Bayesian selection of coarse-grained water models beyond thermodynamic observables and Lennard-Jones interactions

Zavadlav, Arampatzis and Koumoutsakos (Scientific Reports 9, 99, 2019) rank coarse-grained (CG) water models — combinations of a mapping $M$ (water molecules per CG bead), a number of interaction sites, a charge distribution, and rigidity — by Bayesian model evidence against five experimental thermodynamic quantities of interest, with transferability tested only across temperatures. Their own stated exclusions define the open problem: structural observables (radial distribution functions) and dynamical observables (self-diffusion coefficients) cannot serve as calibration targets for mappings $M>1$ because no experiment measures a corresponding quantity; the Lennard-Jones non-bonded form demonstrably cannot fit surface tension and shear viscosity for some mappings, and alternative short-range forms such as Born–Mayer–Huggins were never explored in a Bayesian search; and chargeless and charged model classes cannot be compared because the dielectric constant is undefined for chargeless models. The problem is to extend evidence-based selection over the CG water model space to these excluded axes — admitting structural and dynamical targets across mappings with a principled protocol for the absence of experimental counterparts at $M>1$, non-Lennard-Jones functional forms, and principled comparison of model classes with non-overlapping observable spaces — and to produce the resulting evidence-based ranking. Later literature offers partial neighbors (Bayesian calibration with structural observables at atomistic resolution; gradient-based top-down, i.e., observable-fitting, learning of a single CG water potential; kinetic-criterion mapping selection) but no resolution.

Background

Coarse-grained (CG) molecular models replace groups of atoms by single interaction sites so that molecular dynamics simulations can reach the length and time scales relevant to soft matter and biomolecular systems. For water, a CG model is fixed by three choices: the mapping $M$ (how many water molecules are represented by one CG bead), the number and geometry of interaction sites per bead (single-, two-, or three-site; rigid or flexible), and the Hamiltonian — in particular whether sites carry partial charges and which short-range pair potential describes the non-bonded interactions, conventionally the Lennard-Jones (LJ) form $U(r)=4\epsilon[(\sigma/r)^{12}-(\sigma/r)^{6}]$ with size parameter $\sigma$ and energy parameter $\epsilon$. Established CG water models (for example MARTINI, a widely used biomolecular CG force field; BMW; and the GROMOS CG model) each fix these choices a priori; the BMW model, for instance, groups four water molecules into three sites carrying a fixed dipole and quadrupole and uses the Born–Mayer–Huggins (BMH) short-range potential, an exponential repulsive-plus-attraction form borrowed from ionic-liquid models, instead of LJ.

Zavadlav, Arampatzis and Koumoutsakos (Scientific Reports 9, 99, 2019) replaced this ad-hoc design by data-driven selection. They define a space of 1-, 2-, and 3-site rigid and flexible water models for mappings $M=1$ to 12, with LJ non-bonded interactions and, in multi-site models, partial charges, and rank every model by its Bayesian evidence — the probability of the data under the model,

$$p(\boldsymbol{d}|\mathcal{M})=\int p(\boldsymbol{d}|\phi,\mathcal{M})\,p(\phi|\mathcal{M})\,d\phi,$$

which integrates over the model parameters $\phi$ and automatically penalizes unnecessary complexity (an Ockham's-razor effect, so that a more flexible model must earn its extra parameters through fit). The target data are experimental values of five thermodynamic quantities of interest (QoI) — density $\rho$, dielectric constant $\epsilon_{\mathrm{diel}}$, surface tension $\gamma$, isothermal compressibility $\kappa$, and shear viscosity $\eta$ — combined through a multiplicative Gaussian error model in which each datum's standard deviation is proportional to its value. A hierarchical extension of the framework merges QoI from several thermodynamic conditions into a single evidence value, which the authors use to test transferability across temperatures $T=283$, $298$, and $323$ K; they find, for example, that the rigid three-site model with a particular charge distribution (their model 3S*) is the most transferable choice at mapping $M=4$. Bayesian selection, calibration and validation of CG models through model evidence had been formulated in general terms earlier by Farrell, Oden and Faghihi (Journal of Computational Physics 295, 189–208, 2015); the 2019 work is its systematic instantiation for water.

The same paper states three exclusions that define the open problem. First, structural observables such as the site–site radial distribution function (RDF — the probability of finding a pair of sites at a given separation, relative to a uniform ideal gas, the standard probe of liquid structure) and dynamical observables such as the self-diffusion coefficient (the long-time growth rate of the mean-square displacement of a particle) were excluded from calibration, because for mappings $M>1$ no experiment measures a corresponding quantity — a CG bead does not exist in the laboratory. This exclusion is not a technicality: coarse-graining smooths the free-energy landscape, so CG models propagated with ordinary memoryless (Markovian) molecular dynamics typically diffuse much faster than the real liquid, and matching structure and dynamics simultaneously is itself a known difficulty that any treatment of dynamical QoI must confront. Second, for some mappings there are no LJ parameters $\sigma$ and $\epsilon$ that fit $\gamma$ and $\eta$ well within the liquid state, and the authors suggest replacing the LJ interaction by another short-range form such as BMH — but their Bayesian search never included such forms. Third, the evidence of chargeless models (single-site, no electrostatics) cannot be compared with that of charged models, because observables such as the dielectric constant are undefined for chargeless models: the two model classes live in different observable spaces, and no principled evidence-based comparison across such classes exists. Beyond these limitations, the framework's transferability assessment covered only the temperature axis; the environments in which CG water models are actually deployed — under pressure, at interfaces, or around solutes — were never evaluated within the evidence framework. The paper does not state non-temperature transferability as an open problem.

Problem Statement

Extend the evidence-based (Bayesian) selection framework for coarse-grained models of liquid water of Zavadlav, Arampatzis and Koumoutsakos (2019) so that the model evidence ranks CG water models on an equal footing across the model space, by removing its stated exclusions:

1. Structural and dynamical observables: admit site–site radial distribution functions and self-diffusion coefficients, together with the five thermodynamic quantities of interest used in the source framework (density, dielectric constant, surface tension, isothermal compressibility, and shear viscosity), as calibration targets across mappings $M\ge 1$, with an explicit, principled protocol for the fact that no experimental counterpart exists for $M>1$ and with a stated treatment of the accelerated dynamics of coarse-grained models that otherwise prevents a model from matching both structure and diffusion.

2. Functional forms: include alternative short-range non-bonded forms beyond Lennard-Jones, at least the Born–Mayer–Huggins form, in the candidate model space, since Lennard-Jones demonstrably cannot fit surface tension and shear viscosity for some mappings.

3. Cross-class comparability: define and justify a principled, evidence-based comparison between model classes whose observable spaces do not overlap — chargeless single-site models, for which the dielectric constant is undefined, versus charged multi-site models.

The three extensions are facets of a single objective: they jointly remove the biases that currently distort the evidence ranking — thermodynamic-only targets, an LJ-only interaction family, and the exclusion or incomparable treatment of chargeless classes each restrict which model can win — so that the ranking is on an equal footing across the model space.

References

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