Original: E3B IND : A N E ND - TO -E ND E QUIVARIANT N ETWORK FOR P ROTEIN -L IGAND D OCKING
Revised: E3B IND : A N E ND - TO -E ND E QUIVARIANT N ETWORK FOR P ROTEIN -L IGAND D OCKING

Original: A BSTRACT
Revised: A BSTRACT

Original: In silico prediction of the ligand binding pose to a given protein target is a crucial but challenging task in drug discovery.
Revised: In silico prediction of the ligand binding pose to a given protein target is a crucial but challenging task in drug discovery.

Original: This work focuses on flexible blind selfdocking, where we aim to predict the positions, orientations and conformations of docked molecules.
Revised: This work focuses on flexible blind selfdocking, where we aim to predict the positions, orientations and conformations of docked molecules.

Original: Traditional physics-based methods usually suffer from inaccurate scoring functions and high inference costs.
Revised: Traditional physics-based methods usually suffer from inaccurate scoring functions and high inference costs.

Original: Recently, data-driven methods based on deep learning techniques are attracting growing interest thanks to their efficiency during inference and promising performance.
Revised: Recently, data-driven methods based on deep learning techniques are attracting growing interest thanks to their efficiency during inference and promising performance.

Original: These methods usually either adopt a two-stage approach by first predicting the distances between proteins and ligands and then generating the final coordinates based on the predicted distances, or directly predicting the global roto-translation of ligands.
Revised: These methods usually either adopt a two-stage approach by first predicting the distances between proteins and ligands and then generating the final coordinates based on the predicted distances, or directly predicting the global roto-translation of ligands.

Original: In this paper, we take a different route.
Revised: In this paper, we take a different route.

Original: Inspired by the resounding success of AlphaFold for protein structure prediction, we propose E3Bind, an end-to-end equivariant network that iteratively updates the ligand pose.
Revised: Inspired by the resounding success of AlphaFold for protein structure prediction, we propose E3Bind, an end-to-end equivariant network that iteratively updates the ligand pose.

Original: E3Bind models the protein-ligand interaction through careful consideration of the geometric constraints in docking and the local context of the binding site.
Revised: E3Bind models the protein-ligand interaction through careful consideration of the geometric constraints in docking and the local context of the binding site.

Original: Experiments on standard benchmark datasets demonstrate the superior performance of our end-to-end trainable model compared to traditional and recently-proposed deep learning methods.
Revised: Experiments on standard benchmark datasets demonstrate the superior performance of our end-to-end trainable model compared to traditional and recently-proposed deep learning methods.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: For nearly a century, small molecules, or organic compounds with small molecular weight, have been the major weapon of the pharmaceutical industry.
Revised: For nearly a century, small molecules, or organic compounds with small molecular weight, have been the major weapon of the pharmaceutical industry.

Original: They take effect by ligating (binding) to their target, usually a protein, to alter the molecular pathways of diseases.
Revised: They take effect by ligating (binding) to their target, usually a protein, to alter the molecular pathways of diseases.

Original: The structure of the proteinligand interface holds the key to understanding the potency, mechanisms and potential side effects of small molecule drugs.
Revised: The structure of the proteinligand interface holds the key to understanding the potency, mechanisms and potential side effects of small molecule drugs.

Original: Despite huge efforts made for protein-ligand complex structure determination, there are by far only some 10 4 protein-ligand complex structures available in the protein data bank (PDB) (Berman et al., 2000), which dwarfs in front of the enormous combinatorial space of possible complexes between 10 60 drug-like molecules (Hert et al., 2009; Reymond & Awale, 2012) and at least 20,000 human proteins (Gaudet et al., 2017; Consortium, 2019), highlighting the urgent need for in silico protein-ligand docking methods.
Revised: Despite huge efforts made for protein-ligand complex structure determination, there are by far only some 10 4 protein-ligand complex structures available in the protein data bank (PDB) (Berman et al., 2000), which dwarfs in front of the enormous combinatorial space of possible complexes between 10 60 drug-like molecules (Hert et al., 2009; Reymond & Awale, 2012) and at least 20,000 human proteins (Gaudet et al., 2017; Consortium, 2019), highlighting the urgent need for in silico protein-ligand docking methods.

Original: Furthermore, a fast and accurate docking tool capable of predicting binding poses for molecules yet to be synthesized would empower mass-scale virtual screening (Lyu et al., 2019), a vital step in modern structure-based drug discovery (Ferreira et al., 2015).
Revised: Furthermore, a fast and accurate docking tool capable of predicting binding poses for molecules yet to be synthesized would empower mass-scale virtual screening (Lyu et al., 2019), a vital step in modern structure-based drug discovery (Ferreira et al., 2015).

Original: It also provides pharmaceutical scientists with an interpretable, information-rich result.
Revised: It also provides pharmaceutical scientists with an interpretable, information-rich result.

Original: Being a crucial task, predicting the docked pose of a ligand is also a challenging one.
Revised: Being a crucial task, predicting the docked pose of a ligand is also a challenging one.

Original: Traditional docking methods (Halgren et al., 2004; Morris et al., 1996; Trott & Olson, 2010; Coleman et al., 2013) rely on physics-inspired scoring functions and extensive conformation sampling to obtain the predicted binding pose.
Revised: Traditional docking methods (Halgren et al., 2004; Morris et al., 1996; Trott & Olson, 2010; Coleman et al., 2013) rely on physics-inspired scoring functions and extensive conformation sampling to obtain the predicted binding pose.

Original: Some deep learning methods focus on learning a more accurate scoring function (McNutt et al., 2021; M´endez-Lucio et al., 2021), but at the cost of even lower inference speed due to their adoption of the sampling-scoring framework.
Revised: Some deep learning methods focus on learning a more accurate scoring function (McNutt et al., 2021; M´endez-Lucio et al., 2021), but at the cost of even lower inference speed due to their adoption of the sampling-scoring framework.

Original: Distinct from the above methods,
Revised: Distinct from the above methods,

Original: TankBind (Lu et al., 2022) drops the burden of conformation sampling by predicting the proteinligand distance map, then converting the distance map to a docked pose using gradient descent.
Revised: TankBind (Lu et al., 2022) drops the burden of conformation sampling by predicting the proteinligand distance map, then converting the distance map to a docked pose using gradient descent.

Original: The optimization objective is the weighted sum of the protein-ligand distance error with respect to the predicted distance map and the intra-ligand distance error w.r.t.
Revised: The optimization objective is the weighted sum of the protein-ligand distance error with respect to the predicted distance map and the intra-ligand distance error w.r.t.

Original: the reference intra-ligand distances.
Revised: the reference intra-ligand distances.

Original: This two-stage approach might run into problems during the distance-to-coordinate transformation, as the predicted distance map is, in many cases, not a valid Euclidean distance matrix (Liberti et al., 2014).
Revised: This two-stage approach might run into problems during the distance-to-coordinate transformation, as the predicted distance map is, in many cases, not a valid Euclidean distance matrix (Liberti et al., 2014).

Original: Recently, St¨ark et al.
Revised: Recently, St¨ark et al.

Original: (2022) proposed EquiBind, an equivariant model that directly predicts the coordinates of the docked pose.
Revised: (2022) proposed EquiBind, an equivariant model that directly predicts the coordinates of the docked pose.

Original: EquiBind updates the ligand conformation with a graph neural network, then roto-translates the ligand into the pocket using a key-point alignment mechanism.
Revised: EquiBind updates the ligand conformation with a graph neural network, then roto-translates the ligand into the pocket using a key-point alignment mechanism.

Original: It enjoys significant speedup compared to the popular docking baselines (Hassan et al., 2017; Koes et al., 2013) and provides good pose initializations for them, but on its own the docking performance is less satisfactory.
Revised: It enjoys significant speedup compared to the popular docking baselines (Hassan et al., 2017; Koes et al., 2013) and provides good pose initializations for them, but on its own the docking performance is less satisfactory.

Original: This is probably because, after the one-shot roto-translation, the ligand might fall into an unfavorable position but its conformation could not be further refined.
Revised: This is probably because, after the one-shot roto-translation, the ligand might fall into an unfavorable position but its conformation could not be further refined.

Original: In this paper, we move one step forward in this important direction and propose E3Bind, the first end-to-end equivariant network that iteratively docks the ligand into the binding pocket.
Revised: In this paper, we move one step forward in this important direction and propose E3Bind, the first end-to-end equivariant network that iteratively docks the ligand into the binding pocket.

Original: Inspired by
Revised: Inspired by

Original: AlphaFold2 (Jumper et al., 2021), our model comprises a feature extractor named Trioformer and an iterative coordinate refinement module.
Revised: AlphaFold2 (Jumper et al., 2021), our model comprises a feature extractor named Trioformer and an iterative coordinate refinement module.

Original: The Trioformer encodes the protein and ligand graphs into three information-rich embeddings: the protein residue embeddings, the ligand atom embeddings and the protein-ligand pair embeddings, where the pair embeddings are fused with geometry awareness to enforce the implicit constraints in docking.
Revised: The Trioformer encodes the protein and ligand graphs into three information-rich embeddings: the protein residue embeddings, the ligand atom embeddings and the protein-ligand pair embeddings, where the pair embeddings are fused with geometry awareness to enforce the implicit constraints in docking.

Original: Our coordinate refinement module decodes the rich representations into E(3)-equivariant coordinate updates.
Revised: Our coordinate refinement module decodes the rich representations into E(3)-equivariant coordinate updates.

Original: The iterative coordinate update scheme feeds the output pose of a decoder block as the initial pose of the next one, allowing the model to dynamically sense the local context and fix potential errors (see Figure 3).
Revised: The iterative coordinate update scheme feeds the output pose of a decoder block as the initial pose of the next one, allowing the model to dynamically sense the local context and fix potential errors (see Figure 3).

Original: We further propose a self-confidence predictor to select the final pose and evaluate the soundness of our predictions.
Revised: We further propose a self-confidence predictor to select the final pose and evaluate the soundness of our predictions.

Original: E3Bind is trained end-to-end with loss directly defined on the output ligand coordinates, relieving the burden of conformation sampling or distance-to-coordinate transformation.
Revised: E3Bind is trained end-to-end with loss directly defined on the output ligand coordinates, relieving the burden of conformation sampling or distance-to-coordinate transformation.

