reactant and product, such as cyclopropanone to acetone, require much longer simulation lengths to achieve satisfactory convergence. These results also show that performing one long simulation is better than averaging results from three shortest simulations of the same length using different starting conformations. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1018–1027, 1999     

Calculation of free energy landscapes: A histogram reweighted metadynamics approach

We present an efficient method for the calculation of free energy landscapes. Our approach involves a history‐dependent bias potential, which is evaluated on a grid. The corresponding free energy landscape is constructed via a histogram reweighting procedure a posteriori. Because of the presence of the bias potential, it can be also used to accelerate rare events. In addition, the calculated free energy landscape is not restricted to the actual choice of collective variables and can in principle be extended to auxiliary variables of interest without further numerical effort. The applicability is shown for several examples. We present numerical results for the alanine dipeptide and the Met‐Enkephalin in explicit solution to illustrate our approach. Furthermore, we derive an empirical formula that allows the prediction of the computational cost for the ordinary metadynamics variant in comparison with our approach, which is validated by a dimensionless representation. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Calculation of free energy surfaces using the methods of thermodynamic perturbation theory

A molecular dynamics method for determining the free energy difference between systems separated in configuration space has been developed. With this new approach, which is based on thermodynamic perturbation techniques, potentials of mean force for conformational changes may be calculated. As a test of the method, the potential of mean force and radial distribution function for liquid argon have been computed. The results are in good agreement with those obtained from an ordinary simulation.