cs.LGApr 15, 2026
SaveLEGO-MOF: Equivariant Latent Manipulation for Editable, Generative, and Optimizable MOF Design
Organizations: School of Science and Engineering (SSE), The Chinese University of Hong Kong, Shenzhen (CUHKSZ), Shenzhen, China. · School of Data Science (SDS), The Chinese University of Hong Kong, Shenzhen (CUHKSZ), Shenzhen, China.
Abstract
Metal-organic frameworks (MOFs) are highly promising for carbon capture, yet navigating their vast design space remains challenging. Recent deep generative models enable de novo MOF design but primarily act as feed-forward structure generators. By heavily relying on predefined building block libraries and non-differentiable post-optimization, they fundamentally sever the information flow required for continuous structural editing. Here, we propose a target-driven generative framework focused on continuous structural manipulation. At its core is LinkerVAE, which maps discrete 3D chemical graphs into a continuous, SE(3)-equivariant latent space. This smooth manifold unlocks geometry-aware manipulations, including implicit chemical style transfer and zero-shot isoreticular expansion. Building upon this, we introduce a test-time optimization (TTO) strategy, utilizing an accurate surrogate model to continuously optimize the latent graphs of existing MOFs toward desired properties. This approach systematically enhances carbon capture performance, achieving a striking average relative boost of 147.5% in pure CO2 uptake while strictly preserving structural validity. Integrated with a latent diffusion model and rigid-body assembly for full MOF construction, our framework establishes a scalable, fully differentiable pathway for both the automated discovery, targeted optimization and editing of functional materials.
Figures & tables
Figure 1: Overview of the LEGO-MOF framework. The pipeline is unified by four highly modular stages: (L)atent Representation: Complex MOFs from databases are deconstructed into metal nodes and organic linkers. An SE(3)-equivariant autoencoder compresses variable-length linkers (including virtual anchor atoms) into a fixed-dimensional, disentangled continuous latent space ( ). (E)diting: This continuous manifold enables zero-shot explicit and implicit structural manipulations, such as functional group style transfer and geometric isoreticular expansion, via targeted latent perturbations. (G)eneration: To explore uncharted chemical space, an equivariant latent diffusion model unconditionally samples from a Gaussian prior ( ) to generate novel linker molecules, which are then integrated via rigid-body assembly into de novo MOFs. (O)ptimization: A periodic graph neural network (PBC GNN) trained on coarse-grained latent graphs serves as a differentiable surrogate. Test-Time Optimization (TTO) leverages this surrogate to perform direct gradient ascent on the latent codes of existing MOFs, systematically maximizing target properties (e.g., CO 2 uptake) for downstream validation.
Figure 2: Continuous atomic manipulation via LinkerVAE. Structural evolution of a representative organic linker under targeted atom count perturbations ( ). The three rows represent three different linkers, and the light-blue atoms denote anchor sites (He).
Figure 3: Latent style transfer between structurally distinct linkers. The panel illustrates the successful disentanglement of physical geometry ( ) and chemical features ( ) within the LinkerVAE latent space. (1) Parent 1 ( ) , serving as the source of the initial geometric scaffold, characterized by directly-connected carbon rings structure (highlighted by the orange box). (2) Parent 2 ( ) , the original structural source of the twisted geometric backbone and the iodine substituent. (3) Generated Linker 1 ( ) , a hybrid structure that strictly preserves the 3D spatial backbone of Parent 1 while successfully inheriting the chemical ”style”— an iodine functional group (red circle)—via the interpolated chemical latent representation. (4) Generated Linker 2 ( ) , which maintains the distinct twisted, azo-linked spatial geometry of Parent 2 (highlighted by the blue box) but exhibits an altered, averaged chemical functionalization. Terminal light-blue atoms represent the virtual anchor atoms (He) required for downstream MOF assembly.
