REBIND: Enhancing Ground-state Molecular Conformation Prediction via Force-Based Graph Rewiring

W1. Revision of Manuscript

We thank the reviewer for the comments in improving our manuscript. We have revised our manuscript, addressing your concerns.

Specifically:

• The statement in the Introduction section is modified as below:

To this end, we propose a force-aware self-guidance graph transformer, whereupon the initial conformation predicted from the encoder, a LJ potential-based adjacency matrix is constructed, then utilized in the decoder to predict the final conformation.

• We added the definition of the relative degree in the preliminary section, for clarity.

Additionally, following the reviewer’s suggestion, we will replace Table 5 to our main paper in the subsequent version of our manuscript.

W2 & Q3. Further Ablative Studies

In response to the reviewer’s helpful suggestion, we have extended our ablative studies to include: (1) a comparison between $A_{force}$ and $A_{near}$ in the absence of $A^{bond}$ and $D^{row-sub}$ and (2) additional combinations of these four components. As shown in the table below, incorporating our proposed $A_{force}$ to $A^{bond}$ and $D^{row-sub}$ significantly reduces errors, outperforming those involving $A_{near}$ and $D^{row-sub}$+$A^{bond}$ setting. It is important to note that REBIND is designed to integrate distance and force information in a complementary manner. As such, employing $A_{force}$ alone may yield limited performance improvements due to the absence of distance factors (e.g., atomic pairs with close distances but repulsive forces). However, when combining interatomic forces with $A_{near}$ or $D^{row-sub}$, it achieves substantial improvements in conformation prediction accuracy by effectively modeling both interatomic forces and spatial relationships.

Table 1. Further ablative study of REBIND on the QM9 dataset.

Q2. Computational Burden Assessment

We appreciate the reviewer's insightful comment. To address your concerns, we provide a detailed analysis of both forward and backward computation times (wall-clock time) across all three datasets. Specifically, we compare the time complexity of the following configurations: (1) Utilizing a sole $A^{bond}$, (2) GTMGC [1] that incorporates $A^{bond}$ + $D^{row-sub}$, and (3) REBIND ($A^{bond}$ + $A_{force}$). The results are averaged over 50 gradient steps and calculated separately for each dataset, as they exhibit varying average molecular sizes (8, 15, and 25 for QM9, Molecule3D, and GEOM-DRUGS, respectively).

As expected, the computation time increases from QM9 to GEOM-DRUGS due to the larger average molecular sizes. On average, our method introduces an additional 0.03 seconds of computational overhead compared to GTMGC, due to the calculation of the Lennard-Jones (LJ) potential matrix. For clarity, the summarized results are provided below.

Table 2. Computation time analysis of REBIND (Ours) and baselines. The experiments are conducted on the machine with NVIDIA GeForce RTX 3090 GPU/Intel(R) Xeon(R) Gold 5215 CPU @2.50GHz.

References

[1] GTMGC: Using Graph Transformer to Predict Molecule’s Ground-State Conformation, 24’ ICLR

We sincerely thank the reviewer for the reassessment and positive opinion for our work! We are extremely glad that our responses have met and addressed your concerns.

W1 & Q1. Regarding Problem Setting & Validity of Observations

We thank the reviewer for these thoughtful and constructive comments, which help improve the clarity of our manuscript. Below, we address the reviewer’s concerns: (1) We validate our observations, by clarifying our problem scenario and showing that there is no “sampling bias” for our task which focuses on predicting the most energetically stable conformation, and (2) Empirical verification that REBIND indeed predicts energetically stable conformations.

