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Graph regularized nonnegative matrix factorization with \(L_{2,1}\) norm regularization terms for drug–target interactions prediction
BMC Bioinformatics volume 24, Article number: 375 (2023)
Abstract
Background
Identifying drug–target interactions (DTIs) plays a key role in drug development. Traditional wet experiments to identify DTIs are costly and time consuming. Effective computational methods to predict DTIs are useful to speed up the process of drug discovery. A variety of nonnegativity matrix factorization based methods are proposed to predict DTIs, but most of them overlooked the sparsity of feature matrices and the convergence of adopted matrix factorization algorithms, therefore their performances can be further improved.
Results
In order to predict DTIs more accurately, we propose a novel method iPALMDLMF. iPALMDLMF models DTIs prediction as a problem of nonnegative matrix factorization with graph dual regularization terms and \(L_{2,1}\) norm regularization terms. The graph dual regularization terms are used to integrate the information from the drug similarity matrix and the target similarity matrix, and \(L_{2,1}\) norm regularization terms are used to ensure the sparsity of the feature matrices obtained by nonnegative matrix factorization. To solve the model, iPALMDLMF adopts nonnegative double singular value decomposition to initialize the nonnegative matrix factorization, and an inertial Proximal Alternating Linearized Minimization iterating process, which has been proved to converge to a KKT point, to obtain the final result of the matrix factorization. Extensive experimental results show that iPALMDLMF has better performance than other stateoftheart methods. In case studies, in 50 highestscoring proteins targeted by the drug gabapentin predicted by iPALMDLMF, 46 have been validated, and in 50 highestscoring drugs targeting prostaglandinendoperoxide synthase 2 predicted by iPALMDLMF, 47 have been validated.
Background
Determining the drug–target interactions (DTIs) is a key step in drug development process [1]. However, identifying the DTIs via wet experiments is time consuming and expensive [2, 3]. To reduce the consumption of expensive wet experiments, a variety of computational prediction models for DTIs have been proposed. The existing models for DTIs prediction mainly fall into two categories [4]. The first category formulates the interaction prediction as a binary classification task [5]. The second category aims to estimate the interaction strength of drug–target pairs [6, 7]. This paper focuses on the first category. The first category of DTI prediction models could be further grouped into ligandbased models, docking simulation based models, and chemogenomics based models [8].
Ligandbased models assume that similar ligands would interact with similar proteins [9].
The ligand based models require that a certain number of binding ligands of a given protein target should be known [10]. Docking simulation based models are based on crystal structures of target binding sites and docking simulations [11]. However, obtaining the crystal structure of a target binding site is challenging. Therefore, docking simulation based models couldn’t apply to large scale DTIs prediction.
To avoid above difficulties, chemogenomics based models use known targetdrug interactions, chemical structures of drugs, genomic sequences of target proteins, and/or other related information of targets and drugs to predict potential targetdrug interactions. The chemogenomics based models [8] usually use a DTI network to present the known drug–target interactions, and adopt machine learning or deep learning to predict DTIs. For example, based on the DTI network, Yamanishi et al. [12] proposed a bipartite graph learning method to predict DTIs by mapping the chemical structure space of drugs and the genomic sequence space of proteins into a unified space. In order to predict target proteins for a given drug, and the drugs targeting a given protein, Bleakley and Yamanishi [13] proposed bipartite local models (BLM), which transformed edgeprediction problems into binary classification problems. RLSWNN [14], BLMNII [15] and WKNKN [16] were proposed by integrating the neighbor information of similarity networks of drugs and targets.
In addition to chemical structures of drugs and genomic sequences of target proteins, some works have incorporated multiple types of information, such as sideeffects [17, 18], proteinprotein interactions [19], drugdisease associations [20], proteindisease associations [21] and gene ontology information [22] for DTIs prediction. In order to integrate multiple types of information, random walk with restart (RWR) [23, 24] was used to capture topological relations between nodes in the heterogeneous network. In addition, 2D structural images of drugs [25] and 3D structures of the proteins [26] were also used as input data for DTIs prediction.