Figure 4: Zero-shot isoreticular expansion via continuous latent scaling. a Structural transformation of a representative MOF before (left) and after (right) the targeted 1.4 expansion process. By anisotropically stretching the spatial latent variable ( ) along the principal axes and proportionally scaling the decoded atom count, the model generates elongated linker analogues (blue inset) that seamlessly extend the structural backbone. The strict geometric optimization guarantees perfectly preserved coordination connectivity at the metal-organic interface (red inset). b Distribution of percentage changes in unit cell volume versus gravimetric accessible volume (AV, cm 3 /g). The logarithmic point density highlights a robust positive trend where volumetric enlargement translates into significant gains in storage capacity. c Log-scaled hexbin plot illustrating the inverse correlation between framework density reduction and the enhancement of AV fraction. The high-density cluster in the upper-left quadrant confirms the consistent generation of lighter, high-porosity architectures via latent space manipulation.
Figure 5: Surrogate model performance on MOF carbon capture properties. (a,b) Hexbin parity plots for the validation set, comparing ground truth versus predicted CO 2 uptake in (a) pure CO 2 gas and (b) binary CO 2 /N 2 gas. The density-colored points demonstrate strong correlation and predictive accuracy ( , SRCC, and MAE values are indicated). (c,d) Overlapping distribution plots showing the fidelity of the model’s predictions in capturing the underlying statistical distribution across the validation set.
Figure 6: Test-Time Optimization (TTO) results for MOF property enhancement. (a,b) Categorized density scatter plots for general TTO, showing original versus optimized property values. Points are colored by density for improved (blue-green) and decreased (red-orange) performance. (a) Optimization for pure CO 2 uptake (Exp. 1). (b) Optimization for binary CO 2 /N 2 condition (Exp. 2). Statistical improvements are detailed in Supplementary Table 5. (c) Ceiling breakthrough analysis for pure CO 2 uptake (Exp. 3), focusing on the top 0.33% of original MOFs (1000 samples). The dashed line represents the original dataset’s maximum performance, which the TTO process successfully surpassed.
Figure 7: Representative cases of TTO pushing the performance limits. The top row displays original MOF structures from the high-performance tail of the training set, while the bottom row shows their corresponding optimized versions. Colored boxes highlight the specific structural refinements identified by the model. These micro-structural adjustments result in substantial increases in CO 2 uptake (e.g., from 4.38 to 7.21 mol/kg), demonstrating the capability of the latent manifold to suggest out-of-distribution structural motifs that extend beyond the distribution of the training dataset.
Figure 8: Topology and interpretability of the global latent manifold. UMAP dimensionality reduction of the 256-dimensional global latent features ( ) extracted from the surrogate model for pure CO 2 uptake. (a) A 3D projection of the entire latent manifold, colored by CO 2 uptake, revealing distinct clusters and a clear property gradient. (b) The same 3D manifold, with points colored by their HDBSCAN cluster assignments. Each cluster is annotated with its dominant topological profile (Node connectivity:Linker count), demonstrating that the latent space naturally organizes MOFs into distinct topological families. (c–f) 2D UMAP projections for four representative clusters. Within each topologically homogeneous cluster, the latent space is smooth, exhibiting clear and consistent property gradients (CO 2 uptake) that enable robust targeted optimization.
Figure 9: Distributional alignment of LLDM-generated linkers. (a) 2D UMAP projection comparing the continuous latent space of the original training set (gray) against the unconditionally generated latents (red). The high degree of overlap indicates that the LLDM successfully captures the true data distribution without mode collapse. (b) Density distributions of four key chemical descriptors (Molecular Weight, Number of Rings, Rotatable Bonds, and Topological Polar Surface Area) comparing the training dataset against the valid generated molecules.