(1) Concern on Sampling Bias

We would like to clarify that the focus of REBIND is to predict the most stable molecular conformation, specifically the ground-state conformation, where the energy is at its global minimum. In all of our evaluation settings, only a single conformation exists as the ground truth — this is the conformation with the lowest possible energy, in-line with protein structure prediction [1, 2, 3, 4] and crystal structure prediction [5, 6, 7] tasks. Our task diverges from traditional molecular conformer generation tasks, which aim to generate a diverse set of low-energy conformations representing multiple local minima on the energy surface. In contrast, our work focuses solely on identifying the global minimum conformation. Consequently, our evaluation metrics are designed to align with this task objective, quantifying the prediction error relative to the ground-state conformation (the lowest-energy conformation). As such, we believe the “sampling bias” mentioned by the reviewer, which pertains to selecting one ground truth from an ensemble of conformations, does not apply to the context of our work. Since our problem is defined around a single ground-state conformation, this bias is inherently absent.

(2) Verification of Predicting “Low-Energy” Conformations

To further substantiate that REBIND is capable of predicting low-energy conformations, as aligned with its design, we conducted an energy-based validation experiment. Specifically, we computed the average energy difference between the predicted conformations and the ground-state conformations across all baseline methods. The energy calculations were performed using the Merck Molecular Force Field (MMFF) in RDKit.

The results in Table 1 below demonstrate that REBIND consistently achieves the lowest energy differences ($|E_{\text{pred}} - E_{\text{ground}}|$) compared to the baseline methods. This finding underscores REBIND’s ability to reliably predict the most stable conformation, which aligns closely with our task objective of identifying the global energy minimum.

Table 1. Energy difference of predicted conformations through different models from the ground-state.

References

[1] Protein structure prediction with energy minimization and deep learning approaches, 23’ Natural Computing

[2] Constructing effective energy functions for protein structure prediction through broadening attraction-basin and reverse Monte Carlo sampling, 19’ BMC Informatics

[3] Rapid sampling of local minima in protein energy surface and effective reduction through a multi-objective filter, 13’ Proteome Science

[4] Protein structure prediction by global optimization of a potential energy function, 99’ Proceedings of the National Academy of Sciences

[5] Crystal Structure Prediction of Energetic Materials, 23’ Crystal Growth and Design

[6] Minima Hopping Method for Predicting Complex Structures and Chemical Reaction Pathways, 18’ Handbook of Molecular Modeling

[7] Crystal structure prediction accelerated by Bayesian optimization, 18’ Physics Review Materials

W2. Upon the Importance of Ground-state Conformation

As stated by the reviewer, real-world cases often involve ensembles of multiple conformations. However, the ground-state conformation (the current target in which REBIND wishes to predict) is of paramount importance for its influence on physical, chemical and biological properties of molecules. For example, the ground-state conformation serves as the starting point for understanding molecular interactions in protein-ligand binding affinity. Misinterpretation of this conformation leads to errors in docking studies and binding energy calculations [1]. Furthermore, predicting the bulk properties of crystalline and polymeric materials, such as mechanical strength, optical properties, and conductivity, are rooted in the accurate spatial arrangement of molecules in their ground-state conformations. Averaging over ensembles may miss critical packing interactions or fail to capture key structural details [2].

References

[1] Leach et al., Molecular Modelling: Principles and Applications, 2001

[2] Morsch et. al., Organic Chemistry, 2024 (compiled version from LibreTexts)

Thanks for your response. I will raise my score.

W1. Regarding Novelty & Contribution

We thank the reviewer for the constructive feedback. However, we respectfully disagree with the statement that our contributions lie only in feature manipulation. We re-emphasize our contributions below.

First, we make novel observations that previous GNN-based conformer prediction approaches fall short on “low-degree atoms”, showing large discrepancies in performance with respect to high-degree counterparts. This problem is crucial, as the number of low-degree atoms increases in larger and more complex molecular systems. We provide empirical results to this, where we have visualized the proportion of low-degree atoms ($\text{deg}_i^{\text{rel}}\in(0, 0.3]$) with respect to their molecular sizes in Figure 6 in Appendix. Combined with the observation of performance discrepancy between low-degree and high-degree atoms, this suggests that inaccuracies in predicting the positions of low-degree atoms are a major contributor to performance degradation in conformation prediction for large molecules. These observations emphasize the necessity of novel methods, such as REBIND, to address the challenges associated with low-degree atoms effectively.

