Regularized Molecular Conformation Fields
Authors: Lihao Wang
Yi Zhou
Yiqun Wang
Xiaoqing Zheng
Xuanjing Huang
Hao Zhou
Venue: NeurIPS 2022 Accept (2022)
Keywords: random fields
conformation generation
molecular fragmentation

Abstract: Predicting energetically favorable 3-dimensional conformations of organic molecules from
molecular graph plays a fundamental role in computer-aided drug discovery research.
However, effectively exploring the high-dimensional conformation space to identify (meta) stable conformers is anything but trivial.
In this work, we introduce RMCF, a novel framework to 
generate a diverse set of low-energy molecular conformations through sampling
from a regularized molecular conformation field.
We develop a data-driven molecular segmentation algorithm to automatically partition each molecule into several structural building blocks to reduce the modeling degrees of freedom.
Then, we employ a Markov Random Field to learn the joint probability distribution of fragment configurations and inter-fragment dihedral angles, 
which enables us to sample from different low-energy regions of a conformation space.
Our model constantly outperforms state-of-the-art models for the conformation generation task on the GEOM-Drugs dataset.
We attribute the success of RMCF to modeling in a regularized feature space and learning a global fragment configuration distribution for effective sampling.
The proposed method could be generalized to deal with larger biomolecular systems.

Regularized Molecular Conformation Fields
Lihao Wang', Yi Zhou?, Yiqun Wang", Xiaoging Zheng', Xuanjing Huang' 3 Hao Zhou'
Fudan University, "ByteDance Al Lab
Shanghai Collaborative Innovation Center Intelligent Visual Computing
Institute for Al Industry Research (AIR), Tsinghua University
{wanglh19, zhengxq, xjhuang/@fudan, edu. cn
{zhouyi. naive, Sagun.wang@bbytedann com, zhouhao@air tsinghua. beiinghu.eedu.com
Abstract
Predicting energetically favorable 3-dimensional conformations of organic molecules from molecular graph plays a fundamental role in computer-aiie drug discovery research. However, effectively exploring the high-dimensional conformation space to identify (meta)stable conformers is anything but trivial. this work, introduce RMCF, novel framework to generate diverse of low- energy molecular conformations through sampling from a regularized molecular conformation field. We develop data-driven molecular segmentation algorithm to automatically partition each molecule into several structural building blocks to reduce the modeling degrees of freedom. Then, employ Markov Random Field to learn the joint probability distribution of fragment configurations and inter- fragment dihedral angles, which enables us to sample from different low-energy regions of a conformation space. Our model constantly outperforms state-of-the-an models for the conformation generation task on the GEOM DDMg--rug dataset. We tribute the success of RMCF to modeling in regularized feature space and learning global fragment configuration distribution for effective sampling. The proposed method could be generalized to deal with larger biomolecular systems
Introduction
The spatial arrangement of atoms within a molecule, also known as molecular conformation, de- termines the molecular physico-chemical properties, which plays an essential role downstream computer-aiie aided drug discovery tasks. However, the nigh-dimensiones conformation space spanned by all atomic degrees of freedom (DoF) makes a great challenge identify the local minima of the associated potential energy surface (PES) of molecule. Recently, many machine learning (ML) models have achieved remarkable success in molecular conformation generation tasks (Mansimov
al.] 2019 Simm and Hernandez-Lobatt 2020 Xu et al. 202 Ganea et al.) 202 Xu et al., 2022]. exhibiting orders magnitude faster conformation prediction speed than traditional computational simulation approaches. This makes ML generative models powerful tool for high-throughput screening in drug discovery, especially large molecules.
To fully exploit the power of ML in molecular conformation generation, we need to tackle a few key challenges. First, molecule exhibits invariance under SE(3) transformations (i.e., translation and rotation) in 3-dimensional (3D) Euclidean space. In other words, each molecular conformation is uniquely defined up rigid motion. Hence a single molecule can adopt an infinite of possible poses. Second, molecules exhibit a variety of dynamics under ambient conditions (e.g., bond rotation, vibration, etc.), leading
The code is available at possibly complex PES landscape of high dimension. This makes
"This work was done when Lihao Wang was research intern at Bytedance AT Lah.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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it particularly challenging for ML models to identify local minima within this space to generate energetically favorable conformations effectively.
Some recently developed ML models tried to solve these problems from different perspectives. For instance, GraphDG (Simm and Hernandez-Looaan 2020] and CGCF [Xu et al., 2021a used distance geometry as invariant features for molecular representation learning. The drawbacks of such representations are that these invariant variables are potentially redundant and exhibit mutual dependency, which may cause numerical instability during training. One can also design equivariant networks to hard-code the invariance/euiivariacee condition into the model [Satorras et al. 2021 Xu et al. 2022]. These models cireumvent using intermediate invariant features, but instead directly learn
from the spatial arrangement of atoms. However, some studies [Cohen et al. 2018 Li et al. 2021]
have suggested that the specialized equivariant layers may cause some loss in the expressiveness
of the neural network. On the other hand, GeoMol [Ganea et al. 202 predicts a local structure for each atom, followed by assembling these atom-based building blocks to form the molecular conformation. These models have difficulty dealing with cyclic graphs (e.g., benzyl group) and may lead to unreasonable conformation predictions.
