Reaction coordinates and rate constants for liquid droplet nucleation: Quantifying the interplay between driving force and memory

Atomic scale insights into NaCl nucleation in nanoconfined environments

In this work we examine the nucleation from NaCl aqueous solutions within nano-confined environments, employing enhanced sampling molecular dynamics simulations integrated with machine learning-derived reaction coordinates. Through our simulations, we successfully induce phase transitions between solid, liquid, and a hydrated phase, typically observed at lower temperatures in bulk environments. Interestingly, while generally speaking nano-confinement serves to stabilize the solid phase and elevate melting points, there are subtle variations in the thermodynamics of competing phases with the precise extent of confinement. Our simulations explain these findings by underscoring the significant role of water, alongside ion aggregation and subtle, anisotropic dielectric behavior, in driving nucleation within nano-confined environments. This report thus provides a framework for sampling, analyzing and understanding nucleation processes under nano-confinement.

Stochastic computer experiments of the thermodynamic irreversibility of bulk nanobubbles in supersaturated and weak gas-liquid solutions

Spontaneous gas-bubble nucleation in weak gas-liquid solutions has been a challenging topic in theory, experimentation, and computer simulations. In analogy with recent advances in crystallization and droplet formation studies, the diffusive-shielding stabilization and thermodynamic irreversibility of bulk nanobubble (bNB) mechanisms are revisited and deployed to characterize nucleation processes in a stochastic framework of computer experiments using the large-scale atomic/molecular massively parallel simulator code. Theoretical bases, assumptions, and limitations underlying the irreversibility hypothesis of bNBs, and their computational counterparts, are extensively described and illustrated. In essence, it is established that the irreversibility hypothesis can be numerically investigated by converging the system volume (due to the finiteness of interatomic forces) and the initial dissolved-gas concentration in the solution (due to the single-bNB limitation). Helium nucleation in liquid Pb17Li alloy is selected as a representative case study, where it exhibits typical characteristics of noble-gas/liquid-metal systems. The proposed framework lays down the bases on which the stability of gas-bNBs in weak and supersaturated gas-liquid solutions can be inferred and explained from a novel perspective. In essence, it stochastically marches toward a unique irreversible state along out-of-equilibrium nucleation/growth trajectories. Moreover, it does not attempt to characterize the interface or any interface-related properties, neither theoretically nor computationally. It was concluded that bNBs of a few tens of He-atoms are irreversible when dissolved-He concentrations in the weak gas-liquid solution are at least ∼50 and ∼105 mol m-3 at 600 and 1000 K (and ∼80 MPa), respectively, whereas classical molecular dynamics -estimated solubilities are at least two orders of magnitude smaller.

Enhanced Sampling of Crystal Nucleation with Graph Representation Learnt Variables

In this study, we present a graph neural network (GNN)-based learning approach using an autoencoder setup to derive low-dimensional variables from features observed in experimental crystal structures. These variables are then biased in enhanced sampling to observe state-to-state transitions and reliable thermodynamic weights. In our approach, we used simple convolution and pooling methods. To verify the effectiveness of our protocol, we examined the nucleation of various allotropes and polymorphs of iron and glycine in their molten states. Our graph latent variables, when biased in well-tempered metadynamics, consistently show transitions between states and achieve accurate thermodynamic rankings in agreement with experiments, both of which are indicators of dependable sampling. This underscores the strength and promise of our GNN variables for improved sampling. The protocol shown here should be applicable for other systems and other sampling methods.

Is the Local Ion Density Sufficient to Drive NaCl Nucleation from the Melt and Aqueous Solution?

Even though nucleation is ubiquitous in different science and engineering problems, investigating nucleation is extremely difficult due to the complicated ranges of time and length scales involved. In this work, we simulate NaCl nucleation in both molten and aqueous environments using enhanced sampling of all-atom molecular dynamics with deep-learning-based estimation of reaction coordinates. By incorporating various structural order parameters and learning the reaction coordinate as a function thereof, we achieve significantly improved sampling relative to traditional ad hoc descriptions of what drives nucleation, particularly in an aqueous medium. Our results reveal a one-step nucleation mechanism in both environments, with reaction coordinate analysis highlighting the importance of local ion density in distinguishing solid and liquid states. However, although fluctuations in the local ion density are necessary to drive nucleation, they are not sufficient. Our analysis shows that near the transition states, descriptors such as enthalpy and local structure become crucial. Our protocol proposed here enables robust nucleation analysis and phase sampling and could offer insights into nucleation mechanisms for generic small molecules in different environments.