Figure 10: Representative de novo generated organic linkers. Three uniquely generated and novel linker molecules produced unconditionally by the LLDM. For each sample, both the decoded 3D point cloud configurations (bottom) and their corresponding 2D chemical graphs (top) are displayed. The model demonstrates the capability to generate sophisticated topological features, including complex fused ring systems and diverse functional groups, while maintaining the precise coordination geometries required for downstream MOF assembly.
| Metric | Mean | Std. Dev. |
|---|---|---|
| Unaligned RMSD (Å) | 0.5662 | 0.9115 |
| Kabsch Aligned RMSD (Å) | 0.5662 | 0.9115 |
| Atom Type Accuracy (w/o anchors) | 94.84% | 8.75% |
| Atom Type Accuracy (all atoms) | 95.26% | 8.10% |
| Anchor Count Difference Error | 0.0000 | 0.0000 |
| Anchor Spatial Deviation (Å) | 0.3072 | 0.5386 |
Table S1 : Reconstruction performance of LinkerVAE on the test set. Metrics are calculated between the ground truth and the sampled reconstruction. Anchor refers to the virtual connection atoms (He) required for MOF assembly.
| (Atoms) | Samples | Validity | RMSD (Å) | Anchor Num Diff | Anchor Dist (Å) |
|---|---|---|---|---|---|
| -3 | 1000 | 100% | – | 0.031 0.198 | 0.476 0.692 |
| -2 | 1000 | 100% | – | 0.052 0.289 | 0.437 0.528 |
| -1 | 1000 | 100% | – | 0.014 0.200 | 0.458 0.678 |
| 0 (Recon) | 1000 | 100% | 0.279 0.360 | 0.016 0.171 | 0.377 0.596 |
| +1 | 1000 | 100% | – | 0.014 0.210 | 0.442 0.682 |
| +2 | 1000 | 100% | – | 0.031 0.346 | 0.453 0.678 |
Table S2 : Evaluation of linker structures under atom count perturbations. represents the induced change in the total number of atoms during the decoding phase. All evaluations preserve a 100% chemical validity.
| Generated Set | Geometric Parent | Validity | Shape Similarity | Anchor Dist (Å) |
|---|---|---|---|---|
| Gen 1 (using ) | Parent 1 | 100% | 0.646 | 0.283 |
| Gen 2 (using ) | Parent 2 | 100% | 0.669 | 0.253 |
Table S3 : Quantitative metrics for structural style transfer. was mixed evenly (0.5/0.5) between parent pairs, while was locked to either Parent 1 (Gen 1) or Parent 2 (Gen 2). Shape similarity is calculated via spatial Tanimoto overlap. Anchor distance tracks the spatial shift of connection nodes relative to the geometric parent.
| Property Metric | Mean Change (%) | Std. Dev. | Min Change (%) | Max Change (%) |
|---|---|---|---|---|
| Unit Cell Volume | +38.66 | 11.92 | +2.89 | +75.01 |
| Framework Density | –11.45 | 10.82 | –48.01 | +25.45 |
| Accessible Vol. Fraction | +5.48 | 24.20 | –47.98 | +164.89 |
| Gravimetric AV (cm 3 /g) | +23.32 | 43.27 | –49.45 | +289.32 |
| Pore Channels Diff ( ) | 0.00 | 0.00 | 0.00 | 0.00 |
Table S4 : Quantitative physical property changes following zero-shot latent expansion (1.4 ). Metrics represent the relative percentage shift from the original MOF to the optimized, expanded generated structure across all valid samples (88.15% of total attempts). The absolute preservation of channel counts confirms the strict retention of topology.
| Exp. | Property Target | Valid Samples | Improved Ratio (%) | Net Mean Rel. Boost (%) | Net Median Rel. Boost (%) |
|---|---|---|---|---|---|
| 1 | Pure CO 2 Uptake | 520 | 74.0 | +147.5 | +23.6 |
| 2 | Binary CO 2 /N 2 Uptake | 492 | 70.3 | +94.8 | +13.7 |
| 3 | Top 0.33% CO 2 Uptake | 908 | 44.9 | +1.1 | –2.6 |
Table S5 : Summary of Test-Time Optimization (TTO) performance across three experiments. Metrics are reported for MOFs that passed filtering criteria. Crucially, the mean and median relative boosts represent the net overall shift , calculated across all valid samples in the given experiment (explicitly accounting for both performance improvements and degradations).