In the meantime, we realize that the high-dimensional PES could be described in a reduced basis by consid- ering only a few sigmificant DoF that contribute to the low-energy conformations. Specifically, consider the well-known example of the potential energy diagram of ethane molecule as shown in FFiuure I] While the
Hat
Ha
Fotental enary
H
molecule exhibits a total of 18 DoF, we could capture the most significant dynamics using single variable, i... the rotation w.r.t. the carbon-carbon bond, quan- tified by the H-C-C-H dihedral angle. Other DoF in the ethane molecule, e.g., the stretching and bending of other bonds, do not have significant impact on the potential energy landscape, such that they can be treated small perturbations to the molecular con- formation within a potential well. In other words, to serve the purpose of effectively sampling local min- ima for low-energy conformations, we could reduce the dimension of the conformation space by tracking only few significant DoF, as long as they adequately shape the corresponding energy landscape (e.g., the "w" shape in Figure T). It then follows naturally to generate new conformers by sampling from the low-dimensional conformation space, which is also physically meaningful. In fact, a similar approach has been widely adopted by the molecular docking community, i.e., incremental construction [Meng et al. 201T). where a ligand molecule is partitioned into multiple fragments connected by rotatable covalent bonds. One may then gradually add ligand fragments with appropriate orientations to fit the ligand into the receptor pocket.
Dihedral
Figure 1: Schematic of the potential energy diagram of an ethane molecule. The upper panel shows the Newman projection of three degenerate eclipsed conformations, the lower panel shows two energetically-favoot favorrble stag- gered conformations. The H-C-C-H dihedral angle alone is sufficient to describe the poten- tial energy change.
This paper presents Regularized Molecular Conformation Fields (RMCF), a novel framework for 3D molecular conformation generation. The novelty of this work lies in (1) developing data- driven molecular segmentation algorithm to partition each molecule into fragments with low internal flexibility, which serves as a regularization term for our framework, and (2) employing Markov Random Field (MRF [Murphy 2012 Cotta et al. 2020)) to capture the joint probability distribution of fragment configurations and interfrragment dihedral angles. Our work is partly inspired by the Ising model [Cipra] 1987) for simulating quantum spin systems, where we make an analogy between the spin state (i.c., spin up or down) and the configuration of each molecular fragment (c.g., chair or boat conformation of cyclohexane). The introduction of molecular fragments effectively reduces the dimension of the conformation space, where we only keep the most significant components (i.e., fragment configuration and inter-fragment dihedral angle) for conformation generation. This molecular segmentation serves as a regularization term for our framework, which is data-driven and has the effect of reducing feature dimensions. Then, we employ MRF as generalized setting of the Ising model (with more spin states and more coupling terms) to capture the relationship between adjacent fragments and model the uncertainty of conformations. By estimating the parameters of the MRF, we can obtain a potential energy surface associated with different conformations. Therefore,
2

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we can later sample from the low-energy region of PES, which is implicitly learned by the MRF, to obtain energetically favorable molecular conformations.
We demonstrate the effectiveness of RMCF using the GEOM-QM9 and GEOM-Drugs dataset Axelnod and Gomeez-Bombrrelii 2022]. where results show that RMCF significantly outperforms state-of- the-art ML models. Specifically, our model can generate a diverse set of low-energy conformations located at distinct local minima of the underlying PES. We attribute the success of our model to the automatic construction of Iow-dimensional conformation space and learning a joint probability distribution using MRF to achieve effective sampling of new conformations. We argue that capturing the governing dynamics of molecular systems could significantly improve the performance of learning models for generative purposes.
2 Related Work
Recently, various machine learning models have been proposed for molecular conformation genera- tion. CVGAE [Mansimov et al., 2019| first used a variational autoencoder (VAE) model generate conformation with atomic coordinates. The model suffers from multi-modality problems due to invariance under SE(3) transformations. Some studies tried to address this problem using two main types of approaches. The first type of approach use intermediate invariant features, such as atomic distance, encode the system, then leverage geometric algorithms to solve a set of atomic coordinates that match the invariant quantities. For example, GraphDG (Simm and Hernandez-
2020]
and CGCF [Xu et al. 2021a] proposed to predict the distance matrix by VAE and Flow, respectively, and solve the geometry through the Distance Geometry (DG) method (Liberti et al.. 2014). However, these invariant features are potentially redundant and could exhibit mutual dependency, which makes these methods numerically unstable and could predict unreasonable conformations. Some studies the
suggest that inconsistencies between training and test are responsible for the poor performance models [Shi et al.) 202T Xu et al. 2021b]. ConfGF [Shi et al. 202T passed the gradient of loss to coordinates and ConfVAL [Xu et al. 2021b used bilevel optimization to alleviate the incon- sistencies. GeoMol [Ganea al. 202 defined another set of invariants, which are bond lengths, bond angles, and dihedral angles. Zhu et al. [2022] proposed a model invariant to roto-translation of coordinates of conformations and permutation of symmetric atoms in molecules. Geomol generated multiple conformations by adding noise to the input, which did not learn a global energy function from which sample the low-energy conformations.
