Independent SE(3)-Equivariant Models for End-to-End Rigid Protein Docking
Authors: Octavian-Eugen Ganea
Xinyuan Huang
Charlotte Bunne
Yatao Bian
Regina Barzilay
Tommi S. Jaakkola
Andreas Krause
Venue: ICLR 2022 Spotlight (2022)
Keywords: protein complexes
protein structure
rigid body docking
SE(3) equivariance
graph neural networks

Abstract: Protein complex formation is a central problem in biology, being involved in most of the cell's processes, and essential for applications, e.g. drug design or protein engineering. We tackle rigid body protein-protein docking, i.e., computationally predicting the 3D structure of a protein-protein complex from the individual unbound structures, assuming no conformational change within the proteins happens during binding. We design a novel pairwise-independent SE(3)-equivariant graph matching network to predict the rotation and translation to place one of the proteins at the right docked position relative to the second protein. We mathematically guarantee a basic principle: the predicted complex is always identical regardless of the initial locations and orientations of the two structures. Our model, named EquiDock, approximates the binding pockets and predicts the docking poses using keypoint matching and alignment, achieved through optimal transport and a differentiable Kabsch algorithm. Empirically, we achieve significant running time improvements and often outperform existing  docking software despite not relying on heavy candidate sampling, structure refinement, or templates.

Published as a conference paper at ICLR 2022
INDEPENDENT SE(3)-EQUIVARIANT MODELS FOR END- TO-END RIGID PROTEIN DOCKING
Octavian-Eugen Ganea**
Xinyuan Huangå
Charlotte Bunne
Yatao Bian' Tencent Al Lab
MIT
ETH Zurich
ETH Zurich
Regina Barzilay MIT
Tommi Jaakkola MIT
Andreas Krause
ETH Zurich
ABSTRACT
Protein complex formation is a central problem in biology, being involved in most of the cell's processes, and essential for applications. e.g. drug design or protein engineering. We tackle rigid body protein-proteit docking, i.c., computationally predicting the 3D structure of protein-proteii complex from the individual un- bound structures, assuming no conformational change within the proteins happens during binding. We design a novel pairwise-inde indeeeedeed -euii-equlvarian graph matching network to predict the rotation and translation to place one of the proteins at the right docked position relative to the second protein. We mathematically guarantee basic principle: the predicted complex is always identical regardless the initial locations and orientations of the two structures. Our model, named EQUIDocK, approximates the binding pockets and predicts the docking poses using keypoint matching and alignment, achieved through optimal transport and a differentiable Kabsch algorithm. Empirically, we achieve significant running time improvements and often outperform existing docking software despite not relying heavy candidate sampling, structure refinement, or templates.
INTRODUCTION
protein Zedopendent inhibitor
In recent breakthrough, ALPHAFOLD 2 (Jumper al., 2021; Senior et al., 2020) provides a solution to grand challenge in biology inforrinnnerrngg pro- tein's diree-dimensional structure from its amino acid sequence (Baek et al., 2021), following the dogma sequence determines structure.
Z-dependent
protein
Besides their complex three-dimensional nature, proteins dynamically alter their function and struc- ture in response to cellular signals, changes in the environment, or upon molecular docking. In par- ticular, protein interactions involved in various biological processes including signal transduction, protein synthesis, DNA replication and repair. Molecular docking is key to understanding protein interactions' mechanisms and effects, and, subsequently, developing therapeutic interventions.
Different views of the 3D structure of a protein complex. a. Surface and b. cartoon view of protein Z and its inhibitor.
Figure 1:
We here address the problem of rigid body protein-protein docking which refers to computationally predicting the 3D structure of a protein-protein complex given the 3D structures of the two proteins in unbound state. Rigid body means no deformations occur within any protein during binding, which a realistic assumption in many biological settings.
Popular docking software (Chen et al., 2003; Venkatraman et al., 2009; De Vries et al., 2010; Torchala al., 2013; Schindler et al., 2017; Sunny and Jayaraj, 2021) are typically computationally expensive,
1 *Corresonnennce to: Octavian Ganea (oct @mit edu) and Yatao Bian (yatac..ll) www.nggaai.cccmmm
* "qqual contribution.
6 Work done during an internship at Tencent AI Lab.
Our code is publicly available: https://g/t thub wttp:///cccmmmttttanneeeeeea
http://hhhh.cccmm/crrrraaa/////e http-//gagneeepepeoooopplpba ic.

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Published as a conference paper at ICLR 2022
ligand B
complex
predict
2 e g
receptor X, e gplir
S013),
2. RX + and 2g
different initial placement swapped ligand
RMSD (2 Zis)
Q e S083 Vl.,,
prodict
R' Soja, o' 2* 2 RXX; 2
restpler
complex
X, - Q.X PR C 2
Xi Ran00-m))
Figure 2: Same output guarantee of EQUIDOCK. We predict rigid transformation place the ligand in the binding location w.r.t the receptor. We mathematically guarantee to output the same complex structure up to SE(3) transformation independently of the initial unbound positions, rotations, or roles of both constituents. (RMSD = Root-mean-squee deviation of atomic positions)
taking between minutes and hours solve single example pair, while not being guaranteed find accurate complex structures. These methods largely follow the steps: i.) randomly sample a large number (e.g., millions) of candidate initial complex structures, ii.) employ scoring function to rank the candidates, iii.) adjust and refine the top complex structures based on an energy model (e.g., force field). We here take a first step towards tackling these issues by using deep learning models for direct prediction of protein complex structures.
Contributions. We design EQUIDoCK, fast, end-to-end method for rigid body docking that directly predicts the SE(3) transformation to place one of the proteins (ligand) at the right location and orientation with respect to the second protein (receptor). Our method is based on the principle that the exact same complex structure should be predicted irrespectively of the initial 3D placements and roles of both constituents (see Fig. 2). We achieve this desideratum by incorporating the inductive biases of pairwise SE(3)-qquivariance and commutativity, and deriving novel theoretical results for necessary and sufficient model constraints (see Section 3). Next, we create EQUIDOCK to satisfy these proper- ties by design, being combination of: i) novel type of pairwise independent 3E(3)-equivarian graph matching networks, ii) an attention- sentative points and aligns them with the binding pocket residues using optimal transport, and iii) differentiable superimposition model to recover the optimal global rigid transformation. Unlike prior work, our method does not use heavy candidate sampling or ranking, templates. task-specific geometric or chemical hand-crafted features, or pre-computed meshes. This enables us to achieve plausible structures with a speed-up of 80-500x compared popular docking software, offering a promising competitive alternative to current solutions for this problem.
keypoint selection algorithm that discovers repre-
2 RELATED WORK
Geometric Deep Learning. Graph Neural Networks (GNNs) are becoming the de facto choice for learning with graph data (Bruna et al., 2013; Defferrard et al., 2016; Kipf and Welling, 2016; Gilmer al., 2017; Xu et al., 2018; Li al., 2019). Motivated by symmetries naturally occurring in different data types, architectures are tailored to explicitly incorporate such properties (Cohen and Welling, 201 6a;b; Thomas et al., 2018; Fuchs et al., 2020: Finzi et al., 2020; Eismann al., 2020; Satorras et al., 2021). GNNs are validated in a variety of tasks such as particle system dynamics or conformation-hant energy estimation (Weiler and Cesa, 2019; Rezende al., 2019).
