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.     

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.     

Bridging coarse-grained models by jump-in-sample simulations

We present an efficient method to construct coarse-grained (CG) models from models of finer resolution. The method estimates the free energies in a generated sample of the CG conformational space and then fits the entire effective potential surface in the high-dimensional CG conformational space. A jump-in-sample algorithm that uses a random jumping walk in the CG sample is used to iteratively estimate the free energies. We test the method in a tetrahedral molecular fluid where we construct the intermolecular effective potential and evaluate the CG molecular model. Our algorithm for calculating the free energy involves an improved Wang-Landau (WL) algorithm, which not only works more efficiently than the standard WL algorithm, but also can work in high-dimensional spaces.     

Data-driven discovery of cardiolipin-selective small molecules by computational active learning

Subtle variations in the lipid composition of mitochondrial membranes can have a profound impact on mitochondrial function. The inner mitochondrial membrane contains the phospholipid cardiolipin, which has been demonstrated to act as a biomarker for a number of diverse pathologies. Small molecule dyes capable of selectively partitioning into cardiolipin membranes enable visualization and quantification of the cardiolipin content. Here we present a data-driven approach that combines a deep learning-enabled active learning workflow with coarse-grained molecular dynamics simulations and alchemical free energy calculations to discover small organic compounds able to selectively permeate cardiolipin-containing membranes. By employing transferable coarse-grained models we efficiently navigate the all-atom design space corresponding to small organic molecules with molecular weight less than ≈500 Da. After direct simulation of only 0.42% of our coarse-grained search space we identify molecules with considerably increased levels of cardiolipin selectivity compared to a widely used cardiolipin probe 10-N-nonyl acridine orange. Our accumulated simulation data enables us to derive interpretable design rules linking coarse-grained structure to cardiolipin selectivity. The findings are corroborated by fluorescence anisotropy measurements of two compounds conforming to our defined design rules. Our findings highlight the potential of coarse-grained representations and multiscale modelling for materials discovery and design.

We present a data-driven approach combining deep learning-enabled active learning with coarse-grained simulations and alchemical free energy calculations to discover small molecules to selectively permeate cardiolipin membranes.     

On‐the‐fly reconstruction of free‐energy profiles using logarithmic mean‐force dynamics

Mean‐force dynamics (MFD), which is a fictitious dynamics for a set of collective variables on a potential of mean‐force, is a powerful algorithm to efficiently explore free‐energy landscapes. Recently, we have introduced logarithmic MFD (LogMFD) (Morishita et al., Phys. Rev. E 2012, 85, 066702) which overcomes difficulties encounterd in free‐energy calculations using standard approaches such as thermodynamic integration. Here, we present a guide to implementing LogMFD calculations paying attention to the practical issues in choosing the parameters in LogMFD. A primary focus is given to the effect of the parameters on the accuracy of the reconstructed free‐energy profiles. A recipe for reducing the errors due to energy dissipation is presented. We also demonstrate that multidimensional free‐energy landscapes can be reconstructed on‐the‐fly using LogMFD, which cannot be accomplished using any other free‐energy calculation techniques. © 2013 Wiley Periodicals, Inc.     

Phase-space reaction network on a multisaddle energy landscape: HCN isomerization

By using the HCN/CNH isomerization reaction as an illustrative vehicle of chemical reactions on multisaddle energy landscapes, we give explicit visualizations of molecular motions associated with a straight-through reaction tube in the phase space inside which all reactive trajectories pass from one basin to another, with eliminating recrossing trajectories in the configuration space. This visualization provides us with a chemical intuition of how chemical species "walk along" the reaction-rate slope in the multidimensional phase space compared with the intrinsic reaction path in the configuration space. The distinct nonergodic features in the two different HCN and CNH wells can be easily demonstrated by a section of Poincare surface of section in those potential minima, which predicts in a priori the pattern of trajectories residing in the potential well. We elucidate the global phase-space structure which gives rise to the non-Markovian dynamics or the dynamical correlation of sequential multisaddle chemical reactions. The phase-space structure relevant to the controllability of the product state in chemical reactions is also discussed.     

Smart resolution replica exchange: an efficient algorithm for exploring complex energy landscapes

A coarse-grained representation of a condensed phase system can significantly reduce the number of system degrees of freedom, making coarse-grained simulations very computationally efficient. Moreover, coarse graining can smoothen the free energy landscape of the system. Thus coarse-grained dynamics is usually faster than its fully atomistic counterpart. In this work, the smart resolution replica exchange method is introduced that incorporates the information from coarse-grained simulations into atomistic simulations in order to accelerate the sampling of rough, complex atomistic energy landscapes. Within this methodology, interactions between particles are defined by a potential energy that interpolates between a fully atomistic potential and a fully coarse-grained effective potential according to a parameter lambda. Instead of exchanging the configurations from neighboring resolutions directly, as has been done in the resolution replica exchange methods [E. Lyman et al., Phys. Rev. Lett. 96, 028105 (2006); M. Christen and W. F. v. Gunsteren, J. Chem. Phys. 124, 154106 (2006)], the configuration described at the coarser resolution is first relaxed before an exchange is attempted, similar to the smart walking method [R. Zhou and B. J. Berne, J. Chem. Phys. 107, 9185 (1997)]. This approach greatly increases the acceptance ratio of exchange and only two replicas, one at the atomistic level and one at the coarse-grained level, are usually required (although more can be implemented if desired). This new method can approximately obtain the correct canonical sampling if the exchange interval is sufficiently large to allow the system to explore the local energy landscape. The method is demonstrated for a two-dimensional model system, where the ideal population distribution can be recovered, and also for an alanine polypeptide (Ala(15)) model with explicit water, where its native structure, an alpha helix, is obtained from the extended structure within 1 ns.     

Inhomogeneous multiscale dynamics in harmonic lattices

We use projection operators to address the coarse-grained multiscale problem in harmonic systems. Stochastic equations of motion for the coarse-grained variables, with an inhomogeneous level of coarse graining in both time and space, are presented. In contrast to previous approaches that typically start with thermodynamic averages, the key element of our approach is the use of a projection matrix chosen both for its physical appeal in analogy to mechanical stability theory and for its algebraic properties. We show that thermodynamic equilibrium can be recovered and obtain the fluctuation dissipation theorem a posteriori. All system-specific information can be computed from a series of feasible molecular dynamics simulations. We recover previous results in the literature and show how this approach can be used to extend the quasicontinuum approach and comment on implications for dissipative particle dynamics type of methods. Contrary to what is assumed in the latter models, the stochastic process of all coarse-grained variables is not necessarily Markovian, even though the variables are slow. Our approach is applicable to any system in which the coarse-grained regions are linear. As an example, we apply it to the dynamics of a single mesoscopic particle in the infinite one-dimensional harmonic chain.     

Coarse master equation from Bayesian analysis of replica molecular dynamics simulations

We use Bayesian inference to derive the rate coefficients of a coarse master equation from molecular dynamics simulations. Results from multiple short simulation trajectories are used to estimate propagators. A likelihood function constructed as a product of the propagators provides a posterior distribution of the free coefficients in the rate matrix determining the Markovian master equation. Extensions to non-Markovian dynamics are discussed, using the trajectory "paths" as observations. The Markovian approach is illustrated for the filling and emptying transitions of short carbon nanotubes dissolved in water. We show that accurate thermodynamic and kinetic properties, such as free energy surfaces and kinetic rate coefficients, can be computed from coarse master equations obtained through Bayesian inference.     

On the computational calculation of surface free energies for the disordered semihexagonal reconstructed Au(100) surface

Previously we developed a general method for calculating the free energy of any surface constrained to a distinct surface excess number/density. In this paper we show how to combine a range of such surfaces, whose free energies have been calculated, to produce an ad hoc semigrand canonical ensemble of surfaces from which ensemble surface properties can be calculated, including the ensemble surface free energy. We construct such an ensemble for the disordered Au(100) semihexagonal reconstructed surface using a Glue model potential at 1000 K and calculate the ensemble surface free energy to be 0.088 18 eVA(2). The ensemble average surface lateral density was found to be 1.375 (with respect to the bulk), which is in agreement with previous grand canonical Monte Carlo studies.     

