One- and multidimensional conformational free energy simulations

A new thermodynamic integration approach to conformational free energy simulations is presented. The method is applicable both to one-dimensional cases (reaction coordinates) and multidimensional situations (free energy surfaces). Analysis of the properties of the thermodynamic integration algorithm is used to formulate methods of calculating multidimensional free energy gradients. The method is applied to calculate the free energy profile for rotation around the central C—C bond of n-butane in the gas and liquid phase and to generate maps of the 18-dimensional free energy gradient with respect to all nine ϕ and nine ψ dihedrals of the decaalanine and deca-α-methylalanine peptides in vacuum. For n-butane essentially no change in the gauche–trans equilibrium between the gas and liquid is predicted within the CHARMM explicit hydrogen model, with the thermodynamic integration, thermodynamic perturbation, and direct simulation methods yielding free energy profiles that are identical within errors. For the decapeptides the right-handed helical region of conformational space is investigated. For decaalanine a minimum on the free energy surface is found in the vicinity of (ϕ, ψ) = (-64.5°, -42.5°) in the α-helix region; no minimum exists for 3₁₀-helix-type conformers. For deca-α-methylalanine free energy minima corresponding to both the α-helix at ( - 55.5°, - 51.5°) and the 3₁₀-helix at ( - 54°, - 29°) are found; the α-helix state is favored by about 4 kcal/mol and the barrier for the concerted 3₁₀-helix → α-helix transition is about 3 kcal/mol. The α-methylation also considerably increases the rigidity of the α-helix with respect to deformations. The computational efficiency, ease of generalization to calculations of multidimensional gradients, and analytical capability due to component analysis of free energy differences make the method a novel, powerful tool to improve the basic understanding of conformational equilibria of flexible molecules in condensed phases. A related scheme for energy minimization in the presence of holonomic constraints is also presented, allowing generation of adiabatic energy surfaces in constrained systems. © 1996 by John Wiley & Sons, Inc.     

Metadynamics in the conformational space nonlinearly dimensionally reduced by Isomap

Atomic motions in molecules are not linear. This infers that nonlinear dimensionality reduction methods can outperform linear ones in analysis of collective atomic motions. In addition, nonlinear collective motions can be used as potentially efficient guides for biased simulation techniques. Here we present a simulation with a bias potential acting in the directions of collective motions determined by a nonlinear dimensionality reduction method. Ad hoc generated conformations of trans,trans-1,2,4-trifluorocyclooctane were analyzed by Isomap method to map these 72-dimensional coordinates to three dimensions, as described by Brown and co-workers [J. Chem. Phys. 129, 064118 (2008)]. Metadynamics employing the three-dimensional embeddings as collective variables was applied to explore all relevant conformations of the studied system and to calculate its conformational free energy surface. The method sampled all relevant conformations (boat, boat-chair, and crown) and corresponding transition structures inaccessible by an unbiased simulation. This scheme allows to use essentially any parameter of the system as a collective variable in biased simulations. Moreover, the scheme we used for mapping out-of-sample conformations from the 72D to 3D space can be used as a general purpose mapping for dimensionality reduction, beyond the context of molecular modeling.     

Alchemical free energy calculations and multiple conformational substates

Thermodynamic integration (TI) was combined with (adaptive) umbrella sampling to improve the convergence of alchemical free energy simulations in which multiple conformational substates are present. The approach, which we refer to as non-Boltzmann TI (NBTI), was tested by computing the free energy differences between three five-atomic model systems, as well as the free energy difference of solvation between leucine and asparagine. In both cases regular TI failed to give converged results, whereas the NBTI results were free from hysteresis and had standard deviations well below +/-0.7 kcal/mole. We also present theoretical considerations that make it possible to compute free energy differences between simple molecules, such as the five-atomic model systems, by numerical integration of the partition functions at the respective end points.     

Parallel algorithm for efficient calculation of second derivatives of conformational energy function in internal coordinates

A parallel algorithm for efficient calculation of the second derivatives (Hessian) of the conformational energy in internal coordinates is proposed. This parallel algorithm is based on the master/slave model. A master processor distributes the calculations of components of the Hessian to one or more slave processors that, after finishing their calculations, send the results to the master processor that assembles all the components of the Hessian. Our previously developed molecular analysis system for conformational energy optimization, normal mode analysis, and Monte Carlo simulation for internal coordinates is extended to use this parallel algorithm for Hessian calculation on a massively parallel computer. The implementation of our algorithm uses the message passing interface and works effectively on both distributed-memory parallel computers and shared-memory parallel computers. We applied this system to the Newton–Raphson energy optimization of the structures of glutaminyl transfer RNA (Gln-tRNA) with 74 nucleotides and glutaminyl-tRNA synthetase (GlnRS) with 540 residues to analyze the performance of our system. The parallel speedups for the Hessian calculation were 6.8 for Gln-tRNA with 24 processors and 11.2 for GlnRS with 54 processors. The parallel speedups for the Newton–Raphson optimization were 6.3 for Gln-tRNA with 30 processors and 12.0 for GlnRS with 62 processors. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1716–1723, 1998     

New method for parallel computation of Hessian matrix of conformational energy function in internal coordinates

A new algorithm for parallel calculation of the second derivatives (Hessian) of the conformational energy function of biomolecules in internal coordinates is proposed. The basic scheme of this algorithm is the division of the entire calculation of the Hessian matrix (called "task") into subtasks and the optimization of the assignment of processors to each subtask by considering both the load balancing and reduction of the communication cost. A genetic algorithm is used for this optimization considering the dependencies between subtasks. We applied this method to a glutaminyl transfer RNA (Gln-tRNA) molecule for which the scalability of our previously developed parallel algorithm was significantly decreased when the large number of processors was used. The speedup for the calculation was 32.6 times with 60 processors, which is considerably better than the speedup for our previously reported parallel algorithm. The elapsed time for the calculation of subtasks, data sending, and data receiving was analyzed, and the effect of the optimization using the genetic algorithm is discussed.     

Minimum free energy pathways and free energy profiles for conformational transitions based on atomistic molecular dynamics simulations

An efficient method for the calculation of minimum free energy pathways and free energy profiles for conformational transitions is presented. Short restricted perturbation-targeted molecular dynamics trajectories are used to generate an approximate free energy surface. Approximate reaction pathways for the conformational change are constructed from one-dimensional line segments on this surface using a Monte Carlo optimization. Accurate free energy profiles are then determined along the pathways by means of one-dimensional adaptive umbrella sampling simulations. The method is illustrated by its application to the alanine "dipeptide." Due to the low computational cost and memory demands, the method is expected to be useful for the treatment of large biomolecular systems.     

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.     

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     

Metadynamics simulation of prion protein: beta-structure stability and the early stages of misfolding

In the present study we have used molecular dynamics simulations to study the stability of the antiparallel beta-sheet in cellular mouse prion protein (PrP(C)) and in the D178N mutant. In particular, using the recently developed non-Markovian metadynamics method, we have evaluated the free energy as a function of a reaction coordinate related to the beta-sheet disruption/growth. We found that the antiparallel beta-sheet is significantly weaker in the pathogenic D178N mutant than in the wild-type PrP(C). The destabilization of PrP(C) beta-structure in the D178N mutant is correlated to the weakening of the hydrogen bonding network involving the mutated residue, Arg164 and Tyr128 side chains. This in turn indicates that such a network apparently provides a safety mechanism for the unzipping of the antiparallel beta-sheet in the PrP(C). We conclude that the antiparallel beta-sheet is likely to undergo disruption rather than growth under pathogenic conditions, in agreement with recent models of the misfolded monomer that assume a parallel beta-helix.     

Dynamic effects of nonequilibrium solvation: potential and free energy surfaces for Z/E isomerization in solvent-solute coordinates

A novel definition of a solvent coordinate associated with a given reaction is formulated in terms of molecular-dynamic trajectories of the solvent and is applied to discuss the topography of potential energy and free energy surfaces of model liquid phase Z/E isomerization reactions in solvent-solute coordinates. It is shown that the arrangement of the reactant and product valleys on these surfaces can vary from consecutive to parallel, depending on the strength of the solvent-solute interactions.     

Kinetic energy functional in hyperspherical coordinates

Hyperspherical coordinates and a generator coordinate representation are employed to find a simple form of the kinetic energy for a general three-particle problem. An expression is developed for the determination of adiabatic hyperangular states in a local potential using the finite element method.     

Monte Carlo simulation of electrodeposition of copper: a multistep free energy calculation

Electrodeposition of copper (Cu) involves length scales of a micrometer or even less. Several theoretical techniques such as continuum Monte Carlo, kinetic Monte Carlo (KMC), and molecular dynamics have been used for simulating this problem. However the multiphenomena characteristics of the problem pose a challenge for an efficient simulation algorithm. Traditional KMC methods are slow, especially when modeling surface diffusion with large number of particles and frequent particle jumps. Parameter estimation involving thousands of KMC runs is very time-consuming. Thus a less time-consuming and novel multistep continuum Monte Carlo simulation is carried out to evaluate the step wise free energy change in the process of electrochemical copper deposition. The procedure involves separate Monte Carlo codes employing different random number criterion (using hydrated radii, bare radii, hydration number of the species, redox potentials, etc.) to obtain the number of species (CuCl(2) or CuSO(4) or Cu as the case may be) and in turn the free energy. The effect of concentration of electrolyte, influence of electric field and presence of chloride ions on the free energy change for the processes is studied. The rate determining step for the process of electrodeposition of copper from CuCl(2) and CuSO(4) is also determined.     

The conformational free energy landscape of beta-D-glucopyranose. Implications for substrate preactivation in beta-glucoside hydrolases

Using ab initio metadynamics we have computed the conformational free energy landscape of beta-D-glucopyranose as a function of the puckering coordinates. We show that the correspondence between the free energy and the Stoddard's pseudorotational itinerary for the system is rather poor. The number of free energy minima (9) is smaller than the number of ideal structures (13). Moreover, only six minima correspond to a canonical conformation. The structural features, the electronic properties, and the relative stability of the predicted conformers permit the rationalization of the occurrence of distorted sugar conformations in all the available X-ray structures of beta-glucoside hydrolase Michaelis complexes. We show that these enzymes recognize the most stable distorted conformers of the isolated substrate and at the same time the ones better prepared for catalysis in terms of bond elongation/shrinking and charge distribution. This suggests that the factors governing the distortions present in these complexes are largely dictated by the intrinsic properties of a single glucose unit.     

Endor detection of conformational changes and internal rotations in organic stable free radicals in solution

The detection of conformational changes and internal rotation of the commonly used free radicals DPPH, PAC and BDPA are reported. Comparison with other spectroscopic techniques shows that ENDOR can be uniquely helpful for obtaining such information.     

New space warping method for the simulation of large-scale macromolecular conformational changes

A space warping method, facilitating the modeling of large-scale conformational changes in mesoscopic systems, is presented. The method uses a set of "global (or collective) coordinates" that capture overall behavior, in conjunction with the set of atomic coordinates. Application of the space warping method to energy minimization is discussed. Several simulations where the method is used to determine the energy minimizing structures of simple central force systems are analyzed. Comparing the results and behavior of the space warping method to simulations involving atomic coordinates only, it is found that the space warping method scales better with system size and also finds lower minima when the potential energy surface has multiple minima. It is shown that the transformation of [Ala16]+ in vacuo from linear to globular is captured efficiently using the space warping method.     

Kinetic energy operators in linearized internal coordinates

It is customary to describe molecular vibrations using as exact kinetic energy operators and as accurate potentials as possible. It has become a standard approach to express Hamiltonians in curvilinear internal displacement coordinates, because they offer a simple and physical picture of vibrational motions, including large amplitude changes in the shape. In the older normal mode model of molecular vibrations, the nuclei are thought to vibrate infinitesimally about the reference configuration, and the shape of the molecule is described using linearized approximations of the true geometrically defined internal displacement coordinates. It is natural to ask how the two approaches are related. In this work, I present a general yet practical way to obtain curvilinear displacement coordinates as closed function of their linearized counterparts, and vice versa. In contrast to the conventional power series approach, the body-frame dependency is explicitly taken into account, and the relations are valid for any value of the coordinates. The present approach also allows one to obtain easily exact kinetic energy operators in linearized shape coordinates.