Euclidean Neural Networks (E(3)-NNs). However, plain GNNs and other deep learning methods do not understand data naturally lying in the 3D Euclidean space. For example, how should the output deterministically change with the input, e.g. when it is rotated ? The recent Euclidean neural networks address this problem. being designed from geometric first-principles They make use of SE(3)- equivariant and invariant neural layers, thus avoiding expensive data augmentation strategies. Such constrained models case optimization and have shown important improvements in biology or chemistry e.g. for molecular structures (Fuchs et al., 2020; Hutchinson et al., 2020; Wu et al., 2021;
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Jumper et al., 2021; Ganea al., 2021) and different types of 3D point clouds (Thomas et al., 2018). Different from prior work, we here derive constraints for pairs of 3D objects via pairwise independent Se(dvieeuuvarriannes, and design a principled approach for modeling rigid body docking.
Protein Folding. Deep neural networks have been used to predict inter-residue contacts, distance and/or orientations (Adhikari and Cheng, 2018; Yang et al., 2020; Senior et al., 2020; Ju et al., 2021), that are subsequently transformed into additional constraints or differentiable energy terms for protein structure optimization. ALPHIAFOLD 2 (Jumper al., 2021) and Rosetta Fold (Back et al., 2021) are state-of-the-ar approaches, and directly predict protein structures from co-evolution information embedded in homologous sequences, using geometric deep learning and E(3)-NNs.
Docking and Interaction. Experimentally determining structures of protein complexes is often expensive and tame-consuming rendering premium on computational methods (Vakser, 2014). Protein docking methods (Chen et al., 2003; Venkatraman et al., 2009; De Vries et al., 2010; Biesiada et al., 2011; Torchala al., 2013; Schindler et al., 2017; Weng et al., 2019; Sunny and Jayaraj. 2021; Christoffer et al., 2021; Yan et al., 2020) typically run several steps: first, they sample thousands or millions of complex candidates; second, they use a scoring function for ranking (Moal et al., 2013; Basu and Wallner, 2016; Launay et al., 2020; Eismann et al., 2020); finally, top-ranked candidates undergo a structure refinement process using energy or geometric models (Verburgt and Kihara, 2021). Relevant to proteinnprotein interaction (PPI) is the task of protein interface prediction where GNNs have showed promise (Fout et al., 2017; Townshend et al., 2019; Liu et al., 2020; Xie and Xu, 2021; Dai and Bailey-Kellogg. 2021). Recently, ALPHA 2 and ROSETTAFOL FOLLD have been utilized as subroutines to improve PPIs from different aspects (Humphreys et al., 2021; Pei et al., 1021:Jovine e.g., combining physics-based docking method CLUSPRO (Kozakov al., 2017; Ghani et al., 2021), or using extended multiple-sequence alignments to predict the structure of heterodimeric protein complexes from the sequence information (Bryant al., 2021). Concurrently to our work, Evans et al. (2021) extend ALPHAFOLD 2 to multiple chains during both training and inference.
Interaction (DTI). DTI aims to compute drug-target binding poses and affinity, playing an essential role in understanding drugs' mechanism of action. Prior methods (Wallach et al., 2015; Li et al., 2021) predict binding affinity from protein-ligant co-crystal structures, but such data is expensive to obtain experimentally. These models are typically based on heavy candidate sampling and ranking (Trott and Olson, 2010; Koes et al., 2013; McNutt et al., 2021; Bao et al., 2021), being tailored for small drug-like like ligands and often assuming known binding pocket. Thus, they not immediately applicable to our use case. In contrast, our rigid docking approach is generic and could be extended to accelerate DTII research part of future work.
3 MATHEMATICAL CONSTRAINTS FOR RIGID BODY DOCKING
We start by introducing the rigid body docking problem and derive the geometric constraints for enforcing same output complex prediction regardless of the initial unbound positions or roles (Fig. 2).
Rigid Protein-Protein Docking Problem Setup. We are given as input pair of proteins forming a complex. They are (arbitrarily) denoted as the ligand and receptor, consisting of n and m residues, respectively. These proteins are represented in their bound (docked) state 3D point clouds Xi E R3 X E 3R3 where each residue's 's location is given by the coordinates of its corresponding a-carbon atom. In the unbound state, the docked ligand is randomly rotated and translated in space, resulting in a modified point cloud X, E paxn, For simplicity and w.l.o.g., the receptor remains in its bound location X₂ X
The task is to predict a rotation R E (3()) and a translation t E RR3 such that RX, + Xi, using input the proteins and their unbound positions X, and X2.
Here, R = R(x, X2) and t = t(X, are functions of the two proteins, where we omit residue identity other protein information in this notation, for brevity.
Note that we assume rigid backbone and side-chains for both proteins. We therefore do not tackle the more challenging problem of flexible docking, but approach offers important step towards it.
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b.
kcypsines obtaited vis. illestion LEGMN
poodiiio ligard
9. - (V).E))
C (25.62)
Neding pocket krypoirt alggmeen
Cass
B
binding pocket partts
Cor
Raand X
nocepior
Xo g
trae deched compler
X2
toue bisding location (Prite lchell,
predicted straciett saper lupasiing tespollay Y,
aisker God/ weboused teput
axxiltary arokding potet
Figure Details on EQUIDOcK's Architecture and Losses. a. The message passing operations in IEGMN guarantee pairwise independent SE(3)-equivariance in Eq. (4), b. We predict keypoints for each protein that are aligned with the binding pocket location using an additional optimal transport (OT) loss, c. After predicting the docked position, we compute MSE loss the ligand, as well as a loss to discourage body intersections.
We desire that the predicted complex structure is independent of the initial locations and orientations of the two proteins, well of their roles see Fig. 2. Formally, wish to guarantee that:
(R(2, 2 + 1(Z,) 22 @ Z = (R(X
X2) X + t(X, X2)) @ X2, (SE(3)-invariance)
(R(X, X + t(x X + 2
(R X X2 x(x xx)) (commutativity)
vQ1, Q2 5O(3), Vgi. g2
R
R 3xn
X₂
R 3xm and Z Q: + +gi,e (1,2).
for any rotations Q1, Q2 and translations gı, g2, where @ is concatenation along columns, and = denotes identity after superimposition, i.e. zero Root-Mean-Square Deviation (RMSD) between the two 3D point sets after applying the Kabsch algorithm (Kabsch, 1976). An immediate question arises:
How do the constraints in Eq. (1) translate into constraints for R(-I-) and t(-l.)
The rotation R and translation t change in systematic way when we apply SE(3) transformations or swap proteins' roles. These properties restrict our class of functions as derived below.
