Data set.

To assess the representational ability of the present persistent homology algorithms on small molecules, we use a high quality data set of 1322 protein-ligand complexes with binding affinity data involving 7 protein clusters introduced earlier (denoted as S1322) [101]. It is a subset of the PDBBind v2015 refined set and its detail is given in the Supplementary material 1 of Ref. [101]. We consider a ligand based approach to predict the binding affinities of protein-ligand complexes in various protein clusters. As such, only the ligand information is used in our topological analysis. The ligand structures are taken from PDBBind database without modification. Numbers of ligands in protein clusters range from 94 to 333.

Models and performance.

Two models, i.e., TopBP-KNN(Ligand) and TopBP-ML(Ligand), are constructed. TopBP-KNN(Ligand) is used to directly assess the representation power of persistent homology for small molecules and TopBP-ML(Ligand) is the final practical model. The results are shown in Table 1. All the gradient boosting trees models take the setup described in Section Methods/Machine learning algorithms/Gradient boosting trees.

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Table 1. Pearson correlation coefficients (RMSE in kcal/mol) of ligand based topological model on the S1322 dataset.

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In TopBP-ML(Ligand), we process the geometry, the shape, and the covalent bond information of the small molecules using alpha complex, and the non-covalent intramolecular interactions using multi-level persistent homology with Rips complex. The features used are A-B012-E-S-GBT and R-B012-M1-S-GBT as described in Section Discussion/Ligand based protein-ligand binding affinity prediction. Gradient boosting trees method is used.

In TopBP-KNN(Ligand), we represent the small molecules with a collection of barcodes from element specific persistent homology calculations. Wasserstein distance with p = 2 is applied to measure similarities between two barcodes. The similarity between each pair of small molecules is then measured by taking the average of the Wasserstein distances between all considered barcodes. K-nearest-neighbor (KNN) regression is then applied to the measured similarity. In detail, the 6 barcodes considered are, R-B0-E-KNN, R-B1-E-KNN, R-B2-E-KNN, R-B0-M1-KNN, R-B1-M1-KNN, and R-B2-M1-KNN as described in Section Discussion/Ligand based protein-ligand binding affinity prediction. Leave-one-out validation within each protein cluster with k = 3 is used for this model.

In Table 1, FFT-BP 5-fold cross validation results were obtained based on multiple additive regression trees and a set of physical descriptors, including geometry, charge, electrostatic interactions, and van der Waals interactions for S1322 set [101]. Since multiple additive regression trees is also an implementation of the GBT used in the present work, it is appropriate to compare the FFT-BP results with the GBT results in this work to assess representation power of topological features. It is interesting to note that judging by RMSE, both sets of current topological descriptors have more predictive power than the physical descriptors built on protein-ligand complexes constructed in our earlier work [101]. These physical descriptors were constructed from sophisticated surface areas, molecular volumes, van der Waals interactions, charges computed by quantum mechanics, and Poisson-Boltzmann theory based electrostatics [101]. The success of topological descriptors implies the existence of an alternative and potentially more powerful description of the complex biomolecular world.

Data sets.

The PDBBind database provides a comprehensive collection of structures of protein-ligand complexes and their binding affinity data [99, 102]. The original experimental data in Protein Data Bank (PDB) [103] are selected to PDBBind database based on certain quality requirements and are curated for applications. As shown in Table 2, this database is expanding on a yearly basis. It has become a common resource for benchmarking computational methods and algorithms for protein-ligand binding analysis and drug design. Popular data sets include version 2007 (v2007), v2013, and v2015. Among them, v2013 core set and v2015 core set are identical. A large number of scoring functions has been tested on these data sets. The latest version, v2016, has an enlarged core set, which contains 290 protein-ligand complexes from 58 protein families. Therefore, this test set should be relatively easier than v2015 core set, whose 195 complexes involve 65 protein families. The core sets are constructed by choosing 3 samples with median, maximum, and minimum binding affinity from each protein family for v2007, v2013, and v2015 sets. The core set for v2016 was constructed similarly but with 5 samples from each protein family.

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Table 2. Description of the PDBBind datasets.

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Model and performance.

Two models TopBP-ML(Complex) and TopBP-DL(Complex) are introduced. The results are shown in Table 3. All the gradient boosting trees models take the setup described in Section Methods/Machine learning algorithms/Gradient boosting trees.

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Table 3. Pearson correlation coefficients (RMSE in kcal/mol) of different protein-ligand complex based approaches on PDBBind datasets.

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In TopBP-ML(Complex), alpha complex is used to describe the arrangement of carbon and heavy atom networks, while Rips complex with different distance matrices is used to describe the protein-ligand interactions from the perspective of interaction distances and strength of electrostatics interactions. In detail, the features used are R-B0-I-C, R-B0-CI-S, A-B12-E-S as described in Section Discussion/Complex based protein-ligand binding affinity prediction, and those used in TopBP-ML(Ligand).

With the idea that a sequence of element combinations ordered by their importance in gradient boosting trees models can make an extra dimension of the description, we build a 2D convolutional neural network with one spatial dimension and one dimension of element combination. We combine this 2D CNN with a 1D CNN with the pairwise interaction inputs. For the construction of 2D input, the reader is referred to Section Feature generation from topological invariants. The 1D image-like inputs consist of two parts both generated by the counts in bins method described in Section Feature generation from topological invariants. For the 0th dimensional barcodes from interactive persistent homology of the 36 pairs of atom types ({C,N,O,S} from protein and {C,N,O,S,P,F,Cl,Br,I} from ligand), the interval [0, 50] Å is divided into equal length subintervals of length 0.25 Å. For the 0th dimensional barcodes from interactive persistent homology for electrostatics of the 50 pairs of atom types ({C,N,O,S,H} from protein and {C,N,O,S,P,F,Cl,Br,I,H} from ligand), the parameter interval of [0, 1] is divided into equal length subintervals of length 0.01. These two 1D image-like features have sizes 200 × 36 and 100 × 50. The network architecture is given in Section Methods/Machine learning algorithms/Deep convolutional neural networks.

The final model TopBP(Complex) takes the average of TopBP-ML(Complex) and TopBP-DL(Complex) with the assumption that the errors made by the two approaches are only partially correlated and thus averaging over them may cancel part of the errors. As a result, TopBP(Complex) delivers the best prediction performance on all four testing sets.

DUD data set.

