Physics-Informed Deep Learning Approach for Reintroducing Atomic Detail in Coarse-Grained Configurations of Multiple Poly(lactic acid) Stereoisomers

Multiscale modeling of complex molecular systems, such as macromolecules, encompasses methods that combine information from fine and coarse representations of molecules to capture material properties over a wide range of spatiotemporal scales. Being able to exchange information between different levels of resolution is essential for the effective transfer of this information. The inverse problem of reintroducing atomistic degrees of freedom in coarse-grained (CG) molecular configurations is particularly challenging as, from a mathematical point of view, it is an ill-posed problem; the forward mapping from the atomistic to the CG description is typically defined via a deterministic operator (“one-to-one” problem), whereas the reversed mapping from the CG to the atomistic model refers to creating one representative configuration out of many possible ones (“one-to-many” problem). Most of the backmapping methods proposed so far balance accuracy, efficiency, and general applicability. This is particularly important for macromolecular systems with different types of isomers, i.e., molecules that have the same molecular formula and sequence of bonded atoms (constitution) but differ in the three-dimensional configurations of their atoms in space. Here, we introduce a versatile deep learning approach for backmapping multicomponent CG macromolecules with chiral centers, trained to learn structural correlations between polymer configurations at the atomistic level and their corresponding CG descriptions. This method is intended to be simple and flexible while presenting a generic solution for resolution transformation. In addition, the method is aimed to respect the structural features of the molecule, such as local packing, capturing therefore the physical properties of the material. As an illustrative example, we apply the model on linear poly(lactic acid) (PLA) in melt, which is one of the most popular biodegradable polymers. The framework is tested on a number of model systems starting from homopolymer stereoisomers of PLA to copolymers with randomly placed chiral centers. The results demonstrate the efficiency and efficacy of the new approach.


Bond Vectors Model
In this Section we illustrate a comparison between the Reference model (λ bv = λ bl = 1 and λ v 0 = λ ba = λ da = 0) and a model where we penalize only the bond vectors (λ bv = 1 and λ v 0 = λ bl = λ ba = λ da = 0).As can be seen in S1, the Reference model matches perfectly the target distributions, illustrating a significant deviation in comparison to the model trained with a loss function that penalizes only the bond vectors.Furthermore, the two models show similar behaviour on the prediction of bond angles and dihedral angles distributions (data not shown here).
Figure S1: Comparison of bond lengths among target (red solid line), initial prediction for the reference model (blue dotted line), and the initial prediction for the model that penalize only the bond vectors (green dash-dotted line) for a 100-mer PLLA configuration.

100-mer
In this Section we present additional results concerning the atomistic structure of the 100-mer PLA systems predicted by the trained Machine Learning models.Specifically, we evaluate the performance of the trained models on three 100-mer systems (PLLA100, PDLA100, and

Copo100RAND and Copo100HOM
In this Section we investigate the transferability of the trained Machine Learning models for PLA 100-mer copolymers prepared by a different backmapping strategy.Copo100HOM has the same sequence of stereoisomers per chain as the target system, but the neural network was trained solely on 100-mer homopolymer systems, while Copo100RAND was trained using all the available 100-mer systems, but has random stereochemistry with the same D content Copo100).A comparison among the target, initial prediction, and the output of runEQ is provided for (a) intra-and intermonomeric dihedral angles; (b) intra-and intermolecular radial distribution functions specific for a number of different particles.

Figure S2 :
Figure S2: Comparison of intramonomeric dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLLA configuration.

Figure S3 :
Figure S3: Comparison of intramonomeric dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PDLA configuration.

Figure S4 :
Figure S4: Comparison of intramonomeric dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLA copolymer configuration, only for the L monomers.

Figure S5 :
Figure S5: Comparison of intramonomeric dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLA copolymer configuration, only for the D monomers.

Figure S6 :
Figure S6: Comparison of the intermonomeric dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLLA configuration.

Figure S7 :
Figure S7: Comparison of the intermonomeric dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PDLA configuration.

Figure S8 :
Figure S8: Comparison of intermonomeric dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLA copolymer configuration, only for the L monomers.

Figure S9 :
Figure S9: Comparison of intermonomeric dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLA copolymer configuration, only for the D monomers.

Figure S10 :
Figure S10: Comparison of intramolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLLA configuration.

Figure S11 :
Figure S11: Comparison of intramolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PDLA configuration.

Figure S12 :
Figure S12: Comparison of intramolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLA copolymer configuration.

Figure S13 :
Figure S13: Comparison of intermolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLLA configuration.

Figure S14 :
Figure S14: Comparison of intermolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PDLA configuration.

Figure S15 :
Figure S15: Comparison of intermolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 100-mer PLA copolymer configuration.

(
55%) in every chain.A comparison among the target data and the initial predictions of Copo100RAND and Copo100HOM is provided for (a) intra-and intermonomeric dihedral angles; (b) intra-and intermolecular radial distribution functions specific for a number of different particles.

Figure S16 :
Figure S16: Comparison of intramonomeric dihedral angles among target (red solid line) and the initial predictions for Copo100RAND (blue dotted line) and Copo100HOM (green dash-dotted line) for a 100-mer PLA copolymer configuration, only for the L monomers.

Figure S17 :
Figure S17: Comparison of intramonomeric dihedral angles among target (red solid line) and the initial predictions for Copo100RAND (blue dotted line) and Copo100HOM (green dash-dotted line) for a 100-mer PLA copolymer configuration, only for the D monomers.

Figure S18 :
Figure S18: Comparison of intermonomeric dihedral angles among target (red solid line) and the initial predictions for Copo100RAND (blue dotted line) and Copo100HOM (green dash-dotted line) for a 100-mer PLA copolymer configuration, only for the L monomers.

Figure S19 :
Figure S19: Comparison of intermonomeric dihedral angles among target (red solid line) and the initial predictions for Copo100RAND (blue dotted line) and Copo100HOM (green dash-dotted line) for a 100-mer PLA copolymer configuration, only for the D monomers.

Figure S20 :
Figure S20: Comparison of intramolecular radial distributions around the given atoms among target (red solid line) and initial predictions of Copo100HOM (blue dotted line) and Copo100RAND (green dash-dotted line) models for a 100-mer PLA copolymer configuration.

Figure S21 :
Figure S21: Comparison of intermolecular radial distributions around the given atoms among target (red solid line) and initial predictions of Copo100HOM (blue dotted line) and Copo100RAND (green dash-dotted line) models for a 100-mer PLA copolymer configuration.

Figure S22 :
Figure S22: Comparison of dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PLLA configuration.

Figure S23 :
Figure S23: Comparison of dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PDLA configuration.

Figure S24 :
Figure S24: Comparison of dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PLA copolymer configuration, only for the D monomers.

Figure S25 :
Figure S25: Comparison of dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PLA copolymer configuration, only for the L monomers.

Figure S26 :
Figure S26: Comparison of dihedral angles among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PLA copolymer configuration, only for the distributions between D and L monomers.

Figure S27 :
Figure S27: Comparison of intramolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PLLA configuration.

Figure S28 :
Figure S28: Comparison of intramolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PDLA configuration.

Figure S29 :
Figure S29: Comparison of intermolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PLLA configuration.

Figure S30 :
Figure S30: Comparison of intermolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PDLA configuration.

Figure S31 :
Figure S31: Comparison of intramolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PLA copolymer configuration.

Figure S32 :
Figure S32: Comparison of intermolecular radial distributions around the given atoms among target (red solid line), initial prediction (blue dotted line), and the output of runEQ (green dash-dotted line) for a 30-mer PLA copolymer configuration.