SE(3)-equivariance Constraints. If we apply any distinct SE(3) transformations on the unbound ligand X, and receptor X2, i.c. we dock Q, + g1 onto Q2X₂ + g2, then the rotation matrix ((Q,X, + gi Q2 + g2) and translation vector t QQı + gg 12 - g2) can be derived from the original R X 22) and t(X, (2) assuming that we always do rotations first. In this case, R + gi X2 + g2) can be decomposed into three rotations: i.) apply Qi undo the rotation Q applied on X1, ii.) apply RXx, iii.) apply Q2 to rotate the docked ligand together with the receptor. This gives R(Q, + gi 32|Q2X2 + g2) Q2R(X, (X2) which in turn constraints the translation vector. We provide a formal statement and prove it in Appendix B.I:
Proposition 1. For any Q1, Q2 SO(3), g1, g2 R3, SE((3)-ivariaane) of the predicted docked complex defined by Eq. (1) is guaranteed iff
R X1 + gi Q2 + g2) = Q2,(X,
t(Q X + g Q2 g2) Q2t(2 X2) Q2)( +
As direct consequence of this proposition, we have the following statement. Proposition 2. Any model satisfying Proposition guarantees invariance of the predicted complex w.r.l any SE(3) transformation on X,, and equivariance w.r.t. any SE(3) transformation on X2;
ZZ X₂ Z1 + t(Z, X2) = | + (2),, where Z Q1 + gi
x Z, X + t(x, Z2) Q2 R X X, X + t(x, [X2] + g2, where Zn = Q2 + g2
vQ Q2 (O(3), Vg..g2 e 3 XX,
3xn
XX2 E mm3m
Commutativity. Instead of docking X, with respect to X2, also dock X2 with respect to X1- In this case, we require the final complex structures to be identical after superimposition, i.c., zero RMSD. This property is named commulativity and it is satisfied follows (proof in Appendix B.2).
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Proposition 3. Commutarivity defined by Eq. (1) is guaranteed iff
x R (x1 t(X - (Xx
(3)
Point Permutation Invariance. We also enforce residue permutation invariance. Formally, both [Xx and (2) should not depend on the order or columns of X, and, resp., of X2.
4 EQUIDOCK MODEL
Protein Representation. A protein is sequence of amino acid residues that folds in a 3D structure. Each residue has a general structure with side-chain specifying its type, allowing us to define a local frame and derive SE(3)-invariant features for any pair of residues see Appendix
We represent a protein as a graph G (V,E), similar to Fout et al. (2017); Townshend et al. (2019); Liu al. (2020). Each node i represents one residue and has 3D coordinates x; E R corresponding to the a-carbon atom's s location. Edges are given by k-nearest-neighhor (k-NN) graph using Euclidean distance of the original 3D node coordinates.
Overview of Our Approach. Our model is depicted in Fig. 3. We first build k-NN protein graphs -== (Vi, E and G2 = (V2, We then design SE(3)-invariant node features F; E 1d F2 E Rdxm and edge features {fz V(i,j) E E U & (see Appendix A).
Next, we apply several layers consisting of functions & that jointly transform node coordinates and features. Crucially, we guarantee, by design, pairwise independent SE(3) equuivaaaae for coordinate embeddings, and invariance for feature embeddings. This double constraint is formally defined:
Given Z1, H1, Zz H2 oX1,, F1, X2, F2)
we have Q, Zi + gi. Hi Q2Z2 + B2, H2 (Q + gi. Fi, Q2X, + g2, F2).
(4)
TQ1, Q2 (O(3), Vg. 82 R
We implement & as novel type of pressage-passingg neural network (MPNN). We then use the output node coordinate and feature embeddings to compute RXX,Xx, and t(X, X2). These functions depend on pairwise interactions between the two proteins modeled as cross-messages, but also incorporate the 3D structure in a pairwise-indepeedent Proposition and Proposition 3. We discover keypoints from each protein based on a neural attention mechanism and softly guide them to represent the respective binding pocket locations via an optimal transport based auxiliary loss. Finally, we obtain the SE(3) transformation by superimposing the two keypoint sets via a differentiable version of the Kabsch algorithm. An additional soft-constraint discourages point cloud intersections. We now detail each of these model components.
way to satisfy Eq. (1),
Independent Equivarann Graph Matching Networks (IEGMNs). Our architecture for + satisfying Eq. (4) called Independent Eq33--qqivvarinn Graph Matching Network (IEGMN) - see Fig. 3. It extends both Graph Matching Networks (GMN) (Li et al., 2019) and B(3)-Equivarian Graph Neural Networks (E(3)-GNN) (Satorras et al., 2021). IEGMNs perform node coordinate and feature embedding updates for input pair of graphs G = (V1, 3,), G2 (V2, Eg), and use inter- and intra- node messages, as well as E(3) equiavaarrr coordinate updates. The /-th layer of IEGMNs transforms (1)
node latent/feature embeddings (h hevuu, and node coordinate embeddings (x ()) Revuv, as
Mj-i = 40 (h h (1)
exp(-1i 1x -x x2" (1)) Vej-i EE
(5)
My-+ aーー Wh ( Vi V1,j E V2 or E V2,j E
(6)
m; = NO1 JEV())
m, m Vi V
(7)
p. = Σ Hy-+i Vi Vı, and
μ. Σ Hj Vi V/2
(8)
JEV/
jeV,
x,
(1+1) = rpx; (0)
+ (1- 7)x + Σ (x x 30" (m.-)..Vi U
(9)
JEN(()
h. ((+)) (1 - B) h (1)
3 * (h (
m, Uis f.), Vi V1 U12
(10)
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where ((t) are the neighbors of node i; 4" is real-valued (scalar) parametric function; W is learnable matrix; s", yp" are parametric functions (MLPs) outputting a vector R.; fy-+s and f; are the original edge and node features (extracted SE(3)- 3E(3)-invariantl from the residues), aj-si is an attention based coefficient with trainable shallow neural networks e/" and t/
exp((19(h))) (1), *5*(** (h)
0j Σ, Expe(((p) (h. ()) w/th (h )))
(11)
Note that all parameters of W, o² pt, ,p, 1h9, gbe can be shared different for different IEGMN layers. The output of several IEGMN layers is then denoted as:
Z1 E R 3xn H1 E Rd Z2 E 3xm H2 E Rdxm = FI
(12)
is then straightforward to prove the following (see Appendix B.3): Proposition 4. IEGMNs satisfy the pairwise independent Je(3)-qquvaaiianc property in Eq. (4).
Keypoints for Differentiable Protein Superimposition. obtain K points for each protein, I,, Y2 E , which we name keypoints. We train them to become representative points for the binding pocket of the respective protein pair (softly-enforeed by an additional loss described later). If this would holds perfectly, then the superimposition of Y and Y2 would give the corresponding ground truth superimposition of X, and X2. Our model is
Next, use multi-head attention to
k
3 22j
YIG
Y2k
where Zi; denotes the i-th column of matrix Z,, and of so softmar, Va Di W (H)))((H2)) are attention scores (similarly defined for 85), with W E BRdd a parametric matrix (different for each attention head), a linear layer plus a LeakyReLU non-linearity, and ju(-) is the mean vector.
Differentiable Kabsch Model. We design the rotation and translation that docks protein into protein to be the same transformation used to superimpose Y, and Y2 - see Fig. 3. For this, compute differentiable version of the Kabsch algorithm (Kabsch, 1976) follows. Let
A Y Y E R3x3 computed using zero-mean keypoints. The singular value decomposition (SVD) is A U,SUJ, where U2, U₁ 03).. Finally, we define the differentiable functions
1
0
2(X,(X2;0) 0) = U2
1
0
0
UT. where d = sign(det (c2((UUUUU))
0
d
(13)
t(X, = ((Y))
(2;;
where pa-) is the mean vector of point cloud. is straightforward prove that this model satisfies all the equivariance properties in Eqs. (1) to (3). From a practical perspective, the gradient and backpropagation through the SVD operation was analyzed by (Ionescu et al., 2015; Papadopoulo and Lourakis, 2000) and implemented in the automatic differentiation frameworks such as PyTorch.