Another type of approach applies equivariant networks or kernels, such that after the input is rotated and translated, the output will be transformed accordingly. [Satorras et al. [202T] proposed an
equivariant normalizing flow, B(n)-flow, and Xu et al. [022 proposed GefDiff, which is diffusion
model (Song and Ermon 2019] with equivariant Markov kernels. Guan et al. [[021] proposed
variant of SEquaaqquvvrriant neural networks to learn the gradient fields of an implicit conformational
energy landscape. Equivariant methods circumvent intermediate invariants and model the coordinates directly, but require the design of specialized equivariant layers, which some studics have suggested would lose the expressive power of the network [Li et al. 2021
In addtion, some software for conformation generation are widely used in biochemical research. For instance, RDKit (Riniker and Landrum 2015 is popular open-source software which generates conformations using ETKDG distance geometry, and OMEGA is a commercial software which assembles the fragments with knowledge-based rules to generate conformations.
3 RMCF: Generating in Regularized Conformation Space
In this section, we present the regularized molecular conformation field (RMCF) in detail. Intuitively, RMCF partitions the original molecular graph into multiple molecular fragments, where adjacent fragments in the graph are connected with associated dihedral angles. In such a case, the molec- ular conformation generation problem can be decomposed into a) generating the conformation of molecular fragments, and b) determining the specific dihedral angles between connected fragments. The RMCF learns the potential energy landscape of molecular conformations by modeling a joint probability distribution of fragment conformation and dihedral angles, from which we can perform inference to draw diverse samples.
3

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3.1 Overview of RMCF
partition
inference
assemble
Molecule
RMCF
Configuration
Conformation
The workflow of RMCF. Starting from 2D molecular graph, we partition the molecule into fragments with least intra-fragment DoF. Blue and red circles denote fragment and dihedral angle configurations. respectively, while the black squares denotes the interaction between neighboring configurations. We then use MRF to model the joint probability distribution fragment and dihedral configurations. The last step to assemble the predicted molecular conformation according to the predicted dihedral angles and fragment conformations.
Figure 2:
We start with an overview of the our proposed RMCF model. Generally, as shown in Figure 2 given molecular graph G, to obtain a generated 3D conformation C, RMCF mainly takes three steps:
G 3
1. G G (Section The main idea of our proposed model is to build a conformation distribution within regularized molecular conformation space, which is achieved by partitioning the molecular graph G into multiple molecular fragments F connected by dihedral angles D. Notably, the graph partition should satisfy the least intra-fragment DoF principle (introduced in Section 3.2) which is crucial for reducing the dimension of the molecular conformation space.
The resulting graph after partitioning (including and D) could be jointly modeled with Markov random field (also called undirected graphical model). Since it is regularized with reduced total DoF, name model Regularized Molecular Conformation Fields.
Formally, regularized conformation field = (F, D. F.) is an undirected graph formed by collection of fragment vertices F = (J1, (NN. .dNn). ). and a collection of edges = (e1, cg. ... ,Nng) fragments and dihedral angles in the graph), where NF, ND. NE are the size of F, D, E, respectively [Wainwright et al. 2008). Each edge consists of a pair vertices f E and D. We associate with each vertex E F random variable X; taking values from set of fragment conformations C (see Section
collection of dihedral vertices D = F D (edges between
d2. .d....dd.......
3.2
each vertex d D random variable X,, representing the
value dihedral angles.
- (Section 3.3 and 3.4 Given G, the conformation generation problem turns into generating the conformation molecular fragments and determining the dihedral angles between them, namely inferring a configuration X' from g. A configuration
= (Xy Xa) of the RMCF contains set of fragment conformation X2 {= l 1, .Nr) and set of dihedral angle values X₂ {Ia, i 1, 2. ... .D ). which could be assembled into the 3D conformation of a molecule. X' could be obtained by maximizing the probability modeled by RMCF:
P(zs,"" 3 Idn. Z
II (ar)) JEP
II (Ta.) II
d.ED
(UAA)ER
where Z is the normalizing factor, v is the compatibility functions, which will be further described in Section 3
X X C (Section Given the configuration X, we assemble the predicted fragment confor- mation and dihedral angle values into the corresponding molecular conformation C.

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3.2 Graph Construction with Least DoF Principle
In this section, we elaborate more about the molecular graph partition with least DoF principle used in our proposed RMCF. This is crucial step to our model since it builds the foundation of the following probabilistic modeling. The entire graph construction process has three main components, namely a) constructing the fragment library with BRICS, b) obtaining fragment conformation vocabulary with clustering, and c) yielding final fragment partition by minimizing the intra-fragement DoF.
Specifically, We use BRICS, a decomposition method based on molecular functional groups, to pre-cut all molecular graphs and their 3D conformations and collect each fragment conformation into set C A sufficient number (i.e., 1,000) of fragment conformations are sampled from Cy for the following clustering steps.
Then, we use the K-Medoids clustering algorithm and use the sout-eean-squre deviation (RMSD) of two fragment conformations as the distance metric between two fragments, which takes rotation and translation invariance into consideration. We adjust the number of cluster centers k from to 10 and use the Silhouette Coefficient [Rousseeuw 1987 to measure the goodness of these clusters and choose the optimal seper-parameter k with clusters having the highest Silhouette Coefficient value. Cluster centroid elements are collected into conformation vocabulary Cy.
