SlowGuess/NeurIPS_2023_OCR huggingface dataset:
- Physics-based potentials calculate binding affinity based on energy functions designed by experts. We consider four popular methods: Glide [8], AutoDock $_{\text{vina}}$  [7], DrugScore $_{2018}$  [6], and MM/GBSA [21]. Among these methods, MM/GBSA is the most accurate but computationally expensive. It takes one hour to calculate energy for just one complex on a 64-core CPU server.
- Unsupervised models. Since unsupervised learning is relatively underexplored in this area, we implement two unsupervised EBMs using the same encoder architecture as NERE but different training objectives. The first baseline is the standard Gaussian DSM (see Section 3.2). The second baseline is contrastive learning [4]. For each crystal structure  $(\mathbf{A},\mathbf{X})$ , we apply  $K$  random rigid transformations to obtain  $K$  perturbed protein-ligand complexes  $\mathbf{X}_1,\dots ,\mathbf{X}_K$  as negative samples. Suppose  $-E(\mathbf{A},\mathbf{X}_i)$  is the predicted energy for  $\mathbf{X}_i$ , we train our EBM to maximize the likelihood of the crystal structure  $\exp (-E(\mathbf{A},\mathbf{X})) / \sum_{i}\exp (-E(\mathbf{A},\mathbf{X}_{i}))$ .
Sup	Crystal	Docked
IGN	✓	0.837	0.780
TankBind	✓	n/a	0.824
PLANET	✓	0.811	0.799
Glide	×	0.467	0.478
Autodockvina	×	0.604	0.578
DrugScore2018	×	0.602	0.540
MM/GBSA	×	0.647	0.629
Contrastive	×	0.625.002	0.623.002
Gauss DSM	×	0.638.017	0.632.016
NERE DSM	×	0.656.012	0.651.011
Sup	Crystal	Docked
FANNab	✓	0.325.014	0.326.019
FANNtransfer	✓	0.350.033	0.359.039
AP_PISA	×	0.323	0.144
ZRANK	×	0.318	0.163
PYDOCK	×	0.248	0.164
ESM-1v	×	0.038	0.038
ESM-IF	×	0.024	0.025
Contrastive	×	0.308.037	0.304.021
Gauss DSM	×	0.335.038	0.330.022
NERE DSM	×	0.361.051	0.360.051
Table 1: Pearson correlation on PDBBind (left) and SAbDab test sets (right). We report standard deviations for all models implemented in this paper. "Sup" indicates whether a method is supervised or unsupervised. "Crystal"/"Docked" means a model is evaluated on crystal or docked structures. The performance of TankBind on crystal structures is not available because it performs docking and affinity prediction simultaneously in one model.
- Supervised models. Most of the existing deep learning models for binding affinity prediction belong to this category. They are typically trained on the entire PDBBind database with over 19000 binding affinity data points. The purpose of this comparison is to understand the gap between supervised and unsupervised models when there is abundant labeled data. For this purpose, we include three top-performing methods (IGN [13], TankBind [19], and PLANET [43]) taken from a recent survey by Zhang et al. (2023).
Results (crystal structures). In our first experiment, we evaluate all methods on the crystal structure of protein-ligand complexes. As shown in Table 1 (left, crystal column), NERE outperforms all physics-based and unsupervised models. We note that NERE is orders of magnitude faster than the second best method MM/GBSA (100ms v.s. 1hr per complex). As expected, the three supervised models perform better than NERE because they are trained on 19000 labeled affinity data points. Nonetheless, NERE recovers almost  $80\%$  of their performance without using any labeled data, which implies that binding affinity is closely related to the geometric structure of an input complex.
Results (docked structures). The previous evaluation considers an idealistic setting because crystal structures are not available in real-world virtual screening projects. To this end, we use AutoDock Vina [7] to dock each ligand in the test set to its binding pocket and evaluate all methods on docked protein-ligand complexes. As shown in Table 1 (left, docked column), NERE consistently outperforms the baselines in this setting. Its performance is quite close to the crystal structure setting because docking error is relatively low (median test RMSD=3.55). In summary, these results suggest that NERE has learned an accurate binding energy function useful for structure-based virtual screening.
