Rigid Protein-Protein Docking via Equivariant Elliptic-Paraboloid Interface Prediction
Authors: Ziyang Yu
Wenbing Huang
Yang Liu
Venue: ICLR 2024 poster (2024)
Keywords: Equivariant Graph Neural Network
rigid body protein-protein docking
interface fitting
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: The study of rigid protein-protein docking plays an essential role in a variety of tasks such as drug design and protein engineering. Recently, several learning-based methods have been proposed for the task, exhibiting much faster docking speed than those computational methods. In this paper, we propose a novel learning-based method called ElliDock, which predicts an elliptic paraboloid to represent the protein-protein docking interface. To be specific, our model estimates elliptic paraboloid interfaces for the two input proteins respectively, and obtains the roto-translation transformation for docking by making two interfaces coincide. By its design, ElliDock is independently equivariant with respect to arbitrary rotations/translations of the proteins, which is an indispensable property to ensure the generalization of the docking process. Experimental evaluations show that ElliDock achieves the fastest inference time among all compared methods, and outperforms state-of-the-art learning-based methods, like DiffDock-PP and Alphafold-Multimer, for particularly antibody-antigen docking.

Published as a conference paper at ICLR 2024
RIGID PROTEIN-PROTEIN DOCKING VIA EQUIVARIANT CLLIPTIC-PARABOT
INTERFACE PREDICTION
Ziyang Yu' Wenbing Huang Yang Liu!?
"Department of Computer Science, Tsinghua University IInstitte for AT Industry Research (ATR), Tsinghua University
& GGaoling School of Artificial Intelligence, Renmin University of China
'Beijing Key Laboratory of Big Data Management and Analysis Methods, Beijing, China yuzlyangee aerlyang20emaali ils tsinguua. cdu, cn, www.bing@122 com, Liuyang2011@tsinngii cdu.
ABSTRACT
The study of rigid protein-protein docking plays an essential role in variety of tasks such as drug design and protein engineering. Recently, several learning-base methods have been proposed, exhibiting much faster docking speed than those computational methods. In this paper, we propose a novel learning-based method called ElliDock, which predicts an elliptic paraboloid protein docking interface. To be specific, our model estimates elliptic paraboloid interfaces for the two input proteins respectively, and obtains the transsannlt transformation for docking by making two interfaces coincide. By its design, ElliDock is independently equivariant with respect to arbitrary rotations/translations of the proteins, which is indispensable property to ensure the generalization of the docking process. Experimental evaluations show that ElliDock achieves the fastest inference time among all compared methods and is strongly competitive with current state-of-the-aee learning- -based models such as DiffDock-PP and Multimer particularly for maibbody-antigen docking.
represent the protein-
INTRODUCTION
Proteins realize their biological functions through molecular binding sites, which depend on the complementary in structure and biochemical properties around key regions of protein complexes. Studying the mechanism of protein interaction is of great significance for drug design and protein engineering. In this work, aim to tackle rigid protein-protes docking, which means we only target at learning rigid body transformation from two unbound structures to their docked poses, keeping the conformation of the protein itself unchanged during docking.
Traditional docking softwares (Chen et al. 2003
Venkatraman al. 2009 De Vries al. 2010
feed
ferrbee friknng
Moal et al. 2013 de Vries et al. 2015 Sunny & Jayaraj. 2021) mainly follow this general paradigm: first sampling numerous candidate poses and then updating the structures with higher scores based on a well-designed energy model. These methods are highly computation- ally expensive, which may be unbearable to dock millions of complex structures in practice.
EquiDack
BliDack
inteficn reincat
dagel
Recently, learning based methods (Gainza et al. 2020; Ganea et al. 2021 Evans et al.) 2021
Figure I: EquiDack:; point cloud registration vs FlliDock: interface fitting.
Ketata et al.] 2023 Luo et al. 2023) have been
applied to the task. Equi (Ganea et al. 2021) a regression- based model that tries to predict the functional pocket of each of the receptor and the ligand proteins, and apply point cloud registration to
obtain docking transformation. However, we believe that rigid protein- protein-protein docking task is not
"Corresponding authors: Wenbing Huang, Yang Liu.

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Published as a conference paper at ICLR 2024
entirely equivalent to local point cloud registration for two reasons: 1. the docking transformation is unable to be derived uniquely if the registered point cloud is iol-posed and degenerated to be, for example, within 2-dimensional planes or even 1-dimensional lines; 2. it will cause miss-alignment the registered point clouds of the two proteins are not exactly the respective docking pockets. Another related work is MaSIF (Gainza al. 2020), which is surface- prrface-based and designed to map the geometric and chemical features of local protein regions into fingerprints. MaSIF can well demonstrate the biochemical properties of the protein monomer surface (e.g.. hydrophilicity) with lack of information interaction during protein-protein protein docking, which may hinder the discovery of motein-speeiife binding sites. More recently, xTrimoDock (Luo et al.. 2023) leverages cross- cross-modal representation learning to enhance the representation ability. Since the model is carefully designed based on specific regions (e.g., heavy chains) on the antibody, lacks generalizability across the entire protein domain. Another diffusion-based generative model named DiffDock-PP (Ketata et al., 2023) is proposed to obtain high-quality docked complex structures, yet at the expense of much longer inference time and careful tuning of many sensitive syper-parameters
Contribution. We propose ElliDock, a novel method tailored for rigid protein-protein docking based on global interface fitting. Our model first predicts pair of elliptical paraboloids with the same shape as the binding interfaces, based on the global information of the two proteins. Then we obtain the roto-translation for docking by calculating the transformation that makes two paraboloids coincide. Nicely, our ElliDock is independently equivariant with respect to arbitrary rotations/transsationn of the proteins, which is an indispensable property to ensure the generalization of the docking process. Meanwhile, by constraining predicted interfaces to spatially separate nodes of the receptor and the ligand, ElliDock manages avoid sterie clashes, which is a hard-core problem faced by some of the previous works. F illustrates the difference between EquiDock and our ElliDock.
We conduct experiments on both the Docking Benchmark database (DB5.5) (Vreven et al., 2015) and the Structural Antibody Database (SAbDab) (Dunbar et al. 2014). mainly exploring the performance of our model in heterodimers like antibody-antiger complexes. Experimentally, ElliDock is strongly competitive with current stae-offttheaa learning-hased models such as DiffDock-PP (Ketata et al.) 2203)) and Multimer (Evans et al.) (202I) particularly for antibody-anttigen docking, with significantly decreased inference time compared with all docking baselines.
