Utilizing Machine Learning to Greatly Expand the Range and Accuracy of Bottom-Up Coarse-Grained Models through Virtual Particles

Abstract

Coarse-grained (CG) models parametrized using atomistic reference data, i.e., "bottom up" CG models, have proven useful in the study of biomolecules and other soft matter. However, the construction of highly accurate, low resolution CG models of biomolecules remains challenging. We demonstrate in this work how virtual particles, CG sites with no atomistic correspondence, can be incorporated into CG models within the context of relative entropy minimization (REM) as latent variables. The methodology presented, variational derivative relative entropy minimization (VD-REM), enables optimization of virtual particle interactions through a gradient descent algorithm aided by machine learning. We apply this methodology to the challenging case of a solvent-free CG model of a 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) lipid bilayer and demonstrate that introduction of virtual particles captures solvent-mediated behavior and higher-order correlations which REM alone cannot capture in a more standard CG model based only on the mapping of collections of atoms to the CG sites.

Figure 1

Example of CG mapping and VCG mapping for a lipid bilayer. The mapping operator relates the solvated AA bilayer (left) to the implicit solvent CG resolution (middle), while the mapping operator relates the VCG resolution (right) which contains virtual particles (purple) to the CG resolution.

Figure 2

Workflow diagram of VD-REM. An initial model is fed into VD-REM which iteratively updates model parameters according to gradient descent. These steps are performed as in REM for real particles, while machine-learned models are learned to approximate virtual particle interactions first.

Figure 3

VCG mapping scheme for DOPC. Hydrogens are omitted for clarity. Each DOPC lipid is mapped to six real sites of four types: HG (blue), MG (red), T1 (gray), and T2 (yellow). A virtual particle, VP (purple), with no atomistic correspondence is then appended to the HG bead.

Figure 4

Subset of pair potentials for real CG particles at beginning and end of training. Initial potentials (red) which were adapted from the REM-6 model and final VD-REM-7 model potentials (blue) are shown.

Figure 5

Three-dimensional radial distribution functions for real CG beads of DOPC. Both reference AA statistics (orange) and VCG statistics (green) are plotted.

Figure 6

Predicted values for the HG-VP potential energy derivative across VCG and AA ensembles. The predicted values for the initial (left) and final (right) iterations are plotted using kernel density estimation with a bin width of 25 kcal/mol Å.

Figure 7

Height fluctuation spectra of VD-REM-7 DOPC as a function of wavenumber. Simulation data (circles) were fit according to eq 34 (solid line) to obtain the bending modulus.

Figure 8

Depiction of the VD-REM-7 DOPC model at initial (a), intermediate (b), and final (c) stages of self-assembly.