Coefficien(Rousseeeuvy
Although BRICS presents solution to partition the molecular graph, it cannot guarantee small DoF within the fragments. Therefore, we define a metric to quantify the Intra-fragmen DoF and design an algorithm to re- re-fragment the molecules to minimize this quantity. We define the fragment DoF as the conformation variance Var(C,) which the sum the squares of root-mean-squan deviation(RMSD) between fragment conformations Cj and their cluster centers C,, and the total DoF of partitioned molecule as the mean of its fragment DoF. The best partition strategy P* should have the least DoF F(P):
F(P) = PP|
Var(C,)
RMSD(C,,C7)
(2)
P
JEP
Cj
p arg min F(P)
We could employ graph dynamic programming to search for the optimal solution and the detailed algorithm is in Appendix B
After partition, we can get the fragment set F f2. f2..... .NN and insert the dihedral vertices between connected fragments to establish the RMCF G on the molecular graph G as shown in the Figure
3.3 Approximated Training of RMCF
Training Objective Training in RMCF is intractable, especially when it contains loops [Murphy
2012). It is difficult to estimate the normalization factor Z in the objective (Eq. . which means have to marginalize all possible conformations. We approximate the likelihood through piece-wise training, which has been proven efficient in training graphical models (Sutton and McCallum 2009
2012
Lin et al., 2016 Qu et al.
2022]. The key idea is to train each node and edge independently.
log P(Tri Iing Tdi. .dda
log px(x)))
(xx,) Σ log
log
( xTa))
(4)
21
Za
Ze
J.FF
(F.d))E
Cc.(f)
(d))
where En and Le denote the node-wise and edge-wise negative log-likelihood. Especially, L. (f. measures how probable a fragment and dihedral d appear the same time.
More concretely, for a pair of nodes in the field e = (f, d) E G. let Ifl and |g| denote the cardinality of the state space, |h| denotes the hidden dimension, yf and ya the ground truth of fragment conformation f and dihedral angle value d. which will introduced calculation in the next paragraph. We calculate the representations from the graph neural networks: Let h2 emb(f) and h2 emb(d)
'On the base of BRICS, we further cut the links of rings and side chains to prevent the combinatorial explosion
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denote the fragment embedding and the dihedral embedding (which is a "dummy" embedding) as the network input, we compute the contextualized fragment representations h E+1 and dihedral representations h by:
h -9 (1/,, ( (eNN, sex,)
h (hhs [1 seXs)
where denotes the layer index, y denotes the massage-passing function of GNN, Ny and NV denotes the neighborhoods of fragment and dihedral respectively. The node-wise negative log-likelihood is defined as:
(f) = log softmax (out(h;)) [usl, (d) log softmax (out(ha)) [va]
(6)
where out(-) [b] > R is an operator parameterized by linear layer. The edge-wise negative log-likelilne ogglikelihood is defined as:
de log softanax (E,W.E.) [uy. va]
where E, E R and E, || are the representations of set of node states, We p/ x is matrix carrying the global information between two nodes, parameterized by neural network W. = ML, ha) [Sun et al. 2019
Obtaining ground truth The oracle configuration y (yy,. V') contains set of ground truth fragment conformations indexes y/8 {y,, 1,2, 1.2, ,NF] and set of dihedral angle indexes Va= = (3, l 1, ,ND) denote the choices of fragments conformations and dihedral angles, respectively. For given fragment f. we pick the configuration s whose corresponding fragment conformation C, in vocabulary Cy. has the minimal RMSD to the actual fragment conformation C, as the ground truth ys.
y arg arg min RMSD(C, C;
(8)
The dihedral angle ye defined as the angle between half-planes of two connected 3D fragments. Since the dihedral angle depends on the set of the atoms involved the calculation, for the same type of fragments, we should keep the same set of atoms to calculate.
P4
opel p (- m
For system where two fragments contain four consecutively- bonded atoms, two half-plane intersect on a rotatable bond. The angle between them the dihedral angle. If the connected points are sequentially numbered and located at positions P1, P2, P3. P4 and the corresponding bond vectors are defined as i1 = P2 P1 = P3 Then we have:
P2
Figure 3: Dihedral angle
P2. = P4 P3-
(u u2 us) atan2 (uua)
(u2 us). (n) u2) (u2 us))
In practice, we quantize the dihedral angles yp into a number of evenly divided bins which has interval length L to discretize the continuous angle values
180
=
x
(10)
3.4 Inference & Sampling
In this section, we give brief overview of how we perform inference and sampling in RMCE.
Inference If a RMCF is acyclic (linear- or tree-structured). the max-a-posterior decoding can be done via dynamic programming (Forney 1973]. Cases are a bit complicated when loops occur in the RMCF. We adopt a more generalized algorithm, namely loopy belief propagation (L.PP) (Murphy aL. 2013]. overcome such obstacles. We refer the reader textbooks for more details about the LBP algorithm (Murphy 2012). It is worth noting that LBP is used only during inference time. We do not perform any inference during training due the issue of training stability (Qu et al. 2022
Sampling Markov Chain Monte Carlo is popular sampling from graphical models Andrieu et al.
