 Research
 Open access
 Published:
Drug repositioning based on individual birandom walks on a heterogeneous network
BMC Bioinformatics volumeÂ 20, ArticleÂ number:Â 547 (2019)
Abstract
Background
Traditional drug research and development is high cost, timeconsuming and risky. Computationally identifying new indications for existing drugs, referred as drug repositioning, greatly reduces the cost and attracts everincreasing research interests. Many networkbased methods have been proposed for drug repositioning and most of them apply random walk on a heterogeneous network consisted with disease and drug nodes. However, these methods generally adopt the same walklength for all nodes, and ignore the different contributions of different nodes.
Results
In this study, we propose a drug repositioning approach based on individual birandom walks (DRIBRW) on the heterogeneous network. DRIBRW firstly quantifies the individual worklength of random walks for each node based on the network topology and knowledge that similar drugs tend to be associated with similar diseases. To account for the inner structural difference of the heterogeneous network, it performs birandom walks with the quantified walklengths, and thus to identify new indications for approved drugs. Empirical study on public datasets shows that DRIBRW achieves a much better drug repositioning performance than other related competitive methods.
Conclusions
Using individual random walklengths for different nodes of heterogeneous network indeed boosts the repositioning performance. DRIBRW can be easily generalized to prioritize links between nodes of a network.
Background
Traditional drug research and development depends on cellbased or targetbased screening of chemical compounds to identify a small subset of â€˜hitsâ€™. The identification process aims to further increase their affinity, efficacy and selectivity, before moving forward to animal tests and clinical trials [1]. Drug development in general is complicated, timeconsuming and expensive with highrisk [2]. In light of these difficulties in traditional drug discovery, identifying new indications for existing drugs, also known as drug repositioning, has attracted increasing interests from both the pharmaceutical industry and research community [3]. Drug repositioning is much more economic compared with traditional approaches, it offers a promising alternative to reduce the cost and time, since the repositioned drug has already passed the required safety tests.
However, most successfully repositioned drugs up to date have been the consequence of incidental observations of unexpected efficacy and side effects in the development or on the market [4]. For example, Sildenafil was originally tested for angina, now is indicated for erectile dysfunction and pulmonary hypertension [2]; Minoxidil was originally tested for hypertension; now is indicated for hair loss [5]. With the influx of big biochemical and phenotypic data, drug repositioning holds great potential for precise medicine. It is profitable and promising to develop computational methods to predict new indications for approved drugs on large scale.
Some computational drug repositioning methods have been proposed and they can be roughly divided into two categories: focusing on the interactions between drugs and the targets; and focusing on exploiting the knowledge of diseases and drugs [6]. To name a few, Bleakley and Yamanishi [7] developed a bipartite local model (BLM) to predict target proteins of a given drug and target drugs of a give protein, and then combine these two predictions to give a final prediction for each candidate drugtarget interaction. Cheng et al. [8] used a drugtarget bipartite network topology similarity and a network based inference algorithm (NBI) to infer new targets for known drugs. Wang et al. [9] used known drugtarget interactions as well as drugdrug and targettarget similarities to construct a heterogeneous network, and then introduced a Heterogeneous Graph Based Inference (HGBI) method to iteratively update the strength between unlinked drugtarget pairs based on all the paths in the network connecting them. These drugtarget prediction methods can be readily adopted for drug repositioning.
Chiang et al. [10] attempted to predict novel associations between drugs and diseases based on the widelyadopted â€˜guiltbyassociationâ€™ principle. This principle assumes that if a drug can treat one of two similar diseases, then it might treat the other also; alternatively a disease can be treated by two similar drugs. Following this principle, Gottlieb et al. [11] measured the similarity between the pertaining drug and disease of drugdisease pairs that are known to be associated based on multiple drugdrug sources and diseasedisease similarity metrics, and then ranked the accumulative evidence for association using a logistic regression scheme to predict novel drug indications. Wang et al. [1] integrated omics data about diseases, drugs and drug targets to construct a heterogeneous network and then applied random walks on the network to replenish missing associations between drugs and diseases. Martinez et al. [6] integrated information on diseases, drugs and targets (proteins) to construct a heterogeneous network and then performed propagation flow on the network to prioritize candidate associations between diseases and drugs according to their interconnections in the network. Luo et al. [12] proposed MBiRW to predict drugdisease associations. MBiRW employs known drugdisease associations to improve the drugdrug and diseasedisease similarity measures; and then integrates the similarity networks and drugdisease associations to build a drugdisease heterogeneous network; after that, it performs birandom walk with restart on the network to predict novel potential drugdisease associations. Liu et al. [13] performed a drugcentric random walk and a diseasecentric random walk to obtain the association confidence between the disease nodes and drug nodes of a heterogeneous network.
Most of these aforementioned methods in essence are random walk based solutions. Although they make use of the network topology from different perspectives, they ignore the different contributions of different nodes on transferring the information on the network and almost all adopt a fixed walklength for all nodes. To overcome this issue, we propose a novel drug repositioning approach (called DRIBRW) that performs birandom walk with restart on a heterogeneous network with quantified individual walklength for each node. DRIBRW uses disease symptom information [14] and drug chemical fingerprints [15] to construct a composite diseasedisease similarity network, drugdrug similarity network. It then quantifies the individual walklength for each node based on the topology of known drugdisease association network. Next, it constructs a heterogeneous network based on these three networks. After that, it performs birandom walks with the quantified walklengths to account for the structural differences of these networks and contribution differences of different nodes (including diseases and drugs), and to predict new associations between drugs and diseases, and thus to accomplish the drug repositioning. We evaluate and compare the performance of DRIBRW on several public datasets. DRIBRW obtains much better performance than other related comparing methods [7â€“9, 12, 16] in identifying new indications for existing drugs, and the quantified individual walklength indeed contributes to an improved prediction performance. We want to remark that the proposed individual birandom walk solution is different from existing personalized random walk solutions [17, 18] that mainly focus on setting different restart probabilities for different nodes.
Materials and methods
Dataset
The datasets used in this work include drugdisease associations, drug fingerprints and disease symptoms. We collected 4219 diseases from MeSH [19] and 322 symptoms for each disease from the supplementary material of [14]. The drugdisease association dataset was obtained from [20], it includes 3250 known drugdisease associations involving 799 drugs and 719 diseases. We also collected 881 fingerprints for each drug from PubChem [15]. Since only 525 diseases can find their relevant symptom information from the supplementary material of [14], the final processed dataset includes 525 diseases, 718 drugs and 2177 drugdisease associations. All these data were collected on November 1st, 2017.
Similarity measures
We separately apply a fourstep measurement to quantify the innersimilarity between diseases and between drugs. The first three steps are based on the comprehensive similarity measurement used by Luo et al. [12]. In the fourth step, we use Gaussian interaction profile kernel similarity [21] to measure the similarity between drugs and diseases. Finally, we combine these similarities to form the composite similarity between diseases and between drugs. The fourstep procedure of measuring the similarity between drugs is briefly introduced as follows.
Step 1: Based on the chemical fingerprints of the drug molecules, we can initially measure the similarity \(\mathbf {S}_{r}^{1} \in \mathbb {R}^{n_{r} \times n_{r}}\) between n_{r} drugs via the widely used Cosine similarity metric [22]. Let r_{i} and r_{j} be the vector forms of the chemical fingerprints of drug r_{i} and r_{j}, the chemical similarity \(\mathbf {S}_{r}^{1}(r_{i},r_{j})\) between drug r_{i} and r_{j} is defined as:
Step 2: Too small similarity provides little information for drug repositioning and can be transformed into zeros for accurate prediction [9, 12]. We partition \(\mathbf {S}_{r}^{1}\) into ten subranges ((0,0.1], (0.1,0.2], etc.) and calculate the average similarity of drug pairs with shared diseases for each subrange. We also randomly shuffle \(\mathbf {S}_{r}^{1}\) and repeat the partition and calculation process again. If the average of the nonshuffled subrange is smaller than that of the respective shuffled subrange, the drug similarities divided into this subrange are viewed as not informative; otherwise, they are informative. We then adopt a logistic function [23] to shrink these noninformative similarities to zero and to enlarge these informative similarities. The logistic function is defined as follows:
where c and d are the parameters can be tuned to control the adjustment of \(\mathbf {S}_{r}^{1}\). c is the upper bound of the first subrange whose average similarity is smaller than that of the respective shuffled subrange, d=log(999). After that, we obtain a updated drug similarity matrix \(\mathbf {S}_{r}^{2}\).
Step 3: Two drugs are more similar if they are grouped into the same cluster. To make use of this assumption, we first construct a new weighted drug sharing network with drugs as nodes and edge weight reflecting the number of common diseases by respective pair nodes. After that, we adopt a graph clustering method, ClusterONE [24], to identify potential drug clusters on the network. We then add the clustering cohesiveness of a cluster with \(\mathbf {S}_{r}^{2}\) if and only if the two drugs belong to that cluster.
where \(W_{in}(\mathcal {C})\) denotes the total weight of edges within a cluster of vertices, \(W_{bound}(\mathcal {C})\) represents the total weight of edges connecting nodes of this cluster to nodes of other clusters, and \(P(\mathcal {C})\) is the penalty term. Suppose that drug r_{i} and drug r_{j} locating in the same cluster \(\mathcal {C}\), the comprehensive drug similarity \(\mathbf {S}_{r}^{3}(r_{i},r_{j})\) between drug r_{i} and r_{j} is defined as \((1+f(\mathcal {C}))*\mathbf {S}_{r}^{2}(r_{i},r_{j})\). In this way, we obtain an improved drug similarity matrix \(\mathbf {S}_{r}^{3}\).
Step 4: Based on the assumption that similar drugs tend to show similar interaction and noninteraction profiles with the diseases, we further use Gaussian interaction profile kernel similarity to measure the similarity between drugs [21, 25, 26]. The interaction profile IP(r_{i}) of drug r_{i} is defined as a binary vector encoding the presence or absence of the known associations between the drug and n_{d} diseases. The Gaussian interaction profile kernel similarity between two drugs (r_{i} and r_{j}) is computed as follows:
where Î¥_{r} is the kernel bandwidth, \(\tilde {\Upsilon }_{r}\) is the average number of associated diseases per drug.
To this end, we combine \(\mathbf {S}_{r}^{3}\) and \(\mathbf {S}_{r}^{KR}\) into the composite similarity matrix S_{r} between n_{r} drugs as follows:
Following the above fourstep, we can also compute the composite similarity \(\mathbf {S}_{d} \in \mathbb {R}^{n_{d} \times n_{d}}\) between n_{d} diseases based on the symptom information of these diseases and drugdisease associations.
Quantifying individual walklength
Networkbased drug repositioning methods generally apply random walk on a network with a fixed walklength for all nodes to explore the network topology [12, 27, 28]. They ignore the different contributions of different nodes to some extent. Given that, we introduce an individual walklength measure and try to make better use of the topology of known drugdisease association bipartite network \(\mathbf {W}_{rd} \in \mathbb {R}^{n_{r} \times n_{d}}\) of n_{r} drugs and n_{d} diseases. W_{rd}(r_{i},d_{j})=1 if the association between the drug r_{i} and disease d_{j} is known; and 0 otherwise.
The walklength of a node generally depends on its influence in the network [29]. We extend the Jaccard index measure introduced by Lu et al. [16] to quantify the individual walklength of nodes. Suppose \(\mathcal {N}^{r}(r_{i})\) denote the set of neighbours of drug r_{i} and \(\mathcal {N}^{d}(d_{j})\) denote the set of neighbours of disease d_{j}, if r_{i} and d_{j} share many common neighbours, they will be more probably influenced with each other. For a randomly selected feature f of either r_{i} or d_{j}, traditional Jaccard index measures the probability that both r_{i} and d_{j} have that feature as follows [30]:
Since there is no relationship between diseases or between drugs in the drugdisease bipartite network, \(\mathcal {N}^{r}(r_{i})\cap \mathcal {N}^{d}(d_{j})\) is an empty set. For this reason, we have to modify the definition of Jaccard index for a bipartite graph. Particularly, we define \(\widehat {\mathcal {N}}^{r}(r_{i})=\cup _{c\in \mathcal {N}^{r}(r_{i})}\mathcal {N}^{d}(c)\) as the set of drugs associated with r_{i}â€™s neighbours. Then the Jaccard index on the bipartite network is defined as follows:
JI(r_{i},d_{i}) represents the influence between drugdisease pair (r_{i},d_{j}). We assume that a node with high quantified influence has more probability to interact with others during the random walk process, and this node should have larger walklength. Based on this assumption, we can measure the walklength of each node as follows:
where \(\mathbf {L}_{r}\in \mathbb {R}^{n_{r}}\) and \(\mathbf {L}_{d}\in \mathbb {R}^{n_{d}}\) store the individual walklengths of n_{r} drugs and n_{d} diseases, respectively.
Individual birandom walk
Based on the inner similarity network (defined by S^{r}) of drugs, the inner similarity network (defined by S^{d}) of diseases, and the drugdisease bipartite network initialized by known drugdisease associations, we can construct a heterogeneous network of drugs and diseases (see Fig.Â 1 for example). We adopt a birandom walk with restart procedure [27] on the heterogeneous network. Compared with traditional random walk with restart, the birandom walk with restart can separately propagate information in different subnetworks, instead of the global network [28]. For this reason, birandom walk can separately account for the inner structure of disease similarity network and of drug similarity network, and also make use of associations between drugs and diseases.
A random walker can take a drug as the starting node, its associated diseases as intermediate nodes, and then traverse to other disease nodes. In this way, we can get probabilistic associations between the drug and new diseases, and thus identify potential new indications of the drug. To mimic this process, we perform random walk with restart starting from drug nodes and then traversing to disease nodes based on the quantified individual walklength and the heterogeneous network topology as follows:
where \(\mathbf {F}_{r}^{t}(r_{i},d_{j})\) is the predicted relevance between drug r_{i} and disease d_{j} in the tth iteration, \(\mathbf {F}_{r}^{0}=\mathbf {W}_{rd}\), Î±>0 controls the probability for a walker staying at the starting point, \(\tilde {\mathbf {S}}_{d}=\mathbf {D}_{d}^{\frac {1}{2}}*\mathbf {S}_{d}*\mathbf {D}_{d}^{\frac {1}{2}}\) is the Laplacian normalized result of S_{d} and D_{d} is a diagonal matrix with \(\mathbf {D}_{d}(d_{j},d_{j})=\Sigma _{k=1}^{n_{d}}\mathbf {S}_{d}(d_{j},d_{k})\). If t>L_{r}(r_{i}), the random walker starting from r_{i} will not jump any more. We want to recomment that unlike traditional random walks and birandom walks that adopt the same walklength for all the nodes, the walklength of a node in Eq. (10) is adaptively set based on its topology relationship with other nodes and is different from the walklengths of other nodes.
Similarly, a random walker can also start from a disease node and then traverse to drug nodes based on known drugdisease relationships and drug similarity network. In this way, we can obtain another probability between the disease and drug. To simulate this process, we perform random walk with restart from the disease node (d_{j}) as follows:
where \(\mathbf {F}_{d}^{t}(r_{i},d_{j})\) is the predicted relevance between drug r_{i} and disease d_{j} in the tth iteration, and the same normalization procedure is applied to S_{r} to construct the normalization matrix \(\tilde {\mathbf {S}}_{r}=\mathbf {D}_{r}^{\frac {1}{2}}*\mathbf {S}_{r}*\mathbf {D}_{r}^{\frac {1}{2}}\), D_{r} is a diagonal matrix with \(\mathbf {D}_{r}(r_{i},r_{i})=\Sigma _{j=1}^{n_{r}}\mathbf {S}_{r}(r_{i},r_{j})\).
After iteratively applying Eqs. (1011) with individual walklengths, we can obtain F_{r} and F_{d}, which separately reflect the association confidences between n_{r} drugs and n_{d} diseases from the perspective of the disease similarity network, and from the drug similarity network, along with the known drugdisease associations. To this end, we integrate them as follows:
Obviously, the larger the value of F(r_{i},d_{j}), the larger the probability that drug r_{i} associated with disease d_{j} is. In this way, we can finally identify new indications for existing drugs. The whole procedure of DRIBRW is described in Algorithm 1.
Results and discussion
Performance comparison with other methods
DRIBRW is compared with five related and recent methods (MBiRW [12], BLM [7], JI (Jaccard Index) [16], HGBI [9] and NBI [8]) on the processed dataset. MBiRW, BLM, HGBI and NBI were introduced in the Introduction, the last four methods are originally developed for predicting drugtarget interactions and can be directly adopted to predict drugdisease associations. Parameters of these comparing methods are set (or optimized) as the authors suggested (or provided) in their respective papers or codes. As to DRIBRW, Î± for random walk restart probability is set to 0.1. To reach a comprehensive evaluation, we use six widely used metrics, namely AUROC, AUPR, MacroF1, MicroF1, Precision, Recall. These metrics are also used by those comparing methods [7â€“9, 12, 16]. The formal definitions of these metrics are omitted here, but interested readers than can find the formal definitions of these metrics in these references and references therein. All these methods follow ten fold crossvalidation experimental protocol, and then report the average results and standard deviation in TableÂ 1. In addition, we also plot the receiver operating characteristic (ROC) curve and precision recall (PR) curve, and the value of area under perspective curve in Fig.Â 2.
We can easily find that DRIBRW achieves better performance than these comparing methods. Although both DRIBRW and MBiRW utilize the drug similarity network, disease similarity network and drugdisease association network to construct a heterogeneous network, and then apply birandom walks with restart to account for the structural difference of this network, DRIBRW still performs significantly better than MBiRW. That is because DRIBRW takes into account the different contributions of different nodes and applies individual walklengths for them, whereas MBiRW equally treats all nodes and applies the same walklength. In addition, DRIBRW uses the Gaussian interaction profile kernel similarity to strengthen the effect of known drugdisease associations.
HGBI also applies random walks with restart on the heterogeneous network, but it does not take into account structural difference between drug similarity network and disease similarity network. BLM tries to build a separate classifier for each drug and each drug, but it is still suffered from biased training data, since there are more negative samples than positive samples (known associations). In fact, a number of negative samples should be positive ones. For this reason, BLM has a high Precision and Recall but with a low AUPR value. JI takes into account the influence of a node in the bipartite network and uses common neighbours to predict drugdisease associations. NBI only utilizes known drugdisease associations to run a twostep diffusion model on the bipartite graph and it can not predict new associations for a drug without known associations. For these reasons, both JI and NBI are outperformed by DRIBRW.
Individual walklength analysis
To study the contribution of our proposed individual walklengths, we also test the performance of DRIBRW with fixed walklengths for all the nodes by varying walklength in the disease network and drug network from 0 to 10, respectively. Fig.Â 3 reveals the AUROC and AUPR of DRIBRW under different combined configurations of L_{r} and L_{d}. From this figure, we can clearly see that the AUROC stops increasing when L_{r} and L_{d} are larger than 2, and the AUROC and AUPR values with a fixed walklength are smaller than those of DRIBRW with individual walklengths. This comparison further corroborates the effectiveness and rationality of individual walklengths.
Drug and disease similarity analysis
As introduced in Section 5, we measure the composite inner similarity between diseases and drugs in four steps. To investigate the impact of these four steps and the contribution of Gaussian interaction kernel profile similarity, we introduce three variants (DRIBRW123, DRIBRW124, DRIBRW134) of DRIBRW. Particularly, DRIBRW123 only uses the first three steps (as done by Luo et al. [12]), or excludes the Gaussian interaction kernel profile similarity, to measure the inner similarity between diseases and between drugs. Similarly, DRIBRW134 excludes the second step without shrinking low similarity and enlarging high similarity. DRIBRW124 follows the same naming rule. The AUROC and AUPR values of DRIBRW and its variants by ten fold crossvalidations are shown in Fig.Â 4. Obviously, the AUROC and AUPR values of DRIBRW123 are lower than those of other methods, which show the contribution of Gaussian interaction profile kernel similarity for drug repositioning. Another interesting observation is that DRIBRW134 has a higher AUPR value than other variants and DRIBRW. The cause is that AUPR and AUROC measure the performance from different perspectives and under varying thresholds. The second step may wrongly shrink low similarity and enlarge high similarity, and thus compromise the performance.
Experiments on another two datasets
We collected another two datasets to further study the performance of DRIBRW. The first dataset (named â€˜Gottliebâ€™s Datasetâ€™), was obtained from [11]. This dataset contains 1933 known drugdisease associations involving 593 drugs registered in DrugBank and 313 diseases listed in the Online Mendelian Inheritance in Man (OMIM). The another dataset (â€˜Luoâ€™s Datasetâ€™) is obtained from [12], it includes 663 drugs registered in DrugBank, 409 diseases listed in OMIM database and 2352 known drugdisease associations. TableÂ 2 reports the results of 10 fold crossvalidation of DRIBRW and comparing methods on these two datasets. The experimental setups are kept the same as in previous experiments. From these tables, we can also find that DRIBRW again obtains much better performance than these comparing methods across different evaluation metrics.
Case study
To further demonstrate that the drugdisease associations predicted by DRIBRW can be confirmed by biological experiments, we apply DRIBRW to prioritize potential drugdisease pairs. Here, we use all the collected drugdisease associations as training samples, and then select the top 10 drugdisease pairs with the largest association probabilities as the predicted drugdisease associations. After that, we manually check these associations by referring to the associations stored in Comparative Toxicogenomics Database(CTD) [31]. Particularly, we use the data of chemicaldisease associations labeled with therapeutic downloaded from CTD. The label therapeutic represents a chemical that has a known or potential therapeutic role in a disease. For the predicted associations cannot find in the CTD, we further manually check them on PubMed and list the supportive PubMed IDs. We highlight the drugdisease associations supported by recent papers in PubMed but not included in CTD in boldface. The currently supported and unsupported associations are listed in TableÂ 3.
From TableÂ 3, 6 out of top 10 predicted associations are supported by associations in CTD, the other two drugdisease pairs are supported by recent papers in PubMed but not included in CTD. For instance, Labetalol is an effective agent in essential hypertension as documented in open studies and controlled studies [32]. For another instance, Greminger et al. confirmed the high efficacy of captopril in treatment of severe hypertension refractory to conventional drugs [33]. Meanwhile, ranolazine therapy is safe and well tolerated in a pilot study involving pulmonary arterial hypertension [34]. Although we can not find the direct evidence for the associations of flurandrenolide and scalp dermatoses, flurandrenolide topical is used to treat the itching, redness, dryness, crusting, scaling, inflammation, and discomfort of various skin conditions [35].
These predicted results confirm the capability of DRIBRW in identifying novel drugdisease associations with high confidence. We want to remark that the 2 unsupported associations should not be viewed as incorrect associations. As more experimental evidence becomes available, they maybe further supported.
We also report the top 10 repositioned examples made by other comparing methods, and then manually check these examples by referring to the associations stored in CTD. We further check the associations that cannot find in the CTD on PubMed and list the supportive PubMed IDs. We highlight the drugdisease associations supported by recent papers in PubMed but not included in CTD in boldface. TablesÂ 4, 5, 6, 7 and 8 list the currently supported and unsupported associations for MBiRW, BLM, JI, HGBI and NBI, respectively.
From TableÂ 4, 5 out of top 10 predicted associations are supported by associations in CTD, the other two drugdisease pairs are supported by recent papers in PubMed but not include in CTD. From TableÂ 5, we can clearly see that 1 out of top 10 predicted associations is supported by CTD and the other six associations are supported by recent papers in PubMed. From TableÂ 6, JI totally finds 6 drugdisease pairs with evidence among the top 10 predicted associations. From TableÂ 7, 5 out of top 10 predicted associations are supported by associations in CTD, the other two drugdisease pairs are supported by recent papers in PubMed but not include in CTD. From TableÂ 8, NBI can find 6 associations with evidence. In summary, DRIBRW can make more confident drugdisease repositioning than these comparing methods.
Quantified individual walk length is reasonable
The drugdisease association prediction task is frequently modeled as a link prediction problem in a heterogeneous graph [36â€“38]. The link prediction relies on calculating the similarity between nodes. The number of paths between nodes and walk lengths are regarded as effective similarity metrics in the social network and biological network [36, 39, 40]. The similarities between drugs and diseases can be measured based on the number of walks that connect drug nodes and disease nodes in the network. Integrating the number of walks and their lengths can more comprehensively quantify the potential association probability of the drugdisease pair. In addition, the contribution of different nodes in the heterogeneous network is different. In other words, the information carried by each node in the heterogeneous work is imbalanced. Therefore, it is an issue to adopt a fixed walklength for all nodes in link prediction.
In order to answer why the choice of quantified individual walk length is reasonable, we calculate the shortest path for each drug and disease node, and measure the difference between shortest path and quantified individual walk length. We use the matrix SP(r_{i},d_{j}) to represent the shortest path from the iâˆ’th drug to jâˆ’th disease, \(\mathbf {SP} \in \mathbb {R}^{(n_{r}+n_{d}) \times (n_{r}+n_{d})}\). To calculate SP, we firstly construct an adjacency matrix W:
where \(\mathbf {W}_{rr} \in \mathbb {R}^{n_{r} \times n_{r}}\) contains the shortest path between each two drug nodes, \(\mathbf {W}_{dd} \in \mathbb {R}^{n_{d} \times n_{d}}\) contains the shortest path between each two disease nodes. W_{rd} is the drugdisease association matrix and W_{dr} is the transpose of W_{rd}. Then, we adopt the Dijkstra algorithm to compute the shortest path between two nodes in matrix W. \(\phantom {\dot {i}\!}\mathbf {P}_{r}=(rp_{1},rp_{2},\ldots,rp_{i},\ldots,{rp}_{n_{r}})\) where rp_{i} represents the longest path in the shortest path between ith drug and all the diseases. \(\phantom {\dot {i}\!}\mathbf {P}_{d}=(dp_{1},dp_{2},\ldots,dp_{j},\ldots,{dp}_{n_{d}})\) where dp_{j} represents the longest path in the shortest path between jth disease and all the drugs. In other words, rp_{i} is the maximum shortest path for drug i, which can include nearly all the path information with diseases. dp_{j} is the maximum shortest path for disease j and it can approximately represent the path between disease j and all the drugs. L_{r} and L_{d} store the quantified individual walklengths of n_{r} drugs and n_{d} diseases. After that, we calculate the margin between P_{r} and L_{r} for drugs, and that between P_{d} and L_{d} for diseases. The statistical results are shown in Fig.Â 5. We can find that nearly 60% nodesâ€™ differences are no larger than one. It can explain that the quantified individual walk lengths of most nodes are inline with the shortest path between the respective nodes. However, the maximum shortest path can only partially represent the path information from a drug node to a disease node. L_{r} can give more emphasize on shorter path between diseases and drugs than maximum shortest path, and it generally has a smaller value than P_{r}. It is recognized that the shorter the distance between two nodes, the larger the similarity between them is. For these reasons, our random walk with individual walk achieves more prominent performance than random walk fixed walk length (as shown in Fig.Â 3)
We also perform the correlation analysis on drug similarity matrix S_{r} and drug shortest path matrix W_{rr}. We firstly partition S_{r} into ten subranges ((0, 0.1], (0.1, 0.2], etc.) and then partition W_{rr} into ten subranges to ensure that all the drug pairs in each subrange of S_{r} falling into the corresponding subrange of W_{rr}. Next, we calculate the average shortest path of each subrange for W_{rr}, and compute the correlation of average shortest paths and drug similarities between W_{rr} and S_{r} in each subrange. Similarly, we conduct the correlation analysis on disease similarity S_{d} and disease shortest path matrix W_{dd} in the same way and report the results in Fig.Â 6. We can clearly observe that the average shortest paths between drug pairs or disease pairs decrease as the increases of their similarities. This observation also differentiates the contribution of different walk lengths based on the assumption that nodes with shorter walk lengths contribute more to the similarity between two nodes.
Conclusion
In this paper, we proposed a computational drug repositioning approach that encodes the drug chemical structure information, disease symptom information and known drugdisease interactions information into a heterogeneous network. Our approach accounts for structural difference of subnetworks of the heterogeneous network by birandom walk, and for the contribution differences of different nodes via specifying quantified individual walklengths to them. Experimental study demonstrates that our approach performs better than other related competitive methods and the individual walk lengths contribute to an improved performance. We want to remark that our proposed approach can be easily generalized to predict links between nodes of a heterogeneous network.
Availability of data and material
The datasets used during the current study are available from the corresponding author on reasonable request.
Abbreviations
 AUPR:

The area under precisionrecall curve
 AUROC:

The area under the receiver operating characteristic curve
 CTD:

Comparative Toxicogenomics Database
 DRIBRW:

Drug repositioning approach based on individual birandom walks
 OMIM:

Online Mendelian Inheritance in Man
 PR:

precisionrecall
 ROC:

receiver operating characteristic
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Acknowledgements
The twopage short paper of this work was an oral presentation in the 14th International Symposium on Bioinformatics Research and Applications (ISBRA 2018).
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This article has been published as part of BMC Bioinformatics, Volume 20 Supplement 15, 2019: Selected articles from the 14th International Symposium on Bioinformatics Research and Applications (ISBRA18): bioinformatics. The full contents of the supplement are available at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume20supplement15
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Publication cost is funded by Natural Science Foundation of China (61741214 and 61872300), Fundamental Research Funds for the Central Universities (XDJK2019B024), Natural Science Foundation of CQ CSTC (cstc2018jcyjAX0228).
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YW and GY conceived, designed and carried out the experiments; GY and MG initialized and conceived of the whole program; YW and GY analyzed the results, drafted and finalized the manuscript; MG, YR and LJ were involved in revising the manuscript. All authors read and approved the final manuscript.
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Wang, Y., Guo, M., Ren, Y. et al. Drug repositioning based on individual birandom walks on a heterogeneous network. BMC Bioinformatics 20 (Suppl 15), 547 (2019). https://doi.org/10.1186/s1285901931176
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DOI: https://doi.org/10.1186/s1285901931176