The directory of useful decoys (DUD) [107, 108] is used to benchmark our topological approach for virtual screening. The DUD data set contains 40 protein targets from six classes, i.e., nuclear hormone receptors, kinases, serine proteases, metalloenzymes, folate enzymes, and other enzymes. A total of 3,961 active ligand-target pairs were identified from literature. The number of ligands for each target ranges from tens to hundreds. At most 36 decoys were constructed for each ligand, from the ZINC database of commercially available compounds [109]. At the first step, the ZINC database of 3.5 million compounds was reduced to a database of 1.5 million compounds with similarity less than 0.9 to the ligands. The similarity was measured by Tanimoto coefficient on CACTVS type 2 fingerprints. The decoys were selected so that they possess similar physical properties to the ligands but have dissimilar molecular topology (topology in the sense of chemistry, not mathematical topology). A total of 32 physical properties were used including molecular weight, partition coefficient, and number of hydrogen bonding groups. This results in a total of 128,374 compound-target pairs. A discrepancy between calculated partial charges for the ligand and decoy sets was reported for the original release 2 of DUD datasets, which makes it trivial for virtual screening methods to distinguish between the two categories using those charges [110]. In this work, we use the data with recalculated Gasteiger charges for both ligand and decoy sets given by Armstrong et al. [110] in AutoDock Vina and our electrostatic persistence.

Data processing.

In structure-based virtual screening, the possible complex structures of the target protein and the small molecule candidate are required. For the DUD dataset, the structures of the 40 protein targets, the ligands, and the decoys are given, and we generate the protein-compound complexes by using docking software. To this end, we first add missing atoms to the proteins by using the profix utility in Jackal software package [111]. The receptors and ligands or decoys are prepared using the scripts prepare_receptor4.py and prepare_ligand4.py provided by the AutoDockTools module in MGLTools package (version 1.5.6) [112]. The bounding box of the binding site is defined as a cube with edge size equal to 27 Å, centered at the geometric center of the crystal ligand. AutoDock Vina (version 1.1.2) [113] is used to dock the ligands or decoys to the receptors. The option exhaustiveness is set to 16 and all the other parameters are set to their default values. In each docking experiment, the pose having the lowest binding free energy reported by AutoDock Vina, is used by the reranking models.

Evaluation.

Two measurements, the enrichment factor (EF) and the area under the receiver operating characteristic curve (AUC), are used to evaluate each method’s ability of discriminating actives from decoys. The AUC is defined as (1) where N_a is the number of active ligands, N_d is the total number of decoys, and is the number of decoys that are higher ranked than the ith ligand [98]. An AUC value of 0.5 is the expected value of a random selection, whereas a perfect prediction results in an AUC of 1. The EF at x% denoted by EF_x% evaluates the quality of the set of top x% ranked compounds, by comparing the percentage of actives in the top x% ranked compounds to the percentage of actives in the entire compound set. It is defined as (2) where is the number of active ligands in the top x% ranked compounds, N^x% is the number of top x% ranked compounds, N is the total number of compounds, and N_a is the total number of active ligands.

To evaluate the performance of various methods on the DUD data set, the entries associated with one protein target are used as the test set in the experiment on this protein target [98]. For the selection of the training set of a given protein target, we follow a procedure given in the literature [107], where the entries associated to the rest of the proteins, excluding those that are within the same class of the testing protein and those that have reported positive cross-enrichment with the testing protein, are taken as the training set. The 40 proteins are split into 6 classes [100]. A detailed list of proteins that are excluded from the training set of each protein is given in Table F in S1 Text.

Topology based machine learning models.

Our topology based machine learning model, called TopVS-ML, relies on manually constructed features and utilizes ensemble of trees methods. For the complex with the small molecules (i.e., ligands and decoys) docked to the receptor, features R-B0-I-BP, R-B0-CI-S, and A-B12-E-S are used (see Section Discussion/Complex based protein-ligand binding affinity prediction), whereas features R-B012-M1-S and A-B012-E-S (see Section Discussion/Ligand based protein-ligand binding affinity prediction) are used for the small molecules. The gradient boosting trees method, random forest method, and extra trees method are employed as voters. The averaged probabilities output by the three methods are used for the classifier to decide the class of the testing samples. The modules GradientBoostingClassifier, RandomForestClassifier, and ExtraTreesClassifier in the scikit-learn package [114] (version 0.17.1) are used. The parameters for the three modules are listed in Table 4. TopVS-ML achieves a performance of AUC = 0.83, EF_2% = 8.6, EF_20% = 3.4. These values are the median values of 10 repeated experiments. Table G in S1 Text lists the result of each single experiment confirming that the performance is consistent across each repeated run.

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Table 4. Parameters used in machine learning.

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Topology based deep learning model.

Our topology based deep learning model, called TopVS-DL, relies on 1D image-like inputs for protein-compound complexes and manually constructed features for the compounds. The 2D representation used in binding affinity problem is not used here due to the intractable data size. The manually constructed features for the compounds are R-B012-M1-S and A-B012-E-S as described in Section Discussion/Ligand based protein-ligand binding affinity prediction. The 1D image-like inputs consisted of three parts are all generated by the counts in bins method described in Section Feature generation from topological invariants. (1) For the 0th dimensional barcodes from interactive persistent homology of the 36 pairs of atom types ({C, N, O, S} from protein and {C, N, O, S, P, F, Cl, Br, I} from ligand), the interval [0, 25] Å is divided into equal length subintervals of length 0.25 Å. The barcodes used here are identical to the barcodes in feature R-B0-I-BP. This results in a 1D image-like feature with size 100 × 36. (2) For the 0th dimensional barcodes from interactive persistent homology for electrostatics of the 50 pairs of atom types ({C, N, O, S, H} from protein and {C, N, O, S, P, F, Cl, Br, I, H} from ligand), the parameter interval of [0, 1] is divided into equal length subintervals of length 0.01. The barcodes used are identical to the barcodes in feature R-B0-CI-S. This results in a 1D image-like feature with size 100 × 50. (3) Alpha complex based persistent homology is applied to all carbon atoms and all heavy atoms. The computation is done on the complex as well as only the protein with a cutoff distance of 12 Å from the ligands. The interval [0, 12] Å is divided into equal length subintervals of length 0.125 Å. Counts in bins method is applied to the 0th, 1st, and 2nd dimensional barcodes. The features are generated for persistent homology computation of the complex and the protein. The features for the complex and the difference between the features for complex and protein are finally used. This results in a 1D image-like feature of size 96 × 32. The detailed network architecture is listed in Section Methods/Machine learning algorithms/Deep convolutional neural networks. A consensus model is constructed by taking the average over 25 single models trained independently. TopVS-DL achieves a performance of AUC = 0.81, EF_2% = 9.1, EF_20% = 3.2.

The final model.

Same as the idea of taking the average output of different ensemble of trees models as the final output in TopVS-ML, we add TopVS-DL as another voter to TopVS-ML to construct a final model, called TopVS. Such consensus approach takes the average over different models with the hope that different models make partially uncorrelated errors which are possible to cancel out when averaged. The performance on each of 40 protein targets is reported in Table 5. We have also generated virtual screening results of AutoDock Vina (ADV) based on the computed binding free energy by ADV and compared them with those of the present TopVS in terms of enrichment factors and the areas under the receiver operating characteristic curve (AUC). A comparison of average AUC with those from a large number of methods is given in Table 6.

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Table 5. Performance on each protein in DUD dataset.

