An algorithm for the uniform sampling of iso-energy surfaces and for the calculation of microcanonical averages

In this article an algorithm is proposed to efficiently perform the uniform sampling of an iso-energy surface corresponding to a fixed potential energy U of a molecular system, and for calculating averages of certain quantities over microstates having this energy (microcanonical averages). The developed sampling technique is based upon the combination of a recently proposed method for performing constant potential energy molecular dynamics simulations [Rapallo, A. J Chem Phys 2004, 121, 4033] with well-established thermostatting techniques used in the framework of standard molecular dynamics simulations, such as the Andersen thermostat, and the Nose-Hoover chain thermostat. The proposed strategy leads to very accurate and drift-free potential energy conservation during the whole sampling process, and, very important, specially when dealing with high-dimensional or complicated potential functions, it does not require the calculation of the potential energy function hessian. The technique proved to be very reliable for sampling both low- and high-dimensional surfaces.     

Improved convergence in dual-topology free energy calculations through use of harmonic restraints

When calculating free energy differences between two molecular systems by means of molecular dynamics simulation, accessory potential functions can help eliminate uninteresting configurational entropy contributions, improve convergence, and facilitate reversibility. In this work, we demonstrate that the use of a harmonic potential function to restrain key portions of a molecular system in a free energy perturbation dual-topology molecular dynamics approach dramatically improves convergence and precision of the calculation. Limitations of this technique are illustrated, and its use in conjunction with a fixed bond-length constraint is developed. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1278–1283, 1998     

An integrate-over-temperature approach for enhanced sampling

A simple method is introduced to achieve efficient random walking in the energy space in molecular dynamics simulations which thus enhances the sampling over a large energy range. The approach is closely related to multicanonical and replica exchange simulation methods in that it allows configurations of the system to be sampled in a wide energy range by making use of Boltzmann distribution functions at multiple temperatures. A biased potential is quickly generated using this method and is then used in accelerated molecular dynamics simulations.     

Metadynamics in essential coordinates: free energy simulation of conformational changes

We propose an approach that combines an extraction of collective motions of a molecular system with a sampling of its free energy surface. A recently introduced method of metadynamics allows exploration of the free energy surface of a molecular system by means of coarse-grained dynamics with flooding of free energy minima. This free energy surface is defined as a function of a set of collective variables (e.g., interatomic distances, angles, torsions, and others). In this study, essential coordinates determined by essential dynamics (principle component analysis) were used as collective variables in metadynamics. First, dynamics of the model system (explicitly solvated alanine dipeptide, Ace-Ala-Nme) was simulated by a classical molecular dynamics simulation. The trajectory (1 ns) was then analyzed by essential dynamics to obtain essential coordinates. The free energy surface as a function of the first and second essential coordinates was then explored by metadynamics. The resulting free energy surface is in agreement with other studies of this system. We propose that a combination of these two methods (metadynamics and essential dynamics) has great potential in studies of conformational changes in peptides and proteins.     

An approximate but fast method to impose flexible distance constraints in molecular dynamics simulations

A fast but approximative method to apply flexible constraints to bond lengths in molecular dynamics simulations is presented and the effects of the approximation are investigated. The method is not energy conserving, but coupling to a temperature bath results in stable simulations. The high frequencies from bond-length vibrations are successfully removed from the system while maintaining the flexibility of the bonds. As a test liquid neopentane is simulated at different pressures. Energetic and dynamic properties are not affected by the new flexible constraint simulation method.     

Essential energy space random walks to accelerate molecular dynamics simulations: convergence improvements via an adaptive-length self-healing strategy