Original: Our contributions can be summarized as follows:
Revised: Our contributions can be summarized as follows:

Original: • We formulate the docking problem as an iterative refinement process where the model updates the ligand coordinates based on the current context at each iteration.
Revised: • We formulate the docking problem as an iterative refinement process where the model updates the ligand coordinates based on the current context at each iteration.

Original: • We propose an end-to-end E(3) equivariant network to generate the coordinate updates.
Revised: • We propose an end-to-end E(3) equivariant network to generate the coordinate updates.

Original: The network comprises an expressive geometric-aware encoder and an equivariant contextaware coordinate update module.
Revised: The network comprises an expressive geometric-aware encoder and an equivariant contextaware coordinate update module.

Original: • Quantitative results show that E3Bind outperforms both traditional score-based methods and recent deep learning models.
Revised: • Quantitative results show that E3Bind outperforms both traditional score-based methods and recent deep learning models.

Original:  R ELATED W ORKS
Revised:  R ELATED W ORKS

Original: Protein-ligand docking.
Revised: Protein-ligand docking.

Original: Traditional approaches to protein-ligand docking (Morris et al., 1996;
Revised: Traditional approaches to protein-ligand docking (Morris et al., 1996;

Original: Halgren et al., 2004; Coleman et al., 2013) mainly adopt a sampling, scoring, ranking, and finetuning paradigm, with AutoDock Vina (Trott & Olson, 2010) being a popular example.
Revised: Halgren et al., 2004; Coleman et al., 2013) mainly adopt a sampling, scoring, ranking, and finetuning paradigm, with AutoDock Vina (Trott & Olson, 2010) being a popular example.

Original: Each part of the docking pipeline has been extensively studied in literature to increase both accuracy and speed (Durrant & McCammon, 2011; Liu et al., 2013; Hassan et al., 2017; Zhang et al., 2020).
Revised: Each part of the docking pipeline has been extensively studied in literature to increase both accuracy and speed (Durrant & McCammon, 2011; Liu et al., 2013; Hassan et al., 2017; Zhang et al., 2020).

Original: Multiple subsequent works use deep-learning on 3D voxels (Ragoza et al., 2017;
Revised: Multiple subsequent works use deep-learning on 3D voxels (Ragoza et al., 2017;

Original: Francoeur et al., 2020; McNutt et al., 2021; Bao et al., 2021) or graphs (M´endez-Lucio et al., 2021) to improve the scoring functions.
Revised: Francoeur et al., 2020; McNutt et al., 2021; Bao et al., 2021) or graphs (M´endez-Lucio et al., 2021) to improve the scoring functions.

Original: Nevertheless, these methods are inefficient in general, often taking minutes or even more to predict the docking poses of a single protein-ligand pair, which hinders the accessibility of large-scale virtual screening experiments.
Revised: Nevertheless, these methods are inefficient in general, often taking minutes or even more to predict the docking poses of a single protein-ligand pair, which hinders the accessibility of large-scale virtual screening experiments.

Original: Recently, methods that directly model the distance geometry between protein-ligand pairs have been investigated (Masters et al., 2022;
Revised: Recently, methods that directly model the distance geometry between protein-ligand pairs have been investigated (Masters et al., 2022;

Original: Lu et al., 2022;
Revised: Lu et al., 2022;

Original: Zhou et al., 2022).
Revised: Zhou et al., 2022).

Original: They adopt a two-stage approach for docking, and generate docked poses from predicted protein-ligand distance maps using post-optimization algorithms.
Revised: They adopt a two-stage approach for docking, and generate docked poses from predicted protein-ligand distance maps using post-optimization algorithms.

Original: Advanced techniques in geometric deep learning, e.g. triangle attention (Jumper et al., 2021) with geometric constraints, have been leveraged to encourage the local geometrical consistency of the distance map (Lu et al., 2022).
Revised: Advanced techniques in geometric deep learning, e.g. triangle attention (Jumper et al., 2021) with geometric constraints, have been leveraged to encourage the local geometrical consistency of the distance map (Lu et al., 2022).

Original: To bypass the error-prone twostage framework, EquiBind (St¨ark et al., 2022) proposes a fully differentiable equivariant model, which directly predicts coordinates of docked poses with a novel attention-based key-point alignment mechanism (Ganea et al., 2021b).
Revised: To bypass the error-prone twostage framework, EquiBind (St¨ark et al., 2022) proposes a fully differentiable equivariant model, which directly predicts coordinates of docked poses with a novel attention-based key-point alignment mechanism (Ganea et al., 2021b).

Original: Despite being more efficient, EquiBind fails to beat popular docking baselines on its own, stressing the importance of increasing model expressiveness.
Revised: Despite being more efficient, EquiBind fails to beat popular docking baselines on its own, stressing the importance of increasing model expressiveness.

Original: Iterative coordinate update inter-message passing ligand
Revised: Iterative coordinate update inter-message passing ligand

Original: [Figure]
Revised: [Figure]

Original: constrained update protein-constrained update intra-message passing coordinate update
Revised: constrained update protein-constrained update intra-message passing coordinate update

Original: Trioformer
Revised: Trioformer

Original: Context-Aware E(3)-Equivariant Network ligandprotein
Revised: Context-Aware E(3)-Equivariant Network ligandprotein

Original: i k i j i j k i k’ j i k residue emb.
Revised: i k i j i j k i k’ j i k residue emb.

Original: atom emb.
Revised: atom emb.

Original: pair emb.
Revised: pair emb.

Original: [Figure]
Revised: [Figure]

Original: protein structure docked ligand coordinates
Revised: protein structure docked ligand coordinates

Original: N O N Cl HO
Revised: N O N Cl HO

Original: N
Revised: N

Original: CF
Revised: CF

Original: Molecular conformation generation.
Revised: Molecular conformation generation.

Original: Deep learning has made great progress in predicting lowenergy conformations given molecular graphs (Shi et al., 2021;
Revised: Deep learning has made great progress in predicting lowenergy conformations given molecular graphs (Shi et al., 2021;

Original: Ganea et al., 2021a;
Revised: Ganea et al., 2021a;

Original: Xu et al., 2022;
Revised: Xu et al., 2022;

Original: Jing et al., 2022).
Revised: Jing et al., 2022).

Original: State-of-the-art models are generally SE(3)-invariant or equivariant, adopting the score-matching (Vincent, 2011; Song & Ermon, 2019) or diffusion (Ho et al., 2020) framework.
Revised: State-of-the-art models are generally SE(3)-invariant or equivariant, adopting the score-matching (Vincent, 2011; Song & Ermon, 2019) or diffusion (Ho et al., 2020) framework.

Original: Though protein-ligand docking also aims to generate the conformation of a molecule, the bound ligand, applying standard molecular conformation generation approaches is not viable because (1) there are much more atoms in the system and (2) the protein context must be carefully considered when generating the docked pose.
Revised: Though protein-ligand docking also aims to generate the conformation of a molecule, the bound ligand, applying standard molecular conformation generation approaches is not viable because (1) there are much more atoms in the system and (2) the protein context must be carefully considered when generating the docked pose.

Original: Here we model the protein at the residue level and design our model to carefully capture the protein-ligand interactions.
Revised: Here we model the protein at the residue level and design our model to carefully capture the protein-ligand interactions.

Original: Protein structure prediction.
Revised: Protein structure prediction.

Original: Predicting protein folds from sequences has long been a challenging task.
Revised: Predicting protein folds from sequences has long been a challenging task.

Original: Wang et al.
Revised: Wang et al.

Original: (2017); Senior et al.
Revised: (2017); Senior et al.

Original: (2020a;b) use deep learning to predict the contact map, the distance map and/or the torsion angles between protein residues, and then convert them to coordinates using optimization-based methods.
Revised: (2020a;b) use deep learning to predict the contact map, the distance map and/or the torsion angles between protein residues, and then convert them to coordinates using optimization-based methods.

Original: Recently, AlphaFold2 (Jumper et al., 2021) takes a leap forward by adopting an end-to-end approach with iterative coordinate refinement.
Revised: Recently, AlphaFold2 (Jumper et al., 2021) takes a leap forward by adopting an end-to-end approach with iterative coordinate refinement.

Original: It consists of an
Revised: It consists of an

Original: Evoformer to extract information from Multiple Sequence Alignments (MSAs) and structural templates, and a structure module to iteratively update the coordinates.
Revised: Evoformer to extract information from Multiple Sequence Alignments (MSAs) and structural templates, and a structure module to iteratively update the coordinates.

Original: Though this problem is very different from docking which models heterogeneous entities – a fixed protein structure and a ligand molecular graph, in this paper we show that some ideas can be extended.
Revised: Though this problem is very different from docking which models heterogeneous entities – a fixed protein structure and a ligand molecular graph, in this paper we show that some ideas can be extended.

Original:  T HE E3B IND M ODEL
Revised:  T HE E3B IND M ODEL

Original: E3Bind tackles the protein-ligand docking task with an encoder for feature extraction and a decoder for coordinate update generation.
Revised: E3Bind tackles the protein-ligand docking task with an encoder for feature extraction and a decoder for coordinate update generation.

Original: Specifically, the protein and ligand graphs are first encoded by standard graph encoders.
Revised: Specifically, the protein and ligand graphs are first encoded by standard graph encoders.

Original: Pair embeddings are constructed between every protein residue - ligand atom pair.
Revised: Pair embeddings are constructed between every protein residue - ligand atom pair.

Original: A geometry-aware Trioformer is proposed to fully mix the protein, ligand and pair embeddings (Section 3.2).
Revised: A geometry-aware Trioformer is proposed to fully mix the protein, ligand and pair embeddings (Section 3.2).

Original: With the rich representations at hand, the iterative coordinate refinement module updates the ligand pose by a series of E(3)-equivariant networks to generate the final docked pose (Section 3.3).
Revised: With the rich representations at hand, the iterative coordinate refinement module updates the ligand pose by a series of E(3)-equivariant networks to generate the final docked pose (Section 3.3).

Original: The model is trained end-to-end and capable of directly generating the docked pose (Section 3.4).
Revised: The model is trained end-to-end and capable of directly generating the docked pose (Section 3.4).

Original: An overview of E3Bind is shown in Figure 1.
Revised: An overview of E3Bind is shown in Figure 1.

Original: 3. P RELIMINARIES
Revised: 3. P RELIMINARIES

Original: Notation and Input Feature Construction.
Revised: Notation and Input Feature Construction.

