EBMDock: Neural Probabilistic Protein-Protein Docking via a Differentiable Energy Model
Authors: Huaijin Wu
Wei Liu
Yatao Bian
Jiaxiang Wu
Nianzu Yang
Junchi Yan
Venue: ICLR 2024 poster (2024)
Keywords: protein-protein docking
energy-based model
geometric deep learning
energy-function
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Protein complex formation, a pivotal challenge in contemporary biology, has recently gained interest from the machine learning community, particularly concerning protein-ligand docking tasks. In this paper, we delve into the equally crucial but comparatively under-investigated domain of protein-protein docking. Specifically, we propose a geometric deep learning framework, termed EBMDock,  which employs statistical potential as its energy function. This approach produces a probability distribution over docking poses, such that the identified docking pose aligns with a minimum point in the energy landscape. We employ a differential algorithm grounded in Langevin dynamics to efficiently sample from the docking pose distribution. Additionally, we incorporate energy-based training using contrastive divergence, enhancing both performance and stability. Empirical results demonstrate that our approach achieves superior performance on two benchmark datasets DIPS and DB5.5. Furthermore, the results suggest EBMDock can serve as an orthogonal enhancement to existing methods.

Published as a conference paper at ICLR 2024
EBMDoCK: NEURAL PROBABILISTIC PROTEIN- PROTEIN DOCKING VIA A DIFFERENTIABLE ENERGY-
BASED MODEL
Huaijin Wult, Wei Liu't, Yatao Bian', Jiaxiang W Nianzu Yang', Junchi Yan "Department of Computer Science and Engineering, Shanghai Jiao Tong University
Tennent AI Lab *XVERSE, China
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attps://githun com/ com/wuhua wwwhawuhaaaa httaiiinin//eeeeeeeee
ABSTRACT
Protein complex formation, pivotal challenge in contemporary biology, has re- cently gained interest from the machine learning community, particularly con- cerning pootein-lggand docking tasks. In this paper, delve into the equally crucial but comparatively under-investigated domain of protein-protele docking. Specifically, propose geometric deep learning framework, termed EBMDock, which employs statistical potential as its energy function. This approach produces probability distribution over docking poses, such that the identified docking pose aligns with a minimum point in the energy landscape. We employ a differential algorithm grounded Langevin dynamics to efficiently sample from the docking pose distribution. Additionally, we incorporate energy- based training using con- trastive divergence, enhancing both performance and stability. Empirical results demonstrate that our approach achieves superior performance on two benchmark datasets DIPS and DB5.5. Furthermore, the results suggest EBMDock can serve an orthogonal enhancement to existing methods.
INTRODUCTION
Protein-protein interactions play prominent role in diverse biological processes, e.g. signal trans- duction (Pawson & Nash, 2000), enzymatic reactions (Frieden, 1971; Tetlow et al., 2004), and im- mune responses (Huang, 2000). Understanding the complex formation between proteins is essential. However, the experimental determination of protein structures is complex and expensive, highlight- ing the potential advantage of computational approaches in studying protein-protein interactions, especially universal and stable ones in need (Porter et al., 2019). We address the problem of protein- protein docking and follow the common rigid docking setting in recent works (Ganea et al., 2022) without considering the deformations of proteins, which is an approximation for data collection and tractability by existing techniques.
Over the decades, a series of score- score-based based approaches (Schindler et al., 2017; Saneeidmman-uuhhowe et al., 2005; Yan et al., 2017) have emerged for the prediction of complex structures. computationally exploring the binding interactions between proteins, providing insights into protein-proteis recog- nition, binding affinity, and structural rearrangements. These methods typically entail a multi-step procedure as follows: Firstly, they generate numerous complex candidates. Subsequently. scoring function is employed to rank these candidates (Moal et al., 2013; Basu & Wallner, 2016; Launay et al., 2020; Eismann et al., 2021). Finally, the top-ranked candidates undergo refinement pro- cess utilizing energy or geometric models (Verburgt & Kihara, 2022). As a result, these methods always rely heavy candidate sampling and a well-tailored score function based the spatial and chemical properties of every single atom, facing the challenge of sampling efficiency and stability.
CCorrespondence author. i equal contribution. This work was partly supported by National Key Research and Development Program of China A NSFC (62222607), Tencent Al Lab, and SITU Trans-med Awards Research (STAR) 20210106. J. Wu's work done when he with Tencent Al Lab.

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Table 1: Comparison of different deep-learning based (rigid) Protein-Protein docking methods.
Methods
Formulation
Input
Accuracy Efficiency Stability
AF-multimer (Evans et al., 2021)
Iterative search
sequence structure
High
low
median
Equidock (Ganea et al., 2022)
One-shot regression
low
high high
DIFFDOCK PP (Ketata et al., 2023)
Generative model
structure
median
median
median
EBMDock (Ours)
Energy model
structure
median
high
high
On the other hand, there are also emerging lines of research in the area of geometric deep learning and molecular representation learning, and these methods have also been extended show promise in learning protein structures. Previous works (e.g. EquiDock (Ganea et al., 2022), HMR (Wang et al., 2023)) have approached protein-protein docking by treating it as a regression problem, uti- lizing Kabsch algorithm (Kabsch, 1976) to align two given proteins and aiming to directly predict the final pose (directly output a rotation matrix R and a translation vector t). While these methods efficient, they are limited to generating single predicted result cone-point solution) without characterizing other possible binding states during the actual docking process. DIFFDOCK-PP (Ke- tata al., 2023) formulates docking generative problem: given two proteins, it estimates the distribution over all potential poses using a diffusion generative model. The generative model sac- rifices efficiency for higher accuracy but lacks stability. Another line of protein structure predic- tion is AlphaFold(AF) (Jumper et al., 2021), which involves taking the primary sequences of the proteins as input and conducting a search for multiple sequence alignments (MSAs). Although AF-multimer (Evans et al., 2021) achieves high accuracy, it suffers low efficiency and reliance on template features. The comparison of different methods is shown in 1.
In this paper, we advocate for learning a differentiable energy function modeled by geometric deep neural networks. It provides probability distribution over docking poses, based on which we can apply efficient sampling methods to improve the solutions, either from scratch or from given point solution obtained from an alternative approach e.g. (Ganea et al., 2022: Wang et al., 2023) (using method plug-in), which also enhances the stability. Specifically, we resort to energy- training methods with Langevin dynamics sampling. This conducts implicit data augmentation, which holds great potential, given the limited availability of existing protein complex structures and their difficulty in obtaining through physical simulations.
We present EBMDock, energy-based learning framework for generating docking poses repre- sented by SE(3) transformations for protein-protein complexes. EBMDock features a learnable energy function based on distance likelihood, employing geometric deep learning extract residue features and filter binding interface residues. Inspired by Deepdock (Méndez-Lucio et al., 2021), we model distance distributions using Gaussian mixture models and compute energy the average negative distance log- ligeliiiisss of binding interface residue pairs. Training involves contrastive di- vergence and Langevin dynamics for generating negative samples to optimize the energy landscape. In inference, we sample points from the energy distribution using Langevin dynamics, selecting the lowest energy point as the docking pose prediction. Evaluation on the Database of Interacting Proteins (DIPS) and the Docking Benchmarks 5.5 demonstrates that our method surpasses exist- ing deep learning-bass based approaches and matches the performance of traditional docking software, significantly speeding up the process. Combining EBMDock with EquiDock shows potential for enhancing existing approaches as a plugin. The highlights of this paper are:
1) We propose a geometric deep learning- leased energy function for (rigid) protein-proted docking which employs statistical potential based on the distance likelihood. This approach formulates the docking problem optimal docking pose through differentiable sampling methods.
optimization problem, allowing for efficient and stable determination of the
2) The probability distribution over docking poses derived from the distance distribution and the energy function acts as the measure of confidence, which enables our methods to function as a plugin to boost other one-point solution methods.
3) We propose an mased learning framework for the protein docking problem for the first time. Trained with contrastive divergence, which conducts implicit data augmentation, the energy function can overcome data imbalance.
4) Quantitative results show that it outperforms recent deep learning- bassed based models and the speed is 100x faster than traditional docking software. Meanwhile, it performs stably across datasets. Also, empirical results show that can boost existing deep-learning docking methods by a large margin.
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2
RELATED WORK
2.1 MOLECULAR DOCKING.
Existing molecular docking methods can be categorized into traditional score-based methods and emerging deep learning- based methods. Score- based methods, exemplified by Attract (Schindler al., 2005) and HDock ( (Yan al., 2017), always involve search algorithm to generate candidates and scoring function to select (Moal et al., 2013; Basu & Wallner, 2016; Launay et al., 2020; Eismann al., 2021). Specifically, Attract uses ran- domized search algorithm and physics-basee based scoring function. The scoring function in PatchDock is based on the geometric shape and electrostatic complementarity of the protein surfaces, while HDock combines both physics-based based and empirical terms to estimate the binding energy. However, these methods are all faced with the problems of expensive computation cost and dependence on complex template libraries (Ruiz-Carmone et al., 2014). In contrast, deep learning- -bsedd approaches aim to enhance efficiency while maintaining accuracy.
al., 2017), PatchDock Schneidman-Duhoveee
While several methods (Méndez-Lucio al., 2021; Stärk al., 2022; Lu et al., 2022; Corso al., 2023; Zhang et al., 2023) have been developed for protein-ligand docking, they may not directly address the protein-protein docking task investigated in this study. Protein-ligand interactions are often characterized by ligands binding to deep clefts on the protein surface (Vajda & Guarnieri, 2006), whereas protein-protein interactions involve larger surface areas (ranging from 700 to 1500 A and relatively flat binding interfaces (Bahadur et al., 2004; Nooren & Thornton, 2003), posing additional challenges. EquiDock (Ganea et al., 2022) and DIFFDOCK-PP (Ketata et al., 2023) are among the few methods specifically designed for protein- prriiin docking. In this paper, we pursue the energy-basse -hased treatment and propose learnable energy function for protein-protein docking equipped with binding interface prediction techniques (Sverrisson et al., 2021; Tubiana al., 2022).