2003 We employ simple variant of it. i.c., Gibbs sampling [George and McCulloch 1993). to draw

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samples from RMCF. Specifically, we resample each fragment or dihedral individually, keeping all other nodes fixed in each iteration MMrpphy 2012 The detailed algorithm can be found in Appendix
Clustering Once the Markov chain achieves detailed balance, randomly draw 10, 000 samples from it. However, these samples can be governed by a few conformations, which means most of them are similar. We measure the distance between two configurations X²(1) and x²) by:
d(x ()) || 11
1 H(X)') xx)))
(11)
where || 1112 is the Euclidean norm measuring the difference between dihedrals, is the Hamming distance (Hamming 1950] counting the number of different fragment configurations. We further run
K-means clustering MacQueen et al. 1967] based on the pair-wise distance to partition all samples
into a fixed number of clusters and randomly sample element from each cluster.
3.5 Assembling
Once we have predicted the dihedral angle žd between two fragments, we need to compute the transformation matrix T align the two fragments in arbitrary position and make their dihedral angle equal to the predicted angle and repeat these steps until the whole conformation is assembled.
1/2
R
11g
V2
Given two fragments with six points, located at pI. Ps. Ps. the bond vectors are Uj = P2 P1. Ps P2.V1 Рs Ps. V2 = P6 Ps. and are the same bond after assembling, so we have u2 [v1l-
P4.
1/
8
R
First we need to rotate the second fragment around some axis so that U2 and V1 will be parallel. This axis is the unit normal vector n towards the plane of u2 and V1, and the rotate angle is equal the cross angle 0.
Figure 4: Transformations for As- sembling
U1
COS 0
(12)
V1
U2 X Vi
By using the Rodrigues' rotation formula, we can obtain the rotation matrix with unit normal vector and rotation angle A. Next, rotate the second fragment around u2 to match with the target dihedral angle. Finally we align the anchor points by calculating the final transformation matrix as composite matrix of two rotations and one translation.
f2
- -T
Bly
R(n, 0) = 7 cos + (1 cos 0)
(nar, ny, n.) + sin
n2
0
77g
(13)
flx
0
LLd
R' R'= R(u2,
puu) u2. ((nn))vz)), t= (p) R R'Rp,))
(14)
180
R'R.
T
(15)
0"
1
4 Experiment
We now demonstrate the effectiveness RMCF on the conformation generation task for drug-like molecules.
4.1 Dataset and Baseline
Following previous work on conformation generation, we benchmark our model performance using the GEOM-QMP and GEOM-Druge dataset, which contains small and mid-sized organic molecules with high quality conformations. Considering the small size of molecules in QM9, we put the model performance Appendix [ and mainly focus on discussing GEOM-Drugs results hereafter. We use the same test set as that in GeoDiff, where the remaining molecules are used for training and validation with a 9:1 ratio. The final training/validatiionness set contains 271,539/30 271,539/30,171/11055 molecules, respectively.
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Our molecular segmentation algorithm eventually returned a vocabulary of 9,081 1 types of 2D fragments and 30,408 types of 3D fragments for the GEOM-Drugs dataset. As for the angular discretization of dihedral angles, the 360 degree interval is evenly divided into 72 bins. We adopt the Message-Passsngg Neural Network (MPNN) [Gimmer al, 2017 as the framework for implementing our graph neural network. Other implementation details are provided in Appendix A.
4.2 Evaluation Metries
To quantitatively measure the quality and diversity of generated molecular conformations, we use the same evaluation metrics as defined in Ganea et al. [2021] and Xu et al. [2022]. Let Sy and S, denote the set of generated and reference contormations, respectively. The coverage score (COV) and matching score (MAT) following the conventional Recall measurement can be defined as:
COV-R COV-R
S. |CEE
E S. |RSD((CC) &, E e
(16)
MAT-R S.)
S. Σ min RMSD(C,C)
CES,
where the threshold 8 in coverage score is set 1.25A for the GEOM-Drugs dataset in work. The corresponding prediction precision metrics, i.c., COV-P and MAT-P, are defined in a similar manner, where the set of generated and reference conformations are swapped in the above definition. We set Sg as twice the size of S, for each molecule for fair comparison with previous work (Ganea al., 2021 Xu et al. 2022]. The precision-assocaate
metrics focus conformations that match those in the reference dataset, while the recall-associated metrics emphasize the structural diversity of generated conformations.
generating accurate
Table 1: Results on the GEOM-Drugs dataset, without FF optimization.
COV-R (%)
MAT-R (A) 4
COV-P (%)
MAT-P (A)
Models
Mean
Median
Mean
Median
Mean
Median
Mean
Median
CVGAE
0.00
0.00
3.070
2.994
GraphDG
8.27
0.00
1.972
1.985
2.08
0.00
2,434
2.410
CGCF
53.96
57.06
1.249
1.225
21.68
13.72
1,857
1,807
ConfVAB
55.20
59.43
1.238
1.142
22.96
14.05
1.829
1,816
GeoMol
67.16
71.71
1.088
1.059
ConfGF
62.15
70.93
1.163
1.160
23,42
15.52
1.722
1.686
GeoDiff
89.13
97.88
0.863
0.853
61.47
64.55
1.171
1.123
DMCG
96.69
100.0
0.722
0.724
RDKit
60.19
64.28
1.219
1.133
69.23
87.63
1.113
0.963
OMEGA
81.64
97.25
0,851
0.771
77.18
96.15
0.951
0.854
RMCF-R
82.25
90.77
0.839
0.789
83.02
98.50
0.812
0.722
RMCF-C
87.12
96.26
0.749
0.709
82.01
95.91
0.835
0.754
* We follow the results reported by Zhu al.
