Regularized Molecular Conformation Fields
Anonymous Author(s)
AffiliationAddress email
Abstract
Predicting energetically favorable 3-dimensional conformations of organicmolecules from molecular graph plays a fundamental role in computer-aideddrug discovery research.
However, effectively exploring the high-dimensionalconformation space to identify (meta)stable conformers is anything but trivial.
Inthis work, we introduce R MCF , a novel framework to generate a diverse set of lowenergy molecular conformations through sampling from a regularized molecularconformation field.
We develop a data-driven molecular segmentation algorithmto automatically partition each molecule into several structural building blocks toreduce the modeling degrees of freedom.
Then, we employ a Markov Random
Field to learn the joint probability distribution of fragment configurations and interfragment dihedral angles, which enables us to sample from different low-energyregions of a conformation space.
Our model constantly outperforms state-of-the-artmodels for the conformation generation task on the GEOM-Drugs dataset.
We attribute the success of R MCF to modeling in a regularized feature space and learninga global fragment configuration distribution for effective sampling.
The proposedmethod could be generalized to deal with larger biomolecular systems.
Introduction
The spatial arrangement of atoms within a molecule, also known as molecular conformation, determines the molecular physico-chemical properties, which plays an essential role in downstreamcomputer-aided drug discovery tasks.
However, the high-dimensional conformation space spannedby all atomic degrees of freedom (DoF) makes it a great challenge to identify the local minima ofthe associated potential energy surface (PES) of a molecule.
Recently, many machine learning (ML)models have achieved remarkable success in molecular conformation generation tasks [Mansimovet al., 2019, Simm and Hernandez-Lobato, 2020, Xu et al., 2021a, Ganea et al., 2021, Xu et al.
, 2022],exhibiting orders of magnitude faster conformation prediction speed than traditional computationalsimulation approaches.
This makes ML generative models a powerful tool for high-throughputscreening in drug discovery, especially large molecules.
To fully exploit the power of ML in molecular conformation generation, we need to tackle a few keychallenges.
First, a molecule exhibits invariance under SE(3) transformations (i.e., translation androtation) in 3-dimensional (3D) Euclidean space.
In other words, each molecular conformation isuniquely defined up to rigid motion.
Hence a single molecule can adopt an infinite set of possibleposes.
Second, molecules exhibit a variety of dynamics under ambient conditions (e.g., bond rotation,vibration, etc.), leading to a possibly complex PES landscape of high dimension.
This makesit particularly challenging for ML models to identify local minima within this space to generateenergetically favorable conformations effectively.
Some recently developed ML models tried to solve these problems from different perspectives.
For instance, GraphDG [Simm and Hernandez-Lobato, 2020] and CGCF [Xu et al., 2021a] useddistance geometry as invariant features for molecular representation learning.
The drawbacks of such representations are that these invariant variables are potentially redundant and exhibit mutualdependency, which may cause numerical instability during training.
One can also design equivariantnetworks to hard-code the invariance/equivariance condition into the model [Satorras et al., 2021, Xuet al., 2022].
These models circumvent using intermediate invariant features, but instead directly learnfrom 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 expressivenessof the neural network.
On the other hand, GeoMol [Ganea et al., 2021] predicts a local structurefor each atom, followed by assembling these atom-based building blocks to form the molecularconformation.
These models have difficulty dealing with cyclic graphs (e.g., benzyl group) and maylead to unreasonable conformation predictions.
Dihedral angle
P o t en t i a l ene r g y
HH
HH HH
HH
HH HH
HH
HH HH
In the meantime, we realize that the high-dimensional
PES could be described in a reduced basis by considering only a few significant DoF that contribute to thelow-energy conformations.
Specifically, consider thewell-known example of the potential energy diagramof ethane molecule as shown in Figure 1. While themolecule exhibits a total of 18 DoF, we could capturethe most significant dynamics using a single variable,i.e., the rotation w.r.t.
the carbon-carbon bond, quantified by the H-C-C-H dihedral angle.
