A PPI network of interest is formally denoted as a four-element tuple G = {V, A, X, Λ}, where V = {v_i}(1 ≤ i ≤ n_V) is a set of all n_V proteins, A = [A_ij] is a n_V×n_V adjacency matrix where A_ij = 1 if two proteins, i.e., v_i and v_j, interact with each other and 0 otherwise, X = {X_i}(1 ≤ i ≤ n_V) consists of the GO information of proteins in V, and Λ = {Λ_m}(1 ≤ m ≤ n_V) denotes a set of total n_Λ GO categories that are available to be associated with proteins. Obviously, A and X describe G from the perspectives of network topology and GO, respectively. In this regard, an instance of G can thus be obtained if A and X are determined.

Regarding X, each element, i.e., X_i = {x_ip}, denotes the set of GO annotations taken by v_i without considering GO categories, and the size of X_i is |X_i|. The combination of X_i and Λ preserves the necessary details to sample the GO information for each protein. Assuming that Λ_ip∈Λ is the GO category of x_ip and dom(Λ_m) is a set of possible GO annotations in Λ_m, we have x_ip∈dom(Λ_m) if Λ_ip = Λ_m. The size of dom(Λ_m) is denoted as |dom(Λ_m)|.

To indicate the functional modules of proteins, we adopt a n_V×1 vector, i.e., C = [C_i](1 ≤ i ≤ n_V, 1 ≤ C_i ≤ K), where C_i represents the functional module label of v_i. Therefore, for an arbitrary protein, i.e., v_i, its C_i is equal to k if it is in the k-th functional module.

For an arbitrary protein denoted as v_i, its functional module label C_i is chosen from a Multinomial distribution, which is defined as (1).

where α_k is the probability of a protein that is assigned to the k-th functional module and ∑k=1Kαk=1. Instead of predetermining the value of each element in α, we consider α as a random variable and sample it by using a Dirichlet distribution with a parameter ζ.

In order to completely retain the relationship between GO categories and their corresponding annotations, we sample the GO annotations of v_i with two steps. Specifically, to obtain x_ip, we first choose its GO category, i.e., Λ_ip, from a Multinomial distribution that is specific to the functional module of v_i. Hence, we have

In the above equation, θ_Ci is a n_Λ-dimensional variable randomly selected from the Dirichlet distribution with a parameter λ_Ci. As the subscript of λ_Ci, C_i indicates that the probability distribution of λ_Ci is conditioned on the functional module label of v_i.

Once the GO category of x_ip is determined, the next step is to select the annotation of x_ip from the domain of Λ_ip. Assuming that Λ_ip is actually the m−th category in Λ, i.e., Λ_m, the value of x_ip is then sampled from a Multinomial distribution defined as:

where val_mt is the t-th annotation in dom(Λ_m). Regarding the subscripts C_i and m, their combination indicates that the Multinomial distribution of x_ip is specific to the functional module of v_i and the GO category Λ_m. In other words, proteins in the same functional module share similar Multinomial distributions of GO annotations, which can differ across different GO categories or functional modules. To generate β_Cim, we also place a Dirichlet distribution over it with a prior parameter μ_Cim. The graphical presentation of generating the GO information of proteins is presented in Figure 1.

As mentioned before, we introduce A to represent the interaction relationships for all pairwise proteins in G. Hence, generating PPIs in a PPI network is identical to generate A. Following the observation that proteins in the same functional module are densely connected, the value of A_ij is dependent on a finite mixture of functional modules labels according to Stochastic BlockModel (Nowicki and Snijders, 2001).

Given two proteins, i.e., v_i and v_j, the probability of v_i interacting with v_j follows a Multinomial distribution described below.

In the above equation, the parameter ε_kl is conditioned on the functional module labels of v_i and v_j. The interaction probabilities between all pairs of functional modules are therefore parameterized by ε, which is a K×K matrix. With (4), proteins in the same functional modules present similar regularities when interacting with other proteins. Similarly, we also place a Dirichlet distribution with a prior parameter τ_k to determine ε_k. The graphical presentation of generating PPIs is presented in Figure 2.

So far, the generative process of G is completed by the above generative process that involves several latent variables α, θ, β, and ε. Regarding the values of these variables, we also define corresponding prior parameters, i.e., ζ, λ, μ, and τ, to sample them in a Bayesian manner.

According to the above generative process, a PPI network, i.e., G, is represented as a collection of proteins, PPIs and GO annotations. To indicate the functional module label of each protein, we need to compute the probability of each possible C conditioning on both A and X, and select the one with the maximum posterior probability as the optimal result. Hence, we can formulate an optimization problem as below.

To address this problem, we apply the solution developed in Hu et al. (2020). Instead of explicitly determining C, this solution yields the optimal membership matrix, i.e., α^=[α^ik] to derive Ĉ. Specifically, for v_i, its functional module label C_i is more likely to be equal to k if α^ik is larger.

To indicate to what extent two proteins are likely to interact, a scoring function is designed by taking into account their membership distributions and network paths simultaneously. The motivation of designing such a function is twofold. First of all, for two proteins, the probability of being grouped in the same functional module is larger if their membership distributions are more similar, and accordingly they are more likely to interact with each other. On the other hand, two proteins are less likely to interact if the network path connecting them is longer. Assuming that L_vivj is a set of all network paths connecting v_i and v_j in G and its size is |L_vivj|, the scoring function is defined as below.

In the above scoring function, weight(L_w) evaluates the strength of L_w in terms of providing evidence to support the interaction between v_i and v_j and its definition is given as:

where k is the value of C_i, |L_w| is the number of proteins in L_w and α^zk is the membership over the k-th function module for the z-th protein along the path L_w. Obviously, the value of weight(L_w) is determined by the likelihood of being group in the function module of v_i for the remaining proteins in L_w.

Regarding decay(L_w), the motivation of introducing this term is that it is less likely to interact with each other if two proteins are located far away from each other in a given PPI network. Hence, the definition of decay(L_w) is given by (8) where φ is the decay coefficient and usually set to be greater than or equal to 1. Since the value of weight(L_w) ranges from 0 to 1, decay(L_w) has a decay effect as an exponentiation. The longer the length of L_w is, the more obvious the decay effect of decay(L_w) has. To achieve a balance between accuracy and time, the value of |L_w| is set to be 3 in our experiments.

For each pair of testing proteins, we propose a novel prediction algorithm, namely NGPM, to calculate their interacting probability. To begin with the prediction, NGPM ranks the scores of all pairs of proteins including known PPIs and newly predicted PPIs. Since a predicted PPI is more likely to be real if it is surrounded by more already known PPIs, a sliding window is set by NGPM by selecting the upper and lower 50 pairs of proteins as a reference for the given pair of proteins. NGPM calculates the percentage of known PPIs to all pairs of proteins in this window, and then regards this percentage as the interacting probability for the given pair of testing proteins.

In the experiments, five independent PPI networks collected from different species are used, and they are denoted as Yeast-Tong (Tong et al., 2004), Yeast-Krogan (Krogan et al., 2006), Human (Rolland et al., 2014; Kovács et al., 2019), Escherichia coli (E. coli) (Gagarinova et al., 2016), and Mouse (Malty et al., 2017) respectively. The first two datasets are obtained from the species of yeast, and the Human dataset is composed of three human PPI networks, i.e., HI-II-14 (Rolland et al., 2014), HI-III (Rolland et al., 2014), and HI-tested (Kovács et al., 2019). The rest datasets are generated from other species as indicated by their names. The statistics of all these PPI networks are presented in Table 1.

In the experiments, a five-fold cross-validation has been conducted to yield convincing results and the performance of NGPM is compared with that of ASNE (Liao et al., 2018) and L3 (Kovács et al., 2019) to demonstrate its superiority in PPI prediction. When generating the negative samples, i.e., non-interacting proteins, we adopt the same strategy as L3 for conducting a fair comparison. In particular, for each PPI network, a total of 244 pairs of non-adjacent proteins are randomly selected as negative samples and 100 pairs of them should contain at least one of proteins listed in the top 500 PPIs predicted by L3.

As the most important parameter involved in NGPM, K determines the number of functional modules observed from a given PPI network. To investigate the sensitivity of NGPM to the change of K, we present the performance of NGPM by varying the value of K from 2 to 20 at a step size of 1. In doing so, we are able to determine the best value of K for each dataset.

Given different values of K, the performance of NGPM is presented in Figure 3. Fluctuations are observed for the AUC and PR curves, while the f-measure cures are more stable for all datasets except Mouse. A possible reason for that phenomenon is that f-measure is a harmonic mean of Precision and Recall. Since the increase in the score of K results in opposite changes of Precision and Recall, the fluctuation in the curve of f-measure is alleviated.

Among all kinds of curves in Figure 3, we also note that the robustness of NGPM in terms of AUC is the worst, as the AUC curves are more extensively fluctuated when compared with other curves. After investigating the experimental results, we find that the false-positive rates obtained by NGPM with different values of K are different, thus having a significant impact to the change of AUC curves. Another point worth noting is that the AUC curves are below the PR and f-measure curves for all datasets except Mouse. The reason for the unsatisfactory performance of AUC is due to the imbalance between positive and negative samples in the testing datasets.

According to Figure 3, the best values of K for Yeast-Tong, Yeast-Krogan, Human, E. coli, and Mouse are 14, 6, 10, 18, and 2, respectively. Hence, in the following experiments, we use the best performance of NGPM obtained by using these values for comparison.