2 RELATED WORK
Equivariant Graph Neural Networks (EGNNs). EGNNs are tailored for data lying in the Euclidean space (e.g., the coordinate of an atom), which is unable for plain GNNs to achieve equivariance from different coordinate transformation (Satorras et al.) [221]] Keriven & Peyré 2019). EGNNs propose solution by designing equivariant layers and utilizing the equivariant information in the graph. Therefore they are suitable for tasks such as molecular dynamical systems, interaction affinity
prediction and protein structure generation (Fuchs et al. 2020; Ganea et al. 202T Jing et al. 202 Hutchinson et al.) 1 Wu et al., 12021] Kong al. 2022 Jiao et al. 2022) Due to the particularity of the protein-proteeeooo protein docking task (with random unbound structure, the interaction between proteins cannot be effectively constructed), we specially design an pairwise (e3))--quiva message passing network with global-level interaction, which sets it apart from the previous work.
Protein-Protein Docking. Protein complexes data acquired in the laboratory is not only expensive but also inefficient to produce, so we resort to finding efficient computational methods to discover protein-protein docking patterns (Vakser) 2014). Traditional docking methods (Chen et al. 2003 Venkatraman et al., 2009) De Vries el al.,
et al., 202T Biesiada et al.] 201 1] Schindler al. 2017 Sunny & Jayaraj (021) share similar pattern with relatively high computational cost: sample millions of candidate complex poses at first, rank them with well-designed scoring function, then update the complex structures among the top-ranked candidates based on an energy model. Deep learning methods are recently proposed to tackle protein-pract protein docking task based on molecular
dynamics (Desta et al. 2020 Ghani et al. 202I), surface property deciphering (Gainza et al. 2020),
keypoints alignment (Ganea al. 202 protein structure prediction (Jumper et al. [02 1 Evans
et al. 2021), multiple sequence al alignment (Bryant et al. (022), diffusion (Ketata et al.) 2023) or cross-modal representation learning (Luo al. 2023). Different from prior work, we propose novel method that converts rigid protein docking to predicting binding interfaces as a pair of elliptic paraboloids and calculating the transformation based on interface fitting.
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3 METHOD
As defined in Section 1 the rigid protein-protein task aims to predict a (3) transformation (rotation and translation) of the unbound structures of an interacting protein pair to get the docked complex. To address this, we first develop a pairwise-indeeendent SEu3)-equivariant GNN, named EPIT, to characterize both the intra-protein and inter-proteni interactions in the complex. Then, upon the invariant and equivariant ouiputs of EPIT, we derive the SE(3) transformation via elliptic paraboloid interface prediction and alignment.
3.1 PROTEIN REPRESENTATION
We model protein as graph by regarding each residue a node, and denote Gi &)) G2 = (V2,&) E2) as the ligand graph and the receptor graph, respectively. We associate each node vy with tuple (h,, FFi) where h, E R denotes the SE(3)-invariant feature and o; E R is the 3D coordinate of the q-carbon atom in residue i. The collection of all node representations yields HI R X X E IR N. for the ligand graph G, and H2 R RN "Pxx,XX .X, E RN for the receptor graph G2. The edges are constructed the k-NN manner, that is, selecting nearest nodes in the Euclidean distance from the same protein as the neighbors for each node u;. More details of the graph representation are provided in Appendix
Task Formulation We fix the conformation of the receptor G2, and predict rotation matrix Q SO(3) and translation vector i R-3 such that the transformation of the ligand G1, namely x XiQ Q + f'is at the docking position to the receptor.
3.2 EQUIVARIANT POLARIZABLE INTERACTION TRANSFORMER (EPIT)
We adopt PAINN (Schitt al. 2021 the backbone of model, owing to its promising performance for molecular representation learning, Basically, our model takes as input the node
features H Hp. coordinates X (0) Xx and all-zero vectors V.(0) 0 E xxx3 and
outputs SE(3) features H (L) E R as well as SO(3)-equivariant vectors V(L)) E X after L-layer message passing (p (1,2)).
Upon PAINN, we further introduce a Graph Transformer Layer (GTL.) into each message passing block, enhance the ability of detecting protein pockets. Given pair of adjacent nodes i, with corresponding node features h; f and edge feature Ej-i. we obtain a message from to i through a fully-connected layer:
mj+ FC(h,, ey-x)) E R.".
(1)
Then construct the graph transformer layer follows:
q pj
k j- v j-r FC E R.,,, j. No. ususU,
(2)
j= softmax, Ja (,,,)) E ER,
(3)
(v) (1) (1) R
0 aj
(4)
m -i FC(concat, (o/ R R
(5)
1
m
m m/ ER"
W
JEN,
Here, U denotes the number of attention heads, FC(-) Linear o SiLU Linear(-) denotes fully-connected layer, (--) denotes the inner product operation, M,, collects the neighbors of node i, and concat,, (o) indicates the concatenation of o in terms of all values of u. The aggregated message m will be used to update H and V following the pipeline in PAINN. The intra-part architecture of EPIT is illustrated in Appendiix
So far, all the message passing processes are conducted only through intra intra-protein edges, without modeling the interaction between the receptor and the ligand. As they in unbound states, it makes
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no sense to construct inter-proteni edges based on 3D coordinates similar to those intra-proted edges. Instead, we propose dense edge connections between every node pair of the receptor and the ligand, and only allow the invariant node features H2 (not vi()) ) to be passed along the intra-protein edges. By defining W E # learnable matrix of layer 1, we derive the following updates:
3 sigmoid ( H (1) W H 7P )) E (0,
(1)))
(7)
p H (1) H + ( o FC(H, ER
(8)
where -p denotes the protein other than p. pa()) computes the average along the second dimension of the input matrix, @ implics element wise multiplication, and (C(H)) is conducted on each row of H individually. The updated features Hg (4) considered the input of the next layer.
We have the following conclusion.
Proposition 1 The proposed model EPIT is independent SE(3)-equivariant and translation-
invariant, indicating that if V.L)) H (L) ((L) H EPIT(X,, H1, X2, H2, then, we have V1(h Q1, H, V2 (1) Q2, H L)) = EPIT(X,Q) = fi, H1, X₂Q2
i22 H2, E2), 10,, Q2 SO(3), E, E RR.
The proof is given in Appendix Such property is desirable and it will ensure later docking process to be generalizable to arbitrary poses of the input proteins.
