Predicting the Sensitivity of Multiscale Coarse-Grained Models to their Underlying Fine-Grained Model Parameters

The sensitivity of a coarse-grained (CG) force field to changes in the underlying fine-grained (FG) model from which it was derived provides modeling insight for improving transferability across interaction parameters, transferability across temperature, and the calculation of thermodynamic derivatives. Methods in the literature, such as multi-trajectory finite differences and reweighted finite differences, are either too computationally demanding to calculate within acceptable noise tolerances or are too biased for practical accuracy. This work presents a new reweighting-free, single-simulation formula that allows for practical, high signal-to-noise calculations of CG model sensitivity with respect to FG model interaction parameters and thermodynamic state points. This formula, the self-consistent basis (SCB) single point formula, determines the many-body sensitivity in a single step by approximating the derivative of the many-body potential projected onto the same set of trial functions as the sensitivity. A related diagnostic formula also derived in this paper is the self-consistent iterative (SCI) single point formula, which is useful for identifying the importance of many-body sources of error and verifying CG representability of observables. The SCI formula determines the many-body sensitivity iteratively via a series of partially self-consistent, variational approximations to the complete many-body sensitivity. The new, computationally efficient SCB formula shows substantially less noise than previous methods when applied to single site methanol and solvent-free sodium chloride CG models, though bias can remain a problem. It represents a novel method for calculating alchemical transferability across interaction parameters at low computational cost and with high fidelity, and the results point to new understanding of the current limits of CG model transferability.