Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Authors: Ke Liu
Hao Chen
Chunhua Shen
Venue: ICML 2025 poster (2025)
Keywords: Protein-ligand binding
Primary area: applications->everything_else

Abstract: Developing models for protein-ligand interactions holds substantial significance for drug discovery. Supervised methods often failed due to the lack of labeled data for predicting the protein-ligand binding energy, like antibodies. Therefore, unsupervised approaches are urged to make full use of the unlabeled data. To tackle the problem, we propose an efficient, unsupervised protein-ligand binding energy prediction model via the conservation of energy (CEBind), which follows the physical laws. Specifically, given a protein-ligand complex, we randomly sample forces for each atom in the ligand. Then these forces are applied rigidly to the ligand to perturb its position, following the law of rigid body dynamics. Finally, CEBind predicts the energy of both the unperturbed complex and the perturbed complex. The energy gap between two complexes equals the work of the outer forces, following the law of conservation of energy. Extensive experiments are conducted on the unsupervised protein-ligand binding energy prediction benchmarks, comparing them with previous works. Empirical results and theoretic analysis demonstrate that CEBind is more efficient and outperforms previous unsupervised models on benchmarks.

Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Ke Liu
Hao Chen
Chunhua Shen
Abstract
case of small molecules, plenty of labeled samples exist; thus, supervised methods are quite appropriate (Öztürk et al., 2018; Monteiro al., 2022; Pei et al., 2023; Lu al., 2022). However, for ligands like antibodies, few samples with an affinity label are available, which causes the failure of su- pervised approaches.
Developing models for protein-ligand interactions holds substantial significance for drug discovery. Supervised methods often failed due to the lack of labeled data for predicting the protein-ligand binding energy, like antibodies. Therefore, unsu- pervised approaches are urged to make full use of the unlabeled data. To tackle the problem, we propose an efficient, unsupervised protein-ligand binding energy prediction model via the conser- vation of energy (CEBind), which follows the physical laws. Specifically, given protein-ligand complex, we randomly sample forces for each atom in the ligand. Then these forces are applied rigidly to the ligand to perturb its position, fol- lowing the law of rigid body dynamics. Finally, CEBind predicts the energy of both the unper- turbed complex and the perturbed complex. The energy gap between two complexes equals the work of the outer forces, following the law of conservation of energy. Extensive experiments are conducted on the unsupervised protein- ligand binding energy prediction benchmarks, compar- ing them with previous works. Empirical results and theoretic analysis demonstrate that CEBind is more efficient and outperforms previous unsu- pervised models on benchmarks.
To address the problem of data scarcity, Feng et al. (2024) constructed a self-supervised reconstruction task in their BindNet to learn binding representations. BindNet does not provide the binding affinity but requires subsequent fine- tuning on downstream tasks. Furthermore, in DSMbind, Jin et al. (2024) proposed the unsupervised protein-ligande binding energy prediction problem. The problem can be defined as given only the quaternary structure of protein- ligand complexes without any affinity labels for training, the model predicts the binding energy. Model performance is evaluated in terms of the correlation between predicted bind- ing energy and ground truth. DSMbind solves the problem by considering the log- lideeiod of SE(3) denoising score matching as binding affinity. They train an energy model by matching the forces, i.e., the gradient of the predicted energy with respect to atom coordinates, with the likelihood. Specifically, they perturbed the protein of the complex with randomly sampled rotation and translation. Then they calculate the rotation caused by the predicted forces and match the predicted force and rotation with the sampled translation and rotation, respectively. However, DSMBind made some approximations, and training DSMBind model is memory-intensiv.
1. Introduction
In this work, we propose a memory-efficient and physics- aware training framework dubbed Conservation of Energy based Binding (CEBind), which tackles the problem of un- supervised protein-ligan ligand binding energy prediction via the law of conservation of energy. We focus on the unsupervised protein-ligand binding prediction, including antibodies and small molecules, following the settings of DSMBind (Jin et al., 2024). Specifically, given a protein-ligande complex, we randomly sample forces for each atom in the ligand, which is then applied rigidly to the ligand following the rigid dynamics. Then the energy model predicts the energy of both the perturbed and unperturbed complexes. Accord- ing to the energy conservation law, the difference between the two energies is equal to the work of the outer forces. Be- sides, we also match the gradient of the energy with respect
Predicting protein-ligand binding crucial to the discovery of drugs (Zheng et al., 2020; Ballester & Mitchell, 2010; de Freitas & Schapira, 2017). A protein is highly likely to be bonded with a ligand if its conformation, i.e., qua- ternary structure, holds a high affinity or low binding en- ergy (Chothia& & Janin, 1975). Therefore, the ligand binding prediction comes down to the affinity or energy prediction of the protein complex. The ligands encompass proteins, such as antibodies, and small molecules. In the
'Zhejiang University, Hangzhou, China. Correspondence to: Hao Chen <haochen.cam cad @ @ww..cuu.n...........
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Energy conservation law
Energy Model
ppay ARa0t
Sample forces
Apply forces
following
rigid dynamics
aE
Force matching
Figure 1. Overview CEBind framework: given a protein- ligand complex, CEBind first samples forces for each atom in the ligand. Then the forces applied the ligand following rigid dynamics, get perturbed complex. Next, the energy of both the unperturbed and perturbed complexes is predicted by the energy model. Finally, we match the gradient with respect to atom coordinates with inner forces and follow the law of conservation of energy match the work of outer forces and energy gap.
to atom coordinates with the inner forces. CEBind has the following features:
2.2. Antibody-antige Binding Prediction
Data scarcity is extremely high in antibody-antigen binding prediction. Only 566 samples exist the largest binding affinity dataset of antibodies, SAbDab (Schneider al., 2022). Few works focus supervised antibody-antige binding prediction. Only Myung et al. (2022) predicted the tribodd--ntiigen binding affinity with expensive handcraft feature engineering. Jin et al. (2024) compared their DSM- Bind with Atom3d model (Townshend et al., 2021) and Frame Average model (Puny et al., 2022) they implemented for supervised antibody-antigen binding affinity prediction. Antibody design models often include modules for model- ing the antibody, antigen, and the interaction between them (Viswanath et al., 2013; Alford et al., 2017), We also adapt the antibody design model architecture and compare it with these unsupervised approaches.
Physics-aware. CEBind follows the law of conserva- tion of energy and rigid dynamics.
2. Memory-efficient The process of calculating the rigid dynamic of the complex does not require a gradient.
3. Better performance. Extensive experiments are con- ducted, and empirical results demonstrate that CEBind outperforms previous models.
2. Related Works
2.1. Protein Molecule Binding Prediction
2.3. Unsupervised Binding Energy Prediction
lot of approaches have been proposed for protein-ligand binding prediction (Lu et al., 2022: Jin al., 2024; Somnath al., 2021; Zhang et al., 2023; Wang et al., 2022; Dittrich al., 2018). For supervised methods, graph neural networks are often constructed to encode protein-ligand complexes into representation vector, and a subsequent neural net- work is performed to predict the binding affinity (Lu et al., 2022: Somnath et al., 2021; Zhang et al., 2023; Wang et al., 2022). Unsupervised approaches based on physics often require expensive computation (Miller III al., 2012). The most related works to CEBind are DrugScore (Dittrich et al., 2018) and DSMBind (Jin et al., 2024). DrugScore predicts the energy of each atom pair independently, using atom types and distances. DSMBind further includes infor- mation on the ligand complex context. Our CEBind learns a more physics-awars and efficient energy model.
Jin et al. (2024) introduced the task of unsupervised protein- ligand binding energy prediction for both protein-smat small molecules and antibody-antigens. Operating only with the quaternary structures, namely, the prevailing potein-llgantl complexes, model is trained to predict the complexes' binding affinity. Due to the non-availability of binding affin- ity labels for training, the model's 's performance is evaluated through the correlation between the predicted affinity and ground truth affinity. Our work parallels the task setting in DSMBind (Jin et al., 2024). However, we put forth a physics-aware and more streamlined training framework.
3. Preliminaries and Notations
Protein-Ligand structures. A protein-ligand complex is described as its atom features A = [Ap, Ar.) and posi-
2