MSE Loss. During training, we randomly decide which protein the receptor (say protein 2), keep it in the docked position (i.e., X, = X2), predict the SE(3) transformation using Eq. (13) and use it to compute the final position of the ligand as X ,X)x + t(X, (2)). The mean squared error (MSE) loss is then LMSE >i X x;
Optimal Transport and Binding Pocket Keypoint Alignment. As stated before, we desire that Y1 and Y2 are representative points for the binding pocket location of the respective protein pair. However, this needs to be encouraged explicitly, which we achieve using additional loss.
We first define the binding pocket point sets, inspiring from previous PPI work (Section 2). Given the residues' a-carbon locations of the bound (docked) structures, Xi and X,, select all pairs of residues less than Euclidean distance (r SÅ in our experiments). We assume these are all interacting residues. Denote these pairs as {xi,, x²,), E 1,..., across data pairs. We compute midpoints of these segments, denoted P P IR3xS where
.....,,), S). where S is variable

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Pi. = = = 0.5 (xi= + x(g). We view Pi and P2 binding pocket points. In the unbound state, these sets are randomly moved in space together with the respective protein residue coordinates X1 and X2. We denote them as PI, P2 paxs For clarity, if x QXi + g. then P1 QPi +g
We desire that Y1 is a representative set for the 3D set P1 (and, similarly, Y2 for P2). However, while training time know that every point P1, corresponds to the point p2s (and, similarly, VIA aligns with Y2k- by assumption), we unfortunately do not know the actual alignment between points in Y and Pt, for every E {1,2). This can be recovered using an additional optimal transport loss:
for
min (E)(S,K)
(T, C)2 where C., 1|
Pis|
1| P2sl1?,
(14)
where U(S, K) is the set of S X K transport plans with uniform marginals. The optimal transport plan is computed using an Earth Mover's Distance and the POT library (Flamary et al., 2021), while being kept fixed during pack-propagttion and optimization when only the cost matrix is trained.
Note that our approach assumes that Y1k corresponds to Y2k, for every k E (1...., K). Intuitively, each attention head & will identify specific geometric/chemical local surface feature of protein by y Ie, and match its complementary feature of protein 2 by Y2k-
Avoiding Point Cloud Intersection. In practice, our model does not enforce useful inductive bias, namely that proteins forming complexes are never "intersecting" with each other. To address this issue, we first state a notion of the "interior" of protein point cloud. Following previous work (Sverrisson et al., 2021; Venkatraman et al., 2009), we define the surface of a protein point cloud X E R3*= as fx E R3 G(x) = 7). where G(x) -o In In((E exp(-I|x x; 1 /o)) The parameters a and y are chosen such that there exist no "holes" inside protein (we found y = 10, a = 25 to work well, see Appendix E). As a consequence, the interior of the protein is given by [x E R3, G(x) < <.). Then, the condition for non-intersecting ligand and receptor can be written as (x2,) > 7, Vj E l....,m m and G2 > y, Vi E I,...,n. n. As loss function, this becomes
1 CNi -
1 10x + m (x22))).
G1
(15)
Surface Aware Node Features. Surface contact modeling is important for protein docking. We here design novel surface feature type that differentiates residues closer to the surface of the protein from those in the interior. Similar to Sverrisson et al. (2021), we prioritize efficiency and avoid pre-computing meshes, but show that our new feature is a good proxy for residue's 's depth (i.c. distance to the protein surface). Intuitively, residues in the core of the protein are locally surrounded in all directions by other residues. This is not true for residues on the surface, e.g., neighbors are in a half-space if the surface is locally flat. Building k-NN protein graph, we compute the norm of the weighted average of its neighbor forces, which can be interpreted as the normalized gradient of the G(x) surface function. This SE(3)-invariant feature is
this intuition, for each node (residue) in the
Ax(x; = 1|ENEN wiy - xx]]l EveN, W. x
cxp(-IIx; x2 113/2 ΣjEN, exp( | - X,) 112/2*
where WER'
(16)
Intuitively, as depicted in Fig. 8, residues in the interior of the protein have values close to since they are surrounded by vectors from all directions that cancel out, while residues near the surface have neighbors only in a narrower cone, with aperture depending on the local curvature of the surface. We show in Appendix C that this feature correlates well with more expensive residue depth estimation methods, e.g. based on MSMS, thus offering a computationally appealing alternative. We also compute an estimation of this feature for large dense point clouds based on the local surface angle.
5 EXPERIMENTS
Datasets. We leverage the following datasets: Docking Benchmark 5.5 (DB5.5) (Vreven et al.,
2015) and Database of Interacting Protein Structures (DIPS) (Townshend et al., 2019). DB5.5 is gold standard dataset in terms of data quality, but contains only 253 structures. DIPS is larger protein complex structures dataset mined from the Protein Data Bank (Berman et al., 2000) and tailored for rigid body docking. Datasets information is given in Appendix D. We filter DIPS to only
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Table Complex Prediction Results. As in the main text, the proprictary baselines might internally use parts of the test sets (e.g. to extract templates or features), thus their numbers might be optimistic.
DIPS Test Set
DB5.5 Test Set
Complex RMSD
Interface RMSD
Complex RMSD
Interface RMSD
Methods
Median Mean
Std
Median Mean
Std
Median
Mean
Std
Median Mean
Std
ATTRACT
17.17
14.93
10.39
12.41
14.02
11.81
9.55
10.09
9.88
7.48
10.69
10.90
HDоCK
6.23
10.77
11.39
3,90
8.88
10.95
0.30
5.34
12.04
0.24
4.76
10.83
CLUSPRO
15.76
14.47
10.24
12.54
13.62
11.11
3.38
8.25
7.92
2.31
8.71
9.89
PATCHDoCK
15.24
13.58
10.30
11.44
12.15
10.50
18.26
18.00
10.12
18.88
18.75
10.06
EQUIDOCK
13.29
14.52
7.13
10.18
11.92
7.01
14.13
14.72
5.31
11.97
13.23
4.93
C-RMED ditributions (IPS seaf)
I-9M52 distittttion D35 tea))
C-RMED vs H-FMME scatter piar IDPS need)
Meshed HBOOX
QUSPRO
Complte PMSD
Interboe RASD
HOURDOCK
16
ATTRCTT
1000X
GaSARo
sOURbOOK
ATTACC
HOOCK
NITHHOOX
eqUIDOO
Complex FMBD
Figure 4: Complex-RMSH distributions (DIPS test set); b. test set); c. scatter plot for C-RMSD vs I-RMSD (DIPS test set).
distributions (DIPS
keep proteins with at most 10K atoms. Datasets are then randomly partitioned in train/val/test splits of sizes 203/25/25 (DB5.5) and 39,937/974/965 (DIPS), For DIPS, the split is based on protein family to separate similar proteins. For the final evaluation in Table 1, we use the full DB5.5 test set, and randomly sample 100 pairs from different protein families from the DIPS test set. distrbbttinn (DIPS
Baselines. We compare our EQUIDOCK method with popular state-of-the-art docking software CLUS- PRO (PIPER) (Desta et al., 2020; Kozakov et al., 2017), (Schindler et al., 2017: de Vries et al., 2015), PATCHDOCK (Mashiach et al., 2010; Schneidman- Duhovny et al., 2005), and HDoCK (Yan et al., 2020; 2017b;a; Huang and Zou, 2014; 2008). These baselines provide freenfriendly local packages suitable for automatic experiments or webservers for manual submissions.
