ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery
Anonymous Author(s)
AfﬁliationAddress email
Abstract
In computer-aided drug discovery (CADD), virtual screening (VS) is used forcomparing a library of compounds against known active ligands to identify thedrug candidates that are most likely to bind to a molecular target.
Most VS methodsto date have focused on using canonical compound representations (e.g., SMILESstrings, 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 signiﬁcantimprovements in VS performance, these methods suffer from reduced performancewhen scaling to large virtual compound datasets.
The performance of these methods has shown only incremental improvements in the past few years.
To addressthis 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 newtopology-based graph ranking problem by partitioning a compound into chemicalsubstructures informed by the periodic properties of its atoms and extracting theirpersistent homology features at multiple resolution levels.
We show that the marginloss ﬁne-tuning of pretrained Triplet networks attains highly competitive resultsin differentiating between compounds in the embedding space and ranking theirlikelihood of becoming effective drug candidates.
We further establish theoreticalguarantees for the stability properties of our proposed MP signatures, and demonstrate that our models, enhanced by the MP signatures, outperform state-of-the-artmethods 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).
Introduction
Drug discovery is the early phase of the pharmaceutical R&D pipeline where machine learning (ML)is making a paradigm-shifting impact [30, 82].
Traditionally, early phases of biomedical researchinvolve the identiﬁcation of targets for a disease of interest, followed by high-throughput screening(HTS) experiments to determine hits within the synthesized compound library, i.e., compounds withhigh potential.
Then, these compounds are optimized to increase potency and other desired targetproperties.
In the ﬁnal phases of the R&D pipeline, drug candidates have to pass a series of rigorouscontrolled tests in clinical trials to be considered for regulatory approval.
On average, this processtakes 10-15 years end-to-end and costs in excess of ⇠ 2 billion US dollars [9].
HTS is highly timeand cost-intensive.
Therefore, it is critical to ﬁnd good potential compounds effectively for the HTSstep for novel compound discovery, but also to speed up the pipeline and make it more cost-effective.
To address this need, ML augmented virtual screening (VS) has emerged as a powerful computationalapproach to screen ultra large libraries of compounds to ﬁnd the ones with desired properties andprioritize them for experimentation [59, 38].
[Table]
[Figure]
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 topologicalﬁngerprints of compounds as 2 D or 3 D vectors (same size matrices or arrays) by using TDAtools, i e., multidimensional persistent homology.
Then, we show that Triplet networks, (wherestate-of-the-art pretrained transformer-based models and modernized convolutional neural networkarchitectures serve as the backbone and distinct topological features allow to represent supportand query compounds), successfully identify the compounds with the desired properties.
We alsodemonstrate that the applicability of topological feature maps can be successfully generalized totraditional ML algorithms such as random forests.
The distinct advantage of TDA tools, in particular persistent homology (PH), is that it enables effectiveintegration of the domain information such as atomic mass, partial charge, bond type (single, double,triple, aromatic ring), ionization energy or electron afﬁnity, which carry vital information regarding thechemical properties of a compound at multiple resolution levels during the graph ﬁltration step.
Whilecommon PH theory allows only one such domain function to be used in this process, with our novelmultipersistence approach, we show it is possible to use more than one domain function.
Topologicalﬁngerprints can effectively carry much ﬁner chemical information of the compound structure informedby the multiple domain functions embedded in the process.
Speciﬁcally, multiparameter persistencehomology decomposes a 2 D graph structure of a compound into a series of subgraphs using domainfunctions and generates hierarchical topological representations in multiple resolution levels.
At eachresolution stage, our algorithm sequentially generates ﬁner topological ﬁngerprints of the chemicalsubstructures.
We feed these topological ﬁngerprints to suitable ML/DL methods, and our ToDDmodels achieve state-of-the-art in all benchmark datasets across all targets (See Table 1 and 2).
The key contributions of this paper are:
• We develop a transformative approach to generate molecular ﬁngerprints.
Using multipersistence, we produce highly expressive and unique topological ﬁngerprints for compoundsindependent of scale and complexity.
This offers a new way to describe and search chemicalspace relevant to both drug discovery and development.•
We bring a new perspective to multiparameter persistence in TDA and produce a computationally efﬁcient multidimensional ﬁngerprint of chemical data that can successfully
incorporate more than one domain function to the PH process.
These MP ﬁngerprintsharness the computational strength of linear representations and are suitable to be integratedinto a broad range of ML, DL, and statistical methods; and open a 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 singlepersistence. • We perform extensive numerical experiments in VS, showing that our ToDD models outperform all state-of-the-art methods by a wide margin (See Figure 1).
Related Work
Virtual Screening A key step in the early stages of the drug discovery process is to ﬁnd activecompounds that will be further optimized into potential drug candidates.
One prevalent computationalmethod that is widely used for compound prioritization with desired properties is virtual screening(VS).
There are two major categories, i.e., structure-based virtual screening (SBVS) and ligand-basedvirtual screening (LBVS) [20].
SBVS uses the 3 D structural information of both ligand (compound)and target protein as a complex [11, 50, 77].
SBVS methods generally require a good understandingof 3 D -structure of the target protein to explore the different poses of a compound in a binding pocketof the target.
This makes the process computationally very expensive.
On the other hand, LBVSmethods compare structural similarities of a library of compounds with a known active ligand [70, 62]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 ﬁngerprintsof 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 ofthe target receptor is not known [54].
In the last 3 decades, various LBVS methods have been developed with different approaches andthese can be categorized into 3 classes depending on the ﬁngerprint they produce: SMILES [72]and SMARTS [28] are examples of 1 D -methods which produce 1 D -ﬁngerprints, compressingcompound information to a vector.
RASCAL [69], MOLPRINT2D [8], ECFP [71], SWISS [92] and
SIMCOMP [35] are among 2 D methods which uses 2 D -structure ﬁngerprint and graph matching.
Finally, examples of 3 D -methods are ROCS [36], USR [7], PatchSurfer [42] which use the 3 Dstructure of compounds and their conformations ( 3 D -position of the compound) [75].
On the otherhand, while ML methods have been actively used in the ﬁeld for the last two decades, new deeplearning methods made a huge impact in drug discovery process in the last 5 years [79, 50, 73].
Further discussion of state-of-the-art ML/DL methods are given in Section 6 where we compare ourmodels and benchmark against them.
Earlier TDA methods in VS are also discussed in Appendix A.2.
Topological Data Analysis (TDA) and tools of persistent homology (PH) have recently emergedas powerful approaches for ML, allowing us to extract complementary information on the observedobjects, especially, from graph-structured data.
In particular, PH has become popular for various ML tasks, ranging from bioinformatics [51, 86, 63], ﬁnance [53, 32], material science [43, 61]to network analysis [76, 41, 19] (See the literature overviews [64, 5, 22] and TDA applicationslibrary [33]).
Furthermore, it has become a highly active research area to integrate PH methods intogeometric deep learning (GDL) in recent years [39, 40, 89, 18].
Most recently, the emerging conceptsof 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 granulartopological descriptors of the data.
However, the MP theory is not sufﬁciently mature as it suffersfrom the nonexistence of the barcode decomposition relating to the partially ordered structure of theindex set { ( ↵ i ,  j ) } [55, 80].
The existing approaches remedy this issue via slicing technique bystudying one-dimensional ﬁbers of the multiparameter domain [17], but choosing these directionssuitably and computing restricted SP vectorizations are computationally costly which makes theapproach inefﬁcient in real life applications.
There are several promising recent studies in thisdirection [10, 84, 23], but these approaches fail to provide a practical topological summary such as“multipersistence diagram”, and an effective MP vectorization to be used in real life applications.
Background
We ﬁrst provide the necessary TDA background for our machinery.
While our techniques areapplicable to various forms of data, e.g., point clouds and images (for details, see Section B.2), herewe focus on the graph setup in detail with the idea of mapping the atoms and bonds that make up acompound into a set of nodes and edges that represent an undirected graph.
3.1 Persistent Homology
Persistent homology (PH) is a key approach in TDA, allowing us to extract the evolution of subtlerpatterns in the data shape dynamics at multiple resolution scales which are not accessible with more conventional, non-topological methods [16].
In this part, we go over the basics of PH machinery ongraph structured data.
For further background on PH, see Appendix A.1 and [26, 29].
For a given graph G , consider a nested sequence of subgraphs G 1 ✓ . . .
✓ G N = G .
For each
G i , deﬁne an abstract simplicial complex b G i , [Equation]  i  N , yielding a ﬁltration , a nested sequenceof simplicial complexes
[Equation] b G N .
This step is crucial in the process as one can injectdomain 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.1).
After getting a ﬁltration, one can systematically keep track of the evolution of topologicalpatterns in the sequence of simplicial complexes { b G i } Ni =1 .
A k -dimensional topological feature 136 (or k -hole) may represent connected components ( 0 -dimension), loops ( 1 -dimension) and cavities( 2 -dimension).
For each k -dimensional topological feature  , PH records its ﬁrst appearance inthe ﬁltration sequence, say b G b  , and ﬁrst disappearence in later complexes, b G d  with a unique pair 139 ( b  , d  ) , where b  < d   N .
We call b  the birth time of  and d  the death time of  .
We call 140 d   b  the life span (or persistence) of  .
PH records all these birth and death times of the topologicalfeatures in persistence diagrams .
Let 0  k  D where D is the highest dimension in the simplicialcomplex b G N .
Then k th persistence diagram PD k
( G ) =
{ ( b  , d  ) |  2 H k ( b
G i ) for b   i < d  } .
Here, H k ( b G i ) represents the k th homology group of b G i which keeps the information of the k -holes inthe simplicial complex b G i .
Most common dimensions used in practice are 0 and 1 , i.e., PD 0 ( G ) and 145 PD 1 ( G ) .
For sake of notations, further we skip the dimension (subscript k ).
With the intuition thatthe topological features with long life span (persistent features) describe the hidden shape patterns inthe data, these persistence diagrams provide a unique topological ﬁngerprint of G .
We give the furtherdetails of the PH machinery and how to integrate domain information to the process in Appendix A.1.
3.2 Multidimensional Persistence
MultiPersistence (MP) signiﬁcantly boosts the performance of single parameter persistence techniquedescribed in Appendix A.1.
The reason for the term “single” is that we are ﬁltering the data inonly one direction G 1 ⇢ · · · ⇢ G N = G .
As explained in Appendix A.1, the construction of theﬁltration is the key step to inject domain information to process and to ﬁnd the hidden patterns of thedate.
If one uses a function f : V ! R which has valuable domain information, then this induces asingle parameter ﬁltration as above.
However, various data have more than one very natural domainfunctions to analyze the data, and using them simultaneously would give a much better understandingof the hidden patterns.
For example, if we have two functions f, g : V ! R (e.g., atomic massand partial charge) with very valuable complementary information of the network (compound), MPidea is presumed to produce a unique topological ﬁngerprint combining the information from bothfunctions.
These pair of functions f, g induces a multivariate ﬁltering function F : V ! R 2 with
F ( v ) =
( f ( v ) , g ( v )) .
Again, one can deﬁne a set of nondecreasing thresholds { ↵ i } m 1 and {  j } n 1 for 162 f and g respectively.
Let V ij = { v r 2 V | f ( v r )  ↵ i , g ( v r )   j } , i.e., V ij = F ( v r )  ( ↵ i ,  j ) .
163 Deﬁne
G ij to be the induced subgraph of G by V ij , i.e., the smallest subgraph of G generated by
V ij .
Then, instead of a single ﬁltration of complexes { b G i } , we get a biﬁltration of complexes{ b G ij | 1  i  m, 1  j  n } which is a m ⇥ n rectangular grid of simplicial complexes.
Again,the MP idea is to keep track of the k -dimensional topological features in this grid { b G ij } by using thecorresponding homology groups { H k ( b G ij ) } (MP module).
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 featurein MP grid), and there is no effective way to facilitate this promising idea in real life applications.
Inthe following, we overcome this problem by producing highly effective ﬁngerprints by utilizing theslicing idea in the MP grid in a structured way.
New Topological Fingerprints of the Compounds with Multipersistence
We now introduce our MP Fingerprinting framework, which produces ﬁngerprint of the data asmultidimensional vectors (matrices or arrays) by expanding single persistence (SP) ﬁngerprints(Appendix A.1).
While our construction is applicable and suitable for various forms of data, here wefocus on graphs, and in particular, compounds for virtual screening.
We obtain a matrix (or 3 D array) for each compound as its ﬁngerprint.
We explain how to generalize our framework to other types ofdata in Appendix B.2.
In Appendix B.4, we construct the explicit examples of MP Fingerprints formost popular SP Vectorizations, e.g., MP Betti, MP Landscape and MP Images.
Our framework basically expands a given SP vectorization to a multidimensional vector by utilizing
MP approach.
In technical terms, by using the existing SP vectorizations, we produce multidimensional vectors by effectively using the one of the ﬁltering direction as slicing direction in themultipersistence module.
In our construction, we will use distance (VR-ﬁltration) as our naturalslicing direction.
The motivation for this choice is that VR-ﬁltration captures ﬁne intrinsic structureof the graph by using the pairwise distances between the nodes (atoms).
Here, MP approach allow usto change these atom pairs for VR-ﬁltration with respect to the chosen domain functions.
Here, we describe our construction for 2 D multipersistence.
The construction can easily be extendedto 3 D or higher dimensions (See Appendix B.3).
This is a three step process.
The ﬁrst step is to deﬁnethe ﬁltrations and induced simplicial complexes.
Let G =
( V , E ) be a graph, and let f : V ! R 191 be a ﬁltering function (e.g., atomic mass, partial charge, bond type, electron afﬁnity, ionizationenergy) with threshold sets { ↵ i } mi =1 .
Let V i
= { v r 2 V | f
( v r )  ↵ i } .
This deﬁnes a hierarchy
V 1 ⇢ V 2 ⇢ · · · ⇢ V m =
V among the nodes with respect to the function f .
Before constructingsimplicial complexes, compute the distances between each node in graph G , i.e., d ( v r , v s )
=
d rs .
Let K = max d rs .
Then, for each 1  i 0  m , deﬁne VR-ﬁltration for the vertex set V i 0 with thedistances d ( v r , v s ) =
d rs as described in Section 3.1, i.e.,  i 0 {+
+}
✓  i 0 1 . . .
✓  i 0 K .
This gives 197 m ⇥ ( K + 1) simplicial complexes {  ij } where 1  i  m and 0  j  K .
This is called the 198 bipersistence module .
One can imagine increasing sequence of {V i } as vertical direction, and induced 199 VR-complexes {  ij } as the horizontal direction.
In our construction, we ﬁx the slicing directionas the horizontal direction (VR-direction) in the bipersistence module, and obtain the persistencediagrams in these slices.
The second step is to obtain persistence diagrams.
For 1  i 0  m , we have a single ﬁltration 203  i 0 {+
+}
✓  i 0 1 . . .
✓  i 0 K (horizontal slice in the bipersistence module).
Each such single ﬁltration 204 induces a persistence diagram PD ( V i ) = { ( b j , d j ) | {+
 +} b j < d j  K } .
This produces m 205 persistence diagrams { PD ( V i ) } .
Note that one can consider PD ( V i ) as the persistence diagram of 206 the "point cloud" V i where the pairwise distances induced by the original graph G .
The ﬁnal step is to use the vectorization on these m persistence diagrams.
Let ' be a singlepersistence vectorization, e.g., Betti, Silhouette, Entropy, Persistence Landscape or Persistence Image.
By applying the chosen SP vectorization ' to each PD, we obtain a function ' i = ' ( PD ( V i )) wherein most cases it is a single variable function on the threshold domain [0 , K ] , i.e., ' i : [0 , K ] ! R .
Note that for a collection of graphs, K is chosen at the beginning of the process for all and do notvary with the graph, e.g. it can be chosen as the largest diameter of the graphs or sufﬁciently largenumber close to it.
As most such vectorizations are induced from a discrete set of points PD ( G ) , theycan naturally be expressed as a 1 D vector of length r .
In most cases, r = K or r = 2 K dependingon the vectorization chosen, e.g., r = K for Betti Curve and r = 2 K for Persistence Landscape.
Inthe examples in Section B.4, we explain this conversion explicitly.
Then, let ~' i be the corresponding1 ⇥ r vector for the function ' i .
Now, we are ready to deﬁne our topological ﬁngerprint M ' whichis a 2 D -vector (a matrix)
[Equation] where M i ' is the i th -row of M ' .
Hence, M ' is a 2 D -vector of size m ⇥ r .
Each row M i ' is thevectorization of the persistence diagram PD ( V i ) via the SP vectorization method ' .
We use the ﬁrstﬁltering function f to get a ﬁner look at the graph as it deﬁnes the subgraphs G 1 ✓ . . .
✓ G m =
G 222 where
G i is the subgraph induced by the vertices V i .
Then, we use the distance function to obtain thetopological signatures ~' i = ~' ( PD ( V i )) of these subgraphs {G i } deﬁned by the domain function f .
224 One can consider the subgraphs
G i in G induced by the vertices V i are the substructures deﬁned by 225 the domain function f .
Then our ﬁngerprints are indeed topological signatures of these substructuresinduced by f .
In a way, we look at G with a 2 D resolution and we keep track of the evolution of topological featuresin the induced biﬁltration {  ij } .
The main advantage of this technique is that the outputs are ﬁxedsize matrices (or arrays) for each dataset which are quite suitable for various ML/DL models.
4.1 Stability of the MP Fingerprints
We further show that when the source single parameter vectorization ' is stable, then so is its induced
MP Fingerprint M ' .
(We give the details of stability notion in persistence theory and proof of thefollowing theorem in Section B.1.)
Theorem: Let ' be a stable SP vectorization.
Then, the induced MP Fingerprint M ' is also stable, i.e., with the notation introduced in Section B.1, there exists b C ' > 0 such that for any pair of graphs
G + and G  , we have the following inequality.
[Equation]
Datasets
Cleves-Jain: This is a relatively small dataset [25] that contains 1149 compounds.
There are22 different drug targets, and for each one of them the dataset provides only 2-3 template activecompounds dedicated for model training.
All targets { q } are associated with 4 to 10 active compounds{ L q } dedicated for model testing.
Additionally, the dataset contains 850 decoy compounds ( D ).
Theaim is for each target q , by using the templates, to ﬁnd the actives L q among the pool combined withdecoys
L q [ D . In other words, same decoy set D is used for all targets.
DUD-E Diverse: DUD-E (Directory of Useful Decoys, Enhanced) dataset
[60] is a comprehensiveligand dataset with 102 targets and approximately 1.5 million compounds.
The targets are categorizedinto 7 classes with respect to their protein type.
The "Diverse subset" of DUD-E contains targetsfrom each category to give a balanced benchmark dataset for VS methods.
Diverse subset contains116,105 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.1.
Experiments6.1 Setup
Macro Design We construct different ToDD (Topological Drug Discovery) models, namely ToDDViT, ToDD-ConvNeXt and ToDD-RF to test the generalizability and scalability of topological featureswhile employing different ML models and training datasets of various sizes.
Many neural networkarchitectural choices and ML models can be incorporated in our ToDD method.
ToDD-ViT
and
ToDD-ConvNeXt are Triplet network architectures with Vision Transformer (ViT_b_16) [27] and
ConvNeXt_tiny models [58], pretrained on ILSVRC-2012 ImageNet, serving as the backbone of the
Triplet network.
MP signatures of compounds are applied nearest neighbour interpolation to increasetheir resolutions to 224 2 , followed by normalization.
We only use GaussianBlur with kernel size5 2 and standard deviation 0.05 as a data augmentation technique.
Transfer learning via ﬁne-tuning ViT_b_16 and ConvNeXt_tiny models using Adam optimizer with a learning rate of 5e-4, no warmupor layerwise learning rate decay, cosine annealing schedule for 5 epochs, stochastic weight averagingfor 5 epochs, weight decay of 1e-4, and a batch size of 64 for 10 epochs in total led to signiﬁcantlybetter performance in Enrichment Factor and AUC scores compared to training from scratch.
Theperformance of all models was assessed by 5-fold cross-validation (CV).
Due to structural isomerism, molecules with identical molecular formulae can have the same bonds,but the relative positions of the atoms differ [67].
ViT has much less inductive bias than CNNs,because locality and translation equivariance are embedded into each layer throughout the entirenetwork in CNNs, whereas in ViT self-attention layers are global and only MLP layers are translationally equivariant and local [27].
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 exhibitsimpressive performance.
However, the memory and computational costs of dot-product attentionblocks of ViT grow quadratically with respect to the size of input, which limits its application onlarge-scale datasets [57, 74].
Another major caveat is that the number of triplets grows cubically with [Table]
(cid:57)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:3)(cid:55)(cid:85)(cid:68)(cid:81)(cid:86)(cid:73)(cid:82)(cid:85)(cid:80)(cid:72)(cid:85)(cid:3)(cid:11)(cid:57)(cid:76)(cid:55)(cid:12)(cid:3)(cid:3)(cid:18)(cid:3)(cid:38)(cid:82)(cid:81)(cid:89)(cid:49)(cid:72)(cid:59)(cid:87)(cid:3) (cid:57)(cid:76)(cid:86)(cid:76)(cid:82)(cid:81)(cid:3)(cid:55)(cid:85)(cid:68)(cid:81)(cid:86)(cid:73)(cid:82)(cid:85)(cid:80)(cid:72)(cid:85)(cid:3)(cid:11)(cid:57)(cid:76)(cid:55)(cid:12)(cid:3)(cid:3)(cid:18)(cid:3)(cid:38)(cid:82)(cid:81)(cid:89)(cid:49)(cid:72)(cid:59)(cid:87) the size of the dataset.
Since ConvNeXt depends on a fully-convolutional paradigm, its inherentlyefﬁcient design is viable on large-scale datasets like DUD-E Diverse.
As depicted in Figure 2,
ToDD-ViT and ToDD-ConvNeXt project semantically similar MP signatures of compounds from datamanifold onto metrically close embeddings using triplet margin loss with margin ↵ = 1 . 0 and normp = 2 as provided in Equation 1.
Analogously, semantically different MP signatures are projectedonto metrically distant embeddings.
[Equation]
Sampling Strategy Learning metric embeddings via triplet margin loss on large-scale datasets posesa special challenge in sampling all distinct triplets ( x, x + , x  ) , and collecting them into a singledatabase causes excessive overhead in computation time and memory.
Let P be a set of compounds,x i denotes a compound that inhibits the drug target i , and d ij = d ( x i , x j ) 2 R denotes a pairwisedistance measure which estimates how strongly x i 2 P is similar to x j 2 P .
The distance metric canbe chosen as Euclidian distance, cosine similarity or dot-product between embedding vectors.
Weuse pairwise Euclidian distance computed by the pretrained networks in the implementation.
Sincetriplets ( x, x + , x  ) with d ( x, x  ) >
d ( x, x + ) + ↵ have already negative queries sufﬁciently distantto the anchor compounds from the support set in the embedding space, they are not sampled to createthe training dataset.
We only sample triplets that satisfy d ( x, x  ) <
d ( x, x + ) (where negative queryis closer to the anchor than the positive) and d ( x, x + ) <
d ( x, x  ) <
d ( x, x + ) + ↵ (where negativequery is more distant to the anchor than the positive, but the distance is less than the margin).
Enrichment Factor (EF) is the most common performance evaluation metric for VS methods [81].
EF ↵ % is deﬁned as follows: VS method ' makes a ranking among the compounds to predict theactive ligands.
Let N be the total number of ligands in the dataset and A ' be number of true positivesin the ﬁrst ↵ % of all ligands ( N ↵ = N · ↵ % ).
Then, EF ↵ % =
A ' /N ↵ ↵ / 100 .
In other words, EF ↵ %interprets as how much VS method ' enrich the possibility of ﬁnding active ligand in ﬁrst ↵ % of allligands with respect to the random guess.
This method is also known as precision at k in the literature.
Notice that with this deﬁnition, the maximum score for EF ↵ % is 100 ↵ , i.e., 100 for EF 1% and 20 for EF 5 % .
Baselines We compare our methods against the 23 state-of-the-art baselines including USR [7], ROCS [36], PS [42], GZD [83], PH_VS [48], USR + GZD [75], USR + PS [75], USR + ROCS [75], GZD + PS [75], GZD + ROCS [75], PS + ROCS [75], Findsite [90], Fragsite [91], Gnina [78], GOLD-EATL [87], Glide-EATL [87], CompM [87], CompScore [66], CNN [68], DenseFS [44],
SIEVE-Score [88], DeepScore [85] [Table] and RF-Score-VSv3 [88]. More details about baselines can befound in Appendix {+C.
+}
6.2 Experimental Results
[Table]
[Table]
[Table]
Relative gains are relative to the next best performing model.
Based on the results (mean and standarddeviation of EF scores evaluated by cross-validation) reported in Table 1 and [Table] we observe thefollowing:
• ToDD models consistently achieve the best performance on both Cleves-Jain and DUD-EDiverse datasets across all targets and EF ↵ % levels. • ToDD learns hierarchical topological representations of 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 a model- agnostic method that is extensible to state-of-the-art neural networks as well as ensemble methods like random forests (RF).
• For large-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 small-scale datasets such asDUD-E Diverse, ToDD-RF and ToDD-ConvNeXt exhibit comparable performances exceptfor: CP3A4, GCR and HIVRT.
We conclude that transformer-based models are more robustthan convolutional models despite increased computation time.
6.3 Computational Complexity
Computational complexity (CC) of MP Fingerprint M d depends on the vectorization used andthe number d of the ﬁltering functions one uses.
CC for a single persistence diagram PD k is
O ( N 3 ) [65], where N is the number of k -simplices.
If r is the resolution size of the multipersistencegrid, then CC ( M d ) =
O ( r d · N 3 · C ( m )) where C ( m ) is CC for and m is the number ofbarcodes in PD k , e.g., if is Persistence Landscape, then C ( m )
=
m 2 [12] and hence CC for MP
Landscape with two ﬁltering functions ( d = 2 ) is O
( r 2 · N 3 · m 2 ) .
On the other hand, for MP Bettisummaries, one does not need to compute persistence diagrams, but the rank of homology groupsin the multipersistence module.
Therefore, for MP Betti summary, the computational complexityis indeed much lower by using minimal representations [56, 49].
To expedite the execution time,the feature extraction task is distributed across 8 CPU cores running in parallel in a multiprocessingprocess.
See Appendix C.4 for an additional analysis of computation time to extract MP ﬁngerprintsfrom the datasets.
Furthermore, all ToDD models require substantially fewer computational resourcesduring training compared to current graph-based models that encode a compound through miningcommon molecular fragments, a.k.a., motifs [45].
Training time of ToDD-ViT and ToDD-ConvNeXtfor each individual drug target takes less than 1 hour on a single GPU (RTX 2080 Ti).
6.4 Ablation Study
We tested a number of ablations of our model to analyze the effect of its individual components andto further investigate the effectiveness of our topological ﬁngerprints.
Multimodal Learning We ﬁrst address the question of how adding different domain informationimproves the model performance.
In Appendix C.3, we demonstrate one-by-one the importance ofeach modality (atomic mass, partial charge and bond type) used for graph ﬁltration to the classiﬁcationof each target.
We ﬁnd that their importance varies across targets in a unimodal setting, but theorthogonality of these information sources offers signiﬁcant gain in EF scores when the MP signatureslearned from each modality are integrated into a joined multimodal representation.
Morgan Fingerprints We quantitatively analyze the explainability of our models’ success byreplacing topological ﬁngerprints computed via multiparameter persistence with the most popularﬁngerprinting method: Morgan ﬁngerprints.
Our results in Appendix C.5 show that ToDD engineersfeatures that represent the underlying attributes of compounds signiﬁcantly better than the Morganalgorithm to identify the active compounds across all drug targets.
Network Architecture We investigated ways to leverage deep metric learning by architecting i )a Siamese network trained with contrastive loss, ii ) a Triplet network trained with triplet marginloss, and iii ) a Triplet network trained with circle loss.
Based on our preliminary experiments, theembeddings learned by i and iii provide sub-par results for compound classiﬁcation, hence we use ii .
Conclusion
We have proposed a new idea of the topological ﬁngerprints in VS, allowing for deeper insights intostructural organization of chemical compounds.
We have evaluated the predictive performance ofour ToDD methodology for computer aided drug discovery on benchmark datasets in VS.
Moreover,we have demonstrated that our topological descriptors are model-agnostic and have proven to beexceedingly competitive, yielding state-of-the-art results unequivocally over all baselines.
A futureresearch direction is to enrich ToDD with different VS modalities, and use it on ultra-large virtualcompound libraries.
In addition, we plan to develop formal statistical inference for learning theunderlying ToDD explainability and interpretability.
It is important to note that this new way ofcapturing the chemical information of compounds provides a transformative perspective to everylevel of the pharmaceutical pipeline from the very early phases of drug discovery to the ﬁnal stagesof formulation in development.
Checklist
For all authors...
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[Yes] Dataset links are provided.
See Section 5.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
[Yes] See Section 6.
(c) Did you report error bars (e g., with respect to the random seed after running experi- ments multiple times)?
[No] We reported the standard deviation of our experiments evaluated by 5-fold cross-validation.
See Table 1 and 2 in Section 6.2.
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[-Now, we are ready to prove the stability of MP Fingerprints given in Section 4.
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[-For each V ± i , construct the induced VR-filtration∆ ± i 
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[-Then, byrestricting g to X i 0 , we get a filtering function on X i 0 , i.e., g : X i 
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[-By inducing a simplicial complex (cid:98) X i 0 j for each X i 
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[-j 0 K for the vertex set V i 
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