Machine-guided path sampling to discover mechanisms of molecular self-organization.

Molecular self-organization driven by concerted many-body interactions produces the ordered structures that define both inanimate and living matter. Here we present an autonomous path sampling algorithm that integrates deep learning and transition path theory to discover the mechanism of molecular self-organization phenomena. The algorithm uses the outcome of newly initiated trajectories to construct, validate and-if needed-update quantitative mechanistic models. Closing the learning cycle, the models guide the sampling to enhance the sampling of rare assembly events. Symbolic regression condenses the learned mechanism into a human-interpretable form in terms of relevant physical observables. Applied to ion association in solution, gas-hydrate crystal formation, polymer folding and membrane-protein assembly, we capture the many-body solvent motions governing the assembly process, identify the variables of classical nucleation theory, uncover the folding mechanism at different levels of resolution and reveal competing assembly pathways. The mechanistic descriptions are transferable across thermodynamic states and chemical space.

Solvent reaction coordinate for an SN2 reaction.

We study the prototypical SN2 reaction Cl- + CH3Cl → CH3Cl + Cl- in water using quantum mechanics/molecular mechanics (QM/MM) computer simulations with transition path sampling and inertial likelihood maximization. We have identified a new solvent coordinate to complement the original atom-exchange coordinate used in the classic analysis by Chandrasekhar, Smith, and Jorgensen [J. Am. Chem. Soc. 107, 154 (1985)]. The new solvent coordinate quantifies instantaneous solvent-induced polarization relative to the equilibrium average charge density at each point along the reaction pathway. On the basis of likelihood scores and committor distributions, the new solvent coordinate improves upon the description of solvent dynamical effects relative to previously proposed solvent coordinates. However, it does not increase the transmission coefficient or the accuracy of a transition state theory rate calculation.

Solid-solid phase equilibria in the NaCl-KCl system.

Solid solutions, structurally ordered but compositionally disordered mixtures, can form for salts, metals, and even organic compounds. The NaCl-KCl system forms a solid solution at all compositions between 657 °C and 505 °C. Below a critical temperature of 505 °C, the system exhibits a miscibility gap with coexisting Na-rich and K-rich rocksalt phases. We calculate the phase diagram in this region using the semi-grand canonical Widom method, which averages over virtual particle transmutations. We verify our results by comparison with free energies calculated from thermodynamic integration and extrapolate the location of the critical point. Our calculations reproduce the experimental phase diagram remarkably well and illustrate how solid-solid equilibria and chemical potentials, including those at metastable conditions, can be computed for materials that form solid solutions.

The generic unfolding of a biomimetic polymer during force spectroscopy.

With the help of force spectroscopy, several analytical theories aim at estimating the rate coefficient of folding for various proteins. Nevertheless, a chief bottleneck lies in the fact that there is still no perfect consensus on how does a force generally perturb the crystal-coil transition. Consequently, the goal of our work is in clarifying the generic behavior of most proteins in force spectroscopy; in other words, what general signature does an arbitrary protein exhibit for its rate coefficient as a function of the applied force? By employing a biomimetic polymer in molecular simulations, we focus on evaluating its respective activation energy for unfolding, while pulling on various pairs of its monomers. Above all, we find that in the vicinity of the force-free scenario, this activation energy possesses a negative slope and a negative curvature as a function of the applied force. Our work is in line with the most recent theories for unfolding, which suggest that such a signature is expected for most proteins, and thus, we further reiterate that many of the classical formulae, that estimate the rate coefficient of the crystal-coil transition, are inadequate. Besides, we also present here an analytical expression which experimentalists can use for approximating the activation energy for unfolding; importantly, it is based on measurements for the mean and variance of the distance between the beads which are being pulled. In summary, our work presents an interesting view for protein folding in force spectroscopy.

Nucleation and structural growth of cluster crystals.

We study the nucleation of crystalline cluster phases in the generalized exponential model with exponent n = 4. Due to the finite value of this pair potential for zero separation, at high densities the system forms cluster crystals with multiply occupied lattice sites. Here, we investigate the microscopic mechanisms that lead to the formation of cluster crystals from a supercooled liquid in the low-temperature region of the phase diagram. Using molecular dynamics and umbrella sampling, we calculate the free energy as a function of the size of the largest crystalline nucleus in the system, and compare our results with predictions from classical nucleation theory. Employing bond-order parameters based on a Voronoi tessellation to distinguish different crystal structures, we analyze the average composition of crystalline nuclei. We find that even for conditions where a multiply occupied fcc crystal is the thermodynamically stable phase, the nucleation into bcc cluster crystals is strongly preferred. Furthermore, we study the particle mobility in the supercooled liquid and in the cluster crystal. In the cluster crystal, the motion of individual particles is captured by a simple reaction-diffusion model introduced previously to model the kinetics of hydrogen bonds.

A string reaction coordinate for the folding of a polymer chain.

We investigate the crystallization mechanism of a single, flexible homopolymer chain with short range attractions. For a sufficiently narrow attractive well, the system undergoes a first-order like freezing transition from an expanded disordered coil to a compact crystalline state. Based on a maximum likelihood analysis of committor values computed for configurations obtained by Wang-Landau sampling, we construct a non-linear string reaction coordinate for the coil-to-crystal transition. In contrast to a linear reaction coordinate, the string reaction coordinate captures the effect of different degrees of freedom controlling different stages of the transition. Our analysis indicates that a combination of the energy and the global crystallinity parameter Q6 provide the most accurate measure for the progress of the transition. While the crystallinity paramter Q6 is most relevant in the initial stages of the crystallization, the later stages are dominated by a decrease in the potential energy.

Folding mechanism of a polymer chain with short-range attractions.

We investigate the crystallization of a single, flexible homopolymer chain using transition path sampling. The chain consists of N identical spherical monomers evolved according to Langevin dynamics. While neighboring monomers are coupled via harmonic springs, the non-neighboring monomers interact via a hard core and a short-ranged attractive potential. For a sufficiently small interaction range λ, the system undergoes a first-order freezing transition from an expanded, disordered phase to a compact crystalline state. Using a new shooting move tailored to polymers combined with a committor analysis, we study the transition state ensemble of an N = 128 chain and search for possible reaction coordinates based on likelihood maximization. We find that typical transition states consist of a crystalline nucleus with one or more chain fragments attached to it. Furthermore, we show that the number of particles in the crystalline core is not well suited as a reaction coordinate. We then present an improved reaction coordinate, which includes information from the potential energy and the overall crystallinity of the polymer.