Original: Following Lu et al.
Revised: Following Lu et al.

Original: , the ligand is treated as an atom-level molecular graph G l where the edges denote chemical bonds.
Revised: , the ligand is treated as an atom-level molecular graph G l where the edges denote chemical bonds.

Original: The ligand node features { h l i } 1 ≤ i ≤ n l are calculated by a TorchDrug (Zhu et al., 2022) implementation of the graph isomorphism network (GIN)
Revised: The ligand node features { h l i } 1 ≤ i ≤ n l are calculated by a TorchDrug (Zhu et al., 2022) implementation of the graph isomorphism network (GIN)

Original: (Xu et al., 2018), where n l is the number of atoms in the ligand.
Revised: (Xu et al., 2018), where n l is the number of atoms in the ligand.

Original: Ligand coordinates are denoted as { x l i } 1 ≤ i ≤ n l .
Revised: Ligand coordinates are denoted as { x l i } 1 ≤ i ≤ n l .

Original: we represent the protein as a residue-level K -nearest neighbor graph G p .
Revised: we represent the protein as a residue-level K -nearest neighbor graph G p .

Original: Each protein node j ∈ { 1 , . .
Revised: Each protein node j ∈ { 1 , . .

Original: . , n p } has 3D coordinates x p j (which corresponds to the position of the C α atom of the residue), where n p is the number of residues in the protein.
Revised: . , n p } has 3D coordinates x p j (which corresponds to the position of the C α atom of the residue), where n p is the number of residues in the protein.

Original: The protein node features { h p j } 1 ≤ j ≤ n p are calculated with a geometric-vector-perceptron-based graph neural network (GVP-GNN)
Revised: The protein node features { h p j } 1 ≤ j ≤ n p are calculated with a geometric-vector-perceptron-based graph neural network (GVP-GNN)

Original: Jing et al.
Revised: Jing et al.

Original: For each protein residue-ligand atom pair ( i, j ) , we construct a pair embedding z ij via the outer product module (OPM) which takes in protein and ligand embeddings: z ij =
Revised: For each protein residue-ligand atom pair ( i, j ) , we construct a pair embedding z ij via the outer product module (OPM) which takes in protein and ligand embeddings: z ij =

Original: Linear (cid:0) vec (cid:0) Linear( h l i ) (cid:78) Linear( h p j ) (cid:1)(cid:1) .
Revised: Linear (cid:0) vec (cid:0) Linear( h l i ) (cid:78) Linear( h p j ) (cid:1)(cid:1) .

Original: For notation consistency, throughout this paper we will use i, k to index ligand nodes, and j, k ′ for protein nodes.
Revised: For notation consistency, throughout this paper we will use i, k to index ligand nodes, and j, k ′ for protein nodes.

Original: Problem Definition.
Revised: Problem Definition.

Original: Given a molecular graph of the ligand compound and a fixed protein structure, we aim to predict the binding (docked) pose of the ligand compound { x l i ∗ } 1 ≤ i ≤ n l .
Revised: Given a molecular graph of the ligand compound and a fixed protein structure, we aim to predict the binding (docked) pose of the ligand compound { x l i ∗ } 1 ≤ i ≤ n l .

Original: We focus on blind docking settings, where the binding pocket is not provided and has to be predicted by the model.
Revised: We focus on blind docking settings, where the binding pocket is not provided and has to be predicted by the model.

Original: In the blind re-docking setting, the docked ligand conformation is given, but its position and orientation relative to the protein are unknown.
Revised: In the blind re-docking setting, the docked ligand conformation is given, but its position and orientation relative to the protein are unknown.

Original: In the flexible blind self-docking setting, the docked ligand conformation needs to be inferred besides its position and orientation.
Revised: In the flexible blind self-docking setting, the docked ligand conformation needs to be inferred besides its position and orientation.

Original: The model is thus provided with an unbound ligand structure, which could be obtained by running the ETKDG algorithm (Riniker & Landrum, 2015) with RDKit (Landrum et al., 2013).
Revised: The model is thus provided with an unbound ligand structure, which could be obtained by running the ETKDG algorithm (Riniker & Landrum, 2015) with RDKit (Landrum et al., 2013).

Original: Equivariance.
Revised: Equivariance.

Original: An important inductive bias in protein ligand docking is E(3) equivariance, i.e. , if the input protein coordinates are transformed by some E(3) transformation g · { x p j } 1 ≤ j ≤ n p = {
Revised: An important inductive bias in protein ligand docking is E(3) equivariance, i.e. , if the input protein coordinates are transformed by some E(3) transformation g · { x p j } 1 ≤ j ≤ n p = {

Original: Rx p j + t } 1 ≤ j ≤ n p , ∀ j = 1 . . .
Revised: Rx p j + t } 1 ≤ j ≤ n p , ∀ j = 1 . . .

Original: n p , the predicted ligand coordinates should also be g -transformed:
Revised: n p , the predicted ligand coordinates should also be g -transformed:

Original: F ( g · { x p j } 1 ≤ j ≤ n p ) =
Revised: F ( g · { x p j } 1 ≤ j ≤ n p ) =

Original: g · F ( { x p j } 1 ≤ j ≤ n p ) , where F is the coordinate prediction model.
Revised: g · F ( { x p j } 1 ≤ j ≤ n p ) , where F is the coordinate prediction model.

Original: To inject such symmetry to our model, we adopt a variant of the equivariant graph neural network (EGNN) as the building block in our coordinate update steps (Satorras et al., 2021).
Revised: To inject such symmetry to our model, we adopt a variant of the equivariant graph neural network (EGNN) as the building block in our coordinate update steps (Satorras et al., 2021).

Original: 3. E XTRACTING G EOMETRY -C ONSISTENT I NFORMATION WITH T RIOFORMER
Revised: 3. E XTRACTING G EOMETRY -C ONSISTENT I NFORMATION WITH T RIOFORMER

Original: i k’ j ligand protein fixed distance
Revised: i k’ j ligand protein fixed distance

Original: The E3Bind encoder extracts information-rich ligand atom embeddings h l i , protein residue embeddings h p j and ligand-protein pair representation z ij that captures the subtle protein-ligand interactions.
Revised: The E3Bind encoder extracts information-rich ligand atom embeddings h l i , protein residue embeddings h p j and ligand-protein pair representation z ij that captures the subtle protein-ligand interactions.

Original: Note that this is a non-trivial task since implicit geometric constraints must be incorporated in the representations.
Revised: Note that this is a non-trivial task since implicit geometric constraints must be incorporated in the representations.

Original: As shown in the figure on the right, the protein-ligand distance d ij and d ik ′ can not be predicted independently.
Revised: As shown in the figure on the right, the protein-ligand distance d ij and d ik ′ can not be predicted independently.

Original: They are constrained by the intra-protein distance d jk ′ , as the protein structure is considered to be fixed during docking.
Revised: They are constrained by the intra-protein distance d jk ′ , as the protein structure is considered to be fixed during docking.

Original: In other words, by the triangle inequality, if residues j and k ′ are close, then ligand atom i cannot be both close to j and far from k ′ .
Revised: In other words, by the triangle inequality, if residues j and k ′ are close, then ligand atom i cannot be both close to j and far from k ′ .

Original: The same thing happens when we consider the given intra-ligand distance d ik 1 .
Revised: The same thing happens when we consider the given intra-ligand distance d ik 1 .

Original: While using a simple concatenation of protein and ligand embeddings as the final pair embeddings is common among previous methods (M ´ endez-Lucio et al., 2021; Masters et al., 2022), it dismisses the above geometry constraints and might result in geometrically inconsistent pose predictions.
Revised: While using a simple concatenation of protein and ligand embeddings as the final pair embeddings is common among previous methods (M ´ endez-Lucio et al., 2021; Masters et al., 2022), it dismisses the above geometry constraints and might result in geometrically inconsistent pose predictions.

Original: Trioformer Overview.
Revised: Trioformer Overview.

Original: To tackle the above challenge, we propose Trioformer , an architecture that intensively updates and mixes the protein, ligand and pair embeddings.
Revised: To tackle the above challenge, we propose Trioformer , an architecture that intensively updates and mixes the protein, ligand and pair embeddings.

Original: In each Trioformer block, we first update the protein and ligand node embeddings with multi-head cross-attention using the pair embeddings as attention bias.
Revised: In each Trioformer block, we first update the protein and ligand node embeddings with multi-head cross-attention using the pair embeddings as attention bias.

Original: Next, we use protein and ligand embeddings to update the pair embeddings via the outer product module:
Revised: Next, we use protein and ligand embeddings to update the pair embeddings via the outer product module:

Original: z ij = z ij + OPM (cid:0)
Revised: z ij = z ij + OPM (cid:0)

Original: h l i , h p j (cid:1) .
Revised: h l i , h p j (cid:1) .

Original: The pair embeddings then go through geometry-aware pair update modules described below to produce geometry-consistent representations.
Revised: The pair embeddings then go through geometry-aware pair update modules described below to produce geometry-consistent representations.

Original: The final block outputs are the multi-layer perceptron (MLP)-transitioned protein, ligand and pair embeddings.
Revised: The final block outputs are the multi-layer perceptron (MLP)-transitioned protein, ligand and pair embeddings.

Original: Details of the Trioformer are described in Section B.
Revised: Details of the Trioformer are described in Section B.

Original: Geometry-Aware Pair Updates.
Revised: Geometry-Aware Pair Updates.

Original: To inject geometry awareness to the pair embeddings, we construct intra-ligand and intra-protein distance embeddings ( d ∗ ik and d ∗ jk ′ respectively) and then use them to update the pair embeddings.
Revised: To inject geometry awareness to the pair embeddings, we construct intra-ligand and intra-protein distance embeddings ( d ∗ ik and d ∗ jk ′ respectively) and then use them to update the pair embeddings.

Original: Following (Jumper et al., 2021; Lu et al., 2022), the edge updates are arranged in the form of triangles comprising two ligand-protein edges and one intra-edge (intra-ligand or intra-protein edge) where the distance constraints apply.
Revised: Following (Jumper et al., 2021; Lu et al., 2022), the edge updates are arranged in the form of triangles comprising two ligand-protein edges and one intra-edge (intra-ligand or intra-protein edge) where the distance constraints apply.