2022
Xu al.
[2022
which use the
same data splits as us. GeoMol achieves higher scores in Ganea et al.
2021 's data splits.
COV-R: 22.43/95.10 MAT-R 0.862/0.837, COV-P 78 52/ 94.40, MAT-P 0.933/0.886
4.3 Results and Discussions
The main results are presented in Table
We generated conformations by sampling on the RMCF with two strategies: (1) directly sample specific number of conformations (i.e., S,), and (2) first generate a sufficient number of conformations (we generate 10,000 conformations in this work), then cluster them into Sg clusters and sample one conformation from each cluster. The model performance associated with the above two strategies are named RMCF-R and RMCF-C, respectively. As shown in Table RMCF significantly outperforms all other models in precision metries which means RMCF can generate more accurate and high quality conformations. We also achieve comparable performance recall metries with DMCG (Zhu et al., 2022 model, which the current state-of-ther -he--artt model on recall metrics. We think RMCF and DMCG improve the conformation generation in different aspects. Generally, modeling more freedom in the conformation generation process enables the model to generate diverse outputs (c.g., DMCG), which benefits the recall metric, while reducing modeling
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freedom will help to generate conformations more accurately, leading to better precision metric (e.g.. RMCF).
In addition, we find that the clustering--ffer-saappiig strategy leads to a boost in the structural diversity, while keeping good quality (i.e., energetically favorable conformations) of the generated conformations.
Since we employ discretized treatment for the continuous dihedral angle as well as the fragment 3D conformation, the model performance will inevitably have some discrepancy between the predicted and ground truth conformations, To that end, we investigate the upper and lower bound of our model performance to have a better understanding of its predicting power, as shown in Table 2 Specifically, discretize the ground truth conformations using the molecular segmentation algorithm to obtain the fragment configurations ("gold X²") and inter-fragment dihedral angles ("gold X²""), then evaluate the corresponding metrics. On the other hand, randomly sample some Xy and Xd, whose evaluation results should indicate the performance lower bound using our fragment representation. Surprisingly, we find that although using randomly sampled X3 and X2. we can still outperform some previous models. This is an advantage of using molecular fragment as building blocks for conformation generation, since bypass the need to generate many insignificant variables which may degrade the model performance. We also learn that having accurate dihedral angle predictions (i.e.. X2)) is more significant than good fragment configuration predictions (i.e.. Xy). These experimental results support our claim that capturing few sigmificant DoF within molecule is adequate for generating a diverse set of low-energy conformations, while other DoF could be safely ignored during modeling.
Table 2: The empirical upper and lower bound of RMCF performance on the GEOM-Drugs dataset.
COV-R (%) f
MAT-R (A)
COV-P (%) f
MAT-P (A) 4 Median
RMCF Settings
Mean
Median
Mean
Median
Mean
Median
Mean
gold gold Xr
99.16
100.0
0.302
0.242
99.37
100.0
0.290
0.235
rand gold A'a
96.74
100.0
0.511
0.464
88.93
100.0
0,634
0.561
gold X2, rand Xa
66.21
77.42
1.125
1.092
40.30
34.25
1,429
1.397
rand X rand Na
63.60
72.22
1.156
1.135
37.68
31.25
1.463
1.429
Xy denotes the fragment configuration set, X₂ denotes the inter fragment dihedral angle set "gold" refers the ground truth distribution, "rand" refers to randomly sampled distribution.
At last, we showcase the generated conformations of two example molecules in Figure 5] For each molecule, we take the first three predicted conformations, and align the non-rigid parts for visualization purposes. From FromrFigure S(a), we observe diverse set of conformations mainly driven by the rotation of two single bonds without causing too much steric effect. For the amide bond regions, the model correctly predicts the corresponding dihedral angles to form planar conjugate systems. Meanwhile, for Figure 5/b) see a large planar conjugate system the left side and cyclooctane ring on the right. Again, the model accurately captures the conjugate system and only makes variations in the cyclooctane conformations. Interestingly, the segmentation algorithm tends to only preserve the cyclic groups (e.g., benzyl and furan groups), and even cuts through the amide bonds and other acyclic conjugate systems. This behavior is a direct consequence of balancing the vocabulary size and the variety of local chemical environment.
(a)
(b)
Figure 5: The first three generated conformations of two example molecules. The upper panel shows the 3D atomic arrangement, where the non-rigid fragments are aligned to help visualization. The lower panel shows where the segmentation has been made for each molecule, as indicated by the scissors.

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We show that our model performs reasonable segmentation of Atug-like molecules into small func- tional groups with limited internal DoF, and could correctly predict planar conjugate systems. There- fore, we believe that our generative process is essentially sampling from the local-minima of the learned potential energy surface by RMCF, where those non-essential DoF which contribute to local structural perturbations are omitted in our modeling framework. We argue that both the data-driven segmentation and MRF modeling of the joint probability distribution are essential for the success of our model.
5 Conclusion
We introduce RMCF, a novel framework for 3D molecular conformation generation. Our model physics-motivated, with the central idea to effectively model the joint probability distribution of governing dynamical modes in reduced conformation space to achieve energetically favorable conformation generation. Experimental results show that RMCF outperforms state-of-the-ar models on the GEOM-Drugs dataset predict diverse set of conformations located at distinct local minima of the corresponding molecular potential energy surface. Our methodology can be naturally extended larger biomolecular systems, e.g., proteins, whose conformation prediction is significant topic in the biological research community. We will address this challenge in our future work.
Acknowledgements
We would like to thank the anonymous reviewers for their constructive comments. Hao Zhou is the corresponding author. This work is jointly supported by Vanke Special Fund for Public Health and Health Discipline Development, Tsinghua University (N0.202210800533, Guoqiang Research Institute General Project, Tsinghua University (No. 202IGQG1012) and National Natural Science Foundation of China (No. 62076068).
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13


Reviews:
[{"review_id": "lfM6A8VGy-", "reviewer": "Reviewer_JhnJ", "summary": "The authors propose a new method for 3D conformation generation. The method contains three steps:\n\n(1) the authors use dynamic programming to segment a molecule into several fragments, where the objective is in Eqn.(2);\n\n(2) Convert the generation problem into a classification problem: Given the fragments, predict the conformation and dihedral angles;\n\n(3) Assemble the intermediate results to the conformation of the molecule.\n\nThe method achieves promising results several baselines.", "questions": "1. What is the best choice of $|\\mathcal{V}(f)|$? Any ablation study?\n\n2. I'd like to know more about the fragments you obtained. How many atoms of each atom (average pm std)? If the conformation of a fragment does not lie in a plain, how to determine the dihedral angles?\n", "limitations": "Yes", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Originality:\n\nThe general idea is novel, and make sense. The idea is also similar to how to search the docking pose in molecular docking. The authors should comment on this.\n\n## Significance:\n\n1. The paper miss a reference [A1]. [A1] achieves better results on COV-R, and the mean of MAT-R.\n\n[A1] https://arxiv.org/pdf/2202.01356.pdf\n\n\n## Clarify: \n\nThe paper is hard to follow. This is my biggest concern of this paper.\n\n1. A question to be confirmed: Given a fragment $f$, it should has its own V(f). The way to obtain V(f) is that: first, find the conformations of fragment $f$ in all molecules, and clustering them using some algorithm to obtain |V(f)| clusters. The centroid of each cluster is an element in V(f). Am I right? Do you take roto-translation invariant of the fragment conformations into consideration?\n\n\n2. Line 134: \"Then, we infer a configuration set $\\mathcal{X}$ including the fragment state and the dihedral state\": What do you mean by state? This is very important and should be explained clearly.\n\n3. It is better to use some examples and figures to show what is $\\mathcal{X}_f$, $\\mathcal{X}_d$ and $\\mathcal{V}_f$. Currently, it is hard for me to get it.\n\n4. Eqn.(2): I did not get a clear solution about how to solve it.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "TzQfUJThPLQ", "reviewer": "Reviewer_QFQe", "summary": "This work provides a way to generate small molecule conformations from their molecular graph. The key idea is to find an appropriate fragmentation of a set of small molecules and to learn a Markov Random Field for the joint probability distribution of small molecular fragments. The paper demonstrates the model on the GEOM-Drugs dataset and compares to previously published baselines.\n", "questions": "\nThe presentation of the segmentation algorithm is unclear so I'll focus most of my questions on that topic.  In particular: 1) what is the clustering algorithm used and what is the way to decide the initial number of 3D clusters?  2) What aspect of equation (2) is optimized, i.e. how is the set of molecules re-fragmented (or is it not?) to obtain an optimal solution.  I did not spend much time, but from a quick search of the codebase, I could not find what part of the provided code performs this segmentation optimization.  3) The language in the paper makes me concerned that the authors might have accidentally used the full molecule set (including the validation split) during the construction of the 3D fragments, therefore possibly having some minor leakage.  If that is not the case, please fully clarify the description of this process and point to the relevant code.  4) How does the code deal with out-of-sample fragments?  If the fragmentation of a validation molecule leads to a piece that is not existing in the training set, what happens then?! 5) When there are multiple options, how are the atoms selected that perform the connecting pieces between fragments (in order to reduce the problem to only dihedrals) and does the selection influence the results at all? 6) would it be possible to avoid binning the angles and what if any would be the disadvantages of such an approach?  7) Finally, how do the authors address possible steric clashes from the generated conformations, or does the markov field somehow avoid from ever proposing such a clash even in reasonably complicated small molecules, e.g. a large macrocycle?