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 foridentifying the drug candidates that are most likely to bind to a molecular targetin a large library of compounds.
Most VS methods to date have focused on usingcanonical compound representations (e.g., SMILES strings, Morgan fingerprints)or generating alternative fingerprints of the compounds by training progressivelymore 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 incrementalimprovements in the past few years.
To address this problem, we developed a novelmethod using multiparameter persistence (MP) homology that produces topologicalfingerprints of the compounds as multidimensional vectors.
Our primary contribution is framing the VS process as a new topology-based graph ranking problemby partitioning a compound into chemical substructures informed by the periodicproperties of its atoms and extracting their persistent homology features at multipleresolution levels.
We show that the margin loss fine-tuning of pretrained Tripletnetworks attains highly competitive results in differentiating between compoundsin the embedding space and ranking their likelihood of becoming effective drugcandidates.
We further establish theoretical guarantees for the stability propertiesof our proposed MP signatures, and demonstrate that our models, enhanced by the
MP signatures, outperform state-of-the-art methods on benchmark datasets by awide and highly statistically significant margin (e.g., 93% gain for Cleves-Jain and54% 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, 94].
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 [66, 39]
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In this paper, we develop novel approaches for virtual screening by successfully integrating topologicaldata 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 multiper-
sistence, we produce highly expressive and unique topological fingerprints for compounds
independent of 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
into 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 stability 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 Work2.1 Virtual Screening
A key step in the early stages of the drug discovery process is to find active compounds that willbe further optimized into potential drug candidates.
One prevalent computational method that iswidely used for compound prioritization with desired properties is virtual screening (VS).
Thereare two major categories, i.e., structure-based virtual screening (SBVS) and ligand-based virtualscreening (LBVS) [20].
SBVS uses the 3 D structural information of both ligand (compound) andtarget protein as a complex [11, 52, 89].
SBVS methods generally require a good understanding of3 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 [81, 70]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 [57].
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 [83] and
SMARTS [28] are examples of 1 D -methods which produce 1 D -fingerprints, compressing compoundinformation to a vector.
RASCAL [80], MOLPRINT2D [8], ECFP [82], CDK-graph
[101],CDKhybridization [88],SWISS [106], Klekota-Roth [53], MACSS [28], E-state [35] and SIMCOMP [36]are among 2 D methods which uses 2 D -structure fingerprint and graph matching.
Finally, examplesof 3 D -methods are ROCS [37], USR [7], PatchSurfer [43] which use the 3 D -structure of compoundsand their conformations ( 3 D -position of the compound) [86].
On the other hand, while ML methodshave been actively used in the field for the last two decades, new deep learning methods made a hugeimpact in drug discovery process in the last 5 years [91, 52, 84].
Further discussion of state-of-the-art
ML/DL methods are given in Section 6 where we compare our models and benchmark against them.
2.2 Topological Data Analysis
(TDA) and tools of persistent homology (PH) have recently emerged as powerful approaches for
ML, allowing us to extract complementary information on the observed objects, especially, fromgraph-structured data.
In particular, PH has become popular for various ML tasks, ranging frombioinformatics [54, 98, 74], finance [56, 32], material science [44, 69] to network analysis [87, 42, 19](See the literature overviews [75, 5, 22] and TDA applications library [33]).
Furthermore, it hasbecome a highly active research area to integrate PH methods into geometric deep learning (GDL) inrecent years [40, 41, 103, 18].
Most recently, the emerging concepts of multipersistence (MP) areproposed to advance the success of single parameter persistence (SP) by allowing the use of morethan 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 barcodedecomposition relating to the partially ordered structure of the index set { ( α i , β j ) } [58, 92].
Theexisting approaches remedy this issue via slicing technique by studying one-dimensional fibers ofthe multiparameter domain [17], but choosing these directions suitably and computing restricted SPvectorizations are computationally costly which makes the approach inefficient in real life applications.
There are several promising recent studies in this direction [10, 96, 23], but these approaches fail toprovide a practical topological summary such as “multipersistence diagram”, and an effective MPvectorization to be used in real life applications.
2.3 TDA in Virtual Screening
In [15, 14, 13], the authors obtained successful results by integrating single persistent homologyoutputs with various ML models.
Furthermore, in [50], the authors used multipersistence homologywith fibered barcode approach in the 3 D setting and obtained promising results.
In the past fewyears, TDA tools were also successfully combined with various deep learning models for SBVS andproperty prediction [72, 73].
In [67, 46, 62, 99, 63], the authors successfully used TDA methods togenerate powerful molecular descriptors.
Then, by using these descriptors, they highly boosted theperformance of various ML/DL models and outperformed the existing models in several benchmark datasets.
For a discussion and comparison of TDA techniques with other approaches in virtualscreening and property prediction, see the review article [71].
In this paper, we follow a differentapproach and propose a very general framework by adapting multipersistence homology to VSprocess which produces very fine topological fingerprints which are highly suitable for ML/DLmethods.
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 moreconventional, 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 (cid:98) G i , 1 ≤ i ≤ N , yielding a filtration , a nested sequenceof simplicial complexes (cid:98) G 1 ⊆ . . .
⊆ (cid:98) 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 { (cid:98) G i } Ni =1 .
A k -dimensional topological
feature(or k -hole) may represent connected components ( 0 -dimension), loops ( 1 -dimension) and cavities( 2 -dimension).
For each k -dimensional topological feature σ , PH records its first appearance inthe filtration sequence, say (cid:98) G b σ , and first disappearence in later complexes, (cid:98) G d σ with a unique pair( b σ , d σ ) , where 1 ≤ b σ < d σ ≤ N .
We call b σ the birth time of σ and d σ the death time of σ .
We calld σ − b σ the life span (or persistence) of σ .
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 (cid:98) G N .
Then k th persistence diagram PD k
( G ) =
{ ( b σ , d σ ) | σ
∈ H k
( (cid:98) G i ) for b σ ≤ i < d σ } .
Here, H k ( (cid:98) G i ) represents the k th homology group of (cid:98) G i which keeps the information of the k -holes inthe simplicial complex (cid:98) G i .
Most common dimensions used in practice are 0 and 1 , i.e., PD 0 ( G ) and
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 { α i } m 1 and { β j } n 1 forf and g respectively.
Let V ij = { v r ∈ V | f ( v r ) ≤ α
i , g ( v r ) ≤ β j } , i.e., V ij = F ( v r ) ⪯ ( α i , β j ) .
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 { (cid:98) G i } , we get a bifiltration of complexes{ (cid:98) 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 { (cid:98) G ij } by using thecorresponding homology groups { H k ( (cid:98) 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 → Rbe a filtering function (e.g., atomic mass, partial charge, bond type, electron affinity, ionizationenergy) with threshold sets { α i } mi =1 .
Let V i
= { v r ∈ V | f
( v r ) ≤ α 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 0 ⊆
∆ i 0 1 . . .
⊆
∆ i 0 K .
This givesm × ( K + 1) simplicial complexes { ∆ ij } where 1 ≤ i ≤ m and 0 ≤ j ≤ K .
This is called thebipersistence module .
One can imagine increasing sequence of {V i } as vertical direction, and induced
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∆ i 0 0 ⊆
∆ i 0 1 . . .
⊆
∆ i 0 K (horizontal slice in the bipersistence module).
Each such single filtrationinduces a persistence diagram PD ( V i ) = { ( b j , d j ) | 0 ≤ b j < d j ≤ K } .
This produces mpersistence diagrams { PD ( V i ) } .
Note that one can consider PD ( V i ) as the persistence diagram ofthe "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 φ be a singlepersistence vectorization, e.g., Betti, Silhouette, Entropy, Persistence Landscape or Persistence Image.
By applying the chosen SP vectorization φ to each PD, we obtain a function φ i =
φ ( PD ( V i )) wherein most cases it is a single variable function on the threshold domain [0 , K ] , i.e., φ 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 ⃗φ i be the corresponding
1 × r vector for the function φ i .
Now, we are ready to define our topological fingerprint M φ which isa 2 D -vector (a matrix)
M iφ = ⃗φ i for 1 ≤ i ≤ m, where M iφ is the
i th -row of M φ .
Hence, M φis a 2 D -vector of size m × r .
Each row M iφ is the vectorization of the persistence diagram PD ( V i )via the SP vectorization method φ .
We use the first filtering function f to get a finer look at the graphas it defines the subgraphs G 1 ⊆ . . .
⊆
G m =
G where G i is the subgraph induced by the vertices V i .
Then, we use the distance function to obtain the topological signatures ⃗φ i =
⃗φ ( PD ( V i )) of thesesubgraphs {G i } defined by the domain function f .
One can consider the subgraphs G i in G inducedby the vertices
V
i are the substructures defined by the domain function f .
Then our fingerprints areindeed topological signatures of these substructures induced 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 φ is stable, then so is its induced
MP Fingerprint M φ .
(We give the details of stability notion in persistence theory and proof of thefollowing theorem in Section B.1.)
Theorem: Let φ be a stable SP vectorization.
Then, the induced MP Fingerprint M φ is also stable, i.e., with the notation introduced in Section B.1, there exists (cid:98) C φ > 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 has 1149 compounds.
1 There are 22 differentdrug targets, and for each one of them the dataset provides only 2-3 template active compoundsdedicated for model training, which presents a few-shot learning task.
All targets { q } are associatedwith 4 to 30 active compounds { L q } dedicated for model testing.
Additionally, the dataset contains850 decoy compounds ( D ).
The aim is for each target q , by using the templates, to find the actives
L q among the pool combined with decoys L q ∪ D , i.e., same decoy set D is used for all targets.
DUD-E Diverse: DUD-E (Directory of Useful Decoys, Enhanced) dataset
[68] 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 [64], 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
[Figure]
or 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 ROC-AUC scores compared to training from scratch.
The performance 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 [78].
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 [61, 85].
Another major caveat is that the number of triplets grows cubically withthe 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 α = 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 ) ∈ R denotes a pairwisedistance measure which estimates how strongly x i ∈ P is similar to x j ∈ P .
The distance metric canbe chosen as Euclidean distance, cosine similarity or dot-product between embedding vectors.
Weuse pairwise Euclidean distance computed by the pretrained networks in the implementation.
Sincetriplets ( x, x + , x − ) with d ( x, x − ) >
d ( x, x + ) + α have already negative queries sufficiently distantto the anchor compounds from the support set in the embedding space, they are not sampled to create the 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 + ) + α (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 [93].
VS method φ ranks compounds in the database by their similarity scores.
We measure the similarityscore 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, A φ be the number of true positives (i.e., correctlypredicted active ligands) ranked among the top α % of all ligands ( N α = N · α % ) and N actives be thenumber of active ligands in the whole dataset.
Then, EF α % =
Aφ/N actives α/ 100 .
In other words, EF α %interprets as how much VS method φ enrich the possibility of finding active ligand in the first α %of all ligands with respect to the random guess.
This method is also known as precision at k in theliterature.
With this definition, the max score for EF α % is 100 α , i.e., 100 for EF 1% and 20 for EF 5 % .
6.2 Experimental Results
We compare our methods against the 23 state-of-the-art baselines (see Appendix C.2).
[Table] [Table]
Relative gains are relative to the next best performing model.
Based on the results (mean and standarddeviation of EF scores evaluated by CV) reported in Table 1 and 2, we observe the following:
• ToDD models consistently achieve the best performance on both Cleves-Jain and DUD-EDiverse datasets across all targets and EF α % levels.• ToDD learns hierarchical topological representations of compounds using their atoms’ periodic properties, and captures the complex chemical properties essential for high-throughputVS.
These strong hierarchical topological representations enable ToDD to become a modelagnostic method that is extensible to state-of-the-art neural networks as well as ensemblemethods like random forests (RF).
• For small-scale datasets such as Cleves-Jain, RF is less accurate than ViT despite regularization by bootstrapping and using pruned, shallow trees, because small variations in thedata may generate significantly different decision trees.
For large-scale datasets such as
DUD-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 and RF classifiers despite increased computation time.
6.3 Computational Complexity
Computational complexity (CC) of MP Fingerprint M dψ depends on the vectorization ψ used andthe number d of the filtering functions one uses.
CC for a single persistence diagram PD k is
O ( N 3 ) [76], where N is the number of k -simplices.
If r is the resolution size of the multipersistencegrid, then CC ( M dψ ) =
O ( r d · N 3 · C ψ ( m )) where C ψ ( m ) is CC for ψ and m is the number ofbarcodes in PD k , e.g., if ψ is Persistence Landscape, then C ψ ( m )
=
m 2 [12] and hence CC for MP
Landscape with three filtering functions ( d = 3 ) is O
( r 3 · 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 [59, 51].
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 C.4 for an additional analysis of computationtime to extract MP fingerprints from the datasets.
Furthermore, all ToDD models require substantiallyfewer computational resources during training compared to current graph-based models that encodea compound through mining common molecular fragments, a.k.a., motifs [47].
Training time of
ToDD-ViT and ToDD-ConvNeXt for each individual drug target takes less than 1 hour on a single
GPU (NVIDIA 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.
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope?
[Yes]
(b) Did you describe the limitations of your work?
[Yes] See Appendix D.(c)
Did you discuss any potential negative societal impacts of your work?
[Yes] SeeAppendix D.(d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?
[Yes]
If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results?
[Yes]
(b) Did you include complete proofs of all theoretical results?
[Yes] Given in Appendix B.13.
If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi-
mental results (either in the supplemental material or as a URL)?
[Yes] Dataset linksare provided.
See Section 5.(b)
Did you specify all the training details (e.g., data splits, hyperparameters, how theywere chosen)?
[Yes] See Section 6.1(c)
Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
[No] We reported the standard deviation of our experimentsevaluated by 5-fold cross-validation.
See Table 1 and 2 in Section 6.2.(d)
Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
[Yes] See Section 6.
If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a)
If your work uses existing assets, did you cite the creators?
[Yes](b) Did you mention the license of the assets?
[No] They use public-domain-equivalentlicense.
(c) Did you include any new assets either in the supplemental material or as a URL?
[No](d) Did you discuss whether
and how consent was obtained from people whose data you’reusing/curating?
[N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content?
[N/A]5.
If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, ifapplicable?
[N/A](b)
Did you describe any potential participant risks, with links to Institutional ReviewBoard (IRB) approvals, if applicable?
[N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amountspent on participant compensation?
[N/A]
A Topological Data Analysis (TDA)A.1 Single Parameter Persistent Homology
Here, we give further details on single parameter persistent homology.
To sum up, PH machinery is a3 -step process.
The first step is the filtration step, where one can integrate the domain information tothe process.
The second step is the persistence diagrams , where the machinery records the evolutionof topological features (birth/death times) in the filtration, sequence of the simplicial complexes.
Thefinal step is the vectorization (fingerprinting) where one can convert these records to a function orvector to be used in suitable ML models.
Constructing Filtrations: As PH is basically the machinery to keep track of the evolution oftopological features in a sequence, the most important step is the construction of the sequence(cid:98) G 1 ⊆ . . .
⊆ (cid:98) G N .
This is the key step where one can inject the valuable domain information to the
PH process by using important domain functions (e.g., atomic mass, partial charge).
While there arevarious filtration techniques used for PH machinery on graphs [4, 40], we will focus on two mostcommon methods: Sublevel/superlevel filtration and Vietoris-Rips (VR) filtration .
For a given unweighted graph (compound) G = ( V , E ) with V = { v 1 , . . .
, v m } the set of nodes(atoms) and E = { e rs } the set of edges (bonds), the most common technique is to use a filteringfunction f : V → R with a choice of thresholds
I = { α i } where α 1 = min v ∈V f ( v ) <
α 2 < . . .
<α N = max v ∈V f ( v ) .
For α i ∈ I , let V i
= { v r ∈ V | f
( v r ) ≤ α i } (sublevel sets for f ).
Here, in VSproblem, this filtering function f can be atomic mass, partial charge, bond type, electron affinity,ionization energy or another important function representing chemical properties of the atoms.
Onecan also use the natural graph induced functions like node degree, betweenness, etc.
Let G i be theinduced subgraph of G by V
i , i.e., G i =
( V i , E i ) where E i = { e rs ∈ E | v r , v s ∈ V i } .
This processyields a nested sequence of subgraphs G 1 ⊂ G 2 ⊂ . . .
⊂ G N = G .
To obtain a filtration, next step is toassign a simplicial complex (cid:98) G i to the subgraph G i .
One of the most common techniques is the cliquecomplexes [4].
The clique complex (cid:98) G is a simplicial complex obtained from G by assigning (fillingwith) a k -simplex to each complete ( k + 1) -complete subgraph in G , e.g., a 3 -clique, a complete3 -subgraph, in G will be filled with a 2 -simplex (triangle).
This technique is generally known assublevel filtration with clique complexes.
As f ( v i ) ≤ α i condition in the construction gives sublevelfiltration, one can similarly use f ( v i ) ≥ α i condition to define superlevel filtration .
Similarly, for aweighted graph (bond strength), sublevel filtration on edge weights provides corresponding filtrationreflecting the domain information stored in the edge weights [4].
While sublevel/superlevel filtration with clique complexes is computationally cheaper and morecommon in practise, in this paper, we will essentially use a distance-based filtration technique called
Vietoris-Rips (VR) filtration where the pairwise distances between the nodes play key role.
Thistechnique is computationally more expensive, but gives much finer information about the graph’sintrinsic properties [2].
For a given graph G =
( V , E ) , we define the distance between d ( v r , v s )
=
d rswhere d rs is the smallest number of edges required to get from v r to v s in G .
Then, let Γ n =
( V , E n )be the graph where E n = { e rs | d rs ≤ n } , i.e. E 0 = ∅ and E
1 =
E with Γ
= V and Γ 1 = G .
Inother words, we start with the nodes of G , then for any pair of vertices v r , v s with distance d rs ≤ nin G , we add an edge e rs to the graph Γ n .
Then, define the simplicial complex ∆ n = (cid:98) Γ n , theclique complex of Γ n .
This defines a filtration ∆ 0 ⊂ ∆ 1 ⊂ · · · ⊂ ∆ K where K = max d rs , i.e.the distance between farthest two nodes in the graph G .
Hence, for n ≥ K , ∆ n = ∆ K which isa ( m − 1) -simplex as Γ K is complete
m -graph
where |V| = m .
In particular, in this setting, weconsider the vertex set V as a point cloud where the distances between the points induced from thegraph G .
In graph setting, V R -filtration is also known as power filtration as the graph Γ n is alsocalled G n , n th power of G .
Persistence Diagrams: The second step in PH process is to obtain persistence diagrams (PD)for the filtration ∆ 0 ⊂ ∆ 1 ⊂ · · · ⊂ ∆ K .
As explained in Section 3.1, PDs are collection of2 -tuples, marking the birth and death times of the topological features appearing in the filtration, i.e.
PD k ( G ) = { ( b σ , d σ ) | σ ∈ H k (∆ i ) for b σ ≤ i < d σ } .
This step is pretty standard and there arevarious software libraries for this task [76].
Vectorizations (Fingerprinting): While PH extracts hidden shape patterns from data as persistencediagrams (PD), PDs being collection of points in R 2 by itself are not very practical for statistical and
ML purposes.
Instead, the common techniques are by faithfully representing PDs as kernels [55] or vectorizations [38].
One can consider this step as converting PDs into a useful format to be used in
ML process as fingerprints of the dataset.
This provides a practical way to use the outputs of PH inreal life applications.
Single Persistence Vectorizations transform obtained PH information (PDs) intoa function or a feature vector form which are much more suitable for ML tools than PDs.
Commonsingle persistence (SP) vectorization methods are Persistence Images [3], Persistence Landscapes [12],
Silhouettes [21], Betti Curves and various Persistence Curves [24].
These vectorizations define asingle variable or multivariable functions out of PDs, which can be used as fixed size 1 D or 2 Dvectors in applications, i.e 1 × n vectors or m × n vectors.
For example, a Betti curve for a PD with nthresholds can also be expressed as 1 × n size vectors.
Similarly, Persistence Images is an example of2 D vectors with the chosen resolution (grid) size.
See the examples given in Section B.4 for furtherdetails.
B Multiparameter Persistence (MP) Fingerprints
B.1 Stability of MP Fingerprints
Stability of Single Persistence Vectorizations:
A given PD vectorization φ can be considered as a mapfrom space of persistence diagrams to space of functions, and the stability intuitively represents thecontinuity of this operator.
In other words, stability question is whether a small perturbation in PDcause a big change in the vectorization or not.
To make this question meaningful, one needs to definewhat "small perturbation" means in this context, i.e., a metric in the space of persistence diagrams.
The most common such metric is called Wasserstein distance (or matching distance) which is definedas follows.
Let PD ( X + ) and PD ( X − ) be persistence diagrams two datasets X + and X −
(We omit the dimensions in PDs).
Let PD ( X + ) =
{ q + j } ∪ ∆ + and PD ( X − )
= { q − l } ∪ ∆ − where ∆ ± represents thediagonal (representing trivial cycles) with infinite multiplicity.
Here, q + j = ( b + j , d + j ) ∈ PD ( X + )represents the birth and death times of a topological feature σ j in X + .
Let ϕ : PD ( X + ) → PD ( X − )represent a bijection (matching).
With the existence of the diagonal ∆ ± in both sides, we make surethe existence of these bijections even if the cardinalities |{ q + j }| and |{ q − l }| are different.
Definition B.1 Let PD ( X ± ) be persistence diagrams of the datasets X ± , and M = { ϕ } represent the space of matchings as described above.
Then, the p th Wasserstein distance W p defined as
[Equation]
Now, we define stability of vectorizations.
A vectorization can be considered as an operator fromspace of persistence diagrams P to space of functions (or vectors)
Y , e.g., Ψ : P → Y .
In particular,when Ψ is persistence landscape, Y = C ([0 , K ] , R ) and when Ψ is Betti summary, then Y =
R m(See MP Examples in Section B.4)
Stability of vectorization Ψ basically corresponds to the continuityof Ψ as an operator.
Let d( ., . )
be a suitable metric on the space of vectorizations used.
Then, wedefine the stability of Ψ as follows:
Definition B.2 Let Ψ : P
→ Y be a vectorization for single persistence diagrams.
Let W p , d be the metrics on P and Y respectively as described above.
Let ψ ± =
Ψ( PD ( X ± ))
∈ Y .
Then, Ψ is called stable if
[Equation]
Here, the constant C > 0 is independent of X ± .
This stability inequality interprets as the changesin the vectorizations are bounded by the changes in PDs.
Two nearby persistence diagrams arerepresented by nearby vectorizations.
If a given vectorization φ holds such a stability inequalityfor some d and W p , we call φ a stable vectorization [6].
Persistence Landscapes [12], Persistence
Images [3], Stabilized Betti Curves [48] and several Persistence curves [24] are among well-knownexamples of stable vectorizations.
Now, we are ready to prove the stability of MP Fingerprints given in Section 4.
Let G + =
( V + , E + ) and G − =
( V − , E − ) be two graphs.
Let φ be a stable SP vectorization with thestability equation
[Equation]
for some 1 ≤ p φ ≤ ∞ .
Here, φ ( G ± ) represent the corresponding vectorizations for PD ( G ± ) and
W p represents Wasserstein-
p distance as defined in Section B.1.
Now, let f : V ±
→ R be a filtering function with threshold set { α i } mi =1 .
Then, define the sublevelvertex sets V ± i = { v r ∈ V ± | f ( v r ) ≤ α i } .
For each V ± i , construct the induced VR-filtration∆ ± i
⊂ ∆ ± i 1 ⊂ · · · ⊂ ∆ ± iK as before.
For each 1 ≤ i 0 ≤ m , we will have persistence diagram
PD ( V ± i 0 ) of the filtration { ∆ ± i 0 k } .
We define the induced matching distance between the multiple persistence diagrams as
[Equation]
Now, we define the distance between induced MP Fingerprints as
[Equation]
Theorem B.1 Let φ be a stable SP vectorization.
Then, the induced MP Fingerprint M φ is also stable, i.e., with the notation above, there exists (cid:98) C φ > 0 such that for any pair of graphs G + and G − , we have the following inequality.
[Equation]
[Equation]
Proof: As φ is a stable SP vectorization, by Equation 2, for any 1 ≤ i ≤ m , we haved( φ ( PD ( V + i )) , φ ( PD ( V + i )))
≤ C φ · W p φ ( PD ( V + i ) , PD ( V − i )) for some C φ > 0 , where W p φ is
Wasserstein-
p distance.
Notice that the constant C φ > 0 is independent of i .
Hence,
[Equation]
where the first and last equalities are due to Equation 3 and Equation 4, while the inequality followsfrom Equation 2 which is true for any i .
This concludes the proof of the theorem.
□
B.2 MP Fingerprint for Other Types of Data
So far, to keep the exposition focused on VS setting, we described our construction only in the graphsetup.
However, our framework is suitable for various types of data.
Let X be a an image data or apoint cloud.
Let f : X → R and g : X → R be two filtering functions on X .
e.g., grayscale functionfor image data, or density function on point cloud data.
Let f : X → R be the filtering function with threshold set { α i } m 1 .
Let X i =
f
− 1 (( −∞ , α i ]) .
Then,we get a filtering of X as nested subspaces X 1 ⊂ X 2 ⊂ · · · ⊂ X m = X .
By using the secondfiltering function, we obtain finer filtrations for each subspace
X i where 1 ≤ i ≤ m .
In particular,fix 1 ≤ i 0 ≤ m and let { β j } nj =1 be the threshold set for the second filtering function g .
Then, byrestricting g to X i 0 , we get a filtering function on X i 0 , i.e., g : X i
→ R which produces filtering
X i 0 1 ⊂ X
i 0 2 ⊂ · · · ⊂ X i 0 n = X i 0 .
By inducing a simplicial complex (cid:98) X i 0 j for each X i
j , weget a filtration (cid:98) X i 0 1 ⊂ (cid:98) X i 0 2 ⊂ · · · ⊂ (cid:98) X
i 0 n = (cid:98) X i 0 .
This filtration results in a persistence diagram
(PD) PD ( X i 0 , g ) .
For each 1 ≤ i ≤ m , we obtain PD ( X i , g ) .
Note that after getting {X i } mi =1 viaf , instead of using second filtering function g , one can apply Vietoris-Rips construction based ondistance for each X i 0 in order to get a different filtration (cid:98)
X
i 0 1 ⊂ (cid:98) X i 0 2 ⊂ · · · ⊂ (cid:98) X i 0 n = (cid:98) X i 0 .
By using m PDs, we follow a similar route to define our MP Fingerprints.
Let φ be a singlepersistence vectorization.
By applying the chosen SP vectorization φ to each PD, we obtain afunction φ i =
φ ( PD ( X i , g )) on the threshold domain [ β 1 , β n ] , which can be expresses as a 1 D (or2 D ) vector in most cases (Section B.4).
Let ⃗φ
i be the corresponding 1 × k vector for the functionφ i .
Define the corresponding MP Fingerprint M φ as M iφ = ⃗φ i where M iφ is the i th row of M φ .
Inparticular, M φ is a 2 D -vector (a matrix) of size m × k where m is the number of thresholds for thefirst filtering function f , and k is the length of the vector ⃗φ .
B.3 3 D or higher dimensional MP Fingerprints
If one wants to use two filtering functions and get 3 D -vectors (arrays) as the topological fingerprintof the process, the idea is very similar.
Let f, g : V → R be two filtering functions with thresholdsets { α i } mi =1 and { β j } nj =1 respectively.
Let V ij = { v r ∈ V | f ( v r ) ≤ α i and g ( v r ) ≤ β j } .
Again,compute all the pairwise distances
d ( v r , v s ) = m rs in G before defining simplicial complexes.
Then,for each i 0 , j 0 , obtain a VR-filtration ∆
[Table]
⊆
[Table] [Table]
. ⊆
∆ i
j 0 K for the vertex set V i
j 0 with 809 distances d ( v r , v s )
= m rs in G .
Compute the persistence diagram PD ( V i 0 j 0 ) for the filtration{ ∆ i
j 0 k } .
This gives m × n persistence diagrams { PD ( V ij ) } .
After vectorization, we obtain a3 D -vector (array) of size m × n × r as before.
B.4 Examples of MP Fingerprints
Here, we give explicit constructions of MP Fingerprints for most common SP vectorizations.
As notedabove, the framework is very general and it can be applied to most SP vectorization methods.
In allthe examples below, we use the following setup: Let G =
( V , E ) be a graph, and let f, g : V → R betwo filtering functions with threshold sets { α i } mi =1 and { β j } nj =1 respectively.
As explained above, wefirst apply sublevel filtering with f to get a sequence of nested subgraphs, G 1 ⊆ . . .
⊆
G m = G .
Then,for each G i , we apply sublevel filtration with g to get persistence diagram PD ( G i , g ) .
Therefore,we will have m PDs.
In the examples below, for a given SP vectorization φ , we explain how toobtain a vector ⃗φ ( PD ( G i , g )) , and define the corresponding MP Fingerprint M φ .
Note that weskip the homology dimension (subscript k for PD k ( G ) ) in the discussion.
In particular, for eachdimension k = 0 , 1 , . . .
, we will have one MP Fingerprint M φ ( G ) (a matrix or array) correspondingto { ⃗φ ( PD k
( G i , g )) } .
The most common dimensions are k = 0 and k = 1 in applications.
MP Landscapes Persistence Landscapes λ are one of the most common SP vectorizations introduced by [12].
For a given persistence diagram PD ( G ) = { ( b i , d i ) } , λ produces a function λ ( G ) by using generating functions Λ i for each ( b i , d i ) ∈ PD ( G ) , i.e., Λ i : [ b i , d i ] → R is a piecewise linear function obtained by two line segments starting from ( b i , 0) and ( d i , 0) connecting to the same point ( b i + d i 2 , b i − d i 2 ) .
Then, the Persistence Landscape function λ ( G ) :
[ ϵ 1 , ϵ q ] → R for t ∈ [ ϵ 1 , ϵ q ] is defined as
[Equation]
where { ϵ k } q 1 are thresholds for the filtration used.
Considering the piecewise linear structure of the function, λ ( G ) is completely determined by itsvalues at 2 q − 1 points, i.e., b i ±
d i∈ { ϵ 1 , ϵ 1 . 5 , ϵ 2 , ϵ 2 . 5 , . . .
, ϵ q } where ϵ k. 5 =
( ϵ k + ϵ k +1 ) / 2 .
Hence,a vector of size 1 × (2 q − 1) whose entries the values of this function would suffice to capture all theinformation needed,
i.e. ⃗λ =
[ λ ( ϵ 1 ) λ ( ϵ 1 . 5 ) λ ( ϵ 2 ) λ ( ϵ 2 . 5 ) λ ( ϵ 3 ) . . .
λ ( ϵ q )]
Considering we have threshold set { β j } nj =1 for the second filtering function g , ⃗λ i = ⃗λ ( PD ( G i , g ))will be a vector of size 1 × 2 n − 1 .
Then, as M i λ =
⃗ λ i for each 1 ≤ i ≤ m , MP Landscape M λ ( G )would be a 2 D -vector (matrix) of size m × (2 n − 1) .
MP Persistence Images Next SP vectorization in our list is Persistence Images [3].
Different thanthe most SP vectorizations, Persistence Images produces 2 D -vectors.
The idea is to capture thelocation of the points in the persistence diagrams with a multivariable function by using the 2 D
Gaussian functions centered at these points.
For PD ( G ) = { ( b i , d i ) } , let ϕ i represent a 2 D -Gaussiancentered at the point ( b i , d i )
∈ R 2 .
Then, one defines a multivariable function, Persistence Surface ,(cid:101) µ = (cid:80) i w i
ϕ
i where w i is the weight, mostly a function of the life span d i − b i .
To represent thismultivariable function as a 2 D -vector, one defines a k × l grid (resolution size) on the domain of(cid:101) µ , i.e., threshold domain of PD ( G ) .
Then, one obtains the Persistence Image , a 2 D -vector (matrix)⃗µ =
[ µ rs ] of size k × l where µ rs = (cid:82) ∆ rs (cid:101) µ ( x, y ) dxdy and ∆ rs is the corresponding pixel (rectangle)in the k × l grid.
This time, the resolution size k × l is independent of the number of thresholds used in the filtering,the choice of k and l is completely up to the user.
Recall that by applying the first function f , wehave the nested subgraphs {G i } mi =1 .
For each G i , the persistence diagram PD ( G i , g ) obtained bysublevel filtration with g induces a 2 D vector ⃗µ i = ⃗µ ( PD ( G i , g )) of size k × l .
Then, define MP
Persistence Image as M iµ = ⃗µ i , where M iµ is the i th -floor of the array M µ .
Hence, M µ ( G ) wouldbe a 3 D -vector (array) of size m × k × l where m is the number of thresholds for the first function fand k × l is the chosen resolution size for the Persistence Image ⃗µ .
MP Betti Summaries Next, we give an important family of SP vectorizations, Persistence Curves [24].
This is an umbrella term for several different SP vectorizations, i.e., Betti Curves, Life Entropy,
Landscapes, et al.
Our MP Fingerprint framework naturally adapts to all Persistence Curves toproduce multidimensional vectorizations.
As Persistence Curves produce a single variable function ingeneral, they all can be represented as 1 D -vectors by choosing a suitable mesh size depending on thenumber of thresholds used.
Here, we describe one of the most common Persistence Curves in detail,i.e., Betti Curves.
It is straightforward to generalize the construction to other Persistence Curves.
Betti curves are one of the simplest SP vectorization as it gives the count of topological feature at a given threshold interval.
In particular, β k (∆) is the total count of k -dimensional topological feature in the simplicial complex ∆ , i.e., β k (∆)
= rank ( H k (∆)) .
Then, β k ( G ) : [ ϵ 1 , ϵ q +1 ]
→ R is a step function defined as
[Equation]
for t ∈ [ ϵ i , ϵ i +1 ) , where { ϵ i } q 1 represents the thresholds for the filtration used.
Considering this is astep function where the function is constant for each interval [ ϵ i , ϵ i +1 ) , it can be perfectly representedby a vector of size 1 × q as ⃗β ( G ) =
[ β (1) β (2) β (3) . . .
β ( q )] .
Then, with the threshold set { β j } nj =1 for the second filtering function g , ⃗β i = ⃗β
( PD ( G i , g )) will bea vector of size 1 × n .
Then, as M iβ = ⃗β
i for each 1 ≤ i ≤ m , MP Betti Summary M β ( G ) would bea 2 D -vector (matrix) of size m × n .
In particular, each entry M β = [ m ij ] is just the Betti numberof the corresponding clique complex in the bifiltration { (cid:98) G ij } , i.e., m ij = β ( (cid:98) G ij ) .
This matrix M βis also called bigraded Betti numbers in the literature, and computationally much faster than othervectorizations [59, 51].
B.5 MP Vectorization with Other Filtrations
Sublevel filtration in all directions: Depending on the setup, one can skip power filtration (distance)as it is computationally more expensive than the others.
If one wants to use sublevel/superlevelfiltrations in both directions, the idea is as follows.
Let f, g : V → R be two filtering functions withthreshold sets { α i } mi =1 and { β j } nj =1 respectively (e.g., f is atomic mass, and g is partial charge).
Let
V ij = { v r ∈ V | f ( v r ) ≤ α i and g ( v r ) ≤ β j } .
Let G ij be the induced subgraph of G by V ij , i.e. addany edge in G whose endpoints are in V ij .
Then, define (cid:98) G ij as the clique complex of G ij .
Here, themain difference with the previous construction, we replace the distance function (VR-filtration) ( ∆ ik )with another function g and directly take the clique complex of the induced subgraph G ij .
Then, foreach 1 ≤ i 0 ≤ m , one obtains a single filtration (cid:98) G i 0 1 ⊆ (cid:98) G i 0 2 . . .
⊆ (cid:98) G i 0 n in horizontal direction.
Thissingle filtration gives a persistence diagram PD ( G i 0 ) as before.
Hence, one obtains m persistencediagrams { PD ( G i ) } .
Again, by applying a vectorization φ , one obtains m row vectors of fixed sizer , i.e. ⃗φ i =
φ (( G i )) .
This induces a 2 D -vector (matrix) of size m × r as before.
The advantage ofthis vectorization is computationally cheaper, and one can use any two filtering functions (atomicmass, partial charge, bond type, electron affinity, ionization energy) by skipping the distance function.
While distance function gives better descriptors in graph classification problem, one might prefer thiscomputationally cheaper method for very large datasets.
Weight filtration For a given weighted graph G =
( V , E , W ) , it is common to use edge weights
W = { ω rs ∈ R + | ϵ rs ∈ E} to describe filtration.
For example, in our case, one can take bondstrength in the compounds as edge weights.
By choosing the threshold set similarly I =
{ α i } m 1 withα 1 = min { ω rs ∈ W} <
α 2 < . . .
<
α m = max { ω rs ∈ W} .
For α i ∈ I , let E i =
e rs ∈ V |ω rs ≤ α i } .
Let G i be a subgraph of G induced by V i .
This induces a nested sequence of subgraphs
G 1 ⊂ G 2 ⊂ · · · ⊂ G m =
G .-0.
In the case of weighted graphs, one can apply the MP vectorization framework just by replacing the first filtering (via f )with weight filtering.
In particular, let g : V
→ R be a filteringfunction with threshold set { β j } nj =1 .
Then, one can first applyweight filtering to get G 1 ⊂ · · · ⊂ G m =
G as above, and thenapply f to each G i to get a bilfiltration {G ij } ( m × n resolution).
One gets m PDs as PD ( G i , g ) and induce the corresponding
M φ .
Alternatively, one can change the order by applying g first,and get a different filtering G 1 ⊂ G 2 ⊂ · · · ⊂ G n = G inducedby g .
Then, apply to edge weight filtration to any G j , one getsa bifiltration { (cid:98) G ji } ( n × m resolution) this time.
As a result,one gets n PDs as PD ( G i , ω ) and induce the corresponding
M φ .
The difference is that in the first case (first apply weights,then g ), the filtering function plays more important role as M φuses PD ( G i , g ) while in the second case (first apply g , thenapply weights) weights have more important role as M φ isinduced by PD ( G j , ω ) .
Note also that there is a very differentfiltration method for weighted graphs by applying the followingthe following VR-complexes method.
In Figure 3 and 4, we give an example of sublevel bifiltrationof the compound cytosine 3 by atomic number and partial charge functions.
In Figure 3, atomtypes are coded by their color.
Atomic numbers are given in the paranthesis.
White=Hydrogen (1),
Gray=Carbon (6), Blue=Nitrogen (7), and Red=Oxygen (8).
The decimal numbers next to atomsrepresent their partial charges.
Remark:
[Vietoris-Rips Filtrations] VR-filtration plays an important role in our construction, as itdetects the graph distances between atoms, and size of the topological features [60, 2].
In Figure 5,we gave an illustration how VR-filtration works.
With sublevel/superlevel filtrations (Figure 4), weobtain substructures of the compounds induced by chemical functions, e.g., atom weight, chirality,ionic radius, partial charge.
With VR-filtration (horizontal direction in Figure 5), we capture theinformation on distances between atoms and sizes of the rings evolving in these substructures, andtherefore it gives very fine information about the topological characteristics of these substructureswhere the other methods do not provide.
In the toy example in Figure 5, we use a small graph instead of a real compound to keep the expositionsimple.
Our sublevel filtration (vertical direcion) comes from the degree function.
Degree of a nodeis the number of edges incident to it.
In the first column, we simply see the single sublevel filtrationof by the degree function.
In each row, we develop VR-filtration of the subgraph by using the graphdistances between the nodes.
Here, graph distance between nodes means the length of the shortestpath (geodesic) in the graph where each edge is taken as length 1 .
Then, in the second column, weadd the edges for the nodes whose graph distance is equal to 2 .
In the third column, we add the (blue)edges for the nodes whose graph distance is equal to 3 .
Finally, in the last column, we add the (red)edges for the nodes whose graph distance is equal to 4 .
By construction, all the graphs in the lastcolumn must be a complete graph as there is no more edge to add.
Note that VR-filtration when used for unweighted graphs with graph distance is known as “powerfiltration” in the literature.
For further details on VR-filtration, see [4, 26].
A t o m i c
N u m b er
Partial Charge
Complete Graph
[Figure]
Sub l e v e l D e g ree
Graph Geodesic
C Further Experimental Results
C.1 Dataset Statistics
[Table] [Table]
C.2 Baselines
We compare our methods against the 23 state-of-the-art baselines including USR [7], ROCS [37],
PS [43], GZD [95], PH_VS [50], USR + GZD [86], USR + PS [86], USR + ROCS [86], GZD +
PS [86], GZD + ROCS [86], PS + ROCS [86], Findsite [104], Fragsite [105], Gnina [90], GOLDEATL [100], Glide-EATL [100], CompM [100], CompScore [77], CNN [79], DenseFS [45], SIEVEScore [102], DeepScore [97], and RF-Score-VSv3 [102].
In particular„ we compare our methods against the well-known 3 D -methods Ultrafast Shape Recognition (USR) [7], shape-based, ligand-centric method (ROCS)
[37], PatchSurfer (PS) [43], Zernike(GZD)
[95] and PH_VS [50] in Cleves-Jain dataset.
In Table 1, we report the performances of allthese 3 D methods with 50 conformations [86] except PH_VS with 1 conformation [50].
In Table 2, we compare our models against other state-of-the-art VS methods on DUD-E Diversedataset.
All of these ML methods came in recent years.
Among these, CNN [79] uses a convolutionalneural network based framework with GPU-accelerated layer (i.e., MolGridDataLayer) to define ascoring function for protein-ligand interactions.
DenseFS
[45] improves the previous model CNNby a densely connected convolutional neural network architecture.
Later, Sieve-Score proposes aneffective scoring function: similarity of interaction energy vector score (SIEVE)
[102] where theyextract van der Waals, Coulomb and hydrogen bonding interactions to represent docking poses for
ML.
Random Forest Based Protein-ligand Scoring Function (RF-Score)
[102] is another VS methodproposed in the same work.
Compscore [77] uses genetic algorithms to find the best combination of scoring components.
Recently, energy auxiliary terms learning (EATL)
[100] proposes an approachbased on the scoring components extracted from the output of the scoring functions implementedin several popular docking programs like Genetic Optimisation for Ligand Docking (GOLD) [49],
Molecular Operating Environment (MOE) [1] and Grid-based Ligand Docking with Energetics(GLIDE) [31, 34].
In the same work, they also combine these 3 methods and produced comprehensive
EATL models, CompF and CompM. DeepScore [97] defines a deep learning based target specificscoring function.
Findsite [104] proposes a threading/structure based method, where they improved itlater with Fragsite [105] by using tree regression ML framework.
Finally, Gnina [90] is a recentlyreleased molecular docking software, which uses deep convolutional networks to score protein-ligandstructures.
Note that all methods use 5-fold CV except Findsite-Fragsite (3-fold CV) and Gnina.
C.3 Model performance across different modalities
Table 5, 6, 7, 8 show detailed ablations of the modalities used in the graph filtration step of ToDD.
The success of each periodic property varies per drug target and trained ML model.
However merging
MP fingerprints derived from each one of these domain functions used for graph filtration has alwaysimproved the performance.
Our results also show that MP fingerprints ensure making successful predictions while training fewlabeled data such as Cleves-Jain (with 2-3 labeled compounds per drug target) using a transformerbased model.
RF shows worse performance than ViT on small-scale dataset such as Cleves-Jain, despite regularization by bootstrapping and using pruned, shallow trees.
Additionally, RF is more sensitive to the smallvariations in the training set, and imbalanced data can hamper its accuracy.
In order to effectivelyhandle the large-scale datasets that have long-tailed distributions, we undersample from the majorityclass (decoys).
Specifically, while training RF for the binary classification task on the drug targets of
DUD-E Diverse (class distributions are summarized in Table 4), we use 80% of the active compoundsand the same number of randomly chosen decoys for training.
Undersampling decoys to avoid heavyclass imbalance achieves better trade-offs between the accuracies of active compounds and decoys
[Table]
[Table] [Table]
[Table] [Table]
[Table]
C.4 Computation Time
See 6.3 in the main text for details.
[Table] [Table]
[Table]
[Table]
D Societal Impact and Limitations
D.1 Societal Impact
[Table] We perform in silico experiments and use high-throughput screening to recognize active compounds [Table] that can bind to a drug target of interest, e.g., an enzyme or protein receptor without conducting research on any human or animal subjects.
Our overarching goal is to augment the capabilities of AIto enhance the in silico virtual screening and drug discovery processes, develop new drugs that haveless side effects but are more effective to cure diseases, and minimize the participation of human andanimal subjects as much as possible to ensure their humane and proper care.
D.2 Limitations
We discuss in detail the computational complexity of our model in 6.3.
Our model is versatile andcan be scaled for large libraries by customizing the allocated computational resources.
Please notethat the analysis in C.4 shows the execution time of our computation pipeline when the featureextraction task is distributed across 8 cores of a single Intel Core i7 CPU.
It is possible to parallelizecomputationally costlier operations such as VR-filtration by allocating more CPU cores on the HPCplatform and optimize array operations (e.g., numpy) via the joblib library.
Furthermore, all ToDDmodels require substantially fewer computational resources during training compared to currentgraph-based models that encode a compound through mining common molecular fragments, a.k.a.
,motifs [47].
For instance, training a motif based GNN on GuacaMol dataset which has approximately1.5M drug-like molecules takes 130 hours of GPU time [65].
In contrast, once we generate thetopological fingerprints via Vietoris-Rips filtration, training time of ToDD-ViT and ToDD-ConvNeXtfor each individual drug target takes less than 1 hour on a single GPU (NVIDIA RTX 2080 Ti).