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Recovering molecules from coarse-grained beads: free-energy-conditioned generative backmapping across chemical space

Luis Itza Vazquez-Salazar, Tristan Bereau

arXiv:2609.04432v1physics.chem-phcond-mat.softcs.LGphysics.bio-ph

TL;DR

Compositional backmapping is difficult because bead representations can correspond to multiple molecules, and enumerating chemical space is impractical. Juniper uses graph diffusion conditioned on partition free energy to generate molecules for MARTINI representations; its generated free-energy distributions track the conditioning target, with deviations in the tails.

  • Problem

    Compositional backmapping remains difficult because bead representations are non-injective and correspond to multiple molecules, while chemical space is too vast for enumeration.

    Source: §1 · pp. 1–2

  • Method

    Juniper uses graph diffusion conditioned on CG partition free energy to generate molecules for MARTINI bead representations, with training molecules of up to 9 heavy atoms mapped to one or two beads.

    Source: §2.2 · p. 5, §4 · p. 15

  • Results

    Juniper's generated ∆G_W→O tracks ∆G^CG_W→O linearly, with deviations in the hydrophobic and hydrophilic tails attributed to predictor limitations.

    Source: §4 · p. 15

  • Takeaways & Limitations

    Promising bead combinations from CG screening can be converted into candidate molecules for atomistic inspection or simulation.

    Source: §4 · p. 15

    The reported results concern dimers with at most 9 heavy atoms; unimer samples were too few for a dedicated analysis, and backmapping beyond two beads requires topology information.

    Source: §4 · p. 15

Abstract

from arXiv · show

Transferable coarse-grained (CG) force fields compress chemical space: by aggregating atoms into a reduced set of interaction beads, models such as MARTINI reduce the number of distinguishable compounds by roughly three orders of magnitude, making high-throughput screening of thermodynamic properties tractable across soft matter, with drug--membrane permeability as a well-developed example. The compression is lossy and, so far, one-way: a screen returns a combination of beads, with no established route back to the compounds it stands for. Recovering those compounds--compositional backmapping--is a one-to-many inverse map, distinct from the better-studied conformational problem of rebuilding atomic coordinates from a known mapping. Here we formulate compositional backmapping as conditional graph generation by introducing juniper, a discrete denoising diffusion model over molecular graphs conditioned on the octanol--water partition free energy $\Delta G_{\mathrm{W} \mapsto \mathrm{O}}$, the principal driver of MARTINI bead type assignment and hence a proxy for bead identity. Trained on molecules of up to 9 heavy atoms mapped onto one or two beads, juniper generates molecules that are 93\% valid and 92\% unique for two-bead targets, and whose $\Delta G_{\mathrm{W} \mapsto \mathrm{O}}$ distributions track the target $\Delta G^{\mathrm{CG}}_{\mathrm{W} \mapsto \mathrm{O}}$ linearly ($r^{2} \geq 0.96$), departing only in the hydrophobic and hydrophilic tails. Although the model receives no chemical information beyond a single scalar, the functional groups shift systematically with the imposed free energy, from branched hydrocarbons at the apolar end to amides, imides, and isocyanates at the polar end. A bead combination flagged by a CG screen can therefore be turned into candidate molecules for atomistic study or synthesis.

1 Introduction

Coarse-graining makes chemical-space searches more tractable but discards molecular detail, leaving compositional backmapping from bead combinations to molecules unresolved. The paper frames this as conditional graph generation with juniper, using partition free energy to connect molecules to MARTINI representations.

1 Introduction

  • Coarse-graining reduces chemical-space size by roughly three orders of magnitude, aiding high-throughput searches while losing the chemical detail needed to act on their results.The constrained chemical space remains too vast for enumeration, making property-guided search important.Source: §1 · pp. 1–2
  • Compositional backmapping asks which molecules correspond to a bead combination, a one-to-many inverse problem distinct from rebuilding coordinates when the mapping is known.Because bead mappings are non-injective, a CG screen's bead combination cannot by itself specify a unique molecule for atomistic study or synthesis.Source: §1 · p. 2
  • Generative machine learning can learn a bead-to-molecule distribution from curated examples and produce multiple compounds from a single label.Earlier generative backmapping work focused on spatial coordinates, including molecular liquids, polymer melts, peptides, and proteins.Source: §1 · p. 2
  • The study restricts molecules to at most 9 heavy atoms mapped to one or two MARTINI beads, allowing the compositional task to be formulated in graph space.The chosen free-energy descriptor is the principal parameter controlling MARTINI's compression of chemical space.Source: §1 · pp. 2–3
  • Although partition free energy is widely used for molecule generation and drives MARTINI bead assignment, it imperfectly separates chemical space and can correspond to multiple bead combinations.The paper treats the CG free energy approximately as the mean of the generated molecular distribution, not a value each individual molecule must hit.Source: §1 · p. 3
  • Juniper addresses compositional backmapping with a graph diffusion model conditioned on the free energy associated with a MARTINI bead representation.Unlike prior conditional generation aimed at optimizing an individual molecule for a target property, the goal is a distribution of molecules consistent with the bead representation.Source: §1 · pp. 3–4

