Bottom-up Coarse-Graining: Principles and Perspectives

Abstract

Large-scale computational molecular models provide scientists a means to investigate the effect of microscopic details on emergent mesoscopic behavior. Elucidating the relationship between variations on the molecular scale and macroscopic observable properties facilitates an understanding of the molecular interactions driving the properties of real world materials and complex systems (e.g., those found in biology, chemistry, and materials science). As a result, discovering an explicit, systematic connection between microscopic nature and emergent mesoscopic behavior is a fundamental goal for this type of investigation. The molecular forces critical to driving the behavior of complex heterogeneous systems are often unclear. More problematically, simulations of representative model systems are often prohibitively expensive from both spatial and temporal perspectives, impeding straightforward investigations over possible hypotheses characterizing molecular behavior. While the reduction in resolution of a study, such as moving from an atomistic simulation to that of the resolution of large coarse-grained (CG) groups of atoms, can partially ameliorate the cost of individual simulations, the relationship between the proposed microscopic details and this intermediate resolution is nontrivial and presents new obstacles to study. Small portions of these complex systems can be realistically simulated. Alone, these smaller simulations likely do not provide insight into collectively emergent behavior. However, by proposing that the driving forces in both smaller and larger systems (containing many related copies of the smaller system) have an explicit connection, systematic bottom-up CG techniques can be used to transfer CG hypotheses discovered using a smaller scale system to a larger system of primary interest. The proposed connection between different CG systems is prescribed by (i) the CG representation (mapping) and (ii) the functional form and parameters used to represent the CG energetics, which approximate potentials of mean force (PMFs). As a result, the design of CG methods that facilitate a variety of physically relevant representations, approximations, and force fields is critical to moving the frontier of systematic CG forward. Crucially, the proposed connection between the system used for parametrization and the system of interest is orthogonal to the optimization used to approximate the potential of mean force present in all systematic CG methods. The empirical efficacy of machine learning techniques on a variety of tasks provides strong motivation to consider these approaches for approximating the PMF and analyzing these approximations.

Figure 1

Broad summary of bottom-up CG modeling. Based on FG reference statistics, CG modeling is composed of two steps. (1) CG mapping (often performed on configurational variables) involves real or virtual CG particles at molecular resolution or includes mesoscopic mapping at coarser resolutions. (2) CG mechanics are defined by the specific consistency criteria and design principles that determine the CG equation of motion and CG interactions. CG equations of motions are generally chosen based on the target dynamical information, e.g., with or without fluctuation forces. CG interactions are determined by the designed CG Hamiltonian, which may suffer from an imperfect basis set and transferability issues. Bottom-up parametrization methodologies are then applied to yield effective CG interactions that optimally approximate the level of physics specified by the consistency criteria and design principles.

Figure 2

UCG models are designed to capture the chemical or physical changes “beneath” the CG resolution (illustrated upper middle left for ATP hydrolysis in F-actin). Practical design principles of UCG models are based on the relaxation time of internal state dynamics. (1) In the slowest limit (SST), UCG state dynamics can be treated as a kind of surface hopping. An example of this is the gauche- and anti- configurations from 1,2-dichloroethane (top panel). Based on the target system, distinct UCG states are identified, and the UCG models are built by parametrizing the state-wise interactions and optimizing the kinetic rates described by the Metropolis-Hastings algorithm. (2) The internal states at the fastest switching limit (RLE) can be thought to be in quasi-equilibrium, and the Ehrenfest dynamics idea can describe the internal states by mixing them with the state probability. The UCG models are then constructed by identifying the rapidly varying states with the corresponding order parameters. Then, bottom-up CG methodologies can be applied to determine the UCG state-wise interactions. As an example, we depict the solvated peptide here exhibiting folded and unfolded states determined by the optimal CV (bottom panel).

Figure 3

Summary of the excess entropy-based approach to achieve dynamical representability under Hamiltonian mechanics. The dynamical representability of CG models can be addressed by having a dynamical correspondence between the CG and FG models. In this case, without correct fluctuations, CG dynamics is spuriously accelerated compared to the reference FG dynamics (e.g., diffusion coefficients in this figure). Therefore, the ultimate goal in dynamical correspondence would be to address both directions across the FG and CG systems: (1) Predict the accelerated CG dynamics from the FG information and (2) correct the fast CG dynamics to match the original FG dynamics by observing the missing degrees of freedom upon the coarse-graining process. We depict the molecular liquids (water in this case) at the single-site CG resolution as an example, where the hard sphere (HS) mapping theory can achieve (1) and incorporating the missing rotational information can address (2) (recovered FG).

Figure 4

A summary of machine learning methods related to CG modeling. Kernel methods utilize a covariance function to specify a random process which can be used as a nonlinear estimator for CG potentials (left panel, top). Artificial neural networks (ANNs) can also be used to generate nonlinear force fields. A variety of ANN architectures have been developed with applications in CG modeling, such as autoencoders which compress the data stream before expanding it, which has natural connections to CG methods, and graph convolutions, which apply filters over graphs such as molecular topologies (left panel, bottom). Machine learning has also found uses in analyzing CG models and trajectories. Neural networks can be used to backmap CG configurations (right panel, left). Classifiers, especially interpretable ones, can be used to differentiate between configurations generated by different models (such as a CG model and the reference data it was parametrized from) and also provide specific information about how the ensembles are different (right panel, right).