2.2 ENERGY-BASER MODELING
Energy- based learning (LeCun et al., 2006) is learning paradigm that revolves around modeling and learning the underlying distribution of data. Unlike traditional supervised learning where explicit labels are provided in training, energy-based learning focuses on estimating and optimizing the energy associated with different configurations results in various domains, including data generation (Du & Mordatch, 2019; Subail et al., 2021; Xie et al., 2021), outtof-distribution detection (Liu et al., 2020), molecular structure design (Hataya et al., 2021) and combinatorial optimization (Li et al., 2023). There are several systematic approaches to train energy-basee models such as score matching (Vincent, 2011) and ratio matching, while gradient-based MCMC methods (Welling & Teh, 2011; Titsias & Dellaportas, 2019) are widely used inference. This paper employs the contrastive divergence approach (Hinton, 2002) with a Langevin dynamics sampling (Welling & Teh, 2011) method to train the energy function.
states of the system and has shown promising
3 EBMDOCK METHODOLOGG
3.1 PRELIMINARIES AND PROBLEM SETUP
We start by giving the formal formulation of the rigid docking problem. We are given an unbounded pair of proteins denoted as ligand (Lig) and receptor (Rec) arbitrarily. with n residues and m residues respectively. For simplicity, we denoted the unbounded pair as C == (Lig, Rec). The locations of residues are represented by the coordinates of its corresponding a-carbon atom, denoted as X,. R3xn and X E 3x We aim to predict the conformation their bounded (docked) state with the structure of each part unchanged during the docking process, that is, view each protein as a rigid body, Keeping the position of receptor fixed, ligand rotate with R E SO(3) and translate with t E Is can lead to a docking pose (RX, + t, xz), which means the complex they form is completely determined by a SE(3) transformation applied to the ligand. Following the traditional score-based docking method, we consider that the underlying process of ascertaining (R*, t²) can be modeled by a score function Fo t, C) parameterized by 0 and the criteria:
R "tt" arg max (R,,, RES
(1)
3.2 ENERGY-BASEL MODELING FOR RIGID DOCKING
To tackle the optimization problem, i.c. Eq. 1, we first need to construct a proper complex probability density function po (R, t. C)) positively related to the function Fo (R, t, C). Here employ
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the energy-based treatment, where Eg (R, t, C) -Fo C) and the higher score of function (o t, C)) represents lower energy:
exp ( E R. 0)) .z:=1 == cxp((Eo(RR,,,))
po RR t C)
(2)
Z
Since the observed data only illustrate the ground-truth docking pose, the energy eaaegy-based modeling is superior due to its maximum entropy (i.e., minimum prior) property (Jeffreys, 1946).
In contrast to the majority of prior research that trains energy models for generative modeling, our focus is to determine the optimal docking pose of a protein pair and its corresponding (R, t). For this purpose, the decision-making the energies associated with various docking poses corresponding to different (R, t), and selecting the pose with the lowest energy. Therefore, our primary concern is the relative energies, and we can avoid the need to estimate the partition function or compute expectations by training with carefully designed loss. This allows to parameterize the energy function using any neural architecture, and a thorough discussion on the formulation of the energy loss can be found in (LeCun & Huang, 2005).
process (inference) of our energy- model involves comparing
3.3 ENERGY MODEL ARCHITECTURE
In this section, we will formally introduce our proposed method, which is how to obtain the corre- sponding energy from given docking pose that is specific to a protein-protein complex.
Protein Representation Proteins consist of several sequences of amino acid residues that fold into 3-dimensional structures in space. In line with Ganea et al. (Ganea et al., 2022), we represent protein as a graph g = (V. E). Due to the limitation of computation resources, we work on residue level (our method can be extended to atom level smoothly), which means each node in the graph is an amino acid residue in the protein and each node has a 3D coordinate x; E IR3 which is the 3D coordinate of a-carbon atom of the residue. We use k-nearest-neighhor (KNN) build the graph of protein and the initial node feature is the embedding of residue type.
Protein Feature Extractor After we build the ligand graph Gc EVe,Ec), receptor graph GR = (VR, εR) and get the initial node embedding F, E _dxx, E Ædxm, we apply a message-passing neural network (MPNN) as our node feature extractor. Here we applied a mod- ified Equiformer as our backbone (Liao & Smidt, 2022), referred to as EquiformerPP. Equiformer leverages the strength of Transformer architectures and incorporates -quuivaaau features, and we add a SE(3)-equivariant cross-attention module to aggregate interactions between the ligand and the receptor. Specifically, the transformation process can be formulated as:
Zr E R Hr E Rd Zn E R 3xx .n E d EquiformerPP (X,, Ft, FR), (3) where Hy, Hg are the output node representations and is the output dimension. Zz, ZR are the transformed coordinates of residues but we do not need them in the following.
Energy based Distance Likelihood Potential based on distance likelihood has shown promise in protein-smal molecular conformation (Méndez-Luce et al., 2021). Inspired by that, here we use a deep learning approach learn distance distribution that is specific for node pair (one on ligand and one on receptor). which can be used to compute the energy of a certain docking pose later.
After message passing, we get the output representation of each node, (hr,..... .......,,,,.). ). "br.,....,h,.... ), then we employ a pairwise approach where we concatenate the features of each ligand node with each receptor node. The concatenated features are passed through a mixture den- sity network (MDN) (Bishop, 1994), which comprises a feedforward neural network that generates set of means μu, standard deviations a,, and mixing coefficients a to parametrize a Gaussian mixture model for each ligand-receptor node pair. The mixture model depicts the conditional probability density function of distance for each ligand-receptor node pair, enabling us to estimate the likeli- hood of locating ligand node separated from a receptor node j by any distance di,j- Specifically, for ligand node feature hy, and receptor node feature hr5 (where i,j means the index of node respectively). the MDN applies an MLP to get hidden state hyy:
h = MLP (cat(hi,, h,,)).
Then the hidden state is used to calculate the parameters required for the K = 6 mixed Gaussians:
Ms. ELU (Linear (h,r)) + 1;
(5)
oi.j = ELU (Lincar (hv,)) + 1;
(6)
ai Softmax (Lincar (has)).
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Feature Pairwise concatenation
Food fordffoowrrdddd N Neural Network
H2
Binding Region Prediction
Distance Prob Distribution
EquiformerPP
Lig
Interaction
HR
Selected Residucs
Rec
Transform ligand with R, t
Pairwisc Distance
Eg (R, t,
Figure 1: Construction of EBMDock's energy function. The ligand and receptor are represented as k-NN graphs where each node is one residue. Features of the residues are extracted from the graphs via EquiformerPP. Based on these features, residues on the binding interface are predicted, and the distance distributions between these residues are modeled with Gaussian mixture models parameterized by a, o, μ. Given a SE(3) transformation (R, t) of ligand, the pairwise distance between residues can be calculated and the energy is the mean of their negative log-likelisoods.
The negative la-liielihood of distance can be further defined as
log P (di,) hr, hr,) = - log anjNNN (da.) Miisks oz.jk))
(8)
Finally, we model the energy as a statistical potential, which is computed by averaging all the inde- pendent negative log-likelihood values computed for each agand-receptor node pair on the binding interface, For any R E SO(3), t E 13, we can get the corresponding distance between each node of ligand and receptor, and then we can calculate the energy by:
Es (R C)
1
Σ (KJ)BBI
log P (dgs hi, hrg)
(9)
[BI
where BT is the of residue pairs the binding interface and [B]] is its cardinality. The docking pose, in which each pair of residues is at the most possible distance from each other, corresponds to the state of minimum energy.
Binding Interface Prediction Studies have shown that proeein-protein interfaces display compara- ble chemical and geometric patterns, implying that two proteins may interact if their surfaces exhibit similar shapes and chemical functions (Gainza et al., 2020), thus predicting the functional sites is crucial for comprehending how proteins interact with each other (Sverrisson et al., 2021). Also, when taking the distance between all the receptor and ligand nodes into consideration, it may be greatly affected by the average distance the centroid distance between the ligand and receptor.
Taking the above factors into account, we first predict the interface where binding might occur, then we only predict the distance distribution those node pairs on the binding interface. We employ two different approaches, one is predicting the binding site in ligand and receptor respectively, and other predicting which pairs of nodes may interact, namely contact prediction. We model both predictions as binary classification problems. To improve the accuracy of the binding interface pre- diction, we calculate accessllcceessible surface area (SASA) (Lee & Richards, 1971; Mitternacht, 2016) for each protein and incorporate this
extra feature. See Appendix A for details.
As for the binding site prediction, we classify the nodes in the ligand whose shortest distance to the receptor is smaller than SÁ as positive samples and classify the remaining nodes negative samples. We do the same for nodes in the receptor. The probability of one ligand node (or receptor node) v, (or U.)) be on the binding site be formulated as:
P: (MLP((()))) Pj a (MLMM(())))
(10)
where MI. denotes multi-layer perceptron and as is implemented as the sigmoid function.
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For contact prediction, the node pairs with distance within the threshold 15Á are classified positive samples. We use MLP together with a sigmoid to get the contact probability of node v and by:
Pij = a o (ML hj)).
(11)
Note that both classification tasks have label-imbalance issues, thus apply weighted focal loss (Qin et al., 2018; Lin et al., 2017) to prevent models from easily classifying all samples as negative ones. The loss function takes the form:
foeet (p) -w(1 - p)7 log(p),
(12)
where w and y are hyperparameters.
