Coupled matrix–matrix and coupled tensor–matrix completion methods for predicting drug–target interactions

Abstract

Predicting the interactions between drugs and targets plays an important role in the process of new drug discovery, drug repurposing (also known as drug repositioning). There is a need to develop novel and efficient prediction approaches in order to avoid the costly and laborious process of determining drug–target interactions (DTIs) based on experiments alone. These computational prediction approaches should be capable of identifying the potential DTIs in a timely manner. Matrix factorization methods have been proven to be the most reliable group of methods. Here, we first propose a matrix factorization-based method termed ‘Coupled Matrix–Matrix Completion’ (CMMC). Next, in order to utilize more comprehensive information provided in different databases and incorporate multiple types of scores for drug–drug similarities and target–target relationship, we then extend CMMC to ‘Coupled Tensor–Matrix Completion’ (CTMC) by considering drug–drug and target–target similarity/interaction tensors. Results: Evaluation on two benchmark datasets, DrugBank and TTD, shows that CTMC outperforms the matrix-factorization-based methods: GRMF, -GRMF, NRLMF and NRLMF. Based on the evaluation, CMMC and CTMC outperform the above three methods in term of area under the curve, F1 score, sensitivity and specificity in a considerably shorter run time.

1 Introduction

One major class of in silico drug–target interaction (DTI) prediction methods is machine learning methods that compensate for the lack of 3D structures of drugs and targets in order to identify any potential binding events. Although in vitro experiments are the ultimate step in the drug discovery process, computational predictions are essential to avoid expensive and laborious lab experiments early in the process. To this end, machine learning and other prediction methods have been developed since the early pharmacological DTI predictions [3].

Many DTI prediction methods incorporate drug–drug or target–target structural relationships via ‘Similarity/Distance-Based Methods’. The main disadvantages of this approach are that they are sensitive to the fact that only a small percent of drugs have known interactions and some data sets are of binary nature, even though drug–target binding affinities are continuous in nature. Another family of solutions are the ‘Network-Based Methods’ that utilize graph-based techniques to perform DTI prediction. Although some methods use three networks of protein–protein similarity, drug–drug similarity and known DTIs in a heterogeneous network, they tend to perform poorly in DTI discovery; this may be due to the fact that the properties of DTI networks are not favorable for such methods [6, 13]. ‘Feature-Based Methods’ have recently been used in DTI prediction tasks; these include support vector machines, tree-based methods and other kernel based methods used with 3D protein structures. Any drug–target pair can be represented in terms of a feature vector, often with binary labels, and a machine learning method used to classify the pair vectors into positive or negative interacting proteins prevents extracting the main features disadvantaging performance. To deal with high-dimensional and noisy data in DTI predictions, several ‘Deep Learning Methods’ have been proposed (e.g. [27, 33]). The main disadvantages of these methods are that a great deal of training data and high computational power are required to train the complex model. Additionally, they lack the transparency in interpreting results and performance issues. The ‘Matrix Factorization Methods’, as another family of methods used in DTI, aim to find two matrices, and , whose multiplication gives the interaction matrix with . It is assumed that the drugs and targets lie in the same distance space such that the distance among drugs and targets can be used to measure the strength of their interactions. Therefore, drugs and targets can be embedded in a common low-dimensional subspace (see [16, 28]). Matrix factorization and matrix completion have been reported to be the most reliable methods among all the other methods based on their performance ([1, 8]); however, they are not able to incorporate all the available information about the drugs and targets.

In the current Big Data era, there exist numerous examples of the ‘matrix completion problem’: impute/predict the missing values of a matrix, given an incomplete matrix with values that are noisy and potentially corrupt [5]. A common application is for ‘recommender systems’ such as the ‘Netflix Prize’ [21]. Prediction of the missing values has been an active area of research, resulting in methods such as ‘Singular Value Thresholding’ [4], ‘Fixed Point Continuation’ [18] and ‘Matrix Factorization’ [30]. However, these methods ignore supporting data that could be integrated with the main matrix.

A main challenge in DTI prediction using matrix-based method is to perform the completion task of the sparse matrices of drugs, , targets, , along with their interactions, , (shown in Fig. 1) that are central to the field of drug repositioning (a.k.a. drug repurposing). To address the true structure of many real applications, the proposed matrix completion method has been expanded using the following steps: (1) ‘Coupled Matrix–Matrix Completion’ (CMMC): The matrix completion problem is expanded to cases where the matrix is coupled with additional structural information on the attributes X and Y involved in the matrix, such as a matrix expressing functional similarities of different drugs, and a matrix expressing relations among the targets. It is highly desirable to directly integrate the drug–drug similarity and target–target relation matrices (which may also be sparse themselves) in completion of the sparse DTIs (see Figure 1).

Figure 1 .

An illustration of a sparse coupled drug–drug, , drug–target, and targettarget, , matrices representing the interactions.

(2) ‘Coupled Tensor–Matrix Completion’ (CTMC): The similarity matrices and can often be calculated in complementary ways based on different criteria, resulting in multiple ’s and ’s (see Figure 2). For instance, the drug–drug similarities can be assessed using different structural and functional characteristics and in different chemical environments. When completing the matrix in these situations, instead of matrices and , one must deal with tensors (in this case 3-dimensional arrays) and where and represent the number of different contexts for and , respectively. A major current challenge in data science is that existing algorithms fail to use the highly important structural correlations within tensors. Therefore, prediction/estimation of the missing values in or while considering all structural relations is a much more practically important problem and can be termed as CTMC. To evaluate the proposed methods, we use cross-validation to compare them with three other state-of-the-art methods, namely GRMF [9], NRLMF [2] and -GRMF [7].

Figure 2 .

An illustration of a sparse coupled drug–drug tensor, , drug–target matrix, and target–target tensor, , matrices representing the interactions.

The rest of the manuscript is organized as follows: brief descriptions of the three competing state-of-the-art methods are provided in Section 2. Section 3 describes information about the datasets used in our work, followed by Section 4, which explains our proposed methods. We then present the experimental results of our work and provide relevant discussion in Section 5 and conclude in Section 6.

2 Related work

In Big Data applications it is common that data are sparse (mostly zeros) and partially missing. Missing data imputation, especially in the context of sparse noisy data, is therefore a central problem. A common situation is a matrix with missing entries under the assumption that the completed matrix has low rank. The low-rank matrix completion problem is NP hard and highly non-convex [11], but there are various algorithms that work under certain assumptions on the data; for instance, one approach to low-rank matrix completion is to use the nuclear norm as a convex relaxation of the matrix rank and use semi-definite programming to find a completion that minimizes the nuclear norm (see [5, 10]). Other approaches use matrix factorization with non-convex optimization such as alternating minimization ([14]) or gradient descent ([24]).

Here, we have considered four methods, two of which are based on graph regularization that are generally used in order to fully consider the internal structure of the drug–drug and target–target similarity matrices while keeping them unchanged; another two use specific probability, in particular distribution, functions in order to perform the task of DTI prediction. Moreover, a preprocessing step has been employed in order to deal with the sparsity of the interaction matrices.

2.1 GRMF

GRMF is a two-step method proposed in [9] using weighted nearest known neighbors (WKNKN) as a preprocessing step and graph regularized matrix factorization (GRMF) for predicting DTIs. WKNKN is used to transform the binary into interaction likelihood values in the given drug–target matrix. Given the drug–target matrix , where and denote the number of drugs and targets, respectively, the algorithm returns the nearest known neighbor in descending order based on their similarities to the th drug, , or the th target, . Next, the authors derived a -nearest neighbor graph from the drug similarity matrix, , and target similarity matrix, . Based on the given , a -nearest neighbor graph is then generated in the form:

(1)

for any and , where denotes the set of nearest neighbor to drug . GRMF minimizes the objective function

(2)

where and are two low-rank latent features matrices for drugs and targets, respectively, which approximates the decomposition matrix . For more explanation of the method, we refer the reader to [9]. It follows by a regularization step to prevent overfitting and a normalization step to enhance the performance.

2.2 L_2,1-GRMF

-GRMF is an improved GRMF method to address the issue that the datasets are often located at or near a low-dimensional nonlinear manifold, in combination with the previous matrix-decomposition method. To this end, authors in [7] use the Euclidean distance, , to calculate the nearest neighbor. Next, the interaction matrix is decomposed into two low-rank latent feature matrices and such that and the objective function is written as follows:

(3)

where denotes the Frobenius norm with the number of potential features of and .

2.3 NRLMF

NRLMF [17] is one of the drug–target prediction methods based on a matrix factorization technique and is one of the state-of-the-art method. NRLMF method focuses on predicting the probability that a drug would interact with a target. Specifically, the properties of a drug and a target are represented by two latent vectors in the shared low-dimensional latent space, respectively. As such, the properties of a drug and a target are described via two latent vectors where represents the dimension of the shared latent space to which both drugs and targets are mapped. The authors in [17] model the interaction probability of the drug–target pair, using the following logistic function:

(4)

The final DTI prediction model then is formulated by considering the DTIs as well as the neighborhood of drugs and targets.

2.4 NRLMF β

NRLMF [2] is an algorithm that assigns to any score of NRLMF (see Section 2.3) a new score, based on the expected value of the beta distribution defined by

(5)

where denote the shape parameters of the beta distribution, and represents the beta function [31]. The beta distribution is determined based on interaction information and current NRLMF score and is also known as the conjugative prior for the Bernoulli distribution [31] used in NRLMF and can reflect the amount of interaction information for the NRLMF score. Likewise, GRMF and -GRMF, given the interaction matrix , new scores are utilized to calculate , and , drug similarity and target similarity matrices.

3 Data

For evaluation of our proposed methods, we used two benchmark datasets: 1) Data extracted from DrugBank [32], one of the most popular databases that is widely used as a drug reference resource. This database was first released in 2006, and a database both in bioinformatics and cheminformatics, DrugBank contains detailed drug data with comprehensive drug–target information. The DTI relationships in DrugBank are originally collected from textbooks, published articles and other electronic databases. All data can be freely downloaded from DrugBank. 2) Data extracted from Therapeutic Target Database (TTD) [29], which provides therapeutic proteins, nucleic acid targets and corresponding drug information. This database was first described in 2002 and data in TTD were mainly collected from the literature.

3.1 DrugBank

In order to establish the DTI matrix, a total of six matrices were created. In doing so, a total of drugs with at least one polypeptide target and polypeptide targets that are targeted by those drugs were extracted. They form a matrix of size whose density is . This represents a sparsity of %.

As it is thoroughly discussed in Section 4, multidimensional arrays (tensors) are used to evaluate the performance of the proposed methods. In order to create higher order arrays in the form of drug–drug tensors, the following slices were created:

1. Drug–drug interaction of every pair of the drugs, as available in the DrugBank Database,

2. Drug–drug similarity as calculated by the ‘Morgan Fingerprint’ score [25] provided by the RDKit Python package [15],

3. Drug–drug similarity as calculated by the topological torsion score [22] also provided by the RDKit Python package.

The information for the target–target interaction are obtained based upon the assumption that the interactions between targets are transitive; i.e. if protein is similar to protein , which interacts with protein , then protein may also interact with protein . The following matrices are used in order to create the target–target tensor array:

1. The binary target–target interaction information of every pair of the polypeptide targets, as provided by BioGrid [26],

2. The target–target similarity score of every pair of the polypeptides, as defined by the inverse of ‘Jukes–Cantor’ distance [12].

Jukes–Cantor distance computes the maximum likelihood estimate of the number of substitutions between two sequences with the method -distance, which is proportion of sites at which the two sequences are different. is close to 1 for poorly related sequences and is close to 0 for similar sequences. The similarity score was taken as an inverse of Jukes–Cantor distance [12].

The entries of all the matrices are min–max normalized to . All the matrices are symmetric with respect to the main diagonal. Therefore, each entry of the main diagonal of each matrix is 1. Table 1 summarizes the characteristics of the dataset created and all the information pertaining to the dataset are made available in supplementary materials (see Section 7).

Table 1 .

Summary of datasets

Matrices	Notation	Dimension	Source
Drug–Target Interaction			DrugBank [32]
Drug–Drug Interaction			DrugBank [32]
Drug–Drug Similarity: Morgan fingerprint			Structure: DrugBank [32] Score: [25]
Drug–Drug Similarity: topological torsion			Structure: DrugBank [32] Score: [22]
Target–Target Interaction			Biogrid [26]
Target–Target Similarity			Sequence: DrugBank [32] Score: Inverse of Jukes–Cantor [12]
Drug–Target Interaction			TTD [29]
Drug–Drug Similarity: Morgan fingerprint			Structure: TTD [29] Score: [25]
Target–Target Similarity			TTD [29] Score: Inverse of Jukes–Cantor [12]

3.2 TTD

We also evaluated our proposed CMMC method on TTD [29]. A total of drugs were selected, along with protein targets, with DTIs and from a matrix of density . Similar to those for DrugBank dataset, we calculated drug–drug similarity score using the Morgan Fingerprint score [25] and the target–target similarity score using inverse of Jukes–Cantor distance [12].

4 Methods

In the proposed method, we develop the theoretical framework necessary to create scalable algorithms for coupled matrix–matrix and tensor–matrix completion. These algorithms are applicable to the general case in which the coupled matrices/tensors are sparse themselves. The algorithms are tested against DTI databases for which the details are provided in Section 3. It is noteworthy that the performance of these algorithms in the task of drug repositioning should be evaluated by expert clinicians and their reliability of the results should be confirmed. To this end, we start by introducing the representation theory of reductive groups [20], which provides the basis for the proposed completion algorithms, as well as providing theoretical guarantees on the optimality of our solutions.

A reductive group in general is a linear algebraic group over a field satisfying certain conditions. Let be a real or complex vector space, then the general linear group, , and special linear group, , are reductive groups and so are the products of reductive groups. In general, is the set of invertible matrices over , together with the ‘matrix multiplication’ operation and is a subset of consisting of those elements whose determinants are 1. An real matrix can be thought of as an element in

where and are vector spaces of dimension and , respectively. The group acts on the space by

where is the transpose of matrix . The group acts by linear transformations, meaning that is a ‘representation’ of the reductive group . The space of symmetric matrices can be identified with the space of symmetric tensors . The group acts on a symmetric matrix by

An tensor (i.e. a multi-dimensional array) is an element in the representation of the reductive group , where . For a coupled matrix–tensor, we get the representation

of the group . Using the above framework, the CMMC problem depicted in Figure 1 can be identified with the representation

whereas the CTMC problem depicted in Figure 2 can be represented by

of the group . These reformulation of the problem induces metrics with which the sparse matrices could be optimally completed. For the remainder of the section, we assume that the data under consideration lie in a representation of a reductive group .

4.1 Determining optimal metrics for CMMC and CTMC

Most methods for matrix and tensor completion rely upon the choice of a fixed metric, such as the Euclidean or nuclear norm. If there is a high correlation between the rows/columns in a matrix, or between different tensor slices, then a different metric given by the data itself could be adopted. For a machine learning problem including data points in -dimensional space, , ‘Mahalanobis distance’ [19], which is computed from the covariance matrix of the data, could also be utilized. Equivalent to the Mahalanobis distance is using the Euclidean metric ‘after’ a linear change of coordinates that normalizes the covariance matrix of the data to the identity. A proper action on group could perform the change of coordinates in the vector space such that it preserves the mathematical structure of the data. The ‘Kempf–Ness’ theorem (see below, [23]) shows that there is essentially a unique change of coordinates that is optimal in a certain sense. It is known that the group has a unique maximal compact subgroup . The space has some Euclidean metric and without loss of generality one may assume that is contained in the orthogonal group .

Theorem 1.

Consider the map given by , then either does not have a critical point, or every critical point is a minimum and the set of critical points is a coset, , for some .

The theorem implies that there is a unique optimal metric, which is the Euclidean metric after the change of coordinates given by . The action of does not change the metric. To avoid a degenerated case, in the absence of any critical point, one may choose a slightly smaller reductive group instead (e.g. instead of ) or utilize a regularization that is compatible with the representation theory setup. Thus, the choice of determines the optimal metric that can be used to solve the CMMC and CTMC problems.

The next step is to determine the optimal metric for CMMC and CTMC. Assuming data points in with respective mean and an invertible covariance matrix , then and the function , defined by , has a critical point, namely . The optimal metric is exactly the Mahalanobis distance. However, if the data points are not thus distributed, then a better choice of yields a more optimal metric. Determining an optimal choice of for CMMC induces a metric and regularization terms that are directly used in the algorithm. Given a tensor , one can optimize

(6)

for , using alternating optimization: first optimizing for while fixing and followed by optimizing having and fixed and lastly, optimizing while fixing and , until the desired convergence. Each optimization step reduces to the case of data points in with mean and an invertible covariance matrix , which was discussed above. It can be shown that this procedure converges to an optimal solution and in practice only a few iterations are needed.

In the CTMC case, there are more potential choices for that may yield a more optimal metric. For example or .

4.2 Developing the CMMC and CTMC algorithms

Assuming the actual data for are not known yet , where is a projection map, is given. In order to estimate the missing data, is minimized over all and with the constraint . However, a unique optimal solution is no longer guaranteed for this optimization, because even the low-rank matrix completion problem does not always have a unique optimal solution. An approach to finding an optimal and is to use alternating optimization. Starting with the element as the identity, one can find with , such that is minimal. An optimal can be now found such that is minimal, and this procedure is repeated until a desired convergence is obtained. In some cases, such as for the CTMC case, finding an optimal is in itself an iterative procedure. In that case one can alternate a fixed number of iteration steps for with an optimization step for .

In order to improve the algorithms for CMMC and CTMC, we start by assuming that two symmetric matrices and are given in a way that they are coupled with an incomplete matrix , where and . Without loss of generality, one may assume that and are nonnegative definite. Suppose that the only known entries of are at positions . This constraint can be written as where is some fixed vector, and maps a matrix to . One may use the matrices and as regularization of the matrix completion problem of . For some fixed regularization parameters , the objective function to minimize is defined by

(7)

over all triples with , and with the constraint

Here is the square of the Frobenius norm. for some arbitrary matrix .

The drug–drug and target–target interaction matrices, and , respectively, may be incomplete as well, which in that case the following constraints are imposed

as well as the convex constraints that implied and being nonnegative definite. As a result, besides , both drug–drug and target–target matrices, and , are updated and hence the problem narrows down to a quadratic programming with convex constraints.

For the CTMC method, the drug–drug and target–target interactions/similarities tensors, which are obtained from several sources, are given by matrices of size and of size . In that case, there exist two additional transformations and , which are diagonal matrices with determinant and positive entries on the diagonal. For some fixed regularization parameters , the objective function to minimize hence becomes

(8)

over all triples with , and . Moreover, the objective function also minimizes all the number of layers added to , denoted as , and number of layers added to , denoted as . Here denotes the -th slice of the tensor .

The transformations are used to balance the various sources of drug–drug and target–target interactions, and just like and , and are updated iteratively. If entries of the tensors and have missing entries, certain constraints are adopted in addition to the one which assumes that all the slices are nonnegative definite.

To further explain the CMMC and CTMC methods, we consider the matrix representing the interaction between drugs and targets . Entries are typically within the interval . Only a small percent of the entries of matrix are non-zero and many are unknown. Without loss of generality one may assume the interaction matrix is symmetric and is considered together with two other matrices: drug–drug similarity/interaction () and target–target similarity/interaction () matrices. It is noteworthy that both rows and columns of matrix have the same labels as the rows of and both rows and columns of matrix have the same labels as the columns of .

After forming the coupled structures, the next step is to determine the optimal metric for CMMC/CTMC methods as most methods for matrix (tensor) completion rely upon the choice of a fixed metric, such as the Euclidean metric (nuclear norm). Here, the optimal metric will be determined using Algorithm 1.

Intuitively, minimizing the objective function given in Eq. (7) (and same for Eq. (8)) results in finding an optimal metric under which the distance between interaction matrix and the matrix is minimal. This also applies to two other matrices, and , as well. It is worth mentioning again that and are symmetric invertible matrices whose determinants equal to 1. To this end and to help better understanding the methods, the self-contained executable codes for the two proposed methods, CMMC and CTMC, have made publicly available (see Section 7).

4.3 Scalable algorithms for CMMC and CTMC

A main challenge to the prediction of DT interaction lies in the fact that only a small fraction of the entries in the tensors and matrices are known [1]. It appears that the output, i.e. the completed data, is many times larger then the input of the known entries. In fact, dealing with the large sized tensors may become intractable due to the lack of memory or computational power. It seems possible, however, that the output and the intermediate results can be compressed because of the following observation in a special case. Suppose that is a matrix with missing entries and . Assuming that only the entries in the positions are known with , evaluation of at the positions defines a map where only is known. The optimal solution to minimizing where and satisfying has a very special form, namely with and with the property that the only nonzero entries of are , . Therefore, instead of storing the matrix with entries, it is only needed to remember the matrix and the nonzero entries of , a total of numbers.

5 Results

For the demonstration of all the methods except CTMC, three matrices are created. For the first step, three matrices consisting of drug–drug interaction, DTI and target–target interaction are created (see Section 3). Next, all the methods are evaluated using drug–drug similarity, DTI and target–target similarity matrices. For the demonstration of CTMC, which is capable of handling multiple sources of information in a preserved tensor form, additional layers in the form of similarity scores and/or interaction information are added. Detailed information about different layers and similarity information are provided in Table 1.

5.1 CMMC performance evaluations using DrugBank

For performance evaluations, we consider the CMMC algorithm along with three other algorithms outlined in Section 2. For every iteration, a subset of the interaction matrix, , is created by randomly selecting approximately 10% of the rows and columns of . This results in a matrix, , of size , which corresponds to 1% of the total number of elements of . Next, 10% of the entries are randomly selected and replaced by 0.5, as a surrogate for a value that is neither 0 nor 1, and all four algorithms are used to predict those values. We then average the performance of all algorithms over 100 iterations.

The comparison is divided into two parts; first, we consider drug–drug and target–target interaction matrices coupled with the interaction matrix for which Table 2 represents the results. Next, the methods are compared using the coupled drug–drug and target–target similarity matrices, and respectively, coupled with the interaction matrix . The results are shown in Table 3. The methods are compared based on the total runtime, area under the curve (AUC), F1 score, sensitivity, specificity and accuracy. The threshold columns represent the most appropriate threshold for calling a predicted value either positive or negative to optimize the F1 score calculated over the 100 iterations.

Table 2 .

Metrics of results produced by the algorithms using binary interaction matrices obtained from DrugBank.

	CMMC		WKNKN + CMMC
	Mean	SD	Mean	SD
Runtime (s)	0.337	0.026	0.551	0.033
AUC	0.664	0.072	0.664	0.072
F1	0.184	0.110	0.184	0.110
Sensitivity	0.164	0.085	0.164	0.085
Specificity	0.997	0.011	0.997	0.011
	GRMF		WKNKN + GRMF
	Mean	SD	Mean	SD
Runtime (s)	1302.294	251.657	1299.383	252.889
AUC	0.629	0.078	0.645	0.083
F1	0.061	0.068	0.072	0.078
Sensitivity	0.120	0.085	0.114	0.076
Specificity	0.986	0.025	0.988	0.031
	L-GRMF		WKNKN + L-GRMF
	Mean	SD	Mean	SD
Runtime (s)	1288.877	261.152	1279.952	254.770
AUC	0.636	0.078	0.648	0.076
F1	0.062	0.071	0.074	0.078
Sensitivity	0.117	0.076	0.104	0.063
Specificity	0.986	0.026	0.990	0.027
	NRLMF		WKNKN + NRLMF
	Mean	SD	Mean	SD
Runtime (s)	1.551	0.086	1.546	0.076
AUC	0.597	0.077	0.602	0.080
F1	0.050	0.062	0.051	0.063
Sensitivity	0.116	0.079	0.115	0.090
Specificity	0.976	0.047	0.980	0.062
	NRLMF		WKNKN + NRLMF
	Mean	SD	Mean	SD
Runtime (s)	37.938	0.570	37.883	0.665
AUC	0.596	0.077	0.602	0.081
F1	0.050	0.062	0.051	0.063
Sensitivity	0.116	0.079	0.116	0.090
Specificity	0.976	0.047	0.980	0.063

Table 3 .

Metrics of results produced by the algorithms using similarity matrices obtained from DrugBank.

Algorithm	Runtime (s)		AUC		F1		Sensitivity		Specificity
	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD
CMMC	0.374	0.088	0.761	0.078	0.078	0.060	0.167	0.101	0.994	0.014
GRMF	1289.686	145.483	0.631	0.079	0.062	0.069	0.115	0.079	0.987	0.023
WKNKN + GRMF	1292.219	142.135	0.650	0.076	0.075	0.071	0.116	0.105	0.985	0.071
L21GRMF	1290.000	136.695	0.637	0.078	0.064	0.072	0.115	0.073	0.987	0.026
WKNKN + L21GRMF	1289.366	143.215	0.648	0.081	0.076	0.076	0.111	0.078	0.988	0.040
NRLMF	1.593	1.106	0.601	0.079	0.053	0.061	0.119	0.096	0.974	0.072
WKNKN + NRLMF	1.582	0.079	0.615	0.091	0.062	0.070	0.127	0.128	0.973	0.085
NRLMF	38.537	1.468	0.600	0.079	0.053	0.061	0.119	0.096	0.974	0.072
WKNKN + NRLMF	39.080	2.036	0.615	0.615	0.062	0.070	0.127	0.130	0.973	0.085

All the methods have been tested over the same dataset with and without the pre-processing step, called WKNKN [8] (see Section 2.1). This allows us to replace a given binary values with an interaction likelihood value in any of the matrices. Authors in [8] reported notable improvement in their method using the so called pre-processing step. However, as it is shown in Tables 2 and 3, although WKNKN improves the average value for AUC, F1 score, sensitivity and specificity as well as accuracy, it results in higher standard deviation (SD) values as well. Therefore, the pre-processing step WKNKN may also affect the robustness of the methods. On the other hand, it is noteworthy that the results under the proposed methods, CMMC and CTMC, are not affected by WKNKN and it shows the proposed methods do not necessarily require any pre-processing step. The reason lies in the fact that these methods only use the known interactions given in the original DTI matrix to iterate whereas drug–drug and target–target similarity/interaction matrices, and , respectively, to converge to the completed matrix; moreover, WKNKN only affects the values that are marked 0.5 as a surrogate for the ‘missing’ values, hence it does not affect the results for CMMC. As a result, it also shows the robustness of the CMMC algorithm.

The best performances in terms of AUC, F1 score, sensitivity, specificity and accuracy across different algorithms are highlighted in Tables 2 and 3, based upon which, one may observe the following: Performance based on AUC: The average value of AUC was calculated for each method with and without employing the pre-processing step, WKNKN. The AUCs for CMMC are reportedly higher than all the other methods. The highest average values of AUC was calculated for the three methods outlined in Section 2 based on similarity and interaction matrices are 0.637 and 0.636, respectively. These values are remarkably smaller than those of CMMC, which are 0.761 and 0.664, respectively. The reason that using similarity matrices for both drug–drug and target–target yields a higher AUC lies in the fact that similarity matrices contain more useful information as oppose to interaction matrices that are binary and often times sparse.

Performance based on F1 scores: In terms of F1 scores, although the average scores reported for CMMC method, using similarity and interaction matrices, correspond to small numbers, 0.078 and 0.062, respectively, they still represent higher values than those of other methods.

Performance based on sensitivity and specificity: As shown in Table 3, reported average sensitivity and specificity values for CMMC are recorded as 0.167 and 0.994, respectively, using similarity information, and 0.164 and 0.997 while utilizing interaction information based on Table 2. These values are higher compared to other methods even after using the pre-processing step, WKNKN.

Performance based on runtime: The main advantage of CMMC algorithm over the others methods described in Section 2 is the total time that it takes to perform the method over the dataset. The runtime is obtained by averaging the total running time over each iteration. As shown in Tables 2 and 3 recorded runtime for CMMC algorithm is notably smaller than those of other methods, which represents a faster process.

5.2 CMMC performance evaluations using TTD dataset

To further evaluate the performance of the proposed method CMMC algorithm, we consider another database TTD as described in Section 3.2. Based on the results provided in Section 5.1, since the CMMC algorithm performed better using similarity scores given in Table 3 than interaction information shown in Table 2, we consider using similarity scores in order to evaluate the performance of CMMC algorithm over the TTD dataset. Specifically, the drug–drug similarity matrix, , and target–target similarity, , along with the interaction matrix, (see Table 1). The performance of CMMC method along with four other methods, GRMF, -GRMF, NRLMF and NRLMF, based on the average AUC, F1 score, sensitivity and specificity over TTD dataset are shown in Table 4. Best results are marked bold. CMMC method obtains the best results in terms of average AUC, F1 score, sensitivity and specificity compared to the other method during a shorter period of time.

Table 4 .

Metrics of results produced by the algorithms using similarity matrices obtained from TTD.

Algorithm	Runtime (s)		AUC		F1		Sensitivity		Specificity
	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD
CMMC	3.378	0.307	0.846	0.037	0.084	0.070	0.122	0.093	0.996	0.008
GRMF	4176.739	136.152	0.701	0.064	0.031	0.029	0.095	0.067	0.990	0.013
WKNKN + GRMF	4143.534	120.312	0.683	0.057	0.083	0.074	0.091	0.086	0.991	0.036
L21GRMF	4148.855	121.843	0.699	0.064	0.030	0.031	0.095	0.071	0.989	0.018
WKNKN + L21GRMF	4156.066	111.406	0.690	0.057	0.083	0.076	0.087	0.071	0.993	0.025
NRLMF	4.228	0.639	0.621	0.066	0.030	0.025	0.072	0.057	0.990	0.019
WKNKN + NRLMF	4.579	0.351	0.651	0.063	0.054	0.052	0.077	0.051	0.996	0.009
NRLMF	65.611	3.659	0.621	0.066	0.030	0.025	0.072	0.057	0.990	0.019
WKNKN + NRLMF	67.278	3.810	0.651	0.063	0.054	0.052	0.077	0.051	0.996	0.009

5.3 CTMC performance evaluations

In order to evaluate the performance of the CTMC algorithm, multidimensional arrays of drug–drug and target–target similarity/interaction were created using the information provided in Table 1. The results are shown in Table 6. In order to perform the evaluation, we incorporate both similarity and interaction information between drugs and targets in order to form the drug–drug and target–target tensors, and , respectively. As shown in Table 6, CTMC outperforms all the methods including CMMC in terms of average AUC and sensitivity. The results in terms of F1 score, specificity and accuracy remain the same as CMMC while using similarity information, as shown in Table 3. As stated before, the difference is more remarkable when the similarity scores are used for coupling, most likely because the similarity matrices are rather complete, whereas the interaction matrices are sparse.

Table 6 .

Metrics of results produced by CTMC using data from DrugBank.

CTMC Method
Runtime (s)		AUC		F1
Mean	SD	Mean	SD	Mean	SD
0.513	0.045	0.775	0.080	0.169	0.110
Sensitivity		Specificity		Accuracy
Mean	SD	Mean	SD	Mean	SD
0.169	0.082	0.997	0.011	0.997	0.011

Comparing the performance of the two proposed methods CMMC (coupled with similarity matrices) and CTMC, shown in Table 3 and Table 6, respectively, it is notable that CTMC slightly outperforms CMMC in terms of average values of AUC, F1 score, sensitivity, specificity and accuracy.

Table 7 demonstrates the improvement of performance as more layers are added. Initially, the interaction matrices are coupled for CTMC. This specific case is equivalent to CMMC as one could consider matrices as a two-way tensors. The evaluation results in comparable, albeit better, AUCs. As another layer is added in the drug–drug tensor, , namely the drug–drug similarity scores from Morgan Fingerprint, the AUC improves by approximately 10% as it is shown in Table 7. Similarly, adding another layer to the target–target tensor, the AUC improves by approximately 5%. Adding the third layer to drug–drug tensor, however, does not improve the performance. It is likely due to the fact that the similarity information, calculated by different algorithms from the same database (DrugBank [25]), does not provide any new information hence does not improve the results.

Table 7 .

Performance of the CTMC algorithm adding slices obtained from DrugBank.

		# of Slices	Runtime (s)		AUC		F1		Sensitivity		Specificity
			Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD
		2	0.154	0.013	0.664	0.072	0.184	0.110	0.164	0.085	0.997	0.011
,		3	0.171	0.016	0.723	0.076	0.179	0.109	0.168	0.080	0.995	0.029
,	,	4	0.180	0.014	0.778	0.078	0.180	0.107	0.164	0.071	0.998	0.005

Metrics of results produced by the CTMC algorithm, stratified by the number of slices used in each coupled tensor. Here, denotes binary drug–drug interaction matrix, and represents drug–drug similarity matrix with Morgan fingerprint scores. As for the targets, denotes binary target–target interaction matrix and represents target–target similarity matrix as an inverse of Jukes–Cantor distance of amino acid sequences [12] (see Table 1 for more details on matrices , , and ).

Lastly, the recorded runtime for the CTMC algorithm, which incorporates more information and carries out more calculations, is nonetheless faster than the other algorithms.

5.4 Sensitivity analysis

The optimal regularization parameters for and included in Eqs. (7) and (8) are chosen based on the performance of the algorithm during the execution of CMMC and CTMC methods. In order to determine how sensitive the proposed methods are based on the changes of the arbitrary-then-fixed parameters and , as well as studying the roles of these parameters, the results under CMMC method have been compared from Tables 2, 3 and 4 against different variants of and . The results are shown in Table 5. As setting either parameters or to zero simply means to ignore the role of one of the drug–drug or target–target matrices, it negatively impacts the performance. We have set and to zero and the results are shown in Table 5.

Table 5 .

AUC Results for different and .

CMMC	DrugBank	TTD
;	0.6300	0.4902
;	0.7005	0.8372
;	0.7545	0.8445
;	0.7580	0.8464
;	0.7610	0.8460

While mainly depending on the nature of the database in use, specifically the drug–drug and target–target similarity/interaction matrices, and , it was found that the smaller the values of and , the better the prediction performance.

6 Conclusion

In this manuscript, two methods, CMMC and CTMC, for prediction of DT interactions inspired by matrix-factorization methods are presented. The experiments were performed with and without considering the preprocessing step WKNKN. The algorithm was first used to help with the sparsity of the similarity/interaction matrices. Using this, certain unknown interactions, i.e. 0’s values, were replaced by the likelihood values using nearest neighbor method. Next, experiments were performed over coupled drug–drug, drug–target and target–target matrices, considering drug–drug similarity scores and target–target interactions. In order to test the CMMC method, we considered three matrices consisting of drug–drug similarity (calculated using Extended–Connectivity Fingerprint), drug–target and target–target interactions. For the CTMC method, in addition to the matrices, extra layers for drug–drug tensors were assigned to drug–drug interaction. In forming the target–target tensor, we included target–target similarity scores in addition to their interactions. Ten percent of the entire profile of the known DTI was intentionally left out and the two methods were run and tested in terms of predicting the known interactions. CMMC and CTMC showed strong ability in order to predict new DTI.

As future work, one may incorporate additional interaction and similarity information as well as different similarity scores utilizing different datasets. For instance, in order to form the drug–drug tensor, in addition to considering any possible interaction between drugs, various similarity scores could be calculated using different databases and based on distinct ways of calculating similarity scores, namely, Morgan fingerprint and ‘Avalon fingerprint’ while using different databases. Additionally, a pre-processing step to perform tensor completion ahead of applying CTMC would likely further improve the performance.

 

Key points

• Matrix-Factorization Methods: A group of machine-learning-based methods that is used to help predict missing data using matrix factorization and matrix completion.

• CMMC: A novel matrix-factorization-based method, Coupled Matrix–Matrix Completion, which outperforms several methods in the same category in a shorter runtime.

• CTMC: A novel tensor-based method, Coupled Tensor–Matrix Completion, which is capable to incorporate more information in terms of different similarity scores as well as interaction details from various sources.

Maryam Bagherian is a postdoctoral research fellow at the Department of Computational Medicine and Bioinformatics. Her PhD degree is in applied mathematics and her research includes mathematical physics and mathematical biology.

Renaid B. Kim is an MD/PhD student in the Medical Scientist Training Program at Medical School, University of Michigan, Ann Arbor, pursuing a PhD in Bioinformatics.

Cheng Jiang is a master’s student in the College of Engineering, University of Michigan, Ann Arbor.

Maureen A. Sartor is an associate professor at the Department of Department of Biostatistics, School of Public Health, University of Michigan, Ann Arbor.

Harm Derksen is a professor at the Department of Mathematics, University of Michigan, Ann arbor.

Kayvan Najarian is a professor at the Department of Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor. His research focuses on signal/image processing and machine learning methods for medical applications.