Second, our approach introduces a force-based graph rewiring framework that goes beyond simple feature addition. Unlike prior works that rely on ground-truth interatomic distances or 3D coordinates, we propose a self-guided mechanism using Lennard-Jones (LJ) potentials to dynamically identify and integrate critical non-bonded interactions. This rewiring is performed in a degree-compensating manner, mitigating the biases against low-degree atoms by selectively augmenting graph connectivity where it is most impactful. Our framework shares similarities in novelty with prior works on edge-rewiring to remedy oversquashing [1, 2, 3], edge-rewiring to enhance graph transformers [4, 5] and augmenting the graph itself for graph transformers [6], which modify input graphs to tackle specific tasks or issues.

Finally, our force-based graph rewiring framework is easily applicable to other backbones other than transformers as well - demonstrating its versatility in bringing performance gains. On Table 4 in Appendix, we have shown that when incorporated with our framework, all baseline methods (GINE, GATv2, GraphGPS) show significant error reductions, up to 15% in RMSD.

References

[1] Locality-Aware Graph Rewiring in GNNs, 24’ ICLR

[2] DRew: Dynamically Rewired Message Passing with Delay, 23’ ICML

[3] FoSR: First-order spectral rewiring for addressing oversquashing in GNNs, 23’ ICLR

[4] Global Self-Attention as a Replacement for Graph Convolution, KDD '22

[5] Path-Augmented Graph Transformer Network, 19’ ICML workshop

[6] Gophormer: Ego-Graph Transformer for Node Classification, 21’ Arxiv preprint

W2. Regarding Task Definition & Evaluation

We understand the reviewer’s point. In response, we clarify that: (1) our work is about ground state conformer prediction, not about molecular conformer generation. (2) we already showcased all metrics used in prior work [1] that share our problem scenario, and (3) validation on adoption of E-RMSD.

(1) Clarification upon the problem scenario

The focus of REBIND is on predicting a single, most stable molecular conformation (the ground-state conformation) from a given 2D molecular graph. This problem setting is different from traditional conformer generation approaches, which aim to generate a diverse set of plausible low-energy conformations. Thus, unlike prior generation works of which was evaluated upon coverage or precision, our evaluation metric is the “prediction error with respect to the single lowest energy conformation”. Our problem definition is first proposed by [1], and its importance has been recently acknowledged in the research community. For instance, our ground-state conformer prediction task is recognized as an important problem in tasks such as crystal structure prediction [2-5], which aims to discover minimum-energy arrangement of constituent atoms in molecular crystals.

(2) Clarification upon utilized metrics

As mentioned, our task definition aligns with the benchmark proposed in GTMGC [1], where the goal is to predict the ground-state conformation characterized by the lowest energy. As such, evaluation metrics like D-MAE, D-RMSE, and C-RMSD (which we have included in our manuscript), which measure the accuracy of a single predicted conformation against the ground-truth conformation, are most suitable for our work.

(3) Validation of E-RMSD

As highlighted above, we consider a scenario where we specifically focus on predicting the single ground-state conformation. Given that our target conformation is the one exhibiting the lowest energy, it is crucial for models to not only minimize geometric errors (e.g., RMSD) but also ensure that the predicted conformation is chemically plausible and energetically stable. The E-RMSD metric addresses this by penalizing models that predict low-error conformations but do not adhere to chemical stability principles. This weighting by energy helps distinguish between: (1) predictions that achieve low geometric error but correspond to chemically unstable conformations and (2) predictions that achieve both low geometric error and energetic stability. In summary, E-RMSD is a metric that incorporates energy-based factors to evaluate the energetic stability along with geometric errors, emphasizing our focus on the prediction of most stable conformation.

References

[1] GTMGC: Using Graph Transformer to Predict Molecule’s Ground-State Conformation, 24’ ICLR

[2] Optimality guarantees for crystal structure prediction, 23’ Nature Computer Science

[3] Crystal structure prediction of energetic materials and a twisted arene with Genarris and GAtor, 21’ CrystEngComm

[4] Crystal structure prediction accelerated by Bayesian optimization, 18’ Physics Review Materials

[5] Crystal Structure Prediction of Energetic Materials, 23’ Crystal Growth and Design

Q1. On Correction of Evaluation Results

We specify each metric’s correction process below.

