PMC:4331677 / 12964-17515 JSONTXT

{"project":"2_test","denotations":[{"id":"25707321-18676415-14839520","span":{"begin":4168,"end":4169},"obj":"18676415"},{"id":"25707321-19605421-14839521","span":{"begin":4170,"end":4172},"obj":"19605421"},{"id":"25707321-21506615-14839522","span":{"begin":4173,"end":4175},"obj":"21506615"}],"text":"Construction of fragment interaction model\nAs described above, all binding sites could be described as a matrix S = (s1, s2,⋯,sm)T, where m is the number of binding sites. si = (si1, si2,⋯,sip)T denotes the i-th binding site, where sik corresponds the k-th fragment of the i-th binding site and p is the number of site features/fragments. Meanwhile, all ligands could be described as a matrix L = (l1, l2,⋯,ln)T, where n is the number of ligands. lj = (lj1, lj2,⋯,ljq)T denotes the j-th ligand, where ljk indicates the presence or absence of ligand the k-th substructures of the j-th ligand and q is the number of ligand features. The interaction data set is denoted as D = {(i1, j1), (i2, j2),⋯,(ic, jc)}, where ik = 1, 2,⋯,m; jk = 1, 2,⋯,n; (ik, jk) is the k-th site-ligand pair and c is the number of site-ligand interaction pairs. sik and ljk are the site and ligand vectors in the k-th interaction pair respectively. y = (y1, y2,⋯,yc) denotes the labels of interaction pairs, where yk ∈ {1, −1} indicate the positive and negative interaction respectively. Because fragments of targets and ligands are in a physically interaction distance (within 8 angstrom), it is reasonable to assume that there exist inherent chemical interactions between target and ligand features, and the sum of feature interactions determinates the target-ligand interaction. Therefore, the proposed approach is called fragment interaction model (FIM, Figure 1) and it can be expressed as the following equation:\n(3) F ( s * , l * ) = s * T M l *\nwhere s∗ represents a binding site and l∗ represents a ligand. s∗ and l∗ might be unseen to the data set. M represents genomic and chemical spaces interaction matrix/network. If sign(F (s∗, l∗)) is 1, we predicted a positive interaction, otherwise we predicted a negative interaction (sign(·) is the sign function, return −1 and 1).\nIt is easy to solve parameters M in Equation 3 by logistic regression. However, it is inconvenient to expand to include more information. Therefore, mathematics transformation is conducted, and Equation 3 changes into Equation 4.\n(4) F ( s * , l * ) = w T ( s * ⊗ l * )\nwhere w is a vector vision of features interaction matrix M, and ⊗ denotes tensor product. Obviously, Equation 3 and Equation 4 are equivalent. For convenience, we denoted:\n(5) ψ ( s * , l * ) = s * ⊗ l *\nIt is easy to fit w through support vector machine (SVM). Based on the Lagrange dual theory, Equation 4 can be rewritten as its dual form on the data set.\n(6) F ( s * , l * ) = ∑ k = 1 c α k y k ψ ( s i k , l j k ) T ψ ( s * , l * )\nWhere αk is dual variable and yk is the label of the interaction pair (ik, jk) Equation 6 demonstrates that for a given site-ligand pair (s∗, l∗), it only relates with inner product of the support site-ligand pairs (αk ≠ 0). Therefore, we should only care 0 about the inner product of support site-ligand pairs and the site-ligand pair to predict.\n(7) ψ ( s * 1 , l * 1 ) T ψ ( s * 2 , l * 2 ) = ( s * 1 , ⊗ l * 1 ) T ( s * 2 ⊗ l * 2 ) = s * 1 T s * 2 l * 1 T l * 2\nwhere (s∗1, l∗1) and (s∗2, l∗2) are two pairs of site-ligands. Equation 7 is very important because it transforms tensor product into inner product. On one hand, we avoided calculating the tensor product, which always means large computing load. One the other hand, inner product facilitates us to design kernels and add more information by kernel trick. For convenience, we denoted:\n(8) K l o c ( s * 1 , s * 2 ) = s * 1 T s * 2 , K l i g ( l * 1 , l * 2 ) = l * 1 T l * 2\nThen Equation 7 can be changed as:\n(9) K l o c - l i g ( s * 1 , l * 1 ; s * 2 , l * 2 ) = K l o c ( s * 1 , s * 2 ) K l i g ( l * 1 , l * 2 )\nBecause we utilized linear kernel, therefore, the above Equations are invertible and the genomic and chemical interaction matrix M, on the data set, can be described as:\n(10) M = ∑ k = 1 c α k y k s i l j T\nAlthough Equation 7 have been mentioned in many papers [9,11,23], the kernels in those works were nonlinear and irreversible (because of kernel trick), thus we known little about how the genomic space interact with chemical space. In this paper, we adopted linear kernel without kernel trick, so that the genomic and chemical interaction matrix could be calculate through Equation 10, which rendered the model to be chemical interpretable."}