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Table 6. AUC comparison of different methods on DUD dataset.

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Feature vectors for gradient boosting trees.

In this test, Rips complex based and alpha complex based persistent homology computations up to 2nd dimension are performed for a variety of atom collections with different element types using the Euclidean metric and multi-level distance defined in Eq (3). Two types of features are generated and are denoted by F^C, which is a combination of , , and , and F^S, which is a combination of , , and . The construction of features F^C and F^S are described in Section Feature generation from topological invariants. For sets of the 0th dimensional bars, only and are computed. In each protein cluster, 10-fold or 5-fold cross validation is repeated 20 times for each subset of feature vectors depending on selected element type. The median Pearson correlation coefficients and the root-mean-square error (RMSE) in kcal/mol are reported. For Rips complex, both level 0 computation with distance matrix M and level 1 computation with distance matrix as defined in Eq (4) are performed. A comparison of these results is shown in S1 Text Table B. The results corresponding to alpha complex are shown in S1 Text Table A. The average performance for alpha complex and Rips complex has a Pearson correlation coefficient of 0.987.

Barcode space metrics for k-nearest neighbor regression.

The barcodes generated using Rips complex with distance matrices M and are collected and the distance between each pair of barcodes are measured using the Wasserstein metric d². Leave-one-out prediction for every sample is performed with k-nearest neighbor regression with k = 3 within each protein cluster based on the Wasserstein metric. The results are shown in S1 Text Table C. The performance of the best performing and the worst performing protein clusters is shown in Fig 3. The better the performance, the closer the lines are to the semicircle.

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Fig 3. An illustration of similarities between ligands measured by their barcode space Wasserstein distances.

Ligands are ordered according to their binding affinities and are represented as dots on the semicircle. Specifically, a sample of binding free energy x is plotted at the angle θ = π(E_max − x)/(E_max − E_min) where E_min and E_max are the lowest and the highest energy in the dataset. Each dot is connected with two nearest neighbors based on their barcode space Wasserstein distances. An optimal prediction would be achieved if lines stay close to the semicircle. The majority of the connections stay near the boundary to the upper half sphere demonstrating that barcode space metric based Wasserstein distance measurement reflects the similarity in function, i.e., the binding affinity in this case. The protein clusters with the best and the worst performance are shown. Left: Protein cluster 2. Right: Protein cluster 3.

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The experiments done for this section are summarized in Table 7.

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Table 7. Experiments for ligand-based protein-ligand binding affinity prediction of 7 protein clusters and 1322 protein-ligand complexes.

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Performance of multi-component persistent homology.

It can be noticed from Table 8 that topological features generated from barcode statistics typically outperform those created from counts in bins. R-B012-E-S-GBT and R-B012-M1-S-GBT perform similarly in the majority of the protein clusters whilst R-B012-M1-S-GBT which is based on significantly outperforms R-B012-E-S-GBT which is based on Euclidean distance in protein cluster 3 and 6. To assess in what circumstances does the multi-level persistent homology improve the original persistent homology characterization of small molecules, we analyze the statistics of the size of ligands in Fig 2. It turns out that protein cluster 3 has the smallest average number of heavy atoms and protein cluster 6 has the smallest standard deviation of the number of heavy atoms. This observation partially answers the question that in the cases where the small molecules are relatively simple and are relatively of similar size, multi-level persistent homology is able to enrich the characterization of the small molecules which further improves the robustness of the model. Such enrichment or improvement over the original persistent homology approach is mainly realized in higher dimensional barcodes, i.e. the 1st and 2nd dimensions. In Table 8, the results with ID through 7 to 12 confirm that the 0th dimensional features from computation with are inferior to the results with Euclidean distance whilst the 1st and 2nd dimensional features based on outperforms the best result with Euclidean distance in most cases.

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Table 8. Performance of different approaches on the S1322 dataset.

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It is interesting to note that although Wasserstein metric based KNN methods are not as accurate as GBT approaches, the consensus result obtained by averaging over various predictions with Wasserstein metric on different sets of barcodes is quite accurate.

Robustness of topological learning models.

Certain elements such as Br are very rare in the data sets studied in this work. Considering only the elements of high occurrence will not hurt the performance on the validations performed. However, omitting the low occurrence elements will sacrifice the capability of the model to handle new data in which such elements play an important role. Therefore, we decide to keep the rare elements that result in a large number of features and redundancy in features. For example, the element combinations CBrH and CH will probably deliver the same performance for most of the samples in the data sets studied in this work. To test whether this redundancy causes degenerated results of the model, the features of one element combination is added to the model at a step and the model is validated with an accumulation of the added features at each step. The performance of the model is measured with Pearson correlation coefficient and is plotted against number of element combinations involved in Fig 4. For most cases in Fig 4, the model is robust against the inclusion of more element combinations.

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Fig 4. Plot of performance against number of element combinations used.

The topological learning model performance against the number of element combinations involved in feature construction for 7 protein clusters in S1322. The horizontal axis corresponds to the number of element combinations used for the features. From left to right, one extra element combination is added at a step. The features are then used in gradient boosting trees method to test if the model is robust against redundant information. The results related to alpha complex are marked in red and Rips complex in blue. The median Pearson correlation coefficient between predicted and experimental results is reported of 10-fold cross-validation within each protein cluster repeated 20 times are reported.

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Groups of topological features and their performance in association with GBT.

The experiments of protein-ligand-complex-based protein-ligand binding affinity prediction for the PDBBind datasets are summarized in Table 9.

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Table 9. Experiments for protein-ligand-complex-based protein-ligand binding affinity prediction for the PDBBind datasets.

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Robustness of GBT algorithm against redundant element combination features and potential overfitting.

It is intuitive that combinations of more than 2 element types are able to enrich the representation especially in the case of higher dimensional barcodes. However, the consideration of combination of more element types rapidly increases the dimension of feature space. In the high dimensional feature space, it is almost inevitable that there exists nonessential and redundant features. Additionally, the importance of a feature varies across different problems and data sets. Therefore, it is preferable to keep all the potentially important features in a general model which is expected to cover a wide range of situations. To test the robustness of the model against unimportant features, we select a total of 128 element combinations (i.e., all possible paired choices of one item from {C, N, O, CN, CO, NO, CNO, CNOS} in protein and another item from {C, N, O, S, CN, CO, CS, NO, NS, OS, CNO, CNS, COS, NOS, CNOS, CNOSPFClBrI} in ligand). The 0th, 1st, and 2nd dimensional barcodes are computed for all combinations using alpha complex with Euclidean distance. Features are generated following the barcode statistics method.

A general model with all the features is generated in the first place. The element combinations are then sorted according to their importance scores in the general model. Starting from the most important element combination, one element combination is added to the feature vector each time and then the resulting feature vector is passed to the machine learning training and testing procedure. The order of adding element combinations is based on their importance scores and thus that a less important feature is added each step.