| Metric | Count | Percentage |
|---|---|---|
| Total Generated Samples | 1,000 | 100.00% |
| Valid (V) | 975 | 97.50% |
| Unique (U) | 777 | 79.69% (of Valid) |
| Novel (N) | 638 | 82.11% (of Unique) |
| Overall Unique & Novel | 638 | 63.80% (Absolute) |
Table S6 : Evaluation metrics for generated linkers. Validity (V) denotes the fraction of generated samples that are successfully decoded into chemically valid molecules. Uniqueness (U) is defined as the proportion of non-redundant structures within the valid set, while Novelty (N) quantifies the percentage of unique molecules not present in the training database.
| Dataset Metric | Count | Ratio | Atom Type ( ) | Total Count | Global Ratio (%) |
| Total MOFs | 308,961 | – | (H) | 13,080,711 | 22.47 |
| Total Building Blocks (BBs) | 2,175,295 | 100% | (He) | 11,461,142 | 19.69 |
| Organic Linkers | 1,511,159 | 69.5% | (C) | 20,952,837 | 35.99 |
| Metal Nodes | 664,136 | 30.5% | (N) | 1,789,812 | 3.07 |
| Metal Species Coverage | V, Cr, Ni, Cu, Zn, Ba | (O) | 8,211,113 | 14.11 | |
| Non-Metal Species | H, He, C, N, O, F, P, S, Cl, Br, I | Others ( ) | 2,751,210 | 4.67 | |
Table S7 : Overview of the MOF building block dataset and global atom distribution. The dataset is decomposed into organic linkers and metal nodes. Atomic species are represented by their atomic number ( ). Note that (Helium) denotes virtual anchor atoms, and denotes Hydrogen.
| Rank | Node Core Composition | Common Structural Motif | Frequency | Percentage (%) |
| 1 | { Zn: 2 } | Zn 2 dimer | 189,327 | 28.55 |
| 2 | { Cu: 2 } | Cu 2 paddlewheel | 177,294 | 26.73 |
| 3 | { V: 1 } | V Mononuclear | 91,484 | 13.80 |
| 4 | { O: 1, Zn: 4 } | Zn 4 O cluster (basic zinc acetate) | 88,982 | 13.42 |
| 5 | { Ni: 1 } | Ni Mononuclear | 48,952 | 7.38 |
| 6 | { Ba: 1 } | Ba Mononuclear | 39,050 | 5.89 |
Table S9 : Classification and frequency of metal node species. Unique node species are identified by their core composition (excluding virtual He and H atoms). Frequencies reflect the occurrences across the 308,961 MOFs in the dataset.
| Neighboring Nodes ( ) | Frequency (Linkers) | Percentage (%) |
|---|---|---|
| 2 | 1,321,408 | 87.44 |
| 3 | 5,701 | 0.38 |
| 4 | 142,170 | 9.41 |
| 6 | 41,873 | 2.77 |
| Others | 7 | 0.01 |
| Mean Connectivity | 2.30 nodes per linker | |
Table S10 : Structural connectivity statistics. Distribution of the number of neighboring metal nodes coordinated to a single organic linker across the framework library.
Figure S1 : Residual analysis of the surrogate property predictor. Scatter plots showing the prediction residuals (Predicted Ground Truth) as a function of the ground truth gas uptake for a pure (unary) and b binary mixture (0.15:0.85 bar). The dashed horizontal line represents zero error, while the dotted grey lines indicate the standard deviation bounds. The uniform distribution of residuals around zero across the entire range of uptake values indicates that the SchNet-based surrogate model is free from significant systematic bias and maintains high reliability even in high-performance regimes.
Figure S2 : Detailed correlation between volumetric expansion and accessible volume (AV) enhancement. Jointplot illustrating the relationship between the percentage change in unit cell volume and the resulting change in gravimetric AV ( ) following 1.4 latent geometric scaling. The scatter plot is overlaid with Kernel Density Estimation (KDE) contours, highlighting the high-density clustering of successfully expanded MOFs. The marginal histograms show the distribution for each metric, confirming a robust positive correlation where controlled macroscopic expansion leads to significant improvements in porous capacity across a diverse range of topologies.
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