Kinetics from Metadynamics: Principles, Applications, and Outlook

Metadynamics is a popular enhanced sampling algorithm for computing the free energy landscape of rare events by using molecular dynamics simulation. Ten years ago, Tiwary and Parrinello introduced the infrequent metadynamics approach for calculating the kinetics of transitions across free energy barriers. Since then, metadynamics-based methods for obtaining rate constants have attracted significant attention in computational molecular science. Such methods have been applied to study a wide range of problems, including protein-ligand binding, protein folding, conformational transitions, chemical reactions, catalysis, and nucleation. Here, we review the principles of elucidating kinetics from metadynamics-like approaches, subsequent methodological developments in this area, and successful applications on chemical, biological, and material systems. We also highlight the challenges of reconstructing accurate kinetics from enhanced sampling simulations and the scope of future developments.

Extending and validating bubble nucleation rate predictions in a Lennard-Jones fluid with enhanced sampling methods and transition state theory

We calculate bubble nucleation rates in a Lennard-Jones fluid through explicit molecular dynamics simulations. Our approach—based on a recent free energy method (dubbed reweighted Jarzynski sampling), transition state theory, and a simple recrossing correction—allows us to probe a fairly wide range of rates in several superheated and cavitation regimes in a consistent manner. Rate predictions from this approach bridge disparate independent literature studies on the same model system. As such, we find that rate predictions based on classical nucleation theory, direct brute force molecular dynamics simulations, and seeding are consistent with our approach and one another. Published rates derived from forward flux sampling simulations are, however, found to be outliers. This study serves two purposes: First, we validate the reliability of common modeling techniques and extrapolation approaches on a paradigmatic problem in materials science and chemical physics. Second, we further test our highly generic recipe for rate calculations, and establish its applicability to nucleation processes.

Exploring the Energy Landscape of Riboswitches Using Collective Variables Based on Tertiary Contacts

Messenger RNA regulatory elements, such as riboswitches, can display a high degree of flexibility. By characterizing their energy landscapes, and corresponding distributions of 3D configurations, structure-function relationships can be elucidated. Molecular dynamics simulation with enhanced sampling is an important strategy used to computationally access free energy landscapes characterizing the accessible 3D conformations of RNAs. While tertiary contacts are thought to play important roles in RNA dynamics, it is difficult, in explicit solvent, to sample the formation and breakage of tertiary contacts, such as helix-helix interactions, pseudoknot interactions, and junction interactions, while maintaining intact secondary structure elements. To this end, we extend previously developed collective variables and metadynamics efforts, to establish a simple metadynamics protocol, which utilizes only one collective variable, based on multiple tertiary contacts, to characterize the underlying free energy landscape of any RNA molecule. We develop a modified collective variable, the tertiary contacts distance (Q_TC), which can probe the formation and breakage of all or selectively chosen tertiary contacts of the RNA. The SAM-I riboswitch in the presence of three ionic and substrate conditions was investigated and validated against the structure ensemble previously generated using SAXS experiments. This efficient and easy to implement all-atom MD simulation based approach incorporating metadynamics to study RNA conformational dynamics can also be transferred to any other type of biomolecule.

Quantifying Energetic and Entropic Pathways in Molecular Systems

When examining dynamics occurring at nonzero temperatures, both energy and entropy must be taken into account to describe activated barrier crossing events. Furthermore, good reaction coordinates need to be constructed to describe different metastable states and the transition mechanisms between them. Here we use a physics-based machine learning method called state predictive information bottleneck (SPIB) to find nonlinear reaction coordinates for three systems of varying complexity. SPIB is able to correctly predict an entropic bottleneck for an analytical flat-energy double-well system and identify the entropy- and energy-dominated pathways for an analytical four-well system. Finally, for a simulation of benzoic acid permeation through a lipid bilayer, SPIB is able to discover the entropic and energetic barriers to the permeation process. Given these results, we thus establish that SPIB is a reasonable and robust method for finding the important entropy, energy, and enthalpy barriers in physical systems, which can then be used to enhance the understanding and sampling of different activated mechanisms.