Fig 5 depicts the changes of Pearson correlation coefficient and RMSE (kcal/mol) with respect to the increase of element combinations in predicting four PDBBind core sets. In all cases, the inclusion of top combinations can readily deliver very good models. The behavior of the present method in PDBBind v2007 is quite different from that in other data sets. The performance of the present method improves almost monotonically as the element combination increases. However, in other three cases, the improvement is unsteady. Nevertheless, the performance fluctuates within a small range, which indicates that the present method is reasonably stable against the increase in element combinations. From a different perspective, the increase in element combinations might lead to overfitting in machine learning. Since the model parameters are fixed before the experiments, it shows that GBT algorithms are not very sensitive to redundant features and are robust against overfitting.

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Fig 5. Feature robustness tests on PDBBind datasets.

The performance of the topological learning model against the number of included element combinations for predicting on PDBBind core sets and training on PDBBind refined sets minus the core sets. The 1st and 2nd dimensional barcodes computed with alpha complex is used. Features are generated following barcode statistics method. Element combinations are all possible paired choices of one item from {C, N, O, CN, CO, NO, CNO, CNOS} in protein and another item from {C, N, O, S, CN, CO, CS, NO, NS, OS, CNO, CNS, COS, NOS, CNOS, CNOSPFClBrI} in ligand, which result in 128 element combinations. The horizontal straight lines represents the performance of the 2D representation with deep convolutional neural network (row 10 in Table 10). The blue and red colors correspond to Pearson correlation coefficient and RMSE (kcal/mol) respectively. Each experiment is done by training on refined set minus the core set with the median result of 20 repeated runs reported.

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Usefulness of more than 2 element types for interactive 0th dimensional barcodes.

While using element combinations with more than 2 element types with higher dimensional barcodes enriches characterization of geometry, it remains to assess whether interactive 0th dimensional characterization will benefit from element combinations with more element types. As an example, we denote interactive 0th dimensional barcodes for carbon and nitrogen atoms from protein and oxygen atoms from ligand by B_CN−O, barcodes for carbon atoms from protein and oxygen atoms from ligand by B_C−O, and barcodes for nitrogen atoms from protein and oxygen atoms from ligand by B_N−O. In the case of persistent homology barcode representation, B_CN−O is not strictly the union of B_C−O and B_N−O. However B_CN−O might be redundant to B_C−O and B_N−O. To address this concern, we test features from interactive 0th dimensional barcodes with the 36 element combinations (i.e., {C, N, O, S} for protein and {C, N, O, S, P, F, Cl, Br, I} for ligand) and features for the 160 selected element combinations (i.e., {C, N, O, S, CN, CO, NO, CNO} for protein and {C, N, O, S, P, F, Cl, Br, I, CN, CO, CS, NO, NS, OS, CNO, CNS, COS, NOS, CNOS} for ligand), which are listed as feature group 2 and feature group 1 in Table 10. In all the four cases, the features of the 36 combinations (feature group 2) slightly outperforms or performs as well as the features of the 160 combinations (feature group 1) suggesting that element combinations with more than 2 element types are redundant to all the combinations with 2 element types in the case of interactive 0th dimensional characterization.

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Table 10. Performance of different protein-ligand complex based approaches on the PDBBind datasets.

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Importance of atomic charge in electrostatic persistence.

In element specific persistent homology, atoms of different element types are characterized separately, which offers a rough and implicit description of the electrostatics of the system. However, such implicit treatment of electrostatics may lose important information because atoms behave differently at different oxidation states. Therefore, we explicitly embed atomic charges in interactive 0th dimensional barcodes as described in Eq (10). The resulting topological features are given in feature group 4 in Table 10. It can be seen from Table 10 that the combination of feature group 4 and the Euclidean distance based interactive 0th dimensional barcodes (listed as feature group 6 and 7) generally outperforms the results obtained with only Euclidean distance based features. This observation suggests that electrostatics play an important role and should be taken care of explicitly for the protein-ligand binding problem. Additionally, the inclusion of physical interactions in topological invariants opens a promising new direction in topological analysis.

Relevance of elements that are rare with respect to the data sets.

Since the majority of the samples in both training and testing sets only contain atoms of element types, C, N, O, and H, the performance of the model on the samples with rare occurring elements with respect to data sets is hardly reflected by the overall performance statistics. For simplicity, we refer to such rarely occurring elements with respect to data sets simply by rarely occurring elements in the discussion follows. To assess the aspects of the model that potentially affect the performance on the samples containing rarely occurring elements, we picked the samples containing each rarely occurring element from the original testing set as a new testing set. Three experiments are carried out to address two questions: “Are the training samples containing the same rarely occurring element crucial?” and “Are features addressing the rarely occurring element important?”. A short answer is yes to both according to the results shown in Fig 6. Specifically, for each rarely occurring element, the exclusion of samples containing this element in training set and the exclusion of features addressing this element will both cause degenerated results. It is also shown that the exclusion of samples of the rarely occurring element leads to much worse results. Since both modifications of the model deliver worse results, we conclude that including the samples in the training set with similar compositions to the test sample is crucial to the success of the model on this specific test sample. Even the inclusion of features of more element types or element combinations does not deliver better results in the general testing sets, such features should still be kept in the model in case that a sample with a similar element composition comes in as a test sample.

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Fig 6. Assessment of performance of the model on samples with elements that are rare in the data sets.

For the four data sets PDBBind v2007, v2013, v2015, and v2016 [99], and for each element, the testing set is the subset of the original core sets with only ligands that contain atoms of the particular element type. The features used are features with ID = 7 in Table 10. The reported RMSE is the average taken over the four data sets. Experiment 1: Training set is the original training set and all the features are used. Experiment 2: Training set is the original training set and only features that do not involve the particular element are used. Experiment 3: Training set is the original training set excluding the samples that contain atoms of the particular element type and all features are used. For most of the elements, experiment 1 achieves the best result and experiment 3 yields the worst performance.

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2D persistence for topological deep convolutional neural networks.

Deep learning is potentially more powerful than many other machine learning algorithms when the data size is sufficiently large. In the present work, it is natural to construct a 2D topological representation by incorporating the element combination as an additional dimension, resulting in 16 channels as defined in Section Feature generation from topological invariants. Here 128 element combinations (i.e., all possible paired choices of one item from {C, N, O, CN, CO, NO, CNO, CNOS} in protein and another item from {C, N, O, S, CN, CO, CS, NO, NS, OS, CNO, CNS, COS, NOS, CNOS, CNOSPFClBrI} in ligand) are used for 2D analysis. The advantage of introducing this extra dimension with convolutional neural networks is to prevent unimportant features from interacting with important ones at the lower levels of the model whilst generally unimportant features are still kept in the model in case that they are essential to specific problems or a certain portion of the data set. Fig 7 illustrates the mean value and the standard deviation of the PDBBind v2016 refined set. The existence of significant standard deviations for relatively unimportant element combinations indicates that these features might still contribute to the overall prediction.