3.3 ELLIPTIC PARABOLOID
We briefly discuss the rationale for choosing elliptical paraboloid the interface. On the one hand, using planes, spheres or other symmetric 2D manifolds to model the interface will easily cause the docking ambiguity issue since there are usually different ways to transform 2D manifold from one pose to another. On the other hand, using complex manifolds may introduce bias and higher computational complexity to obtain the docking solutions. Thus we choose elliptic paraboloids to model the binding interface, which are appropriately defined in 3D space and their SE(3) transformations can devised closed form. We provide mathematical details below.
For each point / = [21, 22, 235 E R3 located on the surface of a elliptic paraboloid, we have the standard form (without loss of generality, we might assume that it is symmetric about the z axis):
ax? + buz + CT3 0, a, b E R_,c R. The general form obtained by SE(3) transformations from the standard form belongs to a family of quadric surfaces:
A + (b, σ) +c 0, A R b E R c R.
(10)
Actually any SE(3) transformation of an elliptical paraboloid results the change of the coefficients given by the following proposition.
Proposition 2 we transform the elliptical paraboloid in Eq. Eq. 10 hे = Qu+t via a rotation matrix Q E SO(3) and translation vector t E R²,, then the elliptical paraboloid becomes (A's', ,') (b', F²) + = 0, where A' = QAQ b' = Qb Q A A QQTTQ ,' c t Q QQQTE iQ Qb.
To fulfil our task, construct function d to predict elliptic paraboloid of the general form for each of the input proteins:
A1, bj, C1, A2, b2, = 4 H,, X2, H2)
(11)
We further assume that (A1, b1, e1) shares the same standard form as (Ag, by, 02) since the ligand and the receptor do share the same interface after docking.
Specifically, the function should require hard constraints achieve the following property:
Independent equivariance For simplicity, we only take the ligand for example while the conclusion holds similarly for the receptor. For any SEE() transformation of the input ligand, namely, VQ SO(3), E E IR3 we obtain the new predicted elliptic paraboloids by A', bq. c, A2, bz, C2 T (VVQQ H,,&, Vz H2, E2). It should be guaranteed that (Aq, b/. cf) is transformed from (A, c1) in the same way as stated in Proposition 2
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3. rotation velineuneet
(Q.)
genetal staodard (lignid) (o), (Q(-)(()) Q(())
docked ligand
4. inverse: suandard geveral
(22 + (b2r)
(0a.F)
unbound ligand
fixed roceptoc
2. general to standard (recepuor)
(Q5.-0F (..0
(4j v) + (af m
predict elliptic paraboloid (is the interface) of ligind
1. prodict alliptic paraholoid receptor
Figure 2: Docking process via interface fitting. We decompose the docking process into four steps: 1) predict clliptic paraboloid interfaces for two proteins respectively, 2) conduct general to standard transformations based on the ligand interface, 3) rotation refinement, 4) conduct standard to general transformations based on the receptor interface.
3.4 INTERFACE PREDICTION AND DOCKING VIA EPIT
In this subsection, we introduce how to implement the function 4 through our EPIT. The whole process of docking via interface fitting illustrated in Figure 2 Based on Proposition 2 we can decompose the prediction of an elliptic paraboloid interface by first regressing its standard coefficients and then estimating the SE(3) transformation from the standard from.
We first obtain equivariant vectors (L(L) M (LL) E Nxx33 and invariant features H1") E N ,H!")
R Na from the output of EPIT, then we aggregate the information along all nodes of the receptor (ligand) to get graph-level outputs:
Np Σ
F FC(H(
(ig (H. ()) ER, w' e R pe {1,2},
(12)
Ep LinearNoBias (H(
( () e (M+1)x M MEN (1,2),
(13)
where M (in our experiment, M = 3) is a hyperparameter, HD and (. denote the j-th rows of H and V2(4, respectively. proved that Fp still satisfies SE(3)-invariance and E, satisfies (3(3) equivvarnnce
Predicting Standard Forms and F2 by:
The standard coefficients are obtained with the invariant outputs Fi
((F) + F2(O)
0
A A =
[[][+]]F
bj b;
(14)
F 2
F
where ((-) denotes the softplus operation. The standard form of the predicted elliptic paraboloid is given by:
(A σ + (b σ)) = 0. e (1,2).
(15)
Predicting SE(3) Transformations We first use the equivariant output E, to generate intermediate variables R2 Pyp p E [1,2} that maintain (ou)---qiivariicce::
R Σ E3 30 1)] E 1 3x3 E, E,13M] 13 PE (1,2).
(16)
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Here M aims at ensuring R, to be non-singular. We should make sure det(R,, > 0 to derive a rotation matrix instead of a reflection matrix, otherwise let Rp = -Rp. We then generate a
-qqiivaraa rotation matrix Qp based on Proposition 3
Proposition 3
Given (0(3) eequivaraa matrix R 3333 which satisfies det(R) 0, we can
generate a -3(3)--quivarrin rotation matrix Q from
U = (RR (RR -
As t,, l2 only satisfy SO(3) -equvvariuvaa we generate -qquivvauia vectors by simply adding the coordinate center of the ligand and č of the receptor:
& f
R f2 = E,
(17)
Now obtain the transformations (Q1. {i) for the ligand, and (Q2, fy) for the receptor, and their corresponding elliptic paraboloids of the general form based on Proposition 2
Ap QpA bp - Q, 0, C - Q,A -t, Qub De (1,2). (18)
In our task formulation, we fix the receptor and only dock the ligand towards the receptor. Since (Q1, (2) and (Q2, f2) are the transformations from the general interface to the standard interface, we re-formulate their values as the relative transformation between the ligand and the receptor based on Proposition 4
Proposition 4 Denote Q1+2 E SO(3), f1 E IR3 as the relative transformation of the unbound ligand to the docked pose with the receptor fixed, then we have
Q -222 81-2 = E2 - Q
3.5 ROTATION REFINEMENT
According Section 3.3 we only restrict quadratic coefficients of the elliptic paraboloid with inequality such that nodes should be inside (or outside) the interface utmostly, which considered to be a loose constraint and leads to unexpected degrees of freedom in x-y directions, after the interface of the ligand being transformed to the standard form. So we need an additional rotation to refine x-y angles. Specifically, denote 33 [3] R. we compute the rotation matrix for refinement:
cos
(in 0
Q, =
sin(O) cos(
0 E SO(3).
(19)
0
Then the final docking transformation with rotation refinement conducted is given by:
Q1-2 = Q2Q.Q] E (O(3), h = tg Q2Q.Q R
(20)
3.6 TRAINING OBJECTIVE
Fitness LOss Suppose X1,X; the coordinates in the docked state, from which select all pairs of nodes Yi Yg- kx such that || Y < SÅ, vI < k S K, where K denotes the number of such point pairs. Then we regard the midpoints of those pairs to be functional sites, P = P [ Yz"). In the unbound state, those point sets should follow transformations of corresponding proteins, we denote them as P. and p. For clarity, if X, = =x:Q + then P P Q + f. Naturally, we expect those pockets to be exactly located on our predicted interface. Furthermore, we propose stricter constraint represent how interface acts with each other, that is, we choose to minimize the distance between pockets and the tangent plane at the peak of the interface. Formally, denote Fp E R3 the peak of the predicted interface and ni E R the normal vector of the tangent plane at the peak (p {1, 2)), we enforce the optimization objective as follows:
K
Lit K
1
Σ
Σ
M.
P (A) + (bp, PA(B) + 1pll + (M( Ep, m.))
(21)
PE(122 1=1

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Overlap Loss. One of the crucial issues in previous works is that they encounter steric clashes during the docking process. We address this issue by restricting receptor and ligand to either side of the interface in the standard form, such that the interface spatially isolates protein pairs. Denote xp(+) = (Ap's) :) + (bp. -) + Cp, E (1,2), we can tell whether a node o is inside or outside of the interface p by checking the sign of the formula op ( from which we define the training objective:
N2
Loverlap = min(
(eT
(X, (T)
V
ReLUU-d02
2b2 (x2(1))
NI
2
j=1
(22)
N2
N Σ (V Re
ReLU(o)((xxii)))
22
Refinement Loss. As described in Section 3.5 we conduct a rotation refinement when the interface is transformed to the standard form. We expect to minimize the distance between the docked pockets
in the standard form (denoted as P. and P2) only in x and y dimensions, which can be measured
by Kabsch algorithm (Kabsch 1976.. For clarity, we have P1 [K] Q [i QI f. Pa[k]
Q2 [a] Q l2 (1 < k K) and the refinement loss is defined
Lee = Q- 2, 2] Kabsch P: [: 2). 2))12.
(23)
Dock Loss. Since the degrees of freedom of rigid docking is limited, avoid applying the MSE loss between predicted coordinates and the ground truth to prevent a large number of floating- point calculation errors. During the training process, as we randomly sample the rotation matrix Qge SO(3) and the translation vector Eg2 R to get the unbound ligand pose X = Qge Egt, S i < N1, it's possible to directly evaluate the transformation error by
Ldock ||Q1-22 Quill + ||t + Egr
(24)
4 EXPERIMENTS
4.1 EXPERIMENTAL SETUP
Baselines We mainly compare model ElliDock with one of the-state-of-thhear protein complex structure prediction model Alphafold-Multime (Evans et al. 2021). the template-based docking server HDock?] P Yan et al. 2020). the regression based docking model EquiDock (Ganea et al. 2021) and the diffusion-basse
docking model DiffDock-PPf PF Ketata et al. 2023). Note that, we train EquiDock and DiffDock-PP from scratch with their recommended hyperparameters before testing on the corresponding dataset, while we use pre-trained models of other baselines.
Datasets
We utilize the following two datasets for experiments:
Docking benchmark version 5. We first leverage the Docking benchmark version 5 database (DB5.5) (Vreven al. 2015). which is gold standard dataset with 253 high-quality complex structures. We use the data split provided by EquiDock, with partitioned sizes 203/25/25 for vainn/vlidation/test..
The Structural Antibody Database. To further simulate real drug design scenarios and study the performance of our method on heterodimers, we select the Structural Antibody Database (SAbDab) (Dunbar et al. 2014) for training. SAbDab is database of thousands of antibody-antige complex structures that updates on a weekly basis. Datasets are split based on sequence similarity measured by MMseqs2 (Steinegger & Söding 2017). with partitioned sizes 1,781/300M for train/validationntss.. We then use the RAbD database design with 54 complexes (note that PDB 5d96 will result in NaN IRMSD, so has been removed here), as the test set of antibody-antige sequence correlation greater than 0.8 into the same cluster and ensure that no clusters included in RAbD appear in the training set and the validation set.
Adolf-Bryfogle et al.) (018)) 8). a curated benchmark on antibody
complexes. To prevent data leakage, we put proteins with
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https://itthu..cm//ttaaainn httm://taaianngaacaaqqiuda.a
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Table Complex prediction results (DB5.5 test), Note that means we use the pre-trained model, otherwise we train from scratch on the corresponding dataset before testing.
CRMSD (4)
IRMSD
DockQ (1)
Methods
median
mean
median
median
std
Inference time
HDock
0.327
3.745
7.139
0.289
3,548
6.842
0,981
0.791
0,386
11478.4
Multimer
1.987
7,081
7.258
1.759
7.141
7.889
0.629
0.482
0.418
56762.5
EquiDock
14.136
14.726
5.312
11.971
13.233
4.931
0.036
0,044
0.034
60.1
DifiDock-PP
14.109
15.419
8.160
15,060
16.881
11.397
0.025
0.035
0.033
2103.1
ElliDock
12.995
14.413
6.780
11.134
12.480
4.966
0.037
0.060
0.060
36.7
Table 2: Complex prediction results (SAbDab test). Note that means we use the pre-trained model, otherwise we train from scratch on the corresponding dataset before testing.
CRMSD (L)
IRMSD (L)
DockQ (1)
Methods
median
mean
std
median
mean
std
median
mean
std
Inference time
HDock* 0.323
2.792
6.798
0,262
2.677
6.803
0.982
0.861
0.310
37328.8
Multimer*
13,598
14.071
6,091
12.969
12.548
5,435
0,050
0.104
0.172
197503.1
EquiDock
14,301
15.032
5.548
12.700
12.712
5.390
0.034
0.055
0.067
274.5
DillDock-PP
11.764
12,560
6,241
12.207
12,401
6.353
0.045
0.076
0.090
8308.7
ElliDock
11.541
13.402
6,306
11.319
11.550
4.681
0.054
0.082
0.084
91.2
We apply Complex Root Mean Squared Deviation (CRMSD), Interface Root Mean Squared Deviation (IRMSD) and DockQ (Basu Wallner 2016) evaluate the predicted complex quality. See Appendix for specific definitions.