Machine Learning-Assisted Discovery of Hidden States in Expanded Free Energy Space

Collective variables (CVs) are crucial parameters in enhanced sampling calculations and strongly impact the quality of the obtained free energy surface. However, many existing CVs are unique to and dependent on the system they are constructed with, making the developed CV non-transferable to other systems. Herein, we develop a non-instructor-led deep autoencoder neural network (DAENN) for discovering general-purpose CVs. The DAENN is used to train a model by learning molecular representations upon unbiased trajectories that contain only the reactant conformers. The prior knowledge of nonconstraint reactants coupled with the here-introduced topology variable and loss-like penalty function are only required to make the biasing method able to expand its configurational (phase) space to unexplored energy basins. Our developed autoencoder is efficient and relatively inexpensive to use in terms of a priori knowledge, enabling one to automatically search for hidden CVs of the reaction of interest.

Toward Automated Sampling of Polymorph Nucleation and Free Energies with the SGOOP and Metadynamics

Understanding the driving forces behind the nucleation of different polymorphs is of great importance for material sciences and the pharmaceutical industry. This includes understanding the reaction coordinate that governs the nucleation process and correctly calculating the relative free energies of different polymorphs. Here, we demonstrate, for the prototypical case of urea nucleation from the melt, how one can learn such a one-dimensional reaction coordinate as a function of prespecified order parameters and use it to perform efficient biased all-atom molecular dynamics simulations. The reaction coordinate is learnt as a function of the generic thermodynamic and structural order parameters using the "spectral gap optimization of order parameters (SGOOP)" approach [Tiwary, P. and Berne, B. J. Proc. Natl. Acad. Sci. U.S.A. (2016)] and is biased using well-tempered metadynamics simulations. The reaction coordinate gives insights into the role played by different structural and thermodynamics order parameters, and the biased simulations obtain accurate relative free energies for different polymorphs. This includes an accurate prediction of the approximate pressure at which urea undergoes a phase transition and one of the metastable polymorphs becomes the most stable conformation. We believe the ideas demonstrated in this work will facilitate efficient sampling of nucleation in complex, generic systems.

SGOOP-d: Estimating Kinetic Distances and Reaction Coordinate Dimensionality for Rare Event Systems from Biased/Unbiased Simulations

Understanding kinetics including reaction pathways and associated transition rates is an important yet difficult problem in numerous chemical and biological systems, especially in situations with multiple competing pathways. When these high-dimensional systems are projected on low-dimensional coordinates, which are often needed for enhanced sampling or for interpretation of simulations and experiments, one can end up losing the kinetic connectivity of the underlying high-dimensional landscape. Thus, in the low-dimensional projection, metastable states might appear closer or further than they actually are. To deal with this issue, in this work, we develop a formalism that learns a multidimensional yet minimally complex reaction coordinate (RC) for generic high-dimensional systems. When projected along this RC, all possible kinetically relevant pathways can be demarcated and the true high-dimensional connectivity is maintained. One of the defining attributes of our method lies in that it can work on long unbiased simulations as well as biased simulations often needed for rare event systems. We demonstrate the utility of the method by studying a range of model systems including conformational transitions in a small peptide Ace-Ala₃-Nme, where we show how two-dimensional and three-dimensional RCs found by our previously published spectral gap optimization method "SGOOP" [Tiwary, P. and Berne, B. J. Proc. Natl. Acad. Sci. 2016, 113, 2839] can capture the kinetics for 23 and all 28 out of the 28 dominant state-to-state transitions, respectively.

Nucleation rates from small scale atomistic simulations and transition state theory

The evaluation of nucleation rates from molecular dynamics trajectories is hampered by the slow nucleation time scale and impact of finite size effects. Here, we show that accurate nucleation rates can be obtained in a very general fashion relying only on the free energy barrier, transition state theory, and a simple dynamical correction for diffusive recrossing. In this setup, the time scale problem is overcome by using enhanced sampling methods, in casu metadynamics, whereas the impact of finite size effects can be naturally circumvented by reconstructing the free energy surface from an appropriate ensemble. Approximations from classical nucleation theory are avoided. We demonstrate the accuracy of the approach by calculating macroscopic rates of droplet nucleation from argon vapor, spanning 16 orders of magnitude and in excellent agreement with literature results, all from simulations of very small (512 atom) systems.