Multigraining: an algorithm for simultaneous fine-grained and coarse-grained simulation of molecular systems

A method to combine fine-grained and coarse-grained simulations is presented. The coarse-grained particles are described as virtual particles defined by the underlying fine-grained particles are described as virtual particles defined by the underlying fine-grained particles. The contribution of the two grain levels to the interaction between particles is specified by a grain-level parameter lambda. Setting lambda = 0 results in a completely fine-grained simulation, whereas lambda = 1 yields a simulation governed by the coarse-grained potential energy surface with small contributions to keep the fine-grained covalently bound particles together. Simulations at different lambda values may be coupled using the replica-exchange molecular dynamics method to achieve enhanced sampling at the fine-grained level.     

Activation processes with memory

We propose a mathematical treatment of the activated processes governed by stochastic Langevin dynamics with a colored random force, corresponding to a noise generated by an Ornstein-Uhlenbeck process. Such non-Markovian dynamics take place in a variety of chemical and biological systems. Using the path integral approach, we constructed the conditional probability for passing between two stationary states in configurational space. Our relations can be used for Monte Carlo sampling of evolution trajectories for systems with many degrees of freedom as well as for determining the reaction coordinate used in transition state theory. On the basis of our relation for a conditional probability, we generalize the method of determining the most probable path to the case of colored random force. Using the simple three-hole potential, we examine numerically the effect of nonzero correlation time (memory) on the evolution of the most probable path for a finite temperature.     

Finite-Temperature Dimer Method for Finding Saddle Points on Free Energy Surfaces

The dimer method and its variants have been shown to be efficient in finding saddle points on potential surfaces. In the dimer method, the most unstable direction is approximately obtained by minimizing the total potential energy of the dimer. Then, the force in this direction is reversed to move the dimer toward saddle points. When the finite-temperature effect is important for a high-dimensional system, one usually needs to describe the dynamics in a low-dimensional space of reaction coordinates. In this case, transition states are collected as saddle points on the free energy surface. The traditional dimer method cannot be directly employed to find saddle points on a free energy surface since the surface is not known a priori. Here, we develop a finite-temperature dimer method for searching saddle points on the free energy surface. In this method, a constrained rotation dynamics of the dimer system is used to sample dimer directions and an efficient average method is used to obtain a good approximation of the most unstable direction. This approximated direction is then used in reversing the force component and evolving the dimer toward saddle points. Our numerical results suggest that the new method is efficient in finding saddle points on free energy surfaces. © 2019 Wiley Periodicals, Inc.     

Sensitivity field for nonautonomous molecular rotors

We propose a numerical approach to quantify the control of a nonautonomous molecular rotor motion. Unlike straightforward molecular dynamics simulations in an explicitly time-dependent framework, our method is based on the theory of geometric phases. This theory allows us to define a sensitivity field (SF) in control parameter space that characterizes average motion of a molecule induced by a cyclic perturbation. We show that the SF can be obtained using only equilibrium free energy sampling techniques. A density plot of the SF quantifies response of a molecule to an arbitrary cyclic adiabatic evolution of parameters. For demonstration, we numerically find the SFs for two surface mounted molecular rotor molecules that can be driven, in practice, by strong time-dependent electric fields of a STM tip.     

Nonequilibrium multiscale computational model

A computational multiscale method is proposed to simulate coupled, nonequilibrium thermomechanical processes. This multiscale framework couples together thermomechanical equations at the coarse scale with nonequilibrium molecular dynamics at the fine scale. The novel concept of distributed coarse scale thermostats enables subsets of fine scale atoms to be attached to different coarse scale nodes which act as thermostats. The fine scale dynamics is driven by the coarse scale mean field. A coarse-grained Helmholtz free energy is used to derive macroscopic quantities. This new framework can reproduce the correct thermodynamics at the fine scale while providing an accurate coarse-grained result at the coarse scale.