Inference
10" 20
Huntim (60c)
30
10a
ATTRACT
HDOCK
CLUSPRD
PATCHOOCK EQUlbOk
Evaluation Metrics. To measure prediction's quality, we report Complex Root Mean Square Deviation (CRMSD) and Interface Root Mean Square Deviation (IRMSD), de-
Figure 5: Inference running time distri- butions (log10 10 scale).
fined below. Given the ground truth and predicted complex structures, Z' E 33x(a+m) and Z 33x(++m)) (n+m), we first superimpose them using the Kabsch algorithm (Kabsch, 1976), and then Z ZIZ We compute I-RMSD similarly, but using only the coor-
compute C-RMSD
dinates of the interface residues with distance less than 8A to the other protein's residues. For fair comparison among baselines, we use only the q-carbon coordinates to compute both metrics.
Training Details. We train our models on the train part of DIPS first. using Adam (Kingma and Ba, 2014) with learning rate 2c-4 and early stopping with patience of 30 epochs. We update the best validation model only when it achieves score of less than 98% of the previous best validation score, where the score is the median of Ligand RMSD on the full DIPS validation set. The best DIPS validated model is then tested on the DIPS test set. For DB5.5, we fine tune the DIPS pre-trained
CClusPro: https: /ttp/rlssoooooo//o///oo...e....e edu/. Attrac:: www www.att ract .ar.ct.ph..oe tum. de services/ APTRACT/ATIRR vdi .g, PatchDock: https: http::////oooffd/// cs tww.tau.aaa.... 1/ttcc.../ttccccccc//// HDOCK: http://htpp ab. phys. www.h.uuuuu............ cn httt///fwwaeeeeeoo HHOCK//
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EquiDock
PatchDock
HDOCK
ClusPro
Ground Truth
CRMSD= 7.13
CRMSD= 17.57
CRMSD= 18,00
CRMSD 18.56
PDB ID: 3DMP
Figure 6: Visualization of a protein complex successfully predicted by EQUIDOCK, Note that all other methods find the binding interface on the wrong side of the black protein.
model on the DB5.5 training set using learning rate le-4 and early stopping with 150 epochs patience. The best DB5.5 validated model is finally tested on DB5.5 test set. During training, we randomly assign the roles of ligand and receptor. Also, during both training and testing, we randomly rotate and translate the ligand in space (even though our model invariant to this operation) for all baselines.
Complex Prediction Results. Results are shown in Table 1, Fig. 4 and Appendix E. We note that our method is competitive and often outperforms the baselines. However, we do not use heavy candidate sampling and re-ranking, we do not rely task-specific hand-crafted features, and we currently do not perform structure fine-tuning, aiming to predict the SE(3) ligand transformation in direct shot. Moreover, we note that some of the baselines might have used part of our test set in validating their models, for example to learn surface templates, thus, their reported scores might be optimistic. Notably, HDOCK score function more appropriate comparison would require us to re-build these baselines without information from possible without open-source implementations.
validated on DB4 which overlaps with DB5.5. A
test sets, a task that is currently
Note that EQUIDOCK between 80-500 times faster than the baselines. This is especially important for intensive screening applications that aim to scan over vast search spaces, e.g. for drug discovery. In addition, it is also relevant for de novo design of binding proteins (e.g. antibodies (Jin al., 2021)) for cases when protein docking models are just a component of significantly larger end-to-end architectures targeting more involved biological scenarios, e.g., representing a drug's mechanism of action or modeling cellular processes with single model as opposed to a multi cilti-ppppelnee architecture.
Computational Efficiency. We show inference times in Fig. and Table 4.
Visualization. We show in Fig. successful example test DIPS protein pair for which our model significantly outperforms all baselines.
CONCLUSION
We have presented extremely fast, end-to-end rigid protein docking approach that does not rely on candidate sampling, templates, task-specific features pre-computed meshes. Our method smartly incorporates usefull rigid protein docking priors including commutativity and pairwise independent SE(3)-equivariaacee, thus avoiding the computational burden of data augmentation.
We look forward to incorporating more domain knowledge in EQUIDOcK and extend for flexible docking and docking molecular dynamics, as well as adapt it to other related tasks such as drug binding prediction. On the long term, we envision that fast and accurate deep learning models would allow to tackle more complex and involved biological scenarios, for example model the mechanism of action of various drugs or to design de novo binding proteins and drugs to specific targets (e.g. for antibody generation). Last, hope that our architecture can inspire the design of other types of biological 3D interactions.
First, our presented model does not incorporate protein flexibility which is necessary for various protein families, e.g., antibodies. Unfortunaelly both DB5 and DIPS datasets are biased towards rigid body docking Second, only prevent steric clashes using soft constraint (Eq. (15)) which has limitations (see Table 6). Future extensions would card-constrain the model to prevent such artifacts.
Limitations.

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ACKNOWLEDGEMENTS
The authors thank Hannes Stärk, Gabriele Corso, Patrick Walters, Tian Xie, Xiang Fu, Jacob Stern, Jason Yim, Lewis Martin, Jeremy Wohlwend, Jiaxiang Wu, Wei Liu, and Ding Xue for insightful and helpful discussions. OEG is funded by the Machine Learning for Pharmaceutical Discovery and Synthesis (MLPDS) consortium, the Abdul Latif Jameel Clinic for Machine Learning Health, the DTRA Discovery of Medical Countermeasures Against New and Emerging (DOMANE) threats program, and the DARPA Accelerated Molecular Discovery program. This publication was created part of NCCR Catalysis (grant number 180544). National Centres of Competence in Research funded by the Swiss National Science Foundation. RB and TJ also acknowledge support from NSF Expeditions grant (award 1918839): Collaborative Research: Understanding the World Through Code.
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Appendix
CONTENTS
A Representing Proteins as Graphs
15
B Proofs of the Main Propositions
16
B.1 Proof of Proposition 1.
16
B.2 Proof of Proposition 3.
17
B.3 Proof of Proposition 4.
17
C Surface Features
17
D Datasets
18
E More Experimental Details and Results
20
A REPRESENTING PROTEINS AS GRAPHS
protein comprised amino acid residues. The structure of amino acid residue shown in Figure Fig. 7. Generally, an amino acid residue contains amino (-NH-), a-carbon atom and carboxyl (-CO-), along with side chain (R) connected with the a-carbon atom. The side chain (R) is specific to each type of amino acid residues.
H
O
N
u1
n
t;
a-Carbon
R
side chain
Figure 7: Representation of amino acid residue and its local coordinate system.
We work on residue level (our approaches be extended to atom level well). A protein is represented by a set of nodes where each node an amino acid residue in the protein. Each node has 3D coordinate x; E R3 which is the 3D coordinate of a-carbon atom of the residue.
