Original: Equivariant 3D-Conditional Diffusion Models for Molecular Linker Design
Revised: Equivariant 3D-Conditional Diffusion Models for Molecular Linker Design

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Fragment-based drug discovery has been an effective paradigm in early-stage drugdevelopment.
Revised: Fragment-based drug discovery has been an effective paradigm in early-stage drug development.

Original: An open challenge in this area is designing linkers between disconnected molecular fragments of interest to obtain chemically-relevant candidate drugmolecules.
Revised: An open challenge in this area is designing linkers between disconnected molecular fragments of interest to obtain chemically-relevant candidate drug molecules.

Original: In this work, we propose DiffLinker, an E(3)-equivariant 3D-conditionaldiffusion model for molecular linker design.
Revised: In this work, we propose DiffLinker, an E(3)-equivariant 3D-conditional diffusion model for molecular linker design.

Original: Given a set of disconnected fragments,our model places missing atoms in between and designs a molecule incorporatingall the initial fragments.
Revised: Given a set of disconnected fragments, our model places missing atoms in between and designs a molecule incorporating all the initial fragments.

Original: Unlike previous approaches that are only able to connectpairs of molecular fragments, our method can link an arbitrary number of fragments.
Revised: Unlike previous approaches that are only able to connect pairs of molecular fragments, our method can link an arbitrary number of fragments.

Original: Additionally, the model automatically determines the number of atoms in the linkerand its attachment points to the input fragments.
Revised: Additionally, the model automatically determines the number of atoms in the linker and its attachment points to the input fragments.

Original: We demonstrate that DiffLinkeroutperforms other methods on the standard datasets generating more diverse andsynthetically-accessible molecules.
Revised: We demonstrate that DiffLinker outperforms other methods on the standard datasets generating more diverse and synthetically-accessible molecules.

Original: Besides, we experimentally test our methodin real-world applications, showing that it can successfully generate valid linkersconditioned on target protein pockets.
Revised: Besides, we experimentally test our method in real-world applications, showing that it can successfully generate valid linkers conditioned on target protein pockets.

Original:  Introduction
Revised:  Introduction

Original: The space of pharmacologically-relevant molecules is estimated to exceed 10 60 structures [53] andsearching in that space poses significant challenges for drug design.
Revised: The space of pharmacologically-relevant molecules is estimated to exceed 10 60 structures [52] and searching in that space poses significant challenges for drug design.

Original: A successful approach to reducethe size of this space is to start from fragments , smaller molecular compounds that usually have nomore than 20 heavy (non-hydrogen) atoms.
Revised: 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-hydrogen) atoms.

Original: This strategy is known as fragment-based drug design(FBDD)
Revised: This strategy is known as fragment-based drug design (FBDD) [11].

Original: Given a protein pocket (a part of the target protein that employs suitable properties forbinding a ligand), computationally determining fragments that interact with the pocket is a cheaperand more efficient alternative to experimental high-throughput screening methods [12].
Revised: Given a 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 a cheaper and more efficient alternative to experimental high-throughput screening methods [11].

Original: Once therelevant fragments have been identified and docked to the target protein it remains to combine theminto a single, connected molecule.
Revised: Once the relevant fragments have been identified and docked to the target protein, it remains to combine them into a single, connected molecule.

Original: Among various strategies such as fragment linking, merging andgrowing [31], the former has been preferred as it allows to boost rapidly the binding energy of thetarget and the compound [27, 16].
Revised: Among various strategies such as fragment linking, merging, and growing [30], the former has been preferred as it allows to boost rapidly the binding energy of the target and the compound [26, 15].

Original: This work addresses the fragment linking problem.
Revised: This work addresses the fragment linking problem.

Original: Early computational methods for molecular linker design were based on database search and physical simulations [44], both of which are computationally intensive.
Revised: Early computational methods for molecular linker design were based on database search and physical simulations [43], both of which are computationally intensive.

Original: Therefore, there is increasing interest for machine learning methods that can go beyond the available data and design diverse linkers more efficiently.
Revised: Therefore, there is increasing interest for machine learning methods that can go beyond the available data and design diverse linkers more efficiently.

Original: Existing approaches are either based on syntactic pattern recognition [57] or on autoregressive models [24, 25, 21].
Revised: Existing approaches are either based on syntactic pattern recognition [56] or on autoregressive models [23, 24, 20].

Original: While the former method operates solely on SMILES [54] the latter take into account 3D positions and orientations of the input fragments.
Revised: While the former method operates solely on SMILES [53], the latter take into account 3D positions and orientations of the input fragments.

Original: However these methods are not equivariant with respect to the permutation of atoms and can only combine pairs of fragments
Revised: However, these methods are not equivariant with respect to the permutation of atoms and can only combine pairs of fragments.

Original: [Figure]
Revised: [Figure]

Original: Linker design depends on the target protein pocket, and using this information correctly can improvethe affinity of the resulting overall compound.
Revised: Linker design depends on the target protein pocket, and using this information correctly can improve the affinity of the resulting overall compound.

Original: To date, however, there is no computational methodfor molecular linker design that takes the pocket into account.
Revised: To date, however, there is no computational method for molecular linker design that takes the pocket into account.

Original: In this work, we introduce DiffLinker, a conditional diffusion model that generates molecular linkersfor a set of input fragments represented as a 3D atomic point cloud.
Revised: In this work, we introduce DiffLinker 1 , a conditional diffusion model that generates molecular linkers for a set of input fragments represented as a 3D atomic point cloud.

Original: First, our model generatesthe size of the prospective linker and then samples initial linker atom types and positions from thenormal distribution.
Revised: First, our model generates the size of the prospective linker and then samples initial linker atom types and positions from the normal distribution.

Original: Next, the linker atom types and coordinates are iteratively updated using a neuralnetwork that is conditioned on the input fragments.
Revised: Next, the linker atom types and coordinates are iteratively updated using a neural network that is conditioned on the input fragments.

Original: Eventually, the denoised linker atoms and theinput fragment atoms form a single connected molecule, as shown in Figure 1.
Revised: Eventually, the denoised linker atoms and the input fragment atoms form a single connected molecule, as shown in Figure 1.

Original: DiffLinker enjoys several desirable properties: it is equivariant to translations, rotations, reflectionsand permutations, it is not limited by the number of input fragments, it does not require informationon the attachment atoms and generates linkers with no predefined size.
Revised: 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 not require information on the attachment atoms and generates linkers with no predefined size.

Original: Besides, the conditioningmechanism of DiffLinker allows to pass additional information about the surrounding protein pocketatoms, which makes the model applicable in structure-based drug design applications [7].
Revised: Besides, the conditioning mechanism of DiffLinker allows to pass additional information about the surrounding protein pocket atoms, which makes the model applicable in structure-based drug design applications [6].

Original: We empirically show that DiffLinker is more effective than previous methods in generating chemicallyrelevant linkers between pairs of fragments.
Revised: We empirically show that DiffLinker is more effective than previous methods in generating chemicallyrelevant linkers between pairs of fragments.

Original: Our method achieves the state-of-the-art results insynthetic accessibility and drug-likeness which makes it preferable for using in drug design pipelines.
Revised: Our method achieves the state-of-the-art results in synthetic accessibility and drug-likeness which makes it preferable for using in drug design pipelines.

Original: Besides, DiffLinker remarkably outperforms other methods in the diversity of the generated linkers.
Revised: Besides, DiffLinker remarkably outperforms other methods in the diversity of the generated linkers.

Original: We further propose a more challenging benchmark and show that our method is able to successfullylink more than two fragments, which cannot be done by the other methods.
Revised: We further propose a more challenging benchmark and show that our method is able to successfully link more than two fragments, which cannot be done by the other methods.

Original: We also demonstrate that
Revised: We also demonstrate that

Original: DiffLinker can be conditioned on the target protein pocket: our model respects geometric constraintsimposed by the surrounding protein atoms and generates molecules that have minimum clashes withthe corresponding pockets.
Revised: DiffLinker can be conditioned on the target protein pocket: our model respects geometric constraints imposed by the surrounding protein atoms and generates molecules that have minimum clashes with the corresponding pockets.

Original: To the best of our knowledge, DiffLinker is the first method that is notlimited by the number of input fragments and accounts the information about pockets.
Revised: To the best of our knowledge, DiffLinker is the first method that is not limited by the number of input fragments and accounts the information about pockets.

Original: The overallgoal of this work is to provide practitioners with an effective tool for molecular linker generation inrealistic drug design scenarios.
Revised: The overall goal of this work is to provide practitioners with an effective tool for molecular linker generation in realistic drug design scenarios.

Original:  Related Work
Revised:  Related Work

Original: Molecular linker design has been widely used in the fragment-based drug discovery community [44].
Revised: Molecular linker design has been widely used in the fragment-based drug discovery community [43].

Original: Various de novo design methods refer to the fragment linking problem [52, 36, 41, 38, 48, 50, 22].
Revised: Various de novo design methods refer to the fragment linking problem [51, 35, 40, 37, 47, 49, 21].

Original: Early fragment linking methods were based on search in the predefined libraries of linkers [6, 34], genetic algorithm, tabu search [14] and force field optimization [9].
Revised: Early fragment linking methods were based on search in the predefined libraries of linkers [5, 33], genetic algorithm, tabu search [13] and force field optimization [8].

Original: Having been successfully used inmultiple application cases [28, 46, 43], these methods are however computationally expensive andsubstantially limited by the available data.
Revised: Having been successfully used in multiple application cases [27, 45, 42], these methods are however computationally expensive and substantially limited by the available data.

Original: Hence, there has recently been interest in developing learning-based methods for molecular linkerdesign.
Revised: Hence, there has recently been interest in developing learning-based methods for molecular linker design.

Original: Yang et al.
Revised: Yang et al.

