Fig 1.
Dowker complex based representation for the atomic interactions between two chains of DNA 1D77.
Only the Phosphor (P) atoms of the DNA are considered. A bipartite graph is constructed between the two DNA chains, i.e., chain A and chain B, using a cutoff distance of 16.5 Å. The corresponding Dowker complex is generated and consists of two disjoint components, one from chain A and the other from chain B. The cutoff distance can be used a filtration parameter and two persistent barcodes are obtained. It can be seen that the β1 persistent barcodes are exactly the same for the two types of DCs from chain A and chain B. The β0 persistent barcodes are different because the bipartite graph are not always connected during the filtration process.
Fig 2.
Persistent combinatorial Laplacian matrixes for Dowker complex from C-C pair of PDBID 2POG.
As in the picture, based on the filtration process of bipartite graphes, a filtration of Dowker complexes can be generated and further divided into two disjoint filtration processes in protein and ligand. Then for each filtration process, two sequence of laplacian matrixes in dimension 0 and 1 are depicted. The cutoff extracting the binding core region is 5Å, filtration values are 3.5Å, 4Å, 4.2Å, 4.5Å and 5Å. For 0-D laplacian matrixes, with the increase of filtration value, the matrix size is always same, off-diagonal entries decrease from 0 to -1 until all become -1 and diagonal entries increase until all up to the number of 0-simplexes minus 1. For 1-D laplacian matrixes, the matrix size increase consistently until up to a constant, and off-diagonal entries have nonzero values 1 and -1 due to their oriention and the number of off-diagonal nonzero entries increase at early stage and then decrease until all go to zero, and diagonal entries increase until all up to the number of 0-simplexes.
Table 1.
Detailed information of the three PDBbind databases, i.e., PDBbind-v2007, PDBbind-v2013 and PDBbind-v2016.
Table 2.
The parameters for our DC-based gradient boosting tree (GBT) models.
Table 3.
The PCCs and RMSEs (pKd/pKi) for our DC-GBT models in three test cases, i.e., PDBbind-v2007, PDBbind-v2013 and PDBbind-v2016.
Three DC-GBT models are considered with features from different types of bipartite graphs. The DC-GBT(Dist) model uses features from distance-based bipartite graphs; The DC-GBT(Chrg) model uses features from electrostatic-based bipartite graphs; The DC-GBT(Dist+Chrg) model uses features from both distance-based bipartite graphs and electrostatic-based bipartite graphs.
Fig 3.
Preformance comparison between our models and other models.
The comparison of PCCs between our model and other molecular descriptor based models, for the prediction of protein-ligand binding affinity. The PCCs are calculated based on the core set (test set) of PDBbind-v2007, PDBbind-v2013 and PDBbind-v2016.
Table 4.
The comparison of our DC-GBT model with advanced-mathematical based machine learning models [14, 18, 20, 30, 32, 33].
Note that values marked with * uses PDBbind-v2016 core set (N = 290).
Table 5.
The performance in terms of PCCs and RMSEs (pKd/pKi) for recently-proposed models using different training sets [54–64].
Note that values marked with * uses PDBbind-v2016 core set (N = 290), and the values marked with + uses PDBbind-v2013 core set(N = 180) and PDBbind-v2016 core set(N = 276).
Fig 4.
A bipartie graph and its associated Dowker complex.
It can be seen that there are two disjoint components in DC, one is from the black points and the other is from the green points. Note that a triangle (2-simplex) is formed among the black point set in DC, as the corresponding three black vertices have a common neighbor blue vertex in the bipartite graph.