As a kind of machine learning method, matrix factorization has also been used to predict DTIs and has achieved better performance than other machine learning methods [2]. In DTIs prediction, a DTI matrix is usually used to represent the known drug–target interactions. Matrix factorization decomposes the interaction matrix into two low rank matrices, which represent the feature matrices of drugs and targets. The optimization object of matrix factorization based DTIs prediction methods is that the product of the feature matrices of drugs and targets approximates the interaction matrix of drugs and targets as closely as possible. For example, Gönen [27] proposed a kernelized Bayesian matrix factorization with twin kernels method to predict DTIs. Bolgár and Antal [28] proposed a fusion method, called a variational Bayesian multiple kernel logistic matrix factorization method, which used graph Laplacian regularization, multiple kernel learning, and a variational Bayesian inference process to infer interactions. In order to learn the values of missing entries in DTI matrix, a variety of methods with regularization terms were proposed based on matrix factorization, such as MSCMF [29], NRLMF [30], GRMF [31], \(L_{2,1}\)GRMF [32] and SRCMF [33]. Recently, Ding et al. [34] proposed a multiple kernelbased triple collaborative matrix factorization (MKTCMF) method. MKTCMF used Multikernel learning (MKL) to integrate different similarities of drugs and targets, and used triple collaborative matrix factorization to decompose the original DTI matrix into three matrices: a latent feature matrix of drugs, latent feature matrix of targets and a biprojection matrix.
To solve matrix factorization problems, the above methods used either the alternating least squares algorithm [35] or the multiplicative update algorithm [36]. However, it is difficult to guarantee that the above algorithms converge to a stationary point [37]. Recently, Pock and Sabach [38] proposed an inertial version of the Proximal Alternating Linearized Minimization algorithm (iPALM), which can be used to solve nonnegative matrix factorization, and iPALM has been proven to converge to a stationary point.
In this paper, we propose a novel method iPALMDLMF. iPALMDLMF models DTIs prediction as a problem of nonnegative matrix factorization with graph dual regularization terms and \(L_{2,1}\) norm regularization terms. The graph dual regularization terms are used to integrate the information from the drug similarity matrix and the target similarity matrix, and \(L_{2,1}\) norm regularization terms are used to ensure the sparsity of the matrices obtained by nonnegative matrix factorization. To solve the model, nonnegative double singular value decomposition (NNDSVD) [39] is used to initialize the nonnegative matrix factorization, and an inertial Proximal Alternating Linearized Minimization iterating process is used to obtain the final matrix factorization.
The main contributions of iPALMDLMF are as follows:

1.
Improving the nonnegative matrix factorization model by adding graph dual regularization terms and \(L_{2,1}\) norm regularization terms.

2.
\(L_{2,1}\) norm regularization terms ensure sparsity of the matrices obtained by nonnegative matrix factorization.

3.
The inertial proximal alternating linearized minimization algorithm with fast convergence is used to solve the matrix factorization.
Extensive experimental results show that iPALMDLMF has better performance than other stateoftheart methods. In case studies involving the drug gabapentin and the target prostaglandinendoperoxide synthase 2, 46 of the 50 highestscoring highestscoring targets predicted to interact with gabapentin and 47 of the 50 highestscoring drugs predicted to interact with prostaglandinendoperoxide synthase 2 have been validated by wet experiments. The case studies show that, for drugs that do not have any known target proteins and for proteins that are so far not approved as drug targets, iPALMDLMF also has good prediction performance.
Materials
In order to evaluate prediction performance of the proposed iPALMDLMF, we used the same four benchmark datasets as used by most similar works. The information of the four datasets are shown in Table 1. Each dataset contains three types of information: known drug–target interactions, drug chemical structures and target protein sequences. The datasets correspond to different target protein types, including nuclear receptors (NR), G proteincoupled receptors (GPCR), ion channels (IC) and enzymes (E). Accordingly, the four datasets are called NR, GPCR, IC and E. The four datasets were built by Yamanishi et al. [12] from public databases BRENDA [40], KEGG BRITE [41], SuperTarget [42] and DrugBank [43], and are publicly available at http://web.kuicr.kyotou.ac.jp/supp/yoshi/drugtarget/. The known interactions between n drugs and m proteins are recorded by a \(n \times m\) DTI matrix Z. If the ith drug is approved to target the jth protein, \(Z_{i,j}=1\); otherwise \(Z_{i,j}=0\).
The structural similarities between drugs are calculated using SIMCOMP [44] according to the size of the common substructures between two drugs. The similarity information of n drugs are stored in a \(n\times n\) matrix \(S^d\).
The normalized version of the SmithWaterman score is used to calculate the sequence similarity of the target proteins [45]. Let \(p_1\) and \(p_2\) represent two proteins. The SmithWaterman score of the standardized version of \(p_1\) and \(p_2\) is \(s({p_1},{p_2}) = \frac{{SW({p_1},{p_2})}}{{\sqrt{SW({p_1},{p_1})} \sqrt{SW({p_2},{p_2})} }}\), where SW(., .) be the original SmithWaterman alignment score. The similarity information of m target proteins are denoted by a \(m\times m\) matrix \(S^t\).
Methods
iPALMDLMF models DTIs prediction problem as a nonnegative factorization problem with graph dual regularization terms and \(L_{2,1}\) norm regularization terms. iPALMDLMF takes the DTI matrix Z, drug similarity matrix \(S^d\) and target similarity matrix \(S^t\) as inputs, uses \(S^d\) and \(S^t\) to construct graph dual regularization terms, and solve nonnegative matrix factorization problem of Z with graph dual regularization terms and \(L_{2,1}\) norm regularization terms to obtain the feature matrices of drugs and targets. Finally the feature matrices are utilized to predict DTIs. A brief flow chart of iPALMDLMF is shown in Fig. 1.
Nonnegative matrix factorization
In DTIs prediction, the nonnegativity matrix factorization (NMF) of the DTI matrix is widely used to obtain lowdimensional feature representations of drugs and targets in the DTI space. The general form of the NMF is as follows:
where X and Y represent the latent feature matrices of drugs and targets, respectively. k is the rank of X and Y, \(k \ll \min (m,n)\), \(X \in {\mathbb {R}^{n \times k}}, Y \in {\mathbb {R}^{m \times k}}\). The nonnegativity constraint terms are adopted to ensure nonnegativity of X and Y.
Graph dual regularized nonnegative matrix factorization
As a embedding model, the learning performance of NMF can be greatly improved if the geometrical information has been taken into account [46]. Cai et al. [47] used a graph regularization item to integrate the geometric information. Furthermore, Shang et al. [48] introduced graph dual regularization items based on both data manifold and feature manifold.
In order to obtain geometric information of drugs and targets, two Knearest neighbor graphs \(N^d\) and \(N^t\) of drugs and targets respectively are constructed based on \(S^d\) and \(S^t\), respectively.
For two drugs \(d_i\) and \(d_j\), the weight of the edge between vertices i and j in graph \(N^d\) is defined as follows.
where \(\mathcal {N}_K(i)\) denotes the sets of K most similar drugs of drugs \(d_i\) according to \(S^d\). Based on \(N^d\) and \(S^d\), a sparse matrix \({\hat{S}}_{ij}^d\) is computed as follows.
\({\hat{S}}^d\) is a weight matrix representing the drug neighbor graph. The graph Laplacian of \({\hat{S}}^d\) is \({\mathcal {L}_d} = {D^d}  {{\hat{S}}^d}\), where \(D^d\) is a diagonal degree matrix with \(D_{ii}^d = \sum \limits _r {{\hat{S}}_{ir}^d}\).
Similarly, the weight matrix \({\hat{S}}^t\) corresponding to the target neighbor graph is computed as follows.
The graph Laplacian of \({\hat{S}}^t\) is \({\mathcal {L}_t} = {D^t}  {{\hat{S}}^t}\), where \(D^t\) is diagonal degree matrix with \(D_{jj}^t = \sum \limits _q {{\hat{S}}_{jq}^t}\).
The normalized graph Laplacian forms of \(\mathcal {L}_d\) and \(\mathcal {L}_t\) are as follows.
The optimization model of graph dual regularization nonnegative matrix factorization (GDNMF) of the drugprotein interaction matrix Z is formulated as follows.
where \(\lambda _d\) and \(\lambda _t\) are regularization parameters.
GDNMF with \(L_{2,1}\)norm regularization terms
In order to ensure sparsity of the matrices obtained by nonnegative matrix factorization, we introduce the \(L_{2,1}\)norm of X and Y into GDNMF optimization model, and the optimization model of GDNMF with \(L_{2,1}\)norm regularization terms is formatted as follows.
where \(\lambda _l\) is a regularization parameter, \({\left\ X \right\ _{2,1}}\) and \({\left\ Y \right\ _{2,1}}\) represent \(L_{2,1}\) norms of matrix X and Y, respectively, and \({\left\ X \right\ _{2,1}} = {\sum \limits _i {({{\sum \limits _j {({x_{ij}})} }^2})} ^{1/2}}\), \({\left\ Y \right\ _{2,1}} = {\sum \limits _i {({{\sum \limits _j {({y_{ij}})} }^2})} ^{1/2}}\).
Algorithm
Nonnegative double singular value decomposition
To provide better and explainable initial component matrices for matrix factorization, nonnegative double singular value decomposition (NNDSVD) [39] is adopted to obtain initial value of matrix factorization. NNDSVD is an algorithm based on SVD of Z. \(Z = \sum _{i = 1,.., k}{\sigma _i u_i v_i^T}\), where Z equals to the sum of k leading singular factors, \(u_i\) and \(v_i\) denote the left and right singular vectors corresponding to \(\sigma _i\), respectively, and \(\sigma\) denotes singular value of Z.
For a vector or matrix z, \(z^+=max(0,z)\) represents nonnegative section of z, \(z^=max(0,z)\) represents nonpositive section of z, \(z=z^+z^\). \(Z = \sum _{i = 1,.., k}{\sigma _i u_i v_i^T}\) can be transformed to the following form:
If \(\left\ {u_i^ + } \right\ \left\ {v_i^ + } \right\ > \left\ {u_i^  } \right\ \left\ {v_i^  } \right\\), \(\sqrt{\sigma _i u^+_{i}  v^+_{i} } ( u^+_{i} / u^+_{i} )\) is used to obtain initial value of ith column of X. \(\sqrt{\sigma _i\left\ u_i^{+}\right\ \left\ v_i^{+}\right\ }\left( v_i^{+} /\left\ v_i^{+}\right\ \right)\) is used to obtain initial value of ith column of Y. Otherwise, \(\sqrt{\sigma _i\left\ u_i^{}\right\ \left\ v_i^{}\right\ }\) \(\left( u_i^{} /\left\ u_i^{}\right\ \right)\) and \(\sqrt{\sigma _i\left\ u_i^{}\right\ \left\ v_i^{}\right\ }\left( v_i^{} /\left\ v_i^{}\right\ \right)\). The detailed steps of NNDSVD are shown in the Additional file 1: Table S1, 2.
Proximal alternating linearized minimization
Bolte et al. [49] proposed a Proximal Alternating Linearized Minimization method (PALM), which can be regarded as a blockwise application of the proximal forwardbackward algorithm [50, 51] in the nonconvex setting.
Model (8) can be transformed to the following form:
where \(R(X)={\lambda _d}{\text {Tr}}({X^T}\widetilde{\mathcal {L}}_d X)+{\lambda _l}\left\ X \right\ _{2,1}\), \(R(Y)={\lambda _t}{\text {Tr}}({Y^T}\widetilde{\mathcal {L}}_t Y) +{\lambda _l} \left\ Y \right\ _{2,1}\). The nonnegative constraint of formula (10) can be transformed to the following form:
Then the model (10) is transformed into the following form:
GaussSeidel method is adopted to solve model (13). The schemes are as follows,
Let \(G(X,Y)=\frac{1}{2}\left\ {Z  X{Y^T}} \right\ _F^2+R(X)+R(Y)\). We remove the constant terms by plugging \(Y^i\) into \(\psi (X,Y)\) and get \({X^{i + 1}} \in \arg \min \{ {\delta _X} +R(X)+ \left\ {Z  X{Y^T}} \right\ _F^2\}\), where \(G(X,Y^i)\) is smooth function. After removing the constant term, the secondorder Taylor series of \(G(X,Y^i)\) at a point \(X^i\) is given by:
where \({\nabla _X}G\) is the partial derivative of G with respect to X.
Define the proximal map of f: \(prox_t^{f} = \arg \min \{ f (u)\) \(+ \frac{1}{2t}{\left\ {u  x} \right\ _{F}^2},u \in \mathbb {R}{^d}\}\), where \(f: \mathbb {R}{^d} \rightarrow (  \infty , + \infty ]\) is the lower semicontinuous function to ensure nonnegativity, x is a fixed point, t is a constant, \(x\in \mathbb {R}{^d}\), \(t>0\). According to the definition of proximal map, the solution of formula (16) is as follows (the detailed derivation processes are shown in Appendix):
Similarity, \({Y^{i + 1}} \in prox_{c_2^i}^{\delta _Y}({Y^i}  \frac{1}{{c_2^i}}{\nabla _Y}G({X^{i + 1}},{Y^i}))\), where \(\left\{ \begin{array}{l} c_1^i = {\nabla _X}({\nabla _X}G({X^i},{Y^i})) = {\left\ {{Y^i}{{({Y^i})}^T}} \right\ _F}, \\ c_2^i = {\nabla _Y}({\nabla _Y}G({X^i},{Y^i})) = {\left\ {{X^i}{{({X^i})}^T}} \right\ _F}. \\ \end{array} \right.\)
Let
The formula (17) is translated to
where \(prox_{{c _1^i}}^{\delta _X}U^i\) is a map, which project on \(\mathbb {R}_{+}^{m\times n}\). Similarity, we have
For a sequence \({(X^i, Y^i)}_{i \in \mathbb {N}}\), parameters \(c _1^i\) and \(c_2^i\), we can get
Inertial terms
Alvarez and Attouch [52] first proposed the ideal of inertia in 2001, which was applied in an proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Polyak showed that inertial terms can speed up convergence for the standard gradient method, while the cost of each iteration stays basically unchanged [53, 54]. In PALM, the optimization scheme is an firstorder gradient descent method. In order to accelerate the PALM, inertial terms are used.
Inertial proximal alternating linearized minimization
We uses G to denote the object function of the model (8), i.e.
The partial derivative of function G for X is
The partial derivative of function G for Y is
where \(\frac{{\partial {{\left\ X \right\ }_{2,1}}}}{{\partial X}}\)
\(= \left[ {\begin{array}{*{20}{c}} {\frac{1}{{\left\ {{X^1}} \right\ _2}}} &{} {} &{} {} &{} {} \\ {} &{} {\frac{1}{{\left\ {{X^2}} \right\ _2}}} &{} {} &{} {}&{} {} \\ {} &{} {} &{} \ddots &{} {} &{} {} \\ {} &{} {}&{} {} &{} {\frac{1}{{\left\ {{X^i}} \right\ _2}}} &{} {} \\ {} &{} {} &{} {}&{} {}&{} \ddots &{} {} &{} {} \\ {} &{} {} &{} {}&{} {}&{} {} &{} {\frac{1}{{\left\ {{X^n}} \right\ _2}}} \\ \end{array}} \right] X,\)
\(\frac{{\partial {{\left\ Y \right\ }_{2,1}}}}{{\partial Y}}\)
\(= \left[ {\begin{array}{*{20}{c}} {\frac{1}{{\left\ {{Y^1}} \right\ _2}}} &{} {} &{} {} &{} {} \\ {} &{} {\frac{1}{{\left\ {{Y^2}} \right\ _2}}} &{} {} &{} {} \\ {} &{} {} &{} \ddots &{} {} &{} {} \\ {} &{} {} &{} {} &{} {\frac{1}{{\left\ {{Y^j}} \right\ _2}}} &{} {} \\ {} &{} {} &{} {}&{}{} &{} \ddots &{} {} &{} {} \\ {} &{} {} &{} {} &{} {} &{}{} &{}{\frac{1}{{\left\ {{Y^m}} \right\ _2}}} \\ \end{array}} \right] Y,\)
For sequences \({(X^i, Y^i)}_{i \in \mathbb {N}}\), \({(m_{1}^{i}, m_{2}^{i})}_{i \in \mathbb {N}}\), \({(n_{1}^{i}, n_{2}^{i})}_{i \in \mathbb {N}}\), parameters \(c _1^i\), \(c_2^i\), \(\beta _1^{i}\) and \(\beta _2^{i}\), we can get
The pseudocode of the algorithm (iPALMDLMF) is shown in Algorithm 1.
Experiments
To evaluate the performance of DTIs prediction algorithms, 5 repetitions of 10fold crossvalidation are performed for all prediction methods. The averages 5 repetitions of 10fold crossvalidation results are used as the final test results.
The crossvalidation experiments are conducted under the following two scenarios [55].

1.
\(CV_d\): The drugs are divided in ten folds, each fold is selected in turn as the test dataset and the other remained 9 folds are used as the training dataset. If the ith drug is in the test dataset, the elements in the ith row of Z are all set 0, which means the known interactions with tested drugs are removed from the input DTI matrix. It aims to evaluate the targeted protein prediction performance for the drugs without any known interactive targets.

2.
\(CV_t\): The targets are divided in ten folds, each fold is selected in turn as the test dataset and the other remained 9 folds are used as the training dataset. If the jth target in the test dataset, the elements in the jth column of Z are all set 0, which means the known interactions with tested targets are removed from the input DTI matrix. It aims to evaluate the targeting drug prediction performance for the targets without any known interactive drugs.
We use the area under receiver operating characteristic curve (AUC) and area under the precisionrecall curve (AUPR) to evaluate performance of methods.
Comparison with stateoftheart methods
iPALMDLMF are compared with the following eight methods, namely BLMNII [15], WKNKN [16], RLSWNN [14], GRMF [31], WGRMF, CMF [29], SRCMF [33] and MKTCMF [34], where WGRMF is a weighted form of GRMF. Among them, BlMNII, WKNKN and RLSWNN use the neighborhood information of graph to predict DTIs, while the others are model based on matrix factorization.
Parameter settings
According to the original literature [31, 33, 34] and the source code of GRMF [31], we set parameters to obtain results of relevant methods. For iPALMDLMF, according to previous research [31], grid search [56] are used to choose parameters based on the AUPR value. the regularization parameter \(\lambda _l\) is selected from \(\{2^{2}, 2^{1}, 2^{0}, 2^{1}\}\). \(\lambda _d\) and \(\lambda _t\) are selected from \(\{0, 10^{4}, 10^{3}, 10^{2}, 10^{1}\}\). The numbers of maximum iterations are 2. k is 26 on NR. k is 49 on GPCR. rank k is selected from \(\left\{ {50, 100} \right\}\) on IC and E. For inertial parameters \(\alpha _{1}^i=\alpha _{2}^i=0.2, \beta _{1}^i=\beta _{2}^i=0.4\). \(c_{1}^{i}={\left\ {{Y^i}{{({Y^i})}^T}} \right\ _F}\), \(c_{2}^{i}={\left\ {{X^i}{{({X^i})}^T}} \right\ _F}\).
In order to explore the effect of performance of iPALMDLMF with different values of K, we change the values of K and show the corresponding AUC and AUPR of iPALMDLMF under the \(CV_d\) and \(CV_t\) scenario in Fig. 2. We can find from these four figures that with the increase of the values of K, the performance of iPALMDLMF can not maintain stability on different datasets. As shown in Fig. 2, iPALMDLMF is very sensitive to the value of K. Therefore, based on [31], we set \(K=5\).
Prediction results
Under the \(CV_d\) scenario, iPALMDLMF performs better than other methods in terms of AUC and AUPR on NR, GPCR, IC, and E datasets. The AUC values of iPALMDLMF are 0.886132, 0.87153, 0.814679 and 0.834224 on NR, GPCR, IC, and E datasets, respectively. The AUPR values of iPALMDLMF are 0.549245, 0.398948, and 0.399354 on NR, IC, and E datasets, respectively. On the GPCR dataset, WGRMF achieve the highest AUPR values, which are 0.410652. The AUPR value of iPALMDLMF is 0.392701. The AUC and AUPR values of the different algorithms on the four datasets are shown in Tables 2 and 3, respectively. The AUC and AUPR histograms with error bars of different algorithms are shown in Fig. 3a and b, respectively. The receiver operating characteristic (ROC) curves and the precisionrecall (PR) curves of different methods on the four datasets are shown in Figs. 4 and 5, respectively.
Under the \(CV_t\) scenario, the AUC of iPALMDLMF are higher than the other methods on the four datasets. The AUC values of iPALMDLMF are 0.797695, 0.886124, 0.948157 and 0.938395 on NR, GPCR, IC, and E datasets, respectively. The AUPR values of iPALMDLMF on NR and GPCR datasets are 0.474567 and 0.590447, respectively. On the IC and E dataset, WGRMF achieve the highest AUPR values, which are 0.800896 and 0.799641, respectively. The AUPR value of iPALMDLMF is 0.776349 and 0.772684 on the IC and E dataset, respectively. The AUC values and AUPR values of different algorithms on the four datasets are shown in Table 4 and Table 5, respectively. The AUC and AUPR histograms with error bars of different algorithms are shown in Fig. 6a and b, respectively. ROC and PR curves of different algorithms are shown in Fig. 7 and Fig. 8 on the four datasets, respectively.
Ablation experiments
In order to determine the effect of several techniques on performance in our proposed iPALMDLMF, we separately assess the performance of iPALMDLMF, iPALMDLMF (without NNDSVD, i.e. using SVD in the initialization stage of matrix factorization), iPALMDLMF ( \(\lambda _d\)=0, i.e. the graph regularization term for drugs is not used), iPALMDLMF ( \(\lambda _t\)=0, i.e. the graph regularization term for targets is not used), iPALMDLMF ( \(\lambda _l\)=0, i.e. \(L_{2,1}\) norm graph regularization is not used) and PALMGRMF (i.e. inertial forces is not used). The results of above settings are shown in Tables 6, 7, 8, and 9.
In Tables 6, 7, 8, and 9, iPALMDLMF have better performance than other settings. In \(CV_d\), when NNDSVD are used in the initialization stage of matrix factorization, the AUC values have increased by 0.6%, 1.8%, 2% on NR, GPCR and E datasets, respectively, and the AUC values have decreased by 1.3% on IC datasets. The AUPR values have increased by 2.2%, 11.5%, 6%, 4% on NR, GPCR, IC and E datasets, respectively. In \(CV_t\), using NNDSVD, The AUC values have increased by 6%, 6%, 3%, 1.6% on NR, GPCR, IC and E datasets, respectively. The AUPR values have increased by 6.5%, 9.7%, 1.5% and 3.4% on NR, GPCR, IC and E data sets, respectively. Experimental results show that using NNDSVD in the initial stage of matrix factorization can improve the ability of the algorithm to predict DTIs.
When we use regularization terms for drugs and targets, iPALMDLMF has the good prediction performance in \(CV_d\) and \(CV_t\). In \(CV_d\), when \(\lambda _d=0\), the values of AUC and AUPR of iPALMDLMF are significantly decreased. The AUC values have decreased by 30%, 27%, 27%, 34% on NR, GPCR, IC and E datasets, respectively. The AUPR values have decreased by 75%, 82%, 87%, 96% on NR, GPCR, IC and E datasets, respectively. Similarly, in \(CV_t\), if the graph regularization terms for targets is not used, the performances of iPALMDLMF is significantly decreased too. When \(\lambda _t=0\), the AUC values have decreased by 37%, 38%, 39%, 42% on NR, GPCR, IC and E datasets, respectively. The AUPR values have decreased by 79%, 92%, 91%, 98% on NR, GPCR, IC and E datasets, respectively. When \(\lambda _t=0\), these results show that regularization terms for drugs and targets contribute the improvement of DTIs prediction performance of iPALMDLMF significantly.
In \(CV_d\), when \(\lambda _l=0\), the values of AUC and AUPR of iPALMDLMF are decreased. The AUC values have decreased by 3%, 2%, 0.4%, 1.1% on NR, GPCR, IC and E datasets, respectively. The AUPR values have decreased by 1.9%, 7.2%, 4.8%, 7% on NR, GPCR, IC and E datasets, respectively. Similarly, in \(CV_t\), when \(\lambda _l=0\), the AUC values have decreased by 8.6%, 6.4%, 3.5%, 1.8% on NR, GPCR, IC and E datasets, respectively. The AUPR values have decreased by 0.5%, 11%, 1.1%, 2.8% on NR, GPCR, IC and E datasets, respectively. When \(\lambda _l=0\), these results show that \(L_{2,1}\) regularization terms for drugs and targets contribute the improvement of DTIs prediction performance of iPALMDLMF.
When inertial terms is not used in iPALMDLMF, the values of AUC and AUPR of iPALMDLMF are decreased under \(CV_d\) scenario. The AUC values have decreased by 3.8%, 0.7%, 1.1%, 1.8% on NR, GPCR, IC and E datasets, respectively. The AUPR values have decreased by 0.3%, 3.8%, 8.4%, 5.2% on NR, GPCR, IC and E datasets, respectively. Similarly, the AUC values have decreased by 7.8%, 6.5%, 4.8%, 1.1% on NR, GPCR, IC and E datasets in \(CV_t\), respectively. The AUPR values have decreased by 9.3%, 12.8%, 1.3%, 0.3% on NR, GPCR, IC and E datasets, respectively. These results show that inertial terms contribute the improvement of DTIs prediction performance of iPALMDLMF.
Case studies
To further evaluate the ability of iPALMDLMF to find new targets for a drug and new drugs for a target in practice, two case studies concerning the drug gabapentin and the target prostaglandinendoperoxide synthase 2 were conducted. Furthermore, we also conducted experiments according to [23].
In the first case study, we predicted targets that interact with the drug gabapentin on the IC dataset using iPALMDLMF. Gabapentin (GBP) is an antiepileptic drug, which is an amino acid. In the mechanism of action, gabapentin (GBP) is different from other anticonvulsant drugs which makes identifying interaction target for GBP more complicated [57]. The known interactions of gabapentin with targets were deleted from the training dataset, and the candidate targets of gabapentin predicted by iPALMDLMF were prioritized according to the prediction scores. At last, the top 50 highestscoring predicted targets were picked out to be validated using the original database [12]. The results showed that 46 targets had evidences to drug GBP among the predicted 50 drugs. The detailed results of the predictions are shown in Table 10.
In the second case study, we predicted candidate drugs for the target prostaglandinendoperoxide synthase 2 (PTGS2) on the E dataset and aimed to assess the ability of iPALMDLMF to predict candidate drugs for targets with no known targeting drugs. PTGS2 expression has been validated to be associated with colorectal cancer. However, PTGS2 and prostaglandinendoperoxide synthase 1 are confused in colorectal cancer pathology and therapy. The known interactions of PTGS2 with drugs is essential in clinic [58]. The known interactions of PTGS2 with drugs were removed from the training dataset, and the candidate drugs of PTGS2 predicted by iPALMDLMF were prioritized according to the prediction scores. The top 50 highestscoring predicted drugs were selected to be validated against original database [12] and literatures. Among the predicted 50 drugs, 47 drugs had evidences to target PTGS2, where pentoxifylline, mesalamine, suprofen, mofezolac and sulfinpyrazone have been validated to interact with PTGS2 by literature [59,60,61,62,63], respectively. This means that iPALMDLMF have good performance for new predicted interactions. The detailed results of the case study are shown in Table 11.
According [23], the whole heterogeneous network (in which drug and targets have at least one known interacting pair) was regarded as training data on the E dataset. We removed 80000 proteinprotein interactions from the target proteins network in training data. Among the top 200 highestscoring predictions, we found that all of them can also be supported by the original database [12]. Networks of the predicted drug–target interactions are shown in Fig. 9.
Conclusion
It is important to ensure sparseness of the matrices obtained by nonnegative matrix factorization to find the novel usage of drugs in drug research. In this paper, we propose a matrix factorization based method, iPALMDLMF, to predict interactions between drugs and targets. iPALMDLMF uses graph dual regularization terms to capture structural information from the drug similarity matrix and the target similarity matrix. At the same time, \(L_{2,1}\) norm regularization terms is used to ensure sparseness of the matrices obtained by nonnegative matrix factorization. Finally, an inertial proximal alternating linearized minimization algorithm is used to solve the matrix factorization with graph dual regularization terms and \(L_{2,1}\) norm regularization terms. Extensive experiments show that iPALMDLMF outperforms the stateoftheart methods in predicting DTIs.
As a kind of gradient descent methods, iPALMDLMF can converge to KKT point. In the future, we are interested in using the idea of multiobjective particle swarm optimization [64] and fixedpoint iterative method [65] to obtain a accurate solution in DTIs prediction models. At that time, more attention should be paid to synergistic drug combinations prediction problem [66].
Availability of data and materials
iPALMDLMF is implemented in Matlab and freely available to the public on https://github.com/zhang340jj/iPALMDLMF. The contents of the appendix include some description of symbols, the detailed steps of NNDSVD and the derivation of formula 17.
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This work was supported by the National Natural Science Foundation of China under Grant 62172028 and Grant 61772197.
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J.Z. and M.X. conceived this work and designed the experiments. J.Z. collected the data and carried out the experiments. M.X. analyzed the results. J.Z. and M.X. wrote the manuscript, and M.X. revised it. All authors read and approved the final manuscript.
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Zhang, J., Xie, M. Graph regularized nonnegative matrix factorization with \(L_{2,1}\) norm regularization terms for drug–target interactions prediction. BMC Bioinformatics 24, 375 (2023). https://doi.org/10.1186/s12859023054966
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DOI: https://doi.org/10.1186/s12859023054966