• RMSD, E-RMSD Correction: We ensure that both RMSD and E-RMSD are corrected for molecular atom perturbations. Specifically, we apply alignment (e.g., correcting for rotations and translations) to the predicted and ground-truth conformations before calculating the RMSD, following the same practice in [1]. This ensures that the RMSD metric is invariant to such transformations.
• MAE and RMSE: For MAE and RMSE, these metrics evaluate the errors in pairwise atomic distances, which are inherently perturbation-invariant. Therefore, following conventional practices and as done in prior works [1], we do not apply additional correction for these metrics.

References

[1] GTMGC: Using Graph Transformer to Predict Molecule’s Ground-State Conformation, 24’ ICLR

Thank you for the response. A few comments:

• I would recommend the authors to simplify the explanation of their method. I believe the method can be explained far more simply and the current explanation in the paper (and the response) is overly complicated.
• I still find it hard to understand the goal of the E-RMSD metric and I would recommend the authors report the energy metrics separately from the RMSD metrics
• From the description in the response the authors are correcting for translation and rotation invariance but not for permutation invariance, is this correct? Similarly, pairwise atomic distances are not inherently permutation invariant do the authors account for that?

We thank the reviewer once more for the additional feedback in improving our manuscript. For the comments you've raised, we will address them as follows.

(1) Simplification of Method Section.

We will revise the method section of our manuscript, with focus on simplicity and conciseness.

(2) Concerns on E-RMSD.

We would like to clarify once more on the validity of our proposed metric, with experimental results for substantiation.

The goal of REBIND is to predict the most stable molecular conformation (i.e. conformation prediction task) - where only a single conformation of energy-wise global minima exists. Note that this is in-line with protein structure prediction[1, 2, 3, 4] and crystal structure prediction tasks [5, 6, 7]. This diverges from traditional conformation generation tasks, which aim to generate a set of conformations of similar energy levels, i.e. set of conformations of energy-wise local minimas. Since our task focuses on a single conformation, all our evaluation metrics focus on quantifying the errors between prediction (only a single prediction is made for a single input) and ground truth. E-RMSD incorporates “energy” into the quantification of geometrics errors, penalizing predictions that are energetically invalid. We emphasize once again, that since we consider a single output and single ground truth conformation for each 2D molecular graph, and thus “optimizing the model to learn weighted geometric average of different conformations” does not happen.

To (1) substantiate the assertion that our work focuses on predicting the most stable conformation and (2) to acknowledge the reviewer’s suggestion, we present the energy-based results below in Table 1. Specifically, we computed the average energy difference between predicted conformations and ground-state conformations across all baselines methods. The energy calculations were performed using the Merck Molecular Force Field(MMFF) in RDKit. It can be seen that REBIND consistently achieves to predict most stable conformations compared to baselines. We will add the following results in our revised manuscript.

(3) Concerns on Permutation.

We verify that our method along with GNN-based methods, maintain the ordering of atoms. Thus, it does not need corrections of ordering (permutation) during the computation of MAE and RMSE. We acknowledge the confusion, and will revise our manuscript to clarify this.

Table 1. Energy difference of predicted conformations through different models from the ground-state.

References

[1] Protein structure prediction with energy minimization and deep learning approaches, 23’ Natural Computing

[2] Constructing effective energy functions for protein structure prediction through broadening attraction-basin and reverse Monte Carlo sampling, 19’ BMC Informatics

[3] Rapid sampling of local minima in protein energy surface and effective reduction through a multi-objective filter, 13’ Proteome Science

[4] Protein structure prediction by global optimization of a potential energy function, 99’ Proceedings of the National Academy of Sciences

[5] Crystal Structure Prediction of Energetic Materials, 23’ Crystal Growth and Design

[6] Minima Hopping Method for Predicting Complex Structures and Chemical Reaction Pathways, 18’ Handbook of Molecular Modeling

[7] Crystal structure prediction accelerated by Bayesian optimization, 18’ Physics Review Materials

The authors' comment:

(3) Concerns on Permutation. We verify that our method along with GNN-based methods, maintain the ordering of atoms. Thus, it does not need corrections of ordering (permutation) during the computation of MAE and RMSE. We acknowledge the confusion, and will revise our manuscript to clarify this.

is makes the understanding of the method even more confusing to me. In particular: what is the output of the model? How is that translated into coordinated for the molecule conformation?