Original: [57] proposed SyntaLinker, a SMILES-based deep conditional transformer neuralnetwork that solves a sentence completion problem [58].
Revised: [56] proposed SyntaLinker, a SMILES-based deep conditional transformer neural network that solves a sentence completion problem [57].

Original: This method inherits the drawbacks of
Revised: This method inherits the drawbacks of

Original: SMILES, which are the absence of 3D structure and the lack of consistency (atoms that are close inthe molecule can be far away in the SMILES string).
Revised: 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).

Original: Imrie et al.
Revised: Imrie et al.

Original: [24] overcome these limitations byintroducing an autoregressive model DeLinker and its extension DEVELOP [25] that uses additionalpharmacophore information.
Revised: [23] overcome these limitations by introducing an autoregressive model DeLinker and its extension DEVELOP [24] that uses additional pharmacophore information.

Original: Although these methods operate on 3D molecular conformations, theyuse a very limited geometric information and require input on the attachment atoms of the fragments.
Revised: Although these methods operate on 3D molecular conformations, they use a very limited geometric information and require input on the attachment atoms of the fragments.

Original: Recently, Huang et al.
Revised: Recently, Huang et al.

Original: [21] have proposed another autoregressive method 3DLinker that does notrequire one to specify attachment points and leverages the geometric information to a much greaterextent.
Revised: [20] have proposed another autoregressive method 3DLinker that does not require one to specify attachment points and leverages the geometric information to a much greater extent.

Original: It makes this approach more relevant for connecting docked fragments.
Revised: It makes this approach more relevant for connecting docked fragments.

Original: As both DeLinker and3DLinker are autoregressive models, they are not permutation equivariant which limits their sampleefficiency and ability to scale to large molecules [11, 40].
Revised: 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 [10, 39].

Original: Besides, these methods are capable ofconnecting only pairs of fragments and cannot be easily extended to larger sets of fragments.
Revised: Besides, these methods are capable of connecting only pairs of fragments and cannot be easily extended to larger sets of fragments.

Original: Outside of the linker design problem, several recent works proposed denoising diffusion models formolecular data in 3D. Conformer generation methods GeoDiff [56] and ConfGF [45] condition themodel on the adjacency matrix of the molecular graph.
Revised: Outside of the linker design problem, several recent works proposed denoising diffusion models for molecular data in 3D. Conformer generation methods GeoDiff [55] and ConfGF [44] condition the model on the adjacency matrix of the molecular graph.

Original: Since they have access to the connectivityinformation, they can compute torsion angles between atoms and optimize them [29].
Revised: Since they have access to the connectivity information, they can compute torsion angles between atoms and optimize them [28].

Original: Equivariant
Revised: Equivariant

Original: Diffusion Model (EDM) [19] generates 3D molecules from scratch and is able to be conditioned onthe predefined scalar properties.
Revised: Diffusion Model (EDM) [18] generates 3D molecules from scratch and is able to be conditioned on the predefined scalar properties.

Original: Another diffusion model has been recently proposed for designingprotein scaffolds given protein motifs[51].
Revised: Another diffusion model has been recently proposed for designing protein scaffolds given protein motifs

Original: Having the conditional sampling procedure, this model ishowever trained in an unconditional setup.
Revised: Having the conditional sampling procedure, this model is however trained in an unconditional setup.

Original: Finally, Luo et al.
Revised: Finally, Luo et al.

Original: [35] proposed a model for antibodydesign which combines discrete diffusion for the molecular graphs and continuous diffusion on the3D coordinates.
Revised: [34] proposed a model for antibody design which combines discrete diffusion for the molecular graphs and continuous diffusion on the 3D coordinates.

Original: The conditioning mechanism proposed in this work is the closest to ours, but theirmodel is targeted at generating chains of amino acids rather than atomic point clouds.
Revised: The conditioning mechanism proposed in this work is the closest to ours, but their model is targeted at generating chains of amino acids rather than atomic point clouds.

Original:  Preliminaries
Revised:  Preliminaries

Original: Diffusion Models Diffusion models [47] are a class of generative methods that consist of twocomponents.
Revised: Diffusion Models Diffusion models [46] are a class of generative methods that consist of two components.

Original: The first component, a diffusion process , progressively distorts a data point x mappingit to a noise.
Revised: The first component, a diffusion process , progressively distorts a data point x mapping it to a noise.

Original: The probability of any intermediate state z t for t = 0 , . .
Revised: The probability of any intermediate state z t for t = 0 , . .

Original: , T is defined as q ( z t | x ) =
Revised: , T is defined as q ( z t | x ) =

Original: N ( z t ; α t x , σ 2 t I ) , where α t ∈ R + controls how much signal is retained and σ t ∈ R + controls howmuch noise is added.
Revised: N ( z t ; α t x , σ 2 t I ) , where α t ∈ R + controls how much signal is retained and σ t ∈ R + controls how much noise is added.

Original: Transition probabilities are given by q ( z t | z t − 1 ) =
Revised: Transition probabilities are given by q ( z t | z t − 1 ) =

Original: N ( z t ; α t z t − 1 , σ 2 t I ) , whereα t = α t /α
Revised: N ( z t ; α t z t − 1 , σ 2 t I ) , where α t =

Original:  t − 1 and σ 2 t = σ 2 t − α 2 t σ 2 t − 1 .
Revised: α t /α t − 1 and σ 2 t = σ 2 t − α 2 t σ 2 t − 1 .

Original: The reversed true denoising process that reconstructs the datapoint from noise is defined by q ( z t − 1 | x , z t ) =
Revised: The reversed true denoising process that reconstructs the data point from noise is defined by q ( z t − 1 | x , z t ) =

Original: N ( z t − 1 ; µ t ( x , z t ) , ς 2 t I ) with distribution parametersµ t ( x , z t ) and ς t can be derived analytically and are provided in Appendix A.1.
Revised: N ( z t − 1 ; µ t ( x , z t ) , ς 2 t I ) with distribution parameters µ t ( x , z t ) and ς t can be derived analytically and are provided in Appendix A.1.

Original: The second component of a diffusion model is a generative denoising process that learns to invert thediffusion trajectory having the data point x unknown.
Revised: The second component of a diffusion model is a generative denoising process that learns to invert the diffusion trajectory having the data point x unknown.

Original: The generative transition distribution is definedas p ( z t − 1 | z t ) 
Revised: The generative transition distribution is defined as p ( z t − 1 | z t ) =

Original: = q ( z t − 1 | ˆ x , z t ) , where ˆ x = (1 /α t )
Revised: q ( z t − 1 | ˆ x , z t ) , where ˆ x = (1 /α t )

Original: z t − ( σ t /α t )ˆ ϵ t is an approximation of the datapoint x , and ˆ ϵ t is a prediction of the Gaussian noise computed by a neural network φ [17].
Revised: z t − ( σ t /α t )ˆ ϵ t is an approximation of the data point x , and ˆ ϵ t is a prediction of the Gaussian noise computed by a neural network φ [16].

Original: To generate a new data point, one first samples the Gaussian noise: z T ∼ N (0 , I ) .
Revised: To generate a new data point, one first samples the Gaussian noise: z T ∼ N (0 , I ) .

Original: Then, fort = T, . ..
Revised: Then, for t = T, . .

Original: , 1 , one should iteratively sample z t − 1 ∼ p ( z t − 1 | z t ) and finally sample x ∼ p ( x | z 0 ) .
Revised: , 1 , one should iteratively sample z t − 1 ∼ p ( z t − 1 | z t ) and finally sample x ∼ p ( x | z 0 ) .

Original: For training the neural network φ , we use the objective L ( t ) =
Revised: For training the neural network φ , we use the objective L ( t ) =

Original: ||
Revised: ||

Original: ϵ − ˆ ϵ t || 2 that can be optimized bymini-batch gradient descent using an estimator E t ∼U (0 ,...,T )
Revised: ϵ − ˆ ϵ t || 2 that can be optimized by mini-batch gradient descent using an estimator E t ∼U (0 ,...,T )

Original: [ T · L ( t )]
Revised: [ T · L ( t )]

Original: Molecule Representation
Revised: Molecule Representation

Original: We consider now diffusion models that can operate on moleculesrepresented as 3D atomic point clouds.
Revised: We consider now diffusion models that can operate on molecules represented as 3D atomic point clouds.

Original: A data point x , which is an attributed point cloud consistingof M atoms, is represented by atom coordinates r = ( r 1 , . .
Revised: A data point x , which is an attributed point cloud consisting of M atoms, is represented by atom coordinates r = ( r 1 , . .

Original: , r M ) ∈ R M × 3 and the corresponding feature vectors h
Revised: , r M ) ∈ R M × 3 and the corresponding feature vectors h

Original: = ( h 1 , . .
Revised: = ( h 1 , . .

Original: , h M ) ∈ R M × nf which are one-hot encoded atom types.
Revised: , h M ) ∈ R M × nf which are one-hot encoded atom types.

Original: We willtherefore denote point cloud x as a tuple x = [ r , h ] .
Revised: We will therefore denote point cloud x as a tuple x = [ r , h ] .

Original: Categorical Features Along with continuous atom coordinates, a molecular diffusion model has tooperate on atom types that are discrete variables.
Revised: Categorical Features Along with continuous atom coordinates, a molecular diffusion model has to operate on atom types that are discrete variables.

Original: While categorical diffusion models do exist [18, 2],we follow a simpler strategy [19] that is based on lifting the atom types to a continuous space: weconsider a one-hot encoding of the discrete variables, and add Gaussian noise on top of it.
Revised: While categorical diffusion models do exist [17, 1], we follow a simpler strategy [18] that is based on lifting the atom types to a continuous space: we consider a one-hot encoding of the discrete variables, and add Gaussian noise on top of it.

