Problem formulationLet \(\text{D}=\left\{{\text{d}}_{1}, {\text{d}}_{2}{ \cdots ,\text{ d}}_{{\text{n}}_{\text{d}}}\right\}\), and \(\text{T}=\left\{{\text{t}}_{1}, {\text{t}}_{2}{, \cdots ,\text{ t}}_{{\text{n}}_{\text{t}}}\right\}\) , respectively, represent the set of drugs and the set of targets, where \({\text{n}}_{\text{d}}\) is the number of drugs, and \({\text{n}}_{\text{t}}\) is the number of targets. Let \({\text{H}}_{\text{d}}=\left\{{\text{S}}_{\text{d},1}, {\text{S}}_{\text{d},2}, \cdots , {\text{S}}_{\text{d},{\text{m}}_{\text{d}}}\right\}\) and \({\text{H}}_{\text{t}}=\left\{{\text{S}}_{\text{t},1}, {\text{S}}_{\text{t},2}, \cdots , {\text{S}}_{\text{t},{\text{ m}}_{\text{t}}}\right\}\) respectively represent a set containing \({\text{m}}_{\text{d}}\) drug similarity matrices and a set containing \({\text{m}}_{\text{t}}\) target similarity matrices, where \({\text{S}}_{\text{d},\text{i}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{d}}\times {\text{n}}_{\text{d}}}\left(\text{i}=\text{1,2},\cdots ,{\text{m}}_{\text{d}}\right)\), \({\text{S}}_{\text{t},\text{j}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{t}}\times {\text{n}}_{\text{t}}}\left(\text{j}=\text{1,2},\cdots ,{\text{ m}}_{\text{t}}\right)\). The matrix \(\widehat{\text{A}}\) represents the interaction between drug and target,\(\widehat{\text{A}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{d}}\times {\text{n}}_{\text{t}}}\), where if there is an interaction between drug i and target j, \({\widehat{\text{A}}}_{\text{ij}}=\) 1, and if it is unknown whether there is an interaction between drug i and target j, \({\widehat{\text{A}}}_{\text{ij}}=\) 0. A drug (or target) is referred to as a new drug (or target) if there isn’t a known interaction for it. That is, if \({\widehat{\text{A}}.}_{i}=0\), the drug is termed a new drug, where \({\widehat{\text{A}}.}_{i}\) is the \(i\)th row of \(\widehat{\text{A}}\); and if \({\widehat{\text{A}}.}_{j}=0\), the target is termed the new target, where \({\widehat{\text{A}}.}_{j}\) is the \(j\)th column of \(\widehat{\text{A}}\). A medication (or target) is considered known if it has at least one pair of known interactions. Using drug, target, and interaction data from established drug target pairs, the DTI prediction model seeks to estimate and identify novel drug target pairings with possible interactions.Similarity integrationThe goal of similarity integration (also known as similarity integration) is to fuse the similarity matrices of the upper and middle drugs (or the similarity matrices of the middle drugs) into one matrix, capture the contribution values of each similarity matrix from different perspectives, and provide a more concise and informative input for downstream DTI prediction tasks, reducing the redundancy and interference of noise information to a certain extent. Formally, similarity integration can be defined as a mapping \(\text{f}:\left\{{\text{ R}}^{\text{n}\times \text{n}},\cdots ,{\text{ R}}^{\text{n}\times \text{n}}\right\}\to {\text{R}}^{\text{n}\times \text{n}}\), where the fused drug and target similarity matrices are obtained by f (\({\text{S}}_{\text{d},1}, \cdots , {\text{S}}_{\text{d},{\text{m}}_{\text{d}}}\)) =\({\text{S}}_{\text{d}}\) and f (\({\text{S}}_{\text{t},1}, \cdots , {\text{S}}_{\text{t},{\text{m}}_{\text{t}}}\)) = \({\text{S}}_{\text{t}}\), respectively, where \({\text{S}}_{\text{d}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{d}}\times {\text{n}}_{\text{d}}}\), \({\text{S}}_{\text{t}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{t}}\times {\text{n}}_{\text{t}}}\).Many similarity integration methods are also used for DTI prediction tasks, such as average