5.2 Antibody-Antigen Binding
Data. Our training and test data come from the Structural Antibody Database (SAbDab) [31], which contains 4883 non-redundant antibody-antigen complexes. Our test set is a subset of SAbDab (566 complexes) that has binding affinity labels. Our training set has 3416 complexes (without binding affinity labels) after removing antigen/antibody sequences that appear in the test set. Our validation set comes from Myung et al. [23], which has 116 complexes after removing antibodies or antigens overlapping with the test set.
Baselines. Similar to the previous section, we consider three sets of baselines for comparison:
- Physics-based models. We consider eight biophysical potentials implemented in the CCharPPI webserver [22] and report the top three models (full results are listed in the appendix).
- Unsupervised models. Besides contrastive learning and Gaussian DSM baselines used in Section 5.1, we consider two state-of-the-art protein language models (ESM-1v [20] and ESM-IF [11]). These two models have been successful in predicting mutation effects or binding affinities for
general proteins and we seek to evaluate their performance for antibodies. The input to ESM-1v is the concatenation of an antibody and an antigen sequence and we take the pseudo log-likelihood of antibody CDR residues as the binding affinity of an antibody-antigen pair. The input to ESM-IF is the crystal structure of an antibody-antigen complex and we take the conditional log-likelihood of antibody CDR residues given the backbone structure as its binding affinity.
- Supervised models. We also compare NERE with supervised models trained on external binding affinity data. The model is based on the same frame-averaging neural network (FANN) [28] and pre-trained ESM-2 residue embeddings used by NERE. Since all labeled data from SAbDab are in the validation and test sets, we draw additional data from an protein-protein binding mutation database (SKEMPI) [12]. We obtain 5427 binding affinity data points after removing all complexes appeared in the test set, from which 273 instances are antibody-antigen complexes. We explore two training strategies: 1) training on 273 antibody-antigen data points only ( $\mathrm{FANN}_{\mathrm{ab}}$ ); 2) training on 5427 data points first and then finetuning on 273 antibody-antigen data points ( $\mathrm{FANN}_{\mathrm{transfer}}$ ).
Results (Crystal structure). We first evaluate all models on crystal structures and report the Pearson correlation between true and predicted binding energy. As reported in Table 1 (right, crystal column), NERE significantly outperforms physics-based methods and protein language models. ESM-1v and ESM-IF have almost zero correlation because they only model the likelihood of the antibody sequence. ESM-IF models the conditional likelihood of a sequence given its structure while NERE models the likelihood of the entire protein complex. It is crucial to model the likelihood of structures because binding affinity depends on the relative orientation between an antibody and an antigen. We also find that NERE outperforms the supervised models, even with a small gain over  $\mathrm{FANN}_{\mathrm{transfer}}$  trained on 5427 labeled data points. Overall, these results highlight the advantage of unsupervised learning in low-data regimes.
Results (Docked structure). In our second experiment, we evaluate all methods on docked complexes to emulate a more realistic scenario. We use the ZDOCK program [26] to predict the structure of all antibody-antigen complexes in the test set (its median RMSD is 19.4). As reported in Table 1 (right, docked column), our model still outperforms all the baselines in this challenging setting. The performance of EBMs are quite robust to docking errors regardless of the training algorithm (contrastive learning, Gaussian DSM, or NERE DSM). In summary, our model is capable to predict antibody-antigen binding even when crystal structure is not available.
5.3 Ablation Studies and Visualization
Visualizing energy landscape. First, we study how the learned energy changes with respect to ligand orientations. Given an input complex, we perform a grid search of ligand rotation angles  $\omega = [\omega_{1},\omega_{2},\omega_{3}]$  and plot the predicted energy for each pose. As 3D contour plots is hard to visualize, we decompose it into three 2D contour plots by fixing one of the three axis  $(\omega_{1},\omega_{2},\omega_{3})$  to zero. Ideally, the crystal structure  $(\omega = [0,0,0])$  should be the local minimum because it is physically the most stable conformation. Figure 3a-b show contour plots for small molecules and antibody-antigen complexes. We find that their crystal structures are located relatively near the local minimum.