Original: In the endof the denoising process, once z 0 is sampled, the continuous values corresponding to the atom typesshould be converted back to discrete values.
Revised: In the end of the denoising process, once z 0 is sampled, the continuous values corresponding to the atom types should be converted back to discrete values.

Original: We consider the argmax over the different categories andinclude it in the final transition from z 0 to x .
Revised: We consider the argmax over the different categories and include it in the final transition from z 0 to x .

Original: For more details on the structure of the final transitiondistribution p ( x | z 0 ) and likelihood computation in this setting, we refer the reader to [19].
Revised: For more details on the structure of the final transition distribution p ( x | z 0 ) and likelihood computation in this setting, we refer the reader to [18].

Original: Equivariance Processing 3D molecules requires operations that respect data symmetries.
Revised: Equivariance Processing 3D molecules requires operations that respect data symmetries.

Original: In thiswork, we consider the Euclidean group E(3) that comprises translations, rotations and reflectionsof R 3 and the orthogonal group O(3) that includes rotations and reflections of R 3 .
Revised: In this work, we consider the Euclidean group E(3) that comprises translations, rotations and reflections of R 3 and the orthogonal group O(3) that includes rotations and reflections of R 3 .

Original: A functionf : R 3 → R 3 is E(3)-equivariant if f ( Rx + t ) =
Revised: A function f : R 3 → R 3 is E(3)-equivariant if f ( Rx + t ) =

Original: R f ( x ) + t for any orthogonal matrix R ∈ R 3 × 3 ,det R = ± 1 , for any translation vector t ∈ R 3 and for any x ∈ R 3 .
Revised: R f ( x ) + t for any orthogonal matrix R ∈ R 3 × 3 , det R = ± 1 , for any translation vector t ∈ R 3 and for any x ∈ R 3 .

Original: Note that for simplicitywe use notation Rx =
Revised: Note that for simplicity we use notation Rx =

Original: ( Rx 1 , . . .
Revised: ( Rx 1 , . . .

Original: , Rx M ) ⊤ .
Revised: , Rx M ) ⊤ .

Original: A conditional distribution p ( x | y ) is E(3)-equivariantif p ( Rx + t |
Revised: A conditional distribution p ( x | y ) is E(3)-equivariant if p ( Rx + t |

Original: Ry + t )
Revised: Ry + t )

Original: =
Revised: =

Original: p ( x | y ) for any x , y ∈ R 3 .
Revised: p ( x | y ) for any x , y ∈ R 3 .

Original: A function f and a distribution p are O(3)-equivariant if f ( Rx ) =
Revised: A function f and a distribution p are O(3)- equivariant if f ( Rx ) =

Original: R f ( x ) and p ( Rx | Ry ) =
Revised: R f ( x ) and p ( Rx | Ry ) =

Original: p ( x | y ) respectively.
Revised: p ( x | y ) respectively.

Original: We call the function ftranslation invariant if f ( x + t 
Revised: We call the function f translation invariant if f ( x + t ) =

Original: [Equation]

Original: = f ( x 
Revised: f ( x ) .

Original: [Equation]  DiffLinker: Equivariant 3D-conditional Diffusion Model for Molecular Linker Design 4.1 Equivariant 3D-Conditional Diffusion Model
Revised:  DiffLinker: Equivariant 3D-conditional Diffusion Model for Molecular Linker Design 4.1 Equivariant 3D-Conditional Diffusion Model

Original: Unlike other diffusion models for molecule generation [19, 56], our method is conditioned on 3Ddata.
Revised: Unlike other diffusion models for molecule generation [18, 55], our method is conditioned on 3D data.

Original: More specifically, we assume that each point cloud x has a corresponding context u , whichis another point cloud consisting of all input fragments and (optionally) protein pocket atoms thatremain unchanged throughout the diffusion and denoising processes, as shown in Figure 1.
Revised: More specifically, we assume that each point cloud x 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 1.

Original: Hence,we consider the denoising generative process p to operate on point cloud x while being conditionedon the fixed corresponding context: 
Revised: Hence, we consider the denoising generative process p to operate on point cloud x while being conditioned on the fixed corresponding context: p ( z t − 1 | z t , u ) =

Original: [Equation]
Revised: q ( z t − 1 | ˆ x , z t ) , where ˆ x = (1 /α t ) z t − ( σ t /α t ) φ ( z t , u , t ) .

Original: Initial noise z T is sampled from p ( z T | u ) =
Revised: Initial noise z T is sampled from p ( z T | u ) =

Original: N ( z T ; f ( u ) , I ) , where function f transforms the pointcloud u into a single point that is considered as a mean of the Gaussian noise.
Revised: N ( z T ; f ( u ) , I ) , where function f transforms the point cloud u into a single point that is considered as a mean of the Gaussian noise.

Original: The choice of thefunction f highly depends on the problem being solved and the available priors.
Revised: The choice of the function f highly depends on the problem being solved and the available priors.

Original: In our experiments,we consider two cases.
Revised: In our experiments, we consider two cases.

Original: First, following Imrie et al.
Revised: First, following Imrie et al.

Original: [24], we make use of the information about atomsthat should be connected by the linker.
Revised: [23], we make use of the information about atoms that should be connected by the linker.

Original: We call these atoms anchors and define f ( u ) as the anchors’center of mass.
Revised: We call these atoms anchors and define f ( u ) as the anchors’ center of mass.

Original: However, in real-world scenarios it is unlikely to know which atoms should be theanchors.
Revised: However, in real-world scenarios it is unlikely to know which atoms should be the anchors.

Original: In this case we define f ( u ) as the center of mass of the whole context u .
Revised: In this case we define f ( u ) as the center of mass of the whole context u .

Original: As shown in Appendix A.2, the generative distribution p ( z 0 | u ) is O(3)-equivariant if both f and φare O(3)-equivariant.
Revised: As shown in Appendix A.2, the generative distribution p ( z 0 | u ) is O(3)-equivariant if both f and φ are O(3)-equivariant.

Original: In order to make the generative process also independent of translations, weconstruct the network φ to be additionally translation invariant.
Revised: In order to make the generative process also independent of translations, we construct the network φ to be additionally translation invariant.

Original: Then, instead of sampling the initialnoise from N ( f ( u ) , I ) we center the data at f ( u ) and sample from N (0 , I ) .
Revised: Then, instead of sampling the initial noise from N ( f ( u ) , I ) we center the data at f ( u ) and sample from N (0 , I ) .

Original: 4.2 Equivariant Graph Neural Network
Revised: 4.2 Equivariant Graph Neural Network

Original: The learnable function φ that models the dynamics of the diffusion model is implemented as amodified E(3)-equivariant graph neural network (EGNN) [42].
Revised: The learnable function φ that models the dynamics of the diffusion model is implemented as a modified E(3)-equivariant graph neural network (EGNN) [41].

Original: Its input is the noisy version of thelinker z t at time t and the context u .
Revised: Its input is the noisy version of the linker z t at time t and the context u .

Original: These two parts are modeled as a single fully-connected graphwhere nodes are represented by coordinates r and feature vectors h that include atom types, time t ,fragment flags and (optionally) anchor flags.
Revised: These two parts are modeled as a single fully-connected graph where nodes are represented by coordinates r and feature vectors h that include atom types, time t , fragment flags and (optionally) anchor flags.

Original: The predicted noise ˆ ϵ includes coordinate and feature components:
Revised: The predicted noise ˆ ϵ includes coordinate and feature components:

Original: ˆ ϵ =
Revised: ˆ ϵ =

Original: [ˆ ϵ r ,
Revised: [ˆ ϵ r ,

Original: ˆ ϵ h ] .
Revised: ˆ ϵ h ] .

Original: In order to make function φ invariant to translations, we subtract the initialcoordinates from the coordinate component of the predicted noise following Hoogeboom et al.
Revised: In order to make function φ invariant to translations, we subtract the initial coordinates from the coordinate component of the predicted noise following Hoogeboom et al.

Original: [Equation]
Revised: [Equation]

Original: EGNN consists of the composition of Equivariant Graph Convolutional Layers r l +1 , h l +1 =
Revised: EGNN consists of the composition of Equivariant Graph Convolutional Layers r l +1 , h l +1 =

Original: EGCL [ r l , h l ] which are defined as follows:
Revised: EGCL [ r l , h l ] which are defined as follows:

Original: [Equation]
Revised: [Equation]

Original: where d ij = ∥
Revised: where d ij = ∥

Original: r li − r lj ∥ and ϕ e , ϕ h are learnable functions parametrized by fully connected neuralnetworks (see Appendix A.5 for details).
Revised: r li − r lj ∥ and ϕ e , ϕ h are learnable functions parametrized by fully connected neural networks (see Appendix A.5 for details).

Original: The latter update for the node coordinates is computed by the learnable function
Revised: The latter update for the node coordinates is computed by the learnable function

Original: ϕ vel .
Revised: ϕ vel .

Original: Note that ourgraph includes both a noisy linker z t and a fixed context u , and φ is intended to predict the noisethat should be subtracted from the coordinates and features of z t .
Revised: Note that our graph includes both a noisy linker z t and a fixed context u , and φ is intended to predict the noise that should be subtracted from the coordinates and features of z t .

Original: Therefore, it is natural to keep thecontext coordinates unchanged when computing dynamics and to apply non-zero displacements onlyto the linker part at each EGCL step.
Revised: Therefore, it is natural to keep the context coordinates unchanged when computing dynamics and to apply non-zero displacements only to the linker part at each EGCL step.

Original: Hence, we model the node displacements as follows,
Revised: Hence, we model the node displacements as follows,

Original: [Equation]
Revised: [Equation]

Original: where ϕ r is a learnable function parametrized by a fully connected neural network.
Revised: where ϕ r is a learnable function parametrized by a fully connected neural network.