Evaluation metries
4.2 COMPARISONS OF THE INFERENCE TIME
Given real world applications (for example, simulating the binding process of artificially edited proteins), faster inference time will nicely support large number of simulated docking processes within acceptable time, such that acid space. Here we conduct the inference time test for different methods on two test sets and results shown in Table accordingly. For fairness, all models are tested under the same environment on one single GPU (instead of HDock, which can only run on CPU).
locate the desirable protein candidate from its huge amino
As expected, the traditional template-based docking method HDock takes an extremely long time to run because it needs to generate tens of thousands of structure templates for further update. Multimer a deep learning model, yet requires multiple sequence alignment during inference with GB level parameters, which may attribute to its longest inference time. In comparison, generative- -based and regression-bactt based deep learning models are 10~1,000 times faster. Among them, DiffDook- runs slower because it requires dozens of diffusion steps, each step requires re-running the model. On contrary, ElliDock and EquiDock are both regression models without an order of magnitude difference in speed. We show ElliDock is 2-3 times faster than EquiDock, which is more likely due to the amount of model parameters, number of model layers and so forth.
BMRD
Malbow
DI ao auboo squlooo
WEROOO ELUDOCE
mulbock
BESCE MELTER
Figure of antihody antigen dock- ing on SabDab test set.
Figure 4: IRMSD antibody- antigen docking on S.Abaab test set.
Figure comparison DockQ ElliDock and Alphafold-Multinee on SAbDab test set.
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4.3 RESULTS OF THE COMPLEX PREDICTION
The complex prediction results on the DB5.5
and RAbD are illustrated Table Sable[aand and Table
2 respectively, with statistically processed data shown in Figure 3
We show that, ElliDock outperforms other regression-based model, EquiDock, on both DB5.5 and SAbDab based on IRMSD and CRMSD. Meanwhile, Our method beats DiffDock-PP on all metrics DB5.5, while the CRMSD metric slightly inferior to the latter on SAbDab. We assume that this is because the training objective of Difk PP is completely based on the entire molecules, on the contrary, our method focuses more on pockets in localized regions of the protein. For the comparison with AlphafoldMMultine
4 5
illustrate in Figure 5 that ElliDock has better prediction results most samples on SAbDab, yet Multimer achieves extremely amazing accuracy on specific samples, which may be attributed the powerful representation ability of Alphafold. For pre-trained models like HDock, we respect for its promising performance, yet we have some concerns about its possible data leakage.
4.4 ABLATION STUDIES
Training objectives. To illustrate contributions of different training objectives, we conduct mul- tiple experiments and rule out some of the train- ing objectives each time. Models are trained and tested on DB5.5 only.
Table 3: Ablation studies. We report CRMSD median and IRMSD median of the corresponding best validated model DB5.5.
Methods
CRMSD (1)
IRMSD (U)
Without rotation refinement. With a view to verify the validity of the rotation refinement, removed Q, in the calculation of the docking transformation Q(x, | il larly, we train and test the model on DB5.5.
Full model
13,00
11.13
Lis
13,20
13.02
Luverlup
13.95
12.35
{ (22)) Simi-
Lret
13.23
12.49
Lite Loverlap
13.98
13.24
Ablation study results are shown in Table We conclude that Lit and the rotation refinement influence the docking process by searching for better binding interface, with great impact on IRMSD. Another training objective, Loverlap. designed for tackling conformation overlap, does help to find proper docking positions for ligand, which results in lower CRMSD.
Cis Lief
14.71
13.94
Lovetlap Lref
14.48
11.71
Qr
13.05
13.22
CONCLUSION
We propose new method for rigid protein-protein docking based on elliptic paraboloid interfaces prediction and fitting, which is strongly competitive with state-of-the-art learning-basee methods for particularly antibody-antigon docking. Mcanwhile, we design a global level, pairwise SE(3)- equivariant message passing network called EPIT, serving as solution message passing and updating through nodes when the graph cannot be efficiently constructed as connected.
In the future, we look forward incorporate more domain knowledge to our method, for example, chemical properties on the protein surface. We also hope the interface-basee docking paradigm can give an insight into rigid docking or further flexible docking, and provide more possible research directions.
Limitations. First, our model targeted on rigid protein-protein docking regardless of conformational changes of receptor and ligand, which may not be applicable in all scenarios. A potentially more valuable follow-up direction is to study flexible docking, which takes the conformation changes during docking process into consideration. Moreover, that the prediction results may not fully meet all properties that we require. Hard constraints can be further implemented in the model architecture for improvement.
still use soft constraints in model, so

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ACKNOWLEDGMENTS
This work is supported by the National Science and Technology Major Project under Grant 2020AAA0107300, the National Natural Science Foundation of China (No. 61925601, 62376276), and Beijing Nova Program (20230484278).
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APPENDIX
A SOURCE CODE
Our source code is available at altttp/s///thbbb..cc////lldddu//
B DETAILS OF PROTEIN REPRESENTATION
We construct the protein graph at residue level, each node in the graph corresponds to a residue that makes up the protein and each edge between two nodes represents an interaction. Here we show details on how to extract node features and edge features from the raw protein data.
B. NODE FEATURES
Residue Type Embedding. The key to different amino acid types lies their side chains, which plays an important role in biological functions of the protein. Since most natural proteins composed of only 20 amino acids, we can establish an embedding of residue types to D dimension vectors to capture side chain features.
Relative Positional Embedding. A motif is fundamental structure representing specific biological functions with group of adjacent residues. So the relative position of residues reflect protein functionality. Given a protein chain composed of residues, we construct relative positional embedding for residue i (i {0, 1, N 13) follows:
D RPP: = concat, (100000 34 cos cos((00000 e RR, de (0,1, 1.. - -1.
(25)
B.2
EDGE FEATURES
We adopt protein representation methods in EquiDock construct relative position, relative orien- tation and distance based edge features. Here we introduce new SE(3)-invariant edge feature to better understand protein structure-function relationship.
N-order Resultant Force. The structure of protein is essentially due to the interactions between residues and atoms. Based on this, we establish the force relationship between residues. Denote 5; as the vector starting at node j and ending at node nija norm Σ -| , as the direction of the resultant o--rderr force from neighbors node i. Then calculate the inner product between resultant forces of node and j as the (E(3)-invrriant edge feature
fea (M ī.ja)) E E. (2,
(26)
C MODEL ARCHITECTURES
We show the intra-part architecure of our model EPIT in Figure
D PROOFS OF THE MAIN PROPOSITIONS
Proof of Propasition 1 In the work of PAINN (Schütt et al.) 2021). it has been proved that if V. H' PAINN(X, H), we have VQ, H' PAINN(XQ f. H). VQ SO(3), Ee R here we only need to prove that the graph transformer layer and the iiter-partt architecture have the same properties.