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
tions X = px,, XL). where Ap and Xp denote the pro- tein and AL. and X I. denote the ligand, i.e., C [A, x]. The atom features consist of individual atom types A = (a1, ,g....,a..) a.,l, where a; E CDS denotes the atom type and n denotes the number of atoms. X = ( .. ,nn) where nn E R2 denotes the 3D atom position. The proteins in protein-ligand complexes can also be described with their amino acids, i.e., = Ph. : Psl. where each amino acid consists of atoms P: (at; Od.+1,''.. ,ag+m]- The small molecules in the complexes can be described with their molecule graphs, i.e,, G = (Ap, F, Xp), where E denotes the edges between atoms.
AI), where Al is the total duration that the forces apply to the ligand.
The translation of each atom in the ligand consists of three parts: (1) the translation of the ligand caused by combined forces; (2) the ligand rotation, which causes the translation of all the atoms in the ligand; and (3) the rotation of bonds that can rotate.
The translation of ligands caused by combined forces can be easily calculated according to Newton's 's laws of motion, as follows:
AXLI = 1 f, AP,
(2)
2 M
Problem formulation. Given only the structure of protein-ligand complex C = (A, X), an unsupervised protein-ligand binding energy model is trained to predict Since
where N and M indicate the number of atoms in the ligand and the mass of the ligand, respectively.
The translation caused by the ligand rotation is obtained through rigid dynamics. First of all, we calculate the inertia matrix IN R3x3 and torque E R of the ligand as include the correlation between predicted ggeudo-bindingg follows: energy and the ground truth.
the pscudo-binding energy of the complex the object of training does not involve the ground truth of binding energy, the metrics for evaluation and testing only
N
IN
x 11l2 (x. -x) -4)
(3)
Rotation. Following Jin et al. (2024), we the rotation vector w ( Wa] and the rotation matrix Rw for rotating the atoms. The rotation matrix Rw is applied to the atom coordinates æ; through:
N
T
a 11) x f:
(4)
where l is the center of the ligand and calculated as follows:
R. = + CiW vz + o2w x (w x æ.), (Rotation)
1 E,EX,
where
Σ x
(5)
sin ||
cos W|ll
C1
w ||
C2 =
|
(1)
Then we apply the torque to the ligand and get the atom translation caused by the ligand rotation as follows:
4. Methodology
AX = Rot( 2 I TA'" X1. 14) XT.,
(6)
In this section, describe our binding energy prediction method, which follows the law of conservation of energy and rigid dynamics. The overview of our CEBind is shown in Fig. 1. Given a protein-ligand complex, we sample forces for each atom in the ligand, which are then applied to the ligand following rigid dynamics. Finally, we calculate the energy of the perturbed complex and the unperturbed com- plex. The energy gap between them equals the work of the outer forces, and the gradient of the perturbed complex with respect to atom coordinates matches the inner forces.
where the operation Rot is descrihed in Eq. Rotation. More details can be found in the Appendix. A.1.
For the bonds that can rotate, we also model the rotation of them by taking the fixed atoms as the origin and performing rotating on it. Here we take the side chain for example. The side chain consists of its atom types Aş and atom positions Xs. i.e., Cs = (As, Xs) without the backbone atoms, i.e.. Os CCCC C. The fixed position for the side chain rotation is a-carbon, the positions of which are denoted as X Similarly, we have the inertia matrix and torque of the side chains as follows:
4.1. Rigid Dynamics
Given protein- ligaand complex, C (), we randomly sample forces f E R as the combined forces for each atom in the ligand. Then we apply these forces to the ligand, following rigid dynamics to preserve the structure of ligands. To remove the effect of kinetic energy, we perform the force f during (0, 1 Af), and perform the force ff during (,,
Is = x Xsal 11 2 - Xxn) - -Ss) (7)
Ts =
[x; Xssa) x f::
(8)
1=1
3

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Protein Encoder
Protein
Energy-based basedd Module
Energy
Ligand Encoder
Ligand
Small Molecule
Protein
Figure Model architecture: our energy model consists of protein encoder, a ligand encoder, and an energy-based module. The protein encoder and the ligand encoder learn representations for each pair of protein-ligand complexes. Energy-based module model the interaction between ligand and protein and predict their binding energy.
The translation caused by the side chain rotation is obtained as:
According to Jammer (1999), work is the inner product of force and displacement. The work outer forces is obtained through the inner product of force and displacement, as follows:
A s Rotí AsA? Xs X Sa
Xs-
W Xo+aX
Finally, the translation of each atom is formulated through combining all the translations in Eq. 2. Eq. 6, and Eq. 9 as follows:
f fatter dx = -AXX?
(12)
X.
where the translation X is obtained through Eq. 10. The derivation of Eq. 12 can be found in the Appendix. A.3.
AXL -XL1 AX + AX)
(10)
where [0, AXs] indicates the noise the backbone atoms is 0 and the noise the side chain atoms is AX s-
The energy of the protein-ligand complex C(A,X) is esti- mated by the energy model k EE (A, X). For the perturbed complex C(A, X), the energy is kE X). Then we match the change in energy with the work done by the outer forces as follows:
The perturbed protein-ligand complex is Č(A, X) Č(A, [Xp, XL)), where X2 is perturbed with Eq. 10 as follows:
= X, = *AX A) (11)
Le Ell A, X) kE (A, X)
W||
(13)
= KE Eg A, X) EA X) 2 AX, 21191. (14)
The details of the force- pereepperfommin process are illustrated in appendix. A.2.
4.3. Force Score Matching
4.2. Energy Conservation Law
The gradients of energy with respect atom coordinates are forces that move the ligand to position with lower energy. Therefore, the predicted inner forces by the energy model are calculated as follows:
According the law of conservation of energy, the total energy of isolated system remains constant and the en- ergy change in the system is equal to the work done on that system from the outside. Taking the protein-ligand complex as an isolated system, the binding energy of the complex re- mains constant. After perturbing the ligand with the sampled forces f, the changes in the binding energy of the complex equal the work of the outer forces f outeer- Here, we model the inner force between the ligand and the protein be proportional to their distance, i.e. finner X (X, Xo) and f = f finerr + foter (Without loss of generalization, any other force field modeling can be applied here with- out changing our framework). With a scaler k, we have inner 1(X, Xo).
akE A, X)
f inner
(15)
ax,
Following Jin al. (2024), we also calculate the gradients of the perturbed complex as the force match the inner forces via a denoising score matching follows:
C = E[I inner
V
Enner logp(f ( )1
(16)
EE
A.
X
AX L
KE|l
(17)
aX1

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Algorithm 1 Training procedure (single data point)
2022) for evaluation, and 258 complexes from the PDBbind core set with labels of binding energy (Su al., 2018) for test. The protein is cropped to its binding pocket in the protein-small molecule dataset. The pocket is defined as the residues with a distance less than 10A from the ligand. The antibody-antigen dataset includes 3416 antibody-antigen complexes from the structural antibody database (SAbDab) (Schneider et al., 2022) for training. 116 complexes from CSM-sb (Myung et al., 2022) for evaluation, and 566 com- plexes with labels of binding affinity from SAbDab for test. All test samples never exist in the training datasets. Each entry in the training sets contains only the protein-ligand complexes. The binding affinity labels exist only in the validation and test sets.
Require: A protein-ligand complex C(A,
1: Sample a noise level ar
2: Sample forces for each atom in ligand f ~ NO, d
3: Calculate the translation of each atom 1 with the forces via Eq. 10.
4: Calculate the work of outer forces W via Eq. 12.
5: Apply the noise to the original complex to get the per- turbed complex C = (A, x) via Eq. 11.
6: Predict the energy of both complexes
and 7: Minimize the energy loss Le and force matching loss
Eg A, X) via energy model.
L from Eq. 13 and Eq. 16.
Evaluation metrics. We mainly employ the Pearson cor- relation coefficient between the ground truth protein-ligand binding affinity and the predicted binding energy, which is calculated as follows:
Different from DSMBind (Jin al., 2024), the self- supervision is on each atom instead of the combined trans- lation and rotation of the whole ligand (Song et al., 2024a; Yim et al., 2023; Song et al., 2024b). More details can be found in Appendix. A.4.
Pp E,(Fo(A,)) Fl A, X ( -T)
V VX(E)(A))); -E( (A. E, (y: 7)
4.4. Model Architecture
(18)
where y, is the ground truth binding energy of the i-th protein-ligand complex. We train the model with five dif- ferent random seeds and report the mean and standard de- viation of rp. The complete results can be found in the Appendix. C.I.
The whole model consists of a protein encoder, a ligand encoder, and an energy-based module, as shown in Fig. 2. We use the encoders to encode the protein and ligand into representation vectors. Then the vectors are input into the energy model to model the interactions between protein and ligand. Finally, the energy-base module outputs the bind- ing energy of the complex. The training procedure for our CEBind is described in the Algorithm 1. The first five steps can be done in the dataloader and these operations do not require gradient calculation in the backward propagation. Therefore, CEBind is more memory-efficient Besides, CE- Bind follows physics laws, including rigid dynamics and the law of energy conservation.
Compared approaches. For protein-small molecule bind- ing, we mainly compare our CEBind with physics- based potentials (including Glide (Friesner et al., 2006), AutoDockyina (Eberhardt
al., 2021), MM/GBSA (Miller III et al., 2012), and DrugScore2018 (Dittrich et al., 2018)), unsupervised models (including standard Gaussian denoising score matching (Gauss DSM) (Jin et al., 2024), Contrastive learning (Chen et al., 2020), and DSMBind (Jin et al., 2024)), and supervised approaches (including IGN (Jiang et al., 2021) and PLANET (Zhang et al., 2023)) following Jin et al. (2024). The supervised approaches were trained with over 19,000 samples with binding affinity from the PDBBind database. For antibody bind- ing, we also compare our CEBind with physics-based mod- els (including all the approaches in the CCharPPI server (Moal et al., 2015)), unsupervised models (including con- trastive learning (Chen et al., 2020), Gauss DSM (Jin et al., 2024), ESM-1v (Meier et al., 2021). ESM-IF (Hsu et al., 2022), and DSMBind (Jin et al., 2024)), and supervised approaches, frame- averaging neural network (FANN) (Puny et al., 2022). FANN is trained on extra 273 antibody- antigen samples with binding affinity labels from protein- protein binding mutation database (SKEMPI) (Jankauskaité et al., 2019). BANNeransfer is trained on extra 5427 samples from SKEMPI (Jankauskaité et al., 2019) and then fine-
5. Experiment & Discussion
We trained the energy model with our CEBind framework. We first compare the performance of our CEBind with other approaches. Then we analyze our CEBind in terms of effi- cacy. Finally, we propose an atom-only variant of CEBind to explore the generalization of the unsupervised binding energy prediction approaches.
5.1. Setup
Dataset. The datasets used in this work consist of protein- small molecule dataset and antibody-antigen dataset fol- lowing (Jin et al.. 2024). The protein-small molecule dataset contains 4806 protein-ligand complexes from PDB- bind V2020 database for training (Su et al., 2018), 357 complexes randomly sampled from PDBbind in (Stärk al.,
5