The neighborhood of a node is the set of Is (k = 10 in our experiments) nearest nodes where the distance is the Euclidean distance between 3D coordinates.
Node feature is a one dimension indicator (one-ho hot encoding) the type of amino acid residue. This dimension indicator will be passed into embedding layer.
Local Coordinate System. Similar to Ingraham et al. (2019) and Jumper et al. (2021), we introduce a local coordinate system for each residue which denotes the orientation residue. Based on this, further design SE(3)-invariant edge features. As shown in Figure 7, for residue i,
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denote the unit vector pointing from a-carbon atom nitrogen atom u,. We denote the unit vector pointing from d-carbon atom to carbon atom of the carboxyl (-CO-) as t, u, and t, together define plane, and the normal of this plane is n, u,xt, [t, Finally, define n, x u, Then n, u, and v; together form the basis of residue i's local coordinate system. They together encode the orientation of residue i.
Then we introduce the edge features of an edge j E These features describe the relative position of with respect to i, the relative orientation of with respect to i and the distance between j and i.
Relative Position Edge Features position of j with respect to i:
First we introduce the edge features Pj-v which describe relative
Pj-1
u,
X;
V;
Relative Orientation Edge Features As we mention above, each residue has orientation which carries information. Here we introduce the edge features qj-si kj-si and Eー-> which describe relative orientation of j with respect to i:
u
n))
k--1
[uy]
u
[vg]
Distance-Based Edge Features tion of distance edge features:
Distance also carries information. Here we use radial basis func-
f--rr
1,
Where and scale parameters (ar hirgn are hyperparameters. experiments, the set of scale
parameters used is (1.5" 0, 1,2, ...; 14). So for each edge, there are 15 distance-based edge features.
Surface Aware Node Features We additionally compute 5 surface aware node features defined in Eq. (16) using E (1., 2., 5., 10., 30.).
B PROOFS OF THE MAIN PROPOSITIONS
B. PROOF OF PROPOSITION 1.
Proof. Denote the predicted ligand position by ([XX/X22) + t(x X2) = X,
Assume first that SE(3) invarianve of the predicted docked complex defined by Eq. (1) is satisfied.
Then the transformation dock Q1 + gi with respect Q2 + g2 is the same as the transfor- mation change Q X1 + gı into Q₂Xı g2. We use the notation: R (X, (R())[X²)) Then, we have the following derivation steps:
R(X, + (X,) X,
X, R Xx) (2) R (X,
X, (X R Xx, Q Q2
-
X, + (X = X
X, 0/0
Q2R(X, ( (Q:X++ - 42t Q
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From the last equation above, one derives the transformation of Q1X, g1 into Q2X, + 82, which is, by definition of the functions R and t, the same as the transformation to dock Q1X1 + +g with respect to Q2 + 82- This transformation is
R Q X1 gi Q2 B2) Q Q
( + +1 Q2 + g2) Q2t(X, x2) X2 gi + -2
which concludes the proof.
Conversely, assuming constraints in Eq. (2) hold, we derive that Q1 + g1 transformed into Q2X, + g2. which then is trivial to check that satisfies SE(3)-invariance of the predicted docked complex defined by Eq. (1).
B.2 PROOF OF PROPOSITION 3.
(R(Xi As in Appendix B.1, denote 2 + tt((, X2) X1 Then the transformation dock X2 with respect to X1 is the same as the transformation to change X₃ back X1, which is
Proof. We use the notation R (x1XX||s
R X2 X1 + t(x X2) = X1
X; R [ X (X,, Xx) = R T (X,
X; R XX X2)X; RR (X) (x2)
From the last equation above, we derive the transformation to change X back to X,, which is the same as the transformation to dock X₂ with respect to X₁.
B.3 PROOF OF PROPOSITION 4.
Proof. Let X (1+1) H (++1 x((1)
((+1) H (1+1) = IEGMN(X ( x " be the output of an
IEGMN layer. Then, any matrices Q1, Q2 E S0(3) and any translation vectors g1; g2 R we
want to prove that IEGMN satisfy the pairwise independent SE(3)-equivarianc) property:
QiX(++)) +g,, H ((+I), Q2X ((+1)
H ((+1)
( +gi, H Q2X," +g, H
+g2,
where each column of ( R H Rdxa ,x!") R3xm and !!! E Rdxm represent individual node's coordinate embedding or feature embedding.
We first note that the equations of our proposed IEGMN layer that compute messages mj-vis Mj-vis m; and μ are -Evaa-inn Indeed, they depend on the initial features which are SE(3)-invrriant by design, the current latent node embeddings (h Jie well as the Euclidean distances
the current node coordinates (x! (1) Jev Thus, we also derive that the equation that computes the
new latent node embeddings h, ((+1) is SE(3)-invariant. Last, the equation that updates the coordinates
x: (1+1) is SE(3) with respect to the 3D coordinates of nodes from the same graph but SÉ(3)-invariant with respect to the 3D coordinates of nodes from the other graph since only uses invariant transformations of the latter.
C SURFACE FEATURES
Visualization. We further discuss surface features introduced in Eq. (16). We first visualize their design intuition in Fig. A synthetic experiment shown in Fig. 9.
Correlation with MSMS features. Next, we analyze how accurate are these features compared to established residue depth estimation methods, e.g. based on the MSMS software (Sanner et al., 1996). We plot the Spearman rank-orde, correlation of the two methods in Fig. 10. We observe concentrated distribution with a mean of 0.68 and a median of 0.70, suggesting a strong correlation with the MSMS depth estimation.
17

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Published as a conference paper at ICLR 2022
Closed form expression. Finally, we prove that for points close to the protein surface and rounded by (infinitely) many equally-distanced and equally-spaced points, one can derive closed form expression of the surface features defined in Eq. (16). See Fig. 11. We work in 2 dimensions, but extensions to dimensions are straightforward. Assume that the local surface at point x; has angle a. Further, assume that x; is surrounded by N equally-distanced and equally-spacel points denoted by x-. Then, all a will be identical. Then, the summation vector in the numerator of Eq. (16) will only have non-zero components on the direction that bisects the surface angle, as the other components will cancel-out. Then, under the limit N - J0, we derive the closed form expression:
pe(x;;))=
1 N
Σ KEN:
2 N ==0
cos(
ja
No
2 fa/2
cos(0)d0 sin(a/22
Xx
X
a
(17)
Normalized weighted force mean voctor of longth p(
residue foroa vector m2 xp)
Protein surface
Figure 8: Intuition behind surface features defined in Eq. (16). a. Residues in the core (interior) of protein are likely to have small weighted average directionally spread neighboring forces, while b. residues close to the surface receive vector contributions from narrower space subset and, thus, have larger p feature values.
100
1.00
07
1.0
875
075
06
075
858
050
059
025
029
025
800
600
-025
-050
02
-0.75
81
01
-0.75
42
-.00
-1.00
-3.0
-0.
-1.
-05
0.3
05
15
-05
0.0
0.5
= 0.01
= 0.1
Figure 9: Distribution of our surface feature values defined in Eq. (16) for 500 points uniformly distributed in the unit circle. One can notice strong correlation with the depth (i.e. distance to surface) which is further quantified in Fig. 10. Note that the scale for À in this synthetic experiment differs from that of real proteins.