Even if the output of some of the baselines is permutation equivariant (which I find weird but really depends on the questions asked above), the conformations of the molecule are not permutation equivariant, therefore the evaluation metric should account for this.

We thank the reviewer for the additional feedback. To clarify, when we state that our method and the baseline methods “maintain the ordering of atoms,” this refers to the fact that the atom ordering in the input graph is preserved throughout the forward pass, and the predicted output aligns with this ordering. We provide additional clarification below.

The input to the model includes:

• Adjacency matrix ($\mathbf{A} \in \mathbb{R}^{N \times N}$): represents bond structure where element (i, j) is 1 when there exists a bond between node $i$ and $j$, and 0 otherwise.
• Node features ($\mathbf{X} \in \mathbb{R}^{N \times d}$): represents features of a molecule with $N$ atoms where $i$ th row corresponds to node $i$.

The output of the model is:

• Predicted Conformation matrix ($\hat{C} \in \mathbb{R}^{N \times 3}$): where each row $\hat{C_i} \in \mathbb{R}^{3}$ represents the predicted 3D coordinates of node $i$, and $\hat{C_i}$ and $\mathbf{X}_i$ reference the same node $i$. Note that, the ground truth conformation $C$ is also already aligned with the ordering of $\mathbf{X}$, and consequently, $\hat{C_i}$ and $C_i$ corresponds to the same node $i$.

From the predicted conformation $\hat{C}$ of a single molecule, we calculate MAE and RMSE with respect to the ground-truth interatomic distance $D_{ij} = ||C_i - C_j||_2$. The errors with respect to each node $i$ is defined as follows:

$MAE(i)=\frac{1}{N-1}\sum_{ j\in \mathcal V\backslash \{i\} } |\hat D_{ij} - D_{ij}|_1,$

$RMSE(i) = \sqrt{\frac{1}{N-1}\sum_{j \in \mathcal V\backslash \{i\}} ( \hat D_{ij} - D_{ij} )^2},$

where $\hat D_{ij} = ||\hat C_i - \hat C_j||_2$ is a predicted distance. During the whole process, the assignment of atom to node (i.e., Nitrogen is assigned to node $i$) is the same between ground truth conformation $C$ and predicted conformation $\hat{C}$. Thus, it does not require “re-ordering” to match the ordering of atoms, or assignments.

I thank the authors for the additional reply but this still does not answer my original comment. The critical component that the authors seem to be missing from the replies is that molecules often have several symmetries. For example a simple molecule like benzene has a symmetry group with 12 elements (12 different permutations of the atoms resulting in the same molecule). This leads back to my question/comment:

1. Is the method permutation invariant? If not where is the permutation invariance broken?
2. The possible different permutations should be taken into account during evaluation to compute all the metrics.

We thank the reviewer for providing specific examples. We believe that there was a misunderstanding between us, and the example provided by the reviewer has alleviated this confusion.

Q1. GNNs, including graph transformers with Laplacian encoding are inherently permutation equivariant [1, 2]. Our model, which is composed of transformer architecture with Laplacian encoding and adjacency matrix, is also permutation equivariant as well. The permutation invariance breaks when incorporating adjacency matrix as a bias term.

Q2-1. Metrics involving pairwise distances (MAE, RMSE)

We believe the reviewer’s question can be interpreted in the two following ways, regarding MAE and RMSE. 1) that the ordering of different atoms (i.e. Carbon, Nitrogen) that form the molecule will affect the results of MAE and RMSE. 2) that the symmetry of molecules (i.e. among the ordering that preserve the adjacency matrix of molecules, like the example of benzene with 12 symmetries) may affect the results of MAE and RMSE. We clarify upon both 1) and 2) below.

Scenario #1 - General permutation that changes the adjacency matrix

In our evaluation setting, permutations of atoms that change the adjacency matrix is not of consideration, for such ordering is fixed during evaluation. Consider the molecule H-C≡N-H. The ordering of the atoms can vary, but only a single input graph and atom ordering exists in our evaluation scheme, given by the RDKit’s .GetAtoms(). For example, the ordering is fixed to be C, N, H (connected to C), H (connected to N). Since all our baselines and REBIND preserves the ordering of geometrically different atoms, the output conformation aligns with the ground truth conformations, and thus does not require additional “alignment”. Thus, during evaluation, we only need to focus on handling molecular symmetries - alternations of atom orderings that do not alter the geometry nor the adjacency matrix - which is specified below.

Scenario #2 - molecular symmetry

In permutation scenarios that do not alter the adjacency matrix - i.e., the symmetry in benzene, we provide detailed examples to demonstrate that, for RMSE and MAE, the results are the same.

For simplicity, we consider ethane ($C2H6$).

Ethane has two carbons ($C_1$ and $C_2$) connected by a single bond.

• Each carbon is bonded to three hydrogens:
• $C_1$ has $H_1$, $H_2$, $H_3$
• $C_2$ has $H_4$, $H_5$, $H_6$
• The order of atoms are: $C_1, C_2, H_1, H_2, H_3, H_4, H_5, H_6$.

By symmetry, the hydrogens $H_1$ and $H_2$ around $C_1$, or $H_4$ and $H_6$ around $C_2$, can be swapped without changing the molecule’s geometry.

Let’s say the original ground-truth pairwise distance matrix $D$ is as follows.

Then given the molecular graph, the predicted pairwise distance matrix $\hat D$ can be as follows.

Consider a scenario where the hydrogen mentioned above are swapped; $H_1$ ↔ $H_2$, $H_4$ ↔ $H_6$, in the prediction matrix. The swapping results in the matrix $\hat D_{swapped}$ is as below.

Now, when we compute the MAE for both cases, we can see that the MAE is the same for both cases of $MAE(\hat D - D)$ and $MAE(\hat D_{swapped} - D)$, since the errors are averaged in whole (see the definition of MAE in our previous response). Thus, as demonstrated from the exemplar, we can see that for such symmetries that do not alter the adjacency matrix of the molecular graph, the MAE remains the same. This holds for RMSE as well, which is computed in a similar manner.

As the discussion period is coming to a close, we await your response.

We have carefully addressed your concerns, and hope our responses have resolved any issues or questions.

If you have any additional comments or concerns, please don't hesitate to let us know.

Sincerely,

Authors

W1. Computational Burden Assessment

We thank the reviewer for this valuable observation. To address this, we report the wall-clock computation time during the forward and backward pass of the training process (including gradient computation) in our implementation. We compare the time complexity between 1. incorporating $A^{bond}$ alone, 2. $A^{bond}$ + $D^{row-sub}$ (i.e., GTMGC [1]), 3. $A^{bond}$ + $A_{near}$ (i.e., adjacency matrix constructed via close distance), and 4. REBIND. Note that, the results are averaged upon 50 gradient update steps for each dataset, for they have varying molecular sizes (8, 15, and 25 average molecular size for each dataset).

The results are summarized in Table 1 below. On average, our method only sacrifices an additional 0.03 seconds of computational overhead for computing the Lennard-Jones (LJ) potential matrix, compared to the GTMGC.

Table 1. Wall-clock time analysis. The comparison is conducted on the machine with NVIDIA GeForce RTX 3090 GPU/Intel(R) Xeon(R) Gold 5215 CPU @2.50GHz.

References

[1] GTMGC: Using Graph Transformer to Predict Molecule’s Ground-State Conformation, 24’ ICLR