It should be emphasized that both approaches potentially contribute to the identification of the bind- ing interface. Detailed comparisons can be located in Appendix B. There are approaches designed especially to find the protein-protein interaction interface (Tubiana et al., 2022; Sverrisson et al., 2021), we can also utilize their results in our pipeline for energy calculation.
3.4 ENERGY- BASED LEARNING FRAMEWORK FOR DOCKING POSE PREDICTION
In this section, we describe our proposed energy-based learning framework for protein-protein rigid docking pose prediction. We take the docking process as an optimization problem by sampling in continuous space which has six degrees of freedom (three of translation and three of rotation), and we use three Euler angles to characterize the rotation specifically.
Given specific ligand-receptor pair, we use MDN to predict the corresponding distance distribution of each selected node pair (either from binding site prediction or contact prediction), under which we can calculate the energy of given docking pose, thus explicitly provide the energy distribution of all docking poses. Here, we train our energy model with contrastive divergence:
Le Eo (R+ ,t (t,,C)= E, ,,,C),
(13)
where for a specific protein complex, (R+, t+) the ground-truth docking pose and (R (R))t7) is docking pose sampled from the complex probability density function PR. t C), which we call negative sample. The negative sample can be generated with Markov chain Monte Carlo (MCMC) sampling. Since our energy function is fully differentiable with respect to (R, t), we employ Langevin dynamics (Nijkamp et al., 2019; Grathwohl et al., 2019; Xie et al., 2016) as our MCMC transition kernel following recent developments in Energy-BBssee Models (EBMs). Specifi- cally, we start from a random sampled docking pose (R", tt), and perform:
X (R,,)
R +++1 R
(14)
2
> 7 1 = t"
(15)
where (RT,t") represents the docking pose after T iterations and e" is sampled from a normal distribution (0, A). The process is similar to gradient descent with an added Gaussian noise and we can arrive at low-energy docking pose through a series of steps. It is worth noting that the energy is a statistical potential based on the distance likelihood and no neural network parameters are involved in energy computation from (R. t). Given a pair of unbounded ligand and receptor, we get the predicted distance distribution based on only one forward propagation of the geometric neural network. Thus given a docking pose (R, t), we can compute the distance log-likelihoods quickly which yield the energy and its derivatives. This means our iterative Langevin dynamics sampling is very efficient in practice.
3.5 ENERGY- BASED TRAINING AND INFERENCE
The overall training loss is composed of three parts, including the contrastive divergence Les the L2 regularization loss E, on energy values, and weighted focal loss L-ocal(-) of binding interface prediction. The 1.2 regularization loss L,, = Fgg (R+, t+ C)² + (R, tt=, ,C)² is used to avoid the energy values get arbitrarily large causing gradient overflow problem. The total loss is:
Liotal L. + Rr + 22
(16)

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Table 2: Statisties of two datasets for experiments.
Items
DIPS-Het
DB5.5
avg. # of residues per receptor / ligand
259.4
265.1
avg. # of residues on binding pocket per receptor / ligand
25.5
26.9
avg. # of pocket residue pairs per complex #train / #valid #test
1079.5
779.8
9,093 / 1,146 1,153
/ - 253
where  and X2 hyperparameters. The energy-based training is shown Fig. 2.
Contrastive Divergence
Sample R²
For inference, we add intersection loss Lis RR, t. C) as part of the energy to avoid atom clash, the inference energy is:
dyname ieettions
Eo R.
Baneer Eg + aLis,
(17)
where is hyperparameter (see Appendix C for details). Then we start from random (Ro, t9) and use the Langevin dynamics sam- pling which approximately solves the problem in an iterative manner find (R'; t*) corre- sponded to the lowest energy. The iteration for- mula is the same as Eq. 14 & 15, and we will take enough steps till convergence.
Prodiction F
Grounnd cruth
Energy- training. EBMs are trained with contrastive divergence, where the ergy of the ground truth complex (green dot) de- creases and the energy of the sampled complex (blue dot) increases. Blue/green arrows indicate forward computation while dashed arrows indi- cate gradient back baeckpropagation.
Figure 2:
As the node features from EquiformerPP satisfy SE(3) equivariance, and the energy calculation process as well as the Langevin dynamics sam- pling also exhibits equivariance, our framework maintains SE(3) equivariance.
4
EXPERIMENTS
In this section, we conduct experiments comprehensively evaluate our EBMDock. All deep learning-based methods run on a machine with 19-10920X CPU, RTX 3090 GPU, and I28G RAM.
4.1 EXPERIMENTAL SETTING
Datasets We conduct numerical experiments on two datasets: Database of Interacting Protein Struc- tures (DIPS) (Townshend et al., 2019) and Docking Benchmarks 5.5 (DB5.5) (Vreven et al., 2015). DIPS is a comprehensive dataset of protein complex structures, sourced from the Protein Data Bank (Berman et al., 2000) and DB5.5 is benchmark dataset renowned for its high-quality data manually curated by domain experts. Similar to (Wang et al., 2023), we collect a modified version of DIPS, referred to as DIPS-Hett to focus interactions between different proteins. Both datasets are tailored for rigid body docking. The statistics of datasets are demonstrated in Table 2.
Baselines We compare our EBMDock method with several state-of-the-arr docking methods, in- cluding score-based docking software HDock (Yan et al., 2017), PatchDock (Schneidman-Duhne)) Duhhvvy al., 2005) as well as deep bearning-based algorithm EquiDock (Ganea al., 2022), DIFFDOCK- (Evans et al., 2021). We test HDock with the local
PP (Ketata et al., 2023) and AlphaFold-Multine packages they provided and test PatchDock the webserver. We test AlphafooldMMutimee through the colab interface they provided. For DIADDCC-PP we use their best JPPS-validated model in all related experiments. It should be noticed that HDock, PatchDock and AlphaFold ApphafoldMuulimerr use different training data compared to our models. They might have utilized parts of our test sets for extracting templates or as training examples, which may lead to an optimistic result.
Evaluation Metrics To measure the quality of the predicted docking pose, we report Complex Root Mean Square Deviation (CRMSD), Interface Root Mean Square Deviation (IRMSD), as well as the overall quality of docking evaluation metric DockQ (Basu & Wallner, 2016). For CRMSD, we use the Kabsch algorithm (Kabsch, 1976) to superimpose the predicted and ground-truth complex before computing the RMSD between them. To compute IRMSD, employ a similar approach but only use the coordinates of the interface residues with a distance of less than 8Å to the other protein's 's residues in the ground-truth complex. DockQ is computed following the settings in (Basu
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Table 3: Rigid protein-protein docking results on 100 samples from the DIPS-Het test set. The
number of initial poses sampled from the diffusion or energy model is in parentheses. EBMDock- interface means that we assume the binding interface is given. Note that DIFFDOCK-PP fails' on some samples (6 out of 100). The results we presented are obtained after removing these outliers.
Metric
Complex RMSD
Interface RMSD
D
Inference
Methods
Mean
Median
std
Mean
Median
std
Mean
Median
std
Time (sec)
PatchDuck
19.34
17.95
10.30
17.16
16.17
10.35
0.04
0.02
0.20
2232
I[Dock
2.04
0.23
5.98
2.67
0.25
8.51
0.88
0,98
0.28
782
AlphafooldMMutimeer
6.45
4.66
8.17
6.14
2.12
8.43
0.48
0.52
0.31
1940
EquiDook
11.73
10.94
7.19
11.43
10.82
6.60
0.12
0.05
0.18
3.9
DIFFDOCK-PPCS
12.09
11.87
9.88
13.79
12.50
11.72
0.24
0.04
0.35
30.9
EquiDock+plugin(]]
10.16
9.13
8.09
10.23
9.13
8.08
0.17
0.08
0.20
5.0
EBMDock (5)
9.09
7.95
6.83
8.98
6.93
6.20
0.22
0.12
0.22
8.3
EBMMDock(5-interfface
2.89
1.56
3.97
2.05
1,46
1.54
0.63
0.68
0.76
8.5
Table 4: Rigid protein-protein docking results on DB5.5. EBMDock* means using another inter-
face prediction method to roughly find the interaction interface. Note that DIFFDOCK-PP fails on some samples (13 out of 253). The results we presented are obtained after removing these outliers.
Metric
Complex RMSD 1
Interface RMSD
DockQ t
Inference
Methods
Mean
Median
std
Mean
Median
std
Mean
Median
std
Time (sec)
PatchDock
16.46
16.52
7.56
15.60
15.29
6.03
0.05
0.02
0.01
951
HDock
5.55
0.42
9.42
5.19
0.31
8.99
0.72
0.97
0.42
703
Mullaeeld-Mulimer
7.65
4.86
8.03
6.41
1.69
7.66
0.48
0.55
0.10
2540
EquiDock
17.15
15.90
5.31
14.56
14.10
4.91
0.03
0.02
0.04
3.8
DIFFDOCK-PP(S
17.56
17.21
7.87
17.76
17.12
8.47
0.04
0.01
0.09
37.3
EquiDock+plugin(1)
16.78
15.94
5.11
13.76
12.73
4.56
0.05
0.02
0.04
4.9
EBMDock (5)
14.79
15.64
5.05
12.47
10,99
5.06
0.05
0.04
0.04
7.4
EBMDDock*
12.61
13.26
6.52
10.27
11.56
5.70
0.16
0.05
0.20
10.4
EBMDock(5)inntrrfae
3.71
2.52
3.84
2.25
1.82
1.48
0.57
0.582
0.23
7.6
& Wallner, 2016). For a fair comparison, we utilize only the a-carbon coordinates for computing all the metrics. In order to have a direct comparison of efficiency, we also report the inference time.