D DATASETS
The overview of datasets is in Table 2. DB5.5 is obtained from https::tpth/tt http::////llaaa...mmeeeee umassmed. edu/ benchmark/, while DIPS downloaded from https: //tpt/bghhhbbb com/ http//dorllbbDDeBe While DIPS contains only the bound structures, thus currently being only suitable for rigid docking, DB5.5 also includes unbound protein structures, however, mostly showing rigid structures Fig. 12.
see
18

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Published as a conference paper at ICLR 2022
Pesquency in the DIPS t:zt 505
5-
0.0
0.4
0.8
order confriciant
Figure 10: Distribution of the Spearman rank-order coefficient computed per each protein as the correlation between MSMS residues' depths and our surface features defined in Eq. (16) (for À = 30). Histogram computed over the ligands in the DIPS test (100 proteins).
2)x; 2 (an(
Angle
Protein
For points close to the protein surface where the local surface angle is we can derive closed form expression for the surface feature defined in Eq. (16) under the assumption of being surrounded by infinitely many points at approximately equal distances and equally-spaced. A similar derivation is possible 3D.
Figure 11:
imddian 1.2
A
Note: mean distance between every two nearest Ca atoms is -3.8
120
Frequency D85.5
80
40
RMSD (A) bound unbound structures (of the Co atomic coordinates)
Figure 12: Distance (RMSD) between unbound and bound structures of the DB5.5 dataset reveals that most of the proteins are relatively rigid. Thus, better datasets are needed to tackle the docking conformational change problem.
19

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Published as a conference paper at ICLR 2022
Table 2:
Overview of Datasets. For DIPS, the statistics of number of residues and atoms per protein is based on subset consisting of 200 proteins.
Dataset # Pairs of Proteins # Residues per Protein Atoms per Protein
DIPS DB5.5
41876
276 (# 189)
2159 (= 1445) 2089 (+ (694)
253
268 (#2215)
E MORE EXPERIMENTAL DETAILS AND RESULTS
Baseline Failures. On the test sets, ATTRACT fails for 'IN2C' in DB5.5, 'oi_4oip jo_4ooippddb__" "oi coi_4oip-pbbl_.. 3' and [p7_4p7s.pdbl__ DIPS. For such failure cases, we use the unbound input structure as the prediction for metrics calculation.
Hyperparameters. We perform hyperparameter search over the choices listed in Table and select the best hyperparameters for DB5.5 and DIPS respectively based on their corresponding validation sets.
Table Hyperparameter choices. LN stands for layer normalization, BN stands for batch normaliza- tion.
Hyperparameter
Choice
Node degree (for k-NN)
10
Weight the intersection loss
0.0, 1.0
Normalization for h; of IEGMN layers Normalization for others
No, LN
No, BN, LN
Number of attention heads (K)
25, 50, 100
Slope of leaky relu
0.1, 0.01
Dimension of h, of IEGMN layers Dimension of residue type embedding
32,
32, 64
Number of IEGMN layers If IEGMN layers except the first one share parameters True, False
5, 8
Y) of coordinates skip connection
0.0, 0.25
Weight decay
0, lc-5, lc-4, le-3
Detailed Running Times. In addition to the main text, we show Table 4 detailed running times of all methods. Hardware specifications are follows: ATTRACT 2.2 GHz CPU; HDOCK was run on a single Intel Xeon Gold 6230 2.1 GHz CPU; EQUIDOCK was on single Intel Core 19-9880H 2.3 GHz CPU. CLUSPRO and PATCHDocK have been manually run using their respective web servers.
6-Core Intel Core i7
Plots for DB5.5. show the corresponding plots for DB5.5 results in Fig. 13.
Table Inference time comparison (in seconds). Note: ClusPro and PatchDock were run manually using the respective public webservers, thus their runtimes are influenced by their cluster load.
Runtime on DIPS Test Set
Runtime on DB5.5 Test Set
Methods
Mean Median
Min
Max
Std
Mean
Median
Min
Max
Std
ATTRACT (LOCAL)
1285
793
62
8192
793
570
524
180
1708
373
HDOCK (LOCAL)
778
635
145
3177
570
615
461
210
2593
459
CLUSPRO (WEB)
10475
9831
2632
22654
4512
15507
14393
9207
28528
4126
PATCHDOCK (WEB)
7378
6900
600
16560
3979
3290
2820
1080
14520
2459
EQUIDOCK (LOCAL)
3
22
5
1
53
10
20

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Published as a conference paper at ICLR 2022
Ablation Studies. To highlight contributions of different model components, we provide ablation studies in Table 5. One can note that, as expected, removing the pocket loss results in lower interface RMSD scores compared to removing other components.
Analysis of the Intersection Loss. We further analyze the intersection Ioss introduced in Eq. (15) with parameters y 10 and a 25 (chosen on DB5 validation set). We show in Table that this loss achieves almost perfect values for the ground truth structures, being important to softly constrain non-intersecting predicted proteins.
Camplex
Distribttims of Interface IMSD en DBS
Scatter May Interface resprroomplr 1M65 DBS taut set
ATTRACT
NECHIDCCK
ATRACT
<bock
Gusiko
Acciboox toabocx
Figure 13: DB5.5 test results: a. Complex-RMSD distributions; Interface-RMSD distributions: scatter plot for C-RMSD vs I-RMSD.
Table 5: Ablation studies. We show DIPS test median C-RMSD and I-RMSD values for the corresponding best validation models. Abbreviations: "intersection loss" = intersection loss in Eq. (15), "pocket loss" pocket loss in Eq. (14), "surface feas" surface features in Eq. (16).
Model
C-RMSD I-RMSD
Full model
13.29
10.18
without pocket loss
15.91
12.01
without pocket loss, intersection loss
16.43
12.92
without pocket loss, surface feas
14.80
13.10
without pocket loss, intersection loss, surface feas without surface feas
15.19
11.38
13.73
10.65
without intersection loss without intersection loss, surface feas
15.49
11.09
15.04
10.94
Table Values of the intersection loss defined in Eq. (15) and evaluated on the DIPS validation set in different scenarios. "Centered structures" means that both ground truth ligand and receptor point clouds have been centered (0-mean), without any other modifications.
Centered EquiDock trained
EquiDock trained
Ground truth complexes
structures with intersection loss without intersection loss
56.42
12.68
21.03
1.16
21


Reviews:
[{"review_id": "zkx3zjwbm5", "reviewer": "Reviewer_ydqb", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": " Pros:\nThe whole paper is constructed and written very well.\nThe authors nicely show the proof of SE(3)-invariant and commutativity for the rigid body docking problem.\nFollowing the invariance property, the paper introduces the novel graph matching network and avoids sampling step based on it.\nUsing a soft loss function avoids point cloud intersection.\n\nCons:\nAs much as we appreciated the approach presented here the performance just seems unacceptable compared to other methods\nThe author uses two baselines which were developed 4 years ago. The reason they didn’t use other methods, no local packages, is somewhat problematic too. Still, their performance is much worse than one of the methods and hypothesizing the test data was used to train those models does not resolve this issue.\nTheir runtime is much faster. However, HDOCK, which is the best performance model among those three, uses a bearable amount of time and achieves much better performance. \nThree models are ran on different hardware, which makes runtime comparison problematic.\n\nLack of result analysis\nIt may have been helpful to show the method proposed can provide comparable results on more downstream tasks. Even though its rigid docking prediction is not good enough, maybe the prediction has other properties which can help the other perspectives. \n\nWe also noted the method uses soft constraints to discourage point cloud intersections. However, the authors do not show/discuss if it really prevents this from happening and why not use hard constraints directly (stability during training?).\n\nOther Comments:\nIt is not clear how the number of keypoints K was chosen in the “Keypoints for Differentiable Protein Superimposition” section.\n\nAlso we noticed that the whole section of proofs for propositions lacks citations but they are not claimed as novelty at the beginning. Are they part of the contributions or not? How does these parts relate to previous works?\n\nUpdate:\n===========\nThe improvements implemented in the revised manuscript and clarifications supplied by the authors elevated many of the above concerns and we therefore updated the evaluation to reflect this.", "summary_of_the_paper": "This paper proposes an end-to-end deep learning architecture to model the rigid body protein docking problem. By incorporating the inductive biases of SE(3)-invariance of the final docking position and commutativity, the proposed method avoids the millions of sampling and achieves a competitive performance with much faster speed. In addition, they discover and align keypoints by the attention-based selection algorithm and use optimal transport to predict the binding pocket location based on those selected points. The main contribution of this paper is the combination of the novel graph matching networks and keypoint selection algorithm to predict the 3D position of the docking model. Results are shown on the task of 1) Protein docking complex prediction 2) Runtime", "summary_of_the_review": "Overall we found the paper to be clearly written and offer a nice formulation with a matching modeling approach for the important problem of rigid body protein docking. In that sense we definitely liked the work. However, the results are just unimpressive and no other insights/utility are offered beyond improved running time compared to standard techniques available.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "k5CK23kJ2Yu", "reviewer": "Reviewer_ArSS", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "## Strength\nThe proposed method is quite novel. Built upon Graph Matching Networks [1] and E(3)-Equivariant Graph Neural Networks [2], they propose a SE(3) equivariant graph matching network. The keypoint selection with optimal transport and rotation prediction with Kabsch algorithm are also interesting to me. The proposed method achieves a promising speedup of 80-500x.\nThe proposed method seems general and can be applied to other rigid docking scenarios, i.e., predicting a rotation and translation matrix. The algorithm can potentially be adapted for protein-ligand docking, which is also an important task in computational chemistry. \nThe paper is well written and the mathematical formulation of SE(3) equivariant rigid protein docking is very nice. The authors also promise they will release the code and datasets after reviewing the process.\n\n## Weakness\nAlthough the proposed method achieves a significant speedup compared to previous docking software, it does not always outperform the baselines as acknowledged by the authors. It is also unclear how structure refinement, or templates can be combined with the current method to further improve the performance, which are used in many other protein structure prediction algorithms, e.g., alphafold2.\nThe DIPS dataset does not contain protein structures in their unbounded form. It is unclear how the authors tackle this bias in the dataset.\nThe authors include Surface Aware Node Features into their model. I agree that surface contact modeling is very important for protein docking. It is therefore interesting to see an ablation study over the features.\n\nReference\n[1] Graph matching networks for learning the similarity of graph structured objects\n[2] E(n)-equivariant graph neural networks.\n", "summary_of_the_paper": "The paper proposes a SE(3) equivariant graph matching network for end-to-end rigid protein docking. They propose a novel optimal transport loss to approximate binding pocket and a differentiable Kabsch algorithm to predict the docking pose. They achieve significant running time improvements over existing protein docking software with competitive results, and do not rely on heavy sampling, structure refinement, or templates \n", "summary_of_the_review": "I vote for accepting this manuscript for the reasons listed above. However, several important issues remain to be addressed.\n", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "6fd3Jc52w-", "reviewer": "Reviewer_VRJP", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The authors have put a significant amount of time into designing, developing, and testing their proposed method, and it introduces and adapts several interesting ideas. In particular, the core contribution is their proposed transformation, which guarantees pairwise independent SE(3)-equivariance for the two sets of 3D coordinates. My comments are as follows:\n\n- Since no conformational changes are allowed, the residues buried in the hydrophobic core of the proteins play no role in predicting the binding conformation. Calculating solvent-accessible surface area is one of the most well-studied problems in structural genomics. The authors could use one of such methods, e.g., one based on the Voronoi procedure [PMID: 12376381], to significantly reduce the problem size and render identifying keypoints easier. \n- The authors have a loss term to avoid point cloud intersections, but there are no terms for preventing steric clashes by enforcing Van der Waals radii of proximal atoms. Have the authors checked for steric clashes in the binding pocket?\n- Analyzing performance metrics needs more elaboration. For example, the distribution of C/I RMSDs could be plotted, and/or scatter plots could be added. Moreover, statistical tests could be performed to reliably show whether the proposed method outperforms the other two. The same applies to inference times and a box/violin plot might be a better option. \n- Judging by the results in Table 1, one might infer that HDOCK performs better than IEGMN. It is true that IEGMN has significantly lower inference times, but in practice, experimentally determining protein structures is time-intensive (in the order of months) and costly, and run times in the order of minutes or even hours do not matter.\n- It is good to see that the authors have used two different RMSD definitions; however, the authors should consider looking at CAPRI criteria (critical assessment of predicted interactions criteria) [PMID: 12784359] and standardize their evaluation method.\n- The authors employ several ideas and their contribution to the final performance is not clear. For example, what would happen if the “surface-aware node features” are not used? \n\nSome minor notes:\n- SE(3)-equivariance and SE(3)-invariance are used interchangeably. \n- In p6, what does “normalized keypoints” mean, zero-mean?\n", "summary_of_the_paper": "In this paper, the authors propose a method for the “rigid body” docking of protein-protein complexes, i.e., a complex in which conformational changes in protein structures are not allowed. This method, called independent E(3)-equivariant graph matching networks (IEGMNs), finds the optimal rotation and translation to place proteins in a manner that the distances between residues in the binding site is minimized. A core feature of this method is SE(3) invariance, which means the optimal solution is invariant to the rotations and translations of the two proteins. This is achieved in an elaborate manner employing several ideas and technics:\n\n- A k-NN graph is built for each protein using its CA coordinates and a set of additional features are also extracted\n- Message-passing neural network (MPNN) is used for graph-matching\n- Binding pocket residues or “keypoints” are identified\n- A differentiable Kabsch model is used for superimposing is the “keypoints”\n- Optimal transport is used for the alignment of residues in the binding pocket \n\nThis method, as reported by the authors, runs very fast and can identify the protein complex efficiently. \n", "summary_of_the_review": "The proposed method is interesting, but it combines too many ideas together without quantifying their contribution. Moreover, performance evaluation and comparisons with the existing methods are lacking.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper introduces a novel SE(3) equivariant graph matching network, along with a keypoint discovery and alignment approach, for the problem of protein-protein docking, with a novel loss based on optimal transport. The overall consensus is that this is an impactful solution to an important problem, whereby competitive results are achieved without the need for templates, refinement, and are achieved with substantially faster run times.
Decision: Accept (Spotlight)