Original: For protein-constrained attentive pair updates, each pair ( i, j ) attends to all neighboring pairs { ( i, k ′ ) } 1 ≤ k ′ ≤ n p with a common ligand node i .
Revised: For protein-constrained attentive pair updates, each pair ( i, j ) attends to all neighboring pairs { ( i, k ′ ) } 1 ≤ k ′ ≤ n p with a common ligand node i .

Original: We compute a geometry-informed attention weight a ( h ) ijk ′ for each neighbor ( i, k ′ ) , by adding an attention bias t ( h ) jk ′ =
Revised: We compute a geometry-informed attention weight a ( h ) ijk ′ for each neighbor ( i, k ′ ) , by adding an attention bias t ( h ) jk ′ =

Original: Linear ( h ) ( d jk ′ ) computed from the intra-protein distance embeddings per attention head h .
Revised: Linear ( h ) ( d jk ′ ) computed from the intra-protein distance embeddings per attention head h .

Original: This geometry-informed attention weight is a key ingredient of Trioformer that differentiates it from the AlphaFold2’s Evoformer Jumper et al.
Revised: This geometry-informed attention weight is a key ingredient of Trioformer that differentiates it from the AlphaFold2’s Evoformer Jumper et al.

Original: We then perform a standard multi-head attention with H heads to aggregate information from neighboring pairs,
Revised: We then perform a standard multi-head attention with H heads to aggregate information from neighboring pairs,

Original: [Equation]
Revised: [Equation]

Original: where g ( h ) ij = σ (Linear ( h ) ( z ij )) is a per-head output gate, q
Revised: where g ( h ) ij = σ (Linear ( h ) ( z ij )) is a per-head output gate, q

Original: [Equation]
Revised: [Equation]

Original: h ) ij
Revised: h ) ij

Original: [Equation]
Revised: [Equation]

Original: k ( h ) ij , v ( h ) ij , b ( h ) ij are linear projections of the pair embedding z ij , and t ( h ) jk ′ is a distance-based bias described above.
Revised: k ( h ) ij , v ( h ) ij , b ( h ) ij are linear projections of the pair embedding z ij , and t ( h ) jk ′ is a distance-based bias described above.

Original: The ligand-constrained attentive pair updates are designed similarly.
Revised: The ligand-constrained attentive pair updates are designed similarly.

Original: Here, for pair ( i, j ) , the neighboring pairs { ( k, j ) } 1 ≤ k ≤ n l are those sharing the same protein node j .
Revised: Here, for pair ( i, j ) , the neighboring pairs { ( k, j ) } 1 ≤ k ≤ n l are those sharing the same protein node j .

Original: Multi-head attention is performed across all such neighbors with constraints from intra-ligand distances d ik .
Revised: Multi-head attention is performed across all such neighbors with constraints from intra-ligand distances d ik .

Original: 3. I TERATIVE C OORDINATE U PDATE WITH C ONTEXT
Revised: 3. I TERATIVE C OORDINATE U PDATE WITH C ONTEXT

Original: [Equation]
Revised: [Equation]

Original: WARE E(3)
Revised: WARE E(3)

Original: [Equation]
Revised: [Equation]

Original: E QUIVARIANT L AYER
Revised: E QUIVARIANT L AYER

Original: The coordinate update module iteratively adjusts the current ligand structure { x l i } 1 ≤ i ≤ n l towards the docked pose based on the extracted representations.
Revised: The coordinate update module iteratively adjusts the current ligand structure { x l i } 1 ≤ i ≤ n l towards the docked pose based on the extracted representations.

Original: This module is designed to satisfy the following desiderata: (1) protein context awareness : the model must have an in-depth understanding of the interaction between the ligand and its protein context in order to produce a protein-ligand complex with optimized stability ( i.e. lowest energy); (2) self (ligand) context awareness : the model should honor the basic geometric constraints of the ligand so that the predicted ligand conformation is physically valid; (3)
Revised: This module is designed to satisfy the following desiderata: (1) protein context awareness : the model must have an in-depth understanding of the interaction between the ligand and its protein context in order to produce a protein-ligand complex with optimized stability ( i.e. lowest energy); (2) self (ligand) context awareness : the model should honor the basic geometric constraints of the ligand so that the predicted ligand conformation is physically valid; (3)

Original: E(3)-equivariance : if both the protein and the input ligand pose are transformed by an E(3) transformation, the ligand should dock into the same pocket with the same pose, with its coordinates transformed with the same E(3) transformation.
Revised: E(3)-equivariance : if both the protein and the input ligand pose are transformed by an E(3) transformation, the ligand should dock into the same pocket with the same pose, with its coordinates transformed with the same E(3) transformation.

Original: Compared with methods that generate the final pose in one shot, iterative refinement allows the model to dynamically sense the local context and correct potential errors as the ligand gradually moves toward its final position.
Revised: Compared with methods that generate the final pose in one shot, iterative refinement allows the model to dynamically sense the local context and correct potential errors as the ligand gradually moves toward its final position.

Original: We start from an unbound ligand structure 2 .
Revised: We start from an unbound ligand structure 2 .

Original: At iteration step t = 0 , . . .
Revised: At iteration step t = 0 , . . .

Original: , T − 1 , the module updates the protein, ligand and pair representations (summarized by h ( t ) ) and the ligand coordinates { x ( t ) i } 1 ≤ i ≤ n l with a context-aware E(3)-equivariant layer,
Revised: , T − 1 , the module updates the protein, ligand and pair representations (summarized by h ( t ) ) and the ligand coordinates { x ( t ) i } 1 ≤ i ≤ n l with a context-aware E(3)-equivariant layer,

Original: [Equation]
Revised: [Equation]

Original: where { x j } 1 ≤ j ≤ n p are the fixed protein coordinates.
Revised: where { x j } 1 ≤ j ≤ n p are the fixed protein coordinates.

Original: We now introduce DecoderLayer ( t ) in detail.
Revised: We now introduce DecoderLayer ( t ) in detail.

Original: Context-Aware Message Passing.
Revised: Context-Aware Message Passing.

Original: To ensure the coordinate update module captures both the protein and ligand context, we construct a heterogeneous context graph comprising both protein and ligand nodes.
Revised: To ensure the coordinate update module captures both the protein and ligand context, we construct a heterogeneous context graph comprising both protein and ligand nodes.

Original: In the graph, inter-edges connect protein and ligand nodes, while intra-edges connect two ligand nodes (Figure 1).
Revised: In the graph, inter-edges connect protein and ligand nodes, while intra-edges connect two ligand nodes (Figure 1).

Original: We perform inter- and intra-edge message passing on the graph to explore the current context:
Revised: We perform inter- and intra-edge message passing on the graph to explore the current context:

Original: [Equation]
Revised: [Equation]

Original: Note
Revised: Note

Original: [Table]
Revised: [Table]

Original: the three types
Revised: the three types

Original: [Table]
Revised: [Table]

Original: messages are generated
Revised: messages are generated

Original: [Table]
Revised: [Table]

Original: parameters.
Revised: parameters.

Original: Equivariant Coordinate Update.
Revised: Equivariant Coordinate Update.

Original: We use the equivariant graph convolution layer (EGCL) to process current context geometry and update ligand coordinates in an E(3)-equivariant manner.
Revised: We use the equivariant graph convolution layer (EGCL) to process current context geometry and update ligand coordinates in an E(3)-equivariant manner.

Original: Specifically, we compute messages from the node (and edge) representations and distance information using MLPs ϕ m and φ m ,
Revised: Specifically, we compute messages from the node (and edge) representations and distance information using MLPs ϕ m and φ m ,

Original: [Equation]
Revised: [Equation]

Original: We generate equivariant coordinate updates using the following equation,
Revised: We generate equivariant coordinate updates using the following equation,

Original: [Equation]
Revised: [Equation]

Original: where ϕ x and φ x are gated MLPs that evaluate the message importance.
Revised: where ϕ x and φ x are gated MLPs that evaluate the message importance.

Original: Ligand and protein node embeddings are updated using equations 5 and 6, respectively.
Revised: Ligand and protein node embeddings are updated using equations 5 and 6, respectively.

Original: Note that our decoder is also compatible with other equivariant graph layers, such as the geometric vector perceptron (GVP)
Revised: Note that our decoder is also compatible with other equivariant graph layers, such as the geometric vector perceptron (GVP)

Original: (Jing et al., 2020) or vector neuron (Deng et al., 2021).
Revised: (Jing et al., 2020) or vector neuron (Deng et al., 2021).

Original: We choose EGCL as it is a powerful layer for molecular modeling and conformation generation (Satorras et al., 2021;
Revised: We choose EGCL as it is a powerful layer for molecular modeling and conformation generation (Satorras et al., 2021;

Original: Huang et al., 2022; Hoogeboom et al., 2022).
Revised: Huang et al., 2022; Hoogeboom et al., 2022).

Original: Self-Confidence Prediction.
Revised: Self-Confidence Prediction.

Original: At the end of our decoder, an additional self-confidence module is added to predict the model’s confidence for its predicted docked pose (Jumper et al., 2021).
Revised: At the end of our decoder, an additional self-confidence module is added to predict the model’s confidence for its predicted docked pose (Jumper et al., 2021).

Original: The confidence is a value from zero to one calculated by the equation ˆ c =
Revised: The confidence is a value from zero to one calculated by the equation ˆ c =

Original: σ (cid:16) MLP (cid:16)(cid:80) n l i =1 h ( T ) i (cid:17)(cid:17) .
Revised: σ (cid:16) MLP (cid:16)(cid:80) n l i =1 h ( T ) i (cid:17)(cid:17) .

Original: 3. T RAINING AND I NFERENCE
Revised: 3. T RAINING AND I NFERENCE

Original: End-to-End Training.
Revised: End-to-End Training.

Original: E3Bind directly generates the predicted ligand coordinates along with a confidence score.
Revised: E3Bind directly generates the predicted ligand coordinates along with a confidence score.

Original: We define a simple coordinate loss and train the model end-to-end:
Revised: We define a simple coordinate loss and train the model end-to-end:

Original: L coord = (cid:80) n l i =1 ( x i − x ∗ i ) 2 , where x i is the predicted coordinates and x ∗ i is the ground truth.
Revised: L coord = (cid:80) n l i =1 ( x i − x ∗ i ) 2 , where x i is the predicted coordinates and x ∗ i is the ground truth.