Other DoF inthe ethane molecule, e.g., the stretching and bendingof other bonds, do not have a significant impact onthe potential energy landscape, such that they can betreated as small perturbations to the molecular conformation within a potential well.
In other words, toserve the purpose of effectively sampling local minima for low-energy conformations, we could reducethe dimension of the conformation space by trackingonly a few significant DoF, as long as they adequatelyshape the corresponding energy landscape (e.g., the“ ω ” shape in Figure 1).
It then follows naturally togenerate new conformers by sampling from the low-dimensional conformation space, which is alsophysically meaningful.
In fact, a similar approach has been widely adopted by the molecular dockingcommunity, i.e., incremental construction
[Meng et al., 2011], where a ligand molecule is partitionedinto multiple fragments connected by rotatable covalent bonds.
One may then gradually add ligandfragments with appropriate orientations to fit the ligand into the receptor pocket.
This paper presents Regularized Molecular Conformation Fields (R MCF ), a novel framework for3D molecular conformation generation.
The novelty of this work lies in (1) developing a datadriven molecular segmentation algorithm to partition each molecule into fragments with low internalflexibility, which serves as a regularization term for our framework, and (2) employing a Markov
Random Field (MRF [Murphy, 2012, Cotta et al., 2020]) to capture the joint probability distributionof fragment configurations and inter-fragment dihedral angles.
Our work is partly inspired by the Ising model [Cipra, 1987] for simulating quantum spin systems, where we make an analogy betweenthe spin state (i.e., spin up or down) and the configuration of each molecular fragment (e.g., chairor boat conformation of cyclohexane).
The introduction of molecular fragments effectively reducesthe 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.
Thismolecular segmentation serves as a regularization term for our framework, which is data-driven andhas the effect of reducing feature dimensions.
Then, we employ MRF as a generalized setting of the Ising model (with more spin states and more coupling terms) to capture the relationship betweenadjacent 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,we can later sample from the low-energy region of PES, which is implicitly learned by the MRF, toobtain energetically favorable molecular conformations.
We demonstrate the effectiveness of R MCF using the GEOM-QM9 and GEOM-Drugs dataset[Axelrodand Gomez-Bombarelli, 2022], where results show that R MCF significantly outperforms state-ofthe-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 tothe automatic construction of low-dimensional conformation space and learning a joint probabilitydistribution using MRF to achieve effective sampling of new conformations.
We argue that capturingthe governing dynamics of molecular systems could significantly improve the performance of learningmodels for generative purposes.
Related Work
Recently, various machine learning models have been proposed for molecular conformation generation.
CVGAE [Mansimov et al., 2019] first used a variational autoencoder (VAE) model to generateconformation with atomic coordinates.
The model suffers from multi-modality problems due toinvariance under SE(3) transformations.
Some studies tried to address this problem using two maintypes of approaches.
The first type of approach use intermediate invariant features, such as atomicdistance, to encode the system, then leverage geometric algorithms to solve a set of atomic coordinatesthat match the invariant quantities.
For example, GraphDG [Simm and Hernandez-Lobato, 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 makesthese methods numerically unstable and could predict unreasonable conformations.
Some studiessuggest that inconsistencies between training and test are responsible for the poor performance of themodels [Shi et al., 2021, Xu et al., 2021b].
ConfGF [Shi et al., 2021] passed the gradient of lossto coordinates and ConfVAE [Xu et al., 2021b] used a bilevel optimization to alleviate the inconsistencies.
GeoMol [Ganea et al., 2021] 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 ofcoordinates of conformations and permutation of symmetric atoms in molecules.
Geomol generatedmultiple conformations by adding noise to the input, which did not learn a global energy functionfrom which to sample the low-energy conformations.
Another type of approach applies equivariant networks or kernels, such that after the input is rotatedand translated, the output will be transformed accordingly.
Satorras et al.
[2021] proposed anequivariant normalizing flow, E(n)-flow, and Xu et al.