weighting. There are both linear and nonlinear methods. In this paper, we fuse the similarity matrix by fine-grained selection of similarity integration, which differs from previous global fusion methods. This method starts from the similarity utility of each drug (target) and linearly summarizes the similarity of selected information from different perspectives at a finer granularity. Figure 4 describes the process of a complete fine-grained selection similarity integration and finally obtaining the fused similarity matrix.Figure 4The process of a complete fine-grained selection similarity integration.Calculation of consistency of local drug (target) interactionsBased on the “guide by association” principle, If there exist drugs and targets exhibiting high similarity to a pair of known DTIs, they are more likely to possess interaction relations. Therefore, the local interaction consistency matrix is computed to depict the interaction distribution of neighboring drugs (or targets). Let \({\text{P}}_{\text{h}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{d}}\times {\text{n}}_{\text{t}}}\) be the local interaction consistency matrix of \({\text{S}}_{\text{d},\text{h}}\). In the similarity perspective \({\text{S}}_{\text{d},\text{h}}\), \({\text{P}}_{\text{h}}\left(\text{i},\text{j}\right)\) stands for the consistency of the local interaction between drug \({\text{d}}_{\text{i}}\) and target \({\text{t}}_{\text{j}}\).Let \({\text{Q}}_{\text{s}}\left(\text{i},\text{j}\right)\) be the local interaction consistency matrix of \({\text{S}}_{\text{t},\text{s}}\). In this case, \({\text{Q}}_{\text{s}}\left(\text{i},\text{j}\right)\) stands for the consistency of the local interaction in the similarity view \({\text{S}}_{\text{t},\text{s}}\) between target \({\text{t}}_{\text{i}}\) and drug \({\text{d}}_{\text{j}}\),where \(\text{h}=\text{1,2},\cdots ,{\text{m}}_{\text{d}}\), \(\text{s}=\text{1,2},\cdots ,{\text{m}}_{\text{t}}\).$${\text{P}}_{\text{h}}\left(\text{i},\text{j}\right)=\frac{1}{\sum_{{\text{d}}_{\text{l}}\upepsilon {\mathcal{N}}_{{\text{d}}_{\text{i}}}^{\text{k}}}{\text{S}}_{\text{d},\text{h}}(\text{i},\text{j})}\sum_{{\text{d}}_{\text{l}}\upepsilon {\mathcal{N}}_{{\text{d}}_{\text{i}}}^{\text{k}}}{\text{S}}_{\text{d},\text{h}}(\text{i},\text{j})\cdot {\text{I}}_{\text{A}}\left({\widehat{A}}_{\text{lj}}={\widehat{A}}_{\text{ij}}\right)$$
(1)
$${\text{Q}}_{\text{s}}\left(\text{i},\text{j}\right)=\frac{1}{\sum_{{\text{t}}_{\text{l}}\upepsilon {\mathcal{N}}_{{\text{t}}_{\text{j}}}^{\text{k}}}{\text{S}}_{\text{t},\text{h}}(\text{i},\text{j})}\sum_{{\text{t}}_{\text{l}}\upepsilon {\mathcal{N}}_{{\text{t}}_{\text{j}}}^{\text{k}}}{\text{S}}_{\text{t},\text{h}}(\text{i},\text{j})\cdot {\text{I}}_{\text{A}}\left({\widehat{A}}_{\text{il}}={\widehat{A}}_{\text{ij}}\right)$$
(2)
\({\mathcal{N}}_{{\text{d}}_{\text{i}}}^{\text{k}}\) and \({\mathcal{N}}_{{\text{t}}_{\text{j}}}^{\text{k}}\) represent k nearest neighbors (KNNs) of drug \({\text{d}}_{\text{i}}\) and target \({\text{t}}_{\text{j}}\) retrieved based on \({\text{S}}_{\text{d},\text{h}}\) and \({\text{S}}_{\text{t},\text{h}}\), \({\text{I}}_{\text{A}}(\cdot )\) respectively, and are indicative functions.Construct an initial fine-grained weight matrixThe local interaction consistency matrix can be polymerized to generate the initial fine-grained weight matrix \({\text{W}}_{\text{d}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{d}}\times {\text{m}}_{\text{d}}}\) 和 \({\text{W}}_{\text{t}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{t}}\times {\text{m}}_{\text{t}}}\) of drugs and targets, since it can accurately depict the interaction distribution of nearby drugs (or targets).$${W}_{d}\left(i,h\right)=\sum_{j=1}^{{n}_{t}}{P}_{h}\left(i,j\right)\cdot A\left(i,j\right)$$
(3)
$${W}_{t}\left(j,h\right)=\sum_{i=1}^{{n}_{d}}{Q}_{h}\left(i,j\right)\cdot A\left(j.i\right)$$
(4)
Filling of zero-weight vectors of known drugs (targets)Since the DTI prediction model needs to predict potential DTI from drug and target information of known interaction pairs, if the initial fine-grained weight vector of a known drug is 0, that is, \({W}_{d}.i=0\), then all similarity views corresponding to the drug will lose value, which is an inadequate mining of useful information and will result in an ineffective integration of similarity information. Consequently, this work uses vectors \({V}_{d}=\sum_{i=1}^{{n}_{d}}{W}_{d}.i\) and \({V}_{t}=\sum_{i=1}^{{n}_{t}}{W}_{t}.i\) to express the usefulness of similarity at each perspective for all medications (targets), respectively, in order to avoid this issue. In the original fine-grained weight matrix, replace the matching zero-weight vector with \({V}_{d}\)(\({V}_{t}\)) when the known drug (target) similarity weight is 0.$${W}_{d}.i=\left\{\begin{array}{c}{W}_{d}.i\quad if\,\,{W}_{d}.i\ne 0\\ { V}_{d}\quad if\,\,{W}_{d}.i=0\end{array}\right.$$
(5)
$${W}_{t}.i=\left\{\begin{array}{c}{W}_{t}.i\quad if\,\,{W}_{t}.i\ne 0\\ {V}_{t}\quad if\,\,{W}_{t}.i=0\end{array}\right.$$
(6)
where \({W}_{d}.i\) represents the i-th row vector in the initializes the fine-grained weight matrix of the drug and \({W}_{t}.i\) represents the i-th row vector in the initializes the fine-grained weight matrix of the target.Filling of weight vectors for new drugs (new targets)Since the new drug (new target) has no known target (drug) with which it interacts, their initial fine-grained weight vectors are also 0 according to formula (3) and (4). However, the zero-weight vector filling method for known drugs (targets) is no longer suitable for new drugs (new targets), because different new drugs (new targets) may have undiscovered potential interactions, and the use of uniform weight vectors \({V}_{d}\)(or \({V}_{t}\)) cannot reveal their uniqueness. Therefore, the principle of guilt-by-assocoation is reconnected here, assuming that the similar weight vector of a drug (target) is also approximate to its neighboring drug (target). That is, if there is a new drug dx, the weight of dx can be filled by summarizing the weight vector of the K known drugs closest to it from each similarity perspective. The same is true for new targets:$${W}_{d}.x=\left(\sum_{{\text{d}}_{\text{i}}\upepsilon {\mathcal{N}}_{{\text{d}}_{\text{x}}}^{\text{k},1}}{W}_{d}\left(i,1\right),\sum_{{\text{d}}_{\text{i}}\upepsilon {\mathcal{N}}_{{\text{d}}_{\text{x}}}^{\text{k},2}}{W}_{d}\left(i,2\right),\cdots ,\sum_{{\text{d}}_{\text{i}}\upepsilon {\mathcal{N}}_{{\text{d}}_{\text{x}}}^{\text{k},{\text{m}}_{\text{d}}}}{W}_{d}\left(i,{\text{m}}_{\text{d}}\right)\right)$$
(7)
$${W}_{t}.x=\left(\sum_{{\text{t}}_{\text{i}}\upepsilon {\mathcal{N}}_{{\text{t}}_{\text{x}}}^{\text{k},1}}{W}_{t}\left(i,1\right),\sum_{{\text{t}}_{\text{i}}\upepsilon {\mathcal{N}}_{{\text{t}}_{\text{x}}}^{\text{k},2}}{W}_{t}\left(i,2\right),\cdots ,\sum_{{\text{t}}_{\text{i}}\upepsilon {\mathcal{N}}_{{\text{t}}_{\text{x}}}^{\text{k},{\text{m}}_{\text{t}}}}{W}_{t}\left(i,{\text{m}}_{\text{t}}\right)\right)$$
(8)
where \({\mathcal{N}}_{{\text{d}}_{\text{x}}}^{\text{k},\text{h}}\left(\text{h}=\text{1,2},\cdots ,{\text{m}}_{\text{d}}\right)\) represents the k known drugs of the new drug \({\text{d}}_{\text{x}}\) in the h-class similarity perspective, and \({\mathcal{N}}_{{\text{t}}_{\text{x}}}^{\text{k},\text{h}}\left(\text{h}=\text{1,2},\cdots ,{\text{m}}_{\text{t}}\right)\) represents the k known targets of the new target \({\text{t}}_{\text{x}}\) in the h-class similarity perspective.Fine-grained similarity selectionIf the local interaction consistency is low, it indicates that similar drugs usually interact with different targets, which is inconsistent with the principle of “guilt-by-assocoatio”n, and this further indicates that there is noise information in the corresponding similarity view. This research ,to reduce noise interference, employs a fine-grained selection program that filters the redundant similarity of each drug (target) by thinning the weight matrix. To be more precise, substitute zero for the the lowest weight in each row of weight matrix .Because the lower local interaction consistency of a drug (or target) correlates with the lowest weights in each row, and this lower local interaction consistency corresponds to a noisy similarity perspective. Therefore, it is desirable to set the weight to zero and discard the similarity information of the drug (target) containing noise in this similarity perspective, so as to achieve the effect of reducing redundant information and reducing noise.$$\text{min}\left\{{W}_{d}.i\left(i,j\right)\right\}=0 (\text{i}=\text{1,2},\cdots {,\text{n}}_{\text{d}};\text{j}=\text{1,2},\cdots ,{\text{m}}_{\text{d}})$$
(9)
$$\text{min}\left\{{W}_{t}.i\left(i,j\right)\right\}=0 (\text{i}=\text{1,2},\cdots {,\text{n}}_{\text{t}};\text{j}=\text{1,2},\cdots ,{\text{m}}_{\text{t}})$$
(10)
The similarity with zero weight values in the weight matrix is discarded. In contrast, the similarity with non-zero weight values is ultimately used to integrate the final similarity matrix. This is different from the existing coarse-grained similarity selection method that deletes and integrates the entire similarity matrix. In this paper, similarity is evaluated at a finer granularity, that is, noise similarity information is retained or eliminated from the level of each drug (target) under each similarity view without affecting the useful similarity information of other drugs (target) under this view.Normalization of weightsNormalize the similarity weight matrix obtained above row by row, keeping the similarity range unchanged:$${W}_{d}.i\left(i,j\right)=\frac{{W}_{d}.i\left(i,j\right)}{\sum_{h=1}^{{\text{m}}_{\text{d}}}{W}_{d}.i\left(i,h\right)}(\text{i}=\text{1,2},\cdots {,\text{n}}_{\text{d}})$$
(11)
$${W}_{t}.i\left(i,j\right)=\frac{{W}_{t}.i\left(i,j\right)}{\sum_{h=1}^{{\text{m}}_{\text{t}}}{W}_{t}.i\left(i,h\right)}(\text{i}=\text{1,2},\cdots {,\text{n}}_{\text{t}})$$
(12)
Fusion similarity matrix$${S}_{d}.i=\sum_{j=1}^{{\text{m}}_{\text{d}}}{W}_{d}\left(i,j\right){\text{S}}_{\text{d},\text{j}}.i$$
(13)
$${S}_{t}.i=\sum_{j=1}^{{\text{n}}_{\text{d}}}{W}_{t}\left(i,j\right){\text{S}}_{\text{t},\text{j}}.i$$
(14)
where \({{S}}_{{d},{j}}.{i}\) represents the i-th row of the drug’s j-th class similarity view matrix, and \({{S}}_{{t},{j}}.{i}\) represents the i-th row of the j-th class similarity view matrix of the target.Bidirectional random walk prediction of potential DTI.Bidirectional random walk algorithm with restart on heterogeneous networksFirstly, the drug target heterogeneous network was constructed using the drug similarity matrix \({S}_{d}\), target similarity matrix \({S}_{t}\) and drug target interaction matrix A, where \({\text{S}}_{\text{d}}\) and \({\text{S}}_{\text{t}}\) are obtained after fine-grained selection similarity integration. The possible drug target pairs in the heterogeneous network are then predicted using the bidirectional random walk algorithm (BIRW), and the predicted scores are updated in the drug target interaction matrix. The full procedure for a two-way random walk is as follows.A random walk is performed on the drug network \({S}_{d}\) to obtain the update matrix \({A}_{d}\):$${A}_{d}=\mu \times {S}_{d}\times A+\left(1-\mu \right)\times \widehat{A}$$
(15)
A random walk is performed on the target network \({S}_{t}\) to obtain the update matrix \({A}_{t}\):$${A}_{t}=\mu \times A\times {S}_{t}+\left(1-\mu \right)\times \widehat{A}$$
(16)
The interaction matrix obtained by random walk updating on two networks of drug and target is fused:$${A}^{\prime}=\left({A}_{d}+{A}_{t}\right)/2$$
(17)
Finally, by comparing the integrated and fused matrix with the initial drug target interaction matrix, the drug target interaction matrix A updated by a complete bidirectional random walk algorithm is calculated.$$\text{A}=\text{elementwise max}(\widehat{A},\text{A}^{\prime})$$
(18)
where \({A}_{d}\) and \({A}_{t}\) are is the probability matrices for predicting drug–target interactions, \({A}_{d}\left(i,j\right)\) and \({A}_{t}\left(i,j\right)\) represents the possiblity of interaction between drugs \({\text{d}}_{\text{i}}\) and target \({\text{t}}_{\text{j}}\), and \(\mu\) is the restart parameter. The above process is a one-time BIRW algorithm.Similarity selection strategyBased on the principle of “guilt-by-association”, we reselect the similarities in the drug (target) similarity network to extract more reliable similarity relationships. Specifically, only the top K neighbors of each drug (target) are retained, with the similarity values for all other positions set to zero. Algorithm 1 outlines the entire similarity selection process. The nearest neighbors of each drug (target) are arranged in descending order of similarity, and \(\uptau\) is a decay factor used to assign appropriate weight coefficients to these neighbors. Following this, the complete BIRW process(described by formulas (15) through (18)), is implemented \({l}_{1}\) times on new drug target heterogeneous network which is made up of the drug similarity matrix after similarity selection, target similarity matrixafter similarity selection, and the updated drug–target interaction matrix A . Note that A here refers to the drug–target interaction matrix after each update.Algorithm 1Similarity selection strategy.Model correctionThe previous prediction findings are corrected utilizing the entire information of the drug and the target since the chemical fingerprint information of the drug molecule and the amino acid sequence information of the protein target are complete. To be more precise, a new drug similarity matrix \({\text{S}}_{\text{d}}^{\text{s},\text{p}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{d}}\times {\text{n}}_{\text{d}}}\) and target similarity matrix \({\text{S}}_{\text{t}}^{\text{s},\text{p}}{{ \upepsilon R}}^{{\text{n}}_{\text{t}}\times {\text{n}}_{\text{t}}}\) were obtained by similarity selection of these two similarity matrices using algorithm 1. Specifically, a complete drug similarity matrix \({\text{S}}_{\text{d}}^{\text{s}}\upepsilon {\text{ R}}^{{\text{n}}_{\text{d}}\times {\text{n}}_{\text{d}}}\) and target similarity matrix \({\text{S}}_{\text{t}}^{\text{s}}{{ \upepsilon R}}^{{\text{n}}_{\text{t}}\times {\text{n}}_{\text{t}}}\) were first calculated by SMILEs of drug molecules and protein sequence information . Lastly, the complete BIRW process(described by formulas (15) through (18)), is implemented \(2\) times on \({\text{S}}_{\text{d}}^{\text{s}}\), \({\text{S}}_{\text{t}}^{\text{s}}\), and updated A until it converges. At last, the probability matrix A for the drug target interaction is anticipated.
Drug–target interaction prediction through fine-grained selection and bidirectional random walk methodology