Original: We train the self-confidence module with L confidence =
Revised: We train the self-confidence module with L confidence =

Original: MSE (ˆ c, c ∗ ) , where MSE is the mean squared error loss and c ∗ is the self-confidence prediction target, i.e. the (detached) root-mean-square deviation (RMSD) of the predicted coordinates.
Revised: MSE (ˆ c, c ∗ ) , where MSE is the mean squared error loss and c ∗ is the self-confidence prediction target, i.e. the (detached) root-mean-square deviation (RMSD) of the predicted coordinates.

Original: Details are deferred to Section C.1.
Revised: Details are deferred to Section C.1.

Original: Intuitively, we want our model to predict a low ˆ c for high-RMSD predictions.
Revised: Intuitively, we want our model to predict a low ˆ c for high-RMSD predictions.

Original: The final training loss is the combination of the coordinate loss and the confidence loss
Revised: The final training loss is the combination of the coordinate loss and the confidence loss

Original: L = L coord + β L confidence , where β is a hyperparameter.
Revised: L = L coord + β L confidence , where β is a hyperparameter.

Original: This loss aligns with the goal of docked pose prediction and avoids the time-consuming sampling process and potentially error-prone distance-to-coordinate transformation.
Revised: This loss aligns with the goal of docked pose prediction and avoids the time-consuming sampling process and potentially error-prone distance-to-coordinate transformation.

Original: Inference Process.
Revised: Inference Process.

Original: In practice, the target protein may contain multiple binding sites, or be extremely large where most parts are irrelevant for ligand binding.
Revised: In practice, the target protein may contain multiple binding sites, or be extremely large where most parts are irrelevant for ligand binding.

Original: To solve this problem, we use
Revised: To solve this problem, we use

Original: P2Rank (Kriv´ak & Hoksza, 2018) to segment protein to less than 10 functional blocks (defined as a 20 ˚A graph around the block center), following TankBind (Lu et al., 2022).
Revised: P2Rank (Kriv´ak & Hoksza, 2018) to segment protein to less than 10 functional blocks (defined as a 20 ˚A graph around the block center), following TankBind (Lu et al., 2022).

Original: We then initialize the unbound/bound (depending on the setting) ligand structure with random rotation and translation in each block, dock the ligand, and select the predicted docked pose with the highest self-confidence.
Revised: We then initialize the unbound/bound (depending on the setting) ligand structure with random rotation and translation in each block, dock the ligand, and select the predicted docked pose with the highest self-confidence.

Original: Note that the initial pose might clash with the protein, but these clashes are likely to be repaired during the iterative refinement process.
Revised: Note that the initial pose might clash with the protein, but these clashes are likely to be repaired during the iterative refinement process.

Original: Different from TankBind which selects the final pose through binding affinity estimation for all functional block’s predictions, our pose selection is based on selfconfidence.
Revised: Different from TankBind which selects the final pose through binding affinity estimation for all functional block’s predictions, our pose selection is based on selfconfidence.

Original: As a result, our model only feeds on protein-ligand complex structures, leaving the potential of including more structures without paired affinity for training.
Revised: As a result, our model only feeds on protein-ligand complex structures, leaving the potential of including more structures without paired affinity for training.

Original:  E XPERIMENTS 4. F LEXIBLE S ELF D OCKING
Revised:  E XPERIMENTS 4. F LEXIBLE S ELF D OCKING

Original: As E3Bind is designed to model the flexibility of ligand conformation, it is natural to evaluate it in the flexible blind self-docking setting.
Revised: As E3Bind is designed to model the flexibility of ligand conformation, it is natural to evaluate it in the flexible blind self-docking setting.

Original: We defer the blind re-docking results to Appendix Section A.
Revised: We defer the blind re-docking results to Appendix Section A.

Original: [Table]
Revised: [Table]

Original: Data.
Revised: Data.

Original: We use the PDBbind v2020 dataset (Liu et al., 2017) for training and evaluation.
Revised: We use the PDBbind v2020 dataset (Liu et al., 2017) for training and evaluation.

Original: We follow the time dataset split from (St ¨ ark et al., 2022), where 363 complex structures uploaded later than 2019 serve as test examples.
Revised: We follow the time dataset split from (St ¨ ark et al., 2022), where 363 complex structures uploaded later than 2019 serve as test examples.

Original: After removing structures sharing ligands with the test set, the remaining 16739 structures are used for training and 968 structures are used for validation.
Revised: After removing structures sharing ligands with the test set, the remaining 16739 structures are used for training and 968 structures are used for validation.

Original: Baselines.
Revised: Baselines.

Original: We compare E3Bind with recent deep learning (DL) models and a set of traditional scorebased methods.
Revised: We compare E3Bind with recent deep learning (DL) models and a set of traditional scorebased methods.

Original: For recent deep learning models, TankBind (Lu et al., 2022) and EquiBind (St¨ark et al., 2022) are included.
Revised: For recent deep learning models, TankBind (Lu et al., 2022) and EquiBind (St¨ark et al., 2022) are included.

Original: For score-based methods, QVina-W (Hassan et al., 2017), GNINA (McNutt et al., 2021), SMINA (Koes et al., 2013) and GLIDE (Halgren et al., 2004) are included.
Revised: For score-based methods, QVina-W (Hassan et al., 2017), GNINA (McNutt et al., 2021), SMINA (Koes et al., 2013) and GLIDE (Halgren et al., 2004) are included.

Original: We additionally distinguish between the uncorrected and corrected versions of the recent deep learning models following EquiBind.
Revised: We additionally distinguish between the uncorrected and corrected versions of the recent deep learning models following EquiBind.

Original: The corrected versions adopt post optimization methods (e.g. gradient descent in TankBind (Lu et al., 2022), fast point cloud fitting in EquiBind (St ¨ ark et al., 2022)) to further enforce the intra-ligand geometry constraints when generating the predicted structure.
Revised: The corrected versions adopt post optimization methods (e.g. gradient descent in TankBind (Lu et al., 2022), fast point cloud fitting in EquiBind (St ¨ ark et al., 2022)) to further enforce the intra-ligand geometry constraints when generating the predicted structure.

Original: The uncorrected versions (suffixed with -U), which do not perform post-optimization, reflect the model’s own capability for pose prediction.
Revised: The uncorrected versions (suffixed with -U), which do not perform post-optimization, reflect the model’s own capability for pose prediction.

Original: Specifically, EquiBind-U and E3Bind-U outputs are directly generated by the neural networks and TankBind-U outputs are optimized from the predicted distance map without the ligand configuration loss.
Revised: Specifically, EquiBind-U and E3Bind-U outputs are directly generated by the neural networks and TankBind-U outputs are optimized from the predicted distance map without the ligand configuration loss.

Original: For a fair comparison, E3Bind results are refined from the E3Bind-U predicted coordinates by the post-optimization method of TankBind.
Revised: For a fair comparison, E3Bind results are refined from the E3Bind-U predicted coordinates by the post-optimization method of TankBind.

Original: Metrics.
Revised: Metrics.

Original: We evaluate the quality of the generated ligand pose by the following metrics: (1) Ligand
Revised: We evaluate the quality of the generated ligand pose by the following metrics: (1) Ligand

Original: RMSD , the root-mean-square deviation of the ligand’s Cartesian coordinates, measures at the atom level how well the model captures the protein-ligand binding mechanisms; (2) Centroid Distance , defined as the distance between the average coordinates of predicted and ground-truth ligand structures, reflects the model’s capacity to find the binding site.
Revised: RMSD , the root-mean-square deviation of the ligand’s Cartesian coordinates, measures at the atom level how well the model captures the protein-ligand binding mechanisms; (2) Centroid Distance , defined as the distance between the average coordinates of predicted and ground-truth ligand structures, reflects the model’s capacity to find the binding site.

Original: All metrics are calculated with hydrogen atoms discarded following previous work.
Revised: All metrics are calculated with hydrogen atoms discarded following previous work.

Original: Performance in Flexible Self-Docking.
Revised: Performance in Flexible Self-Docking.

Original: Table 1 summarized the quantitative results in flexible blind self-docking.
Revised: Table 1 summarized the quantitative results in flexible blind self-docking.

Original: Our model achieves state-of-the-art on most metrics.
Revised: Our model achieves state-of-the-art on most metrics.

Original: Specifically, E3Bind shows exceptional power in finding ligand poses with high resolution,
Revised: Specifically, E3Bind shows exceptional power in finding ligand poses with high resolution,

Original: where the percentage of predicted poses with RMSD < 2 ˚ A increases by 33% compared to the previous state-of-the-art DL-based model TankBind.
Revised: where the percentage of predicted poses with RMSD < 2 ˚ A increases by 33% compared to the previous state-of-the-art DL-based model TankBind.

Original: E3Bind achieves 2.1 ˚A for the 25-th percentile of the ligand RMSD, ourperforming all previous methods by a large margin.
Revised: E3Bind achieves 2.1 ˚A for the 25-th percentile of the ligand RMSD, ourperforming all previous methods by a large margin.

Original: These results verify that our model is able to better capture the protein-ligand interactions and generate a geometrically-consistent binding pose.
Revised: These results verify that our model is able to better capture the protein-ligand interactions and generate a geometrically-consistent binding pose.

Original: Notably, among uncorrected deep learning models, E3Bind-U enjoys more significant performance improvement, showcasing its low dependency on additional post-optimization.
Revised: Notably, among uncorrected deep learning models, E3Bind-U enjoys more significant performance improvement, showcasing its low dependency on additional post-optimization.

Original: E3Bind-U also outperforms traditional docking softwares by orders of magnitude in inference speed (Section G), demonstrating its potential for high-throughput virtual screening.
Revised: E3Bind-U also outperforms traditional docking softwares by orders of magnitude in inference speed (Section G), demonstrating its potential for high-throughput virtual screening.

Original: Performance in Flexible Self
Revised: Performance in Flexible Self

Original: Docking for Unseen Protein.
Revised: Docking for Unseen Protein.

Original: We further evaluate our model’s capacity on a subset of the above test set containing 144 complexes with the protein unseen in training.
Revised: We further evaluate our model’s capacity on a subset of the above test set containing 144 complexes with the protein unseen in training.

Original: As shown in Table 2, E3Bind and E3Bind-U show better generalization ability than other DL-based model.
Revised: As shown in Table 2, E3Bind and E3Bind-U show better generalization ability than other DL-based model.