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
a. Protein and Ligand Encoders
embodding
embedding
ESM
GNN
atoen positions
Protein
Small Molocule
positions
h. General Protein and Ligand Encoder
Energy based Module
atom embedding
Proteinn
protein ligand
interaction modeling
Energy
posstions
Ligind
Figure Implementation of CEBind. (a) Protein encoder and small molccule encoder. The protein encoder consists of the embodding module, ESM module, and atom positions. The small molecule encoder contains graph neural network (GNN). atom embedding module, atom positions. They both output combination of the three features. ESM module outputs the representation each residue in the protein. (b) A general encoder for both small molecules and proteins. The general encoder outputs the representation of ligands with combination their atom emhedding atom positions. (c) The architecture the energy-hased model. The energy model includes an equivariant GNN and protein- -igand interaction modeling module. indicates the multiplication.
tuned on the 273 samples. The comparison mainly follows DSMBind (Jin et al., 2024).
To explore the gap between small molecules and proteins, implement general ligand encoder for both small molecules and proteins as shown in Fig. 3(b). The gen- eral encoder consists of an atom embedding module. By concatenating the atom positions and atom embedding as
5.2. Implementation Details
The implementation of our CEBind is illustrated in Fig. We implement variants of the ligand encoder. Following previous works (Jin et al., 2024; Schneider et al., 2022; Eber- hardt et al., 2021), we first implement the ligand encoder for protein and small molecules, respectively, as shown in Fig. 3(a). The protein encoder consists of an atom embed- ding module to represent each atom in the protein as a vector and an ESM module model the interaction inside the pro- tein and embed each residue into vector. Then, the vectors and atom positions concatenated to represent each atom as follows:
R Embed((A)) @ XL.
(21)
the general encoder can be used for both proteins and small molecules.
The energy-based module contains equivariant graph neural network and a protein-ligand interaction modeling module following Jin et al. (2024) as shown in Fig. 3(c). With the two modules, the representations for each atom in the ligand obtained as [Ra.', Rp') ((Ri, Rp)), where Rp obtained through Eq. 19 Eq. 21. Then multiplication is performed between the atom representation Ry and Rp' get the energy of each atom pair. The binding energy of complex is the sum of the energies of all the atom pairs within a distance threshold.
RLP Embed((:)) Ar.) @ ESM(P1.) @ Xt., (19)
where @ AL. and Pt. indicate the concatenation, atom representation, and protein residues.
The small molecule encoder consists of GNN to model the interaction inside the molecule and embed each atom into a vector. With the atom representation from the atom embed- ding module and the atom positions, the atom representation of the small molecule is obtained through:
5.3. Experimental Results
The comparison of CEBind with previous methods demon- strates the efficacy of CEBind. Then we demonstrate the efficiency CEBind both theoretically and empirically. Fi- nally, we conduct experiments with the general atom-only protein and ligand encoder explore the gap between small
RLM Embed(AL) @ GNN(G) XL.
(20)
6

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Table Pearson correlation on protein-small molecule and antibody-antigen datasets. denotes that results are reproduced from their official code. Tp denotes the Pearson correlation. The best results in each category of approaches are labeled in bold.
Protein-sman molecule
Antibody-antigen
Method
Pp
Method
Pp t
PLANET
0.811
FANN
0.325 0.014
Supervised
IGN
0.837
FANN FANNuaaseer: transfer
0.350 0.033
Glide
0.467
ZRANK
0.318
Autodockvina
0.604
CP_PIE
0.234
Physics-bass basses
DrugScore2018
0.602
PYDOCK
0.248
MM/GBSA
0.647
AP PISA
0.323
ESM-Iv
0.024
ESM-IF
0.024
Contrastive
0.625 + 0.002
Contrastive
0.308 = 0.037
Unsupervised
Gauss DSM
0.636 + 0.008
Gauss DSM
0.360 0.016
DSMBind
0.644 + 0.003
DSMBind
0.365 0.011
CEBind
0.652 + 0.005
CEBind
0.374 0.009
Table 2. Pearson correlation protein-small molecule and muuibody-antigen datasets. CEBindatom denotes the general encoder for Cross Val denotes that CEBindatom trained and tested on different datasets.
ligands.
Method
DSMBind
CEBind
CEBindatom
Cross Val
Protein-small molecule
0.644
0.652+00.005 0.374=0.009
0.648+0.003 0.346:00.004
0.646++00001 0.320=10.002
Antibody-antigon
0.365=10.011
Table Ablation study. Pearson correlation on the protein-small molecule and antibody- datasets. L. and Cr denote the energy conservation and force matching, respectively.
L
Lt
ESM
Protein-molecule(r, t)
Antibody-anntgen(r,, t)
0.646 0.002
0.361 + 0.016
V
0.646 0.001
0.362 1 0.025
V
0.650 + 0.004
0.346 + 0.010
V
V
0.652 = 0.005
0.374 + 0.009
molecules and proteins the unsupervised protein-ligand binding energy prediction task.
FANNtransfer is pre-trained on the 5427 samples from SKEMPI (Jankauskaité et al., 2019) and fine-tuned with 273 antibody- samples with binding affinity labels. FANN is trained directly the 273 samples with binding affinity labels. The training for supervised models is done because there are only 566 samples with binding affinity labels in SAbDb and all of them are in the test set.
We compare CEBind with previous methods on the protein- small molecule dataset and antibody-antigen dataset. CE- Bind outperforms previous unsupervised models in terms of the Pearson correlation coefficient on both datasets, as shown in Table 1. For protein-smal molecule binding en- ergy prediction, supervised methods outperform unsuper- vised approaches, including CEBind, since they are trained with plenty of labeled data. Although the best physics- based approach, MM/GBSA, comparable with CEBind, it is computationally intensive.
5.4. Ablation Study
To evaluate the effectiveness of our conservation of energy and force matching, we conducted an ablation study as shown Table. 3. Both force matching and conservation of energy are effective, with which CEBind achieves an improvement 2.2% and 1.,% respectively. ESM plays an important role antibody-antiggn binding energy predic- tion. However, it is not essential for protein-smal molecules
For antibody-antigen binding energy prediction, unsuper- vised CEBind outperforms all previous approaches, includ- ing supervised models, since the labeled data for super- vised model training is searce. The supervised model