Training and Inference Details Our models are trained and evaluated on the training and validation set of Ditt-Het, respectively. As for the training loss Liotal. we set \₁ and X2 as 0. and 50. When computing the contrastive divergence Le, we randomly sample three groups of (R. t?)) 50 steps of Langevin dynamics are then applied to them independently and the average of three energies serves as the second term in Eq. 13 to ensure the stability of training. We use Adam with learning rate 3e-4 and weight decay le-4 as the optimizer. More implementation details can be found in Appendix F.
We evaluate on the full DB5.5 and 100 samples randomly selected from the DIPS-Het test set, provide the pdb id of the chosen 100 samples in Appendix F. Again, we randomly sample 5 groups of (R², t?) and use Langevin dynamics sampling with 100 steps. The (R.00, t!!0) with the lowest energy is the predicted SE(3) transformation. In addition, we use deep learning-based method EquiDock generate an initial (R², tº) for each complex and optimize it with our energy function and Langevin dynamics sampling. Since other methods themselves are time-consuming. here we only use our method as a plug-in to boost Equi We also try to equip our method with another interface prediction method on DB5.5, utilizing their results in our pipeline for energy calculation.
4.2 RESULTS OF DOCKING POSE GENERATION
Results are shown in Table 3 & 4 and Fig. 4 & 5. It can be observed that EBMDock consistently out- performs EquiDock and PatchDock across both datasets. Compared DIFFDOCK-PE, EBMDock demonstrates superior performance across various metrics, except for a slight disadvantage in the av- erage DockQ metric. The results show that both EBMDock and EquiDock achieve very low standard deviations, indicating that EBMDock exhibits stability compared to all other methods. Utilizing an existing interface prediction method (Tubiana et al., 2022) can further enhance our results, named as EBMDock* Providing the ground-truth binding interface, FBBMDock consistently achieves favorable conformation of the complex and delivers competitive results compared to HDock, named as (B These results indicate that when we have a rough knowledge of the
We consider cases with CRMSD metric exceeding 50A as failure cases.
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1.0
Complex RMSD
Interfoc RMSD
0.8
55.00
10
0,6
10
Figure 4: Distributions of three evaluation metrics DIPS-Heet test set (100 selected samples). Here we choose to compare the results of the deep learning- cearning-based method.
PatchDock DockQ:0.047
EquiDock DockQu0.038
Equidock DeckQ:0.187
Groumd Truth
DockQ:0.559
Figure 5: Visualization of the protein complex 2ebg, which is successfully predicted by EBMDock.
interaction interface, our energy function and sampling technique exhibit remarkable effectiveness. Additionally, when taking the predicted docking poses of EquiDock as the initial points, EBMDock can improve the poses by moving them to the local minimum nearby, leading to large improve- ment beyond EquiDock, named as EquiDock+plugin(() In terms speed, EBMDock is times faster than traditional software and AF-multimer, and also nearly 5 times faster than DIFFDOCK-PP This result demonstrates efficiency of our sampling approach and also aligns with the statements mentioned in Section 3.4. More experiment results shown in Appendix E.
To further showcase sampling from the probability distribution, here show the Langevin dynamics sampling process of the top candidates for one cho- protein complex in Fig. 3. For each candidate, the DockQ increases as the energy decreases, indi- cating the effectiveness of our energy function and Langevin dynamics sampling. However, low en- ergy solution does not necessarily correspond high DockQ because the energy landscape may have more than one local optimum, some of which may be far from the ground truth, as the 'No.3' candidate so- lution shows: even though the 'No.3' candidate just has slightly higher energy than the 'No.2' candidate after iterations, might correspond to local op- timum that is far away from the global optimum.
No.1
DockQ Energy
No.2
Energy
Iteration
Figure 3: Sampling the top candidates with Langevin dynamics for a protein complex.
Moreover, compared with HDock. PathDock, and AF-multimer, we only consider the residue types and coordinates as the initial features and do not rely any finely designed task-specific features.
CONCLUSION
We have presented EBMDock, energy-based learning framework for protein-protein docking. By leveraging advanced geometric neural networks and distance likelihood modeling, it provides an energy probabilistic distribution over docking poses, enabling efficient sampling for improved solutions. The framework incorporates energy-based training with Langevin dynamics sampling and demonstrates superior performance compared existing deep learning- based methods. EBMDock also serves plugin to enhance the capabilities of other docking methods. Future work will explore multiple protein docking which involves graph matching methods for multiple graphs (Jiang et al., 2021).

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Published as a conference paper at ICLR 2024
REFERENCES
Ranjit Prasad Bahadur, Pinak Chakrabarti, Francis Rodier, and Joël Janin. A dissection of specific and non-specific protein-proteii interfaces. Journal of molecular biology, 336(4):943-955. 2004.
Sankar Basu and Björ Wallner. Dockq: quality measure for protein-protein docking models. PloS one, 1 11(8):001118 61879, 2016.
Helen M Berman, John Westbrook, Zukang Feng, Gary Gilliland, Talapady N Bhat, Helge Weissig, Ily N Shindyalov, and Philip Bourne. The protein data bank. Nucleic acids research, 28(1):
235-242, 2000.
Christopher M Bishop. Mixture density networks. 1994.
Gabriele Corso, Hannes Stärk, Bowen Jing, Regina Barzilay, and Tommi S. Jaakkola. Diffdock: Diffusion steps, twists, and turns for molecular docking. In The Eleventh International Conference on Learning Representations, 2023.
Yilun Du and Igor Mordatch. Implicit generation and modeling with energy based models. Advances in Neural Information Processing Systems, 32, 2019.
Stephan Eismann, Raphael JL Townshend, Nathaniel Thomas, Milind Jagota, Bowen Jing, and Ron O Dror. Hierarchical, rotation-equivariant neural networks to select structural models of protein complexes. Proteins: Structure, Function, and Bioinformatics, 4955)-93-- 2021.
Richard Evans, Michael O'Neill, Alexander Pritzel, Natasha Antropova, Andrew Senior, Tim Green, Augustin Zidek, Russ Bates, Sam Blackwell, Jason Yim, et al. Protein complex prediction with alphafold-multimer. biorxiv, pp. 2021-10, 2021.
CARL Frieden. Protein-protein interaction and enzymatic activity. Anmual review of biochemistry, 40(1):653-696 1971.
Pablo Gainza, Freyr Sverrisson, Frederico Monti, Emanuele Rodola, D Boscaini, MM Bronstein, and BE Correia. Deciphering interaction fingerprints from protein molecular surfaces using geo- metric deep learning. Nature Methods, 17(2):184-192 2020.
Octavian-Eugen Ganea, Xinyuan Huang, Charlotte Bunne, Yatao Bian, Regina Barzilay, Tommi S Jaakkola, and Andreas Krause. Independent se (3)-equivariant models for end-to-end rigid protein docking. In International Conference Learning Representations, 2022.
Johannes Gasteiger, Florian Becker, and Stephan Günnemann. Gemnet: Universal directional graph neural networks for molecules. Advances Neural Information Processing Systems, 34:6790-
2021.
Will Grathwohl, Kuan-Chich Wang, Jömn-Henrin Jacobsen, David Duvenaud, Mohammad Norouzi, and Kevin Swersky. Your classifier is secretly an energy based model and you should treat it like one. arXiv preprint arXiv: 032663, 2019.
Zhichun Guo, Wenhao Yu, Chuxu Zhang, Meng Jiang, and Nitesh V Chawla. Graseq: graph and sequence fusion learning for molecular property prediction. In Proceedings of the 29th ACM international conference information & knowledge management, pp. 435 433-443, 2020.
Ryuichiro Hataya, Hideki Nakayama, and Kazuki Yoshizoe. Graph energy- based model for sub- structure preserving molecular design. arXiv preprint arXiv:2102.04600 2021.
Geoffrey E Hinton. Training products of experts by minimizing contrastive divergence. Neural computation, 14(8):1771-1800, 2002.
Emiel Hoogeboom, Victor Garcia Satorras, Clément Vignac, and Max Welling. Equivariant dif- fusion for molecule generation in 3d. In International Conference on Machine Learning, pp.
8867-8887, 2022.
Ziwei Huang. Structural chemistry and therapeutic intervention of protein-protein interactions in immune response, human immunodeficiency virus entry, and apoptosis. Pharmacology & Thera- peutics, 86(3):201-215, 2000.
10

---

Published as a conference paper at ICLR 2024
Harold Jeffreys. An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 186(1007):453-44) 4614 1946.
Zetian Jiang, Tianzhe Wang, and Junchi Yan. Unifying offline and online multi-graph matching via finding shortest paths on supergraph. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43(10))3648-36633 2021. doi: 10.1 100/TPPMM.2020.29999999
John Jumper, Richard Evans, Alexander Pritzel, Tim Green, Michael Figurnov, Olaf Ronneberger, Kathryn Tunyasuvunakool, Russ Bates, Augustin Zidek, Anna Potapenko, al. Highly accurate protein structure prediction with alphafold. Nature, 596(7873):583- -899,,55899008 2021.
Wolfgang Kabseh. A solution for the best rotation to relate two sets of vectors. Acta Crystallo- graphica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography, 32 (5):922-923, 1976.
Mohamed Amine Ketata, Cedrik Laue, Ruslan Mammadov, Hannes Stark, Menghua Wu, Gabriele Corso, Céline Marquet, Regina Barzilay, and Tommi Jaakkola. Diffdock-pp: Rigid protein- protein docking with diffusion models. In ICLR 2023-Machine Learning for Drug Discovery workshop, 2023.
Guillaume Launay, Masahito Ohue, Julia Prieto Santero, Yuri Matsuzaki, Cécile Hilpert, Nobuyuki Uchikoga, Takanori Hayashi, and Juliette Martin. Evaluation of consrank-like scoring functions for rescoring ensembles of ponein-proteiin docking poses. Frontiers molecular biosciences, 7: 559005, 2020.
Yann LeCun and Fu Jie Huang. Loss functions for discriminative training of energy-based models.
In International workshop on artificial intelligence and statistics, pp. 206-213, 2005.
Yann LeCun, Sumit Chopra, Raia Hadsell, M Ranzato, and Fujie Huang. tutorial on energy-based learning. Predicting structured data, 1(0), 2006.
Byungkook Lee and Frederic M Richards. The interpretation of protein structures: estimation of static accessibility. Journal of molecular biology, 55(3):379- 1971.
Yang Li, Jinpei Guo, Runzhong Wang, and Junchi Yan. T2t: From distribution learning in training to gradient search in testing for combinatorial optimization. In Advances in Neural Information Processing Systems, 2023.
Yi-Lun Liao and Tess Smidt. Equiformer: Equivariant graph attention transformer for 3d atomistic graphs. In The Eleventh International Conference Learning Representations, 2022.
Tsung-Yi Yi Lin, Priya Goyal, Ross Girshick, Kaiming He, and Piotr Dollár. Focal loss for dense object detection. In Proceedings of the IEEE international conference on computer vision, pp- 2980-2988, 2017.
Weitang Liu, Xiaoyun Wang, John Owens, and Yixuan Li. Energy-based out-of-distribution detec- tion. Advances in neural information processing systems, 33:21464-2144 2020.
Yi Liu, Limei Wang, Meng Liu, Yuchao Lin, Xuan Zhang, Bora Oztekin, and Shuiwang Ji. Spherical message passing for 3d molecular graphs. International Conference on Learning Representa- tions (ICLR), 2022.
Wei Lu, Qifeng Wu, Jixian Zhang, Jiahua Rao, Chengtao Li, and Shuangjia Zheng. Tankbind: Trigonometry-wwan neural networks for drug-protein binding structure prediction. In Advances in Neural Information Processing Systems, 2022.
Yujie Luo. Shaochuan Li, Yiwu Sun, Ruijia Wang, Tingting Tang, Beiqi Hongdu, Xingyi Cheng, Chuan Shi, Hui Li, and Le Song. xtrimodock: Rigid protein docking via cross-modal representa- tion learning and spectral algorithm. bioRxiv, pp. 2023-02, 2023.
Oscar Méndez-Lucio, Mazen Ahmad, Ehecatl Antonio del Rio- Chanona, and Jörg Kurt Wegner. A geometric deep learning approach to predict binding conformations of bioactive molecules. Nature Machine Intelligence, 3(12):1033-1039, 2021.

---

Published as a conference paper at ICLR 2024
Simon Mitternacht. Freesasa: An open source e library for solvent accessible surface area calcula- tions. F1000Research, 5, 2016.
lain H Moal, Mieczyslaw Torchala, Paul A Bates, and Juan Fernnndez-Res Recio. The scoring of poses in protein-protenn docking: current capabilities and future directions. BMC bioinformatics. 14(1):
1-15, 2013.
Erik Nijkamp. Mitch Hill, Song-Chun Zhu, and Ying Nian Wu. Learning non-convergent non- persistent short-run meme toward energy- based model. Advances Neural Information Process- ing Systems, 32, 2019.
Trene MA Nooren and Janet M Thornoon. Structural characterisation and functional significance of transient protein-protein interactions. Journal of molecular biology, 325(5);911-1018, 2003.
Tony Pawson and Piers Nash. Protein-protein interactions define specificity in signal transduction.
Genes & development, 14(9):027-1047,, 2000.
Kathryn A Porter, Israel Desta, Dima Kozakov, and Sandor Vajda. What method to use for protein- protein docking? Current Opinion in Structural Biology, 55:1-7, 2019.
Ruoxi Qin, Kai Qiao, Linyuan Wang, Lei Zeng, Jian Chen, and Bin Yan. Weighted focal loss: An effective loss function to overcome unbalance problem of chest x-ray14 14. In 10P Conference Series: Materials Science and Engineering, volume 428, pp. 012022. IOP Publishing, 2018
Danilo Jimenez Rezende, Séhastien Racanière, Irina Higgins, and Peter Toth. Equivariant hamilto- nian flows. arXiv preprint UrXIv:1909.13739, 2019.
Sergio Ruiz-Carmona, Daniel Alvarez-Gareia, Nicolas Foloppe, A Beatriz Garmendia-Doval. Szil- veszter Juhos, Peter Schmidtke, Xavier Barril, Roderick Hubbard, and David Morley. rdock: fast, versatile and open source program for docking ligands to proteins and nucleic acids. PLoS computational biology, 100351,, 2014.
Victor Garcia Satorras, Emiel Hoogeboom, and Max Welling. E (n) equivariant graph neural net- works. In International conference on machine learning, pp. 9323 22232,93,,,2 2021.
Christina EM Schindler, Isaure Chauvot de Beauchêne, Sjoerd J de Vries, and Martin Zacharias. Protein-protein and geptide-protedd docking and refinement using attract in capri. Proteins: Struc- ture, Function, and Bioinformatics, 85(3):391-398, 2017.
Dina Saineidmman-Dhhony Yuval Inbar, Ruth Nussinov, and Haim J Wolfson. Patchdock and symmdock: servers for rigid and symmetric docking, Nucleic acids research, 33(suppl_2):W363- W367, 2005.
Andrew Shrake and John A Rupley. Environment and exposure to solvent of protein atoms. lysozyme and insulin. Journal of molecular biology, 79(2):351-371, 1973.
Hannes Stärk, Octavian Ganea, Lagnajit Pattanaik, Regina Barzilay, and Tommi Jaakkola. Equibind: Geometric deep learning for drug binding structure prediction. In International Conference on Machine Learning, pp. 20503-20521, 2022.
Mohammed Suhail, Abhay Mittal, Behjat Siddiquie, Chris Broaddus, Jayan Eledath, Gerard
Medioni, and Leonid Sigal. Energy-based learning for scene graph generation. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 13936-13945, 2021.
Freyr Sverrisson, Jean Feydy, Bruno E Correia, and Michael M Bronstein. Fast end-to-end learning on protein surfaces. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 152211 2021.
Ian Tetlow, Robin Wait, Zhenxiao Lu, Rut Akkasaeng, Caroline G Bowsher, Sergio Esposito, Behjat Kosar-Hashemi. Matthew K Morell, and Michael J Emes. Protein phosphorylation in amyloplasts regulates starch branching enzyme activity and protein-protein interactions. The Plant Cell, 16(3):694-708, 2004.
12

---

Published as a conference paper at ICLR 2024
Michalis Titsias and Petros Dellaportas, Gradient-basee adaptive markov chain monte carlo. Ad- vances Neural Information Processing Systems, 32, 2019.
Raphael Townshend, Rishi Bedi, Patricia Suriana, and Ron Dror. End-to-end learning on 3d protein structure for interface prediction. Advances in Neural Information Processing Systems, 32, 2019.
Jéröme Tubiana, Dina Schneidman-Duhont and Haim Wolfson. Scannet: an interpretable geo- metric deep learning model for structure-based protein binding site prediction. Nature Methods, 19(6):730-739, 2022.
Sandor Vajda and Frank Guarnieri. Characterization of pooeiin-ligany interaction sites using experi- mental and computational methods. Current Opinion in Drug Discovery and Development, 9(3): 354, 2006.
Jacob Verburgt and Daisuke Kihara. Benchmarking of structure refinement methods for protein complex models. Proteins: Structure, Function, and Bioinformatics, 90(1):83-99)) 95, 2022.
Pascal Vincent. connection between score matching and denoising autoencoders. Neural compu- tation, 23(7); 1664-1646, 2011.
Thom Vreven, lain H Moal, Anna Vangone, Brian G Pierce, Panagiotis L Kastritis, Mieczyslaw Torchala, Raphael Chaleil, Brian Jiménez-García, Paul A Bates, Juan Fernandez-Recio, et al. Updates to the integrated protein-protein interaction benchmarks: docking benchmark version and affinity benchmark version Journal of molecular biology, 127(11):3031-304) 2015.
Yiqun Wang, Yuning Shen, Shi Chen, Lihao Wang, YE Fei, and Hao Zhou. Learning harmonic molecular representations on riemannian manifold. In The Eleventh International Conference on Learning Representations, 2023.
Max Welling and Yee W Teh. Bayesian learning via stochastic gradient langevin dynamics. In Proceedings of the 28th international conference on machine learning (ICML-I1), pp. 681-688, 2011.
Jianwen Xie, Yang Lu, Song-Chun Chun Zhu and Yingnian Wu. A theory of generative convnet. In International Conference on Machine Learning, pp. (635-2644, PMLR, 2016.
Jianwen Xie, Yifei Xu, Zilong Zheng, Song-Chun Zhu, and Ying Nian Wu. Generative pointnet: Deep energy-based learning unordered point sets for 3d generation, reconstruction and classifi- cation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 14976-14985, 2021.
Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? International Conference on Learning Representations, 2019.
Yumeng Yan, Di Zhang, Pei Zhou, Botong Li, and Sheng You Huang. Hdock: web server for protein-protein and wrotein-dna/mm docking based on hybrid strategy. Nucleic acids research, 45(W1)::365-W373, 2017.
Nianzu Yang, Huaijin Wu, Kaipeng Zeng. Yang Li, and Junchi Yan. Molecule generation for drug design: graph learning perspective. arXiv preprint 200202220..0 2022a.
Nianzu Yang, Kaipeng Zeng, Qitian Wu, Xiaosong Jia, and Junchi Yan. Learning substructure
invariance for diitr-i-ittiittionn molecular representations. In Advances in Neural Information Processing Systems, 2022b.
Nianzu Yang, Kaipeng Zeng, Qitian Wu, and Junchi Yan. Molerec: Combinatorial drug recommen- dation with substructure-aware molecular representation leaming. In Proceedings of the ACM Web Conference 2023, pp. 4075-4085, 2023.