During the comparison, since ASNE can use different measurements to calculate the similarity between two proteins and determine their interacting probability accordingly, two most commonly used measurements including Euclidean similarity and cosine similarity are chosen in our experiments, and they are denoted as eASNE and cASNE, respectively. The results of performance comparison are shown in Figures 4, 5 and Table 2 where Figures 4, 5 show the ROC and PR curves of L3, NGPM, and ASNE obtained in each dataset, and Table 2 records the exact scores yielded by each prediction algorithm.

When compared with ASNE, NGPM obtains a better performance on each metric across all the datasets except for Human and E. coli. On average, NGPM performs better by 6.28, 17.28, 12.08, 49.50, and 15.32% in terms of Precision, Recall, f-measure, AUC, and PR-AUC, respectively than eASNE while cASNE yields the worst performance among them. However, both NGPM and ASNE do not perform well on the Human dataset in terms of AUC. A main reason for that phenomenon is due to the serious imbalance between interacting and non-interacting proteins in the Human dataset. As mentioned before, the strategy of selecting negative samples in NGPM is as same as in L3, but it leads to the imbalance of interacting samples and non-interacting samples. Since the Human dataset is the largest one, it has more than 30,000 positive samples, while the negative sample is only 244. Thus it suffers the disadvantage of imbalance seriously and smaller AUC scores are obtained by all algorithms when compared with the other datasets.

In order to more specifically illustrate the advantage of NGPM compared to ASNE in PPI prediction, we take the prediction results of NGPM on the Human dataset as an example. In particular, the nodes in Figure 6 represent proteins, and an edge connecting two nodes represents the interaction between them. Regarding the two proteins UBE2D3 and CLNS1A, they are classified as the negative sample in the testing dataset and thus there is no edge between them. However, ASNE predicts that they can interact with a probability as high as 0.76, thus leading to a wrong conclusion. NGPM accurately predicts the true relationship between UBE2D3 and CLNS1A. In the prediction result of NGPM, the interacting probability between these two proteins is < 0.4. Hence, NGPM is believed to be more reliable than ASNE when predicting PPIs. In addition, PPIs indicated by red lines are all successfully predicted by NGPM but incorrectly predicted by ASNE. These interactions have been verified by the BioGRID database (Chatr-Aryamontri et al., 2017) and can provide help for understanding the biological processes in the cell. Among them, CLNS1A, SNRPD1, EPB41, SNRPG, SNRPD3, and LSM6 are all important components of the cytoplasm, they can form protein complexes together. It is for this reason that NGPM is able to provide a precise prediction result for these proteins. Besides, all the proteins except EPB41 can participate in the process of RNA molecular interaction. Proteins UBE2D3 and RNF115 can also add ubiquitin groups to the proteins in cells to help them form ubiquitin chains, so that they can complete the catalysis of the ubiquitin reaction due to the interaction between them.

In addition to correctly predict known PPIs, NGPM is also capable of predicting novel PPIs that are not found in the testing dataset. Since NGPM allows each protein to be associated with a membership distribution and also finds the path between two proteins, the interacting probability can be determined by NGPM for any pair of proteins in a PPI network given such information. As indicated by Figure 7, several pairs of proteins extracted from the Yeast-Tong dataset are presented. PPIs represented by the edges are novel PPIs predicted by NGPM and these interactions have been confirmed by BioGRID database (Chatr-Aryamontri et al., 2017). In this regard, the ability of NGPM in predicting novel PPIs could thus be verified.

In order to verify whether NGPM can effectively eliminate the negative impact imposed by noise data such as false positives and false negatives after combining gene ontology and network topology, we compare the performance of NGPM on five PPI network with L3. From Table 2, NGPM obtains the best Recall and f-measure scores on all datasets. Specifically, when compared with L3, the performance of NGPM is better by 174.57, 79.42, 1.68, and 1.83% in terms of Recall, f-measure, AUC, and PR-AUC, respectively, and hence NGPM can reduce the negative impact caused by the noise data for PPI prediction. However, NGPM does not achieve the best performance on Precision, there are several reasons for this phenomenon. First of all The performance of NGPM is constrained by the existence of network paths. If there is no path between two proteins, NGPM can not predict the interaction between them and hence it will consider their interacting probability as 0. In doing so, a part of PPIs in the testing dataset are able to be predicted as non-interacting protein pairs, thus increasing the false negatives in the prediction result. Secondly, when predicting the interacting probability for proteins pairs, the longest path is set to be 3 in experiments, which is constrained by the computational efficiency of NGPM. A longer path will consume more time and we may be unable to obtain the prediction result after an acceptable period. Although the longer a path is, the less impact it has on determining the interacting probability between two proteins and consequently some PPIs are falsely predicted by NGPM. In this regard, the number of false positive samples obtained by NGPM is larger than the other algorithms, thus reducing the prediction accuracy of NGPM.