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
since atom-scale interaction is more important in protein- small molecule interaction. More visual results can be found in Appendix. C.4.
Table 4. Pearson correlation on protein-small molecule and mulibody-antigge datasets. "Only ligand" and "Both" indicate forces applying on the ligand and forces applying on the ligand and protein, respectively.
5.5. Discussion
Model
Protein-small molecule
Antibody- antigen
5.5.1. EFFICIENCY OF CEBIND
We mainly compare CEBind with the second-best unsuper- vised approach, DSMBind, in terms of training time and GPU memory consumption. All the comparisons are per- formed with the same settings. The average training time of each batch for DSMBind and CEBind is 0.0152 seconds and 0.0116 seconds, respectively. CEBindd is faster than DSMBind since the process of performing rigid dynamies is done before training in data processing, i.e., the first five steps in Algorithm 1. The average peak GPU memory uti- lization for CEBBndd and DSMBind is 4.89 GB and 6.43 GB, respectively. CEBind requires less GPU memory for train- ing than DSMbind, which sheds light on co-training with other models, like designing a complex with high binding affinity. The low memory requirement of CEBind is due to the fact that fewer operations require gradient calculation. More details can be found in the Appendix. C.2.
Only ligand Both
0.652++0.005 0.665+0.011
0.374
0.362-+0.015
comparable on the protein-small molecule dataset, while on the antibody-antigen dataset, CEBindatom performs worse than CEBind due to the absence of ESM. Although both GNN and ESM model the interaction inside ligands, ESM is critical for proteins because of its large-scale pre-training, while GNN is not essential for molecules since the interac- tion modeling is also performed in the energy-based model. Although the performance of CEBindaton decreases on the cuibbddy-nttiged dataset, it is still comparable with the best supervised model (rp = 0.350) and better than the best physics-based approach (rp = 0.323).
In the Cross Val, CEBindatom trained on the antibody- antigen dataset well generalized to the protein-small molecule dataset, with performance better than all previous unsupervised models. Although generalizing CEBind trained on the protein-small molecule dataset to the antibody--nntigen dataset is hard since the size of molecules and proteins differs, it is still comparable with the best physics- model. The size range of samples in the antibody-antigen training set encompasses almost all the sample sizes in the protein-small molecule test set. How- ever, the size ranges of the protein-small molecule training set and the salibody-antigge test set overlap only marginally. More details can be found in Appendix. C.3.
5.5.2. BIAS OF FORCE APPLICATION
CEBind applies the sampled forces exclusively to ligands, not to the target protein, which could introduce bias. How- ever, any other forces can be applied to the complex without altering our framework. We also explore the application of forces on both protein and ligand in a complex. The results of applying the forces to both protein and ligand and applying forces only to ligand are comparable on both the protein-ligand and antibody-antigen datasets as shown in Table. 4. On the protein-small molecule dataset, by apply- ing forces on both ligand and protein, the performance of CEBind increases slightly since the side chains are included in the rigid dynamics. On the Antibody-antigen dataset, the performance decreases slightly since too many atoms are included. For both datasets, the results are comparable since all relative positions of the proteins and ligands can be obtained by moving only the ligands if the side chains are not taken into account.
With the results that CEBindatom is generalizable in both datasets, our CEBind and AlphaFold3 (Abramson et al., 2024) come to the same conclusion that a universal ap- proach to all biomolecules. including both proteins and small molecules, is possible.
6. Conclusion
5.5.3. GAP BETWEEN PROTEIN-SMALL MOLECULE AND ANTIBODY-ANTIGEN
In this work, we propose an efficient training framework for unsupervised protein-ligand binding energy prediction via the conservation of energy, named CEBind. By sampling forces on each atom and applying them rigidly the ligand, CEBind follows physies laws, including rigid dynamics and the energy conservation law. Experiments demonstrate that CEBind is memory- and outperforms previous methods. Furthermore, we implement a general encoder for both antibodies and small molecules to explore the gap between them. The generalization of the general encoder
To explore the gap between small molecules and proteins as ligands in protein-ligand binding, we implement a gen- eral encoder for both protein and small molecule ligands, as shown in Fig. 3(b). We explore the gap through two ways: (1) Evaluate an atom-only model with the general encoder, CEBindatom- (2) Train CEBind on one dataset and evaluate it on the other dataset. The Pearson correlation results are shown in Table. 2. CEBind and CEBind are
8

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
indicates that potential universal approach exists for all biomolecules, including both proteins and small molecules, which is the same AlphaFold3.
Drugscore2018. Journal of chemical information and modeling, 59(1):509-52) 2018.
Eberhardt, J., Santos-Marting D., Tillack, A. F., and Forli, S. Autodock vina 1.2. O: New docking methods, expanded force field, and python bindings. Journal of chemical information and modeling, (8)33338,,,389,, 2021.
Impact Statement
The goal of this work to advance the field of Machine Learning and Computational Biology. While there are many potential societal consequences of our work, we believe that none of which must be specifically highlighted here.
Feng, S., Li, M., Jia, Y., Ma, W.-Y., and Lan, Y. Protein- ligand binding representation learning from fine-grained interactions. In The Twelfth International Conference on Learning Representations, 2024. URL https: openreview net
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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
A. Method Details
(3)
OH
OH
Translation and rotation of the whole ligand
First rotatable bond
Second rotatable bond
NH2
NHz
NH2
HH2
Figure Force performing process. (1) Calculate the rotation translation the whole ligand. (2) rotatable bond, we fix the substructure connected to the bond and calculate the new position with the sample forces according to the kinematic theory. (3) For second rotatable bond, we repeat the process (2).
A.1. Rigid Dynamics Derivation
Given a protein-ligand complex, C = (A, X), we randomly sample forces f E R$ each atom the ligand, Then we apply these forces to the ligand, following the rigid dynamics. The translation of each atom in the ligand consists of three parts: (I) the translation of the ligand caused by combined forces; (2) the ligand rotation, which causes the translation of all the atoms in the ligand; and (3) the rotation of bonds that can rotate.
(1) Ligand translation:
AX 1 1E S M
A12
(22)
The translation caused by the ligand rotation is obtained through rigid dynamics. First of all, we calculate the inertia matrix I E RR3x3 and torque T E R3 of the ligand as follows:
(2) Ligand rotation:
IN
x nll (x. - (x) - p))
(23)
N
x - 1)) fi:
(24)
1=1
where ja is the center of the ligand and calculated as follows:
a2 x
p
Σ
(25)
N
Then apply the torque the ligand and the acceleration is: with the acceleration, we have the angular displacement as:
= I T.
(26)
I TAt2.
(27)
With the rotation equation in Eq. Rotation, we have the atom coordinates as:
X = Rot(w, XL, 11).
(28)
Finally, we have the translation of the atoms caused by the rotation as:
AX, X X = Rot 2 I
TA', XI. - 11) X
(29)
where the operation Rot is described Eq. Rotation.

---

Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
(3) Side chain rotation. The derivation of translation caused by the side chain rotation is similar the ligand rotation as above.
A.2. Force Performing Process
The process of performing sampled forces illustrated in Fig. 4. First, calculate the translation and rotation of the whole ligand. Then for each rotatable bond, we fix the two substructures connected to the bond and calculate the rotation of each substructure rigid. Finally, we sum the position changes calculated for all the rotatable bonds and the whole ligand as the final position of a ligand.
A.3. Work of Outer Forces (the Energy Change).
Given the combined forces f. inner force f inner, and outer force fouter: f= f snner + f outer- We perform the f during (0, Ar) and f during (2 At, At). In a small range of AX, we model the inner force between the ligand and the protein to be proportional to their distance, i.e. J inner & (X; - on) (More accurate force field modeling can be adapted here without changing our framework). With a scaler k, the force is represented as { iinner = k(x, Xo). The work of the combined forces equal to zero since the displacement X during (0, 1 equals the displacement X2 during (1 At) as follows:
W combine fX (-fX2)
Inner and outer forces do work of the same magnitude and in opposite directions as follows:
f
X,
fotter dX = { Sx. f fdx
f inner dX +
[x.
Xo
fX1 + (-
0.
Therefore we have:
f iiner dX = =- =- outer dX
X.
Xo
i...
W inner W outer-
(30)
Therefore, to calculate the work of outer forces, we can calculate the work of inner forces.
W outer
W inner
Xo+Ax
f finner d.X
Xo
Xotax
k(X xo) dx
xo)
Xx
2AX²
The energy model is trained to predict the energy of 1 E(C). Therefore, the energy of complex C is (C). For energy conservation, have the work of outer forces equals the energy changes follows:
1 FAX²
(C) kEE W outer
i.e.
E
Therefore, the loss for energy conservation is:
L. = E, - 2 A' (
12