Original: After the sequence of EGCLs is applied, we have an updated graph with new node coordinatesˆ r =
Revised: After the sequence of EGCLs is applied, we have an updated graph with new node coordinates ˆ r =

Original: [ u r , ˆ z rt ] and new node features ˆ h =
Revised: [ u r , ˆ z rt ] and new node features ˆ h =

Original: [ˆ u h , ˆ z ht ] .
Revised: [ˆ u h , ˆ z ht ] .

Original: Since we are interested only in the linker-relatedpart, we discard the coordinates and features of context nodes and consider the tuple [ˆ z rt , ˆ z ht ] to bethe EGNN output.
Revised: Since we are interested only in the linker-related part, we discard the coordinates and features of context nodes and consider the tuple [ˆ z rt , ˆ z ht ] to be the EGNN output.

Original: 4.3 Linker Size Prediction
Revised: 4.3 Linker Size Prediction

Original: To predict the size of the missing linker between a set of fragments, we represent fragments as afully-connected graph with one-hot encoded atom types as node features and distances betweennodes as edge features.
Revised: 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 as node features and distances between nodes as edge features.

Original: From this, a separately trained GNN (see Appendix A.6 for details) producesprobabilities for the linker size.
Revised: From this, a separately trained GNN (see Appendix A.6 for details) produces probabilities for the linker size.

Original: Our assumption is that relative fragment positions and orientationsalong with atom types contain all the information essential for predicting most likely size of theprospective linker.
Revised: 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.

Original: When generating a linker, we first sample its size with the predicted probabilitiesfrom the categorical distribution over the list of linker sizes seen in the training data, as shown in
Revised: When generating a 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

Original: Figure 1.
Revised: Figure 1.

Original: 4.4 Protein Pocket Conditioning
Revised: 4.4 Protein Pocket Conditioning

Original: In real-world fragment-based drug design applications, it often occurs that fragments are selected anddocked into a target protein pocket [23].
Revised: In real-world fragment-based drug design applications, it often occurs that fragments are selected and docked into a target protein pocket [22].

Original: To propose a drug candidate molecule, the fragments haveto be linked.
Revised: To propose a drug candidate molecule, the fragments have to be linked.

Original: When generating the linker, one should take the surrounding pocket into account andconstruct a linker that has no clashes with protein pocket atoms (in other words, the configurationof the linker and pocket atoms should be physically-realistic) and keeps the binding strength high.
Revised: When generating the linker, one should take the surrounding pocket into account and construct a linker that has no clashes with protein pocket atoms (in other words, the configuration of the linker and pocket atoms should be physically-realistic) and keeps the binding strength high.

Original: To add pocket conditioning to DiffLinker, we represent a protein pocket as an atomic point cloudand consider it as a part of the context u (cf.
Revised: To add pocket conditioning to DiffLinker, we represent a protein pocket as an atomic point cloud and consider it as a part of the context u (cf.

Original: Section 4.1).
Revised: Section 4.1).

Original: We also extend node features with anadditional binary flag marking atoms that belong to the protein pocket.
Revised: We also extend node features with an additional binary flag marking atoms that belong to the protein pocket.

Original: Finally, as the new contextpoint cloud contains much more atoms, we modify the joint representation of the data point z t andthe context u that are passed to the neural network φ .
Revised: Finally, as the new context point cloud contains much more atoms, we modify the joint representation of the data point z t and the context u that are passed to the neural network φ .

Original: Instead of considering fully-connected graphs,we assign edges between nodes based on a 4 Å distance cutoff as it makes the resulting graphs lessdense and counterbalances the increase in the number of nodes.Input Fragments
Revised: Instead of considering fully-connected graphs, we assign edges between nodes based on a 4 Å distance cutoff as it makes the resulting graphs less dense and counterbalances the increase in the number of nodes.

Original: Pocket-Conditioned Samples True Molecule

Original: Unconditioned Samples

Original: Numbers of clashes in DiffLinker samples with different pocket representations Reference Full Pocket Backbone No Pocket 0 10 20 30 40 50 N u m b e r o f c l a s h

Original: e s

Original: Numbers of clashes in DiffLinker samples with different pocket representations

Original: [Figure]

Original:  Experiments 
Revised:  Experiments 5.1 Datasets

Original: 5.1 Datasets

Original: ZINC and CASF
Revised: ZINC and CASF

Original: We follow Imrie et al.
Revised: We follow Imrie et al.

Original: [24] and consider a subset of 250,000 molecules randomlyselected by Gómez-Bombarelli et al.
Revised: [23] and consider a subset of 250,000 molecules randomly selected by Gómez-Bombarelli et al.

Original: [15] from ZINC database [26].
Revised: [14] from ZINC database [25].

Original: First, we generate 3D conformersusing RDKit [32] and define a reference 3D structure for each molecule by selecting the lowest energyconformation.
Revised: First, we generate 3D conformers using RDKit [31] and define a reference 3D structure for each molecule by selecting the lowest energy conformation.

Original: Then, these molecules are fragmented by enumerating all double cuts of acyclic singlebonds that are not within functional groups.
Revised: Then, these molecules are fragmented by enumerating all double cuts of acyclic single bonds that are not within functional groups.

Original: The resulting splits are filtered by the number of atomsin the linker and fragments, synthetic accessibility [13], ring aromaticity, and pan-assay interferencecompounds (PAINS) [4] criteria.
Revised: The resulting splits are filtered by the number of atoms in the linker and fragments, synthetic accessibility [12], ring aromaticity, and pan-assay interference compounds (PAINS) [3] criteria.

Original: One molecule can therefore result in various combinations oftwo fragments with a linker in between.
Revised: One molecule can therefore result in various combinations of two fragments with a linker in between.

Original: The resulting dataset is randomly split into train (438,610examples), validation (400 examples), and test (400 examples) sets.
Revised: The resulting dataset is randomly split into train (438, examples), validation (400 examples), and test (400 examples) sets.

Original: Another evaluation benchmarkused by Imrie et al.
Revised: Another evaluation benchmark used by Imrie et al.

Original: [24] is taken from the CASF-2016 dataset [49].
Revised: [23] is taken from the CASF-2016 dataset [48].

Revised: In contrast to ZINC, where

Revised: Input Fragments

Revised: Pocket-Conditioned Samples True Molecule

Revised: Unconditioned Samples

Revised: Numbers of clashes in DiffLinker samples with different pocket representations Reference Full Pocket Backbone No Pocket 0 10 20 30 40 50 N u m b e r o f c l a s h

Original: In contrast to ZINC, wheremolecule conformers were generated computationally, CASF includes experimentally verified 3Dconformations.
Revised: e s Numbers of clashes in DiffLinker samples with different pocket representations [Figure] molecule conformers were generated computationally, CASF includes experimentally verified 3D conformations.

Original: Following the same preprocessing procedures as for the ZINC dataset, Imrie et al.[24] obtained an additional test set of 309 examples, which we use in our work.
Revised: Following the same preprocessing procedures as for the ZINC dataset, Imrie et al. [23] obtained an additional test set of 309 examples, which we use in our work.

Original: GEOM ZINC and CASF datasets used in previous works only contain pairs of fragments.
Revised: GEOM ZINC and CASF datasets used in previous works only contain pairs of fragments.

Original: However,real-world applications often require connecting more than two fragments with one or more linkers[23].
Revised: However, real-world applications often require connecting more than two fragments with one or more linkers [22].

Original: To address this case, we construct a new dataset based on GEOM molecules [3] which wedecompose into three or more fragments with one or two linkers connecting them.
Revised: To address this case, we construct a new dataset based on GEOM molecules [2] which we decompose into three or more fragments with one or two linkers connecting them.

Original: To achievesuch splits, we use RDKit implementations of two fragmentation techniques –– an MMPA-basedalgorithm [10] and BRICS [8] –– and combine results removing duplicates.
Revised: To achieve such splits, we use RDKit implementations of two fragmentation techniques –– an MMPA-based algorithm [9] and BRICS [7] –– and combine results removing duplicates.

Original: Overall, we obtain 41,907molecules and 285,142 fragmentations that are randomly split in train (282,602 examples), validation(1,250 examples) and test (1,290 examples) sets.
Revised: Overall, we obtain 41,907 molecules and 285,142 fragmentations that are randomly split in train (282,602 examples), validation (1, examples) and test (1,290 examples) sets.

Original: Pockets Dataset
Revised: Pockets Dataset

Original: In order to assess the ability of DiffLinker to generate valid linkers given additionalinformation about protein pockets, we use the protein-ligand dataset curated by Anonymous [1] from 
Revised: In order to assess the ability of DiffLinker to generate valid linkers given additional information about protein pockets, we use the protein-ligand dataset curated by ? ] from Binding

Original: Binding MOAD [20].
Revised: MOAD [19].

Original: To define pockets, we consider amino acids that have at least one atom closerthan 6 Å to any atom of the ligand.
Revised: To define pockets, we consider amino acids that have at least one atom closer than 6 Å to any atom of the ligand.

Original: All atoms belonging to these residues constitute the pocket.
Revised: All atoms belonging to these residues constitute the pocket.

Original: Wesplit molecules into fragments using RDKit’s implementation of MMPA-based algorithm [10].
Revised: We split molecules into fragments using RDKit’s implementation of MMPA-based algorithm [9].

Original: Werandomly 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.[Table]
Revised: We 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.

Original: 5.2 Evaluation
Revised: 5.2 Evaluation

Original: Metrics First, we report several chemical properties of the generated molecules that are especiallyimportant in drug design applications: average quantitative estimation of drug-likeness (QED) [5],average synthetic accessibility (SA) [13] and average number of rings in the linker.
Revised: Metrics First, we report several chemical properties of the generated molecules that are especially important in drug design applications: average quantitative estimation of drug-likeness (QED) [4], average synthetic accessibility (SA) [12] and average number of rings in the linker.