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1fy -1
b
P61
G)-4
Vid
120
W ++b
POC)
V
stack
120
RBF
1:20
250
SIJU
W +
128
128
384
FCC-)
SILU
/-
120
384
120
++
(1
1211
128
GTL
Message
Update
0/-4
3P
AH(1)
aH(#]
+8[]]
Figure 6: The intra-part architecture of EPIT with the Graph Transformer Layer (GTL) (a) well as the message (b) and update blocks (c) of the equivariant message passing. The message block and the update block together form one EPIT layer.
Given SE(3)-invariance invariance (u) (w) (a) still satisfy (F(33-invariance since feature mj-ii from Equation 2 know that g) k 3-ss: Uj
First, we prove the graph transformer layer satisfies
mj-si remains the same for any roto-translation transformation. Thus GTL satisfies -iivaaanne,,, which means the intra-part architecture of EPIT satisfies -Evvaaine for features and SO(3)- equivariance for vectors well.
Then we prove the inter-part architecture satisfies (iv33-iinariance. From Equation 7 we know that g H satisfy SF(3)-invariance since H the output of the intra-part layer of EPIT, satisfying (F(3))invarian) We have proved that the inter-part architecture of EPIT satisfies SF(3) -ivaaiiine
Proof of Proposition 2
From E²' = QT +t twwe have Eे TT (H' -t). Substitute á into the surface (
equation before transformation:
(F tT QQQQQ (H' - E) T f +QQ+++==+ c
(27)
E' ETQQQ F' - D (T t) + QAQ t f- Qb c=
(28)
It illustrates that o² is on the elliptic paraboloid with parameters:
A' QAQ b' = Qb Q(A A Q TT c T QAQ E Qb.
(29)
Proof of Proposition 3 Denote L R.T (then L)2,,, we can easily show that is
positive semidefinite matrix.
Vo L RR 0
R ū) 2.
(30)
For clarity. Since L is symmetric, denote L SAST where S is orthogonal and A is diagonal, then U = / = S T is defined as the matrix square root of L, which is unique since L satisfies positive semidefinite.
For vW SO(3), denote R' W L' R UU' 1/2 have U' (R' ) (W T> = (WSAS W TT (WS) TUWT.
(31)
Then we can derive that Q satisfies (0uu)-equivarraanc::
WU wWW
WQ
(32)
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Next, we show that Q is orthogonal:
Q Q (U R UU' R
ST (SA
SA
(33)
L-'R
R (RR') )
I.
Finally, since det R > 0, (ddt(t))) = det (RR = det RR)² > 0, we have
(et(Q) det (ee((UU1) (U- det(UU-)) R) det(L) 1//a det(R) > 0.
(34)
So we assert det (Q) = 1, which means Q is rotation matrix with - -qquivarrrr.
Proof of Proposition Denote 1i 1 the corresponding standard form elliptic paraboloid surfaces of ligand and receptor, we can derive the transformation from general form to standard form:
Q(T,IT)) = Qp ((T))))) Ep. Q(12TT) = Q, ((r;)4)) -0, Ep. {1,2}.
(35)
When the receptor fixed, the transformation of the ligand is the superposition of 1) the transfor- mation from general to standard of its predicted elliptic paraboloid and 2) the transformation from standard to general of receptor's s predicted elliptic paraboloid. We show that:
Q1-2 Q(22(1)))) o Q(T14) = Q2QT,
(36)
= [(T1[T)) + ((2(15)) = -2--
(37)
E MORE EXPERIMENTAL DETAILS
Training details Our models are trained and tested on an Intel(R) Xeon(R) Gold 5218 CPU @ 2.30GHz and NVIDIA GeForce RTX 2080 Ti GPU. We save the top- 10 models with the lowest loss evaluated on the whole validation set and apply early stopping with the patience of epochs. After the training process, test CRMSD and IRMSD the test set of all 10 best validated models and record the best performance. During training process, we randomly select SO(3) rotation and R translation to apply ligand coordinates and get the unbound structures the input for the model.
Metrics
We give specific definitions of CRMSD, IRMSD and DockQ below.
CRMSD. Given the ground truth and predicted complex structure X X Nx3 N = Nı + No. we first conduct point cloud registration by Kabsch algorithm (Kabsch 1976), and compute CRMSD
=V I 12.
IRMSD. Interface RMSD is similar to CRMSD but only computed among pocket points. We have already obtained the ground truth pockets Pr Pj for ligand and receptor, then we select nodes with the same index P P3 from the predicted complex structure, denoted P/, Py. After concatenating Pr P3 and P/, P respectively we get the pocket nodes of the complex, Pr P' E x3 We compute IRMSD the same way as CRMSD with ground truth and predicted pocket nodes 1. and P.
DockQ. DockQ is a weighted average three terms including contact accuracy, interface RMSD and ligand RMSD, which is quality measure for protein prretin docking models. We compute DockQ with the public too
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Test sets All test samples can be found in PDB (Burley al. 2017), here only list the entry IDs of complexes in the corresponding test set.
DB5.5 test set. IAVX, IHIV, IHCF, LIRA, длT, IJPS, IMLC, IN2C, INW9, 1QA9, IVFB, IZHI, 2AIA, 2A9K, 2AYO, 2SIC, 2SNI, 2VDB, 3SZK, 4POU, 5C7X, 5HGG, 5JMO, 6BOS, 7CEI.
SAbDab test set (RAbD). 3MXW, 302D, 3K2U, 3BN9, 3HI6, 4DVR, 4G6J, 1FE8, 4ETQ. 4FFV, INCB, 3W9E, 3RKD, 5BV7, 5H14, 5J13, 4CMH, IN82, 3L95, 3S35, 1W72, 3NID, 4XNQ. 1A2Y, 4K15, 1IC7, 4OT1, IA14, 5GGS, 1UJ3, 4DTG, 3CX5, 2ADF, 2B2X, 4G6M, 2XWT, 4FQJ, 2DD8, 5L6Y, 5MES, 2XQY, 2VXT, 4LVN, SNUZ, 4H8W, 4YDK, 5B8C, 5EN2, 1OSP, 4QCI, 5F90, IIQD, 2YPV.
should be noted that HDock is unable to conduct docking on PDB 5HI4 and PDB 5MES, thus the results of HDock reported in Table have excluded the samples.