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Fig 7. Heat map plot of the 16 channels.

The mean value (left image) and the standard deviation (right image) of each digit over the PDBBind v2016 refined set are shown. The top 8 maps are for protein-ligand complex and the other 8 maps are for the difference between protein-ligand complex and protein only. For each map, the vertical axis is the element combinations ordered according to their importance and the horizontal axis is the dimension of spatial scales.

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As shown in Fig 5, for all the data sets except the PDBBind v2007 set, the 2D topological deep learning with convolutional neural networks performs significantly better. The inferior performance of convolutional neural networks in v2007 might be a result of the small data size. Note that v2007 training set has 1105 protein-ligand complexes, whereas other training sets have more than 2700 complexes. Consequently, topological deep convolutional neural networks are able to outperform the topological GBT algorithm in predicting v2013, v2015 and v2016 core sets. Indeed, topological deep convolutional neural networks have advantages in dealing with large data sets.

Simplicial complex.

A (geometric) k-simplex denoted σ^k is the convex hull of k + 1 affinely independent points in . The convex hull of each nonempty subset of the k + 1 points forms a subsimplex and is regarded as a face of σ^k. The points are also called vertices of σ^k.

A set of simplices K is a simplicial complex if all faces of any simplex in K are also in K and the intersection of any pair of simplices in K is either empty or a common face of the two simplices.

Homology.

A k-chain of a simplicial complex K denoted by c_k is a formal sum of k-simplices in K. Here, we take the field for the coefficients of the formal sum. Under the addition operation of , a group of k-chains is called a chain group and denoted C_k(K) which has the basis as the set of k-simplices in K.

A boundary operator denoted by ∂_k: C_k(K) → C_k−1(K) maps a k-chain which is a linear combination of k-simplices to the same linear combination of the boundaries of the k-simplices. With a k-simplex σ^k = [v₀, …, v_k] where v_i are the vertices of σ^k, the boundary operator is defined as , where is a (k-1)-simplex which is a face of σ^k with the ith vertex being absent. Since we are working with the coefficients, we omit the orientations of the simplices.

A k-cycle is a k-chain whose image under the boundary operator ∂_k is the empty set. The collection of all the k-cycles forms a group denoted Z_k(K) which is the kernel of ∂_k: C_k(K) → C_k−1(K).

The image of ∂_k+1: C_k+1(K) → C_k(K) is called the boundary group and is denoted by B_k(K). B_k(K) is a subgroup of Z_k(K) following the property of the boundary operator that ∂_k ∘ ∂_k+1 = 0.

The kth homology group is the quotient group defined as H_k(K) = Z_k(K)/B_k(K). Its ranks are the Betti numbers of K and its generators (equivalence classes) are also of interest.

The kth Betti number β_k is defined and often computed as rankH_k(K) = rankZ_k(K) − rankB_k(K). Intuitively, Betti numbers count the number of k-dimensional holes that can not be continuously deformed to each other. Analogous to the continuous case, in simplicial topology, two cycles (elements of Z_k(K)) that defer by the boundary of a chain (an element of B_k(K)) are considered to be able to deform continuously to each other and are thus representing the same element in H_k(K).

Persistent homology.

A filtration of a simplicial complex K is a nested sequence of subcomplexes of K such that ⌀ = K⁰ ⊂ K¹ ⊂ … ⊂ K^m = K. Each Kⁱ is itself a simplicial complex. We are interested in tracking the birth and death of homology generators along filtration.

Given a simplicial complex K with its filtration, the p-persistent kth homology group of Kⁱ is defined as . Intuitively, this records the homology classes of Kⁱ that are persistent at least until K^i+p. Persistent homology allows us to not only compute n-dimensional holes at a specific setup, but also compute the parameter values corresponding to the birth and death of the n-dimensional holes along the filtration.

A generator (equivalence class) in H_k(Kⁱ) which does not exist in H_k(Kⁱ⁻¹) or H_k(K^j) and lasts until H_k(K^j−1), under the mappings along the sequence of homology groups induced by inclusion maps along the sequence of simplicial complexes, is associated with the interval [x_i, x_j), where x_i and x_j are the filtration levels associated to Kⁱ and K^j. A collection of these intervals tracks the appearing and disappearing of homology generators along the filtration process. Such collections of intervals can be visualized by stacking horizontal line segments (barcodes) or by plotting in a plane (persistence diagrams). The collection of intervals associated to the kth homology group is called the kth dimensional barcodes.

Simplicial complexes and filtration.

Given a finite set of points X and a non-negative scale parameter r, the Vietoris-Rips complex and alpha complex are constructed as follows.

With a predefined distance function d(⋅, ⋅) in X, a subset X′ of X forms a simplex if d(x_i, x_j) ≤ r for all x_i, x_j ∈ X′. The collection of all such simplices is the Vietoris-Rips complex of the finite metric space X with scale parameter r denoted by Rips(X, r). It is obvious that Rips(X, r) ⊆ Rips(X, r′) for r ≤ r′.

With Alpha(X, r) being the alpha complex of X with the scale parameter r and given the Delaunay triangulation induced by the Voronoi diagram of X, a simplex in the Delaunay triangulation belongs to Alpha(X, r) if all its 1-faces (1-simplex as subset of the simplex) have length no greater than 2r. Similar to Rips complex, alpha complex also has the property that Alpha(X, r) ⊆ Alpha(X, r′) for r ≤ r′.

Covalent bonds.

Covalent bonds are formed via shared electron pairs or bonding pairs. The lengths and the number of covalent bonds can be easily detected from 0th dimensional barcodes. For macromolecules, the same type of covalent bonds have very similar bond lengths and thus 0th dimensional barcode patterns.

Non-covalent interactions.

Non-covalent interactions play a critical role in maintaining the 3D structure of biomolecules and mediating chemical and biological processes, such as solvation, binding, protein-DNA specification, molecular self-assembly, etc. Physically, non-covalent interactions are due to electrostatic, van der Waals forces, hydrogen bonds, π-effects, hydrophobic effects, etc. The ability to characterize non-covalent interactions is an essential task in any methodological development. The 1st and 2nd dimensional barcodes are suitable for the characterization of the arrangement of such interactions in a larger scale. Additionally, we propose multi-level persistence and electrostatic persistence to reveal local and pairwise non-covalent interactions via 0th dimensional barcodes as well.

Chirality, cis effect and trans effect.

Chirality, cis and trans effects are geometric properties of many molecules. Among them, chirality is a symmetry property such that a chiral molecule cannot be superposed on its mirror image. Cis and trans effects are due to molecular steric and electronic effects. Chirality, cis and trans effects often play a role in molecular kinetics, activity and catalysis, and thus their characterization is an important issue in developing topological methods. These effects should be reflected from barcodes of various dimensions.

Multi-leveled protein structures.