2 Methods

Juniper models molecules as graphs and learns to denoise them conditioned on a hydrophobicity label. Its methods define the training data, generation settings, and metrics used to evaluate generated molecules and their distributions.

2.1 Diffusion models in graphs

  • Juniper represents molecules as atom-and-bond graphs labeled by hydrophobicity and trains a discrete diffusion transformer to denoise them conditioned on that scalar.Noise is added to nodes and edges over T steps; classifier-free guidance uses a learnable null condition during training.Source: §2.1 · pp. 4–5, Figure 2 · p. 4

2.2 Training

  • Training used molecules of up to 9 heavy atoms mapped to 28 MARTINI v2 beads, with partition free energies as labels for individual beads or bead-pair sums.Single-bead and two-bead datasets were curated with different heavy-atom thresholds; training ran for 1,000 epochs with 500 diffusion steps per epoch.Source: §2.2 · p. 5

2.3 Generation

  • Generation starts from random noise, conditions on target free energy and atom count, and samples 100 molecules per bead combination at γ values of 0.5, 1, or 2.The analysis treats bead labels as permutation-invariant and ignores bead size.Source: §2.3 · p. 5

2.4.1 Basic metrics

  • Evaluation counts valid, unique, and novel molecules, plus repetition rate and training-set coverage; validity requires connected fragments and successful graph reconstruction by RDKit.Uniqueness removes duplicates among valid molecules, while novelty counts unique valid molecules absent from the training set.Source: §2.4.1 · p. 6

2.4.2 Distribution analysis

  • Hydrophobicity analysis uses ALOGPS log P values to derive partition free energy and compares distributions with Jensen–Shannon divergence and Gaussian-kernel maximum mean discrepancy.The paper also reports Wildman–Crippen estimates in the Supporting Information.Source: §2.4.2 · pp. 6–7

2.4.3 Functional-group and diversity analysis

  • Chemical-space analysis identifies functional groups with Ertl’s algorithm and assesses generated-versus-reference embeddings with Fréchet ChemNet distance and molecular diversity with internal chemical diversity.The ChemNet distance compares mean and covariance of embeddings; internal chemical diversity uses pairwise Jaccard–Tanimoto similarity.Source: §2.4.3 · p. 7

3 Results

Juniper generates valid, unique molecules conditioned on ΔG_W7→O, with generated values tracking the target linearly but deviating in the hydrophobic and hydrophilic tails. Unconditional sampling spans the training space, while conditional samples show systematic chemistry shifts across bead families.

3.1 Unconditional generation

  • In unconditional generation, juniper outperforms both CVAE models on all reported metrics except novelty, with valid and unique molecule percentages about 20% higher.Source: §3.1 · p. 8
  • Unconditional samples cover low-density regions and span high and low ΔG_W7→O values and the full range of SASA, whereas the CVAE models favor populated regions.Source: §3.1 · p. 9

3.2 Conditional generation with ∆GW7→O

  • 93% of conditionally generated graphs are valid and 92% are unique, with low repetition rates.Source: §3.2 · p. 9
  • Conditional samples shift across chemical space: C1-C1 is hydrophobic with larger SASA, Na-Nda spans a central ΔG_W7→O range, and P3-P4 is more hydrophilic with smaller SASA.Source: §3.2 · p. 9, Figure 4 · p. 10
  • At γ = 0.5, P5-P5 has the largest deviation overall at 2.9 kcal mol−1; increasing γ reduces tail deviations but can reduce diversity and validity.Source: §3.2 · pp. 11, 13
  • Most bead combinations have small DJS values, averaging 0.07, while FCD values are small overall and increase for low-population ΔG_W7→O values.Source: §3.2 · pp. 11, 15
  • r2 ≥0.96 for the relationship between ΔGCG_W7→O and generated mean ΔGAA_W7→O, with deviations in both hydrophobic and hydrophilic tails.Source: §3.2 · p. 11, Figure 5 · p. 12
  • Functional groups vary by bead family, from branched hydrocarbons in C-C to increasingly heteroatom-rich fragments in N-P and P-P, while C-P shows diverse chemistry.Source: §3.2 · p. 13, Figure 6 · p. 14

4 Conclusions

Juniper is a graph diffusion model that uses coarse-grained partition free energy as a proxy for MARTINI bead mapping to generate atomistic molecules. Its generated samples follow the target free energy and show bead-family-linked chemical composition, within a scope limited to small two-bead molecules.