---

Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
A.4. Force score matching
The force score matching follows the score matching of score-based diffusion model (Dittrich et al., 2018). The object of score matching is
arg min 2 panl(x) (1))0()) V. log Pdata (ar [=]
where sx) is the prediction model .i.e. f Jinner in our model. V log Pitaa (x) is the derivatives of data distributions. The distribution innner our work is approximately a Gaussian distribution as I iiner N(O,o²), Therefore, we have:
f inner
((fin)r)
exp
(- 20²
22T
Taking the natural logarithm of pf inner) gives:
f inner
2
log (f
inner) log
2
202
f inner 20
2 log(2TT0)
Then take the derivative f inner of log inner) (nnee)))) and have:
i
f inner
log (nnee)
0f inner
r²
Therefore, we have the object:
Cj
inner
f inner
log pf inner 1ll
OkEE
A,
X)
E|I
+ Tiner
OXL
o²
Flg
A, X)
kAXi,
E|ll
+
L
|139
0XL
a²
A,
AX
+
|:3]
0x
σ²
Therefore, for the total loss of training, the scaler l can be ignored.
B. Training Details
B.I. Hardware
We train CEBind, Gauss DSMBind, and DSMBind for 10 epochs. All our experiments are conducted on a computing cluster with GPUs of NVIDIA GeForce RTX 4090 24GB and CPUs of AMD EPYC 7763 64-Core of 3.52GHz. All the inferences conducted on a single GPU of NVIDIA GeForce RTX 4090 24GB.
B.2. Hyper-parameter
We train the models with the same hyperparameters following DSMBind (Jin et al., 2024). The batch size, learning rate, and hidden vector size are 4, le-3, and 256, respectively. We use the pre-trained ESM of version esm2 .36 3B BB.URSOD for protein residue embedding. We use the SRU (Lei et al., 2017) DSMBind. We assign the duration of At as random number from 0 to 1.
protein-ligand interaction modeling model following
C. A Additional Results
C.1. Results with Different Random Seed
The Pearson correlation on the autibody-antigen and protein-sman small molecule datasets with different random seeds is shown in Table. 5, where r* denotes that no positive Pearson correlation available with this random seed. The results of DSMBind
13

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Table Pearson correlation -nttign and protein-small molecule dataset with different random seeds, Gauss DSM indicates the Gaussian DSMbind method. Mean and dey denote the average value and standard deviation, respectively.
antibody-antigen Gauss DSM
protein-smad molecule
seed
DSMBind
CEBind
DSMBind
Gauss DSM
CEBind
0
0.3551
0.2619
0.3405
0.5937
0.5747
0.6489
1
0.3273
0.3381
0.3478
0.6455
0,6248
0,6510
2
0.3873
0.3724
0.6328
0.6441
3
0.2943
0.3249
0.3360
0.6061
0.4877
4
0.3572
0.3431
0.3384
0.6200
0.4781
5
0.3595
0,3639
0.3365
0.3397
0.1667
0.6431
6
0.3736
0.3339
0.3423
0.6453
0.6376
0.6467
7
0.3190
0.3207
0,2347
0.6249
0,4780
0.6457
8
0,2897
0.3356
0.3556
0.6388
0,6358
0.6456
9
0.3433
0.3465
0.3249
0.6327
0.5630
0.6461
10
0.3402
0,3830
0.6422
0.3282
0.6494
11
0.3315
0.3389
0.3784
0.6470
0.5366
0.6629
12
0.3821
0,2402
0.6066
0.6492
0.5312
13
0.3515
0.3595
0.3784
0.4507
0.6417
14
0.3171
0.3474
0.6039
0.2564
0.6273
Top-5 mean
0.3655
0,3600
0.3735
0.6438
0.6361
0.6518
Top-5 dev
0.0105
0.0157
0.0096
0.0029
0.0079
0.0052
Top-10 10 mean
0.3500
0.3487
0.3584
0,6309
0.5820
0.6483
Top-10 dev
0.0190
0.0160
0.0168
0.0149
0.0609
0.0054
and Gauss DSMBind are produced through their official codes. CEBind much better than previous models terms of the Pearson correlation both antibody-antigen datasets and protein-smal molecule datasets. Besides, CEBind is more stable than DSMBind, as shown in Table 5: the standard deviation of DSMbind is smaller than DSMBind. Fig. 5 shows the robustness of DSMBind over previous models intuitively.
Boxplot of Pearson correlation
Boxplot of Pearson correlation
0.38
Pearson correlation
Pearson correlation
0.34
0.5
0.30
0.28
0.3
0.26
0.24
DSMBind
Gauss DSMBind Method
CEBind
DSMBind
Gauss DSMBind Method
CEBind
Figure Boxplot Pearson correlation on the antibody-antigen datasets.
Figure Boxplot Pearson correlation on the protein-small molecule dataset.
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---

Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Algorithm 2 Training procedure of CEBind (single data point)
Require: A protein-ligand complex C)A,
1: Sample a noise level a.
2: Sample forces for each atom in ligand f ~ ~~(00
3: Calculate the translation of each atom A with the forces via Eq. 10.
4: Calculate the work of outer forces W via Eq. 12.
5: Apply the noise to the original complex to get the perturbed complex Ö (A, X) via Eq. 11.
6: Predict the energy of both complexes Eg (A, X)) and Ea (A, X via energy model.
7: Minimize the energy loss Le and force matching loss L from Eq. 13 and Eq. 16.
Algorithm 3
Training procedure of DSMBind (single data point)
Require: A protein-ligand complex C)
1: Sample a noise level rotation vector, and translation vector for the whole rigid.
2: Apply the noise to original complex to get the perturbed complex.
3: Predict the energy of the perturbed complex.
4: Calculate the gradient of the energy with respect to the atom coordinate.
5: Calculate the rotation of the rigid with the gradient.
6: Minimize DSM objective.
Table 6. Performance comparison of different training datasets
Dataset
protein-small molecule
antibody
Separately
0.648 + 0.003
0.346 0.004
Cross Val
0.646 + 0.001
0.320 0.002
Union
0.646 + 0.004
0.352 = 0.007
Table Pearson correlation on protein-sman small molecule and antibody antigen datasets with pre- gre-training on protein-protein pair dataset.
Pre-train
Freeze
Protein-molecule(r, f)
Antibody-antigentr, 1)
0.646 + 0.002
0.361 + 0.013
0.646 + 0.002
0.375 + 0.014
0.652 0.005
0.374 + 0.009
C.2. Comparison between DSMBind and CEBind
The training procedures of CEBind and DSMBind are shown in Algorithm. 2 and Algorithm. 3, where the blue color denotes that the process does not require a gradient, while the red color denotes that the process requires a gradient. The process of calculating the dynamics of the ligand requires a gradient in DSMbind. While CEBind does not require a gradient for the rigid dynamic calculation process, the process can be done the data loader, which can not only cut down on GPU memory consumption but also accelcrate the training process.
C.3. Generalization
The distribution of the antibody-antigen training set and protein-small molecule test set shown in Fig. 7. The distribution of the antibody-antigen test set and protein-smal molecule training set is shown in Fig. 8. The size range of samples in the antibody-antigen training set encompasses almost all the sample sizes in the protein-small molecule test set. However, the size ranges of the protein-small molecule training set and the antibody-antigen test set overlap only marginally. This is the reason why the CEBindatom trained on the antibody-antige training set can generalize to the protein-smal molecule test set, while the CEBindatom trained on the protein-small small molecule training set cannot generalize to the antibody-antigen test set. When trained on the union of antibody-antig antigen dataset and protein-small molecule dataset, the results on antibody- antigen are improved compared with the training on each dataset separately, while the results on protein-sman molecule vary a little as shown in Table. 6. The variation in the results is due to the distribution of training data and test data as shown in Fig. 9.
15