Original: Note that in this setting traditional score-based methods performs better than DL-based
Revised: Note that in this setting traditional score-based methods performs better than DL-based

Original: [Table]
Revised: [Table]

Original: 43.8 TankBind-U 4.
Revised: 43.8 TankBind-U 4.

Original: 65.2 E3Bind-
Revised: 65.2 E3Bind-

Original: [Table] [Table]
Revised: [Table] [Table]

Original: [Table]
Revised: [Table]

Original: models on finding high-quality binding poses ( RMSD < 2 ˚ A ), highlighting the need of increasing the out-of-distribution generalization capability for the latter.
Revised: models on finding high-quality binding poses ( RMSD < 2 ˚ A ), highlighting the need of increasing the out-of-distribution generalization capability for the latter.

Original: That said, we observe that E3Bind produces much fewer predictions that are far off-target ( RMSD > 5 ˚ A ).
Revised: That said, we observe that E3Bind produces much fewer predictions that are far off-target ( RMSD > 5 ˚ A ).

Original: Benefit of Iterative Refinement.
Revised: Benefit of Iterative Refinement.

Original: Figure 2 demonstrates the benefits of iterative refinement in improving ligand pose prediction.
Revised: Figure 2 demonstrates the benefits of iterative refinement in improving ligand pose prediction.

Original: As the ligand undergoes more rounds of coordinate refinement, its
Revised: As the ligand undergoes more rounds of coordinate refinement, its

Original: RMSD decreases and self-confidence score increases.
Revised: RMSD decreases and self-confidence score increases.

Original: Figure 3 visualizes how E3Bind’s iterative refinement process successfully identified the correct binding site after 4 iterations and further refined the ligand conformation for maximal interactions with the protein.
Revised: Figure 3 visualizes how E3Bind’s iterative refinement process successfully identified the correct binding site after 4 iterations and further refined the ligand conformation for maximal interactions with the protein.

Original: The final pose is close to the ground truth with an RMSD of 1.42 ˚A.
Revised: The final pose is close to the ground truth with an RMSD of 1.42 ˚A.

Original: In contrast, Equibind, a method that does not adopt the iterative refinement scheme, yields a pose with an RMSD of 4.18 ˚A.
Revised: In contrast, Equibind, a method that does not adopt the iterative refinement scheme, yields a pose with an RMSD of 4.18 ˚A.

Original: [Figure]
Revised: [Figure]

Original: 4. A BLATION S TUDY
Revised: 4. A BLATION S TUDY

Original: A series of ablation studies are done to investigate different factors influencing the performance (Section E.1).
Revised: A series of ablation studies are done to investigate different factors influencing the performance (Section E.1).

Original: (1) Removing geometry constraints in the pair update modules degrades the fraction of good poses (with RMSD <
Revised: (1) Removing geometry constraints in the pair update modules degrades the fraction of good poses (with RMSD <

Original: 2 ˚ A ) from 23.4% to 20.1%.
Revised: 2 ˚ A ) from 23.4% to 20.1%.

Original: (2) Replacing the Trioformer with a simple concatenation of the protein and ligand embeddings hurts the performance even worse (11.7%), highlighting the strength of Trioformer in extracting geometric features for docking.
Revised: (2) Replacing the Trioformer with a simple concatenation of the protein and ligand embeddings hurts the performance even worse (11.7%), highlighting the strength of Trioformer in extracting geometric features for docking.

Original: (3) Decreasing the number of iterations from 32 to 4 makes the fraction of good poses drop to 14.9%, reiterating the benefit of the iterative refinement process.
Revised: (3) Decreasing the number of iterations from 32 to 4 makes the fraction of good poses drop to 14.9%, reiterating the benefit of the iterative refinement process.

Original: The performance degrades to 17.6% when we remove intra-message passing in the coordinate update modules.
Revised: The performance degrades to 17.6% when we remove intra-message passing in the coordinate update modules.

Original: (5) Removing P2Rank and instead segmenting the protein into 30 random blocks for docking has little impact on performance, showing that E3Bind does not rely on P2Rank as much as TankBind for binding site pre-selection.
Revised: (5) Removing P2Rank and instead segmenting the protein into 30 random blocks for docking has little impact on performance, showing that E3Bind does not rely on P2Rank as much as TankBind for binding site pre-selection.

Original: 4. C ASE S TUDY
Revised: 4. C ASE S TUDY

Original: E3Bind correctly identifies the binding site in an unseen large protein.
Revised: E3Bind correctly identifies the binding site in an unseen large protein.

Original: Figure 4a shows a representative case of a challenging example in blind docking.
Revised: Figure 4a shows a representative case of a challenging example in blind docking.

Original: For an unseen protein, E3Bind (shown in magenta) correctly identifies the native binding site with a low RMSD of 4.2 ˚ A, while QVina-W,
Revised: For an unseen protein, E3Bind (shown in magenta) correctly identifies the native binding site with a low RMSD of 4.2 ˚ A, while QVina-W,

Original: [Figure]
Revised: [Figure]

Original: EquiBind and TankBind dock to three distinct binding sites away from the ground truth.
Revised: EquiBind and TankBind dock to three distinct binding sites away from the ground truth.

Original: Further examination of the self-confidence score shows this pose is the only one with confidence above 0.91, while poses at other binding sites all have confidence below 0.86 (Figure S2).
Revised: Further examination of the self-confidence score shows this pose is the only one with confidence above 0.91, while poses at other binding sites all have confidence below 0.86 (Figure S2).

Original: E3Bind strikes a balance between modeling protein-ligand interaction and structure validity.
Revised: E3Bind strikes a balance between modeling protein-ligand interaction and structure validity.

Original: Figure 4b presents the docking result of a small protein.
Revised: Figure 4b presents the docking result of a small protein.

Original: All methods except TankBind identify the correct binding site, with E3Bind prediction aligning better to the ground truth protein-ligand interaction pattern.
Revised: All methods except TankBind identify the correct binding site, with E3Bind prediction aligning better to the ground truth protein-ligand interaction pattern.

Original: Interestingly, TankBind produces a knotted structure that is invalid in nature.
Revised: Interestingly, TankBind produces a knotted structure that is invalid in nature.

Original: This shows that the two-stage approach might generate invalid protein-ligand distance maps that can not be transformed into plausible structures in the distance-to-coordinate optimization stage.
Revised: This shows that the two-stage approach might generate invalid protein-ligand distance maps that can not be transformed into plausible structures in the distance-to-coordinate optimization stage.

Original: [Figure]
Revised: [Figure]

Original: correctly identifies the binding site from the large protein, while the other methods are off-site.
Revised: correctly identifies the binding site from the large protein, while the other methods are off-site.

Original: (b) E3Bind outperformed the other models in predicting the binding pose accurately, while TankBind generated an invalid structure with knotted rings.
Revised: (b) E3Bind outperformed the other models in predicting the binding pose accurately, while TankBind generated an invalid structure with knotted rings.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: Fast and accurate docking methods are vital tools for small molecule drug research and discovery.
Revised: Fast and accurate docking methods are vital tools for small molecule drug research and discovery.

Original: This work proposes E3Bind, an end-to-end equivariant network for protein-ligand docking.
Revised: This work proposes E3Bind, an end-to-end equivariant network for protein-ligand docking.

Original: E3Bind predicts the docked ligand pose through a feature extraction – coordinate refinement pipeline.
Revised: E3Bind predicts the docked ligand pose through a feature extraction – coordinate refinement pipeline.

Original: Geometry-consistent protein, ligand and pair representations are first extracted by Trioformer.
Revised: Geometry-consistent protein, ligand and pair representations are first extracted by Trioformer.

Original: Then the ligand coordinates are iteratively updated by a context-aware E(3)-equivariant network.
Revised: Then the ligand coordinates are iteratively updated by a context-aware E(3)-equivariant network.

Original: Empirical experiments show that E3Bind is competitive against state-of-the-art blind docking methods, especially without post-optimization of the pose.
Revised: Empirical experiments show that E3Bind is competitive against state-of-the-art blind docking methods, especially without post-optimization of the pose.

Original: Interesting future directions include: modeling the protein backbone/side-chain dynamics to better capture the drug-target interaction and exploring better ways of feature extraction, geometry constraint incorporation and coordinate refinement.
Revised: Interesting future directions include: modeling the protein backbone/side-chain dynamics to better capture the drug-target interaction and exploring better ways of feature extraction, geometry constraint incorporation and coordinate refinement.

Original:  A CKNOWLEDGEMENT
Revised:  A CKNOWLEDGEMENT

Original: We would like to express sincere thanks to Bozitao Zhong.
Revised: We would like to express sincere thanks to Bozitao Zhong.

Original: He contributed a lot to this project.
Revised: He contributed a lot to this project.

Original: We would also like to extend our appreciation to Minkai Xu, Chang Ma, Haoxiang Yang, Minghao Xu,
Revised: We would also like to extend our appreciation to Minkai Xu, Chang Ma, Haoxiang Yang, Minghao Xu,

Original: Zhaocheng Zhu for their helpful discussions and insightful comments.
Revised: Zhaocheng Zhu for their helpful discussions and insightful comments.

Original: This project is supported by
Revised: This project is supported by

Original: Twitter, Intel, the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, the Canada CIFAR AI Chair Program,Mila, Samsung Electronics Co., Ltd., Amazon Faculty Research Award, Tencent AI Lab Rhino-Bird Gift Fund, a NRC Collaborative R&D Project (AI4DCORE-06) as well as the IVADO Fundamental Research Project grant PRF-2019-3583139727.
Revised: Twitter, Intel, the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, the Canada CIFAR AI Chair Program, Samsung Electronics Co., Ltd., Amazon Faculty Research Award, Tencent AI Lab Rhino-Bird Gift Fund, a NRC Collaborative R&D Project (AI4D-CORE-06) as well as the IVADO Fundamental Research Project grant PRF-2019-3583139727.

Original: R EPRODUCIBILITY S TATEMENT
Revised: R EPRODUCIBILITY S TATEMENT

Original: All code for data preprocessing, training and inference will be publicly released upon acceptance.
Revised: All code for data preprocessing, training and inference will be publicly released upon acceptance.

Original: Trioformer detail can be found in Section B. Detail of the confidence loss is in Section C.1.
Revised: Trioformer detail can be found in Section B. Detail of the confidence loss is in Section C.1.