[2022] proposed GeoDiff, which is a diffusionmodel [Song and Ermon, 2019] with equivariant Markov kernels. Guan et al.
[2021] proposed avariant of SE(3)-equivariant neural networks to learn the gradient fields of an implicit conformationalenergy landscape.
Equivariant methods circumvent intermediate invariants and model the coordinatesdirectly, but require the design of specialized equivariant layers, which some studies have suggestedwould lose the expressive power of the network [Li et al., 2021].
In addtion, some software for conformation generation are widely used in biochemical research.
Forinstance, RDKit [Riniker and Landrum, 2015] is a popular open-source software which generatesconformations using ETKDG distance geometry, and OMEGA is a commercial software whichassembles the fragments with knowledge-based rules to generate conformations.
R MCF : Generating in Regularized Conformation Space
In this section, we present the regularized molecular conformation field (R MCF ) in detail.
Intuitively,
R MCF partitions the original molecular graph into multiple molecular fragments, where adjacentfragments in the graph are connected with associated dihedral angles. In such a case, the molecular conformation generation problem can be decomposed into a) generating the conformation ofmolecular fragments, and b) determining the specific dihedral angles between connected fragments.
The R MCF learns the potential energy landscape of molecular conformations by modeling a jointprobability distribution of fragment conformation and dihedral angles, from which we can performinference to draw diverse samples.
3.1 Overview of R MCF
We start with an overview of the our proposed R MCF model.
Generally, as shown in Figure 2, given a molecular graph G , to obtain a generated 3D conformation R , R MCF mainly takes three steps: [Equation]
[Figure] G −→ G
Section 3.2): The main idea of our proposed model is to build a conformationdistribution within a regularized molecular conformation space, which is achieved by partitioningthe 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 F and D ) could be jointly modeled with a Markovrandom field (also called undirected graphical model).
Since it is regularized with reduced total
DoF, we name our model Regularized Molecular Conformation Fields.
Formally, a regularized conformation field G = ( F, D, E ) is an undirected graph formed by acollection of fragment vertices F = ( f 1 , f 2 , · · · , f N F ) , a collection of dihedral vertices D =( d 1 , d 2 , · · · , d N D ) , and a collection of edges E
= ( e 1 , e 2 , · · · , e N E ) ⊂
F × D (edges betweenfragments and dihedral angles in the graph), where N F , N D , N E are the size of F , D , E ,respectively [Wainwright et al., 2008].
Each edge consists of a pair of vertices f ∈ F and d ∈ D .
We associate with each vertex f ∈ F a random variable X f taking values from a set of fragmentconformations V ( f ) (see Section 3.2), each vertex d ∈ D a random variable X d , representing thevalue of dihedral angles.2.
G −→ X (Section 3.3 and 3.4): Given G , the conformation generation problem turns intogenerating the conformation of molecular fragments and determining the dihedral angles betweenthem, namely inferring a configuration X from G .
A configuration X =
( X f , X d ) of the R MCFcontains a set of fragment conformation X
f = { x f i | i = 1 , 2 , · · · , N F } and a set of dihedral anglevalues X
d
= x d i | i = 1 , 2 , · · · , N D } , which could be assembled into the 3D conformation of amolecule.
X could be obtained by maximizing the probability modeled by R MCF : [Equation] where Z is the normalizing factor, ψ is the compatibility functions, which will be further describedin Section 3.3. X −→ R
Section 3.5): Given the configuration X , we assemble the predicted fragment confor- mation and dihedral
angle values into the corresponding
molecular conformation R 3.2 Graph Construction with Least DoF Principle In this section, we elaborate more about the molecular graph partition with least DoF principle used inour proposed R MCF .
This is a crucial step to our model since it builds the foundation of the followingprobabilistic modeling. The entire graph construction process has three main components, namely a)constructing the fragment library with BRICS, b) obtaining fragment conformation vocabulary withclustering, and c) yielding final fragment partition by minimizing the intra-fragement DoF. Specifically, before establishing a field, we need to search for a solution P ( P can be converted to Gtrivially) to partition a molecule into fragments and discretize each continuous fragment conformationspace into several fragment conformation representatives, also called the conformation vocabulary V ( f ) of fragment f .