Xiao-Meng Zhang, Li Liang, Lin Liu, and Ming-Jing Tang. Graph neural networks and their current applications in bioinformatics. Frontiers in genetics, 12:690049, 2021.
Yangtian Zhang, Huiyu Cai, Chence Shi, and Jian Tang. E3bind: An end-to-end equivariant network for potein-ligand docking. In The Eleventh International Conference on Learning Representa- fions, 2023.
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A SURFACE FEATURE
This section introduces the method we use to calculate surface-aware node features. We use solvent- accessible surface area (SASA) (Lee & Richards, 1971; Mitternacht, 2016) represent the surface degree of each residue. Specifically, solvent-accessinn area of molecule that accessible to solvent molecules. It provides valuable information about the molecular shape, volume, and interactions with the surrounding environment. SASA can be calculated using various algorithms. Here we adopt the Shrake-Rupley (Shrake Rupley, 1973) algorithm and the general steps involved are as follows:
surface area (SASA) measures the surface
Define the probe radius, denoted as Tprobe.
Generate the solvent-accessilee surface by placing probe sphere of radius r probe at discrete points on the molecular surface. Determine if each point accessible the probe molecule, considering no overlap with the molecule.
Calculate the surface area of each atom using the area of its van der Waals sphere, denoted as A,- Calculate the overlapping area with adjacent atom j's van der Waals spheres, denoted Aij.
Compute the SASA for each atom i by SASA, A; Σj Asy- Then up the SASA values for all atoms within the residue obtain the SASA value of each residue.
We then normalize the calculated SASA values to number between and 1 to represent the surface degree of each residue, higher value indicates greater surface degree. The normalization performed by dividing all SASA values in protein by the largest one in it. We analyze the frequency of unnormalized and normalized SASA values across all proteins in DB5.5, as shown in Fig. S6. The SASA values of residues the binding interface are also analyzed for comparison. It can be observed that the distribution of SASA values of residues on the binding interface is different from that of all residues, indicating that this feature be helpful for binding interface prediction.
All Residues
All Residues
0.125
Residues Binding Regions
Residues Binding Regions
Frequency in DBS.5
Frequency in DB5.5
0.0150
01000
0.0075
01025
8.000
150
710
82
0.6
0.6
0.5
1.0
SASA
SASA
Figure S6: Distribution of unnormalized (left) and normalized (right) residue SASA across pro- teins in DB5.5. Residues on the binding interface are analyzed specially to be compared with all residues.
B BINDING SITE PREDICTION VS CONTACT PREDICTION
In Section 3.3, we present two approaches for predicting the binding interface in protein complexes: binding site prediction (BSP) and contact prediction (CP). BSP involves predicting the binding sites on the ligand and receptor respectively, while CP focuses predicting the residue pairs that interact with each other. To construct the energy function, we select the relevant residue pairs based on the prediction results and consider only those pairs in the calculation. For BSP, we the Cartesian product of the predicted binding sites on the ligand and receptor to identify the interacting residue pairs. On the other hand, CP directly utilizes the predicted residue pairs without further modification.
During the training phase, we employ BSP and CP respectively to compute the loss for binding interface prediction. Through experimentation, we observe that training the energy models with BSP
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loss auxiliary loss results in stabler performances. During the inference phase, evaluate the performance BSP and CP on the DIPS-Het test set. The results, presented in Table S5, demonstrate that CP outperforms BSP across all evaluation metries. This indicates that CP is capable of selecting high-quality residue pairs with better generalization capabilities.
Table S5: Comparison on DIPS-Het of different binding interface prediction methods. BSP means binding site prediction, CP means contact prediction. AUC (area under the receiver operating char- acteristic curve) and AP (average precision) are metrics of the binding interface prediction module.
Complex RMSD
Interface RMSD +
DockQ
Method
AUC f
AP
Mean
Median
Mean
Median
Mean
Median
BSP
0.86
0,67
11.49
10.23 8.98
11.13 9.23
9.46
0.17
0.09
CP
0.95
0.58
9.74
7.90
0.20
0.11
C INTERSECTION Loss
In a natural protein complex, the ligand and receptor should never intersect with each other. In our energy function, however, the optimum points may lead varying degrees of intersection. To alleviate this issue, we add an auxiliary term to the energy function during inference- the inter- section loss Lis(Rs R,, t. C). Following previous work (Sverrisson et al., 2021; Ganea et al., 2022), the surface of protein point cloud X E R3xa is expressed as {x E IR3 G(x) =)) where G(x) = -a In (_n (-p x, x2l²/a)). For a point x R$ outside the protein, it has G(x) > 7. To avoid intersections in the complex, each residue in the ligand should be outside the receptor, so as to the residues in the receptor. As a result, we can write the intersection loss as:
1 = n
m
C,s(R,t,C)
max(0. 7- GR(RXLi + +t)
max(0, -G.A.,)(X))))) (18)
m
where Gn is the surface of receptor and Gl(R,E) is the surface of ligand after transformed with (R. t). We set the parameters = 8 and o = 8 so that the ground truth has Lrs(R* ,,' C²,C) = 0 and no "holes" Ee (R, C) + iL [R, t, C). is parameter that controls the influence of intersection loss on energy function and partly depends on the shape of the interface in the complex. We find 1.5 works well on DB5.5 and a 0.5 works well on DIPS-Het. We conduct experiments on the DIPS- Het test set with different a and the results are shown in Table $6.
inside protein. Finally, the energy function for inference is Ein (R, C)
Table $6: Docking Performance on DIPS-Het with different coefficient of Lis-
Complex RMSD +
Interface RMSD
DockQ
value
Mean
Median
Mean
Median
Mean
Median
0
9.74
8.98
9.23
7.90
0.20
0.11
0.5
9.09
7.95
8.98
6.93
0.22
0.12
9.66
9.13
10.16
9.67
0.19
0.08
50
11.63
11.40
13.00
12.41
0.08
0.04
D ADDITIONAL EXPERIMENTS
D. EXPERIMENTS ON ANTIBODY- ANTIGEN DOCKING
We have claimed that HDock and AlphafolddMultimer use different training data compared to models. They might have utilized parts of our test sets for extracting templates or as training exam- ples, which may lead an optimistic result. For fair comparison, we follow (Luo et al., 2023) and select 68 sabbody-antigen samples from the Protein Data Bank which were released after October 2022 and have not been used to train ApphaFoddMMutimee
templates in HDOCK. The PDB
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Table S7: Ablation studies DIPS-Het.
Complex RMSD 4
Interface RMSD
DockQ
Binding Interface Prediction
CD Loss
Mean
Median
Mean
Median
Mean
Median
18.87
18.50
19.47
19.63
0.03
0.01
13.34
14.94
10.55
11.02
0.14
0.05
9.74
8.98
9.23
7.90
0.20
0.11
Table S8: Rigid protein-protein docking results on 68 antibody-antigen samples. The number of initial poses sampled from the diffusion energy model is in parentheses. EBMDock-interface means that we assume the binding interface is given. Note that DIFFDOCK-PP fails on 4 samples (Complex RMSD metric exceeds 50 Å). The results we presented obtained after removing these outliers.
Metric
Complex RMSD
DockQ
f
Methods
Mean
std
Mean
std
HDock
15.78
6.36
0.09
0.19
AlphaFold-Multimer
13.65
5.89
0.11
0.17
EquiDock
18.47
2.71
0.04
0.02
DIFFDOCK-PPG)
23.82
7.04
0.02
0.06
EBMDock (5)
14.87
4.93
0.05
0.02
EBMDock(S)-interface
6.51
5.13
0.32
0.22
id of these samples are listed here: 8dls, 8dlr, 8dfi, 8dfh, 8dec, 8dad, 7zr8, 7zf8, 7xxl, 7xh8, 7x26, 7wsl, 7wsi, 7ws6, 7ws2, 7wrz, 7wrv, 7wro, 7wrl, 7wrj. 7wog, 7wlc, 7wef, 7wee, 7wed, 7wer, 7wbz, 7urq, 7uaq, 7tty, 7ttx, 7ttm, 7tpj. 7tp4, 7tp3, 7tlz, 7the, 7tc9, 718w, 7t7b, 7101, 7swp, 7sul, 7str, 7sem, 7sd5, 7sbu, 7sbg, 7sbd, 7sa6, 7s5p, 7rxp, 7rxi, 7rbu, 7qtk, 7n0a, 7lo8, 7l07, 7kql, 7fje, 7f7e, 7f6z, 7f6y, 7eng, 7ek0, 7ejz, 7ejy, 7c9p.
The results are shown in Table s8. Compared to Table and 4, both HDOCK and AlphaFold- Multimer exhibit significant decrease in performance on unscen cases. Although EBMDOCK's results also decline, they are close to those of AF-Multimer and surpass HDOCK in CRMSD. Addi- tionally, when EBMDOCK is able to obtain rough docking interface information, the docking results are far superior to both AF-Multimer and HDOCK.
D.2 EXPERIMENTS ON FULL DIPS-HEP TEST SET
In Table 3, report the results on only 100 samples from the DIPS-Het test set. That's because methods like PatchDock, HDock, and AlphaFold-Multing have too long inference time. These baselines limit from conducting fair comparison that our method is highly efficient and relatively stable, allowing scalability to large-scale datasets. Here, present our results the full DIPS-Het test set (1153 samples) in Table $9.
more extensive dataset. We have shown
E ABLATION STUDY
We conduct additional experiments on DIPS-Het to further examine the contributions of each pro- posed technique. The model without binding interface prediction uses all ligand-receptor node pairs to calculate energy. while the model without contrastive divergence simply minimizes the energy of ground-truth complexes. The results in Table S7 show that both techniques improve the perfor- mances. Specifically, the binding interface prediction module mainly improves the Interface RMSD since it can find the residues on the proteinn interfaces, while contrastive divergence con- tributes to the overall performance, which is in line with design purposes.