Original: Other implementation details are in Section D.
Revised: Other implementation details are in Section D.

Original: A E XPERIMENT FOR BLIND RE - DOCKING
Revised: A E XPERIMENT FOR BLIND RE - DOCKING

Original: The setting of blind re-docking allows the model to take the ground-truth conformation as prior knowledge.
Revised: The setting of blind re-docking allows the model to take the ground-truth conformation as prior knowledge.

Original: Note that this is a less realistic setting, since in many real-world applications the bound ligand conformation is not known in advance.
Revised: Note that this is a less realistic setting, since in many real-world applications the bound ligand conformation is not known in advance.

Original: Compared to the flexible blind self-docking, the model is given the full ligand conformation as geometric constraints
Revised: Compared to the flexible blind self-docking, the model is given the full ligand conformation as geometric constraints

Original: [Table]
Revised: [Table]

Original: and only needs to predict the rigid-body motion for the molecule to dock into the binding site.
Revised: and only needs to predict the rigid-body motion for the molecule to dock into the binding site.

Original: Although the nature of rigid docking is somewhat contradictory to the E3Bind model as it allows the full flexibility of the ligand structure
Revised: Although the nature of rigid docking is somewhat contradictory to the E3Bind model as it allows the full flexibility of the ligand structure

Original: [Table]
Revised: [Table]

Original: we are still able to enforce a rigid structure through TankBind-style gradient descent postoptimization (Lu et al., 2022) using the ground truth conformation as configuration loss.
Revised: we are still able to enforce a rigid structure through TankBind-style gradient descent postoptimization (Lu et al., 2022) using the ground truth conformation as configuration loss.

Original: [Table]
Revised: [Table]

Original: Results of blind rigid docking are shown in Table S1.
Revised: Results of blind rigid docking are shown in Table S1.

Original: Although the setting of rigid docking is not very suitable for our E3Bind model, we achieve SOTA in most metrics, especially in centroid distance
Revised: Although the setting of rigid docking is not very suitable for our E3Bind model, we achieve SOTA in most metrics, especially in centroid distance

Original: B D ETAILS OF THE T RIOFORMER
Revised: B D ETAILS OF THE T RIOFORMER

Original: The Trioformer comprises a stack of Trioformer blocks detailed in Algorithm 1.
Revised: The Trioformer comprises a stack of Trioformer blocks detailed in Algorithm 1.

Original: It is a variant of
Revised: It is a variant of

Original: AlphaFold2’s Evoformer with baked-in geometry awareness.
Revised: AlphaFold2’s Evoformer with baked-in geometry awareness.

Original: In each block, the ligand embeddings { h l i } are updated with a multi-head cross attention module, with the transformed pair embeddings as bias ( MHAWithPairBias ):
Revised: In each block, the ligand embeddings { h l i } are updated with a multi-head cross attention module, with the transformed pair embeddings as bias ( MHAWithPairBias ):

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: The protein embeddings { h p j } are updated likewise.
Revised: The protein embeddings { h p j } are updated likewise.

Original: The node-level embeddings then go through a two-layer MLP transition modules and reach the block output state.
Revised: The node-level embeddings then go through a two-layer MLP transition modules and reach the block output state.

Original: On another track, the pair embeddings { z ij } are first updated with the node-level embeddings via the outer product module.
Revised: On another track, the pair embeddings { z ij } are first updated with the node-level embeddings via the outer product module.

Original: Then, geometry-aware attentive pair update modules (detailed in Section 3.2) injects implicit geometric constraints into the pair embeddings.
Revised: Then, geometry-aware attentive pair update modules (detailed in Section 3.2) injects implicit geometric constraints into the pair embeddings.

Original: Finally, the pair embeddings go through MLP transition modules and become the remaining part of block outputs.
Revised: Finally, the pair embeddings go through MLP transition modules and become the remaining part of block outputs.

Original: C D ETAILS OF SELF - CONFIDENCE PREDICTION C. CONFIDENCE LOSS
Revised: C D ETAILS OF SELF - CONFIDENCE PREDICTION C. CONFIDENCE LOSS

Original: We train the self-confidence module with a confidence prediction loss L confidence , which is the mean squared error (MSE) between the predicted self-confidence and its target value c ∗ .
Revised: We train the self-confidence module with a confidence prediction loss L confidence , which is the mean squared error (MSE) between the predicted self-confidence and its target value c ∗ .

Original: c ∗ is dependent
Revised: c ∗ is dependent

Original: Input : ligand embeddings { h l i } , protein embeddings { h p j } , pair embeddings { z ij } , intra-ligand distances { d ik } , intra-protein distances { d jk ′ } /* Cross attention updates ligand and protein with pair */ { h l i } ← { h l i } + MHAWithPairBias( { h l i } , { h p j } , { h p j } , { z ij } ) ; { h p j } ← { h p j } + MHAWithPairBias( { h p j } , { h l i } , { h l i } , { z ij } ) ; /
Revised: Input : ligand embeddings { h l i } , protein embeddings { h p j } , pair embeddings { z ij } , intra-ligand distances { d ik } , intra-protein distances { d jk ′ } /* Cross attention updates ligand and protein with pair */ { h l i } ← { h l i } + MHAWithPairBias( { h l i } , { h p j } , { h p j } , { z ij } ) ; { h p j } ← { h p j } + MHAWithPairBias( { h p j } , { h l i } , { h l i } , { z ij } ) ; /

Original: * Node level transitions */ { h l i } ← { h l i } + MLP( { h l i } ) ; { h p j } ← { h p j } + MLP( { h p j } ) ; /* OPM updates pair embeddings with ligand and protein */ { z ij } ← { z ij } + OPM( { h l i } , { h p j } ) ; /* Geometry-aware pair update modules */ { z ij } ← { z ij } + LigandConstrainedAttentivePairUpdate( { z ij } , { d ik } , { d jk ′ } ) ; { z ij } ← { z ij } + ProteinConstrainedAttentivePairUpdate( { z ij } , { d ik } , { d jk ′ } ) ; /
Revised: * Node level transitions */ { h l i } ← { h l i } + MLP( { h l i } ) ; { h p j } ← { h p j } + MLP( { h p j } ) ; /* OPM updates pair embeddings with ligand and protein */ { z ij } ← { z ij } + OPM( { h l i } , { h p j } ) ; /* Geometry-aware pair update modules */ { z ij } ← { z ij } + LigandConstrainedAttentivePairUpdate( { z ij } , { d ik } , { d jk ′ } ) ; { z ij } ← { z ij } + ProteinConstrainedAttentivePairUpdate( { z ij } , { d ik } , { d jk ′ } ) ; /

Original: * Pair transition */ { z ij } ← { z ij } + MLP( { z ij } ) ; return { h l i } , { h p j } , { z ij } on the root-mean-square deviation (RMSD) of the coordinate prediction: it linearly increases with the RMSD of the predicted pose when the latter is small, and is set to a small value c 0 once RMSD reaches γ :
Revised: * Pair transition */ { z ij } ← { z ij } + MLP( { z ij } ) ; return { h l i } , { h p j } , { z ij } on the root-mean-square deviation (RMSD) of the coordinate prediction: it linearly increases with the RMSD of the predicted pose when the latter is small, and is set to a small value c 0 once RMSD reaches γ :

Original: [Equation]
Revised: [Equation]

Original: (cid:16) { x l i } , { x l i ∗ } (cid:17) if RMSD (cid:16) { x l i } , { x l i ∗ } (cid:17) ≤ γ, cotherwise.
Revised: (cid:16) { x l i } , { x l i ∗ } (cid:17) if RMSD (cid:16) { x l i } , { x l i ∗ } (cid:17) ≤ γ, cotherwise.

Original: [Figure]
Revised: [Figure]

Original: C. SELF - CONFIDENCE PREDICTION RESULTS
Revised: C. SELF - CONFIDENCE PREDICTION RESULTS

Original: We plot the relationship between the predicted confidence score and the ligand RMSD for test examples on the left plot of Figure S1.
Revised: We plot the relationship between the predicted confidence score and the ligand RMSD for test examples on the left plot of Figure S1.

Original: Most data points have high confidence scores and low ligand
Revised: Most data points have high confidence scores and low ligand

Original: RMSDs.
Revised: RMSDs.

Original: We further calculate the correlation metrics between the negative RMSD and the predicted confidence score.
Revised: We further calculate the correlation metrics between the negative RMSD and the predicted confidence score.

Original: The results give a Pearson correlation coefficient of 0.641 and a Spearman correlation coefficient of 0.783, indicating a decent pose-selection capability of our self-confidence prediction module.
Revised: The results give a Pearson correlation coefficient of 0.641 and a Spearman correlation coefficient of 0.783, indicating a decent pose-selection capability of our self-confidence prediction module.

Original: We next investigate the behavior of the self-confidence score for success and failure cases (Figure S1, right).
Revised: We next investigate the behavior of the self-confidence score for success and failure cases (Figure S1, right).

Original: The distribution plot shows that the distributions for these two cases have high divergence.
Revised: The distribution plot shows that the distributions for these two cases have high divergence.

Original: For successful docking examples ( RMSD < 5 ˚ A ), the average confidence score is 0.91, while for failure cases, the average confidence score is 0.61.
Revised: For successful docking examples ( RMSD < 5 ˚ A ), the average confidence score is 0.91, while for failure cases, the average confidence score is 0.61.

Original: [Figure]
Revised: [Figure]

Original: D I MPLEMENTATION D ETAILS
Revised: D I MPLEMENTATION D ETAILS

Original: The dimensions of protein, ligand and pair embeddings are set to 128.
Revised: The dimensions of protein, ligand and pair embeddings are set to 128.

Original: Layer normalization and dropout are applied in each block.
Revised: Layer normalization and dropout are applied in each block.

Original: For the implementation of EGCL, we use the normalized coordinate differences instead of the original coordinate differences.
Revised: For the implementation of EGCL, we use the normalized coordinate differences instead of the original coordinate differences.

Original: Specifically, all coordinates are divided by 5 before fed into EGCL and the final coordinates are multiplied by 5.
Revised: Specifically, all coordinates are divided by 5 before fed into EGCL and the final coordinates are multiplied by 5.

Original: SiLU activation function is used in the EGCL layers and LeakyReLU is used in the Trioformer blocks.
Revised: SiLU activation function is used in the EGCL layers and LeakyReLU is used in the Trioformer blocks.