Hyperparameters We show hyperparameter choices for training in Table and bold the best performing ones after evaluating on the corresponding validation set.
Visualization The comparison of the predicted complex structures from different methods on PDB 1W72 has been illustrated Figure 7 We also provide visualization of elliptic paraboloid interfaces predicted by ElliDock protein complex samples in Figure g
Table 4: Hyperparameters of the training process. The bold marks represent the optimal parameters among their candidates based on validation results.
Params
DB5.5
SAbDab
learning rate
2e-4
2e-4, 3e-4, 5e-4
layers of EPIT T.
2.
2, 3,
embed size D
64
64
hidden size H
128
128
attention heads U
16
4, 8, 16,
neighbors
10
10
RBF size
20
radial cut (A)
3.0
1.0, 2.0,
dropout rate
0.1
0.1
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ElliDock vs groundtruth Dock0=0.003
EquiDock vs groundtruth DockQ-0.027
DiffDoc--PP vs groundtruth Dock0=0.063
Multimer groundtruth DockQ-0.016
HDDck groundtruth
Dock0-0.014
Figure 7: Visualization of the predicted complex structures from different methods on PDB 1W72 (from RAbD).
PDB: 1IRA
PDB: 2A9K
PDB: 2SNI
Figure 8: Visualization of elliptic paraboloid interfaces predicted by ElliDock on protein complex samples (from DB5.5). Red: ligand, green: receptor, blue: the predicted interface.
16


Reviews:
[{"review_id": "OwngG4tEZD", "reviewer": "Reviewer_raHF", "summary": "This paper proposed a deep learning method called ElliDock on solving rigid proteins docking. Firstly, they use a pair of elliptic paraboloids to estimate docking interfaces and then use equivariant neural network to learn rotation and translation. The first part is interesting. They compared with classical docking method HDock and deep learning based methods, EquiDock, DiffDock-PP, multimer. Experimental results show some minor improvement on these old deep learning baselines and much worse than HDock. My major concern are: 1) they didn't compare with the latest method, 2) the setting is not realistic. see weaknesses i listed as follow.", "questions": "- HDock performs much better than ElliDock. I guess the major reason is the input rigid structure is native unbounded structure. However, using native unbound structure as docking method's input is not realistic. What's the performance if you dock predicted structures.\n\n- the pair of elliptical paraboloids are with the same shape. How to handle large protein or samll proteins?", "limitations": "", "rating": 3, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- this paper predicted a pair of elliptical paraboloids as two proteins' binding interfaces. This idea is interesting.\n\n- this method compared both classical docking method, e.g., HDock and deep-learning based docking method, e.g., EquiDock, Multimer.\n\n- it's glade to see experimental comparison on both general protein complex and antibody-antigen complexes. The last one is important in the field of drug design.", "weaknesses": "- so small font size in figures, for example, Figure 1.\n\n- the improvement is minor on DB5.5 when comparing with EquiDock and DiffDock-PP and much worse than Multimer and HDock.\n\n- [1] and [2] are stronger baselines on solving antibody-antigen complex. \n\n[1] Yujie Luo, et.al. xTrimoDock: Rigid Protein Docking via Cross-Modal Representation Learning and Spectral Algorithm.\n\n[2] Ambrosetti, Francesco, et.al. Modeling antibody-antigen complexes by information-driven docking.\n\n- This paper claims ElliDock solves steric clashes well. I guess one reason is this experiment uses native unbound structure as the input. There is a trade-off between steric clashes and dockq if you use predicted structures. When using predicted structures, there is no perfect pocket to fit ligand on receptor.", "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": "", "extra_scores": {}}, {"review_id": "kSvLJqltur", "reviewer": "Reviewer_5pax", "summary": "This paper proposes a rigid protein-protein docking model, ElliDock, that functions by predicting a pair of paraboloids on the ligand and receptor with the same shape and aligning them via roto-translation such that they coincide. The paper's main competitors are EquiDock, DiffDock-PP, and AlphaFold-Multimer, all of which rely on different methods. ElliDock outperforms all of these models in terms of inference time. It outperforms EquiDock and DiffDock-PP on the DB5.5 dataset, and all models on the SAbDab dataset.", "questions": "1. The inference time is reported in Table 1, but not in Table 2. Could you please report the values for Table 2?\n2. Could you please report the inference time in Table 3. It would be interesting to see which losses could be removed for even more speed-up, but without accuracy loss.\n3. Since DiffDock-PP is a diffusion-based model it is capable of modelling multiple conformations. How does ElliDock fare when the ligand-protein pair has multiple possible conformations?\n4. Could you argue a bit about why ElliDock's training objective is more suitable than DiffDock-PP's?\n5. Why do you think ElliDock outperforms AlphaFold-Multimer on SAbDab?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The is high-quality and presented very well.\n2. The argument as to why EquiDock suffers compared to ElliDock is developed well.\n3. The method paraboloid matching in the context of protein-protein docking is seemingly novel and mathematically developed very well in the paper.\n4. The results are convincing.", "weaknesses": "1. The paraboloid formulation is elegant, but I would like to see some discussion as to why this is the right choice compared to other considered options.\n2. The argument of why ElliDock's formulation is better than EquiDock seems to be developed sufficiently well, but I would like to see some more discussion contrasting with DiffDock-PP and AlphaFold-Multimer.", "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": "", "extra_scores": {}}, {"review_id": "B3gqvIZU8m", "reviewer": "Reviewer_SQcb", "summary": "The paper introduces a novel supervised learning approach for protein docking, focusing on predicting the interface between ligand and receptor proteins in the form of an elliptic-paraboloid. This method operates under the assumption that the true docked pockets are precisely situated along our predicted interface. The primary aim of this learning process is to identify elliptic-paraboloids that align with these ground-truth docked pockets. This objective is complemented by additional constraints, such as the loss associated with elliptic-paraboloid overlap, which ensures spatial separation of protein pairs at the interface, and the standard dock loss, quantifying the difference between predicted and ground-truth rotation and translation matrices.\n\nThe authors illustrate that the key advantage of this method lies in its efficiency during inference. Once the elliptic-paraboloid interface is predicted, it enables the efficient calculation of docking poses. Furthermore, this proposed method exhibits equivariance to arbitrary rotations and translations of the proteins, a crucial feature ensuring the generalization of the docking process.\n\nTo evaluate the performance of their approach, the authors conducted experiments using two specific benchmark datasets: \"Docking benchmark version 5\" and \"The Structural Antibody Database.\" They compared their method to four baseline machine learning-based docking techniques: HDock (pretrained), Multimer (pretrained), EquiDock (trained from scratch), and DiffDock-PP (trained from scratch).