To that end, we first collect all kinds of fragments from the entire dataset toobtain the conformation vocabulary V ( f ) for each fragment. Then, we define a metric to quantifythe internal DoF within the discrete space V ( f ) , and employ dynamic programming to search for apartition strategy P with least total DoF for each molecule.
We use existing functional group segmentation method BRICS to pre-cut all molecular graphs andtheir 3D conformations and collect each fragment conformation into a set R ( f ) 1 .
After obtainingthese molecular fragments, a sufficient number (i.e., 1,000) of fragment conformations r are sampledfrom R ( f ) , and we cluster these samples to obtain V ( f ) .
We use the K-Medoids clustering algorithmand use the root-mean-square deviation (RMSD) of two fragment conformations as the distancebetween two fragments, which takes rotation and translation invariance into consideration. Weadjust the number of cluster centers k from 1 to 10 and use the Silhouette Coefficient to measure thegoodness of these clusters. We choose the optimal hyper-parameter k with clusters having the highest Silhouette Coefficient value.
Cluster centroid elements are collected into conformation vocabulary V ( f ) to denote the discrete configurations of f .
While BRICS presents a solution to partition the molecular graph, it cannot guarantee a small DoFwithin the fragments. Therefore, we define a metric to quantify the fragment DoF and design analgorithm to re-fragment the molecules to minimize this quantity. As a general rule of thumb, weshould minimize the intra -fragment DoF while capturing the significant dynamic modes using interfragment dihedral angles.
We define the fragment f DoF as the maximum RMSD between all pairsof fragment conformations in V ( f ) , and the total DoF of a partitioned molecule as the sum of itsfragment DoF. The best partition strategy P ∗ should have the least DoF:
[Equation]
This problem is analogous to dictionary-based word segmentation or word break.
We could employgraph dynamic programming to search for the optimal solution: first search for a substructure fromthe fragment collection C , then split the molecule into several substructures to minimizes the sum of
DoF of all substructures.
Every substructure is also a nested sub-problem until the substructure canno longer be split.
The detailed algorithm is in Appendix D. After partition, we can get the fragmentset F = ( f 1 , f 2 , · · · , f N F ) and insert the dihedral vertices between connected fragments to constructthe R MCF G on the molecular graph G .
3.3 Approximated Training of R MCF
Training Objective Training in R MCF is intractable, especially when it contains loops [Murphy,2012]. It is difficult to estimate the normalization factor Z in the objective (Eq.
1),which means wehave to marginalize all possible conformations. We approximate the likelihood through piece-wisetraining , 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.
[Equation]
(3) where L n and L e denote the node-wise and edge-wise negative log-likelihood.
Especially, L e ( f, d )measures how probable a fragment f and a dihedral d appear at the same time .
More concretely, for a pair of nodes in the field e = ( f, d ) ∈ G , let | f | and | g | denote the cardinalityof the state space, | h | denotes the hidden dimension, y f and y d are the ground truth of fragmentconformation f and dihedral angle value d , which will introduced calculation in the next paragraph. We calculate the representations from the graph neural networks: Let h 0 f = emb( f ) and h 0 d = emb( d )denote the fragment embedding and the dihedral embedding (which is a “dummy” embedding) as the network input, we compute the contextualized fragment representations h ℓ +1 f and dihedralrepresentations h ℓ +1 d by: [Equation]
where ℓ denotes the layer index, g denotes the massage-passing function of GNN, N f and N d denotesthe neighborhoods of fragment and dihedral respectively. The node-wise negative log-likelihood isdefined as: [Equation] where out( · ) : R | h | → R | f | is an operator parameterized by a linear layer.