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Artiiccty erogen
Protetvomal molecc
00
93
63-
100
125
200
239 230
750
Alom
Figure 7. The distribution of the number of atoms in the training antibody and test molecules data.
Figure 8. The distribution of the number of atoms in the training molecules and test antibody data.
Figure 9. The distribution of the number atoms in the union the protein-smal molecule and antibody-antigen training dataset and test data.
C.4. Visualization
The heatmap of epitope and paratope binding is shown in Fig. 10 where each pixel indicates the binding energy and the darker the higher. The Spearman correlation between experimental binding energy affinity and predicted binding energy of the Equibind dataset shown in Fig. 11. The heatmap of pocket and small molecule binding is shown in Fig, 12. The Spearman correlation between experimental binding energy affinity and predicted binding energy on the SAbDab dataset shown in Fig. 13.
Equibind test correlation: Spearman R 0.6629
Expermmental bind ing offinity (logw)
-5
Epitape: residues
-10
-55
-8
-30
-12
OFTROGYA NOCEOSTTOEELLLA LLPEO SYISO LLWWEEAOSSPSSSS
-45000 -40000 -35000 -30000 -25000 200000 -15000 10000
Paratope residues
Predicted binding energy
Figure 10. The heatmap of the epitope and paratope binding.
Figure 11. The Spearman correlation between experimental binding energy affinity and predicted binding energy of the Equibind dataset.
C.5. Pre-train CEBind with large-scale protein dataset
We try to pre-train CEBind with a large-scale protein dataset. The protein pre-training dataset contains 27,692 protein-protein pairs. The results are shown in Table 7, where freeze denotes that the parameters of the encoder are frozen in training, With pre-training, the Pearson correlation on protein-small molecule gets even worse and the results on antibody-antigen get slightly better. For the protein-small molecule dataset, the pre- trainedd model with frozen encoder performs worse than the model without pertaining for two reasons: (1) the model is pre-trained on the protein-protein pairs instead of the protein-small molecule. (2) the parameters of the encoder are frozen, which limits the learning ability of the model. With the pre-trained model as the initial parameters, the model without freezing the encoder parameters still performs worse, which indicates that the model with a bad initialization would also trapped the local optimal. For the antibody-anne anttien dataset, the pre-trained model performs better without the encoder parameters freezing. The better performance owes the
16

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
SAbDab test correlation: Spearman 0.3830
Experimental bindlng
anoreda maleculc
-00
-10
-100
-12
CAA.TT/ACCCCNEEEEERREEEEEEEEEEEE TENGESOAYY
-16000-1400-1200
Pocket residues
Predicted binding energy
Figure 12. The heatmap of the Pocket and Small molecule binding.
Figure 13. The Spearman correlation between experimental binding energy affinity and predicted binding energy on the SAbDab dataset.
knowledge obtained through the protein-protein pair pre-training.
17

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
Distribution of Affinity
Distribution of Affinity
400
50
40
Requency
30
200
20
100
10
Aftinity
Affinity
Figure 14. The distribution of affinity of original SAbDab Figure 15. The distribution of log affinity of SAbDab dataset dataset
Distribution Affinity
Distribution of Affinity
350
30
250
20
Requency
15
150
50
0.4
-10
Affinity
Affinity
Figure 16. The distribution of affinity original Equibind Figure The distribution log affinity of Equibind dataset dataset
18

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Physics Aware Neural Networks for Unsupervised Binding Energy Prediction
SAbDab test correlation: Spearman 0.3830
Embedeings
Experimental binding
-12
-16000-14000-222 14000-12000-011111 10000--1000--0000 0000 -6000 -4000 -0000 Predicted binding energy
E-SC Conponet
Figure 19. T-SNE visualization of embedding, where colors Figure 18. The distribution of affinity and energy on SabDab indicate binding affinity. shown, there appear to two clusters. The left one indicates high affinity and the right one low affinity.
dataset
19