Protein structures are typically described in terms of primary, secondary, tertiary and quaternary levels. The protein primary structure is the linear sequence of amino acids in the polypeptide chain. Protein secondary structure refers to the local 3D structure of protein segments containing mainly α-helix and β-sheets, which are highly regular and can be easily detected by distinct Frenet-Serret frames. A tertiary structure refers to the 3D structure of a single polypeptide chain. Its formation involves various non-covalent and covalent interactions including salt bridges, hydrophobic effects, and often disulfide bonds. A quaternary structure refers to the aggregation of two or more individual folded protein subunits into a 3D multi-subunit complex. Protein structures are further complicated by its functional domains, motifs, and particular folds. The protein structural diversity and complexity result in the challenge and opportunity for methodological developments. Various persistent homology techniques, including multi-component, multi-level, multi-dimensional [119], multi-resolution [90], electrostatic, and interactive [27] persistent homologies have been designed either in our earlier work or in this paper for protein structural diversity and complexity.

Protein-ligand, protein-protein, and protein-nucleic acid complexes.

Topological characterization of proteins is further complicated by protein interactions or binding with ligands (drugs), other proteins, DNA and/or RNA molecules. Although a normal protein involves only carbon (C), hydrogen (H), nitrogen (N), oxygen (O) and sulfur (S) atoms, its protein-ligand complexes bring a variety of other elements into the play, including, phosphorus (P), fluorine (F), chlorine (Cl), Bromine (Br), iodine (I), and many important biometals, such as calcium (Ca), potassium (K) sodium (Na), iron (Fe), copper (Cu), cobalt (Co), zinc (Zn), manganese (Mn), chromium (Cr), vanadium (V), tin (Sn), and molybdenum (Mo). Each biological element has important biological functions and its presence in biomolecules should be treated uniformly as a set of points in the point cloud data. The interaction of protein and nucleic acids can be very intricate. Qualitatively, multiscale and multi-resolution persistent homology demonstrates interesting features in 3D DNA structures [89]. Typically, 3D RNA structures are more flexible and difficult to extract topological patterns. Interactive persistent homology, element specific persistent homology and binned representation for persistent homology outputs were designed to deal with interactions between protein-ligand, protein-protein, and protein-nucleic acid complexes [14, 27, 94]. These approaches worked well in protein-mutation site interactions [14]. Additionally, multi-level persistent homology and electrostatic persistence proposed in this work are useful tools to describe some other specific interactions.

Multi-level persistent homology.

Small molecules such as ligands in protein-ligand complexes usually contain fewer atoms than large biomolecules such as proteins. Bonded atoms stay closer than non-bonded ones in most cases. As a result, the collection of 0th dimensional bars will mostly provide the information about the length of covalent bonds and the higher dimension barcodes will most likely be very sparse. It is difficult to capture non covalent bond interactions among atoms especially hydrogen bonds and van der Waals pairwise interactions in 0th dimensional barcodes. In order to describe non covalent interactions, we propose multi-level persistent homology, by simply modifying the distance matrix, similar to the idea of modifying distance matrix to emphasize on the interactions between protein and ligand [27]. Given the original distance matrix M = (d_ij) with 1 ≤ i, j ≤ N, the modified distance matrix is defined as (3) where d_∞ is a large number which is set to be greater than the upper limit of the filtration value chosen by a persistent homology algorithm. Note that this matrix may fail to satisfy triangle inequality whilst still satisfies the construction principle of Rips complex.

The present multi-level persistent homology is able to describe any selected interactions of interest and delivers two benefits in characterizing biomolecules. Firstly, the pairwise non-covalent interactions can be reflected by the 0th dimensional barcodes. Secondly, such treatment generates more higher dimensional barcodes and the small structural fluctuation among different conformations of the same molecule can be captured. The persistent barcode representation of the molecule can be significantly enriched to better distinguish between different molecular structures and isomers. As an illustration, we take the ligand from the protein-ligand complex with PDB code “1BCD” which only has 10 atoms. A different conformation of the ligand is generated by using the Frog2 web server [120]. The persistent barcodes generated using Rips complex with the distance matrices M are identical and only have 0th dimensional bars due to the simple structure. In this case, the 0th dimensional bars only reflect the length of each bond and therefore fail to distinguish the two slightly different conformations of the same molecule. However, when the modified distance matrices are employed, the barcode representation is significantly enriched and is able to capture the tiny structural perturbation between the conformations. An illustration of the outcome from the modified distance matrix is shown in Fig 8. A general nth level persistence characterization of molecules can be obtained with the distance matrix as, (4) where D(i, j) is the smallest number of bonds to travel from atom i to atom j and d_∞ is some number greater than the upper limit of the filtration value.

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Fig 8. Multi-level persistent homology on simple small molecules.

Illustration of representation ability of in reflecting structural perturbations among conformations of the same molecule. Left: The structural alignment of two conformations of the ligand in protein-ligand complex (PDB:1BCD). Right: The persistence diagram showing the 1st and 2nd dimensional results generated using Rips complex with for two conformations. It is worth noticing that the barcodes generated using Rips complex with M are identical for the two conformations.

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Interactive persistent homology.

In protein-ligand binding analysis and analysis involving interactions, we are interested in the change of topological invariants induced by interactions that are caused by binding or other processes. Similar to the idea of multi-level persistent homology, we can design a distance matrix to focus on the interactions of interest. For a set of atoms, A = A₁ ∪ A₂ with A₁ ∩ A₂ = ∅ where only interactions between atoms from A₁ and atoms from A₂ are of interest [27]. The interactive distance matrix is defined as (5) where M is the original distance matrix induced from Euclidean metrics or other correlation function based distances, a_i and a_j are atoms i and j, and d_∞ is a number greater than the upper limit of the filtration value. In applications, A₁ and A₂ can be respectively a set of atoms of the protein and a set of atoms of the ligand in a protein-ligand complex. In this case, the characterization of interactions between ligand and protein is an important task. In the modeling of point mutation induced protein stability changes, A₁ could be the set of atoms at the mutation site and A₂ could be the set of atoms of surrounding residues close to the mutation site. Similar treatment can be used for protein-protein and protein-nucleic acid interactions.

Correlation function based persistent homology.

For biomolecules, the interaction strength between pair of atoms usually does not align linearly to their Euclidean distances. For example, van der Waals interaction is often described by the Lennard-Jones potential. Therefore, kernel function filtration can be used to emphasize certain geometric scales. Correlation function based filtration matrix was introduced in our earlier work [26]: (6) where Φ(d_ij, η_ij) is a radial basis function and η_ij is a scale parameter. This filtration can be incorporated in the element specific persistent homology (7) Additionally, one can simultaneously use two or more correlation functions characterized by different scales to generate a multiscale representation of biomolecules [106].

Flexibility and rigidity index based filtration matrix.

One form of the correlation function based filtration matrix is constructed by flexibility and rigidity index. In this case, the Lorentz function is used in Eq (7) (8) where d_ij is the Euclidean distance between point i and point j and η_ij is a parameter controlling the scale and is related to radius of two atoms. When distance matrices based on such correlation functions are used, patterns at different spatial scales can be addressed separately by altering the scale parameter η_ij. Note that the rigidity index is given by [121] (9) This expression is closely related to the rigidity density based volumetric filtration [90].

Electrostatic persistence.

Electrostatic effects are some of the most important effects in biomolecular structure, function, and dynamics. The embedding of electrostatics in topological invariants is of particular interest and can be very useful in describing highly charged biomolecules such as nucleic acids and their complexes. We introduce electrostatics interaction induced distance functions in Eq (10) to address the electrostatic interactions among charged atoms. The abstract distance between two charged particles are rescaled according to their charges and their geometric distance, and is modeled as (10) where d_ij is the distance between the two atoms, q_i and q_j are the partial charges of the two atoms, and c is a nonzero tunable parameter. c is set to a positive number if opposite-sign charge interactions are to be addressed and is set to a negative number if same-sign charge interactions are of interest. The form of the function is adopted from sigmoid function which is widely used as an activation function in artificial neural networks. Such function regularizes the input signal to the [0, 1] interval. Other functions can be similarly used. This formulation can be extended to systems with dipole or higher order multipole approximations to electron density. The weak interactions due to long distances or neutral charges result in correlation values close to 0.5. When c > 0, the repulsive interaction and attractive interaction deliver the correlation values in (0.5, 1) and (0, 0.5) respectively. The distances induced by Φ(d_ij, q_i, q_j; c) are used to characterize electrostatic effects. The parameter c is rather physical but chosen to effectively spread the computed values over the (0, 1) interval so that the results can be used by machine learning methods. Another simple choice of charge correlation functions is However, this choice will lead to a different filtration domain. Additionally, a charge density can be constructed (11) where r is a position vector, ∥r − r _j∥ is the Euclidean distance between r and jth atom position r _j and η_j is a scale parameter. Eq (11) can be used for electrostatic filtration as well. In this case, the filtration parameter can be the charge density value and cubical complex based filtration can be used.

Multi-component persistent homology.

Multicomponent persistent homology refers to the construction of multiple persistent homology components from a given object to describe its properties. Obviously, element specific persistent homology leads to multi-component persistent homology. Nevertheless, in element specific persistent homology, the emphasis is given to the appropriate selection of important elements for describing certain biological properties or functions. For example, in biological context, electronegative atoms are selected for describing hydrogen bond interactions, polar atoms are selected for describing hydrophilic interactions, and carbon atoms are selected for describing hydrophobic interactions. Note that in chemical context, an atom may have many sharply different chemical and physical properties, depending on its oxidation states. Whereas, in multicomponent persistent homology, the emphasis is placed on the systematic generation of topological invariants from different combinatorial possibilities and the construction of 2D or high-dimensional persistent maps for deep convolutional neural networks.

Counts in bins.

For a given set of atoms A, we denote its barcodes as B = {I_α}_{α ∈ A} and represent each bar by an interval I_α = [b_α, d_α], where b_α and d_α are respectively the birth and death positions on the filtration axis. The length of each bar, or the persistence of topological invariant is given by p_α = d_α − b_α. To locate the position of all bars and persistences, we further split the set of barcodes on the filtration axis into a predefined collection of N bins with Bin_i = [l_i, r_i], where l_i and r_i are the left and the right positions of the ith bin. We generate features by counting the numbers of births, deaths, and persistences in each bin, which leads to three counting feature vectors, namely, counts of birth , death , and persistence , (12) where ‖ ⋅ ‖ is the cardinality of a set. Note that the above discussion should be applied to three topological dimensions, i.e., barcodes of the 0th dimension (B⁰), 1st dimension (B¹) and 2nd dimension (B²). In general, this approach enables the description of bond lengths, including the length of non-covalent interactions, in biomolecules and was referred to as binned persistent homology in our earlier work [14, 27, 94].

Barcode statistics.

Another method of feature vector generation from a set of barcodes is to extract important statistics of barcode collections such as maximum values and standard deviations. Given a set of bars B = {[b_α, d_α]}_α∈A, we define sets of Birth = {b_α}_α∈A, Death = {d_α}_α∈A, and Persistence = {d_α − b_α}_α∈A. Three statistic feature vectors , , and can then be generated in the sense of the statistics of the collection of barcodes. For example, consists of avg(Birth), std(Birth), max(Birth), min(Birth), sum(Birth), and cnt(Birth), where avg(⋅) is the average value of a set of numbers, std(⋅) is the standard deviation of a set of numbers, max(⋅) and min(⋅) are maximum and minimum values in a set of numbers, sum(⋅) is the summation of elements in a set of numbers, and cnt(⋅) is the count of elements in a set. The generation of is the same by examining the set Death. contains the same information with two extra terms, the birth and death values of the longest bar. Statistics feature vectors are collected from barcodes of three topological dimensions, i.e., the 0th, 1st, and 2nd dimensions.

Persistence diagram slice and statistics.

A more thorough description of sets of barcodes is to first divide the sets into subsets and extract features analogously to the barcode statistics method. As shown in Fig 9, a persistence diagram can be divided into slices in different directions. The barcodes that fall in each slice form a subset. Each subset is described in terms of feature vector by using the barcode statistics method. When the persistence diagram is sliced horizontally, members in each subset have similar death values and the barcode statistics feature vector is generated for the set of birth values. Similarly, members in each subset have similar birth values if the persistence diagram is sliced vertically, and the barcode statistics feature vector is generated for the set of death values. The barcode statistics feature vectors are generate for both set of birth values and set of death values if the persistence diagram is sliced diagonally, where members in each subset have similar persistence. This type of feature vector generation describes the set of barcodes in more detail but will produce longer feature vectors.

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Fig 9. Illustration of dividing set of barcodes into subsets.

The barcodes are plotted as persistence diagrams with the horizontal axis being birth and the vertical axis being death. From left to right, the subsets are generated according to the slicing of death, birth, and persistence values.

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2D representation.

The construction of multi-dimensional persistence is an interesting topic in persistent homology. In general, it is believed that multi-dimensional persistence has better representational power for complex systems described by multiple parameters [43]. Although multidimensional persistence is hard to compute, one can compute persistence for one parameter while fixing the rest of the parameters to a sequence of fixed values. In the case where there are two parameters, a bifiltration can be done by taking turns to fix one parameter to a sequence of fixed values while computing persistence for the other parameter. For example, one can take a sequence of resolutions and compute persistence for distance with each fixed resolution. The sequence of outputs can be stacked to form a multidimensional representation [119].

Computing persistence multiple times and stacking the results is especially useful when the parameters that are not chosen to be the filtration parameter are naturally discrete with underlying orders. For example, the multi-component or element specific persistent homology will result in many persistent homology computations over different selections of atoms. These results can be ordered by the percentage of atoms used of the whole molecule or by their importance scores in classical machine learning methods. Also, multiple underlying dimensions exist in the element specific persistent homology characterization of molecules. This property enables 2D or 3D topological representation of molecules. Based on the observation that the performance of the predictor degenerates when too many element combinations are used, we order the element combinations according to their individual performance on the task using methods of ensemble of trees. Combining the dimension of spatial scale and dimension of element combinations, a 2D topological representation is obtained. Such representation is expected to work better in the case of complex geometry such as protein-ligand complexes. With denoting the collection of element combinations ordered by their individual importance scores on the task and B^k(E_i) being the kth dimensional barcodes obtained with atoms of element combination E_j, eight 2D representations are defined as (13) where with γ = b, d, p is the barcode counting rule defined in Eq (13). For 0th dimensional, since all bars start from zero, there is no need for . These eight 2D representations are regarded as eight channels of a 2D topological image. In protein-ligand binding analysis, 2D topological features are generated for the barcodes of a protein-ligand complex and for the differences between barcodes of the protein-ligand complex and those of the protein. Therefore, we have a total of 16 channels in a 2D image for the protein-ligand complex. This 16-channel image can be fed into the training or the prediction of convolutional neural networks.

In the characterization of protein-ligand complexes using alpha complexes, 2D features are generated from the alpha complex based on persistent homology computations of protein and protein-ligand complex. A total of 128 element combinations are considered. The [0, 12]Å interval is divided into 120 equal length bins, which defines the resolution of topological images. Therefore, the input feature for each sample is a 120×128×16 tensor. Fig 10 illustrates 16 channels of sample 1wkm in PDBBind database. These images are directly used in deep convolutional neural networks for training and prediction.

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Fig 10. The 2D topological maps of the 16 channels of sample 1wkm.

The top 8 maps are for protein-ligand complex and the other 8 maps are for the difference between protein-ligand complex and protein only. For each map, the horizontal axis is the dimension of spatial scale and the vertical axis is element combinations ordered by their importance.

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When there are fewer element combinations considered which can hardly form another axis, the axis of element combinations can be added into the original channels to form 1D representations that can be used in 1D CNN.

K-nearest neighbors algorithm via barcode space metrics.

One of the simplest machine learning algorithms is k-nearest neighbors (KNN) for classification or for regression. In KNN regression, for a given object, its property values is obtained by the average or the weighted average of the values of its k nearest neighbors induced by a given metric of similarity. Then, the problem becomes how to construct a metric on the dataset.

In the present work, instead of computing similarities from constructed feature vectors, the similarity between biomolecules can simply be derived from distances between barcodes generated from different biomolecules. Popular barcode space metrics include the bottleneck distance [122] and more generally, the Wasserstein metrics [95, 96]. The definition of the two metrics is summarized as follows.

Given two bars I₁ = [b₁, d₁] and I₂ = [b₂, d₂] regarded as ordered pairs in , the l^∞ distance between the two bars is defined as Δ(I₁, I₂) = max(|b₂ − b₁|, |d₂ − d₁|). For a single bar I = [b, d], λ(I) is defined as λ(I) = (d − b)/2 which helps reflect the difference between the existence of the bar itself and the void. For two finite barcodes and and a bijection θ from A′ ⊆ A to B′ ⊆ B, the penalty of θ is defined as (14) Intuitively, a bijection θ is penalized for linking two bars with large difference and for ignoring long bars from either set. The bottleneck distance is defined as , where the minimum is taken over all possible bijections from subsets of A to subsets of B.

The Wasserstein metric, a L_p generalized analog to the bottleneck distance can be defined with the penalty [96] (15) and the corresponding distance . It approaches the bottleneck distance by setting p goes to infinity. In this work, we choose p = 2.

Wasserstein metric measures the closeness of barcodes generated from different biomolecules. It will be interesting to consider other distances for metric spaces, such as Hausdorff distance, Gromov-Hausdorff distance [123], and Yau-Hausdorff distance [124] for biomolecular analysis. However, an exhaustive study of this issue is beyond the scope of the present work.

The barcode space metrics can be directly used to assess the representation power of various persistent homology methods on biomolecules without being affected by the choice of machine learning models and hyperparameters. We show in the section of results that the barcode space metrics induced similarity measurement is significantly correlated to molecule functions.

Wasserstein metric measures from biomolecules can also be directly implemented in a kernel based method such as nonlinear support vector machine algorithm for classification and regression tasks. However, this aspect is not explored in the present work.

Gradient boosting trees.

Gradient boosting trees is an ensemble method which ensembles individual decision trees to achieve the capability of learning complex feature target maps and can effectively prevent overfitting by using shrinkage technique. The gradient boosting trees method is realized using the GradientBoostingRegressor module in scikit-learn software package [114] (version 0.17.1). A set of parameters found to be efficient in our previous study on the protein-ligand binding affinity prediction [27] is used uniformly unless specified. The parameters used are n_estimators = 20000, max_depth = 8, learning_rate = 0.005, loss = ‘ls’, subsample = 0.7, max_features = ‘sqrt’.

Deep convolutional neural networks.

The deep convolutional neural networks in this work are implemented using Keras [125] (version 1.1.2) with Theano backend [126] (version 0.8.2).

For TopBP-DL(Complex), a widely used convolutional neural network architecture is employed beginning with convolution layers followed by dense layers. Due to the limited computation resources, parameter optimization is not performed, while most parameters are adopted from our earlier work [94]. Reasonable parameters are assigned manually. The detailed architecture is shown in Fig 11. The Adam optimizer with learning rate 0.0001 is used. The loss function is the mean squared error function. The network is trained with a batch size of 16 and 150 epochs. The training data is shuffled for each epoch.

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Fig 11. The network architecture of TopBP-DL.

The structured layers are shown in boxes/rectangles with sharp corners for 2D/1D image-like content and the unstructured layers are shown in rectangles. The numbers in convolution layers mean the number of filters and filter size from left to right. The dense layers are drawn with number of neurons and activation function. The pooling size of the pooling layers and dropout rate of the dropout layers are listed. The layers that are repeated n times are marked with “×n” sign on the right side of the layer.

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The network architecture of TopVS-DL is shown in Fig 12. The Adam optimizer with learning rate set to 0.0001 is used. The loss function is binary cross-entropy. The network is trained with a batch size of 1024 and 10 epochs. The training data is shuffled for each epoch. The batch size is larger than that used in TopBP-DL due to the much larger training set in this problem. Because of the same reason, the training process converges to a small loss very fast with only a few training steps.

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Fig 12. The network architecture of TopVS-DL.

The 1D image-like layers are shown in sharp-corner rectangles. The numbers in convolution layers mean the number of filters and filter size from left to right. The pooling size of the pooling layers and dropout rate of the dropout layers are listed. The layers that are repeated n times are marked with “×n” sign on the right side of the layer.

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