Recently, accelerated molecular dynamics (AMD) technique was generalized to realize essential energy space random walks so that further sampling enhancement and effective localized enhanced sampling could be achieved. This method is especially meaningful when essential coordinates of the target events are not priori known; moreover, the energy space metadynamics method was also introduced so that biasing free energy functions can be robustly generated. Despite the promising features of this method, due to the nonequilibrium nature of the metadynamics recursion, it is challenging to rigorously use the data obtained at the recursion stage to perform equilibrium analysis, such as free energy surface mapping; therefore, a large amount of data ought to be wasted. To resolve such problem so as to further improve simulation convergence, as promised in our original paper, we are reporting an alternate approach: the adaptive-length self-healing (ALSH) strategy for AMD simulations; this development is based on a recent self-healing umbrella sampling method. Here, the unit simulation length for each self-healing recursion is increasingly updated based on the Wang-Landau flattening judgment. When the unit simulation length for each update is long enough, all the following unit simulations naturally run into the equilibrium regime. Thereafter, these unit simulations can serve for the dual purposes of recursion and equilibrium analysis. As demonstrated in our model studies, by applying ALSH, both fast recursion and short nonequilibrium data waste can be compromised. As a result, combining all the data obtained from all the unit simulations that are in the equilibrium regime via the weighted histogram analysis method, efficient convergence can be robustly ensured, especially for the purpose of free energy surface mapping.     

The bilayer-vesicle transition is entropy driven

Self-assembled bilayer membranes have a remarkable inclination to form closed shells or vesicles. This bilayer-vesicle transition has been shown experimentally and by various kinds of computer simulation techniques. Here we study this transition using coarse-grained molecular dynamics. The advantage of this simulation technique is that it allows for a detailed analysis of the transition, such as changes of the internal energy. Generally it is assumed that the bilayer-vesicle transition is driven by minimization of the edge energy. However, our simulations, which include solvent particles, show an increase in the potential energy of the system during the transition, implicating that the transition is not energy but entropy driven.     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.     

金属Cu晶化与玻璃化行为的分子动力学模拟

A series of simulations of the crystallization and vitrification processes for metal Cu were carried out by means of the molecular dynamics technique. The radial distribution function, common neighbors, internal energy and volume of the system were recorded during the processes. The atomic internal energy, atomic Voronoi volume and atomic stress field of the relax system were analyzed at zero temperature. The interaction between atoms in the system is described using the embedded atom potential as proposed by Mishin. The simulation results show that crystalline and non-crystalline phases form at lower and higher cooling rates respectively. In comparison to nanocrystals, it is found that metallic glass has higher internal energy and larger volume. The intrinsic stress field is induced by distortion of the lattice.     

金属铜升温熔化过程的分子动力学模拟

采用分子动力学方法模拟了金属铜的升温熔化过程.原子间作用势采用FS (Finnis-Sinclair)势，结构分析采用双体分布函数(PCF)、均方位移(MSD)等方法.计算结果表明,在连续升温过程中，金属铜在1444 K熔化，在该熔化点的扩散系数为4.31×10－9 m2•s－1.上述结论与实验值相当接近，并且比之采用EAM镶嵌原子势所作模拟得到的结果更佳，说明FS势可以用来处理象液铜这样较复杂的无序体系.本文指出了升温速率在金属熔化过程中所起的作用.     

Constraint dynamics algorithm for simulation of semiflexible macromolecules

Semiflexible models are often used to study macromolecules containing stable structural elements. Based on rigid body dynamics, we developed a rigid fragment constraint dynamics algorithm for the simulation of semiflexible macromolecules. Stable structural elements are treated as rigid fragments. Rigid fragment constraints, defined as combinations of distance constraints and position constraints, are introduced to limit internal molecular motion to the required mode. The constraint forces are solved separately for each rigid fragment constraint and iteratively until all constraint conditions are satisfied within a given tolerance at each time step, as is done for the bond length constraint in the SHAKE algorithm. The orientation of a rigid fragment is represented by the quaternion parameters, and both translation and rotation are solved by the leap-frog formulation. We tested the algorithm with molecular dynamics simulations of a series of peptides and a small protein. The computation cost for the constraints is roughly proportional to the size of the molecule. In the microcanonical ensemble simulation of polyvalines, the total energy was conserved satisfactorily with time steps as large as 20 fs. A helix folding simulation of a synthetic peptide was carried out to show the efficiency of the algorithm in a conformational search. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1555–1566, 1998     

Implementation of the IMOMM methodology for performing combined QM/MM molecular dynamics simulations and frequency calculations