The edge-wise negativelog-likelihood is defined as:
[Equation] where E f ∈ R | f |×| h | and E d ∈ R | d |×| h | are the representations of a set of node states, W e ∈ R | h |×| h | is a matrix carrying the global information between two nodes, parameterized by a neuralnetwork W e = MLP( h f , h d ) [Sun et al., 2019].
Obtaining ground truth The oracle configuration Y = ( Y f , Y d ) contains a set of ground truthfragment conformations indexes Y f
= { y f i | i = 1 , 2 , · · · , N F } and a set of dihedral angle indexes Y d = { y d i | i = 1 , 2 , · · · , N D } denote the choices of fragments conformations and dihedral angles,respectively.
For a given fragment f , we pick the configuration s whose corresponding fragmentconformation r s in vocabulary V ( f ) , has the minimal RMSD to the actual fragment conformation r ,as the ground truth y f .
[Equation]
The dihedral angle φ is defined as the angle between half-planes of twoconnected 3D fragments.
Since the dihedral angle depends on the set ofthe atoms involved in the calculation, for the same type of fragments, weshould keep the same set of atoms to calculate.
For a system where two fragments contain four consecutively-bondedatoms, two half-planes intersect on a rotatable bond.
The angle betweenthem is the dihedral angle .
If the connected points are sequentiallynumbered and located at positions p 1 , p 2 , p 3 , p 4 and the correspondingbond vectors are defined as u 1 =
p 2 − p 1 , u 2 = p 3 − p 2 , u 3 = p 4 − p 3 .
Then we have:
[Equation]
In practice, we quantize the dihedral angles φ into a number of evenly divided bins which has intervallength L to discretize the continuous angle values
[Equation]
3.4 Inference & Sampling
In this section, we give a brief overview of how we perform inference and sampling in R MCF .
Inference If a R MCF is acyclic (linear- or tree-structured), the max-a-posterior decoding can bedone via dynamic programming [Forney, 1973].
Cases are a bit complicated when loops occur in the
R MCF .
We adopt a more generalized algorithm, namely loopy belief propagation (LBP)
[Murphyet al., 2013], to overcome such obstacles.
We refer the reader to textbooks for more details about the
LBP algorithm
[Murphy, 2012].
It is worth noting that LBP is used only during inference time.
Wedo not perform any inference during training due to the issue of training stability [Qu et al., 2022].
Sampling Markov Chain Monte Carlo is popular in sampling from graphical models [Andrieu et al.
We employ a simple variant of it, i.e., Gibbs sampling [George and McCulloch, 1993], to drawsamples from R MCF .
Specifically, we resample each fragment or dihedral individually, keeping all other nodes fixed in each iteration [Murphy, 2012]. The detailed algorithm can be found in Appendix
E. Clustering Once the Markov chain achieves detailed balance, we randomly draw 10 , 000 samplesfrom it.
However, these samples can be governed by a few conformations, which means most of themare similar.
We measure the distance between two configurations X (1) and X (2) by:
[Equation]
where ∥ · ∥ 2 is the Euclidean norm measuring the difference between dihedrals, H is the Hammingdistance [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 samplesinto a fixed number of clusters.
3.5 Assembling
[Figure] [Equation]
[Figure]
Once we have predicted the dihedral angle ˆ x d between twofragments, we need to compute the transformation matrix Tto align the two fragments in arbitrary position and make theirdihedral angle equal to the given value φ and repeat these stepsuntil the whole conformation is assembled.
Given two fragments with six points, located at p 1 , p 2 , p 3 , p 4 ,p 5 , p 6 , the bond vectors are u 1 = p 2 − p 1 , u 2 = p 3 − p 2 , v 1 =p 5 − p 4 , v 2 = p 6 − p 5 .
u 2 and v 1 are the same bond afterassembling, so we have | u 2 | = | v 1 | .
First we need to rotate the second fragment around some axisso that u 2 and v 1 will be parallel.
This axis is the unit normalvector n towards the plane of u 2 and v 1 , and the rotate angle isequal to the cross angle θ .