Visualizing interaction energy. Since the predicted energy is a summation of all pairwise interactions  $\sum_{i,j:d_{i,j} < 20\AA}\phi_o(h_i,h_j)$ , we seek to visualize the contribution of different residues to predicted binding energy. Figure 3c is one example. Each row and column in this heatmap represent an epitope residue and an antibody CDR residue, respectively. Each entry in the heat map is the interaction energy between two residue  $\phi_o(h_i,h_j)$ . An entry is left blank if the distance  $d_{i,j} > 20\AA$ . Interestingly, we find that the model pays the most attention to CDR-H3 and CDR-L3 residues. In most test cases, their energy is much lower than other CDR residues. This agrees with the domain knowledge that CDR-H3 and CDR-L3 residues are the major determinant of binding.
Ablation studies. Lastly, we perform four ablation studies to understand the importance of different model components. We first replace our SE(3)-invariant protein encoder with a non-invariant neural network where atom 3D coordinates are directly concatenated to atom features in the input layer. As shown in Figure 3d, the model performance drops significantly when the encoder is not SE(3) invariant. In addition, we run NERE DSM by removing the rotation  $\ell_r$  or translation DSM term  $\ell_t$  from the objective  $\ell_{\mathrm{DSM}} = \ell_t + \ell_r$ . We find that removing either of them substantially hurts the model
<sup>1</sup>We have also tried other neural network architectures like 3D convolutional neural networks and graph convolutional networks, but they did not achieve better performance. Their results are shown in the appendix.
C
Figure 3: a-b) Visualizing learned energy landscape for small molecules and antibodies. We perform a grid search of ligand rotation angles  $\omega = [\omega_{1},\omega_{2},\omega_{3}]$  and plot the predicted energy as 2D contour plots with one of the axis  $\omega_{i}$  fixed. The crystal structure is at the origin (marked with *) and supposed to be the local minimum of the energy landscape. c) The heat map of our learned energy function. Each entry represents the binding energy between residues  $(i,j)$  (the darker the stronger). An entry is left blank (grey) if their distance  $d_{ij} > 20\AA$ . Our model correctly puts more attention to CDR-H3/L3 residues. d) Ablation studies on different model components.
performance. Therefore, it is crucial to consider both rotation and translation degree of freedom in unsupervised binding energy prediction. Lastly, we train NERE DSM by replacing ESM-2 amino acid embedding with one-hot encoding (this change only applies to the antibody model). We find that ESM-2 is helpful but NERE DSM is still able to infer binding affinity with reasonable accuracy after removing language model features.
In this paper, we developed an energy-based model for unsupervised binding affinity prediction. The energy-based model was trained under SE(3) denoising score matching where the rotation score was predicted by Neural Euler's Rotation Equation. Our results show that the learned energy correlates with experimental binding affinity and outperforms supervised models in the antibody case.
Limitations. Indeed, there are many ways to improve our model. Our current model for antibodies only considers their backbone structure (C $\alpha$  atoms) and ignores all side-chain atoms. While the model for small molecules include all atoms, we have not incorporated the flexibility of side chains in a protein or rotatable bonds in a ligand. Considering their interaction is crucial for protein-ligand binding prediction. Our future work is to extend our encoder with all-atom structures and our NERE DSM algorithm with side-chain torsional rotations.
Broader Impacts. Our method is applicable to a wide range of biological areas like drug discovery, immunology, and structural biology. We believe our work does not have any potential negative societal impacts since our aim is to accelerate scientific discovery.
Code and Data. Our code and data are available at github.com/wengong-jin/DSMBind.
We would like to thank Divya Nori, Chenyu Wang, and anonymous reviewers for their valuable feedback on the manuscript. Wengong Jin is supported by the BroadIgnite Award and the Eric and Wendy Schmidt Center at the Broad Institute of MIT and Harvard. Caroline Uhler was partially supported by NCCIH/NIH (1DP2AT012345), ONR (N00014-22-1-2116), AstraZeneca, and a Simons Investigator Award.
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A NERE Rotational Equivalence
Proposition 1. Suppose we rotate a protein-ligand complex so that new coordinates become  $\boldsymbol{x}_i' = \boldsymbol{R}\boldsymbol{x}_i$ . The new force  $\boldsymbol{f}'$ , torque  $\tau'$ , inertia matrix  $I_N'$ , and angular velocity  $\omega'$  under the rotated complex are
$$
\boldsymbol {f} _ {i} ^ {\prime} = \boldsymbol {R} \boldsymbol {f} _ {i}, \tau^ {\prime} = \boldsymbol {R} \tau , \boldsymbol {I} _ {N} ^ {\prime} = \boldsymbol {R} \boldsymbol {I} _ {N} \boldsymbol {R} ^ {\top}, \omega^ {\prime} = \boldsymbol {R} \omega ,
$$
Proof. After rotating the whole complex, the energy function  $E(\mathbf{A}, \mathbf{X}') = E(\mathbf{A}, \mathbf{X})$  since the encoder is SE(3)-invariant. Given that  $\pmb{x}_i = \pmb{R}^\top \pmb{x}_i'$  and  $\partial \pmb{x}_i / \partial \pmb{x}_i' = \pmb{R}^\top$ , the new force becomes
$$
\boldsymbol {f} _ {i} ^ {\prime} = \left(\frac {\partial E (\boldsymbol {A} , \boldsymbol {X} ^ {\prime})}{\partial \boldsymbol {x} _ {i} ^ {\prime}}\right) ^ {\top} = \left(\frac {\partial E (\boldsymbol {A} , \boldsymbol {X})}{\partial \boldsymbol {x} _ {i}} \frac {\partial \boldsymbol {x} _ {i}}{\partial \boldsymbol {x} _ {i} ^ {\prime}}\right) ^ {\top} = \left(\frac {\partial \boldsymbol {x} _ {i}}{\partial \boldsymbol {x} _ {i} ^ {\prime}}\right) ^ {\top} \boldsymbol {f} _ {i} = \boldsymbol {R} \boldsymbol {f} _ {i}
$$
Based on the definition of torque and the fact that cross products satisfy  $\mathbf{R}\mathbf{x} \times \mathbf{R}\mathbf{y} = \mathbf{R}(\mathbf{x} \times \mathbf{y})$ , we have
$$
\begin{array}{l} \boldsymbol {\tau} ^ {\prime} = \sum_ {i} \left(\boldsymbol {x} _ {i} ^ {\prime} - \boldsymbol {\mu} ^ {\prime}\right) \times \boldsymbol {f} _ {i} ^ {\prime} = \sum_ {i} \left(\boldsymbol {R x} _ {i} - \boldsymbol {R} \boldsymbol {\mu}\right) \times \boldsymbol {R f} _ {i} \\ = \boldsymbol {R} \sum_ {i} (\boldsymbol {x} _ {i} - \boldsymbol {\mu}) \times \boldsymbol {f} _ {i} = \boldsymbol {R} \tau \\ \end{array}
$$
Likewise, using the fact that  $\pmb{R}\pmb{R}^{\top} = \pmb{I}$ , the new inertia matrix becomes
$$
\begin{array}{l} \boldsymbol {I} _ {N} ^ {\prime} = \sum_ {i} \| \boldsymbol {x} _ {i} ^ {\prime} - \boldsymbol {\mu} ^ {\prime} \| ^ {2} \boldsymbol {I} - (\boldsymbol {x} _ {i} ^ {\prime} - \boldsymbol {\mu} ^ {\prime}) (\boldsymbol {x} _ {i} ^ {\prime} - \boldsymbol {\mu} ^ {\prime}) ^ {\top} \\ = \sum_ {i} \| \boldsymbol {x} _ {i} - \boldsymbol {\mu} \| ^ {2} \boldsymbol {I} - (\boldsymbol {R x} _ {i} - \boldsymbol {R} \boldsymbol {\mu}) (\boldsymbol {R x} _ {i} - \boldsymbol {R} \boldsymbol {\mu}) ^ {\top} \\ = \sum_ {i} \boldsymbol {R} \| \boldsymbol {x} _ {i} - \boldsymbol {\mu} \| ^ {2} \boldsymbol {I} \boldsymbol {R} ^ {\top} - \boldsymbol {R} (\boldsymbol {x} _ {i} - \boldsymbol {\mu}) (\boldsymbol {x} _ {i} - \boldsymbol {\mu}) ^ {\top} \boldsymbol {R} ^ {\top} \\ = \boldsymbol {R} \left(\sum_ {i} \| \boldsymbol {x} _ {i} - \boldsymbol {\mu} \| ^ {2} \boldsymbol {I} - (\boldsymbol {x} _ {i} - \boldsymbol {\mu}) (\boldsymbol {x} _ {i} - \boldsymbol {\mu}) ^ {\top}\right) \boldsymbol {R} ^ {\top} \\ = \boldsymbol {R} \boldsymbol {I} _ {N} \boldsymbol {R} ^ {\top} \\ \end{array}
$$
For angular velocity, we have
$$
\boldsymbol {\omega} ^ {\prime} = C \boldsymbol {I} _ {N} ^ {\prime - 1} \boldsymbol {\tau} ^ {\prime} = C R \boldsymbol {I} _ {N} ^ {- 1} \boldsymbol {R} ^ {\top} \boldsymbol {R} \boldsymbol {\tau} = C R \boldsymbol {I} _ {N} ^ {- 1} \boldsymbol {\tau} = \boldsymbol {R} \boldsymbol {\omega}
$$
Therefore, NERE layer is equivariant under rotation.
Protein encoder architecture. The energy function needs to be SE(3)-invariant and differentiable with respect to  $X$ . Thus, we adopt the frame averaging neural network [28] so that  $E$  directly takes coordinates  $X$  as input rather than a distance matrix. To be specific, our energy function is parameterized as follows
$$
\left\{\boldsymbol {h} _ {1}, \dots , \boldsymbol {h} _ {n} \right\} = \frac {1}{| \mathcal {G} |} \sum_ {g _ {k} \in \mathcal {G}} \phi_ {h} (\boldsymbol {A}, g _ {k} (\boldsymbol {X})) \tag {15}
$$
The encoder  $\phi_h$  is a modified transformer network [17] that learns atom representations  $h_1, \dots, h_n$  based on atom features  $\mathbf{A}$  and coordinates  $\mathbf{X}$ . Specifically, the model first projects the coordinates  $\mathbf{x}_i$  onto a set of eight frames  $\{g_k(\mathbf{x}_i)\}$  defined in Puny et al. [28], concatenate the projected coordinates with atom features  $\mathbf{a}_i$ , encode the vector sequence  $[a_1, g_k(\mathbf{x}_1)], \dots, [a_n, g_k(\mathbf{x}_n)]$  to their hidden representations  $h_1^k, \dots, h_n^k$ , and then average the frame representations  $\mathbf{h}_i = \sum_k \mathbf{h}_i^k / 8$  to maintain SE(3) invariance.
Model hyperparameters. In the small molecule case, our model has two components: molecular graph encoder (MPN) and frame-averaging encoder. For the MPN encoder, we use the default
Crystal	Docked
ZRANK	0.318	0.163
RosettaDOCK	0.064	0.025
PYDOCK	0.248	0.164
SIPPER	-0.138	0.003
AP_PISA	0.323	0.144
FIREDOCK	0.101	-0.052
FIREDOCK_AB	0.199	0.042
CP PIE	0.234	0.120
3D CNN	0.286.021	0.162.017
GNN	0.244.075	0.249.075
NERE (ours)	0.361.051	0.360.051
Table 2: Additional baselines on the SAbDab test set.