Original: Next, following
Revised: Next, following

Original: Imrie et al.
Revised: Imrie et al.

Original: [24], we measure validity, uniqueness and novelty of the samples.
Revised: [23], we measure validity, uniqueness and novelty of the samples.

Original: We then determineif the generated linkers are consistent with the 2D filters used to produce the ZINC training set.
Revised: We then determine if the generated linkers are consistent with the 2D filters used to produce the ZINC training set.

Original: These filters are explained in detail in Appendix A.10.
Revised: These filters are explained in detail in Appendix A.10.

Original: In addition, we record the percentage of theoriginal molecules that were recovered by the generation process.
Revised: In addition, we record the percentage of the original molecules that were recovered by the generation process.

Original: To compare the 3D shapes ofthe sampled and ground-truth molecules, we estimate the Root Mean Squared Deviation (RMSD)between the generated and real linker coordinates in the cases where true molecules are recovered.
Revised: To compare the 3D shapes of the sampled and ground-truth molecules, we estimate the Root Mean Squared Deviation (RMSD) between the generated and real linker coordinates in the cases where true molecules are recovered.

Original: We also compute the SC RDKit metric that evaluates the geometric and chemical similarity between theground-truth and generated molecules [39, 33].
Revised: We also compute the SC RDKit metric that evaluates the geometric and chemical similarity between the ground-truth and generated molecules [38, 32].

Original: Baselines We compare our method with DeLinker [24] and 3DLinker [21] on the ZINC test setand with DeLinker on the CASF dataset.
Revised: Baselines We compare our method with DeLinker [23] and 3DLinker [20] on the ZINC test set and with DeLinker on the CASF dataset.

Original: Besides, we adjust 3DLinker to connect more than twofragments (see Appendix A.7 for details) and evaluate its performane on the GEOM dataset.
Revised: Besides, we adjust 3DLinker to connect more than two fragments (see Appendix A.7 for details) and evaluate its performane on the GEOM dataset.

Original: Toobtain 3D conformations for the molecules generated by DeLinker on ZINC and CASF, we apply apre-trained ConfVAE [55] followed by a force field relaxation procedure.
Revised: To obtain 3D conformations for the molecules generated by DeLinker on ZINC and CASF, we apply a [Table] pre-trained ConfVAE [54] followed by a force field relaxation procedure.

Original: For all methods includingours, we generate linkers with the ground-truth size unless explicitly noted otherwise.
Revised: For all methods including ours, we generate linkers with the ground-truth size unless explicitly noted otherwise.

Original: To obtain
Revised: To obtain

Original: SMILES representations of atomic point clouds generated by our models, we utilize OpenBabel [37]to compute covalent bonds between atoms.
Revised: SMILES representations of atomic point clouds generated by our models, we utilize OpenBabel [36] to compute covalent bonds between atoms.

Original: We also use OpenBabel to rebuild covalent bonds for themolecules in test sets in order to correctly compute the recovery rate, RMSD and SC RDKit scores forour models.
Revised: We also use OpenBabel to rebuild covalent bonds for the molecules in test sets in order to correctly compute the recovery rate, RMSD and SC RDKit scores for our models.

Original: In ZINC and CASF experiments, we sample 250 linkers for each input pair of fragments.
Revised: In ZINC and CASF experiments, we sample 250 linkers for each input pair of fragments.

Original: For the GEOM dataset and in experiments with pocket conditioning, we sample 100 linkers for eachinput set of fragments.
Revised: For the GEOM dataset and in experiments with pocket conditioning, we sample 100 linkers for each input set of fragments.

Original: 5.3 Results 
Revised: 5.3 Results ZINC and CASF

Original: ZINC and CASF

Original: While our models have much greater flexibility and applicability in moreapplications as we show below, they also outperform other methods on standard benchmarks ZINCand CASF in terms of chemical relevance of the generated molecules.
Revised: While our models have much greater flexibility and applicability in more applications as we show below, they also outperform other methods on standard benchmarks ZINC and CASF in terms of chemical relevance of the generated molecules.

Original: As shown in Table 1, moleculessampled by DiffLinker are more synthetically accessible and demonstrate higher drug-likeness, whichis especially important for drug design applications.
Revised: As shown in Table 1, molecules sampled by DiffLinker are more synthetically accessible and demonstrate higher drug-likeness, which is especially important for drug design applications.

Original: Besides, our models generate linkers containing more rings.
Revised: Besides, our models generate linkers containing more rings.

Original: Moreover, our molecules usually share higher chemical and geometric similarity with thereference molecules as demonstrated by the SC RDKit scores in Table 4.
Revised: Moreover, our molecules usually share higher chemical and geometric similarity with the reference molecules as demonstrated by the SC RDKit scores in Table 4.

Original: In terms of validity, our modelsperform on par with the other methods.
Revised: In terms of validity, our models perform on par with the other methods.

Original: Note that both autoregressive approaches employ valencyrules at each generation step explicitly, while our model is shown to be able to learn these rulesfrom the data.
Revised: Note that both autoregressive approaches employ valency rules at each generation step explicitly, while our model is shown to be able to learn these rules from the data.

Original: Remarkably, the validity of the reference molecules from CASF with covalent bondscomputed by OpenBabel is 92.2% while our model generated molecules with 90.2% validity.
Revised: Remarkably, the validity of the reference molecules from CASF with covalent bonds computed by OpenBabel is 92.2% while our model generated molecules with 90.2% validity.

Original: Notably,sampling the size of the linker significantly improves novelty and uniqueness of the generated linkerswithout significant degradation of the most important metrics.
Revised: Notably, sampling the size of the linker significantly improves novelty and uniqueness of the generated linkers without significant degradation of the most important metrics.

Original: Examples of linkers generated by
Revised: Examples of linkers generated by

Original: DiffLinker for different input fragments are provided in Figure 5.
Revised: DiffLinker for different input fragments are provided in Figure 5.

Original: Multiple Fragments
Revised: Multiple Fragments

Original: The major advantage of DiffLinker compared to recently proposed autoregressive models DeLinker and 3DLinker is one-shot generation of the whole linker between arbitraryamounts of fragments.
Revised: The major advantage of DiffLinker compared to recently proposed autoregressive models DeLinker and 3DLinker is one-shot generation of the whole linker between arbitrary amounts of fragments.

Original: This overcomes the limitation of DeLinker and 3DLinker, which can only linktwo fragments at a time.
Revised: This overcomes the limitation of DeLinker and 3DLinker, which can only link two fragments at a time.

Original: Although these autoregressive models can be adjusted to connect pairs offragments iteratively while growing the molecule, the full context cannot be taken into account in thiscase.
Revised: Although these autoregressive models can be adjusted to connect pairs of fragments iteratively while growing the molecule, the full context cannot be taken into account in this case.

Original: Therefore, suboptimal solutions are more likely to be produced.
Revised: Therefore, suboptimal solutions are more likely to be produced.

Original: To illustrate this difference,we adapted 3DLinker to iteratively connect pairs of fragments in molecules where more than twofragments should be connected.
Revised: To illustrate this difference, we adapted 3DLinker to iteratively connect pairs of fragments in molecules where more than two fragments should be connected.

Original: As shown in Table 1, 3DLinker fails to construct valid molecules inalmost 84% of cases and cannot recover any reference molecule while, despite the higher complexityof linkers in this dataset, our models achieve 94% validity and recover more than 50% of the referencemolecules.
Revised: As shown in Table 1, 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, our models achieve 94% validity and recover more than 50% of the reference molecules.

Original: Besides, molecules generated by 3DLinker have no rings in the linkers, have substantiallylower QED and are much harder to synthesize.
Revised: Besides, molecules generated by 3DLinker have no rings in the linkers, have substantially lower QED and are much harder to synthesize.

Original: Examples of linkers generated by DiffLinker fordifferent input fragments are provided in Figure 4.
Revised: Examples of linkers generated by DiffLinker for different input fragments are provided in Figure 4.

Original: An example of the DiffLinker sampling processfor the molecule from the GEOM dataset is shown in Figure 1.
Revised: An example of the DiffLinker sampling process for the molecule from the GEOM dataset is shown in Figure 1.

Original:  Pocket Conditioning To illustrate the ability of DiffLinker to take surrounding pocketsinto account, we trained three models on the Pockets Dataset: these are respectively conditioned onthe full-atomic pocket representation, conditioned on the pocket backbone atoms and unconditioned.
Revised: Protein Pocket Conditioning To illustrate the ability of DiffLinker 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.

Original: Besides the standard metrics reported in Tables 5 and 6, we also compute the number of clashesbetween generated molecules and surrounding pockets.
Revised: Besides the standard metrics reported in Tables 5 and 6, we also compute the number of clashes between generated molecules and surrounding pockets.

Original: We say that there is a clash between twoatoms if the distance between them is lower than the sum of their van der
Revised: We say that there is a clash between two atoms if the distance between them is lower than the sum of their van der

Original: Waals radii.
Revised: Waals radii.

Original: As shown in
Revised: As shown in

Original: Figure 2, the model conditioned on the full-atomic pocket representation generates molecules withalmost the same amount of clashes (in average 7 clashes per molecule) as in the reference complexesfrom the test set (in average 6 clashes per molecule).
Revised: 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 in the reference complexes from the test set (in average 6 clashes per molecule).

Original: There is a clear trend on the number of clashesdepending on the amount of information about pockets DiffLinker is conditioned on: the modelconditioned on pocket backbone atoms generates molecules with 14 clashes in average, and theunconditioned model produces molecules with 21 clashes in average.
Revised: There is a clear trend on the number of clashes depending on the amount of information about pockets DiffLinker is conditioned on: the model conditioned on pocket backbone atoms generates molecules with 14 clashes in average, and the unconditioned model produces molecules with 21 clashes in average.

Original:  Conclusion
Revised:  Conclusion

Original: In this work, we introduced DiffLinker, a new E(3)-equivariant 3D-conditional diffusion model formolecular linker design.
Revised: In this work, we introduced DiffLinker, a new E(3)-equivariant 3D-conditional diffusion model for molecular linker design.

Original: DiffLinker designs realistic molecules from a set of disconnected fragmentsby generating a linker, i.e., an atomic point cloud that interconnects the input fragments.
Revised: DiffLinker designs realistic molecules from a set of disconnected fragments by generating a linker, i.e., an atomic point cloud that interconnects the input fragments.

Original: Whileprevious methods were only capable to connect pairs of fragments, DiffLinker naturally scales to anarbitrary number of fragments.
Revised: While previous methods were only capable to connect pairs of fragments, DiffLinker naturally scales to an arbitrary number of fragments.

Original: Our method does not require to specify the attachment points of thefragments and predicts the distribution of linker size from the fragments.
Revised: Our method does not require to specify the attachment points of the fragments and predicts the distribution of linker size from the fragments.

Original: We show that the proposedmethod outperforms other models on standard benchmarks and produces more chemically-relevantmolecules.
Revised: We show that the proposed method outperforms other models on standard benchmarks and produces more chemically-relevant molecules.

Original: Besides, we demonstrate that our model can be conditioned on protein pockets andgenerate linkers with a minimum number of clashes.
Revised: Besides, we demonstrate that our model can be conditioned on protein pockets and generate linkers with a minimum number of clashes.

Original: We believe that our method will acceleratethe development of prospective drug candidates and has the potential to become widely used in thefragment-based drug design community. 
Revised: We believe that our method will accelerate the development of prospective drug candidates and has the potential to become widely used in the fragment-based drug design community. Acknowledgments

Revised: We thank Arne Schneuing, Yuanqi Du, and Joshua Southern for helpful feedback and insightful discussions.

Revised: Ilia Igashov has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 945363.

Revised: Clément

Revised: Vignac would like to thank the Swiss Data Science Center for supporting him through the PhD fellowship program (grant P18-11).

Original: A.1 Parameters of the True Denoising Process
Revised: A Appendix

Revised: Algorithm Input: linker x , context u , neural network φ Sample t ∼ U (0 , . .

Revised: . , T ) , ϵ t ∼ N (0 , I )

Revised: z t ← α t x + σ t ϵ t ˆ ϵ t ← φ ( z t , u , t )

Revised: Minimize ∥

Revised: ϵ − ˆ ϵ t ∥

Revised: Algorithm

Revised: Input: context u , neural network φ

Revised: Center context u at f ( u )

Revised: Sample z T ∼ N (0 , I ) for t in T, T − 1 , . . .

Revised: Sample ϵ t ∼ N (0 , I )

Revised: ˆ ϵ t ← φ ( z t , u , t )

Revised: z t − 1 ← (1 /α t ) · z t − σ 2 t / ( α t σ t ) · ˆ ϵ t + ς t · ϵ end for

Revised: Sample x ∼ p ( x | z 0 , u )

Revised: A.1 Parameters of the True Denoising Process

Original: The parameters of the true denoising process q ( z t − 1 | x , z t ) =
Revised: The parameters of the true denoising process q ( z t − 1 | x , z t ) =

Original: N ( z t − 1 ; µ t ( x , z t ) , ς 2 t I ) are given as:
Revised: N ( z t − 1 ; µ t ( x , z t ) , ς 2 t I ) are given as:

Original: [Equation]
Revised: [Equation]

Original: A.2 Equivariance of the Conditional Diffusion Model
Revised: A.2 Equivariance of the Conditional Diffusion Model

Original: Proposition 1.
Revised: Proposition 1.

Original: Consider a prior noise distribution p ( z T | u )
Revised: Consider a prior noise distribution p ( z T | u )

Original: =
Revised: =

Original: N ( z T ; f ( u ) , I ) and transitiondistributions p ( z t − 1 |
Revised: N ( z T ; f ( u ) , I ) and transition distributions p ( z t − 1 |

Original: z t , u ) =
Revised: z t , u ) =

Original: q ( z t − 1 | ˆ x , z t ) , where q is an isotropic Gaussian, f : R M × 3{+
+} → R 3 is afunction operating on 3D point clouds, and ˆ x is an approximation computed by the neural network φaccording to Equation (1).
Revised: q ( z t − 1 | ˆ x , z t ) , where q is an isotropic Gaussian, f : R M × {+
+} → R 3 is a function operating on 3D point clouds, and ˆ x is an approximation computed by the neural network φ according to Equation (1).

Original: Let the conditional denoising probabilistic model p be a Markov chaindefined as
Revised: Let the conditional denoising probabilistic model p be a Markov chain defined as

Original: [Equation]
Revised: [Equation]

Original: t

Original: [Equation] p [Equation] If f is O(3)-equivariant and φ is equivariant to joint O(3)-transformations of z t and u , then p ( z 0 | u )is O(3)-equivariant.
Revised: If f is O(3)-equivariant and φ is equivariant to joint O(3)-transformations of z t and u , then p ( z 0 | u ) is O(3)-equivariant.

Original: Proof.
Revised: Proof.

Original: O(3)-equivariance of function f
Revised: O(3)-equivariance of function f

Original: and the fact that q is isotropic Gaussian distribution implies
Revised: and the fact that q is isotropic Gaussian distribution implies

Original: O(3)-equivariance of the prior distribution:
Revised: O(3)-equivariance of the prior distribution:

Original: [Equation]
Revised: [Equation]

Original: Likewise, O(3)-equivariance of function φ and Equation (1) imply O(3)-equivariance of all transitionprobabilities p ( z t − 1 | z t , u ) .
Revised: Likewise, O(3)-equivariance of function φ and Equation (1) imply O(3)-equivariance of all transition probabilities p ( z t − 1 | z t , u ) .

Original: To obtain the distribution p ( z 0 | u ) of data point z 0 , we can consider joint distributionp ( z 0 , z 1 , . .
Revised: To obtain the distribution p ( z 0 | u ) of data point z 0 , we can consider joint distribution p ( z 0 , z 1 , . .

Original: , z T | u ) and marginalize it by z 1 ...T :
Revised: , z T | u ) and marginalize it by z 1 ...T :

Original: [Equation]
Revised: [Equation]

Original: u

Original: [Equation]

Original: d

Original: [Equation] ..T [Equation] Having prior and all transition distributions equivariant, it is now trivial to show O(3)-equivariance ofp ( z 0 | u ) :
Revised: Having prior and all transition distributions equivariant, it is now trivial to show O(3)-equivariance of p ( z 0 | u ) :

Original: [Equation]
Revised: [Equation]

Original: A.3 Problem with Translations
Revised: A.3 Problem with Translations

Original: Consider transition probability p (
Revised: Consider transition probability p (

Original: z t − 1 | z t , u ) =
Revised: z t − 1 | z t , u ) =

Original: q ( z t − 1 | ˆ x , z t ) .
Revised: q ( z t − 1 | ˆ x , z t ) .

Original: Translation equivariance ofp ( z t − 1 | z t , u ) means that
Revised: Translation equivariance of p ( z t − 1 | z t , u ) means that

Original: [Equation]
Revised: [Equation]

Original: More precisely,
Revised: More precisely,

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Therefore, the mean of this distribution can be written as:
Revised: Therefore, the mean of this distribution can be written as:

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Neural network φ is translation invariant meaning that φ ( z t + t , u + t , t ) =
Revised: Neural network φ is translation invariant meaning that φ ( z t + t , u + t , t ) =

Original: φ ( z t , u , t ) .
Revised: φ ( z t , u , t ) .

Original: It means that:
Revised: It means that:

Original: [Equation]
Revised: [Equation]

Original: So we see that ˆ µ t ( z t , u ) is equivariant to translations, however input and output translations are notequal because λ ̸ = 1 .
Revised: So we see that ˆ µ t ( z t , u ) is equivariant to translations, however input and output translations are not equal because λ ̸ = 1 .

Original: It means that equivariance of distributions from Equations (7) and (8) does nothold.
Revised: It means that equivariance of distributions from Equations (7) and (8) does not hold.

Original: More formally,
Revised: More formally,

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: We can also write that
Revised: We can also write that

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Table]

Original: A.4 Diffusion Model
Revised: A.4 Diffusion Model

Original: We trained all DiffLinker models with T = 500 diffusion steps using polynomial noise schedule:
Revised: We trained all DiffLinker models with T = 500 diffusion steps using polynomial noise schedule:

Original: [Equation]
Revised: [Equation]

Original: (21) where s = 10 − 5 is a precision value that helps to avoid numerically unstable situations [19].
Revised: (21) where s = 10 − 5 is a precision value that helps to avoid numerically unstable situations [18].

Original: A.5 Dynamics
Revised: A.5 Dynamics

Original: EGNN takes as input a graph of atoms belonging to the linker z t and its context u represented byfeature vectors i ∈ R in and coordinates i ∈ R 3 .
Revised: EGNN takes as input a graph of atoms belonging to the linker z t and its context u represented by feature vectors i ∈ R in and coordinates i ∈ R 3 .

Original: Feature vector
Revised: Feature vector

Original: i consists of atom types, fragments flagand time step t .
Revised: i consists of atom types, fragments flag and time step t .

Original: If anchors are known, additionally anchor flag is passed.
Revised: If anchors are known, additionally anchor flag is passed.

Original: If the model is conditionedon the protein pocket, additionally pocket flag is passed.
Revised: If the model is conditioned on the protein pocket, additionally pocket flag is passed.

Original: First, atom features are passed to the encoder:
Revised: First, atom features are passed to the encoder:

Original: i → Linear(in, nf) → 0 i .
Revised: i → Linear(in, nf) → 0 i .

Original: Next, as discussed in Section 4.2, L Equivariant Graph Convolutional Layers (EGCL) are sequentiallyapplied.
Revised: Next, as discussed in Section 4.2, L Equivariant Graph Convolutional Layers (EGCL) are sequentially applied.

Original: Learnable components of EGCL ϕ e , ϕ h , ϕ r are implemented as neural networks that includefully-connected layers (FC), batch normalization layers (BN) and activations SiLU.
Revised: Learnable components of EGCL ϕ e , ϕ h , ϕ r are implemented as neural networks that include fully-connected layers (FC), batch normalization layers (BN) and activations SiLU.

Original: Message ϕ e : takes a pair of node embeddings li and lj and the squared distance d 2 ij = ∥
Revised: Message ϕ e : takes a pair of node embeddings li and lj and the squared distance d 2 ij = ∥

Original: i − j ∥ 2between these nodes and outputs a message ij ∈ R nf :concat[ li , lj , d 2 ij ] → {FC(2 · nf + 1, nf) → SiLU→ FC(nf, nf) → SiLU} → ij
Revised: i − j ∥between these nodes and outputs a message ij ∈ R nf : concat[ li , lj , d 2 ij ] → {FC(2 · nf + 1, nf) → SiLU

Revised: → FC

Revised: (nf, nf) → SiLU

Original:  Features update ϕ h : takes as input node embedding li and its aggregated message i =
Revised: } → ij Features update ϕ h : takes as input node embedding li and its aggregated message i =

Original: (cid:80) j ij andreturns the updated node embedding:concat
Revised: (cid:80) j ij and returns the updated node embedding:

Original: [Equation] ,  nf) → BN → SiLU → FC(nf, nf) → BN → add( li )} → l +

Revised: concat[ li , i ] → {FC(2 · nf, nf) → BN → SiLU → FC(nf, nf) → BN → add( li )} → l +

Original: i
Revised: i

Revised: Coordinates update

Revised: ϕ r :

Revised: takes

Revised: the same input as ϕ e and

Original: Coordinates update ϕ r : takes the same input as ϕ e and outputs a scalar valueconcat[ li , lj , d 2 ij ] → {FC(2 · nf + 1, nf) → SiLU → FC(nf, nf) → SiLU → FC(nf, 1) } → output
Revised: outputs a scalar value concat [ li , lj , d 2 ij ] → {FC(2 · nf + 1, nf) → SiLU → FC(nf, nf) → SiLU → FC(nf, 1) } → output

Original: Equivariance
Revised: Equivariance

Original: The equivariance of EGCLs is achieved by construction.
Revised: The equivariance of EGCLs is achieved by construction.

Original: The messages ϕ e and thenode updates ϕ h depend only on scalar node features and distances between nodes that are E(3)-invariant.
Revised: The messages ϕ e and the node updates ϕ h depend only on scalar node features and distances between nodes that are E(3)- invariant.

Original: Coordinate updates ϕ vel additionally depend linearly on the difference between coordinatevectors r li and r lj , which makes them E(3)-equivariant.
Revised: Coordinate updates ϕ vel additionally depend linearly on the difference between coordinate vectors r li and r lj , which makes them E(3)-equivariant.

Original: Training We trained separate models for ZINC, Multi-Frag and Pocket datasets.
Revised: Training We trained separate models for ZINC, Multi-Frag and Pocket datasets.

Original: Hyper parametersof the models and average time required for training one epoch are provided in Table 2.
Revised: Hyper parameters of the models and average time required for training one epoch are provided in Table 2.

Original: All modelswere trained on a single Tesla V100-PCIE-32GB GPU using Adam with learning rate 2 · 10 − 5 andweight decay 10 − 13 .
Revised: All models were trained on a single Tesla V100-PCIE-32GB GPU using Adam with learning rate 2 · 10 − 5 and weight decay 10 − 13 .

Original: A.6 Linker Size Prediction
Revised: A.6 Linker Size Prediction

Original: Graph neural network for predicting probabilities of the number of atoms in the prospective linker fora given set of fragments takes as input a fully-connected graph of atoms belonging to the fragmentsrepresented by feature vectors
Revised: Graph neural network for predicting probabilities of the number of atoms in the prospective linker for a given set of fragments takes as input a fully-connected graph of atoms belonging to the fragments

Revised: [Table]

Original:  i ∈ R in and inter-atomic squared distances d 2 ij = ∥ i − j ∥ 2 , and outputsa vector of probabilities corresponding to the predefined linker sizes ∈ [0 , 1] out .
Revised: represented by feature vectors i ∈ R in and inter-atomic squared distances d 2 ij = ∥ i − j ∥ 2 , and outputs a vector of probabilities corresponding to the predefined linker sizes ∈ [0 , 1] out .

Original: First, node embeddings are computed: i → Linear(in, nf) → 0 i .
Revised: First, node embeddings are computed: i → Linear(in, nf) → 0 i .

Original: Next, a sequence of L Graph Convolutional Layers (GCL) is applied.
Revised: Next, a sequence of L Graph Convolutional Layers (GCL) is applied.

Original: Learnable components of GCLϕ e , ϕ h are implemented in the same way as for EGNN.
Revised: Learnable components of GCL ϕ e , ϕ h are implemented in the same way as for EGNN.

Original: [Table] [Table] Finally, node embeddings Li are projected onto R out , aggregated and normalized resulting in the vectorof label probabilities:
Revised: Finally, node embeddings Li are projected onto R out , aggregated and normalized resulting in the vector of label probabilities:

Original: [Equation]
Revised: [Equation]

Original: Training We trained two models for ZINC and Multi-Frag datasets.
Revised: Training We trained two models for ZINC and Multi-Frag datasets.

Original: Hyper parameters of themodels and average time required for training one epoch are provided in Table 3.
Revised: Hyper parameters of the models and average time required for training one epoch are provided in Table 3.

Original: Both models weretrained using Adam with learning rate 10 − 4 and weight decay 10 − 13 .
Revised: Both models were trained using Adam with learning rate 10 − 4 and weight decay 10 − 13 .

Original: Both models were trained on asingle Tesla V100-PCIE-32GB GPU.
Revised: Both models were trained on a single Tesla V100-PCIE-32GB GPU.

Original: A.7 3DLinker on GEOM Dataset
Revised: A.7 3DLinker on GEOM Dataset

Original: In order to run 3DLinker on GEOM dataset, we had to additionally filter the original test set consistingof 1,290 input fragment sets and remove examples with more than 3 disconnected fragments.
Revised: In order to run 3DLinker on GEOM dataset, we had to additionally filter the original test set consisting of 1,290 input fragment sets and remove examples with more than 3 disconnected fragments.

Original: Forthe remainder test set that included 1,172 input fragment triplets, we ran 3DLinker twice: first, toconnect two randomly selected fragments (10 samples per fragment pair) and then to connect theresulting compound with the third fragment (10 samples per input).
Revised: For the remainder test set that included 1,172 input fragment triplets, we ran 3DLinker twice: first, to connect two randomly selected fragments (10 samples per fragment pair) and then to connect the resulting compound with the third fragment (10 samples per input).

Original: In both steps, we used the half ofthe original linker size.
Revised: In both steps, we used the half of the original linker size.

Original: Overall, we obtained 100 samples for each input fragment triplet.
Revised: Overall, we obtained 100 samples for each input fragment triplet.

Original: We useda pre-trained 3DLinker model available at https://github.com/YinanHuang/3DLinker .
Revised: We used a pre-trained 3DLinker model available at https://github.com/YinanHuang/3DLinker .

Original: Theresults are provided in Tables 1 and 4.
Revised: The results are provided in Tables 1 and 4.

Original: A.8 Evaluation Details
Revised: A.8 Evaluation Details

Original: The principal difference between our and other methods is that we generate 3D point cloud of atomsthat should be further connected with covalent bonds while other methods generate covalent bondsalong with atom types.
Revised: 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.

Original: We emphasize that both DeLinker and 3DLinker employ valency rules ateach generation step which makes is much easier to achieve high validity of samples.
Revised: We emphasize that both DeLinker and 3DLinker employ valency rules at each generation step which makes is much easier to achieve high validity of samples.

Original: In our case,
Revised: In our case,

Original: DiffLinker learns these chemical rules from the data and places atoms at the relevant distances fromeach other.
Revised: DiffLinker learns these chemical rules from the data and places atoms at the relevant distances from each other.

Original: Since the output of DiffLinker is a 3D point cloud, we need to additionally computecovalent bonds between pairs of atoms based on their types and pairwise distances.
Revised: Since the output of DiffLinker is a 3D point cloud, we need to additionally compute covalent bonds between pairs of atoms based on their types and pairwise distances.

Original: For doing that,we use OpenBabel [37].
Revised: For doing that, we use OpenBabel [36].

Original: To be consistent in the evaluation methodology, we recomputed covalent bonds using OpenBabel forall molecules in ZINC and CASF test sets.
Revised: To be consistent in the evaluation methodology, we recomputed covalent bonds using OpenBabel for all molecules in ZINC and CASF test sets.

Original: Next, for each updated molecule, we obtained linkers byremoving irrelevant atoms and saved the resulting molecules and fragments in SDF and SMILESformats.
Revised: Next, for each updated molecule, we obtained linkers by removing irrelevant atoms and saved the resulting molecules and fragments in SDF and SMILES formats.

Original: Molecules saved in SDF format were considered as ground truth and used for 3D comparison(for computing RMSD and SC RDKit metrics).
Revised: Molecules saved in SDF format were considered as ground truth and used for 3D comparison (for computing RMSD and SC RDKit metrics).