[Equation]
By using the Rodrigues’ rotation formula, we can obtain the rotation matrix with unit normal vectorn and rotation angle θ .
Next, we rotate the second fragment around u 2 to match with the targetdihedral angle.
Finally we align the anchor points by calculating the final transformation matrix T asa composite matrix of two rotations and one translation.
Experiment
We now demonstrate the effectiveness of R MCF on the conformation generation task for drug-likemolecules.
4.1 Dataset and Baseline
Following previous work on conformation generation, we benchmark our model performance usingthe GEOM-QM9 and GEOM-Drugs dataset, which contains small and mid-sized organic moleculeswith high quality conformations.
Considering the small size of molecules in QM9, we put themodel performance in Appendix 4, 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 trainingand validation with a 9:1 ratio.
The final training/validation/test set contains 271,539/30,171/1,034molecules, respectively.
Our molecular segmentation algorithm eventually returned a vocabulary of 9,081 types of 2Dfragments and 30,408 types of 3D fragments for the GEOM-Drugs dataset.
As for the angulardiscretization of dihedral angles, the 360 degree interval is evenly divided into 72 bins.
We adopt the Message-Passing Neural Network (MPNN)
[Gilmer et al., 2017] as the framework for implementingour graph neural network.
Other implementation details are provided in Appendix A.
4.2 Evaluation Metrics
To quantitatively measure the quality and diversity of generated molecular conformations, we usethe same evaluation metrics as defined in Ganea et al.
[2021] and Xu et al.
Let S g and S rdenote the set of generated and reference conformations, respectively.
The coverage score ( COV )and matching score ( MAT ) following the conventional Recall measurement can be defined as: [Equation]
[Equation]
where the threshold δ in coverage score is set as 1 . 25 Å for the GEOM-Drugs dataset in our work.
The corresponding prediction precision metrics, i.e., COV-P and MAT-P , are defined in a similarmanner, where the set of generated and reference conformations are swapped in the above definition.
We set S g as twice the size of S r for each molecule for fair comparison with previous work [Ganeaet al., 2021, Xu et al., 2022].
The precision-associated metrics focus more on generating accurateconformations that match those in the reference dataset, while the recall-associated metrics emphasizethe structural diversity of generated conformations.
4.3 Results and Discussions
The main results are presented in Table 1.
We generated conformations by sampling on the R MCFwith two strategies: (1) directly sample a specific number of conformations (i.e., S g ), and (2) firstgenerate a sufficient number of conformations (we generate 10,000 conformations in this work), thencluster them into S g clusters and sample one conformation from each cluster.
The model performanceassociated with the above two strategies are named RMCF-R and RMCF-C, respectively.
As shownin Table 1, R MCF significantly outperforms all other models in all metrics except for the COV-Rmetric, where our model performance is comparable to that of GeoDiff.
In particular, for COV-Pand MAT-R metrics, our model shows a significant performance gain compared to others, whichmeans R MCF can generate more accurate and high quality conformations.
In addition, we find thatthe clustering-after-sampling strategy leads to a boost in the structural diversity, while keeping a goodquality (i.e., energetically favorable conformations) of the generated conformations.
Since we employ a discretized treatment for the continuous dihedral angle as well as the fragment 3Dconformation, the model performance will inevitably have some discrepancy between the predictedand ground truth conformations.
To that end, we investigate the upper and lower bound of our modelperformance to have a better understanding of its predicting power, as shown in Table 2.
Specifically,we discretize the ground truth conformations using the molecular segmentation algorithm to obtain thefragment configurations (“gold X f ”) and inter-fragment dihedral angles (“gold X d ”), then evaluate thecorresponding metrics.
On the other hand, we randomly sample some X f and X d , whose evaluationresults should indicate the performance lower bound using our fragment representation.
Surprisingly,we find that although using randomly sampled X f and X d , we can still outperform some previousmodels.
This is an advantage of using molecular fragment as building blocks for conformationgeneration, since we bypass the need to generate many insignificant variables which may degradethe model performance.
We also learn that having accurate dihedral angle predictions (i.e., X d ) ismore significant than good fragment configuration predictions (i.e., X f ).
These experimental resultssupport our claim that capturing a few significant DoF within a molecule is adequate for generating adiverse set of low-energy conformations, while other DoF could be safely ignored during modeling.
At last, we showcase the generated conformations of two example molecules in Figure 6.
Foreach molecule, we take the first three predicted conformations, and align the non-rigid parts forvisualization purposes.
From Figure 6(a), we observe a diverse set of conformations mainly drivenby the rotation of two single bonds without causing too much steric effect.
For the amide bondregions, the model correctly predicts the corresponding dihedral angles to form planar conjugatesystems.
Meanwhile, for Figure 6(b) we see a large planar conjugate system on the left side and acyclooctane ring on the right.
Again, the model accurately captures the conjugate system and onlymakes variations in the cyclooctane conformations.
Interestingly, the segmentation algorithm tendsto only preserve the cyclic groups (e.g., benzyl and furan groups), and even cuts through the amidebonds and other acyclic conjugate systems.
This behavior is a direct consequence of balancing thevocabulary size and the variety of local chemical environment.
[Table] [Table] [Figure] We show that our model performsreasonable segmentation of druglike molecules into small functionalgroups with limited internal DoF, andcould correctly predict planar conjugate systems.
Therefore, we believethat our generative process is essentially sampling from the local-minimaof the learned potential energy surfaceby R MCF , where those non-essential DoF which contribute to local structural perturbations are omitted in ourmodeling framework.
We argue thatboth the data-driven segmentation and MRF modeling of the joint probabilitydistribution are essential for the success of our model.
Conclusion
We introduce R MCF , a novel framework for 3D molecular conformation generation.
Our modelis physics-motivated, with the central idea to effectively model the joint probability distribution ofgoverning dynamical modes in a reduced conformation space to achieve energetically favorableconformation generation.
Experimental results show that R MCF outperforms state-of-the-art modelson the GEOM-Drugs dataset to predict a diverse set of conformations located at distinct local minimaof the corresponding molecular potential energy surface.
Our methodology can be naturally extendedto larger biomolecular systems, e.g., proteins, whose conformation prediction is a significant topic inthe biological research community.
We will address this challenge in our future work.
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We conducted our experiments on 8 A100 GPUs and took abouthours to train R MCF on GEOM-Drugs
[Table]
[Figure] [Figure] C QM9 Results [Table]
Algorithm
Require: the fragments type collection F c , the molecular graph G m
Ensure: the best molecular graph partition P ∗ function D O F( f ) return max r i ,r j ∈ V ( f ) RMSD( r i , r j ) end function function P ARTITION ( G ) s ← 100 , P ← {} for f ∈ F c do if IsSubstructure( G, f ) then s m ← 0 , z ← 0 , P m ← { f }
F r ← Remove( G, f ) for f ∈ F r do dof, P f ← PARTITION( f ) s m ← s + dof ∗
|P
f | , z ← z + |P f | , P m ← P m ∪ P f end for s m ← s m /z end if if s m <
s then s ← s m , P ← P m end if if P = {} then return DoF( G ) , { G } end if end for end function dof, P ∗ ← PARTITION( G m ) return P ∗
E
Sampling Algorithm
Algorithm
Require: an R MCF G , the number of conformations N C , the number of sampling iterations N S
Ensure: a set S C containing N C conformations
S ← {} initialize a configuration X randomly for i ∈ [1 , 2 , · · · , N S ] do for n ∈ F ∪ D do
U ← { u | ( n, u ) ∈ E
▷ x u is fixed in this iteration
P ← softmax (cid:8)(cid:80) u ∈ U E u [ x u ] W un E T n + out( h n ) (cid:9)
▷ the value can be pre-computed x n ∼ Cat( | n | , P ) end for add X to S end for compute the pairwise distance K with Eq.
S C ← run clustering with k = N C over S based on K return S C