A technique for implementing the integrated molecular orbital and molecular mechanics (IMOMM) methodology developed by Maseras and Morokuma that is used to perform combined quantum mechanics/molecular mechanics (QM/MM) molecular dynamics simulations, frequency calculations and simulations of macromolecules including explicit solvent is presented. Although the IMOMM methodology is generalized to any coordinate system, the implementation first described by Maseras and Morokuma requires that the QM and MM gradients be transformed into internal coordinates before they are added together. This coordinate transformation can be cumbersome for macromolecular systems and can become ill-defined during the course of a molecular dynamics simulation. We describe an implementation of the IMOMM method in which the QM and MM gradients are combined in the cartesian coordinate system, thereby avoiding potential problems associated with using the internal coordinate system. The implementation can be used to perform combined QM/MM molecular dynamics simulations and frequency calculations within the IMOMM framework. Finally, we have examined the applicability of thermochemical data derived from IMOMM framework. Finally, we have examined the applicability of thermochemical data derived from IMOMM frequency calculations. Received: 11 May 1998 / Accepted: 14 August 1998 / Published online: 16 November 1998     

Comparisons of classical chemical dynamics simulations of the unimolecular decomposition of classical and quantum microcanonical ensembles

Previous studies have shown that classical trajectory simulations often give accurate results for short-time intramolecular and unimolecular dynamics, particularly for initial non-random energy distributions. To obtain such agreement between experiment and simulation, the appropriate distributions must be sampled to choose initial coordinates and momenta for the ensemble of trajectories. If a molecule's classical phase space is sampled randomly, its initial decomposition will give the classical anharmonic microcanonical (RRKM) unimolecular rate constant for its decomposition. For the work presented here, classical trajectory simulations of the unimolecular decomposition of quantum and classical microcanonical ensembles, at the same fixed total energy, are compared. In contrast to the classical microcanonical ensemble, the quantum microcanonical ensemble does not sample the phase space randomly. The simulations were performed for CH(4), C(2)H(5), and Cl(-)---CH(3)Br using both analytic potential energy surfaces and direct dynamics methods. Previous studies identified intrinsic RRKM dynamics for CH(4) and C(2)H(5), but intrinsic non-RRKM dynamics for Cl(-)---CH(3)Br. Rate constants calculated from trajectories obtained by the time propagation of the classical and quantum microcanonical ensembles are compared with the corresponding harmonic RRKM estimates to obtain anharmonic corrections to the RRKM rate constants. The relevance and accuracy of the classical trajectory simulation of the quantum microcanonical ensemble, for obtaining the quantum anharmonic RRKM rate constant, is discussed.     

Molecular dynamics simulation of papain-E-64 (N-[N-(L-3-trans-carboxyoxirane-2-carbonyl)-L-leucyl]agmatine) complex

To investigate the possible binding mode of E-64 (N-[N-(L-3-trans-carboxyoxirane-2-carbonyl)-L-leucyl]agmatine), a potent cysteine protease inhibitor, to papain active site, molecular dynamics simulations were applied to two complex forms: R- and S- configurational forms of E-64 C2 atom for the covalent bond formation with the papain Cys-25 SH group. The tertiary structures of the papain-E-64 complexes were built by visual interactive modelling and the energy minimization technique, and were subjected to the dynamics simulations of 10 ps. Although no significant difference was observed between the potential energies of energy-minimized R- and S-complex forms, the molecular dynamics simulations suggested that the hydrogen bonding mode of the former form is more advantageous than that of the latter one. Comparing with the hydrogen bonds observed in the papain-E-64 complex crystal, it could be concluded that the present molecular dynamics simulation reflects well the three-dimensional structure concerning the interaction of E-64 with the papain active site. The conformational characteristics of E-64 and its possible interaction mode with papain were also discussed.     

Accelerated molecular dynamics: a promising and efficient simulation method for biomolecules

Many interesting dynamic properties of biological molecules cannot be simulated directly using molecular dynamics because of nanosecond time scale limitations. These systems are trapped in potential energy minima with high free energy barriers for large numbers of computational steps. The dynamic evolution of many molecular systems occurs through a series of rare events as the system moves from one potential energy basin to another. Therefore, we have proposed a robust bias potential function that can be used in an efficient accelerated molecular dynamics approach to simulate the transition of high energy barriers without any advance knowledge of the location of either the potential energy wells or saddle points. In this method, the potential energy landscape is altered by adding a bias potential to the true potential such that the escape rates from potential wells are enhanced, which accelerates and extends the time scale in molecular dynamics simulations. Our definition of the bias potential echoes the underlying shape of the potential energy landscape on the modified surface, thus allowing for the potential energy minima to be well defined, and hence properly sampled during the simulation. We have shown that our approach, which can be extended to biomolecules, samples the conformational space more efficiently than normal molecular dynamics simulations, and converges to the correct canonical distribution.     

On the application of accelerated molecular dynamics to liquid water simulations

Our group recently proposed a robust bias potential function that can be used in an efficient all-atom accelerated molecular dynamics (MD) approach to simulate the transition of high energy barriers without any advance knowledge of the potential-energy landscape. The main idea is to modify the potential-energy surface by adding a bias, or boost, potential in regions close to the local minima, such that all transitions rates are increased. By applying the accelerated MD simulation method to liquid water, we observed that this new simulation technique accelerates the molecular motion without losing its microscopic structure and equilibrium properties. Our results showed that the application of a small boost energy on the potential-energy surface significantly reduces the statistical inefficiency of the simulation while keeping all the other calculated properties unchanged. On the other hand, although aggressive acceleration of the dynamics simulation increases the self-diffusion coefficient of water molecules greatly and dramatically reduces the correlation time of the simulation, configurations representative of the true structure of liquid water are poorly sampled. Our results also showed the strength and robustness of this simulation technique, which confirm this approach as a very useful and promising tool to extend the time scale of the all-atom simulations of biological system with explicit solvent models. However, we should keep in mind that there is a compromise between the strength of the boost applied in the simulation and the reproduction of the ensemble average properties.     

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.     

Parallel replica dynamics with a heterogeneous distribution of barriers: application to n-hexadecane pyrolysis

Parallel replica dynamics simulation methods appropriate for the simulation of chemical reactions in molecular systems with many conformational degrees of freedom have been developed and applied to study the microsecond-scale pyrolysis of n-hexadecane in the temperature range of 2100-2500 K. The algorithm uses a transition detection scheme that is based on molecular topology, rather than energetic basins. This algorithm allows efficient parallelization of small systems even when using more processors than particles (in contrast to more traditional parallelization algorithms), and even when there are frequent conformational transitions (in contrast to previous implementations of the parallel replica algorithm). The parallel efficiency for pyrolysis initiation reactions was over 90% on 61 processors for this 50-atom system. The parallel replica dynamics technique results in reaction probabilities that are statistically indistinguishable from those obtained from direct molecular dynamics, under conditions where both are feasible, but allows simulations at temperatures as much as 1000 K lower than direct molecular dynamics simulations. The rate of initiation displayed Arrhenius behavior over the entire temperature range, with an activation energy and frequency factor of E(a) = 79.7 kcal/mol and log A/s(-1) = 14.8, respectively, in reasonable agreement with experiment and empirical kinetic models. Several interesting unimolecular reaction mechanisms were observed in simulations of the chain propagation reactions above 2000 K, which are not included in most coarse-grained kinetic models. More studies are needed in order to determine whether these mechanisms are experimentally relevant, or specific to the potential energy surface used.     

Optimization of partial multicanonical molecular dynamics simulations applied to an alanine dipeptide in explicit water solvent

The partial multicanonical algorithm for molecular dynamics and Monte Carlo simulations samples a wide range of an important part of the potential energy. Although it is a strong technique for structure prediction of biomolecules, the choice of the partial potential energy has not been optimized. In order to find the best choice, partial multicanonical molecular dynamics simulations of an alanine dipeptide in explicit water solvent were performed with 15 trial choices for the partial potential energy. The best choice was found to be the sum of the electrostatic, Lennard-Jones, and torsion-angle potential energies between solute atoms. In this case, the partial multicanonical simulation sampled all of the local-minimum free-energy states of the P(II), C(5), α(R), α(P), α(L), and C states and visited these states most frequently. Furthermore, backbone dihedral angles ? and ψ rotated very well. It is also found that the most important term among these three terms is the electrostatic potential energy and that the Lennard-Jones term also helps the simulation to overcome the steric restrictions. On the other hand, multicanonical simulation sampled all of the six states, but visited these states fewer times. Conventional canonical simulation sampled only four of the six states: The P(II), C(5), α(R), and α(P) states.