4 Conclusions

  • Juniper connects MARTINI coarse-grained representations to atomistic molecules through a graph diffusion model conditioned on ΔG^CG_W7→O.The conditioning free energy is used as a proxy for MARTINI’s mapping of chemical space.Source: §4 · p. 15
  • Juniper’s generated samples show a linear relationship between their mean ΔG_W7→O and the conditioning value, with deviations in the hydrophobic and hydrophilic tails attributed to predictor limitations.The generated and training ΔG_W7→O distributions also show strong agreement, and solvent-accessible surface area is recovered from the training dataset.Source: §4 · p. 15
  • Functional groups separate by bead family despite the model receiving no chemical information beyond a single free energy.The authors suggest ΔG_W7→O carries more of the chemistry of MARTINI mapping than its role as one fitting target would imply.Source: §4 · p. 15
  • Larger γ improves generation near the desired property value but reduces diversity and, in distribution tails, validity.The effect is more pronounced in the tails than at the mean.Source: §4 · p. 15
  • The study is limited to molecules with at most 9 heavy atoms mapped to two-bead MARTINI v2 representations, and it does not assess synthetic accessibility.It also omits bonded parameters; the authors expect incorporating them to improve backmapping.Source: §4 · p. 15
  • For the studied dimers, promising bead combinations can be returned to candidate molecules for atomistic inspection or simulation.The authors describe this as a step toward closing the coarse-grained screening or optimisation loop.Source: §4 · pp. 15–16

Conflicts of interest

The authors declare no conflicts of interest.

Conflicts of interest

Data availability

The paper identifies where its code, model weights, generated molecules, and supplementary material can be accessed, while noting that the training sets are not redistributed.

Data availability

  • The source code, molecule-generation scripts, trained model weights, and generated molecules are available through the listed GitHub and Zenodo links.Analysis scripts are also linked in the availability statement.Source: Data availability · p. 16
  • The unimer and dimer training sets are from Ref. 9 and are not redistributed; supplementary figures, tables, and methodological detail are provided in the ESI.Source: Data availability · p. 16

Declaration of AI-assisted tools

The authors report using Claude for language editing, structural revision, and LaTeX preparation, and state that they take responsibility for the manuscript’s content.

Declaration of AI-assisted tools

A Discrete graph diffusion: full formulation

The appendix gives the discrete graph-diffusion construction, using separate noising of node and edge features and a conditioned neural network to reverse that corruption. Its formulation follows DiGress and classifier-free conditioning rather than introducing a new construction.

A Discrete graph diffusion: full formulation

A.1 Forward process (noising)

  • Molecular graphs encode atom and bond categories as one-hot node and edge features, with bond absence represented explicitly.Nodes and edges are diffused separately because handling the complete-graph space at once is too large.Source: §A.1 · p. 21
  • The forward process applies transition matrices to noisy graph features over T steps, starting from the clean graph at t=0.The transition matrices for nodes and edges are chosen to satisfy a condition based on the training-data marginals.Source: §A.1 · p. 21
  • The proposed node and edge transitions converge toward their training-set marginal distributions rather than a uniform distribution.The node marginal mX is the marginal distribution of atoms; transition probability to a category is proportional to its training-set marginal probability.Source: §A.1 · p. 21

A.2 Reverse process (inference)

  • In reverse diffusion, a neural network takes a noisy graph Gt and predicts a clean graph.The model estimates the reverse step Pθ(Gt−1 | Gt), conditioned on the graph’s global guide value y, which remains constant during noising.Source: §A.2 · p. 22
  • The reverse-probability construction is applied to edges as well as nodes.The transition-probability and neural-network terms in the reverse equations have corresponding roles for both feature types.Source: §A.2 · p. 22
  • Classifier-free denoising combines conditional and unconditional predictions to obtain marginal probabilities from which samples are drawn.The displayed reverse-step factors pair a transition probability with a neural-network prediction of node or edge types given the noisy graph and guide value.Source: §A.2 · p. 22

B Conditional variational autoencoder

The paper benchmarks juniper against a recurrent-neural-network CVAE conditioned on molecular properties. It adapts the reference model and trains two variants on this work’s dataset.