Original: All models including variants in ablation study are trained with Adam optimizer with learning rate 0 . 0001 for 300 epochs.
Revised: All models including variants in ablation study are trained with Adam optimizer with learning rate 0 . 0001 for 300 epochs.

Original: The model with the best valid score (measured by the fraction of predicted pose with
Revised: The model with the best valid score (measured by the fraction of predicted pose with

Original: RMSD <
Revised: RMSD <

Original: 2 ˚ A ) is evaluated on the test set.
Revised: 2 ˚ A ) is evaluated on the test set.

Original: We use the TankBind (Lu et al., 2022) pipeline for data preprocessing, protein segmentation and featurization.
Revised: We use the TankBind (Lu et al., 2022) pipeline for data preprocessing, protein segmentation and featurization.

Original: The only modification is that we remove all complexes in the training set sharing ligands with the test set to reduce information leak at test time.
Revised: The only modification is that we remove all complexes in the training set sharing ligands with the test set to reduce information leak at test time.

Original: For the context-aware decoder, instead of using a set of parameters for each EGCL layer, we adopt the recycling technique of AlphaFold2 (Jumper et al., 2021).
Revised: For the context-aware decoder, instead of using a set of parameters for each EGCL layer, we adopt the recycling technique of AlphaFold2 (Jumper et al., 2021).

Original: Specifically, E3Bind’s decoder contains 4 EGCL layers ( i.e. holds parameters for 4 coordinate update iterations).
Revised: Specifically, E3Bind’s decoder contains 4 EGCL layers ( i.e. holds parameters for 4 coordinate update iterations).

Original: For iterations longer than 4, we recycle the decoder parameters by feeding the previously predicted pose to the decoder as the initial pose.
Revised: For iterations longer than 4, we recycle the decoder parameters by feeding the previously predicted pose to the decoder as the initial pose.

Original: During training, we randomly select the number of cycles from Uniform(1 , N cycle ) , and only backpropagate through the last cycle (by detaching the gradient of its initial pose) to reduce the computational cost.
Revised: During training, we randomly select the number of cycles from Uniform(1 , N cycle ) , and only backpropagate through the last cycle (by detaching the gradient of its initial pose) to reduce the computational cost.

Original: N cycle is set to 32 in our experiments.
Revised: N cycle is set to 32 in our experiments.

Original: E F URTHER A BLATION S TUDY R ESULTS E. A BLATION S TUDY OF E3B IND
Revised: E F URTHER A BLATION S TUDY R ESULTS E. A BLATION S TUDY OF E3B IND

Original: [Table]
Revised: [Table]

Original: E. I MPACT OF P OST O PTIMIZATION
Revised: E. I MPACT OF P OST O PTIMIZATION

Original: To investigate the impact on time cost and performance of different post-optimization methods, we plug each model into each post-optimization method and log the performance in Table S3.
Revised: To investigate the impact on time cost and performance of different post-optimization methods, we plug each model into each post-optimization method and log the performance in Table S3.

Original: In practice, fast point cloud fitting is faster but yields worse results.
Revised: In practice, fast point cloud fitting is faster but yields worse results.

Original: This justifies the choice of gradient descent as the post-optimization method for E3Bind.
Revised: This justifies the choice of gradient descent as the post-optimization method for E3Bind.

Original: Note that for the two benchmarked postoptimization methods, our model consistently outperforms EquiBind and TankBind.
Revised: Note that for the two benchmarked postoptimization methods, our model consistently outperforms EquiBind and TankBind.

Original: [Table]
Revised: [Table]

Original: (Lu et al., 2022) minimizes the weighted sum of the protein-ligand distance error w.r.t.
Revised: (Lu et al., 2022) minimizes the weighted sum of the protein-ligand distance error w.r.t.

Original: the predicted distance map and the intra-ligand distance error w.r.t.
Revised: the predicted distance map and the intra-ligand distance error w.r.t.

Original: the reference distances.
Revised: the reference distances.

Original: F S ENSITIVITY TO I NITIALIZATION
Revised: F S ENSITIVITY TO I NITIALIZATION

Original: E3Bind model’s iterative refinement start from some initial structure at the functional block.
Revised: E3Bind model’s iterative refinement start from some initial structure at the functional block.

Original: The initial structure is generated by RDKit and placed in the functional block with random rotation and translation.
Revised: The initial structure is generated by RDKit and placed in the functional block with random rotation and translation.

Original: Here we investigate the influence of initialization for the model’s final performance.
Revised: Here we investigate the influence of initialization for the model’s final performance.

Original: We generate 30 pose predictions for each protein-ligand complex with different translation and rotation.
Revised: We generate 30 pose predictions for each protein-ligand complex with different translation and rotation.

Original: As shown in Figure S3, 80% of the test set examples have their standard deviation of ligand
Revised: As shown in Figure S3, 80% of the test set examples have their standard deviation of ligand

Original: RMSD within 0.196 ˚A and standard deviation of centroid distance within 0.188 ˚A.
Revised: RMSD within 0.196 ˚A and standard deviation of centroid distance within 0.188 ˚A.

Original: The result indicates E3Bind’s low sensitivity to ligand pose initialization and the strong robustness of its iterative refinement process.
Revised: The result indicates E3Bind’s low sensitivity to ligand pose initialization and the strong robustness of its iterative refinement process.

Original: [Figure]
Revised: [Figure]

Original: G I NFERENCE S PEED
Revised: G I NFERENCE S PEED

Original: The inference speed of different methods is shown in Table S4.
Revised: The inference speed of different methods is shown in Table S4.

Original: Here we report the average time cost in seconds for a single prediction.
Revised: Here we report the average time cost in seconds for a single prediction.

Original: As shown in Section G, we can arbitrarily compose deep learning methods and post-optimization methods.
Revised: As shown in Section G, we can arbitrarily compose deep learning methods and post-optimization methods.

Original: Therefore, here we compare the inference speed of their uncorrected versions.
Revised: Therefore, here we compare the inference speed of their uncorrected versions.

Original: For TankBind and E3Bind, P2Rank segmentation time ( ∼ 0 .
Revised: For TankBind and E3Bind, P2Rank segmentation time ( ∼ 0 .

Original: 15 s in parallel) is included.
Revised: 15 s in parallel) is included.

Original: For TankBind, the time for distance-to-coordinate optimization ( ∼ 0 .
Revised: For TankBind, the time for distance-to-coordinate optimization ( ∼ 0 .

Original: 44 s in parallel) is also included.
Revised: 44 s in parallel) is also included.

Original: As shown in Table S4, the inference speed of E3Bind exceeds traditional methods by a large margin.
Revised: As shown in Table S4, the inference speed of E3Bind exceeds traditional methods by a large margin.

Original: Avg.
Revised: Avg.

Original: Sec
Revised: Sec

Original: [Table]
Revised: [Table]

Original: H A DDITIONAL TRAJECTORIES AND CASE STUDIES
Revised: H A DDITIONAL TRAJECTORIES AND CASE STUDIES

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: I E XAMINING THE V ALIDITY OF G ENERATED S TRUCTURES I. B OND L ENGTH D ISTRIBUTION
Revised: I E XAMINING THE V ALIDITY OF G ENERATED S TRUCTURES I. B OND L ENGTH D ISTRIBUTION

Original: Here we examine the validity of our generated structures by comparing the bond length distribution plots between the predicted and ground truth structures.
Revised: Here we examine the validity of our generated structures by comparing the bond length distribution plots between the predicted and ground truth structures.

Original: We can see from Figure S6 that E3Bind outputs have very similar bond distributions compared to the ground truth.
Revised: We can see from Figure S6 that E3Bind outputs have very similar bond distributions compared to the ground truth.

Original: [Figure]
Revised: [Figure]

Original: I. S TERIC C LASH P ROBLEM
Revised: I. S TERIC C LASH P ROBLEM

Original: Steric clash is a common problem in docking, where two non-bonding atoms show an unnatural overlap in the predicted structure.
Revised: Steric clash is a common problem in docking, where two non-bonding atoms show an unnatural overlap in the predicted structure.

Original: While traditional docking methods avoid generating steric clashes by incorporating a repulsion term in the scoring function, deep learning models have yet to solve this issue.
Revised: While traditional docking methods avoid generating steric clashes by incorporating a repulsion term in the scoring function, deep learning models have yet to solve this issue.

Original: In our experiments, we find that E3Bind produces very few steric clashes 3 ( 3.3% of the test set contain at least one steric clash), while EquiBind (St ¨ ark et al., 2022) are more likely to produce severe steric clashes ( 21% of the test set have clashes) as shown in S7b.
Revised: In our experiments, we find that E3Bind produces very few steric clashes 3 ( 3.3% of the test set contain at least one steric clash), while EquiBind (St ¨ ark et al., 2022) are more likely to produce severe steric clashes ( 21% of the test set have clashes) as shown in S7b.

Original: This is mainly because EquiBind docks the ligand into protein by key-point matching in one shot manner, which prohibits further ligand conformational changes to fit in the protein pocket.
Revised: This is mainly because EquiBind docks the ligand into protein by key-point matching in one shot manner, which prohibits further ligand conformational changes to fit in the protein pocket.

Original: On the other hand, E3Bind iteratively refines the ligand’s position, orientation and conformation, which means it could correct its previous errors, resulting in much fewer steric clash problems.
Revised: On the other hand, E3Bind iteratively refines the ligand’s position, orientation and conformation, which means it could correct its previous errors, resulting in much fewer steric clash problems.

Original: It is also observed that our self-confidence prediction module plays an important role in selecting structures with fewer steric clashes.
Revised: It is also observed that our self-confidence prediction module plays an important role in selecting structures with fewer steric clashes.

Original: Without the confidence module, E3Bind with only 4 iterations has a 13.8% change of generating a pose with steric clashes.
Revised: Without the confidence module, E3Bind with only 4 iterations has a 13.8% change of generating a pose with steric clashes.

Original: When we apply the confidence module for filtering, the fraction drops to 6.6% .
Revised: When we apply the confidence module for filtering, the fraction drops to 6.6% .

Original: Subsequent iterations of E3Bind further decrease the fraction to 3.3% .
Revised: Subsequent iterations of E3Bind further decrease the fraction to 3.3% .

Original: [Figure]
Revised: [Figure]