Equivariant 3D-Conditional Diffusion Models for Molecular Linker Design
Authors: Ilia Igashov
Hannes Stärk
Clement Vignac
Victor Garcia Satorras
Pascal Frossard
Max Welling
Michael M. Bronstein
Bruno Correia
Venue: Submitted to ICLR 2023 (2023)
Keywords: Molecules
Drug Discovery
Molecular Linker Design
Equivariant
Diffusion Models

Abstract: Fragment-based drug discovery has been an effective paradigm in early-stage drug development. An open challenge in this area is designing linkers between disconnected molecular fragments of interest to obtain chemically-relevant candidate drug molecules. In this work, we propose DiffLinker, an E(3)-equivariant 3D-conditional diffusion model for molecular linker design. Given a set of disconnected fragments, our model places missing atoms in between and designs a molecule incorporating all the initial fragments. Unlike previous approaches that are only able to connect pairs of molecular fragments, our method can link an arbitrary number of fragments. Additionally, the model automatically determines the number of atoms in the linker and its attachment points to the input fragments. We demonstrate that DiffLinker outperforms other methods on the standard datasets generating more diverse and synthetically-accessible molecules. Besides, we experimentally test our method in real-world applications, showing that it can successfully generate valid linkers conditioned on target protein pockets.

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EQUIVARIANT 3D-CONDITIONS DIFFUSION MODELS FOR MOLECULAR LINKER DESIGN
Anonymous authors Paper under double-blind review
ABSTRACT
Fragment-based based drug discovery has been effective paradigm in early-stage drug development. An open challenge in this area is designing linkers between dis- connected molecular fragments of interest obtain chemically-relevan candi- date drug molecules. In this work, we propose DiffLinker, an E(3) equivivrrn 3D-conditional diffusion model for molecular linker design. Given a set of dis- connected fragments, our model places missing atoms in between and designs a molecule incorporating all the initial fragments. Unlike previous approaches that are only able to connect pairs of molecular fragments, our method can link an arbi- trary number of fragments. Additionally, the model automatically determines the number of atoms in the linker and its attachment points to the input fragments. We demonstrate that DiffLinker outperforms other methods on the standard datasets generating more diverse and synthetically-accessie perimentally test our method in real-world applications, showing that it can suc- cessfully generate valid linkers conditioned on target protein pockets.
molecules. Besides, we ex-
INTRODUCTION
The space of pharrnacooogically-reeeaaar molecules is estimated to exceed 1060 structures (Virshup al. ).13), and searching that space poses significant challenges for drug design. A successful approach to reduce the size of this space is to start from fragments, smaller molecular compounds that usually have no more than 20 heavy (non mon-hyyroggen atoms. This strategy is known as fragment- based drug design (FBDD) (Erlanson et al.) 2016). Given protein pocket (a part of the target protein that employs suitable properties for binding a ligand), computationally determining fragments that interact with the pocket is cheaper and more efficient alternative to experimental high- high-throughput screening methods (Erlanson et al. 2016). Once the relevant fragments have been identified and docked to the target protein, remains combine them into single, connected molecule. Among various strategies such as fragment linking, merging, and growing (Lamoree & Hubbard (017)), the former has been preferred as allows to boost rapidly the binding energy of the target and the compound (Jencks 1981 Hajduk et al. 1997). This work addresses the fragment linking problem.
based on database search and phys- ical simulations (Sheng & Zhang 2013). both of which are computationally intensive. Therefore, there is increasing interest for machine learning methods that can go beyond the available data and design diverse linkers more efficiently. Existing approaches are either based on syntactic pattern recognition (Yang et al., 2020) or on autoregressive models (Imrie al., 2020 202T Huang al.) 2022). While the former method operates solely on SMILES (Weininger 1988). ). the latter take into account 3D positions and orientations of the input fragments, as this information is essential for designing stable molecules in various application (see Appendix [A.]] for details). However, these methods are not equivariant with respect to the permutation of atoms and can only combine pairs of fragments.
Early computational methods for molecular linker design
Linker design depends on the target protein pocket, and using this information correctly can improve the affinity of the resulting overall compound. To date, however, there is no computational method for molecular linker design that takes the pocket into account.
In this work. we introduce DiffLinker, conditional diffusion model that generates molecular linkers for a set of input fragments represented as a 3D atomic point cloud. First, our model generates the size of the prospective linker and then samples initial linker atom types and positions from the

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Linker Size Prediction
Generative Denoising Process
Input Fragments
Graph NN
Linker Size Prob
Ze 12
ro p(zalo
Sample Linker Size
T
= 0.757
t == 0.257
t= 0.17
Figure 1: (Top) Overview of the molecular linker generation process. First, probabilities of linker sizes are computed for the imput fragments. Next, linker atoms are sampled and denoised using our Tragment-conditione equivariant diffusion model. (Bottom) Example of the linker generation process. Linker atoms are highlighted in orange.
normal distribution. Next, the linker atom types and coordinates are iteratively updated using neural network that is conditioned on the input fragments. Eventually, the denoised linker atoms and the input fragment atoms form a single connected molecule, as shown in Figure
DiffLinker enjoys several desirable properties: it is equivariant to translations, rotations, reflections and permutations, it is not limited by the number of input fragments, it does require information on the attachment atoms and generates linkers with no predefined size. Besides, the conditioning mechanism of DiffLinker allows to pass additional information about the surrounding protein pocket atoms, which makes the model applicable in structure-bar drug design applications (Congreve) al. 2005).
We empirically show that DiffLinker is more effective than previous methods in generating chemically-relevant linkers between pairs of fragments. Our method achieves the tftt--of-the-art results in synthetic accessibility and drug-likeness which makes it preferable for using in drug de- sign pipelines. Besides, DiffLLnker remarkably outperforms other methods in the diversity of the generated linkers. We further propose able to successfully link more than two fragments, which cannot be done by the other methods. We also demonstrate that DiffLinker can be conditioned on the target protein pocket: geometric constraints imposed by the surrounding protein atoms and generates molecules that have minimum clashes with the corresponding pockets. To the best of our knowledge, DiffLinkeer is the first method that is not limited by the number of input fragments and accounts the information about pockets. The overall goal of this work is to provide practitioners with an effective tool for molecular linker generation in realistic drug design scenarios.
challenging benchmark and show that method is
model respects
2 RELATED WORK
Molecular linker design has been widely used in the fragment-based drug discovery community
(Sheng & Zhang 2013). Various de novo design methods refer the fragment linking problem
Tschinke & Cohen 1993, Miranker & Karplus 1995 Roe & Kuntz, 1995: Pearlman & Murcko,
1996 [Stahl et al. 2002 Thompson
al. 2008 Ichihara et al., 201 1b. Early fragment linking
methods were based on search in the predefined libraries of linkers (Böhm 1992 Lauri & Bartlett
1994), genetic algorithm, tabu search (Glover 1986) and force field optimization (Dey & Caflisch
2008 Having been successfully used in multiple application cases (Ji al. 2003 Silverman 2209
Sheng & Zhang, 2011), these methods are however computationally expensive and substantially
limited by the available data.
Hence, there has recently been interest in developing learning-based methods for molecular linker design. Yang al. (2020) proposed Syntal.nnker, SMIL.S--aased deep conditional transformer
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neural network that solves sentence completion problem (Zwigg el. 2012). This method inherits the drawbacks of SMILES, which are the absence of 3D structure and the lack of consistency (atoms that are close in the molecule can be far away in the SMILES string). [Imrie et al. (2020) overcome these limitations by introducing an autoregressive model DeLinker and its extension DEVELOP (Imrie et al. 202T) that uses additional pharmacophore information. Although these methods op- crate on 3D molecular conformations, they use very limited geometric information and require input on the attachment atoms of the fragments. Recently, Huang et al. (2022) have proposed an- other autoregressive method 3DLinker that does not require one to specify attachment points and leverages the geometric information to a much greater extent. It makes this approach more relevant for connecting docked fragments. As both DeLinker and 3DLinker are autoregressive models, they are not permutation equivariant which limits their sample efficiency and ability to scale to large molecules (Flesedy& & Zaidi 202T; Rath & Condurache| 2022). Besides, these methods are capable of connecting only pairs of fragments and cannot be easily extended to larger sets of fragments.
Outside of the linker design problem, several recent works proposed denoising diffusion models for molecular data in 3D. Conformer generation methods GeoDiff (Xu et al. 2022) and ConfGF (Shi et al. 2021) condition the model on the adjacency matrix of the molecular graph. Since they have access to the connectivity information, they can compute torsion angles between atoms and optimize them (Jing et al., 2022). Equivariant Diffusion Model (EDM) (Hoogeboom et al., (022) generates 3D molecules from scratch and is able to be conditioned on the predefined scalar prop- erties. Another diffusion model has been recently proposed for designing protein scaffolds given protein motif's (Trpppe et al. (2022). Having the conditional sampling procedure, this model is how- ever trained in an unconditional setup. Finally, Luo al. (2022) proposed model for antibody design which combines discrete diffusion for the molecular graphs and continuous diffusion on the 3D coordinates. The conditioning mechanism proposed this work the closest to ours, but their model is targeted at generating chains of amino acids rather than atomic point clouds.
3 PRELIMINARIES
3.1 DIFFUSION MODELS
Diffusion models (Sohl-Dickstein et al. 2015) are a class of generative methods that consist of a diffusion process, which progressively distorts a data point mapping it to noise, and generative denoising process which approximates the reverse of the diffusion process.
The diffusion process iteratively adds noise to a data point in order to progressively transform it into the Gaussian noise. At a time step t = 0, ...,, T, the conditional distribution of the intermediate data state 2₁ given the previous state is defined by the multivariate normal distribution,
= ((=-,,
(1)
where O₂ E + controls how much signal is retained and a, E RR+ controls how much noise is added. By hypothesis, the full transition model is Markov, so that can be written:
T
q( Zu, 2i,... 2r|z) = ( x:) II (((112(11)
Since the distribution q is normal, a simple formula for the distribution of z, given æ can be derived: a(2, .)) = A(2zloo,,00
where a. au/ar-1 and क: of पه This closed-form expression shows that noise does not need to be added iteratively to x in order to achieve an intermediate state 2t.
Another key property of Gaussian noise is that the reverse of the diffusion process, the true denoising process, also admits a closed- -for solution when conditioned on x:
q(z-11 x 2:) = (2 (z zt), &11,,
(4)
where distribution parameters can be obtained analytically:
0.o%_1
and = 0,01-1
(x 2:)
of
2,
+
of
2
at
(5)
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This formula simply describes that if a diffusion trajectory starts and ends at 27, then the expected value of any intermediate state is an interpolation between x and zT-
The second component of diffusion model is the generative denoising process that learns to invert this trajectory having the data point a unknown. The generative transition distribution is defined as: p(zz11/2) = c(zz-1 | zx),
where d is an approximation of the data point æ computed by a neural network yp. Ho et al. (2020) have empirically shown that it works better to predict the Gaussian noise éç = =p(21, instead, and then estimate the data point è using Equation (3):
Φ = (1/az)= (or/ar)é.
The neural network is trained to maximize evidence lower bound to the likelihood of the data under the model. Up to a prefactor that depends on t. this objective equivalent to the mean squared error between predicted and true noise (Ho al., 2020 Kingma et al., (02TT. We therefore use the simplified objective L(t) = -lle- é,11 that can be optimized by mini-batch gradient descent using an estimator E.-,( -(()).
Finally, once the network is trained, it can be used to sample new data points. For this purpose, one first samples the Gaussian noise: zr (O I). Then, for T, ..... 1, one should iteratively sample 2;-1 p(z(11/z)) and finally sample æ ~ p(z]zo).
3.2 DIFFUSION FOR MOLECULES
Molecule Representation We consider now diffusion models that operate on molecules rep- resented as 3D atomic point clouds. A data point æ, which is an attributed point cloud consisting of M atoms, is represented by atom coordinates r == (r.....,, TMM) MM3 and the correspond- ing feature vectors h (hi,..., har) E IR (g xuf which are one-hot encoded atom types. We will therefore denote point cloud x as a tuple = hr.
Categorical Features Along with continuous atom coordinates, molecular diffusion model has to operate on atom types that are discrete variables. While categorical diffusion models do exist (Hoggeboom et al. 2021 I Austin et al., 2021), we follow a simpler strategy (Hoogeboom et al.) 2022] that based on lifting the atom types to a continuous space: we consider one-hot encoding of the discrete variables, and add Gaussian noise top of it. In the end of the denoising process, once 2g is sampled, the continuous values corresponding to the atom types should be converted back to discrete values. We consider the argmax over the different categories and include it in the final transition from zn to x. For more details on the structure of the final transition distribution p(z) 20) and likelihood computation in this setting, refer the reader to (Hoogeboom et al.) 2022).
Equivariance Processing 3D molecules requires operations that respect data symmetries. In this work, we consider the Fuclidean group E(3) that comprises translations, rotations and reflections of R33 and the orthogonal group 0(3) that includes rotations and reflections of TR3._ A function f: 3 + 3 is Equuvvaviaar if f(R + t) =Rf(x) + for any orthogonal matrix R 3x33 det R = +1, for any translation vector t E R and for any x E R². Note that for simplicity we use notation Rx [Ræ...... RE T A conditional distribution p(æ (y) is E3)-eqqiivariant
if RR + t Ry + t) = p( p(xly) for any y E I... A function f and a distribution p are 0(3)- equivariant if f (Rx)
Rf(xx) and pl Rxx Ry) p(xly) respectively. We call the function translation invariant if f(x + t) f(x).
4 DIFFLINKER: EQUIVARIANT 3D-CONDITIONAL DIFFUSION MODEL FOR MOLECULAR LINKER DESIGN
In this section, we introduce DiffLinkerr a new E(3)-equivariant diffusion model for generating molecular linkers conditioned on 3D fragments. We formulate equivariance requirements for the underlying denoising distributions in Section 4.1 and propose appropriate learnable dynamic function in Section 4.2 We discuss the strategy of sampling the size of a linker and conditioning on protein pockets in Sections 4.3 and 4.4es.ccrppecttvee The full linker generation workflow is schemat- ically represented in Figure and the overview of DiffLinker training and sampling procedures is provided in Algorithms
and correspondingly.
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4.1 EQUIVARIANT 3D- ID-CONIITIONAL DIFFUSION MODEL
Unlike other diffusion models for molecule generation (Hoogeboom et al. 2022] Xu et al. 2022). our method is conditioned three-dimensional data. More specifically, assume that each point cloud æ has a corresponding context u, which is another point cloud consisting of all input fragments and (optionally) protein pocket atoms that remain unchanged throughout the diffusion and denoising processes, as shown in Figure ! Hence, we consider the generative process from Equation (6) to operate on point cloud æ while being conditioned on the fixed corresponding context:
p(zt-1|zz, u) q= 21), where a (1/ax) (01/ar))((z u, u,
(8)
The presence of 3D context puts additional requirements on the generative process as it should be cquivariant to its transformations.
Consider a prior noise distribution p(zr|u)= N(rr; f(u): I) and transition dis- tributions p(zt-1/zı, u) = g(21-1(i,2 2t). where is an isotropic Gaussian, f R Rx3 4 R3 is function operating on 3D point clouds, and & is an approximation computed by the neural network Let the conditional denoising probabilistic model p be a Markov chain
Proposition 1
according to Equation defined as
T
p(zo; 21,..., zr(u) = p(zr u)) II P[zZ-1122, u))
If f is -quuvvavvrr and up equivariant to joint Ormmstaaasssosaaaioon of z, and u, then pl [0) u) is 0(3) -qquivrru..
The choice of the function f highly depends on the problem being solved and the available priors. In our experiments, we consider two cases. First, following (Imrie et al. 2020),, we make use of the information about atoms that should be connected by the linker. We call these atoms anchors and define f(u) ) as the anchors' center of mass. However, in real-world scenarios it is unlikely to know which atoms should be the anchors. In this case we define f(u) the center of mass of the whole context u.
We note that the probabilistic model p is equívariant to translations, shown in Appendix A.3 To overcome this issue and make the generative process independent of translations, we construct the network yp be additionally translation invariant. Then, instead of sampling the initial noise from (f)) I) we center the data at f(u) and sample from (00 I).
4.2 EQUIVARIANT GRAPH NEURAL NETWORK
The learnable function to that models the dynamics of the diffusion model is implemented as a modified E(3)-equivariant graph neural network (EGNN) (Satorras et al.] 2021). Its input is the noisy version of the linker 2, at time f and the context u. These two parts are modeled as a single fully-connected graph where nodes include atom types, time f, fragment flags and (optionally) anchor flags. The predicted noise includes coordinate and feature components: é j e² In order to make function şp invariant to translations, we subtract the initial coordinates from the coordinate component of the predicted noise following Hoogeboom et al. (2022):
represented by coordinates and feature vectors h that
è [2" en1 ((zz, t) EGNN(*,, u, - [z] 0].
(10)
EGNN consists of the composition of Equivariant Graph Convolutional Layers pll h EGCL/r', h²] which are defined as follows:
m:j = dh (hi 11 dby), h, = (h, Σ Em).. r = =r] + Doa(t' h,,i),
(1I)
where dy = | r' and de, dy are learnable functions parametrized by fully connected neural networks (see Appendix A.5 for details).
The latter update for the node coordinates is computed by the learnable function &mnt- Note that our graph includes both noisy linker 2, and fixed context u, and &o is intended predict the noise
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that should be subtracted from the coordinates and features of 2,. Therefore, it is natural to keep the context coordinates unchanged when computing dynamics and to apply non-zers displacements only to the linker part at each EGCL step. Hence, model the node displacements follows,
Eitt
- deft o. (h', 1/y, diy) if node belongs to the point cloud zt. (12) if node belongs to the context u,
Qvet(r', 11, =
where 6, is a learnable function parametrized by a fully connected neural network.
The equivariance of convolutional layers is achieved by construction. The messages de and the node updates dn depend only on scalar node features and distances between nodes that are E(3)-invariant. Coordinate updates &vel additionally depend linearly on the difference between coordinate vectors rf and ry, which makes them Eequevarra.
After the sequence of EGCLS applied, we have an updated graph with new node coordinates i [w", _ and new node features h = ia, 2f). Since we are interested only in the linker-related part, we discard the coordinates and features of context nodes and consider the tuple [2], 2/) to be
the EGNN output.
4.3 LINKER SIZE PREDICTION
To predict the size of the missing linker between a set of fragments, we represent fragments as a fully-connected graph with one-hot encoded atom types node features and distances between nodes as edge features. From this, a separately trained GNN (see Appendix[A.6fftt details) produces probabilities for the linker size. Our assumption is that relative fragment positions and orientations along with atom types contain all the information essential for predicting most likely size of the prospective linker. When generating linker, we first sample its size with the predicted probabilities from the categorical distribution over the list of linker sizes seen in the training data, as shown in Figure
4.4 PROTEIN POCKET CONDITIONING
In real-world fragment-based drug design applications. it often occurs that fragments are selected and docked into a target protein pocket (Igashov al. 2022). To propose a drug candidate molecule, the fragments have to be linked. When generating the linker, one should take the surrounding pocket into account and construct linker that has no clashes with protein pocket atoms (in other words, the configuration of the linker and pocket atoms should be physically-realiste strength high. To add pocket conditioning to DiffLinker, we represent a protein pocket as an atomic point cloud and consider it as part of the context (cf. Scction[.11) We also extend node features with an additional binary flag marking atoms that belong to the protein pocket. Finally, as the new context point cloud contains much more atoms, modify the joint representation of the data point z, and the context u that are passed to the neural network Instead of considering fully-connected graphs, we assign edges between nodes based 4 Å distance cutoff it makes the resulting graphs less dense and counterbalances the increase in the number of nodes.
and keeps the binding
5 EXPERIMENTS
5.1 DATASETS
ZINC We follow Imrie et al. (020 and consider subset of 250.000 molecules randomly se- et al. (2018) from ZINC database (Irwin & Shoichet) 2005). First,
lected by we generate 3D conformers using RDKit (Landrum) (013) and deñnn a reference 3D structure for each molecule by selecting the lowest energy conformation. Then, these molecules are fragmented by enumerating all double cuts of acyclic single bonds that are not within functional groups. The resulting splits are filtered by the number of atoms in the linker and fragments, synthetic accessibil- ity (Ertl & Schuffenhauer 2009), ring aromaticity, and pan-assay interference compounds (PAINS) therefore result in various combinations of two fragments with a linker between. The resulting dataset is randomly split into train (438,610 examples), validation (400 examples), and test (400 examples) sets.
(Baell & Holloway| 2010)) criteria. One molecule

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Input Fragments
Pocket- Conditioned Samples
Numbers of clathes in Diftt different pocket representations
DiffLinker inker samples
tan t
50
40
Nfmber clahhes
True Molecule
Unconditioned Samples
20
Beference
Foll
Backbone
Figure 2: (Left) Examples linkers sampled by DiffLinker conditioned on pocket atoms (top row) and unconditioned (bottom row). Linkers sampled by the unconditioned model have multiple clashes with the protein pocket. (Right) Distribution of numbers of clashes reference test molecules and samples from three DiffLinker models differently conditioned (or unconditioned) pocket atoms.
CASF Another evaluation benchmark used by Imrie al. (2020) taken from the CASF-2016 dataset (Su et al., 2018). In contrast to ZINC, where molecule conformers were generated compu- tationally, CASF includes experimentally verified 3D conformations. Following the same prepro- cessing procedures as for the ZINC dataset, [mrre et al. (2020) obtained an additional test set of 309 examples, which we use our work.
GEOM ZINC and CASF datasets used in previous works only contain pairs of fragments. How- ever, real-world applications often require connecting more than two fragments with one or more linkers [Igashov et al., 2022). To address this casc, construct
dataset based GEOM ments with one two linkers connecting them. To achieve such splits,
molccules (Axclrod & Jemeez-Bombarelii 2022) which we decompose into three or more frag-
RDKit implemen-
tations of two fragmentation techniques an MMPA-bas based algorithm (Dossetter et al.) (013)) and BRICS (Degen et al. 2008) and combine results removing duplicates. Overall, we obtain 41, 11,907 molecules and 285, 140 fragmentations that are randomly split in train (282,602 examples), valida- tion (1,250 examples) and test (1,288 examples) sets.
Pockets Dataset In order to assess the ability of DiffLinker to generate valid linkers given addi- tional information about protein pockets, we use the protein-ligand dataset curated by Schneuing et al. 22022) from Binding MOAD (Hu et al. 2005). To define pockets, we consider amino acids any atom of the ligand. All atoms belonging these
that have at least one atom closer than residues constitute the pocket. We split molecules into fragments using RDKit's implementation of MMPA-based algorithm (Dossetter et al. (013).. randomly split the resulting data into train (185,678 examples), validation (490 examples) and test (566 examples) sets taking into account Enzyme Commission (EC) numbers of the proteins.
5.2 EVALUATION
Metrics First, report several chemical properties of the generated molecules that espe- cially important in drug design applications: average quantitative estimation of drug-likeness (QED) (Bickerton et 2012), average synthetic accessibility (SA) (Ertl & Schuffenhaueri 2009) and aver- age number of rings in the linker. Next, following Imrie et al. (2020). we measure validity, unique- ness and novelty of the samples. We then determine if the generated linkers are consistent with the 2D filters used to produce the ZINC training set. These filters are explained in detail in Appendix A.II addition, we record the percentage of the original molecules that were recovered by the generation process. To compare the 3D shapes of the sampled and ground-truth molecules, we esti- mate the Root Mean Squared Deviation (RMSD) between the generated and real linker coordinates in the cases where true molecules recovered. We also compute the SCROKи metric that evalu- ates the geometric and chemical similarity between the ground -ouuhhtruthhh and generated molecules (Putta) al. 2005 Landrum et al. 2006).

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Table 1: Performance metrics ZINC, CASF and GEOM test sets. The first three metrics assess the chemical relevance of the generated molecules. The last three metrics evaluate the standard generative properties of the methods.
Method
QED
SA
Rings t
Valid,
Unique, %
Novel,
DeLinker + ConfVAE MMFF
0.64
3.11
0.21
98.3
44.2
47.1
3DLinker (given anchors)
0.65
3.11
0.23
99.3
29.0
41.2
ZINC
3DLinker
0.65
3.14
0.24
71.5
29.2
41.9
DiffLinker
0.68
3.01
0.25
93.8
24.0
30.3
Diffl Linker (given anchors)
0.68
3.03
0.26
97.6
22.7
32.4
Diffl linker (sampled size)
0.65
3.19
0.32
90.6
51.4
42.9
Difflinker (given anchors, sampled size)
0.65
3.24
0.36
94.8
50.9
47.7
DeLinker ConíVAE MMFF
0.35
4.05
0.35
95.7
51.6
55.6
CASF
DiffLinker
0.41
4.00
0.34
85.3
40.5
41.8
DiffLinker (given anchors)
0.40
4.03
0.38
90.2
37.3
48.4
DiftLinker inner (sampled size)
0.40
4.06
0.30
63.7
60.0
49.3
DiftLinker (given anchors, sampled size)
0.40
4.10
0.38
68
57.1
56.9
3DLinkka
0.36
3.56
0.00
16.3
73.7
GBOM
DiffLinker
0.48
93.5
36.7
70.7
DiffLinker (given unchors)
0.49
93.4
37.3
DiffLinker (sampled size)
0.46
3,24
87.4
63.1
DiffLinker (given anchors, sampled size)
0.47
3.30
88.8
64.4
Table 2: Metrics assessing the ability of the methods generate molecules that are chemically and geometrically similar to the reference ones.
SCRDka
Method
Filters, Recovery,
RMSD
>>
>>
>
Avg.
DeLinker ConfVAE MMFF
84.88
80.2
3.73
0.61
0.09
0.49
3DI .inker (given anchors)
84.24
94.0
0.10
99.86
97.05
63.78
0.92
ZINC
3DLinker
83.72
93.5
0.11
99.83
96.22
63.63
0.92
DiffLinker
99.72
94.62
67.85
DiffLinker (given anchors)
84.36
87.2
0.32
99.96
97.02
71.73
0.94
Dif'Linker (sampled size)
87.98
70.7
0.34
99.37
90.21
51.90
0.90
Difflinker (given anchors, sampled size)
84.76
77.5
0.35
99.67
95.04
56.35
0.91
DeLinker ConfVAE MMFF
77.25
52.8
11.89
1.24
0.19
0.03
0.39
CASP
DiffI
87.73
42.8
0.44
92.17
79.62
50.14
DiffLinker (given anchors)
82.37
50.2
0.37
95.07
89.05
60.63
0.86
DiffLinker (sampled size)
89.27
40.5
0.34
92.83
79.12
43 43.99
0.82
DiffLinker (given anchors, sampled size)
94.52
87.90
55.25
3DLinker
94.10
60.63
29,30
8.22
0.72
GEOM
Diml inker
47.68
85.6
0.12
95.53
87.87
71.36
0.92
Dif'Linker (given anchors)
49.70
85.2
0.12
95.12
86.79
69.91
0.92
DiffLinker (sampled size)
55.68
68.8
91.74
81.47
57.11
0.88
DiffLinker (given anchors, sampled sizc)
57.90
67.9
90.95
79.57
54.78
0.87
Baselines We compare method with DeLinker (Imrie 2020 and 3DLinker (Huang al., 2022) on the ZINC test set and with DeLinker on the CASF dataset. Besides, we adjust 3DLinker connect more than two fragments (see Appendix Aapendix[A.71ffe (A.7for details) and evaluate its performane on the GEOM dataset. To obtain 3D conformations for the molecules generated by DeLinker on ZINC and CASF, we apply a pre-trained ConfVAE (Xu et al., (02T) followed by force field relaxation procedure. For all methods including ours, we generate linkers with the ground-truth size unless explicitly noted otherwise. To obtain SMILES representations of atomic point clouds generated by our models, utilize OpenBabel (O Boyle et al. 2011) compute covalent bonds between atoms. We also use OpenBabel to rebuild covalent bonds for the molecules in test sets in order to correctly compute the recovery rate, RMSD and SCRDKI scores for models. In ZINC and CASF experiments, we sample 250 linkers for each input pair of fragments. For the GEOM dataset and in experiments with pocket conditioning, sample 100 linkers for each input set of fragments.
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5.3 RESULTS
ZINC and CASF While our models have much greater flexibility and applicability in more appli- cations as we show below, they also outperform other methods on standard benchmarks ZINC and CASF in terms of chemical relevance of the generated molecules. As shown in Table molecules sampled by DiffLinker are more synthetically accessible and demonstrate higher drug-likeness, which is especially important for drug design applications. Besides, our models generate linkers containing more rings. Moreover, our molecules usually share higher chemical and geometric sim- ilarity with the reference molecules as demonstrated by the SCRDKu scores in Table 2 In terms of validity, our models perform on par with the other methods. Note that both autoregressive ap- proaches employ valency rules at each generation step explicitly, while our model is shown to be able to learn these rules from the data. Remarkably, the validity of the reference molecules from CASF with covalent bonds computed by OpenBabel is 92.2% while model generated molecules with 90.2% validity. Notably, sampling the size of the linker significantly improves novelty and uniqueness of the generated linkers without significant degradation of the most important metries. Examples of linkers generated by DiffLinker for different input fragments are provided in Figure 6
Multiple Fragments The major advantage of Diffl innke compared to recently proposed autore- gressive models DeLinker and 3DLinker is one-shot generation of the whole linker between arbitrary amounts of fragments. This overcomes the limitation of DeLinker and 3DLinker, which can only link two fragments at a time. Although these autoregressive models can be adjusted to connect pairs of fragments iteratively while growing the molecule, the full context cannot be taken into ac- count in this case. Therefore, suboptimal solutions are more likely to be produced. To illustrate this difference, we adapted 3DLinker to iteratively connect pairs of fragments in molecules where more than two fragments should be connected. As shown in Table [] 3DLinker fails to construct valid molecules in almost 84% of cases and cannot recover any reference molecule while, despite the higher complexity of linkers in this dataset, models achieve 93% validity and recover more than 85% of the reference molecules. Besides, molecules generated by 3DLinker have no rings in the linkers, have substantially lower QED and are much harder to synthesize. Examples of linkers generated by DiftLinker for different input fragments are provided in Figure 5 An example of the DiffLinker sampling process for the molecule from the GEOM dataset is shown in Figure [
Protein Pocket Conditioning To illustrate the ability of DiffLLnkerr to take surrounding pockets into account, we trained three models on the Pockets Dataset: these are respectively conditioned on the full-atomic pocket representation, conditioned on the pocket backbone atoms and unconditioned. Besides the standard metrics reported in Tables g and [ we also compute the number of clashes between generated molecules and surrounding pockets. We say that there is clash between two atoms if the distance between them is lower than the sum of their van der Waals radii. As shown in Figure 2 the model conditioned on the full-atomic pocket representation generates molecules with almost the same amount of clashes (in average 7 clashes per molecule) as the reference complexes from the test set (in average 6 clashes per molecule). There is clear trend on the number of clashes depending on the amount of information about pockets DiffLinker is conditioned on: the model conditioned pocket backbone atoms generates molecules with 14 clashes in average, and the unconditioned model produces molecules with 21 clashes in average.
CONCLUSION
In this work, we introduced DiffLLikeer, a new E(3)-equivariant 3D-conditional diffusion model for molecular linker design. DiffLinker designs realistic molecules from a set of disconnected fragments by generating linker, i.e., an atomic point cloud that interconnects the input fragments. While previous methods were only capable to connect pairs of fragments, DiffLinker naturally scales to an arbitrary number of fragments. Our method does not require to specify the attachment points of the fragments and predicts the distribution of linker size from the fragments. We show that the proposed method outperforms other models on standard benchmarks and produces more molecules. Besides, we demonstrate that our model can be conditioned on protein pockets and generate linkers with a minimum number of clashes. We believe that method will accelerate the development of prospective drug candidates and has the potential to become widely used in the fragment- based drug design community.
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A
APPENDIX
Algorithm 1 Training
Input: linker æ, context u, neural network s Sample ~ U(O,..., T), c ~ IO,,) 21 6 a,æ oiCs
é + p(2,, u,
Minimize F Ex 11
Algorithm/ Sampling
Input: context u, neural network
Center context u at f(u)
Sample 2T ~ (0, I)
for t in T, T
... 1:
Sample E, I)
é - (z, u.t)
Zt-1 - (1/oz) 2t / è -
end for Sample x plæ u)
A.1 APPLICATIONS
There are several major drug discovery directions in which one of highly relevant or even preferred strategies is design linker for fragments that are placed in the space and have fixed positions.
Fragment Based Drug Discovery (FBDD) By analogy with classical drug discovery methods, one of the common strategies in FBDD is operate on fragments that strongly interact with the target proteins. First, strongly binding fragments are defined and characterized (using high-throughhut screening followed by X-ray / /MRR, or virtual screening and docking). As a result, the exact location and orientation of the fragments in which they bind strongly to the parts of the protein pocket is defined. The next step is to find linker between the fragments that preserves positions and thus the binding strength of the fragments (preferably, adding the linker will boost the binding strength of the whole molecule) Bancet et al. (020). There has been reported a range of successful works which the starting point for a linker design was crystal structure of protein with fragments bound to it [Bancet et al., 2020). To name a few, inhibitors for CK2 (De Fusco et al. 2017). LDH-A (Kohlmann let al. 2013) and Dot1L (Mobitz et al. (017), which are proteins playing crucial roles in progress of various cancers, were designed by linking the fragments that were experimentally observed in a bound state with the corresponding targets.
Proteolysis targeting chimera (PROTAC) PROTAC is a heterohifunctional small molecule de- signed for stimulating degradation of a target protein by connecting it to an E3-ligase. PROTAC consists of two ligands joined by a linker: one ligand recruits and binds a target protein while the other recruits and binds E3 ubiquitin ligase (Békés et al. 2022). For designing PROTACs, one of possible strategies is to dock two proteins (with ligands bound to them) to explore a favorable con- formation of the prospective tertiary complex. This information about the initial docking pose of the proteins and exact positions of bound fragments is further used for designing linker that will stabilize the whole complex (Bai et al. 2021 Farnaby et al. 2019).
Scaffold hopping Scaffold hopping is strategy for designing novel compounds by replacing the central core structure of the known molecule. As shown by Sun et al.] (2012), various scaffold- hopping strategies rely on the experimental 3D data of the initial compound bound to a target com- plex: the information about the geometry of the initial bound molecule important for altering its core with the increase of the binding affinity, potency or selectivity of the whole molecule. In such a case, scaffold- scaffold-hopping of the bound molecule can be considered as a linking problem of several disconnected fragments with fixed known 3D coordinates.
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A.2 PROOF OF PROPOSITION
-eueequuivee of function / and the fact that is isotropic Gaussian distribution implies O(3)- equivariance of the prior distribution:
p[R Ru) = N (N(eRRRr 1) = (N(RRr(RR())) = -<rr)))))((<r( = p(=r/(()))))
Likewise, -quuvvaaiee of function yp and Equation (8) imply O()--equivariance of all transition probabilities p(z(-1|=;; u))
To obtain the distribution p(zolu) of data point zo. we can consider joint distribution p(zo. 21, : zr/u) and marginalize it by 21...r
T-1 p(zolu) / p(zo, 21.............................. 2T zr/u)dz1..1 p(zrlu) II p(z z ())..)
Having prior and all transition distributions equivariant, is now trivial show -33-eqquivariance of p(z))()) u):
pl (Rz Ru) (RZr Ru) II (Rz, R2 Ru)dzi1..r
T-1
/ p(zT|u)
p( R2, R2141, Ru)dzı...r (equivariant prior ((zr(u))
T-1
/ p(zrlu) II
p z 1 u)dzi... (equivariant transition kernels p(z,1 21+1, u))
Am0
21,..., (/) p(zulu))
A.3 PROBLEM WITH TRANSLATIONS
Consider transition probability p(z1-1|Z(;14 = q(21-1100, 21). p(z1-1/21, u) means that
Translation equivariance of
p t/ + u t) p(= Vt R
(13)
More precisely,
M(z + +t; tt t) - (i (1
(14)
where
a,o7-1
a0-10? a,
Au( u) = (á, xz) =
(15)
of
of
and
- 2z
1
Tt p(Z u,
(16)
0t
az
Therefore, the mean of this distribution can written as:
(z( u)
- 2,
p(2), u,t).
(17)
Neural network jo is translation invariant meaning that cp t, s02r, u,
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It means that:
1
oi
(x + t, +t)
(2t +
pzz + t, u - t,t)
(18)
aoo
1
di
(zz
t)
gl U,,)
(19)
1
0
1 -1
2t
plzz; u,
(20)
at
0
at
ji, u) +
(21)
0t
= - u) Aet.
(22)
So we see that (zz, u) is cquivariant to translations, however input and output translations are not equal because À * 1. It means that equivariance of distributions from Equations [3) and (14 does not hold. More formally,
p(z-1 + 1/2 + 1, 1) = N (2.1 u) Xt,5, = -2 u) + (A 1)t, s
(23)
(24)
* (2)
(25)
We can also write that
p(z-11 + 1/ + u p (z + -1- A)1/2, u) # p (2))
(26)
A.4 DIFFUSION MODEL
We trained all DiffLLinee I models with T 500 diffusion steps using polynomial noise schedule:
2s) (1 (t/T)²),
(27)
where 8 10-5 is a precision value that helps to avoid numerically unstable situations (Hoggeboom
al. 2022).
Sampling For all the experiments discussed in the main text, we sampled with the same number of denoising steps T 500 used in training. Sampling time for all the datasets is provided in Table Although the time reported in Table i more than affordable for applying our method in practice, we explored the capability of DiffLinker sample even faster without performance loss. Following Nichol & Dhariwal (202T), we conducted additional evaluation of DiffLinker with the reduced number of denoising steps T 500 in sampling, considering T T/2, T/5, T/10, T/20. 7/50 and 7/100 values. Figure 3 shows how performance metrics obtained on ZINC test set depend on the number of denoising steps performed sampling. In all cases we used DiffLinker pretrained on ZINC with T 500 denoising steps. As shown in Figure 3 our model is robust to significant reduction of the number of denoising steps in sampling resulting in 10-fold gain in sampling speed reduce the sampling speed from 0.365
without any performance degradation. Effectively, to 0.036 seconds per molecule with no significant performance metrics loss.
A.5 DYNAMICS
EGNN takes as input graph of atoms belonging to the linker 2, and its context u represented by feature vectors h; E Ri and coordinates R33 Feature vector h, consists of atom types, fragments flag and time step If anchors are known, additionally anchor flag passed. the model is conditioned on the protein pocket, additionally pocket flag is passed.
First, atom features are passed to the encoder: h; - Linear(in, nf) - hh.
Next, as discussed in Section L Equivariant Graph Convolutional Layers (EGCL) are sequen- tially applied. Learnable components EGCL de, dR, dr are implemented as neural networks that include fully-connected layers (FC), batch normalization layers (BN) and activations SiLU,
Diffl inker code is availahle at: https:://noonyooul 4open. sciercee cancee//// Ffliikker FFliniiineeeee
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Validity, recovery and RMSD for different number of denoising steps
Figure 3: Dependency of validity, re- covery and RMSD on the number of denoising steps in sampling shows that DiffLinker is robust to reducing the number of denoising steps. The robust- ness of Diffl Linker allows for 10-fold gain in sampling speed without any performance degradation. For all ex- periments we used DiffLinker trained ZINC with T 500 steps and performed evaluation on ZINC test set sampling 250 linkers for each input set of fragments.
90
0.37
validity, recovery (%6
80
T=
0.36
70
0.35
60
0.34
50
2
8
log
Table 3: Sampling time for different datasets (with T 500 denoising steps).
Experiments were
performed on single Tesla VI00-PCIE-32GB GPU.
dataset
batch size
time per molecule, s
ZINC
128
0.37
CASF
128
0.63
Multi-FFaammet
64
0.37
Pockets (full atomic representation)
32
0.46
Pockets (backbone representation)
32
0.21
Pockets (no pocket)
32
0.12
Message de: takes a pair of node embeddings h, and h', and the squared distance d? = -||r r1
between these nodes and outputs message maj E Rat:
concat(h!, h/y. dBy) (FC(2- of nf) + SiLU + FC(nf, nf - SiLU) + mij
Features update ds; takes as input node embedding h and its aggregated message m, E; mi and returns the updated node embedding:
concat[h/, m.] + (FC(2 nf, nf) BN + SiLU + FC(nf, nf) - BN + add(h!) - h
Coordinates update d,: takes the same input as de and outputs scalar value
concat(h,, h/, df. + {FC(2 nf I, nf) SiLU FC(nf, nf) + SiLU - FC(nf, 1) + output
Training We trained separate models for ZINC, Multi-Frag and Pocket datasets. Hyper parameters of the models and average time required for training one epoch are provided in Table All models were trained on single Tesla V100-PCIE-32GB GPU using Adam with learning rate 2 10-1 and weight decay 10-13.
A.6 LINKER SIZE PREDICTION
Graph neural network for predicting probabilities of the number of atoms in the prospective linker for given set of fragments takes input a fully-connected graph of atoms belonging to the fragments represented by feature vectors h, E Rin and inter-atomic squared distances d?, = ||r: - 1 and
outputs vector of probabilities corresponding to the predefined linker sizes p E [0, 1
First, node embeddings computed: h; Linear(in, nf) h
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Table 4: Hyper parameters of EGNN models trained on ZINC, Multi--raagment and Pocket datasets.
dataset
given anchors
nf
batch size
epochs
time per epoch, min
ZINC
128
128
300
15.2
ZINC
yes
128
128
300
17.1
Multi- Falii-Fragment
no
128
839
10.6
Multi- Muli--Fragment
yes
128
128
1240
10.1
Pocket (full atomic representation)
yes
128
32
420
20.3
Pocket (backbone representation)
yes
128
32
620
10.7
Pocket (no pocket)
yes
128
32
670
8.4
Table 5: Hyper parameters of SizeGNN models trained ZINC and Multi-Fragment datasets.
dataset
hid
out
batch size
epochs
time per epoch, min
ZINC
8
256
10
5
256
10.6 9.2
Multi-Fragment
9
256
33
5
256
119
Next, sequence of L Graph Convolutional Layers (GCL) is applied. Learnable components of GCL Øe, @% are implemented in the same way as for EGNN.
Finally, node embeddings h are projected onto Ro aggregated and normalized resulting in the vector of label probabilities:
h (FC(nf, out) -> Mean Softmax p
Training We trained two models for ZINC and Multi-Frag datasets. Hyper parameters of the models and average time required training one epoch are provided Table Both models were trained using Adam with learning rate 10-4 and weight decay 10 13, Both models were trained on a single Tesla V100-PCIE-32GI GPU.
A.7 3DLINKER ON GEOM DATASET
In order to run 3DLinker on GEOM dataset, we had to additionally filter the original test set con- sisting of 1,288 input fragment sets and remove examples with more than disconnected fragments. For the remainder test set that included 1,170 input fragment triplets, we ran 3DLiinkr twice: first, connect two randomly selected fragments (10 samples per fragment pair) and then connect the resulting compound with the third fragment (10 samples per input). In both steps, we used the half of the original linker size. Overall, we obtained 100 samples for each input fragment triplet. We used pre-trained 3DLinker model available at wttttts://// htt///githul com wwt. www.i/annnnnnnn........ The results are provided Tables and
A.8 EVALUATION DETAILS
The principal difference between our and other methods is that we generate 3D point cloud of atoms that should be further connected with covalent bonds while other methods generate covalent bonds along with atom types. We emphasize that both DeLinker and 3DLinker employ valency rules each generation step which makes is much easier to achieve high validity of samples. In our case, DiffLinker learns these chemical rules from the data and places atoms at the relevant distances from each other. Since the output of DiffLinker is 3D point cloud, need additionally compute covalent bonds between pairs of atoms based on their types and pairwise distances. For doing that, OpenBabel (O' Boyle al. 2011).
To be consistent in the evaluation methodology, we recomputed covalent bonds using OpenBabel for all molecules in ZINC and CASF test sets. Next, for each updated molecule, obtained linkers by removing irrelevant atoms and saved the resulting molecules and fragments in SDF and SMILES for- mats. Molecules saved in SDF format were considered as ground truth and used for 3D comparison
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Table 6: QED, SA and number of rings for molecules in training, validation and test datasets.
Dataset
QED f
SA +
Rings f
Train
0.734
2.986
0.213
ZINC
Validation
0.736
2.930
0.207
Test
0.729
2.944
0.274
GEOM
Train
0,546
2.720
0,901
Validation
0.556
2.711
0.853
Test
0.537
2.735
0.849
Pockets
Train
0.330
5.330
0.926
Validation
0.418
4.783
0.763
Test
0.356
5.445
1.272
CASF
0.481
3.572
0.271
(for computing RMSD and SCRDKH metrics). Molecules and linkers saved in SMILES format were considered as ground truth and used for 2D comparison (novelty and recovery rates). To evaluate other methods, we used original SMILES representations.
Our samples For each generated point cloud, we computed covalent bonds with OpenBabel, and extracted the largest connected component. Next, we obtained linker by matching the generated molecule with the corresponding fragments (computed with OpenBabel as explained above) and re- moving irrelevant atoms. Finally, we kekulized the resulting linker and saved the generated molecule with recomputed covalent bonds and the corresponding linker in SDF and SMILES formats.
Metries To compute validity, we apply sanitization and additionally check that the molecule contains all atoms from fragments. For all other metrics, we consider only subset of valid samples. To compute novelty, we first preprocess SMILES of the linker by removing stereo- chemistry and canonicalizing tautomer SMILES. Then, we count how many of the resulting gen- erated linker SMILES were represented in the training set. To compute uniqueness, we com- pare SMILES of whole molecules and count number of unique molecules sampled for each input pair of fragments. To compute recovery, pled for a given pair of fragments with SMILES of the corresponding ground-truth molecule. Before comparison,
compare SMILES of each molecule sam-
remove hydrogens and stercochemistry from molecules. To compute RMSD, we consider only recovered molecules and align them with the corresponding ground- truth molecules using RDKit function rdkit. Chem. rdMo 1 Al igg which returns the optimal RMSD for aligning two molecules. To compute quantitative drug-likeness (QED) (Bickerton et al., 2012), we used RDKit function rakit Chem. QED qed. To compute synthetic accessibility, we used function calculateScore provided by Ertl & Schuffenhauer (2009) in the RDKit- compatible package sascorer. py. For calculating the number of rings, we used RDKit function rdkit Chem. rdMo Descriptors. CalcNumRingss Table [ provides mean QED, SA and number rings computed for molecules in training, validation and test datasets.
A.9 ADDITIONAL RESULTS
As explained Section A.7 we had additionally filter the original GEOM test set consisting of 1, 1,288 input fragment sets and remove examples with more than 3 disconnected fragments. For each input fragment set, we obtained 100 samples with 3DLinker. For consistency we report DiffLinker performance on GEOM in Tables and computed in the same setting: 1,170 input examples and 100 samples per input. DiffLinker results computed on the full GEOM test set with 250 samples per input are provided in Tables 8 and
DiffLinker results on the Pockets test set with 100 samples per input are provided in Tables 8 and 9 as well. We note that train/test split of the Pockets dataset performed solely based PDB- codes of the protein- ligand complexes and EC numbers of proteins. In the resulting dataset used for evaluation, therefore have 17 molecules that also represented in the training set but bound to
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different proteins. For the full picture, alternatively provide another reduced test set which does not contain molecules from the training set at all. It includes 453 examples from the initial test set. We provide evaluation metries obtained on the reduced test set in Table 10
A.10 INPAINTING
Table 7: Comparison of DifTLinker trained in 3D-conditioning setting with DiflLinker trained in in- painting setting. Evaluation was performed on ZINC validation set. For each input pair of fragments we sampled 50 linkers.
Method
Valid, %
Recovered, %
RMSD +
SCRDKi 1
QED 1
SA |
Rings 1
3DDconditioning Inpainting
91.0
66.3
0.31
0.92
0.71
3.02
0.21
65.5
65.9
0.79
0.89
0.73
3.09
0.25
An alternative way of conditioned linker generation with diffusion models is impainting strategy (Lugmayr al.) 2022), in which the model is trained to denoise a full molecule, while the infer- ence step the known part of the molecule is generated using the true denoising process. Inpainting strategy for molecular linker design can be easily implemented using vanilla EDM (Hoogeboom et al. 2022) with a slight tweak of the sampling function.
This approach was the first that we tried in our experiments. However, models trained in 3D- conditioning setting, explained in Section 4.1 significantly outperformed the inpainting models, In Table 7 we provide a comparison of DifiLinker trained in -D-conniti setting with DiffLinker trained in inpainting setting. Both models have identical architectures and were trained with identi- cal hyper parameters. For evaluation we used validation ZINC set (400 examples) and sampled 50 linkers for every pair of input fragments.
As shown in Table 7 using 3D-conditioned model reduces the number of invalid molecules (i.e., chemically incorrect with disconnected fragments) by 25%. Besides, inpainting model gener- ates molecules with more than 2-fold higher RMSD. Even though the chemical properties of the molecules generated by both methods are comparable, significantly lower validity indicates that inpainting approach is suboptimal in the molecular linker design task.
A.11 2D FILTERS
2D Filters used by Imric et al. (2020) for constructing ZINC and CASF datasets include synthetic accessibility (Ertl & Schuffenhaueri 2009), ring aromaticity (RA), and pan-assay interference com- pounds (PAINS) (Baell & Holloway| 2010) criteria. RA controls the correctness of the covalent bond orders in the rings of the a linker and PAINS checks a molecule does not contain compounds that often give false-positive results in high-throughput screens (Baell & Holloway| 2010). Even though we used the same datasets as in Imrie et al. (2020) that were created using all three filters, however modify the metric "Passed 2D Filters" by removing SA from it. Instead introduce our own SA-based metric that we report separately.
Problem with SA filter used by Imrie et al. 2020) and Huang et al. (2022) The molecule is considered to pass the synthetic accessibility filter if its SA-score is lower than SA-score of the corresponding pair of fragments. Even though our models performed on par better than DeLinker and 3DLinker according to all other metrics, almost all molecules generated by our models did not pass SA-filter. We investigated this issue and figured out that SMILES of fragments passed to SA filter by DeLinker and 3DLinker contained dummy atoms representing anchors. These atoms did not have any atom type assigned and therefore such molecules were considered hard to be synthesised. For almost all molecules in the test set SA-scores of fragments with dummy atoms were higher than SA-scores of the whole molecules. However, for most of fragments without dummy atoms SA-scores were much lower. Figure 4 shows 4 examples of molecules and fragments with and without dummy atoms and the corresponding SA-scores. We can conclude that SA-filter proposed by authors of DeLinker shows nothing but the fact that molecules with unknown atoms hard to be synthesized. Therefore, we considered this metric to be irrelevant and excluded it from our report. Instead, we report average Synthetic Accessibility score of full generated molecules.
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Table 8: Performance metrics on GEOM (250 samples, full test set) and Pockets test set (100 sam- ples). The first three metrics assess the chemical relevance of the generated molecules. The last three metrics evaluate the standard generative properties of the methods.
Method
QED
SA !
# Rings f
Valid, %
Unique, %
Novel, %
Difflinnkrr
0.48
2.99
0.75
93.4
31.6
68.7
GEOM
Diffl Linker (given anchors)
0.49
3.01
0.79
93.5
32.1
68.5
Diffl_inken (sampled size)
0.45
3.27
0.76
87.1
57.3
76.2
DiffLinker (given anchors, sampled size)
0.46
3.33
0.84
88.6
58.2
76.2
Pockets
DiffLinkee (pocket atoms)
0.45
3.89
1.06
88.7
62.5
73.1
DiffLinker (pocket backbone)
0.45
3.77
0.95
90.4
60.9
72.8
DifflLiner (unconditioned)
0.45
3,83
1.10
93.6
61.1
74.9
Table 9: Metrics assessing the ability of the methods to generate molecules that are chemically and geometrically similar to the reference ones. For GEOM (250 samples, full test set) and Pockets test set (100 samples).
SCRDka
Method
2D Fillers, %
Recovery,
RMSD L
0.7
> 0.8
>> 0.9
Avg
Diffl linker
49.47
89.1
0.11
95.90
88.81
73.31
0.93
GBOM
DiffLinker (given anchors)
51.25
88.0
0.11
95.57
87.77
72.12
0.93
iinkr (sampled size)
56.05
77.5
0.07
92.11
82.11
58.03
0.88
Diffl linker (given anchors, sampled size)
58.16
77.1
0.07
91.38
80.37
55.71
0.88
Pockets
Diffl inker (pocket atoms)
69.47
37.3
0.87
71.61
57.30
34,52
0.78
Diffl iinke (pocket backbone)
66.48
37.8
0.89
65.58
52.77
31.50
0.76
DiffLinker (unconditioned)
59.43
36.5
0.89
50.95
30.59
0.75
Table 10: Performance metrics on the reduced Pockets test set (453 examples, 100 samples).
Method
QED t
SA +
Rings t
Valid, %
Unique,
Novel, %
Difelinker (pocket atoms)
0.50
3.77
1.07
86.2
77.5
90.7
DiffLinkerr (pocket backbone)
0.50
3.62
0.94
88.3
75.6
90.6
DiffLinker (unconditioned)
0.51
3.70
1.13
92.0
75.7
93.0
SCRDki
Method
Filters,
Recovery,
RMSD
>>
> 0.9
Avg f
Diffl inker (pocket atoms)
60.84
31.2
0.86
66.35
48.67
26.62
0.75
DiffLinker (pocket backbone)
57.15
31.1
1.03
58.76
42.81
23.14
0.72
Diffl inker (unconditioned)
48.52
30.1
0.96
57.03
40.57
22.16
0.72
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0:Q
YY
our fragments
onhr
Q
(without dummy atoms)
2.11
2.01
2.11
2.33
full molecules
2.29
2.23
2.88
2.21
PIQ
DeLinker fragments
(with dummy atoms)
3.26
3.49
4.37
3.44
Figure 4:
Synthetic accessibility scores (SA-scores) for fragments without dummy atoms (top row), full molecules (middle row) and fragments with dummy atoms (bottom row).
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GEOM Dataset
Input Fragments
True Molecule
DiffLinker Samples
P
80
Figure 5: Examples of linkers generated by DiffLinker (sampled size) for fragments from GEOM datasets.
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ZINC Dataset
Fragments
True Malecule
Dift Infelinkerrt Samples
CASF Dataset
Ingul Fragments
True Molecule
DiffLinker Samples
Figure 6: Examples of linkers generated by DiffLinker (sampled size) for fragments from CASF and ZINC datasets.
24


Reviews:
[{"review_id": "lE9PUg1zDOc", "reviewer": "Reviewer_oTUZ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "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": "", "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": "The authors propose a method to generate the remainder of a molecule in 3D given some fragments. The generator is an E(3) equivariant denoising diffusion model, conditioned on the positions of the fragment atoms, and optionally also on the protein pocket that the molecule should fit into.", "summary_of_the_review": "The paper combines conditional generation, denoising diffusion, and equivariant GNNs to tackle an important problem in drug discovery. ", "strength_and_weaknesses": "This is a neat application of equivariant diffusion. The authors have chosen their problem well if the aim is to quickly produce something of practical use for drug discovery.  Linker design is a real task in the drug discovery process, it is constrained enough to do well, and existing ML models for the purpose have limitations that the new model addresses.\n\nWhy did the authors choose to train a conditional generator on fragmented molecules, rather than training a model to denoise entire molecules and treating conditional generation as an inpainting task?\n\nThe generated molecules in figure 2 look odd e.g. they contain 3, 4, and 7-member rings. Do you get better molecules after some simple post-filtering? Also in this figure, the bonds in the input fragments and true molecule do not look correct.\n\nIn table 1, please add the QED, SA and #rings scores for the ZINC, CASF and GEOM training data as well as the generated molecules.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "WD92hU_xye", "reviewer": "Reviewer_LoJe", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "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": "", "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": "This paper proposes DiffLinker, which is a diffusion model for molecular linker design that can link an arbitrary amount of fragments. It features an E(3)-equivariant model in the diffusion process and 3-D conditional distribution on input fragments and optional protein pocket. The result shows superior performance on various real-world applications including a newly proposed one in the paper.", "summary_of_the_review": "This paper tackles conditional generation in molecular design using recent diffusion model and equivariant graph neural networks. The generation quality is impressive compared with existing work.", "strength_and_weaknesses": "Strength:\n1. The paper is written very clearly with each step and model overview.\n2. The results on linker design is very impressive and pioneering the study in the field.\n\nWeakness:\n1. It is not very clear on adopting diffusion process in conditional linker design. What's the advantage of VAE-based or Normalizing Flow-based algorithms?\n2. What's the limitation and common failures in the proposed generation process?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "Tl9_Xcu2Jh", "reviewer": "Reviewer_3ysG", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "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": "", "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": "The paper presents a diffusion model approach for predicting molecular linkers between disconnected fragments. This is a two stage process: a model first predicts linker size, and then a diffusion model produces the linker results. Empirical results demonstrates the effectiveness of the proposed method compared with baseline methods. ", "summary_of_the_review": "The paper presents an interesting application of diffusion models to molecule linker design. While the method itself is straightforward, the method demonstrates the utility of diffusion models over autoregressive methods. The idea of the two-stage process is also interesting and could be useful in other applications. ", "strength_and_weaknesses": "Strength:\n- Description of the method is generally clear.\n- This is a novel application of diffusion models on 3d molecule linker design.\n- Code is provided to encourage reproducibility. \n- A thorough ablation study is given to DiffLinker with regards to given anchors and sampled size.\n\nWeakness:\n- The method is described specifically for DDPM, so it inherits all the efficiency issues of DDPM as well. \n- It might help to give some context of some domain-specific settings, especially protein pockets and their representations. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "G-YN6dBXZaB", "reviewer": "Reviewer_TyBL", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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": "", "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": "The paper studied the problem of molecular linker generation, which aims to generate the linker given different individual fragments of a desired molecule/drug. Specifically, the work borrows the idea of the recent equivariant diffusion molecule generative model [1] into this specific \"molecular missing part\" generation problem. Experiments on several human-designed datasets show that the proposed method is competitive or better than existing methods.\n\n[1] Emiel Hoogeboom, Vıctor Garcia Satorras, Clement Vignac, and Max Welling. Equivariant diffusion for molecule generation in 3d. In International Conference on Machine Learning, pp. 8867–8887. PMLR, 2022.", "summary_of_the_review": "The paper introduced a diffusion model for linker design. the idea is not surprising for the ML community and the technical contribution is a little limited. The empirical improvement over existing methods is large, but the evaluation benchmarks are also not reasonable enough from my view.", "strength_and_weaknesses": "[Strength]:\n\n1. The paper is well organized. The tackled linker generation scenario is clearly defined, and the method is easy to follow and understand.\n2. The author nicely provides the source code.\n3. The benchmark table's numerical results are superior to existing methods.\n\n[Weakness]\n\n1. Overall, the paper is not surprising to me. From the ML perspective, this work is an incremental adoption of the equivariant diffusion model in the context of linker design, without significant technical challenges.\n2. Besides, actually diffusion models itself has the capacity for \"inpainting\" missing parts. As shown in the original diffusion model paper [1], the diffusion model can naturally recover the missing patches of images. This further indicates this \"linker design\" is a natural and simple usage of existing diffusion models for molecule generation, making the technical contribution limited.\n3. I'm also a little doubtful about the experimental setup. From the domain-specific (biological or chemical) perspective, considering we already have several fragments and want to design the linker for them: why for the different disconnected fragments we can know their relative positions and even exact coordinates before generating the linker (as shown in Fig.1). I think for this linker design problem, we should take these fragments' positions as known, which, however, is even treated as rigid bodies in this paper.\n4. I'm also concerned about the \"reflection\" equivariance imposed on the proposed model. According to my knowledge, the energy of reflected biostructures should be different, which means they shouldn't have reflection symmetry. However, this is not the case for the proposed model.\n5. [minor point] Considering this is an application paper, actually I'm not sure whether the evaluation itself is sound. For the dataset part, the data is obtained by manually subtracting fragments from molecular structure datasets. This makes me feel like the benchmark itself is human-designed and far away from real-world challenges. I understand this setup may be already adopted in a few existing benchmarks, but this doesn't mean we can just follow them despite the limitation in existing benchmarks. Maybe better and more realistic benchmarks can also be introduced.\n\n[1] Ho, Jonathan, Ajay Jain, and Pieter Abbeel. \"Denoising diffusion probabilistic models.\" Advances in Neural Information Processing Systems 33 (2020): 6840-6851.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject