ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery
Authors: Andac Demir
Baris Coskunuzer
Yulia Gel
Ignacio Segovia-Dominguez
Yuzhou Chen
Bulent Kiziltan
Venue: NeurIPS 2022 Accept (2022)
Keywords: Drug discovery
topology-based graph classification
persistent homology

Abstract: In computer-aided drug discovery (CADD), virtual screening (VS) is used for comparing a library of compounds against known active ligands to identify the drug candidates that are most likely to bind to a molecular target. Most VS methods to date have focused on using canonical compound representations (e.g., SMILES strings, Morgan fingerprints) or generating alternative fingerprints of the compounds by training progressively more complex variational autoencoders (VAEs) and graph neural networks (GNNs). Although VAEs and GNNs led to significant improvements in VS performance, these methods suffer from reduced performance when scaling to large virtual compound datasets. The performance of these methods has shown only incremental improvements in the past few years. To address this problem, we developed a novel method using multiparameter persistence (MP) homology that produces topological fingerprints of the compounds as multidimensional vectors. Our primary contribution is framing the VS process as a new topology-based graph ranking problem by partitioning a compound into chemical substructures informed by the periodic properties of its atoms and extracting their persistent homology features at multiple resolution levels. We show that the margin loss fine-tuning of pretrained Triplet networks attains highly competitive results in differentiating between compounds in the embedding space and ranking their likelihood of becoming effective drug candidates. We further establish theoretical guarantees for the stability properties of our proposed MP signatures, and demonstrate that our models, enhanced by the MP signatures, outperform state-of-the-art methods on benchmark datasets by a wide and highly statistically significant margin (e.g., 93\% gain for Cleves-Jain and 54\% gain for DUD-E Diverse dataset).

ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery
Andac Demir" Novartis
Baris Coskunuzer" University of Texas at Dallas coskumz2@utdlllas edu
andac. wew.irooovattis.comm
Ignacio Segovia-Dominguee
Yuzhou Chen Temple University
Yulia Gel
University of Texas at Dallas Jet Propulsion Laboratory, Caltech
University of Texas at Dallas National Science Foundation
Bulent Kiziltan
Novartis
bulent.. bulent.kiziitannne.. www.kiiilaannnovatiii.cooo com
Abstract
computer-aided drug discovery (CADD), virtual screening (VS) is used for identifying the drug candidates that are most likely to bind to molecular target in a large library of compounds. Most VS methods to date have focused on using canonical compound representations (e.g., SMILES strings, Morgan fingerprints) or generating alternative fingerprints of the compounds by training progressively complex variational autoencoders (VAEs) and graph neural networks (GNNs). A Although VAHS and GNNs led to significant improvements these methods suffer from reduced performance when scaling large virtual com- pound datasets. The performance of these methods has shown only incremental improvements in the past few years. To address this problem, developed novel method using multiparameter persistence (MP) homology that produces topological fingerprints of the compounds multidimensional vectors. Our primary contri- bution is framing the VS process as new topology-basee graph ranking problem by partitioning a compound into chemical substructures informed by the periodic properties of its atoms and extracting their persistent homology features at multiple resolution levels. We show that the margin loss fine-tuning of pretrained Triplet networks attains highly competitive results in differentiating between compounds in the embedding space and ranking their likelihood of becoming effective drug candidates. We further establish theoretical guarantees for the stability properties our proposed MP signatures, and demonstrate that our models, enhanced by the MP signatures, outperform state-of-the- aht methods on benchmark datasets by wide and highly statistically significant margin (e.g., 93% gain for Cleves-Jain and 54% gain for DUD-E Diverse dataset).
VS performance,
Introduction
Drug discovery is the early phase of the pharmaceutical R&D pipeline where machine learning (ML) making a paradigm-shiftngg impact 9I] Traditionally, early phases of biomedical research involve the identification of targets for disease of interest, followed by bigh-throughput screening
Equal contribution.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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50
ToDD-ViT
Enrichment Factor 1%
90
Parichment Factor at
TaDD-RF
ToDD-RE
70
TaDD-CCowNNeet
30
ROCS
GZD-ROUS
PS-ROCS
DeepScore
20
USR+ROCS
50
RF-Score
Sieve-Scorv
Comph
10
30
Cleves-Jain Dataset Truined
DUD-E Diverse Dutaset Trained
Figure 1: Comparison virtual screening performmance. Each bubble's diameter propuctiona to score. ToDD offers significunt gain regardllss of the choice of clossification model such as random forests (RF), vision transformer (VTT) or modemnized ResNet architecture ConvNeXt. The standard performance metric E.a.% 3% defined ina and therefore the maximum attainable value 50 for E 100 for EF
(HTS) experiments to determine hits within the synthesized compound library, i.e., compounds with high potential. Then, these compounds are optimized to increase potency and other desired target properties. In the final phases of the R&D pipeline, drug candidates have to pass series of rigorous controlled tests in clinical trials to be considered for regulatory approval. On average, this process takes (0-15 years end-to-end and costs in excess of ~ billion US dollars [10]. HTS is highly time and cost-intensive. Therefore, is critical to find good potential compounds effectively for the HTS step for novel compound discovery, but also speed up the pipeline and make it more cost-effective. To address this need, ML augmented virtual screening (VS) has emerged as a powerful computational approach screen ultra large libraries of compounds to find the ones with desired properties and prioritize them for experimentation [65] 40
In this paper, we develop novel approaches for virtual screening by successfully integrating topological data analysis (TDA) methods with ML and deep learning (DL) tools. We first produce topological fingerprints of compounds as 2D 3D vectors by using TDA tools, i.e., multidimensional persistent homology. Then, we show that Triplet networks, (where state-of-the-an pretrained transformer- based models and modernized convolutional neural network architectures serve the backbone and distinct topological features allow to represent support and query compounds), successfully identify the compounds with the desired properties. We also demonstrate that the applicability of topological feature maps can be successfully generalized to traditional ML algorithms such as random forests.
The distinct advantage of TTA tools, in particular persistent homology (PH), is that it enables effective integration of the domain information such as atomic mass, partial charge, bond type (single, double, triple, aromatic ring), ionization energy electron affinity, which carry vital information regarding the chemical properties of a compound at multiple resolution levels during the graph filtration step. While common PH theory allows only one such domain function to be used in this process, with novel multipersistence approach, we show it is possible to use more than one domain function. Topological fingerprints can effectively carry much finer chemical information of the compound structure informed by the multiple domain functions embedded in the process. Specifically, multiparameter persistence homology decomposes 2D graph structure of compound into series of subgraphs using domain functions and generates hierarchical topological representations in multiple resolution levels. At each resolution stage, our algorithm sequentially generates finer topological fingerprints of the chemical substructures. We feed these topological fingerprints to suitable ML/DL methods, and our ToDD models achieve state-of-the-art in all benchmark datasets across all targets (See Table T and
The key contributions of this paper are:
We develop a transformative approach to generate molecular fingerprints. Using multiper- sistence, we produce highly expressive and unique topological fingerprints for compounds independent scale and complexity. This offers a new way to describe and search chemical space relevant to both drug discovery and development.
We bring a new perspective to multiparameter persistence in TDA and produce a com- putationally efficient multidimensional fingerprint of chemical data that can successfully incorporate more than one domain function to the PH process. These MP fingerprints harness the computational strength of linear representations and are suitable to be integrated

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into broad range of ML, DL, and statistical methods; and open path for computationally efficient extraction of latent topological information.
We prove that our multidimensional persistence fingerprints have the same important sta- bility guarantees as the ones exhibited by the most currently existing summaries for single persistence.
We perform extensive numerical experiments VS, showing that our ToDD models outper- form all state-of-the-art methods by wide margin (See Figure I).
2 Related Work
2.1 Virtual Screening
A key step in the early stages of the drug discovery process is to find active compounds that will be further optimized into potential drug candidates. One prevalent computational method that is widely used for compound prioritization with desired properties is virtual screening (VS). There two major categories, i.e., structure-based virtual screening (SBVS) and ligand-based virtual screening (LBVS) (20). SBVS uses the 3D structural information of both ligand (compound) and target protein as a complex I2 52 86). SBVS methods generally require good understanding of 3D-structure of the target protein to explore the different poses of a compound in a binding pocket of the target. This makes the process computationally expensive. On the other hand, LBVS methods compare structural similarities of a library of compounds with a known active ligand 179 69] with an underlying assumption that similar compounds are prone to exhibit similar biological activity. Unlike SBVS, LBVS only uses ligand information. The main idea is to produce effective fingerprints of the compounds and use ML tools to find similarities. Therefore, computationally less expensive LBVS methods can be more efficient with larger chemical datasets especially when the structure of the target receptor is not known [56].
In the last 3 decades, various LBVS methods have been developed with different approaches and these can be categorized into 3 classes depending on the fingerprint they produce: SMILES [8] and
SMARTS [29] are examples of D D-methods which produce D- D-fingerprints, compressing compound
information to a vector. RASCAL (78). MOLPRINT2D 99, ECFP 80), CDK-graph (97). CDK-
hybridization JSISS [103, Klekota-Roth (3). MACSS 29 E-state (66 and SIMCOMP (37
are among 2D methods which uses 2-struce fingerprint and graph matching. Finally, examples of 3.D-mehhod are ROCS (38] USR [8) PatchSurfer 42 which use the 3D-structure of compounds and their conformations (3 DDo-iiin of the compound)
84] On the other hand, while ML methods have been actively used in the field for the last two decades, new deep learning methods made a huge Further discussion of state-of-the-a ML/DL methods are given in Section 6 where we compare our models and benchmark against them.
impact in drug discovery process in the last years
[88 52
2.2 Topological Data Analysis
TDA and tools of persistent homology (PH) have recently emerged powerful approaches for MI, allowing us to extract complementary information on the observed objects, especially, from graph-structured data. In partieular, PH has become popular for various ML tasks such clustering, classification, and anomaly detection, with a wide range of applications including material science [68]
43). insurance 999 46), finance (55), and cryptocurrency analytics [3 4 73). (For more details see surveys (2) and TDA applications library (34) Furthermore, it has become a highly active research
area to integrate PH methods into geometric deep learning (GDL) in recent years 44] 100 19 23
Most recently, the emerging concepts of multipersistence (MP) are proposed to advance the success of single parameter persistence (SP) by allowing the use of more than one domain function in the process to produce more granular topological descriptors of the data. However, the MP theory is not sufficiently mature as it suffers from the nonexistence of the barcode decomposition relating to the partially ordered structure of the index set ((,;,3)) [57 89 The existing approaches remedy this issue via slicing technique by studying -imenninsaa fibers of the multiparameter domain (18), but choosing these directions suitably and computing restricted SP vectorizations are computationally costly which makes the approach inefficient real life applications. There are several promising recent studies in this direction [1] 93 24, but these approaches fail to provide practical topological summary such as "multipersistence diagram", and an effective MP vectorization to be used in real life applications.
3

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2.3 TDA in Virtual Screening
the authors obtained successful results by integrating single persistent homology outputs with various ML models. Furthermore, in [50], the authors used multipersistence homology with fibered barcode approach in the 3D setting and obtained promising results. In the past few years, TDA tools were also successfully combined with various deep learning models for SBVS and property prediction [2]
In 16 15
6T 95 62], the authors successfully used TDA methods to generate powerful molecular descriptors. Then, by using these descriptors, they highly boosted the performance of various MLJDL models and outperformed the existing models in several benchmark datasets. For a discussion and comparison of TDA techniques with other approaches in virtual screening and property prediction, see the review article [70]. In this paper, we follow a different approach and propose a framework by adapting multipersistence homology to VS process which produces fine topological fingerprints which are highly suitable for ML/DL methods.
72 In 66
Background
We first provide the necessary TDA background for machinery. While our techniques applicable to various forms of data, e.g., point clouds and images (for details, see Section [B.2). here focus the graph setup in detail with the idea of mapping the atoms and bonds that make up compound into a set of nodes and edges that represent an undirected graph.
3.1 Persistent Homology
Persistent homology (PH) is key approach in TDA, allowing us to extract the evolution of subtler patterns in the data shape dynamics at multiple resolution scales which are not accessible with more conventional, non-topological methods [17]. In this part, we go the basics of PH machinery on graph-structured data. For further background on PH, see Appendix A..I and (7
For given graph G, consider nested sequence of subgraphs 91 .... S 9N G. For each G5 define abstract simplicial complex G., 1 S i S N, yielding a filtration, nested sequence of simplicial complexes 91 C ... C GN- This step is crucial in the process as one can inject domain information to the machinery exactly at this step by using a filtering function from domain, e.g., atomic mass, partial charge, bond type, electron affinity, ionization energy (See Appendix A.T. After getting a filtration, one can systematically keep track of the evolution of topological patterns in the sequence of simplicial complexes J. A k-dimensional topological feature (or k-hole) may represent connected components (O-dimension), loops (1 (1-dimension) and cavities (2-dimension). For each k-dimensional topological feature a, PH records its first appearance in the filtration sequence, say Go,, and first disappearence in later complexes, Gd, with a unique pair (br, d,),, where < by < dz < N. We call by the birth time of a and d, the death time of a. We call de ba the life span (or persistence) of PH records all these birth and death times of the topological features in persistence diagrams. Let O S k S D where D is the highest dimension in the simplicial complex GN. Then 1hth persistence diagram PDx(G) = {(b,, do) a HG: for b, S i < dg Here, (G.) represents the htth homology group of G. which keeps the information of the k-holes in the simplicial complex G. Most common dimensions used practice are 0 and 1, i.e., (g) and (G). For sake of notations, further we skip the dimension (subscript k). With the intuition that the topological features with long life spans (persistent features) describe the hidden shape patterns in the data, these persistence diagrams provide unique topological fingerprint of G. We give the further details of the PH machinery and how to integrate domain information into the process Appendix A.I
3.2 Multidimensional Persistence
MultiPersistence (MP) significantly boosts the performance of the single parameter persistence technique described in Appendix A. The reason for the term "single" is that we are filtering the data in only one direction 91 C ... C GN = G. As explained in Appendix A. the construction of the filtration is the key step to inject domain information to process and to find the hidden patterns of the data. If one uses function ʃ V + which has valuable domain information, then this induces single parameter filtration as above. However, various data have more than one domain function

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to analyze the data, and using them simultaneously would give much better understanding of the hidden patterns. For example, if we have two functions f.g: 9: charge) with valuable complementary information of the network (compound), MP idea is presumed to produce a unique topological fingerprint combining the information from both functions. These pair of functions f.9 g induces multivariate filtering function F: V + R22 with F(u) = (f()), Again, one can define a set of nondecreasing thresholds {a, Im and {,3, 1 for f and g respectively. Let Vi, = fur E V | fV,) I 0i; g(,,) < B,}. i... Vij F(U,)) 3 (av,, 8j). Define Gig to be the induced subgraph of G by Vej. i.e., the smallest subgraph of g generated by Vij. Then, instead of a single filtration of complexes } we get bifiltration of complexes (G.; 1 S i S m, 1 S j S n) which is m x n rectangular grid of simplicial complexes. Again, the MP idea is to keep track of the k d-dimensional topological features in this grid {Giy] } by using the corresponding homology groups (G))) (MP module).
V y I (e.g., atomic mass and partial
As noted in Section 2 because of the technical problems related to partially ordered structure of the MP module, the MP theory has no sound definition yet (e.g., birth/death time of a topological feature in MP grid), and there no effective way to facilitate this promising idea in real life applications. In the following, we overcome this problem by producing highly effective fingerprints by utilizing the slicing idea in the MP grid in structured way.
4 New Topological Fingerprints of the Compounds with Multipersistence
ToDD framework produces fingerprints of compounds as multidimensional vectors by expanding single persistence (SP) fingerprints (Appendix A. While our construction is applicable and suitable for various forms of data, here we focus on graphs, and in particular, compounds for virtual screening. We obtain a 2D matrix (or 3D array) for each compound as its fingerprint employing 2 or 3 functions/weeight (e.g., atomic mass, partial charge, bond type, electron affinity, ionization energy) to perform graph filtration. We explain how to generalize our framework to other types of data in Appendix B.2, In Appendix B..]] we construct the explicit examples of MP Fingerprints for most popular SP Vectorizations, e.g., Betti, Silhouette, Landscapes.
Our framework basically expands a given SP vectorization MP approach. In technical terms, by using the existing SP vectorizations, we produce multidi- mensional vectors by effectively using one of the filtering direction as slicing direction in the multipersistence module. We explain our process in three steps.
multidimensional vector by utilizing
Step 1 Bifiltrations: This step basically corresponds to obtaining relevant substructures from the given compound in an organized way. Here, we give the computationally most feasible method, called sublevel bifiltration with 2 functions. Depending on the task and dataset, the other filtration types or more functions/weightt can be more useful. In Section B.5 we give details for other filtration methods we use in our experiments. i.e., Vietoris- -ipt (distance) and weight filtration.
Let f.g: : V + R be two filtering functions with threshold sets {ax,) and {3,)5=1 respectively (e.g.. I is atomic mass, and g is partial charge). Let V; = (ur E V f(r) S ax) and let Ga be the induced subgraph of G by V., i.c. add any edge in G whose endpoints are in V,, Similarly, let Vi, = (u, E V f(., S a and g(v,) S },} C V. Let G.j be the induced subgraph of G, by Vig. Then, define Gij as the clique complex of Giy (See Section A. In particular, by using the first function (f), we filter G one (say vertical) direction (G.). }. Then, by using the second function (g), we filter each G; in horizontal direction and obtain a bifiltration (G.z). These subgraphs {Giy) represent the induced substructures of the compound G by using the filtering functions f and g.
In Figure 2 and 3 we give an example of sublevel bifiltration of the compound cytosine by atomic number and partial charge functions. In Figure 2 atom types are coded by their color. Atomic numbers are given in the parenthesis. White-Hydrogen (1), Gray=Carbon (6). Blue=Nitrogen (7), and Red=Oxygen (8). The decimal numbers next to atoms represent their partial charges.
Persistence Diagrams: After constructing the bifiltration G.j. the second step is to obtain persistence diagrams for each row. By restricting the bifiltration to single row, for each 1 < ig 5 m, one obtains a single filtration Goot C Gi. C G in horizontal direction. This is called horizontal slice in the bipersistence module. Each such single filtration induces a persistence diagram PD(G.) [(b,, d;) 0 < b, < dj < n). This produces m persistence diagrams (PD(G.)) Notice
Step 2
5

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that one can consider (D( as the single persistence diagram of the "substructure" G3 filtered by the second function g (See Section A.I
Step Vectorization: The final step is to use a vectorization on these m persistence diagrams. Let oo be single persistence vectorization, e-g., Betti, Silhouette, Entropy, Persistence Landscape Persistence Image. Specifically, we use Betti to case computational complexity. By applying the chosen SP vectorization yp each PD, we obtain function (P; = (G)( where in most cases is a single variable function on the threshold domain [0, nl. i.c.,
[1, n. + R. The number of thresholds m, n are important it determines the size of our topological fingerprint. As most such vectorizations are induced from a discrete set of points PD(g). it is common to express them as vector in the form & ((((1) (2(2 ... (pn)]. In the examples in Section B.4 we explain this conversion explicitly for different vectorizations. Hence, we obtain vector 47, of size X 12 for each row S m.
Now, we can define our topological fingerprint Ms which is 2Dvvectorr (a matrix)
0.380
M = F. for isism m,
0.358
-0.571
-0.736
-0.618
where M p is the 1th rrow of M.. Hence, M. is 2eetect of size m x.. Each row M is the vectorization of the persistence diagram PD(G;) via the SP vectorization method s. We use the first filtering function f get a finer look the graph as it defines the subgraphs G1 C S 9m. g. Then, by using the second function g on each Gr record the evolution of topological features in each G; as (D(G.). While this construc- tion gives our 2D (matrix) fingerprints Ma, one can also use 3 functions/weighh for filtration and obtain finer 3D (array) topological fingerprint (Section) B.3).
0.363
0,389
0.653
-0.330
-0.440
0.261
0.067
0.225
Figure 2: Cylosine. types
by their color: White-Hydrogen, Gry-Carbon, Blue=Nitrogen, and Rod=Oxygen. decimal numbers next atoms represent their partial churges.
In way, we look at G with a 2D resolution (functions / and lenses) and keep track of the evolution of topological features in the induced substructures }. y. The main advantage of this technique is that the outputs are fixed size multidimensional vectors for each dataset which are suitable for various ML/DL models.
Atomic Number
Partial Charge
Figure 3:
Sublevel bifiltration of eytosine is induced filtering functions atomic charge f and atomic number 9. In the horizontal direction, thresbolds -0.5. -0.1, +0.3. +0.7 filters the compound into substructures (e) < with respect their partial charge. the vertical direction, threshokss = 1, 6, fillers the compound representing the sabsinuture I", T...0 (f(w) & g(e) S 9). Whenever two nodes
-0.5,
subsiructures g(v) < with respect to atomic nurbbers. Each box.
A...p indexed by their upper right coordinates (atoms) are in the substructure, there is an edge (bond) between them in the original compound, we include the edge the substrucure.

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4.1 Stability of the MP Fingerprints
We further show that when the source single parameter vectorization yp is stable, then so is its induced MP Fingerprint Mg ge (We give the details of stability notion in persistence theory and proof of the following theorem in Section (.Tiion/BBI]
Theorem: Let ip be stable SP vectorization. Then, the induced MP Fingerprint M. is also stable,
i.e., with the notation introduced in Section B. there exists Cip > such that for any pair of graphs
9+ and g we have the following inequality.
(. ()
CC D
(PD(ST))))
5 Datasets
Cleves-Jain: This a relatively small dataset (26 that has 1149 [ompounds There are 22 different drug targets, and for each one of them the dataset provides only 2-3 template active compounds dedicated for model training, which presents a few-shot learning task. All targets {q} are associated with 4 to 30 active compounds } dedicated for model testing. Additionally, the dataset contains 850 decoy compounds (D). The aim is for each target q. by using the templates, to find the actives Ly among the pool combined with decoys Lg U D, ie., same decoy set D is used for all targets.
There
DUD-E Diverse: DUD-E (Directory of Useful Decoys, Enhanced) dataset [67] is comprehensive ligand dataset with 102 targets and approximately 1.5 million compounds] The targets are categorized into classes with respect to their protein type. The "Diverse subset" of DUD-E contains targets from each category to give balanced benchmark dataset for VS methods. Diverse subset contains 116,105 compounds from 8 target and 8 decoy sets. One decoy set is used per target.
More detailed information about each dataset can be found in Appendix C.I
6 Experiments
6.1
Setup
Maero Design We construct different ToDD (Topological Drug Discovery) models, namely ToDD- VIT, ToDD-ConNNeXt and ToDD-RF to test the generalizability and scalability of topological features while employing different ML models and training datasets of various sizes. Many neural network architectural choices and ML models can be incorporated in our ToDD method. ToDD-ViT and ToDD-ConyNeXt are Triplet network architectures with Vision Transformer (ViT b (6)6) (8 and ConvNeXt_ tiny models [63] pretrained on ILSVRC-2012 ImageNet, serving as the backbone of the Triplet network. MP signatures of compounds are applied nearest neighbour interpolation to increase their resolutions to 2242 followed by normalization. We only use GaussianBlur with kernel size 52 and standard deviation 0.05 as a data augmentation technique. Transfer learning via fine-tuning ViT_b 16 and ConvNeXt_nn tiny models using Adam optimizer with learning rate of 5e-4, no warmup or layerwise learning rate decay, cosine annealing schedule for epochs, stochastic weight averaging for 5 epochs, weight decay of le-4, and batch size of 64 for 10 epochs in total led significantly better performance in Enrichment Factor and ROC-AUC scores compared to training from scratch. The performance of all models was assessed by 5-fold validatinn (CV).
Due to structural isomerism, molecules with identical molecular formulac can have the same bonds, but the relative positions of the atoms differ (76). ViT has much less inductive bias than CNNs, because locality and translation equivariance are embedded into each layer throughout the entire network in CNNs, whereas in ViT self-attention layers are global and only MLP layers are translation- ally equivariant and local (28). Hence, ViT is more robust to distinct arrangements of atoms in space, also referred to as molecular conformation. On a small-scale dataset like Cleves-Jain, ViT exhibits impressive performance. However, the memory and computational costs of dot-product attention blocks of ViT grow quadratically with respect to the size of input, which limits its application on large-scale datasets [60 83). Another major caveat is that the number of triplets grows cubically with
"Cleves-Jain dataset:
-rwggaaaaa-aaaeeeeeeraaa 2006.
'DUD-E Diverse dataset: http://dude docking org.subsets//dvvrsse
7

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Har
Vision Transformeer (ViT)
Charge
ConvNeXi
Iead Type
Anchor sample from support set
shared weights & biases
Isrial Charge
Transformer (VIT)
ConvNeXi
Tocal Typs
Positive sample
shared weights biases
Anesi Mast
Pardaal Charge
Vision Transformer (VIT)
ConvNeXt
embeddings
Type
Vietoris- Rips Persistent Homology Features
Negative sample
Graph Filtrations
Figure 4: End-to-end model pipeline. Anchor sample, and positive sample. =t, compounds that bind the same drug targes, whereas negative sample, is decoy, graph representation of each compound decomposed subgraphs induced by the periodic properties: atomic mass, partial charge and boad type. Potcutially these domain functions can be augmented using other periodic properties such ionization energy electron allinity well using molecular information such chirality, orbital hybridization number of Hydrggen honds or numher of conjugated bonds at the cost of computational complexity. Subgraphs may have isolated nodes and edges. Our MP framework establishes Vietoris-Rips complexes
subgraph provides signatures (topological fingerprints) the compounds. Both ToDD-V DTT and ToDD-ConvNeXt can encode the pair distances between positive query and nogative query against anchor sample from the support set.
the size of the dataset. Since ConvNeXt depends on a fully-convolutional paradigm, its inherently efficient design is viable on large-scale datasets like DUD-E Diverse. As depicted in Figure 4 ToDD- ViT and CoDD-ConvNeet project semantically similar MP signatures of compounds from data manifold onto metrically close embeddings using triplet margin loss with margin 1.0 and norm p = 2 as provided in Equation| Analogously, semantically different MP signatures are projected onto metrically distant embeddings.
Mx, xx,xx),))))) (0, + |[f(x) - f(x+ I|, [|f(x) - f(x f(x))lpp)
Sampling Strategy Learning metric embeddings via triplet margin loss large-scale datasets poses a special challenge in sampling all distinet triplets (x, , and collecting them into a single database causes excessive overhead in computation time and memory. Let P be of compounds, o; denotes a compound that inhibits the drug target i, and d,j = d(z;, xr2)) E R denotes a pairwise distance measure which estimates how strongly 24 E P is similar to xy E P. The distance metric can be chosen as Euclidean distance, cosine similarity or dot-product between embedding vectors. We use pairwise Euclidean distance computed by the pretrained networks in the implementation. Since triplets (xc, x+, + x>) with d(x,x") > d(x, x+) + a have already negative queries sufficiently distant the anchor compounds from the support set the embedding space. they are not sampled to create the training dataset. We only sample triplets that satisfy d(z, x < d(z, x (where negative query closer to the anchor than the positive) and d(z, x) < d(x,, x d(x; x + a (where negative query is more distant to the anchor than the positive, but the distance is less than the margin).
Enrichment Factor (EF) the most common performance evaluation metric for VS methods (90). VS method &p ranks compounds in the database by their similarity scores. We measure the similarity score using the inverse of Euclidean distance between the embeddings of an anchor and drug candidate. Let N be the total number of ligands in the dataset, Ag be the number of true positives (i.e., correctly predicted active ligands) ranked among the top a% of all ligands (N, N a%) and N be the number of active ligands in the whole dataset. Then, EF.% = ApiNuatis In other words, EF.% 100
interprets as how much VS method enrich the possibility of finding active ligand in the first a% of all ligands with respect to the random guess. This method also known as precision
in the literature. With this definition, the max score for EFa is 100 i.c., 100 for EF and 20 for EF
8

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6.2 Experimental Results
We compare our methods against the 23 state-of ofrt baselines (see Appendix
Table I: Comparison of EF 2%, 5%, 10% and ROC-AUC values between ToDD and other virtual sercening methods the Cleves-Jain dataset.
Model USR
EF 2% (max. 50)
EF 5% (max. 20)
EF 10% (max. 10)
ROC-AUC
10.0
6.2
4.1
0.76
0.81
PS +2
10.7
6.6
4,9
0.78
ROCS 18
20.1
10.7
6.2
0.83
USR
13.7
7.7
4.7
0.81
USR + PPS 84
13.1
7.9
5.0
0.80
G7D RS
84
16.0
9.1
5.9
0.82
PHLVS
18.6
NA
NA
GZD ROCS S4
20.3
10.8
5.3
0.83
PS + ROCS 84
20.5
10.7
6.4
0.83
ToDD-RF ToDD-ViT
35.2:1:2.3
15.6=21.0
8.1
0.94
39.6+1.4
18.6=0.44
9.9-0.1
0.90
Relative gains
92.9%
83.7%
54.1%
13.3%
Table 2: Comparison of EF 1% (max. 100) between ToDD and other virtual screening methods on 8 targets of the DUD-E Diverse subset.
Model
AMPC
CXCR4
KIFII
CP3A4
GCR
AKTI
HIVRT
HIVPR
Avg.
Findsite
0.0
21.7
34.2
1.2
34.7
16.5
Pragsite 102
4.2
42.5
0.0
32.9
29.1
47.1
2.4
48.7
25.9
Gnina 87
2.1
15.0
38.0
1.2
39.0
4.1
11.0
28.0
17.3
GOI.D-EAT.
25.8
20.0
33.5
17.9
34.6
29.2
28.7
23.4
26.6
Glide EATL
96
35.5
20.8
30.5
15.1
24.0
31.6
29.0
22.0
26.1
CompM
CompScore
06
39.6
51.6
51.3
14,0
27.1
37.6
21.8
18.2
32.7
CNN 77
2.1
5.0
11.2
28.7
12.8
12.2
20.8
DenseFS
1
14.6
5.0
4.3
44.3
20,9
89.4
12.8
8.4
25.0
SIEVE. Score
30.7
61.1
6.7
33.3
42.1
39.8
38.3
38.2
DeepScore
04
28.1
56.8
37.1
40.9
43.8
62.8
47.9
RF-Score-Väv3
08
32,3
60.9
4.5
25.9
32.5
41.9
39.8
63.7
37.9
'oDDD-FF
42.9--4.5
92.3:1:3.2
75.0::5.0
67.6:t334
78.9:2-.0
90,7=
64.1-12.3
92.1:::1.5
73.7
ToDDD.ConNNeXi
46.2.1.366
84.6.12.8
3..5
28.8.12.8
46.0112.0
81.2
37.5.13.6
74.6.1
58.9
Relative gains
16.7%
51.1%
38.1%
52.6%
92.9%
1.5%
46.3%
40.2%
53.9%
Relative gains are relative to the next best performing model. Based on the results (mean and standard deviation of EF scores evaluated by CV) reported in Table and we observe the following:
ToDD models consistently achieve the best performance both Cleves-Jain and DUD-E Diverse datasets across all targets and EF.% levels.
ToDD learns hierarchical topological representations compounds using their atoms' peri- odic properties, and captures the complex chemical properties essential for high-throughput VS. These strong hierarchical topological representations enable ToDD to become model- agnostic method that is extensible to state-of-the-an neural networks as well as ensemble methods like random forests (RF).
For small-scale datasets such as Cleves-Jain, RF is less accurate than VIT despite regular- ization by bootstrapping and using pruned, shallow trees, because small variations in the data may generate significantly different decision trees. For large-scale datasets such as DUD-E Diverse, ToDD-RF and ToDD-ConNNeXt exhibit comparable performances except for: CP3A4, GCR and HIVRT. We conclude that transformer-based models more robust than convolutional models and RF classifiers despite increased computation time.
6.3 Computational Complexity
Computational complexity (CC) of MP Fingerprint M depends on the vectorization & used and the number d of the filtering functions
CC for single persistence diagram PDş is ((13) 744.

---

where N is the number of k-simplices. If is the resolution size of the multipersistence grid, then CC(M) = O(r" NVo C. (m)) where (.m) CC for & and m is the number of barcodes in PD,, e.g., if 1b is Persistence Landscape, then (m)) m² [3] and hence CC for MP Landscape with three filtering functions (d = 3) is O(+ NN3. m²). one does not need to compute persistence diagrams, but the rank of homology groups in the MP module. Hence, for MP Betti summary, the computational complexity is indeed much lower by using minimal representations [88 [5]]. To expedite the execution time, the feature extraction task is distributed across the 8 cores of an Intel Core i7 CPU (100GB RAM) running in a multiprocessing process. See Appendix [.4 for an additional analysis of computation time to extract MP fingerprints from the datasets. Furthermore, all ToDD models require substantially fewer computational resources during training compared to current graph-based models that encode a compound through mining common molecular fragments, a.k.a., motifs 477. Training time of ToDD-ViT and ToDD-ConvNeXi for each individual drug target takes less than hour on single GPU (NVIDIA RTX 2080 Ti).
). On the other hand, for MP Betti summaries,
6.4 Ablation Study
We tested a number of ablations of our model to analyze the effect of its individual components and to further investigate the effectiveness of our topological fingerprints.
Multimodal Learning We first address the question of how adding different domain information improves the model performance. In Appendix C.3 we demonstrate one-by-one the importance of each modality (atomic mass, partial charge and bond type) used for graph filtration to the classification of each target. We find that their importance varies across targets in unimodal setting, but the orthogonality of these information sources offers significant gain in EF scores when the MP signatures learned from each modality are integrated into a joined multimodal representation. Tables and (8 provide detailed results for the performance of cach modality across all drug targets.
6
Morgan Fingerprints We quantitatively analyze the explainability of our models' success by replacing topological fingerprints computed via multiparameter persistence with the most popular fingerprinting method: Morgan fingerprints. Our results in Appendix C.5 show that ToDD engineers features that represent the underlying attributes of compounds significantly better than the Morgan algorithm to identify the active compounds across all drug targets. We provide detailed tabulated results of our benchmarking study across all drug targets in Tables TT
TT
Network Architecture We investigated ways to leverage deep metric learning by architecting i) a Siamese network trained with contrastive loss, ii) Triplet network trained with triplet margin loss, and tii) a Triplet network trained with circle loss. Based on our preliminary experiments, the embeddings learned by i and iéi provide sub-par results for compound classification, hence we use éi.
Conclusion
We have proposed new idea of the topological fingerprints in VS, allowing for deeper insights into structural organization of chemical compounds. We have evaluated the predictive performance of our ToDD methodology for computer aided drug discovery on benchmark datasets. Moreover, we have demonstrated that our topological descriptors are model-agnostic and have proven to be exceedingly competitive, yielding ofate-of-the--ae art results unequivocally over all baselines. A future research direction is to enrich ToDD with different VS modalities, and use it on ultra-large virtual compound libraries. It is important note that this new way of capturing the chemical information of compounds provides a transformative perspective to every level of the pharmaceutical pipeline from the very early phases of drug discovery to the final stages of formulation in development.
8 Acknowledgments
YG, BC, YC and ISD were partially supported by Simons Collaboration Grant # 579977, the National Science Foundation (NSF) under award ECCS 2039701, the Department of the Navy, Office of Part of YG's contribution is also
Naval Research (ONR) under ONR award # based upon work supported by (while serving at) the NSF Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF and/or the ONR.
10

---

References
[1] Molecular operating environment (moe), 2020.09 chemical computing group ule, 1010 sherbooke st. west, suite 910, montreal, qc. canada, h3a 2r7, 2022.
[2] Henry Adams and Baris Coskunuzer. Geometric approaches on persistent homology, arXiv preprint arXiv:2103.06408, 2021.
[3] Henry Adams, Tegan Emerson. Michael Kirby. Rachel Neville, Chris Peterson, Patrick Shipman, Sofya Chepushtanova, Eric Hanson, Francis Motta, and Lori Ziegelmeier. Persistence images: A stable vector representation of persistent homology. Journal of Machine Learning Research, 18, 2017.
[4] Cuneyt Gurean Akcora, Yitao Li, Yulia R Gel, and Murat Kantarcioglu. Bitcoinheist: Topological data analysis for ransomware detection on the bitcoin blockchain. In Proceedings of the 29th IJCAI, 2019.
[5] Mehmet Aktas, Esra Akhas, and Ahmed El Fatmaoui. Persistence homology of networks: methods and applications. Applied Network Science, 4(1):1-28, 2019.
[6] Erik J Amézquita, Michelle Y Quigley, Tim Ophelders, Elizabeth Munch, and Daniel Chitwood. The shape of things to come: Topological data analysis and biology, from molecules to organisms. Developmental Dynamics, 249(7):816-88)) 2020.
[7] Nieves Atienza, Rocío González-Diaz and Manuel Soriano-Trigueros Trigueros. On the stability of persistent entropy and summary functions for topological data analysis. Puttern Recognition, 107:107509, 2020.
[8] Pedro Ballester and Graham Richards. Ultrafast shape recognition to search compound databases for similar molecular shapes. Journal of Computational Chemistry. 28( 28(10):1111-722 2007.
[9] Andreas Bender, Hamse Mussa, Robert C Glen, and Stephan Reiling. Similarity searching of chemical databases using atom environment deseriptors (molprint 2d): evaluation of performance. Journal of Chemical Information and Computer Sciences, (4(5):1708-1718 2004.
[10] Nurken Berdigaliyev and Mohamad Aljofan. An overview of drug discovery and development. Future Medicinal Chemistry, 12(10):939-947, 2020.
[11] Magnus Bakke Botnan, Steffen Oppermann, and Steve Oudot. Signed barcodes for paramer persistence via rank decompositions and rank-exact resolutions. arXiv preprint urXiv:2107.06800 2021.
[12] Natasja Brooijmans and Irwin D Kuntz. Molecular recognition and docking algorithms. Annual Review of Biophysics and Biomolecular Structure, 32(1):335-333) 373, 2003.
[13] P. Bubenik. Statistical topological data analysis using persistence landscapes. Journal of Machine Learning Research, 16(1):77-102, 2015.
[14] Zixuan Cang, Lin Mu, and Guo Wei Wei. Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening. PLoS Computational Biology, 14(1): 10059999, 2018.
[15] Zixuan Cang and Guo-Wei Wei. Topologynct: Topology based deep convolutional and multi-task neural networks for biomolccular property predictions. PLoS Computational Biology, 13(7)e1005699 2017.
[16] Zixuan Cang and Guo Wei Wei. Integration of clement specific persistent homology and machine learning for protein-ligand binding affinity prediction. International Journal for Numerical Methods in Biomedical Engineeringg 34(2):e2914, 2018.
[17] Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308, 2009.
[18] Mathieu Carriere and Andrew Blumberg. Multiparameter persistence image for topological machine learning. In NewrIPS, volume 33, pages 22432-22444, 2020.
[19] Mathieu Carrèère, Frédéric Chazal, Yuichi Ike, Théo Lacombe, Martin Royer, Yuhei Umeda. Perslay: A neural network layer for persistence diagrams and new graph topological signatures. In AISTATS, pages 2786-2796 20062799,,22 2020.
[20] Claudio N Cavasotto and Andrew J W Orry. Ligand docking and structure based virtual screening in drug discovery. Current Topics in Medicinal Chemistry. 1000):10061101 2007.
11

---

[21] Frédéric Chazal. Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, and Larry Wasserman. Stochastic convergence of persistence landscapes and silhouettes. In Proceedings the Thirtieth Annual Symposium on Computational Geometry, pages 483 2014.
[22] Frédéric Chazal and Bertrand Michel. An introduction to topological data analysis: fundamental and practical aspects for data scientists. Frontiers in Artificial Intelligence, 4. 2021.
[23] Yuzhou Chen, Ignacio Segovia, and Yulia R Gel. Z-GCNETs: time zigzags graph convolutional networks for time series forecasting. In ICML. pages 1684-1694 PMLR, 2021.
[24] Yuzhou Chen, Ignacio
Baris Coskunuzer, and Yulia Gel. Tamp-s2genets: Coupling time-aware multipersistence knowledge representation with spatio-supra graph convolutional networks for time-series forecasting, ICLR, 2022.
[25] Yu-Min Chung and Austin Lawson. Persistence curves: A canonical framework for summarizing persistence diagrams. arXiv preprint
2019.
[26] Ann F. Cleves and Ajay N Jain. Robust ligand-hased modeling of the biological targets of known drugs. Journal of Medicinal Chemistry, 49(10))2921-2933 2006.
[27] Tamal Krishna Dey and Yusu Wang. Computational Topology for Data Analysis. Cambridge University Press, 2022.
[28] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010. / 1199,, 2020.
[29] Joseph Durant, Burton Leland, Douglas R Henry, and James G Nourse Reoptimization of mdl keys for use in drug discovery. Journal of Chemical Information and Computer Sciences, 42(1)))),2(()))22) 12(6))1273-12888 2002.
[30] Herbert Edelsbrunner and John Harer. Computational Topology: An Introduction. American Mathematical Society, 2010.
[31] Sean Ekins, Ana C Puhl, Kimberley M Zorn. Thomas R Lane, Daniel P Russo, Jennifer J Klein, Anthony J Hickey, and Alex M Clark. Exploiting machine learning for end-to-end drug discovery and development. Nature Materials, 18(5):435 441, 2019.
[32] Richard A Friesner, Jay L Banks, Robert B Murphy, Thomas A Halgren, Jasna J Klicic, Daniel T Mainz, Matthew P Repasky, Eric H Knoll, Mee Shelley, Jason K Perry, et al. Glide: a new approach for rapid, accurate docking and scoring. 1. method and assessment of docking accuracy. Journal of Medicinal Chemistry, 4737): 1749, 2004.
[33] Marian Gidea and Yuri Katz. Topological data analysis of financial time series: Landscapes of crashes. Physica A: Statistical Mechanics and Its Applications, 491:820-834, 2018
[34] Barbara Giunti. Tda applications lihrary, 2022. tda appplicationss/iirrrr.
https://www. wwwp::/wwwwwtooooooooooo
wwwp://.rorooooo/gge.eo org/ groups/24254122 www.o/rww...gggroo//////2222222
[35] Thomas A Halgren, Robert B Murphy, Richard A Friesner, Hege S
Beard, Leah L Frye, W Thomas Pollard, and Jay Banks. Glide: a new approach for rapid, accurate docking and scoring. enrichment factors in database screening. Journal of Medicinal Chemistry, 4777):1550-1755, 2004.
[36] Lowell H Hall and Lemont B Kier. Electrotopological state indices for atom types: novel combination of electronic, topological, and valence state information. Journal of Chemical Information and Computer Sciences, 35(6):1039 (136))003--144 1045, 1995.
[37] Masahiro Hattori, Yasushi Okuno, Susumu Goto, and Minoru Kanehisa. Development of a chemical structure comparison method for integrated analysis of chemical and genomic information in the metabolic pathways. Journal of the American Chemical Society, 125(39):1 1855-11865. 2003.
[38] Paul CD Hawkins, Geoffrey Skillman, and Anthony Nicholls. Comparison of shape-matehe matching and docking as virtual screening tools. Journal of Medicinal Chemistry, 50(1):74 82, 2007.
[39] Felix Hensel, Michael Moor. and Bastian Rieck. A survey of topological machine learning methods. Frontiers in Artificial Intelligence, 4:52, 2021.
12

---

[40] Jéröme Hert, Peter Willett, David J Wilton, Pierre Acklin, Kamal Azzaoui, Edgar Jacoby, and Ansgar Schuffenhauer. New methods for ligand-based virtual screening: use of data fusion and machine learning enhance the effectiveness of similarity searching. Journal of Chemical Information and Modeling. 46(2):422477) 470, 2006.
[41] Christoph Hofer, Florian Graf, Bastian Rieck, Mare Niethammer, and Roland Kwitt. Graph filtration learning. In JCML, pages
PMLR, 2020.
[42] Bingjie Hu, Xiaolei Zhu, Lyman Monroe, Mark Bures, and Daisuke Kihara, Pl-patchsurfer: novel molecular local surface-based method for exploring protein-ligand interactions. International Journal of Molecular Sciences, 15(9):15122-11 151155, 2014.
[43] Takashi lchinomiya, Ippei Obayashi, and Yasuaki Hiraoka. Persistent homology analysis of craze formation. Physical Review E, 95(1):112544 2017.
[44] Fergus Imrie, Anthony R Bradley, Mihaela van der Schaar, and Charlotte M Deane. Protein -peeii-specific models using deep neural networks and transfer learning improve virtual screening and highlight the need for more data. Journal of Chemical Information and Modeling, 58(11):2319 19-1111:211233311 2330,, 2018.
[45] Peiran Jiang, Ying Chi, Xiao-Shuang Li, Xiang Liu, Xian-Sheng Hua, and Kelin Xia. Molecular persistent spectral image (mol-psi) representation for machine learning models in drug design. Briefings Bioinformatics, 23(1) (3(1)bbab527, 2022.
[46] Tian Jiang, Meichen Huang. Ignacio Segovia-Dominguez Nathaniel Newlands, and Yulia R Gel. Learning space-time crop yield patterns with zigzag persistence- based Istm: Toward more reliable digital agriculture insurance. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 36, pages 12538- 12544, 2022.
[47] Wengong Jin, Regina Barzilay, and Tommi Jaakkola. Hierarchical generation of molecular graphs using structural motifs. In /CML., pages 4839-4848. PMLR, 2020.
[48] Megan Johnson and Jac-Hun Jung. Instability of the betti sequence for persistent homology and a stabilized version of the betti sequence. arXiv preprint arXiv:2109. 09210198,,9, 2021.
[49] Gareth Jones, Peter Willett, Robert Glen, Andrew R Leach, and Robin Taylor Development and validation of genetic algorithm for flexible docking, Journal of Molecular Biology, 267(3):277-748,
1997.
[50] Bryn Keller, Michael Lesnick, and Theodore L Willke. Persistent homology for virtual screening. 2018.
[51] Michael Kerber and Alexander Rolle. Fast minimal presentations of bi-graded persistence modules. In ALENEX, pages 207-220. SIAM, 2021.
[52] Talia B Kimber, Yonghui Chen, and Andrea Volkamer. Deep learning in virtual screening: recent applications and developments. International Journal of Molecular Sciences, 22(5) 2021.
[53] Justin Klekota and Frederick P Roth. Chemical substructures that enrich for biological activity. Bioinfor- matics, 24(21):233 2:211:222222 2008
[54] Nils M Kriege, Fredrik D Johansson, and Christopher Morris. A survey on graph kemels. Applied Network Science, 5(1):1-42, 2020.
[55] Gregory Leibon, Scott Pauls, Daniel Rackmore, and Robert Savell. Topological structures in the equities market network. Proceedings of the National Academy of Sciences, 20589)229289225994 2008.
[56] Christian Lemmen and Thomas Lengauer, Computational methods for the structural alignment of molecules. Journal of Computer-Aided Molecular Design, 14(3):215-32,, 2000. 2000.
[57]
M Lesnick. Multiparameter persistence lecture notes, 2019. https://ww.albant caaibany.ddae edu/1ML644116
AMAT 840 840_Spring._aaaee httpa//0a00//M0MMMaaa00 Cottetooooottee_eeeeeeeee .pf
[58]
Michael Lesnick and Matthew Wright, Computing minimal presentations and bigraded betti numbers of 2- parameter persistent homology. arXiv preprint 1902.00090 2019.
[59] Sunhyuk Lim, Facundo Mcmoli, and Osman Berat Okutan. Victoris-rips persistent homology, injective metric spaces, and the filling radius. arXiv preprint arXiv:2001 0005888,..88,.. 2020.
[60] Liyuan Liu, Xiaodong Liu. Jianfeng Gao, Weizhu Chen, and Jiawei Han. Understanding the difficulty of training transformers. arXiv preprint arX1v:2004.08249, 2020.
13

---

[61] Xiang Liu, Huitao Feng. Jie Wu, and Kelin Xia. Dowker complex based machine learning (deml) models for protein-ligand binding affinity prediction. PLoS Computational Biology. 18(4):e1009943 2022.
[62] Xiang Liu and Kelin Xia. Neighborhood complex based machine learning (neml) models for drug design. Interpretability of Machine Intelligence in Medical Image Computing, and Topological Data Analysis and Its Applications for Medical Data, pages 87-97. Springer, 2021.
[63] Zhuang Liu, Hanzi Mao, Chao- Yuan Wu, Christoph Feichtenhofer, Trevor Darrell, and Saining Xie. A convnet for the 2020s. arXiv preprint arXiv:2201 003545, 2022.
[64] Krzysztof Maziarz, Henry Jackson -lux,, Pashmina Cameron, Finton Sirockin, Nadine Schneider, Niko- laus Stiefl, Marwin Segler, and Mare Brockschmidt. Learning to extend molecular scaffolds with structural motifs, arXiv preprint arXiv:2103.080: 2021.
[65] James L Melville, Edmund K Burke, and Jonathan D Hirst. Machine learning in virtual screening. Combinatorial Chemistry & High Throughput Screeming, 12(4):332-343, 2009.
[66] Zhenyu Meng and Kelin Xia. Persistent spectral-based machine learning (perspect ml) for protein-ligand binding affinity prediction. Science Advances, 7(19)cabc3322 2021.
[67] Michael M Mysinger, Michael Carchia, John Irwin, and Brian K Shoichet. Directory of useful decoys, enhanced (dud-e): better ligands and decoys for better benchmarking. Journal of Medicinal Chemistry. 55(14):6582-655) 6694, 2012.
[68] Takenobu Nakamura, Yasuaki Hiraoka, Akihiko Hirata, Emerson G Escolar, and Yasumasa Nishiura. Per- sistent homology and many-body atomic structure for medium-range order in the glass. Nanotechnology, 36(30):30401,, 2015.
[69] Bruno J Neves, Rodolpho C Braga, Cleber Melo-Filho, José Teófilo Moreira-Filho, Fagene N Muratov, and Carolina Horta Andrade. Qsar-based virtual screening: advances and applicatioos drug discovery. Frontiers in Pharmucology. 9:1275, 2018.
[70] Due Duy Nguyen, Zixuan Cang, and Guo-Wei Wei. A review of mathematical representations of biomolecular data. Physical Chemistry Chemical Physics, 22(8):4343-4367, 2020.
[71] Due Duy Nguyen. Zixuan Cang, Kedi Wu, Menglun Wang, Yin Cao, and Guo-We Wei. Mathematical deep learning for pose and binding affinity prediction and ranking in d3r grand challenges. Journal of Computer-Aided Molecular Design, 33(1)-11-8,, 2019.
[72] Duc Duy Nguyen, Kaifu Gao, Menglun Wang, and Guo-Weii Wei. Mathdl: mathematical deep learning for d3r grand challenge 4. Journal of Computer-Aided Molecular Design, 34(2):131-144 2020.
[73] Dorcas cal anomaly detection in dynamic multilayer blockchain networks. In ECML PKDD, pages 788 804, 2021.
Segovia Dominguez. C Akcora, Murat Kantarcioglu, and Yulia R Gel. Topologi-
[74] Nina Otter, Mason Porter, Ulrike Tillmann, Peter Grindrod, and Heather Harrington. A roadmap for the computation of persistent homology. EPJ Data Science, 6:1-38, 2017.
[75] Yunierkis Perez-Castillo, Stellamaris Sotomayor-Burneo, Karina Jimenes- Vargas, Mario Gonzalez- Rodriguez, Maykel Cruz-Montaauudo, Vinicio Armijos-Jaramillo, M Natália DS Cordeiro, Fernanda Borges, Aminael Sánchez- Sánchez-Rodriguez, and Eduardo Tejera. Compscore: boosting structure-based virtual screening performance by incorporating docking scoring function components into consensus scoring. Journal of Chemical Information and Modeling, 59(9):3655-3666 2019.
[76] Ralph H Petrucci, F Geoffrey Herring, and Jeffry D Madura. General Chemistry: Principles and Modern Applications. Pearson Prentice Hall, 2010.
[77] Matthew Ragoza, Joshua Hochuli, Elisa Idrobo, Jocelyn Sunseri, and David Ryan Koes. Protein-ligand scoring with convolutional neural networks. Journal of Chemical Information and Modeling, (7))9442 957, 2017.
[78] John W Raymond, Eleanor Gardiner, and Peter Willett. Rascal: Calculation of graph similarity using maximum common edge subgraphs. The Compuder Journal, 45(6))631 2002.
[79] Peter Ripphausen, Britta Nisius, and Jürgen Bajorath. State-of-the-arr in ligand-based virtual screening. Drug Discovery Today,
2011.
14

---

[80] David Rogers and Mathew Hahn. Extendedd-comnectiitt fingerprints. Journal of Chemical Information and Modeling, 50(5):442-75,, 2010.
[81] Julian Schwartz, Mahendra Awale, and Jean-Louis Reymond. Smifp (smiles fingerprint) chemical space for virtual screening and visualization of large databases of organic molecules. Journal of Chemical Information and Modeling. 53(8):197--1889, 2013.
[82] Chao Shen, Junjie Ding, Zhe Wang, Dongsheng Cao, Xiaoqin Ding, and Tingjun Hou. From machine learning to deep learning: Advances in scoring functions for protein-ligand docking. Wiley Interdisci- plinary Reviews: Computational Molecular Science, 10(1):e1429, 2020.
[83] Zhuoran Shen, Mingyuan Zhang, Haiyu Zhao, Shuai Yi, and Hongsheng Li. Efficient attention: Attention with linear complexities. In Proceedings of the IEEE/CVF Winter Conference Applications Computer Vision, pages 3531-3539, 2021.
[84] Woong- Hoe Shin, Xiaolci Zhu, Mark Gregory Bures, and Daisuke Kihara. Three- compound comparison methods and their application in drug discovery. Molecules, 20(7): 11111-228111666 2015.
[85] Christoph Steinbeck, Christian Hoppe. Stefan Kuhn, Matteo Floris, Rajarshi Guha, and Egon L Wil- lighagen. Recent developments of the chemistry development kit (cdk)-an open-source java library for chemo-and bioinformatics. Current Pharmaceutical Design, 12(17)211112210, 2006.
[86] Vladimir B Sulimov, Danil C Kutov, and Alexey V Sulimov. Advances in docking. Current Medicinal Chemistry, 26(42):7555-755 7580, 2019.
[87] Jocelyn Sunseri and David Ryan Koes. Virtual screening with gnina 1.0. Molecules, 26(23):7369, 2021.
[88] Dominique Sydow, Lindsey Burggraaff, Angelika Szengel, Herman WT van Vlijmen, Adriaan P Ilzerman., Gerard JP van Westen, and Andrea Volkamer. Advances and challenges in computational target prediction. Journal of Chemical Information and Modeling, 59(5):1728 1742, 2019.
[89] Ashleigh Linnea Thomas. Invariants and Metrics for Multiparameter Persistent Homology. PhD thesis,
Duke University, 2019.
[90] Jean-Frangois Truchon and Christopher Bayly. Evaluating virtual screening methods: good and bad metrics for the "early recognition" problem. Journal of Chemical Information and Modeling, 47(2))488-
508, 2007.
[91] Jessica Vamathevan, Dominic Clark, Paul Czodrowski, Ian Dunham, Edgardo Ferran, George Lee, Bin Li, Anant Madabhushi, Parantu Shah, Michaela Spitzer, et al. Applications of machine learning in drug discovery and development. Nature Reviews Drug Discovery, 18(6)463-477, 2019.
[92] Vishwesh Venkatraman, Padmasini Ramji Chakravarthy, and Daisuke Kihara. Application of 3d zemike descriptors to shape-based ligand similarity scarching. Journal of Cheminformatiis, 1(1):1-19, 2009.
[93] Oliver Vipond. Multiparameter persistence landscapes. Journal of Machine Learning Research, 21:61-1, 2020.
[94] Dingyan Wang. Chen Cui, Xiaoyu Ding. Zhaoping Xiong, Mingyue Zheng, Xiaomin Luo, Hualiang Jiang, and Kaixian Chen. Improving the virtual screening ability of target-specific scoring functions using deep learning methods. Frontiers in Pharmacology, page 924, 2019.
[95] LIU Xiang and Kelin Xia. Persistent tor-algebra based stacking ensemble learning (pta-sel) for protein- protein binding affinity prediction. In ICLR 2022 Workshop on Geometrical and Topological Representa- tion Learning. 2022.
[96] Guo-Li Xiong, Wen-Ling Ye, Chao Shen, Ai-Ping Lu, Ting-Jun Hou, and Dong-Sheng Cao. Improving structure-based virtual screeming performance via learning from scoring function components. Briefings Bioinformatics, 22(3):bbaa094, 2021.
[97] Chun Wei Yap. mndel-descriptors An open source software to calculate molecular descriptors and fingerprints. Journal of Computational Chemistry, 32(7):1466-144 2011.
[98] Nobuaki Yasuo and Masakazu Sekijima. Improved method of structure based virtual screening via interaction-eneery- based learning. Journal of Chemical Information and Modeling, 105-)111 2019.
[99] Monisha Yuvaraj. Asim K Dey, Vyacheslav Lyubchich, Yulia R Gel, and H Vincent Poor. Topological clus- tering of multilayer networks. Proceedings of the National Academy of Sciences, 11881)):201199941188 2021.
15

---

[100] Qi Zhao and Yusu Wang. Learning metrics for persistence-basee summaries and applications for graph classification. In NeurIPS, volume 32, 2019.
[101] Hongyi 7hou, Hongnan Cao, and Jeffrey Skolnick. Findsitecomb2. 0: A new approach for virtual ligand screening of proteins and virtual target screening of biomolecules. Journal of Chemical Information and Modeling, 58(1 (0(11):2343-33555 2018.
[102] Hongyi Zhou, Hongnan Cao, and Jeffrey Skolnick. Fragsite: fragment-based approach for virtual ligand screening. Journal of Chemical Information and Modeling, 61(4)20074 2089, 2021.
[103] Vincent Zoete, Antoine Daina, Christophe Bovigny, and Olivier Michielin. Swisssimilarity: web tool for low to ultra high throughput Ligand-based virtual screening. Journal of Chemical Information and Modeling, 56(8): 1399-1404, 2016.
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Reviews:
[{"review_id": "cJUuBYwgALE", "reviewer": "Reviewer_sEym", "summary": "In this paper, a new type of molecular fingerprint is introduced, and its application in the virtual screening setup is shown. The fingerprint is based on multiparameter persistence homology that produces topological descriptions of molecules in the form of multidimensional vectors. The methodology expands single persistence methods and applies multiple filtering functions, which can be based either on atom distances or chemical properties. Architectures such as Vision Transformer and ConvNeXt are used to process the multidimensional fingerprints. Finally, the triplet loss is used to train these models for the virtual screening task. The method is compared against many baselines on two datasets. Additionally, the computational complexity is calculated, and the stability of the multiparameter persistence fingerprints is formally proven in the supplementary material.", "questions": "1. How were the atomic properties (atomic mass, partial charges) selected for the evaluation? Could you visualize graph hierarchies w.r.t. these filtration functions for some example compounds?\n2. In section 6.1 it is said that “ViT is more robust to distinct arrangements of atoms in space, also referred to as molecular conformation.” Are conformations encoded in the fingerprint? Based on the description in section 4, the fingerprints use only graph edge distances.\n3. How is the triplet loss used to predict the ligand ranking, which is used to calculate EF scores?\n", "limitations": "In Appendix D, only positive societal impacts are presented. The computational complexity is a limitation that was analyzed in Section 6.3, but not mentioned in the section about method limitations. The number of atomic properties used as filtering functions not only increases the cost of calculating the fingerprint, but also requires more memory and processing power of the ML models.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The theoretical foundations of the paper are sound. The persistence homology is thoroughly explained, and the modifications are presented using a clear mathematical notation (though some definitions may be found only in the supplementary materials).\n- The stability of the proposed fingerprint is formally proven.\n- The experimental evaluation on two datasets shows the strong performance of the proposed method. The number of baseline models in the comparison is impressive.\n- A section about computational complexity was included in the paper.\n- The ablation study corroborates the usefulness of multiparameter persistence and explains architectural choices. \n\nWeaknesses:\n- Unfortunately, the method is not presented in a clear way. Some of the details were moved to the supplementary materials, so it is difficult to grasp the idea before reading the appendix and rereading the paper. For example, VR-filtration is not explained in the main text, but this term is used multiple times. Maybe it would be a good idea to add a figure explaining how the fingerprint is calculated (showing VR-filtering on one axis and property-based filtering on the other one).\n- The proposed method shows a great improvement over the other method included in the comparison, but these methods are mostly scoring functions that were trained with a significantly different objective. Please, correct me if I am wrong, but I think that most of these methods use docking poses to score ligands and this is how they build their rankings. The MP Fingerprint, on the other hand, is trained to separate active ligands from decoys only based on the topological features and ignoring the 3D conformations. It would be useful to add a few simple baselines, e.g. a random forest model on Morgan fingerprints (similar to the ablation study, but I cannot find the information about which model was used in this experiment).\n- Section 6.4 mentions a comparison of more contrastive objectives, but the (quantitative) results of the preliminary experiments are not included in the paper.\n- The code is not provided, which makes the results difficult to reproduce.\n\nMinor comments:\n- In the abstract it is said that “VS is used for comparing a library of compounds against known active ligands to identify drug candidates…” This description of VS is inaccurate as it is not the only way to perform virtual screening. For example, there are structure-based approaches to the VS that can be performed even if no active ligands are known.\n- It would be interesting to see the performance of the MP fingerprints in different benchmarks, e.g. ADMET property prediction.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Qrb9bOYNphv", "reviewer": "Reviewer_WV81", "summary": "\nThe authors proposed a new topological fingerprint in virtual screening to understand structural organization of chemical compounds. The method is based on multiparameter persistence homology, which produces topological fingerprints of the compounds as multidimensional vectors. They show that the proposed topological descriptors outperform the state-of-the-art methods on benchmark datasets.\n", "questions": "The authors should use popular performance measures such as AUC scores of ROC curve and precision-recall curve. I'm afraid that researchers in the NeurIPS community are not familiar with the performance measures used in this paper (e.g., enrichment factor alpha %).\n\nThere are many fingerprints and descriptors in chemoinformatics. Examples include the following papers:\nFCFP (Rogers et al, J. Chem. Inf. Model., 2010)\nCDK graph (Yap et al, J. Comput. Chem., 2011)\nCDK hybridization (Steinbeck et al, Curr. Pharm. Des., 2006)\nKlekota-Roth (Klekota et al, Bioinformatics, 2008)\nMACCS (Durant et al, J. Chem. Inf. Comp. Sci., 2002)\nPubChem (Chen et al, J. Chem. Inf. Model., 2009)\nE-state (Hall et al, J. Chem. Inf. Comp. Sci., 1995)\nThe proposed fingerprint is better than the other fingerprints?\n\nThe prediction accuracy in virtual screening depends too much on the number of positive examples (active compounds) in the training set. In my experience, transformer-based methods do not work well when there are few positive examples in the training set.\n\nIt is hard to follow this paper.\nThe explanations on the detailed methods are not clear.\n\nThe output of the propose method is not clear.\nWhat is the dimension/size of the proposed topological fingerprint?\n\nThe number of compounds in the chemical library is often huge. How is the scalability of the proposed method?\n", "limitations": "The paper does not discuss the limitations of the proposed method clearly. \nThe performance is not rigorously verified. \n", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Fingerprinting is an important research topic in drug discovery.\nThe proposed method provided high accuracy.\n\nThere are many previous works on chemical fingerprints and descriptors, but the relationship with related works is not clear.\nThe authors did not compare the performance with many other fingerprints.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "1wtup6iKbMz", "reviewer": "Reviewer_ACvS", "summary": "In this paper, the authors develop multiparameter persistence (MP) homology based topological fingerprints for virtual screening. In their model, a compound is decomposed into chemical substructures at different scales and their topological information is extracted by using MP homology features. These topological fingerprints are combined with learning models. Their models are extensively tested on two benchmark datasets, i.e., Cleves-Jain and DUD-E Diverse dataset. It has been found that their models have better performance than existing models. ", "questions": "The major issue is the missing of important literature in the related work. Even though the authors have mentioned some TDA-based drug design models, i.e., refs 13-15, they have been only very briefly mentioned in the supplementary information. Further, these works are from 2017 to 2018. In fact, recently, there are great progresses in TDA-based drug design (and virtual screening). The authors should discussed them in the related works.  \n\nFor instance,\n\n1) Duc Duy Nguyen, Zixuan Cang, Kedi Wu, Menglun Wang, Yin Cao  and Guo-Wei Wei, Mathematical deep learning for pose and binding affinity prediction and ranking in D3R Grand Challenges, Journal of Computer Aided Molecular Design, 33, 71-82   (2019).\n\n2) Duc Nguyen, Zixuan Cang, and Guo-Wei Wei, A review of mathematical representations of biomolecular data, Physical Chemistry Chemical Physics, 22, 4343-4367 (2020).\n\n3) Duc Duy Nguyen, Kaifu Gao, Menglun Wang, and Guo-Wei Wei, MathDL: Mathematical deep learning for D3R Grand Challenge 4, Journal of Computer Aided Molecular Design,  34, 131-147 (2020).\n\n4) Zhenyu Meng and Kelin Xia, \"Persistent spectral–based machine learning (PerSpect ML) for protein-ligand binding affinity prediction.\" Science Advances, 7 (19), eabc5329 (2021)\n\n5) Peiran Jiang, Ying Chi, Xiao-Shuang Li, Xiang Liu, Xian-Sheng Hua, and Kelin Xia, \"Molecular persistent spectral image (Mol-PSI) representation for machine learning models in drug design.\" Briefings in Bioinformatics, 23 (1), bbab527 (2022)\n\n6) Xiang Liu, Huitao Feng, Jie Wu, and Kelin Xia, \"Dowker complex based machine learning (DCML) models for protein-ligand binding affinity prediction.\" PLOS Computational Biology, 18(4), e1009943 (2022)\n\n7) Xiang Liu, and Kelin Xia, \"Persistent Tor-algebra based stacking ensemble learning (PTA-SEL) for protein-protein binding affinity prediction\", ICLR 2022 Workshop on Geometrical and Topological Representation Learning (2022)\n\n8) Xiang Liu, and Kelin Xia, \"Neighborhood complex based machine learning (NCML) models for drug design.\" In Interpretability of Machine Intelligence in Medical Image Computing, and Topological Data Analysis and Its Applications for Medical Data, pp. 87-97. Springer, Cham (2021).\n  ", "limitations": "None.", "rating": 9, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The models are innovative and the results are great! However, TDA-based learning models for drug design (including virtual screening) has been a hot research areas for the past decade. There is a great amount of works in this area that is missing in the current paper. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}]

Author rebuttal:

Decision: Accept