B Conditional variational autoencoder

  • The benchmark CVAE uses recurrent neural networks with an LSTM cell in both encoder and decoder, retaining the hyperparameters reported in Ref. 39.The original-repository validation conditions on three properties instead of the original paper’s five; generated molecules show high correlation with target property values.Source: B Conditional variational autoencoder · p. 22
  • The CVAE implementation adds 0.2 dropout, weight decay of 10−5, a decay rate of 0.97, and early stopping with patience 10.Source: B Conditional variational autoencoder · p. 22
  • On this work’s dataset, the multiple-property CVAE conditions on partition free energy, hydrogen-bond donors and acceptors, and TPSA, while the single-property variant uses partition free energy alone.Both models stopped at epoch 115 because validation loss had ceased to improve, before the 200-epoch maximum.Source: B Conditional variational autoencoder · p. 22

C Why juniper?

The name juniper comes from an analogy between MARTINI’s beads and gin’s ingredients: the method aims to recover the components represented by the beads.

C Why juniper?

  • The naming analogy treats coarse-grained beads and their types as gin, and the mapping as vermouth.Source: C Why juniper? · p. 23
  • Juniper names the method because juniper berries are described as the main flavouring of gin.Source: C Why juniper? · p. 23
  • The analogy focuses the method on recovering the components of the coarse-grained “gin.”Source: C Why juniper? · p. 23

D Tables

The tables describe molecular-graph generation metrics and distributional comparisons for unguided samples from Juniper and CVAE models.

D Tables

  • Table S1 reports validity, uniqueness, novelty, repetition rate, and coverage for state-of-the-art molecular-graph generation models.Training datasets appear in brackets; validity, uniqueness, and novelty are fractions, while repetition rate and coverage follow the main manuscript’s Eqs. 3 and 4.Source: Table S1 · p. 24
  • For Table S1, compare models across the five reported generation metrics while noting each model’s training dataset.Source: Table S1 · p. 24
  • Table S2 reports Jensen–Shannon divergence for ΔG_W7→O and SASA distributions from unguided Juniper and CVAE samples.Its first half compares training and unconditioned distributions, and its second half compares the models’ distributions.Source: Table S2 · p. 24

E Figures

The supplementary figures visualize bead populations and unguided samples in chemical space, alongside basic generation metrics and their dependence on γ.

  • Figure S1 plots training-dataset bead populations against ΔG_W7→O.Source: Figure S1 · p. 25
  • Figure S2 shows unguided samples and the training distribution across octanol–water partition free energy and solvent-accessible surface area.The generated samples are distinguished by model, with 1D kernel density estimates in the margins.Source: Figure S2 · p. 26
  • Figure S3 reports the fractions of valid, unique, and novel generated dimer samples, with γ increasing from top to bottom.Source: Figure S3 · p. 27
  • Figure S4 plots dimer repetition and novelty rates, with γ increasing from top to bottom.The repetition and novelty rates are defined in Eqs. 3 and 5 of the main manuscript, respectively.Source: Figure S4 · p. 28

A B C

The supplementary figures compare generated samples across bead combinations using chemical-space, free-energy, diversity, and structural-similarity views. The accompanying text describes how conditioning shifts sampled free energy and surface area for hydrophobic and intermediate combinations.

  • Figure S6 maps ΔG_W7→O jointly with SASA for C1-C1, Na-Nda, and P3-P4 samples against training-dataset and bead-specific distributions.Its caption states that conditioning effects are visible in all three cases; Figure S5 provides example molecules for the three combinations.Source: Figure S6 · p. 30
  • Figure S7 compares coarse-grained ΔG_W7→O with the mean atomistic ΔG_W7→O; the associated plots include bead-family coloring and second-order polynomial trend lines by γ.Source: Figure S7 · p. 31, A B C · p. 31
  • Figures S9–S10 depict Jensen–Shannon divergence and squared maximum mean discrepancy for generated versus training free-energy distributions at γ = 1.Figure S10 includes the distributions and witness functions for each combination.Source: Figure S9 · p. 33, Figure S10 · p. 34
  • Figure S11 compares shared, training-only, and generated-only functional groups across groups present in at least 50 bead combinations.The network is based on t-SNE decomposition of SMARTS fragments and ranks fragments by similarity.Source: Figure S11 · p. 35
  • Figures S12–S13 show internal chemical diversity across bead combinations and its change with γ, highlighting the largest increases and decreases.Figure S12 indicates that values closer to one represent greater diversity; Figure S13 also shows the mean change across combinations.Source: Figure S12 · p. 36, Figure S13 · p. 37
  • Figure S14 encodes Fréchet ChemNet distance between training and generated molecules, with larger values indicating greater structural differences.The figure varies γ from top to bottom.Source: Figure S14 · p. 38

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