Original: Molecules and linkers saved in SMILES format were considered as ground truth and used for 2D comparison (novelty and recovery rates).
Revised: Molecules and linkers saved in SMILES format were considered as ground truth and used for 2D comparison (novelty and recovery rates).

Original: To evaluateother methods, we used original SMILES representations.
Revised: To evaluate other methods, we used original SMILES representations.

Original: Our samples For each generated point cloud, we computed covalent bonds with OpenBabel, andextracted the largest connected component.
Revised: Our samples For each generated point cloud, we computed covalent bonds with OpenBabel, and extracted the largest connected component.

Original: Next, we obtained a linker by matching the generatedmolecule with the corresponding fragments (computed with OpenBabel as explained above) andremoving irrelevant atoms.
Revised: Next, we obtained a linker by matching the generated molecule with the corresponding fragments (computed with OpenBabel as explained above) and removing irrelevant atoms.

Original: Finally, we kekulized the resulting linker and saved the generated moleculewith recomputed covalent bonds and the corresponding linker in SDF and SMILES formats. 
Revised: Finally, we kekulized the resulting linker and saved the generated molecule with recomputed covalent bonds and the corresponding linker in SDF and SMILES formats. [Table]

Original: Metrics To compute validity, we apply sanitization and additionally check that the molecule contains all atoms from fragments.
Revised: Metrics To compute validity, we apply sanitization and additionally check that the molecule contains all atoms from fragments.

Original: For all other metrics, we consider only a subset of valid samples.
Revised: For all other metrics, we consider only a subset of valid samples.

Original: To compute novelty, we first preprocess SMILES of the linker by removing stereochemistry andcanonicalizing tautomer SMILES.
Revised: To compute novelty, we first preprocess SMILES of the linker by removing stereochemistry and canonicalizing tautomer SMILES.

Original: Then, we count how many of the resulting generated linker
Revised: Then, we count how many of the resulting generated linker

Original: SMILES were represented in the training set.
Revised: SMILES were represented in the training set.

Original: To compute uniqueness, we compare SMILES ofwhole molecules and count number of unique molecules sampled for each input pair of fragments.
Revised: To compute uniqueness, we compare SMILES of whole molecules and count number of unique molecules sampled for each input pair of fragments.

Original: To compute recovery, we compare SMILES of each molecule sampled for a given pair of fragments with SMILES of the corresponding ground-truth molecule.
Revised: To compute recovery, we compare SMILES of each molecule sampled for a given pair of fragments with SMILES of the corresponding ground-truth molecule.

Original: Before comparison, we removehydrogens and stereochemistry from molecules.
Revised: Before comparison, we remove hydrogens and stereochemistry from molecules.

Original: To compute RMSD, we consider only recoveredmolecules and align them with the corresponding ground-truth molecules using RDKit functionrdkit.
Revised: To compute RMSD, we consider only recovered molecules and align them with the corresponding ground-truth molecules using RDKit function rdkit.

Original: Chem.rdMolAlign which returns the optimal RMSD for aligning two molecules.
Revised: Chem.rdMolAlign which returns the optimal RMSD for aligning two molecules.

Original: To compute quantitative drug-likeness (QED) [5], we used RDKit function rdkit.Chem.QED.qed .
Revised: To compute quantitative drug-likeness (QED) [4], we used RDKit function rdkit.Chem.QED.qed .

Original: Tocompute synthetic accessibility, we used function calculateScore provided by Ertl & Schuffenhauer [13] in the RDKit-compatible package sascorer.py .
Revised: To compute synthetic accessibility, we used function calculateScore provided by Ertl & Schuffenhauer [12] in the RDKit-compatible package sascorer.py .

Original: For calculating the number of rings, weused RDKit function rdkit.
Revised: For calculating the number of rings, we used RDKit function rdkit.

Original: Chem.rdMolDescriptors.
Revised: Chem.rdMolDescriptors.

Original: CalcNumRings .
Revised: CalcNumRings .

Original: A.9 Additional Results
Revised: A.9 Additional Results

Original: As explained in Section A.7, we had to additionally filter the original test set consisting of 1,290input fragment sets and remove examples with more than 3 disconnected fragments.
Revised: As explained in Section A.7, we had to additionally filter the original test set consisting of 1, input fragment sets and remove examples with more than 3 disconnected fragments.

Original: For eachinput fragment set, we obtained 100 samples with 3DLinker.
Revised: For each input fragment set, we obtained 100 samples with 3DLinker.

Original: For consistency we report DiffLinkerperformance on GEOM in Tables 1 and 4 computed in the same setting: 1,290 input examples and100 samples per input.
Revised: For consistency we report DiffLinker performance on GEOM in Tables 1 and 4 computed in the same setting: 1,290 input examples and 100 samples per input.

Original: DiffLinker results computed on the full GEOM test set with 250 samples perinput are provided in Tables 5 and 6.
Revised: DiffLinker results computed on the full GEOM test set with 250 samples per input are provided in Tables 5 and 6.

Original: DiffLinker results on the Pockets test set with 100 samples per input are provided in Tables 5 and 6as well.
Revised: DiffLinker results on the Pockets test set with 100 samples per input are provided in Tables 5 andas well.

Original: A.10 2D Filters
Revised: A.10 2D Filters

Original: 2D Filters used by Imrie et al.
Revised: 2D Filters used by Imrie et al.

Original: [24] for constructing ZINC and CASF datasets include syntheticaccessibility [13], ring aromaticity (RA), and pan-assay interference compounds (PAINS) [4] criteria.
Revised: [23] for constructing ZINC and CASF datasets include synthetic accessibility [12], ring aromaticity (RA), and pan-assay interference compounds (PAINS) [3] criteria.

Original: RA controls the correctness of the covalent bond orders in the rings of the a linker and PAINS checksif a molecule does not contain compounds that often give false-positive results in high-throughputscreens [4].
Revised: RA controls the correctness of the covalent bond orders in the rings of the a linker and PAINS checks if a molecule does not contain compounds that often give false-positive results in high-throughput screens [3].

Original: Even though we used the same datasets as in Imrie et al.
Revised: Even though we used the same datasets as in Imrie et al.

Original: [24] that were created using allthree filters, we however modify the metric "Passed 2D Filters" by removing SA from it.
Revised: [23] that were created using all three filters, we however modify the metric "Passed 2D Filters" by removing SA from it.

Original: Instead weintroduce our own SA-based metric that we report separately.
Revised: Instead we introduce our own SA-based metric that we report separately.

Revised: The first three metrics assess the chemical relevance of the generated molecules.

Revised: The last three metrics evaluate the standard generative properties of the methods.

Revised: [Table]

Revised: [Table]

Original: Problem with SA filter used by Imrie et al.
Revised: Problem with SA filter used by Imrie et al.

Original: [24] and Huang et al.
Revised: [23] and Huang et al.

Original:  The molecule is consideredto pass the synthetic accessibility filter if its SA-score is lower than SA-score of the correspondingpair of fragments.
Revised: [20] 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.

Original: Even though our models performed on par or better than DeLinker and 3DLinkeraccording to all other metrics, almost all molecules generated by our models did not pass SA-filter.
Revised: Even though our models performed on par or better than DeLinker and 3DLinker according to all other metrics, almost all molecules generated by our models did not pass SA-filter.

Original: We investigated this issue and figured out that SMILES of fragments passed to SA filter by DeLinkerand 3DLinker contained dummy atoms representing anchors.
Revised: We investigated this issue and figured out that SMILES of fragments passed to SA filter by DeLinker and 3DLinker contained dummy atoms representing anchors.

Original: These atoms did not have any atomtype assigned and therefore such molecules were considered hard to be synthesised.
Revised: These atoms did not have any atom type assigned and therefore such molecules were considered hard to be synthesised.

Original: For almost allmolecules in the test set SA-scores of fragments with dummy atoms were higher than SA-scores of [Table]
Revised: For almost all molecules in the test set SA-scores of fragments with dummy atoms were higher than SA-scores of the whole molecules.

Original: The first three metrics assess the chemical relevance of the generated molecules.

Original: The last three metrics evaluate the standard generative properties of the methods.

Original: [Table]

Original: that are chemically and geometrically similar to the reference ones.

Original: For GEOM (250 samples, full test set) and Pockets test set (100 samples).

Original: [Table]

Original: the whole molecules.

Original: However, for most of fragments without dummy atoms SA-scores were much[Table] lower.
Revised: However, for most of fragments without dummy atoms SA-scores were much lower.

Original: Figure 3 shows 4 examples of molecules and fragments with and without dummy atoms andthe corresponding SA-scores.
Revised: Figure 3 shows 4 examples of molecules and fragments with and without dummy atoms and the corresponding SA-scores.

Original: We can conclude that SA-filter proposed by authors of DeLinker showsnothing but the fact that molecules with unknown atoms are hard to be synthesized.
Revised: We can conclude that SA-filter proposed by authors of DeLinker shows nothing but the fact that molecules with unknown atoms are hard to be synthesized.

Original: Therefore, weconsidered this metric to be irrelevant and excluded it from our report.
Revised: Therefore, we considered this metric to be irrelevant and excluded it from our report.

Original: Instead, we report average
Revised: Instead, we report average

Original: Synthetic Accessibility score of full generated molecules.
Revised: Synthetic Accessibility score of full generated molecules.

Original: our fragments (without dummy atoms)
Revised: our fragments (without dummy atoms)

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: GEOM Dataset Input Fragments
Revised: GEOM Dataset Input Fragments

Original: True Molecule
Revised: True Molecule

Original: DiffLinker Samples
Revised: DiffLinker Samples

Original: [Figure]
Revised: [Figure]