Reviews:
[{"review_id": "cvjwJF113f", "reviewer": "Reviewer_fEra", "summary": "This paper proposes CEBind, an unsupervised method for predicting protein-ligand binding energy via the conservation of energy. Specifically, this method random samples forces on atoms to move the molecules and predict the energy of both unperturbed and perturbed complex. And the energy gap between two complexes should be the work of outer forces. Experiments are performed on unsuperivised benchmarks and show better performance.\n\n## update after rebuttal\nAfter reviewing the rebuttal and discussion, I still doubt the novelty and effectiveness about this work. The energy conversation loss is the main novelty compared to DSMBind and it brings marginal improvement against DSMBind. But given that this method is indeed SOTA, so I increase my score to 2.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your comments! All your concerns have been carefully addressed as below. The manuscript will be carefully revised accordingly. We sincerely hope our responses fully address your questions.\n\n> **W1**: The novelty of CEBind. Difference between our CEBind and DSMBind.\n\n**A1**: Thanks for your comments. The novelty of our CEBind is threefold: **(1)**\n*We effectively integrated the law of conservation of energy and rigid dynamics into our CEBind*, as demonstrated in Table 3, which the reviewer mentioned. **(2)** The score matching in our CEBind is *more efficient and follows the physics law*. **(a) More efficient:** Our CEBind requires less memory and takes less time to train, as we described in Section 5.5.1. **(b) Physics aware:** We match the derivative of the energy with respect to the coordinates (i.e., the force) with the sampled perturbation force. Instead of matching the sampled distance perturbation with the force in CEBind. As a result, our CEBind is more stable and efficient. **(3)** *Stable and better performance:* our CEBind consistently outperforms DSMBind through our energy conservation loss, as well as the more physically meaningful score matching loss, as shown in Table 5 in Appendix. **(4)** The detailed comparison between CEBind and DSMBind can be found in Appendix C.2.\n\n> **W2**: (1) The improvement of CEBind over DSMBind is too marginal (Table 1). (2) The Spearman correlation also needs to be reported in Table 1.\n\n**A2**: (1) Thanks for your concern. The advantages of our CEBind over DSMBind are twofold. **(a) Performance**: our CEBind outperforms DSMBind consistently as shown in Table 5. **(b) Efficiency**: Training CEBind requires less memory and is faster as we described in Section 5.5.1.\n(2) We also provide the spearman correlation as follows, which also demonstrate that CEBind outperforms previous works.\n\n| Model    | Protein-small molecule | Antibody-antigen |\n|----------|-----------------------|-------------|\n| DSMBind  |   0.640 $\\pm$ 0.003   |   0.358 $\\pm$ 0.011 |\n| **CEBind**   | 0.652 $\\pm$ 0.004   | 0.387 $\\pm$ 0.009  |\n\n> **W3:** This kind of denoising method can only differentiate energy difference for the same complex with different conformers, but it’s hard to capture the energy difference between different complexes.\n\n**A3**: Thanks for your comments. We respectfully disagree. Our CEBind can indeed be applied to capture the energy difference between different complexes. All the results in this paper are the correlation between the ground truth and the prediction of our model for different complexes. **The perturbation is only performed in the training**, while in inference, our model only needs the structure of the complexes to output pseudo-energy values for each complex without any perturbations or different conformations.", "claims_and_evidence": "Please refer to the weakness part", "methods_and_evaluation_criteria": "Please refer to the weakness part", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "Please refer to the weakness part", "relation_to_broader_scientific_literature": "The proposed method may be useful for unsupervised binding energy prediction as high quality labeled data is very limited in this area.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "**Strengths:**\n- This paper propose an unsupervised method via energy conservation law to predict the protein-ligand binding energy.\n\n**Weaknesses:**\n- I doubt the novelty of this method. The proposed method is quite similar to the DSMBind, which uses denoising score matching to perform unsupervised binding energy prediction. DSMBind uses SE(3) noise to perturb complex structures and use predicted energy gradient to calculate rotation and translation, whereas the proposed method just does the similar thing in the other way around, i.e. sample force first and calculate rotation and translation, and then predict energy of perturbed structures and calculate gradient force. We can also see this similarity from Table 3, where after removing energy conservation loss the performance is basically the same as DSMBind.\n- The improvement of this method upon DSMBind is too marginal (table 1), which is expected as the methodology is very similar to DSMBind. So the experimental results cannot convince me this method is more effective than DSMBind. Also, spearman correlation also needs to be reported in Table 1.\n- This kind of denoising method can only differentiate energy difference for the same complex with different conformers, but it’s hard to capture the energy difference between different complexes.", "other_comments_or_suggestions": "Please refer to the weakness part.", "questions_for_authors": "Please refer to the weakness part.", "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": "oUtBxhbs1v", "reviewer": "Reviewer_FmAK", "summary": "The paper proposes CEBind, an unsupervised deep learning model for predicting protein-ligand binding energy based on the conservation of energy principle. It aims to address the challenge of limited labeled data for binding energy prediction, particularly for complex biomolecules like antibodies. Instead of requiring labeled affinity data, CEBind leverages rigid body dynamics and energy conservation laws to estimate binding energy changes.\n\n## Update After Rebuttal\nAfter carefully reading the rebuttal and all the discussion, I now believe the manuscript demonstrates both novelty and effectiveness. In my view, the authors have addressed the concerns of all reviewers. Therefore, I recommend acceptance and have raised my score to 5.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments and kind support! All your concerns have been carefully addressed as below. The manuscript will be carefully revised accordingly. We sincerely hope our responses fully address your questions.\n\n> **W1:** Evaluating CEBind on real docking simulations would strengthen its practical applicability.\n\n**A1:** Thanks for your suggestion. We totally agree that the practical applicability of CEBind could be strengthened by evaluating on real docking simulations. But the evaluation of our CEBind in this paper follows the common protein-ligand binding prediction evaluation pipelines [1,2,3]\n\n[1] NeurIPS'24, Unsupervised protein-ligand binding energy prediction via neural Euler’s rotation equation.\n\n[2] ICLR'24, Proteinligand binding representation learning from fine-grained interactions.\n\n[3] JCIM'23, Planet: a multi-objective graph neural network model for protein–ligand binding affinity prediction.\n\n> **Suggestions:** Typo and caption correction.\n\n**A2:** Thanks for your kind suggestions. We will fix the typos and change the caption of Table 2 to cross-dataset.\n\n> **Questions:** How was the PDBBind dataset split? Was the data leakage considered in the evaluation set?\n\nThanks for your insightful concern. We removed overlapping instances with the validation, test, and training sets. The experiments are fair and the same for all models. We also conducted additional experiments of removing the training samples with a similarity of 40% with validation and test samples. The results are shown below. The average performance (0.513 VS 0.511) of CEBind is not affected by the similarity between the training and test datasets, which proves that information leakage did not occur.\n\n| split | protein-small molecule | antibody-antigen | Average |\n|----------|----------|----------|----------|\n| original | 0.652 $\\pm$ 0.005 | 0.374 $\\pm$ 0.009 | 0.513 |\n| remove similar | 0.645 $\\pm$ 0.003 | 0.377 $\\pm$ 0.012 | 0.511|", "claims_and_evidence": "The paper presents a well-motivated and innovative approach for unsupervised binding energy prediction. The claims made by the authors are well-supported, backed by solid theoretical foundations and comprehensive empirical results.", "methods_and_evaluation_criteria": "The method is sound and the evaluation metrics are well-chosen which align well with the problem of unsupervised protein-ligand binding energy prediction.", "theoretical_claims": "- All major theoretical claims are valid, correctly derived, and align with established physics principles. Energy conservation proof is correct. Rigid body dynamics derivations are well-grounded. Score matching loss is based on proper statistical modeling.", "experimental_designs_or_analyses": "- The experimental design is well-desined and statistically sound.\nThe datasets are well-chosen and representative of real-world binding problems.\nThe evaluation metrics are appropriate and unbiased.\nComparisons to baselines are fair and prove CEBind’s superiority.\nAblation studies confirm the model’s design choices.", "relation_to_broader_scientific_literature": "- This mthod builds upon three major research directions in computational biology and machine learning: unsupervised learning for protein-ligand interactions, physics-aware modeling, and neural energy-based models for binding prediction.", "essential_references_not_discussed": "- CEBind builds upon energy-based learning, particularly methods involving score matching and diffusion models:\n    - Song et al., \"Score-Based Generative Modeling through Stochastic Differential Equations,\" ICLR 2021. They introduced the score-based diffusion models and the force-matching loss in CEBind is a form of score-matching.", "other_strengths_and_weaknesses": "### Strengths\n- The paper is clear, well-structured, and provides all necessary details for reproducibility.\n- The combination of machine learning and physics makes CEBind more interpretable and generalizable, setting it apart from fully data-driven models.\n- This work has the potential to accelerate molecular docking simulations and enhance drug discovery pipelines, making it scientifically and practically significant.\n\n### Weakness\n- Evaluating CEBind on real docking simulations would strengthen its practical applicability.", "other_comments_or_suggestions": "- Equation (3) in Section 3.3: A missing subscript in  $W_{\\text{outer}}$ should be corrected for consistency.\n- Table 2 caption: “Cross-validation results for different datasets”. It should be clarifyed that these are cross-dataset generalization results, as some readers might confuse this with standard k-fold cross-validation.\n- Appendix A.2 notation consistency: The force notation $F_{\\text{inner}}$ and $F_{\\text{outer}}$ should be consistent across equations for clarity.", "questions_for_authors": "- How was the PDBBind dataset split? Was the data leakage considered in the evaluation set?", "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": "fW0dtzshT3", "reviewer": "Reviewer_2j23", "summary": "The paper builds on the paradigm of unsupervised learning of binding affinity introduced in DSMBind but introduces a new method for force-matching that doesn't require computing the gradient of the energy function and that operates on the level of per-atom forces. The energy learned from the force-matching objective is then used as a ranking function and compared to ground-truth experimental binding data on small-molecule and antibody datasets where some gains are observed over DSMBind.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your comments! All your concerns have been carefully addressed as below. The manuscript will be carefully revised accordingly. We sincerely hope our responses fully address your questions.\n\n> **W1:** There is a marginal empirical improvement over DSMBind.\n\n**A1:** Thanks for your concern. The advantages of our CEBind over DSMBind are twofold. **(a) Performance**: our CEBind outperforms DSMBind consistently as shown in Table 5. **(b) Efficiency**: Training CEBind requires less memory and is faster, as we described in Section 5.5.1. \n\nTo demonstrate the effectiveness of CEBind, we also provide more experimental results in **A1** and **A2** of response to reviewer 1arn and **A2** to reviewer fEra.\n\n> **W2:** How is CEBind physics-based other than using the same equations that are used in mechanics?\n\n**A2:** Thanks for your comment. In this work, \"physics aware\" refers to the law of conservation of energy and the rigid dynamics. \n- **(1) For the law of conservation of energy**, we have constructed a new training method using the conservation of energy. As shown in the ablation study in Table 3, this approach is effective.\n- **(2) For rigid dynamics**, we match the derivative of the energy with respect to the coordinates (i.e., the force) with the sampled perturbation force. Instead of matching the sampled distance perturbation with the force in CEBind. As a result, our CEBind is more stable and efficient as described in Section 5.1.1.\n\n> **W3:** This method can't directly discriminate non-binders because their structure is not defined in the same way it's defined for molecules that successfully co-crystalize. Lack of negative data in the evaluation (see the comment above).\n\n**A3:** Thanks for your concern. We respectfully disagree. Our method operates on the quaternary structure of protein-ligand complexes without relying on specific crystallization-derived assumptions. Importantly, the physical principles leveraged by our model, such as energy conservation and force response under rigid dynamics, remain valid regardless of binding success. In fact, for non-binders, the model can still compute an energy landscape, and the predicted energy shift under force perturbation often reflects weaker or unstable interactions. This gives us a continuous signal that allows non-binders to be distinguished from strong binders, even in the absence of traditional co-crystal structures.\n\nAs for the training data, our method follows the commonly used evaluation pipeline for unsupervised protein-ligand binding prediction, where performance is assessed based on the correlation between predicted binding energy and ground truth affinity. This pipeline has been widely accepted in the literature and allows for a fair and consistent evaluation of binding affinity prediction [1,2,3].\n\n[1] NeurIPS'24, Unsupervised protein-ligand binding energy prediction via neural euler’s rotation equation.\n\n[2] ICLR'24, Proteinligand binding representation learning from fine-grained interactions.\n\n[3] JCIM'23, Planet: a multi-objective graph neural network model for protein–ligand binding affinity prediction.\n\n> **W4:** The paper doesn't seem to reference the entire field of molecular dynamics which is a very much physics-based family of methods that is often also unsupervised and used to predict binding affinity.\n\n**A4:** Thank you for your comment. While molecular dynamics (MD) plays a crucial role in binding affinity prediction, our CEBind method differs from traditional MD.\n\n**During training**, CEBind leverages rigid dynamics and energy conservation principles to predict binding energies through perturbations. **For inference, CEBind requires only the structure of the protein-ligand complex without perturbing them**, thus avoiding the extensive computational and memory demands of full molecular dynamics simulations. Compared to MD’s long simulation times and complex force fields, CEBind is more efficient and does not require lengthy molecular simulations or large labeled datasets.\n\nAlthough we did not reference MD literature in our paper, we recognize its importance in binding affinity prediction and plan to include the relevant references in the revised version, including those you mentioned in your review.", "claims_and_evidence": "1. CEBind doesn't require differentiating through the entire energy network.\n\n2. There is a marginal empirical improvement over DSMBind. \n\n3. I don't quite see how CEBind is physics-based other than using the same equations that are used in mechanics. There are plenty of aspects of physics that are involved in the binding process that this paper doesn't even discuss such as quantum effects or even the simplest electrostatic interactions.", "methods_and_evaluation_criteria": "This paper inherits a major problem from Jin et al, 2024 in that the method can't directly discriminate non-binders because their structure is not defined in the same way it's defined for molecules that successfully co-crystalize. I view this as a major limitation of the paper both from the methods and evaluation points of view.", "theoretical_claims": "There are no theoretical claims in the paper.", "experimental_designs_or_analyses": "1. Lack of negative data in the evaluation (see the comment above).\n2. It would be valuable from the practical point of view to measure performance of the method (or maybe consider training) on the task of predicting _relative_ binding affinity [1]\n\n\n[1] Protein-Ligand Binding Free Energy Calculations with FEP. Wang et al, 2019.", "relation_to_broader_scientific_literature": "The paper continues the line of research deep learning based unsupervised binding affinity prediction initiated by DSMBind (Jin et al, 2024).", "essential_references_not_discussed": "The paper doesn't seem to reference the entire field of molecular dynamics which is a very much physics-based family of methods that is often also unsupervised and used to predict binding affinity.", "other_strengths_and_weaknesses": "No.", "other_comments_or_suggestions": "No.", "questions_for_authors": "No questions.", "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": "ZO6QXv1mGo", "reviewer": "Reviewer_1ran", "summary": "This paper proposes an unsupervised learning approach, called CEBind, to binding energy prediction. CEBind includes rigid dynamics and the training loss function is motivated by energy conservation loss. Besides, it outperforms previous works.\n## update after rebuttal\nThe authors have provided satisfactory responses to my queries. I regard this as a high-quality manuscript.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your constructive comments and kind support! All your concerns have been carefully addressed as below. The manuscript will be carefully revised accordingly. We sincerely hope our responses fully address your questions.\n\n> **W1:** The similarity between training set and test set should be considered and explained, which may lead to data leakage.\n\n**A1:** Thanks for your insightful concern. We removed overlapping instances with the validation, test, and training sets. The experiments are fair and the same for all models. We also conducted additional experiments of removing the training samples with a similarity of 40% with validation and test samples. The results are shown below. The average performance (0.513 VS 0.511) of CEBind is not affected by the similarity between the training and test datasets, which proves that information leakage did not occur.\n\n\n| split | protein-small molecule | antibody-antigen | Average |\n|----------|----------|----------|----------|\n| original\t| 0.652 $\\pm$ 0.005\t| 0.374 $\\pm$ 0.009\t| 0.513 |\n| remove similar |\t0.645 $\\pm$ 0.003 |\t0.377 $\\pm$ 0.012\t| 0.511|\n\n> **W2:** The time $\\Delta t$ is randomly selected. How the selection of  $\\Delta t$ impacts the performance of CEBind should be explained.\n\n**A2:** $\\Delta t$ is randomly selected from a uniform distribution $U(0.1,1)$. For your concern, we also conducted experiments to explore the impact of $\\Delta t$, and the results shown below demonstrate that the randomly selected $\\Delta t$ makes CEBind better.\n\n| $\\Delta t$ |\tprotein-small molecule | antibody-antigen|\n|----------|----------|----------|\n| 0.1 | 0.649 $\\pm$ 0.006 | 0.362 $\\pm$ 0.019|\n0.3 | 0.649 $\\pm$ 0.002 | 0.352 $\\pm$ 0.003|\n0.5 | 0.647 $\\pm$ 0.001 | 0.355 $\\pm$ 0.007|\n0.7 | 0.650 $\\pm$ 0.004 | 0.359 $\\pm$ 0.007|\n0.9 | 0.650 $\\pm$ 0.001 | 0.365 $\\pm$ 0.017|\nRandom | 0.652 $\\pm$ 0.005 | 0.374 $\\pm$ 0.009|", "claims_and_evidence": "The claims in the paper are well-supported by theoretical justifications, empirical results, and comparisons with prior methods.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria are appropriate for the problem.", "theoretical_claims": "The theoretical claims in the paper appear well-founded and mathematically consistent, with no major errors identified.", "experimental_designs_or_analyses": "The experimental design and analysis in the paper are well-structured and valid for the problem at hand.  The paper effectively justifies its claims with rigorous testing and fair comparisons.", "relation_to_broader_scientific_literature": "The paper advances prior work in unsupervised protein-ligand binding energy prediction by incorporating physics-based modeling, specifically the conservation of energy, and rigid dynamics.\n- Unsupervised Protein-Ligand Binding Prediction: CEBind improves DSMBind by using rigid dynamics and energy conservation, making it more memory-efficient and physically interpretable.\n- Physics-Based Binding Affinity Prediction: CEBind introduces a new energy-based framework that explicitly follows the conservation of energy, making it a hybrid between physics-based and data-driven methods.\n- Generalization to Biomolecules: CEBind's general encoder approach aligns with AlphaFold3's vision of a universal model for biomolecules.", "essential_references_not_discussed": "The paper cites most of the essential related works, but a few additional references could further strengthen its context and comparison to prior work.\n-  Baek et al., Accurate Prediction of Protein Structures and Interactions Using a Three-Track Neural Network (Science, 2021) (RoseTTAFold). This work is an early step toward multi-scale molecular interaction modeling, relevant to the generalization claims in CEBind. Citing this could position CEBind in the broader trend toward universal molecular modeling.\n-  Hsu et al., ESM-2: Evolutionary Scale Modeling of Protein Structures (2022). The ESM models capture protein sequence and structure relationships, which may improve antibody-antigen binding prediction. If CEBind's ligand encoder were combined with ESM-2 representations, it might further enhance performance.", "other_strengths_and_weaknesses": "This paper is strong in originality, significance, and clarity, with only minor areas for improvement.\n### Strengths\n- Well-written and structured, with clear explanations of theoretical foundations, methodology, and experiments.\n- The physics-aware approach (CEBind) is innovative, explicitly incorporating conservation of energy and rigid dynamics into an unsupervised binding energy prediction framework.\n- This paper bridges the gap between physics-based methods and data-driven unsupervised learning.\n- Efficient and scalable, requiring less GPU memory, making it suitable for large-scale molecular modeling tasks.\n\n### Weakness\n- The similarity between training set and test set should be considered and explained, which may lead to data leakage.\n- The time $\\Delta t$ is randomly selected. How the selection of $\\Delta t$ impacts the performance of CEBind should be explained.", "other_comments_or_suggestions": "- Did the dataset splitting take into account protein sequence similarity during the experiment? If not, there may be an information leakage problem.\n- How does the selection of $\\Delta t$ impact performance?", "questions_for_authors": "Predicting protein-ligand affinity is crucial for drug design. The proposed method is interesting and efficient. My only concern is the data splitting strategy. If the training and test datasets are highly similar, there is a potential risk of data leakage. Clarification on how the data was partitioned would strengthen the validity of the reported results.", "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:

Meta-review: This paper introduces CEBind, a novel approach for unsupervised prediction of protein-ligand binding energy using energy conservation principles. CEBind incorporates rigid dynamics and energy conservation to estimate binding energy changes without labeled data, which directly addresses the lack of high-quality labeled datasets in this domain. Strengths include its theoretical basis and empirical results demonstrating improved performance over prior methods, particularly DSMBind. However, one reviewer expressed concerns about the paper's limited novelty and questioned whether it merely extended DSMBind. The authors' rebuttal clarified substantial differences, such as a more physics-aware score matching mechanism and improved computational efficiency.

Scores were really polarised on this one - 2, 5, 1, 5. Despite reservations about the magnitude of improvement over DSMBind (the main sticking point), all reviewers acknowledged the paper's potential. Post-rebuttal, despite the discussion there wasn't a material shift in scores. Another reviewer was concerned about the practical applicability for distinguishing non-binders, but the authors provided compelling explanations for the method's utility in refining candidates in molecular design processes. I followed up on this myself, and the authors confirmed that the method was best suited to lead optimisation rather than discovering de novo binders. This is not a problem in itself - it just needed to be made clearer. The auithors also cleared up another query I had about log scaling of outputs. To this AC's eyes, the contributions of this work appear solid, with an innovative unsupervised learning methodology offering enhanced computational efficiency and accuracy.

Despite the split in scores, I recommend accepting this paper, as I think there is relevance to the broader scientific community interested in drug discovery and molecular interaction prediction.
Decision: Accept (poster)