\n\nIn the first benchmark dataset, the authors demonstrated that their ElliDock outperforms most of the baseline methods in terms of docking accuracy while achieving a significant reduction in inference time, with the exception of the pretrained model.\n\nIn the second benchmark dataset, the authors showed that ElliDock outperforms most of the baseline methods, except for HDock. The authors noted that HDock's superior performance may be attributed to the possibility of data leakage. They further conducted an ablation study to highlight the importance and effectiveness of each training objective function introduced in the ElliDock method.", "questions": "Would you kindly offer substantiating evidence or theoretical findings to validate the assumption that elliptic-paraboloids accurately align with the ground-truth docked pockets?\n\n    Could you furnish more compelling evidence to substantiate the assertion of potential data leakage in HDock, as stated in the paper?\n\n    Can you provide a more comprehensive explanation regarding the reasons behind the improved inference time compared to other methods?\n\n    Could you consider releasing the splits of the second dataset utilized in your experiments to facilitate reproducibility efforts within the research community?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The concept of streamlining the learning process through the introduction of an inductive bias that transforms a complex docking learning problem into the prediction of coinciding elliptic-paraboloids is truly interesting.\n\nThe experimental findings are particularly compelling, notably the notable enhancements in terms of inference efficiency. The experiments were conducted with great attention to detail, involving thorough comparisons with state-of-the-art machine learning-based docking methods.\n\nThe paper was thoughtfully composed, and it's commendable that the source code is available as open-source, making it accessible for others to explore and implement.", "weaknesses": "Challenging the assumption that elliptic-paraboloids precisely correspond to ground-truth docked pockets is a substantial step. To validate this assumption, it would be beneficial for the authors to consider conducting experiments using an established docking database, providing empirical evidence regarding the validity of this premise.\n\nThe results in comparison to models trained from scratch are indeed remarkable. While the authors suggest that the success of baseline models like HDock may be due to potential data leakage, it can be exceptionally challenging to definitively prove such claims. More compelling evidence supporting the presence of leakage in HDock would be valuable.\n\nThe authors demonstrate a significant improvement in inference time for their method compared to baseline approaches, yet the exact source of this efficiency enhancement remains somewhat unclear. An in-depth complexity analysis would be helpful in shedding light on the reasons behind this improvement. Typically, if a method directly predicts rotation and translation matrices, inference time should be as swift as the execution of these operations.\n\nIn addition to comparisons with machine learning-based docking methods, it would be beneficial to provide an overview of results in comparison to well-established docking methods that do not rely on machine learning techniques.\n\nFurthermore, sharing the splits of the second benchmark dataset used in your experiments would be advantageous, as it would enable others to replicate the results more easily and conduct experiments with the same splits in the future. This transparency promotes reproducibility in the research community.", "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": "", "extra_scores": {}}, {"review_id": "8pTQ4l0lEH", "reviewer": "Reviewer_ptkL", "summary": "The paper studies the problem of rigid-body protein-protein docking, which is important for drug design and protein engineering. The proposed method, ElliDock, models the protein-protein docking interface using an elliptic paraboloid approximation. By aligning the interfaces of two unbound proteins, ElliDock ensures roto-translation equivariance. The experimental results show the method outperforms other deep learning-based methods and achieves the fastest inference speed.", "questions": "1. In Eqs. (3)(4), the coefficient $\\alpha_{j\\to i}$ appears to merely normalize the inner product of $q$ and $k$, without taking into account the keys from other edges. This approach seems to limit the attention mechanism's capacity to capture interactions among edges or nodes. Given this, wouldn't it be more straightforward to directly map the message $m_{j\\to i}$ to a scalar coefficient $\\alpha_{j\\to i}$?\n2. In Sec. 4.3, there's a mention that pre-trained models like HDOCK could be prone to data leakage issues. From my understanding, HDOCK functions as a template-based model and doesn't utilize learning in its scoring function. Could you provide a more detailed explanation regarding the concerns of data leakage?\n3. In Eq. (12), while $F_p$ is identified as a 4-dimensional vector, its fourth element seems not to be used in the main paper.\n\nOverall, I appreciate the method introduced to address the rigid-body docking problem. I’m willing to raise my score if all the concerns listed in the Weakness section can be addressed during rebuttal.", "limitations": "", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper addresses a significant real-world problem and offers a comprehensive discussion of the related work, to the best of my understanding.\n2. The idea of using an elliptic paraboloid to approximate the docking interface is interesting. This paper thoroughly discusses the advantages of this paraboloid-centric method over previous point cloud registration techniques. Furthermore, the introduction of various loss terms to derive meaningful interfaces improves the rationale behind the proposed method.", "weaknesses": "1. The major concern about the method lies in the comparison with traditional template-based methods, e.g., HDOCK. As shown in Tables 1&2, HDOCK outperforms other methods by a significant margin, suggesting it may already effectively address the rigid-body docking challenge. Although the ElliDock’s inference speed is much faster than HDOCK, its performance is unsatisfactory. Given real-world applications, the value of faster inference time seems negligible, especially if the docking challenge doesn't necessitate high-throughput situations. Please correct me if I’m wrong.\n2. Despite the considerable depth in the methods section discussing the rationale and procedure of using an elliptic paraboloid for approximation, it remains unclear how this looks like on real-world data. It would be good if there are some visualization experiments showing the learned paraboloid interfaces.\n3. The paper's notations appear cluttered and confusing. For example, iEq.(8) denotes the other protein as $H_{-p}$, which is unconventional. In Eq. (12), the FC layer maps representation to $4$-dim without explanation. Meanwhile, Eq. (13) introduces the hyperparameter $M$ without elaboration. The bracket notation further complicates understanding. For better clarity, consider excluding the $p$, since most operations apply similarly to both proteins, and employ subscript notation for indexing.\n4. A deeper exploration and connection with surface-based methodologies, such as MaSIF, would be beneficial. It would be insightful to include a comparison with it in the benchmarking section.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)