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Table S9: Rigid protein-protein docking results on full DIPS-Het test set. The number of initial poses sampled from the energy model is in parentheses. EBMDock-interface means that we assume the binding interface is given.
Metric
Complex RMSD +
Interface RMSD +
DockQ t
Methods
Mean
Median
std
Mean
Median
std
Mean
Median
std
EBMDock (5)
9.18
8.12
6.23
9.12
7.23
5.94
0.21
0.11
0.20
EBMDock(S)-imtrfiac
2.93
1.65
3.87
2.14
1.53
1.54
0.62
0.68
0.71
Fsi
Input 3D Graph
Input Graph
Lincar
Lincar
Type 2.
Linear
Litear
Embedding
DTP
Linear
Equivarint Graph Allennion
Reshape
Scale
Scalans
Imoge feature f
Cerky Rel
Gells
Layar Norm
Linear
DTF
Embedking Edlge-Degree)
Feed Forword Networkk
Softrax
Literr
Layer Noen
Lineer
Kenhape
Feed Forword Network
Cassc
Linrar
Linear
(a) Equiformer Architecture
(b) Equivariant Gruph Attention
(d) Feed Forward Network
Figure S7: Architecture of Equiformer (Liao & Smidt, 2022). Equiformer embeds input 3D graphs with atom and edge-degree embeddings and processes them with Transformer blocks, consisting of cquivariant graph attention and feed-forward networks. "g" denotes multiplication, "ø" denotes addition, and "DTP" stands for depth-wise tensor product. Σ within circle denotes summation over all neighbors. Gray cells indicate intermediate irreps features.
F IMPLEMENTATION DETAILS
F.1 THE ARCHITECTURE OF EQUIFORMERPP
The origin architecture Equiformer has been shown in Fig. S7, which directly extracted from the original paper Liao & Smidt (2022). Since the Equiformer is designed for one 3D graph, not capturing any interaction of two graphs, we add a equivaaaui cross-attention module to ag- gregate interactions between the ligand and the receptor. Here will describe this cross-attention module, shown in Fig. $8.
Ligand
Equiformer Layer
Cross Attention
Receptor
Equiformer Layer
Cross Attention
Figure S8: Demonstration of the cross-attention module.
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Given the ligand features yr and receptor features UR, the cross-attention layer enables communi- cation between proteins:
(y W. Wz) (yRWV))
4/2 Gate(
vd
vR = Gatel (uR (aWW))unWW) (yr.Wv),
Vd
where yr. E [Rad M/G Im denotes the features of the ligand/receptor protein, d denotes the dimension of features, n/m denotes the number of nodes in the ligand/receptor graphs. Wq. WK. and Wy are the parameter matrices for the query, key, and value in attention computation, respec- tively. The gate is an equivariant activation function, which can be found in the original paper Liao & Smidt (2022). The modified architecture is named EquiformerPP.
F.2 TEST SAMPLES
Here we provide the pdb id of the selected 100 samples: 6R17, 5W3X, 6J40, SCK3, 2V4I, IM57, 2A4R, 5JH5, IMU2, 7QIH. 5A63, 7ATE, 4ZXS, 4XSS, 4H62, 7RNW, 40Z1, 5MR3, 6B BB, 4Y50, 7ZVY, 2HHF, 6FC1, 7VKB, 4LNO, 3ZV0, 7SZ0, IYRT, IJEQ, 2D5R, 3BPQ, SHQP, 4EEC, 6CWX, EEGP, 3G50, 2P10, 1QOV, 3SND, 4DFC, 18IH, 4IYP, 4HRT, 314Q, 5ZWB, 5YRH, 4Y61, 1TY4, ILPH, 4J2L, 5ZWL, 5D7G, 6NVW, 30JY, 2FBW, SJKC, 7U7N, 7MU2, 2F8V, 6C0Y, 7NNL, SBRK, 3MEU, 5H67, 4UE6, 6ZDX, 6U07, 5QZU, 5IC6, 7NIN, 5DOB, 40YD, 3C5W, 6GZC, 7LTS, 3ASZ, 7KM6, 5ME5, 2DSP, 4I5N, 3SXU, SM5E, 6KMQ, IJLT, 7F5M, SVLL, 6FTO, 4QLB, 5KY7, 3KYS, 7CNR.
F.3 EVALUATION METRICS
We have briefly described the evaluation metrics Section 4.1, and we will give further expla- nations here. We assess the performance of rigid protein docking using three metrics: Com- plex RMSD, Interface RMSD, and DockQ. Let's denote the ground truth and predicted complex structures as Z E (x++m) and Z (3xxx++m), respectively. When calculating the Complex RMSD (C-RMSD), we first align Z with Z" using the Kabsch algorithm and then use the formula a+m Z - P where |- |F denotes the Frobenius norm. For the calculation of Interface RMSD (IRMSD), we follow a similar approach but only consider the coordinates of the interface residues of the ground truth complex. These residues are defined as those within distance of 8A from the other protein's 's residues. DockQ is based on three standardized criteria from the Critical Assessment of Predicted Interactions (CAPRI): Tyrms Trme: and Snat- To compute Tyrms we first superimpose the backbone atoms the receptors in the ground truth and predicted complex, and then calculate the ligand RMSD based on the coordinates of the backbone atoms of corresponding ligands. For Trms. we align the interface residues (residues with a-carbon within a distance of 8 A from any a-carbon in the other protein) and calculate the backbone RMSDD. Lastly, fnat represents the recall in recovering residue-residue contacts between the proteins, where two residues are considered "in contact" if their o-carbon atoms have a distance less than 8 Å. Dock is continuous score ranging from 0 to 1, derived from Lyme, Iyme, and frat. A higher DockQ score indicates better performance in terms of docking predictions. Note that backbone means that all atoms in residues are used for calculation, not only the a-carbon atoms.
F.4 TRAINING DETAILS
Here we provide more details about the training process. We first preprocess the protein complexes by randomly assigning the roles of ligand and receptor and randomly translating and rotating the ligand while keeping the receptor fixed, which are used to examine the SE(3)-qquiaariancc of our method. The calculation of the energy function only considers the residue pairs the binding interface. Note that the information about the binding interface comes from ground truth during training, and binding site prediction or contact prediction during inference. When computing the contrastive divergence L., we randomly sample three groups of (R. t from the distribution of U(- T and N (u, o), respectively, where μ vector from the center of the ligand the center of receptor and a is the average distance between two centers in complexes, set as 40A in our
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0 DockQ:0.004
10 DockQ:0.017
t= 20 DockQ:0.154
Ground Truth
30 DockQ:0.333
40 DockQ:0.533
t 50
DuckQu0.717
Figure 89: Visualization of Langevin dynamies sampling process of the protein complex 1a95.
experiments. Then r steps of Langevin dynamics are applied to them independently and the average of three energies serves as the second term in Eq. 13. Besides, we set bounds for R and t. For R, the bounds are mr) and values out of the range are handled by adding 2kmr, where k: is proper integer. For t, the bounds are [-400, 400), big enough range for protein complexes, and values out of the range are simply clamped. As for the training loss Liotal, we observe that the orders of magnitude of three terms are 1, 10, and 1e-3, respectively. Accordingly, we set A and X as 0. and 50, respectively. More hyperparameters are demonstrated in Table $10. We train EBMDock on machine with 10-20X CPU, RTX 3090 GPU, and 128G RAM for days.
Table S10: Hyperparameter choices of EBMDock and the training phase settings.
Hyperparameters
Values
Node degree (for k-NN)
10
Dimension of h of EquiformerPP layers
128
Network
Number of Mixed Gaussians
6
Dropout Rate
0.2
Binding Interface Prediction Focal Loss Weight
0.25
Binding Interface Prediction Focal Loss Gamma
2
Learning Rate of Langevin dynamics(
0.01
LD
Number of Samples of Langevin dynamics
3
Number of Steps of Langevin dynamics (r)
50
Batch Size
Training
# Epoches
50
Optimizer
Adam
Learning Rate
3e-4
Weight Decay
le44
G MORE VISUALIZATION
To show the effectiveness of Langevin dynamics sampling, we present visual example of protein complex 1a95 in Fig. S9. As the iterations 7 increase, the predicted complexes have higher DockQ and are closer the ground truth.
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H DISCUSSION
In this section, we will discuss EBMDock and the possible improvements in future work further.
Firstly, EBMDock uses residue-level modeling but be easily extended to the atom level. Since internal structures of residues of the same class are relatively similar and modeling on residue level can be computationally efficient, we represent the residues with their alpha carbon. To extend EBM- Dock to the atom level, we only need to modify the nodes in the protein graph from representing a residue to an atom. Instead of learning the statistical potential of residue distances, we would learn the statistical potential of atomic distances. The rest of the network structure and training methods remain unchanged. However, the computing time and memory consumption will increase a lot. we generalize our methods to flexible docking, which is very challenging. we can still use residue- level modeling and rely on existing tools such as Rosetta to complete the side chain conformation based on the predicted backbone structures.
Secondly, we believe energy-based modeling promising for the protein-protein docking problem. By carefully designing the deep learning- based energy function, energy-based modeling has clear physical meaning while maintaining high inference efficiency. The AF-Multimer method, which is based on multiple sequence alignments (MSAs), is difficult to improve in terms of efficiency and has significant limitations when dealing with large-scale data. Although the accuracy of energy- modeling cannot match that of AF-Multimer currentl, has the potential be used for large-scale virtual screening. Besides, the performances of EBMDock are limited by the scarcity of protein structural information while the protein sequence information is easily accessible and abundant. Given sufficient time, we can leverage AF to predict complex structures and generate new training data to further expand our approach.
Finally, the proposed energy-based learning framework has good extensibility. The backbone can be replaced by new representation learning methods such as those based on the protein surface. The energy function can be generalized to contain more terms. As long as an energy term is differen- tiable, it can be incorporated into our framework easily. For example, the HDOCK and AF-Multimer achieve good performance on old proteins because they directly use information from training sets such as templates and sequence alignments. We can design new differentiable energy terms to mea- sure how well the docking poses conform the templates. We can also design energy terms based on physics, for example, van der Waals dispersion/reeulsionn, directional hydrogen bond energy, Coulomb electrostatic energy, and desolvation-free energy. As the calculation of these terms may be expensive, we can approximate them using deep learning methods. We believe they can improve the performance and will explore these possibilities in future work.
I RELATED WORK OF GEOMETRIC DEEP LEARNING
Graph structured data is a ubiquitous type of data and finds particular resonance within the realm of computational biology (Zhang et al., 2021; Yang et al., 2022a). Notably, complex biological en- tities, ranging from molecules proteins, can all be represented through graphs. Moreover, Graph Neural Networks (GNNs), a method commonly used for graph structures, have been widely applied in computational biology (Xu et al., 2019; Guo et al., 2020; Yang al., 20226; 2023). Unlike generic graph data, molecular systems in 3D space exhibit symmetries, such as translations and ro- tations (Satorras et al., 2021; Gasteiger et al., 2021). Recently, many studies have incorporated such inductive biases into GNN design. These novel methods have demonstrated superior performance in tasks including molecular property prediction (Liu et al., 2022), conformation generation (Hooge- boom et al., 2022), particle system dynamics (Satorras et al., 2021), and conformation based energy estimation (Rezende et al., 2019).
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Reviews:
[{"review_id": "4USLVbpljh", "reviewer": "Reviewer_S6B4", "summary": "This work proposes an energy-based model for rigid protein-protein docking.\nThe energy function is parameterized as an averaged negative distance log-likelihood between some residue pairs, which can be trained using standard contrastive divergence.\nRepresented as a SE(3) transformation (R, t) (6 degrees of freedom), a possible docking pose can be generated via Langevin dynamics guided by the energy function. In practice, the authors restrict the calculation of energy function to residue pairs on the binding site.\nThe experimental results show that the proposed method outperforms recent dl-based baselines and run much faster than traditional docking programs.", "questions": "1. Training an EBM with contrastive divergence where negative samples are generated by Langevin dynamics on the fly is quite time-consuming. May I know how much time it takes to train the EBMDock and have you tried any other techniques to reduce the cost?\n\n2. Another straightforward idea of rigid Protein-protein docking is that we first train an inter-residue distance predictor and then recover the docking poses that best fit the predicted inter-residue distance map via gradient descent. Have you tried comparing EBMDock with such a model? \n\n3. Apart from the diverse predictions, can you briefly illustrate the insights or motivation for using EBM to solve rigid protein-protein docking? (As training EBM using CD is so costly while training the diffdock-pp is relatively cheap.)", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is well presented with a straightforward idea, following the standard EBM paradigm.\n2. A binding site prediction module is involved to restrict energy function parameterization on relevant residue pairs, which might be crucial for the success of the method.\n3. The idea seems general and can be applied to other kinds of rigid-docking settings.", "weaknesses": "1. The method only considers the position of alpha carbon in each residue, which can be a crucial limitation. Being able to model the geometry of full backbone atoms or even sidechain atoms is necessary in real-world scenarios.\n\n2. The authors include the DockQ score as one of their evaluation metrics. The original DockQ score includes the percentage of native contacts preserved in the predicted structure as one of criteria. However, the current method ignores all backbone atoms except for alpha carbon. The author should make this point clear.\n\n3. The proposed method falls short of the AlphaFold-Multimer on both evaluation tasks. Although the authors have mentioned that the results of AFM might be optimistic (test set may overlap with the training set of the AFM), it would be better if they could report the results of both models on another benchmark set composed of proteins released after the cutoff date of the AFM's training set.", "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": "k70GXFaU76", "reviewer": "Reviewer_XEYz", "summary": "This paper proposes an energy-based model, called EBMDock, for protein-protein docking. EBMDock learns an energy function $E(R, t)$, where $R$ is a rotation and $t$ is a translation. This energy function is parameterized by a modified Equiformer architecture and trained using contrastive divergence, a standard algorithm for EBM optimization. At test time, EBMDock uses langevin dynamics to find the optimal rotation and translation (the lowest energy). EBMDock is evaluated on standard protein docking benchmarks and compared with a list of prior methods for protein docking.", "questions": "* In practice, how do you calculate the gradient $\\nabla_R E(R, t)$?", "limitations": "", "rating": 3, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* Formulating protein docking as an energy-based model is an interesting idea.\n* The presentation of the method is relative clear.", "weaknesses": "* The performance of EBMDock is much worse than AlphaFold-multimer (AFM) (DockQ: 0.16 vs 0.48). Although EBMDock runs much faster than AFM, the performance is far from AFM. As a result, EBMDock is not going to be useful in practical setting since its prediction is mostly inaccurate.\n* EBMDock is only evaluated on rigid-body docking setting, where the ground truth structure of each protein is given. Therefore, its comparison with AFM is not fair because it does not assume the binding site is known and it folds both proteins from sequence directly.\n* In the case where ground truth is not available, we still need to use AF2 to predict the structure of each protein. As a result, the entire workflow for EBMDock won't be much faster than AFM.\n* Gradient of rotation $\\nabla_R E(R, t)$ is not defined clearly.\n* Equiformer-PP architecture is not described.", "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": "NZZnH32tsp", "reviewer": "Reviewer_AgXx", "summary": "The authors propose EBMDock, an energy-based approach to rigid protein-protein docking. They sample from the distribution of docking poses using the energy function and show competitive performance compared to other ML and classical docking methods. The EBMDock inference time is substantially less than more computationally intensive methods like AlphaFold-Multimer and PatchDock. EBMDock’s flexibility allows for using the method with initial guesses from other methods like EquiDock.", "questions": "1. What is the performance on binding site and contact prediction, before pose prediction and docking? How does this performance compare to binding site and contract prediction classifiers?\n\n2. Do the authors have any ideas about what explains the performance gap to HDock and how they could improve the EBMDock protocol? The results suggest that, based on DockQ scores, DiffDock-PP, EquiDock, and EBMDock without providing the ground truth binding site are not usable for pose prediction. This is a valuable result, but it also motivates proposals for substantial improvements.\n\n3. It would be helpful to define all of the variations (Equidock+plugin, EBMDock (5), etc.) in a single place before the results are presented. The naming conventions could also be made clearer and more specific.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The method is well-motivated and provides a nice contrast to prevailing ML methods in this area. The focus on inference time performance is practical and helpful for practitioners.", "weaknesses": "It is still broadly unclear if ML methods for protein-protein docking have any utility compared to \"classical\" methods. This is still an interesting area of research, with much further progress to be made, but clear discussion of limitations and future directions is particularly warranted.", "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": "SQTGuw8WCb", "reviewer": "Reviewer_94kF", "summary": "The paper proposes EBMDock, an energy-based learning framework for rigid protein-protein docking. It models the docking problem as optimizing the pose that minimizes the energy function.  The energy function is a statistical potential based on the distance likelihood between residue pairs on the binding interface. It is modeled using Gaussian mixture models parameterized by a geometric deep neural network. EBMDock is trained with contrastive divergence using Langevin dynamics for negative sampling. This allows implicit data augmentation to handle limited protein complex data.", "questions": "- It is intriguing to observe that AF-Multimer outperforms rigid PP docking, especially considering that AF2-Multimer relies solely on sequence inputs and does not incorporate any structural information. Could the author provide an explanation for this phenomena?\n- While the energy function is modeled as statistical potential using distance likelihoods, the formulation seems rather simplistic. Could more complex energy formulations further improve accuracy?\n- For the MDN, why choose a 6-component GMM? Is performance sensitive to the number of mixtures?\n- How was the EquiformerPP architecture designed? What modifications were made to Equiformer?\n- How does inference time scale with protein complex size? Could efficiency be an issue for very large complexes?\n- Have the authors considered extending this to flexible protein docking? What changes would be needed?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The energy-based formulation provides a nice probabilistic framework for modeling the docking problem and enables efficient sampling methods. Contrastive divergence training is clever for data augmentation.\n- The method maintains equivariance throughout the pipeline leveraging geometric deep learning. This is important for generalizability.\n- Ablation studies analyzing the impact of different binding site prediction methods add nicely to the empirical analysis.", "weaknesses": "- The empirical results of the proposed method are not as strong as expected, as it fails to outperform traditional methods and AlphaFold-multimer. This raises questions about the effectiveness of the proposed approach in comparison to existing techniques.\n\n- The use of only 100 samples for the DIPS-Het test set is questionable. It is unclear if there are any constraints that prevent the application of the proposed method to larger datasets. The authors should provide more information on this aspect and, if possible, test their method on a more extensive dataset.\n\n- The omission of xTrimoDock [1] from the baselines is a notable oversight. According to [1], xTrimoDock can outperform both HDock and AlphaFold-Multimer, and it is also relatively fast. The authors should consider including xTrimoDock in their comparison to provide a more comprehensive evaluation of their proposed method.\n\n- While the performance of EBMDock-interface is impressive, it relies on the ground-truth binding interface, which might not be a fair comparison to other baselines. This could lead to misunderstandings regarding the method's actual performance. I suggest that the authors clarify this point and consider presenting the results of EBMDock-interface separately in a standalone table to avoid potential confusion.\n\n[1] xTrimoDock: Rigid Protein Docking via Cross-Modal Representation Learning and Spectral Algorithm", "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)