\n\n\nAlthough the following would not be necessary for acceptance of this work, it would be nice if the authors showed the behavior of their model for the dataset in Table 4 as a function of the cutoff parameter delta, if they clarified how they selected the examples for Figure 5, and if they could include a brief comparison with the torsional-diffusion model, whose preprint is here:\n\nhttps://arxiv.org/abs/2206.01729\n\n", "limitations": "\nThe authors did not adequately address any limitations of this work. Perhaps the authors could address how this method would scale to macromolecules, to molecules with fragments outside the training set, or to macrocycles.  Would it be possible to predict the structure of macrocycles if those were outside of a training set?  \n", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "\nThe key idea behind this work is sound and original. The presentation occasionally leaves some open questions (see below).  Although there is some code that allows for the reproducibility of this work and would add to the clarity of this method, I haven't verified if the code actually works as advertised. The problem that this work addresses is significant enough so it warrants some additional consideration in my opinion, despite the possible problems with this first draft, some of which I highlight below.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "HRoW8uAJv0f", "reviewer": "Reviewer_ckMd", "summary": "Representation learning for chemistry and molecules in specific is an emerging area in deep learning. In this work, the authors propose a framework (and then a model) to generate low energy 3d conformations of molecules. The work proposes to segment a given molecule into  fragments (with limited internal flexibility), and then proposes to learn a joint probability distribution over the fragments using a Markov Random Field. As parameter estimation for the MRF is intractable, the authors provide an approximate learning strategy leveraging GNNs and Gibbs sampling to achieve their objective.", "questions": "1. Why not perform evaluation on Geom-QM9? Another dataset considered in the baseline models.\n2. For me, it is not clear how you back propagate through the RMCF procedure. Please add detail\n3. Similarly, please elaborate on the graph dynamic programming and also Eq (3). Specifically What does the GNN output? I was under the opinion it is the edge representations but then Eq(3) has a set on the left.\n4. Also please address the concerns in the weaknesses section.", "limitations": "The authors have discussed the limitations of their work. For example:\n1. Have discussed intractability of parameter estimation in Sec 3.4, and ways to counteract it.\n2. In Sec 4.3, have discussed a few limitations of the segmentation framework\n\nI do not see any potential negative societal impact of their work", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The idea to segment a molecule into fragments, and then use the framework to generate low energy. states is interesting.\n2. The observation that even randomly sampled $\\mathcal{X}_f, \\mathcal{X}_d$ in section 4.3 can outperform some previous works is insightful\n\nWeaknesses:\n1. In my opinion, the writing is not precise and hard to understand. For e.g. a) in section 3.2, it is hard to decipher what is a sufficient number of 3D fragments? b) in sec 3.3 - how is the oracle configuration obtained?  c) in sec 3.5 the sampling procedure, it is not clear what the samples are? the fragments? 3d positions? and how detailed balance is satisfied d) Why are the side chains removed in 3.2, etc. Infact, i felt it was incredibly hard to read the entire of section 3. \n2. Evaluation performed on a single dataset. Also the scores from other works appear to be incorrect (e.g. GeoMol scores are better than numbers presented here, etc) in addition to RMCF not outperforming other benchmarks.\n3. The idea to fragment and compute graph representations have already been performed (e.g. especially energy based models [1]), which have not been cited. The idea to use GNNs to parametrize them has also been done in [1]\n\nReferences:\n1. Cotta, Leonardo, et al. \"Unsupervised joint k-node graph representations with compositional energy-based models.\" Advances in Neural Information Processing Systems 33 (2020): 17536-17547.\n\n\n\nPost Rebuttal Updates:\n\nThank you very much for the rebuttal and answering my questions on the dataset (QM9 has smaller molecules, so impacts performance), backprop and other questions I had raised. I have also read through the other reviews and the authors response. In light of all this, I am updating my score.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "B3TKA-7ZJes", "reviewer": "Reviewer_u5WS", "summary": "This paper proposes a novel fragment-based model for molecular conformation generation. The authors develop a data-driven molecular segmentation algorithm to break each molecule into several fragments. The model learns the joint probability distribution of fragment configurations and dihedral angles between fragments. The proposed method outperforms existing baselines with a clear margin.", "questions": "Questions:\n\n- If I understand it correctly, the BRICS segmentation will give a unique segmentation for each molecule. It seems only clustering is needed after that, but where the \"argmin P\"  in Equation (2) comes from? Also, how do you determine the number of clustering centers?\n- The segmentation algorithm returns a vocabulary of 9k 2D fragments and 30k 3D fragments. How is the frequency of each fragment? Since the number of modes is still very large, I'm curious about whether the model will have some difficulties in learning the long tail modes.\n- Is it hard to train the model without discretizing the continuous angle values into bins?\n\nMissing reference:\nBesides EGNN and GeoDiff, [1] is also an equivariant network which directly learns 3D coordinates of atoms.\n\n[1] Guan, Jiaqi, Wesley Wei Qian, Wei-Ying Ma, Jianzhu Ma, and Jian Peng. \"Energy-Inspired Molecular Conformation Optimization.\" In International Conference on Learning Representations. 2021.", "limitations": "The authors didn't discuss limitations. There is not negative social impact.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The proposed fragment-based model for molecular conformation generation is novel.\n- The paper is well-written and easy to follow.\n- The proposed method shows great empirical performance.\n\nWeaknesses:\n- I have some concerns about the segmentation algorithms. See details in Questions.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept