Based on known protein–domain interactions downloaded above, we can first construct an original network PDI as follows: for any given protein node p i   and domain node   d j , if and only if there is a known interaction between them, there is an edge between p _i and   d j in PDI. Then we can further obtain an adjacency matrix   A M p d as follows: for any given protein p _i and domain   d j , if and only if there is a known interaction between p i and   d j , there is A M p d ( p i , d j ) = 1; otherwise, there is A M p d ( p i , d j )   = 0 . Due to limited known PDI, obviously, A M p d is a sparse matrix. Hence, in order to improve the density of A M p d , we will apply the collaborative filtering algorithm on A M p d according to the following steps:

Step 1: Applying the protein-based collaborative filtering algorithm on PDI as follows:

First, based on A M p d and PDI, we will construct a novel co-occurrence matrix C M P P as follows: for any two given proteins p i and p j , there is   C M P P ( p i , p j )   =   1 , if and only if there is at least one common domain node existing between them; otherwise, there is   C M P P ( p i , p j ) = 0 . Hence, a similarity matrix S M P P between protein and protein can be calculated after normalizing C M P P as follows: SMPP ( p i , p j ) = { | N ( p i ) ∩ N ( p j ) | | N ( p i ) | × | N ( p j ) | : i f   i ≠ j 0 : Otherwise (1) Here, | N ( p i ) | denotes the number of known domains associated to p i in PDI; in other words, it denotes the sum of elements equaling to one in the i t h row of A M p d . | N ( p i ) ∩ N ( p j ) | represents the number of known domains related to both p i and p j simultaneously.

Based on matrices   A M p d and S M P P , we can further obtain a novel recommendation matrix R M P D as follows: R M P D = S M P P × A M p d (2) Next, for any given protein node p i and domain node d j   in PDI, if the interaction between p i and d j is associated already, then for a protein node p k other than p i , it is no doubt that the higher the similarity between p k and p i , the more possibility that there may exist a potential association between   p k and d j . Thereafter, we can define the recommendation standard between protein p k and d j   based on the similarities between proteins as follows: S t d p k   d j = 1 N × ∑ i = 1 N R M P D ( p i , d j ) (3) Here, N denotes the number of proteins in PDI. Based on the above Eq. 3, for any given domain node d j , if there is a protein node   p k   satisfying R M P D (   p k , d j ) > S t d   p k   d j , then we will further recommend the protein p k to the domain d j . Thereafter, we will add a new association edge between p k and d j in A M p d and obtain an update protein–domain adjacency matrix U A M p d .

Step 2: Applying the domain-based collaborative filtering algorithm

Similarly, we can also obtain an original adjacency matrix A M d p and a co-occurrence matrix C M d d . Obviously, as for the matrix A M d p , there is A M d p = A M p d T . However, as for the matrix C M d d , for any two given domains d i and d j , there is   C M d d ( d i , d j ) = 1 , if and only if there is at least one common protein node existing between them; otherwise, there is   C M d d ( d i , d j ) = 0 . After normalizing C M d d , we can calculate the similarity between d i and d j as follows: SMDD ( d i   ,   d j ) = { | N ( d i ) ∩ N ( d j ) | | N ( d i ) | × | N ( d j ) | , i f   k ≠ r 0 , Otherwise , (4) where | N ( d i ) | represents the number of known proteins associated with d i in PDI, and | N ( d i ) ∩ N ( d j ) | represents the number of known proteins related to d i and d j simultaneously.

We can as well define the recommended standard and recommendation matrix as follows: RMDP = SMDD × A M d p (5) S t d d k p j = 1 M × ∑ i = 1 M RMDP ( d i , p j ) (6)

Here, M means the number of domains in PDI . In particular, if there exists a domain node d k in the i t h column of  RMDP satisfying R M D P ( d k , p j ) > S t d d k   p j , then we further recommend the protein d k to domain p j . Thereafter, we also add a new association edge between d k and p j in A M d p and obtain an update association U A M d p .

Step 3: Mutual recommendation between proteins and domains

Based on the updated matrix   U A M p d and U A M d p , the   U A M p d is N × M dimension matrix, and U A M d p is M × N matrix. By transposing the matrix A M d p , it is obvious that we can construct the mutual recommendation matrix MRM as follows: MRM ( p i   ,   d j ) = { U A M p d ( p i   ,   d j ) +   U A M d p T ( p i   ,   d j ) , otherwise 1 , i f     UA M p d ( p i   ,   d j ) = 1   a n d     UA M d p T ( p i   ,   d j ) = 1 (7)

For instance, according to Figure 1 and the given matrix   A M p d = [ 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 1 ] , we can obtain its corresponding matrices C M P P , S M P P , and   R M P D as follows: C M PP = [ 0 1 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 0 ] , SMPP= [ 0 0 .5 0 .71 0 .5 0 0 .5 0 0 0 0 0 .71 0 0 1 0 0 .5 0 0 .71 0 0 .71 0 0 0 0 .71 0 ] , RMPD= [ 0.5 1.21 0.5 0.5 0.5 0.5 0 0 0.71 1.41 0 0.71 0.5 1.21 0 0.71 0 0.71 0 0.71 ]

To be specific, as illustrated in Figure 1, if tanking the domain node d 1 as an instance, then it is obvious that there are two protein nodes p 1 and p 2 associated with d 1 from the matrix A M p d . In addition, according to Eq. 2, we can as well obtain the recommended standard R M P D   ( p 3 , d 1 ) = 0.71   > S t d p 3   d 1 = 0.44 . Hence, we will recommend the protein node p 3 to d 1 . In the same way, the protein node p 4   will be recommended to d 1 as well. On the contrary, R M P D   ( p 2 , d 2 ) = 0.5 and R M P D   ( p 5 , d 2 ) = 0.5 are less than the recommended standard   S t d p 2   d 2 = S t d p 5   d 2 = 1.01. So there is no need to recommend the protein node p 2   and p 5 to d 2 . In addition, according to a previous description, it is obvious that these novel edges between p 3 and d 1 , p 4 and d 1 , p 1 and d 3 , p 3 and d 4 will be added to the original protein–domain association matrix A M p d in the same time. Similarly, we can apply the domain-based collaborative filtering algorithm. Thereafter, we can obtain a recommendation protein–domain adjacency matrix based on PDI. Finally, as shown in Figure 2. We can get the mutual recommendation matrix MRM.

For any two given protein p i and p j , we estimate the relationship between p i and p j by applying the Gaussian kernel interaction profile (van Laarhoven et al., 2011) and further obtain an N × N dimensional weight matrix between proteins W B P based on the mutual recommendation matrix MRM .   W B P ( p i   ,   p j )    represents the relationship between protein p i and p j , and it can be defined as follows: W B P ( p i ,   p j ) = e x p ( − δ p ‖ I P ( d i ) − I P ( d j ) ‖ ) 2 (8) where δ p = δ p ′ 1 N ∑ i = 1 N ‖ I P ( d i ) ‖ 2 (9)

Here, I P ( d i ) and I P ( d j ) represents the vector at the i t h and j t h column of the mutual recommendation matrix M R M separately. δ p is an adjustment coefficient, which controls kernel bandwidth based on normalizing the new bandwidth parameter   δ p ′ .

Previous research has indicated that with similar functions, co-expressed and complex topologies are more likely to be essential proteins. Inspired by them, in this paper, we combine biological and topological features to detect potential proteins by subcellular localizations, gene expression data, and orthologous information and PPI networks.

It is obvious that the location information of a protein in a cell is an important characteristic of essential proteins. First, we analyze the 11 kinds of subcellular location relationship between the known essential proteins, and the Figure 3 statistical distribution of each subcellular location is shown in Figure 4. We can find that essential proteins are not randomly distributed in different subcellular locations, and essential proteins appear more often in the nucleus and mitochondrion, which means that proteins in the nucleus and mitochondrion are more possible to be essential proteins. What is more, from Figure 4, there are more essential proteins in the nucleus and mitochondrion and a few essential proteins in the peroxisome and extracellular, which provides us with convenience.

In order to distinguish the importance of different subcellular locations, let N s u b means the number of all subcellular localizations and   N s u b ( i ) represent the number of proteins associated with the i t h subcellular localization. Then A v e s u b denotes the average number of proteins related to each subcellular localization. The score of the i t h subcellular localization E v e s u b ( i ) can be expressed as follows: A v e s u b = ∑ i = 1 N s u b N s u b ( i ) N s u b (10) E v e s u b ( i ) = N s u b ( i ) A v e s u b (11)

Let S u b ( p k ) represent the set of subcellular localizations associated with the protein p k . Therefore, for a given protein p k , its subcellular localization score P r o s u b ( p k ) is computed as the sum of the scores of all subcellular locations where it appears. P r o S u b ( p k ) = ∑ i ε S u b ( p k ) E v e s u b ( i ) (12) Similar to describing subcellular scores, for any given protein   p k , let   P r o o r t ( p k ) mean the score of orthologous information. Hence, we can define its feature of orthology information score for p k as follows: P r o O r t ( p k ) = O r t ( p k ) m a x p i ε P P I { O r t ( p i ) } (13)

We use the Pearson correlation coefficient (Priness et al., 2007) as a similarity measure of gene expression profiles to calculate the expression intensity of two genes. P C C ( p k , p r ) = 1 n − 1 ∑ i = 1 n ( E x p ( p k , i ) − E x p ( p k ) ¯ σ ( p k ) ) ( E x p ( p r , i ) − E x p ( p r ) ¯ σ ( p r ) ) (14) Here E x p ( p k , i ) represents the expression level of p k at the i t h time node. E x p (   p k ) ¯ is the average gene expression value of protein   p k , and σ ( p k ) is the standard deviation of protein   p k . Thereafter, let   N G ( p k ) denote the set of neighbors of protein   p k . So we can compute its new functional score of protein p k as follows: P r o E x p ( p k ) = e x p ( p k ) m a x p i ε P P I { e x p ( p i ) } (15) where e x p ( p k ) = ∑ p r ε N G ( p k ) P C C ( p k , p r ) (16)

It is a fact that essential proteins are more likely products of complex functions (Dezso et al., 2003). In addition, it is obvious that triangles have stable characteristics. Inspired by this, we further utilize the major triangle topological feature calculated by the original PPI network for obtaining each protein topological feature score. Therefore, for a given protein p k , we can calculate the topological feature score as follows: P r o T r i ( p k ) = ∑ p r ε N G ( p k ) N G ( p k ) ∩ N G ( p r ) N G ( p k ) (17) Based on the above formulas for any given protein p k , we can obtain the main topological and biological feature scores.

In order to effectively solve the problem of multifeature integration, we apply an improved entropy weight method (Dastbaz et al., 2018) to automatically generate the best parameters to integrate biological features. Based on the protein characteristics we have normalized, let { B F i 1 , B F i 2 , ... B F i M } represent all features; then we can further construct an N × M dimensional matrix B F and an M × 1 dimensional matrix P M as follows: B F = [ B F 11 ⋯ B F 1 M ⋮ ⋱ ⋮ B F N 1 ⋯ B F N M ] (18) P M = [ p 1 ⋮ p M ] (19)

Next, based on our normalized biological features, we can obtain the entropy value of each feature separately as follows: e i = − 1 l n N ∑ i = 1 N B F i j . l n ( B F i j ) (20)

Therefore, for the i t h protein biological feature, we can calculate the entropy weight of each feature by the following formula: w j = ( 1 − e i ) ∑ i = 1 M ( 1 − e i ) (21)

Based on the above formula, for a given protein p k , we can further calculate its integrated biological score as follows: p r o B i o ( p k ) = ∑ k = 1 M w j B F k j (22) Finally, according to the above Eq. 18, for any given protein p k , we can further obtain its initial score as follows. p r o s c o r e ( p k ) = λ × p r o B i o ( p k ) + ( 1 − λ ) × P r o T r i ( p k ) (23)

Here, λ is a proportion parameter with a value between 0 and 1.

According to   W B P , our prediction model CFMM can apply improved PageRank to identify potential proteins. Let W P ( p k   ,   p r ) = W B P ( p k   ,   p r ) ( 1 + max ( W B P ( p k   ,   p r ) ) ) 2 , and for any two given proteins p k and p r , we can define the distribution rate possibility matrix as follows: D R P M ( p k   ,   p r ) = { W P ( p k   ,   p r ) × p r o s c o r e ( p r ) ∑ p i ε N G ( p k ) p r o s c o r e ( p i ) i f   W P ( p k   ,   p r ) ≠ 0 0 , O t h e r w i s e (24)

Based on the above distribution rate matrix DRPM, let a possibility vector p r o S c o r e ( t ) , p r o S c o r e ( t + 1 ) mean the score vector of protein at the t t h and t + 1 t h time separately; therefore, we can iteratively compute the protein ranks as follows: p r o S c o r e ( t + 1 ) = α × p r o S c o r e ( t ) × D R P M + ( 1 − α ) × p r o S c o r e ( 0 ) (25)

Here the parameter α ∈ (0, 1) in order to adjust the proportion p r o s c o r e ( t ) and initial score   p r o s c o r e ( 0 ) .

Based on the above descriptions, our prediction method CFMM can be concisely described as follows.

In order to further evaluate the performance of CFMM in this section, two different datasets, the DIP database and the Krogan database, are adopted to compare CFMM with 14 competitive detection models, which include DC (Hahn and Kern, 2005), SC (Estrada and Rodríguez-Velázquez, 2005), BC (Joy et al., 2005), EC (Bonacich, 1987), IC (Stephenson and Zelen, 1989), CC (Wuchty and Stadler, 2003), NC (J. Wang et al., 2012), ION (Peng et al., 2012), Pec (Li et al., 2012), CoEWC (Zhang et al., 2013), POEM ((Zhao et al., 2014), TEGS (Zhang W. et al., 2018), CVIM (S. Li et al., 2020), and NPRI (Z. Chen et al., 2020). For the purpose of observing the accuracy of the experiment more intuitively, we chose to use a bar graph to compare the 1, 5, 10, 15, 20, and top 25% of each method. Figure 5 shows that the comparison of the identifying results of different algorithms on the DIP and Krogan database separately. From Figure 5A, the newly put forward CFMM method detected a larger number of essential proteins in the top 1–25% compared with 14 other competitive methods. It is obvious that CFMM can reach the accuracy of 92.16, 83.14, 71.37, 63.87, 55.84, and 52.43% in the top 1, 5, 10, 15, 20, and 25% predicted candidate key proteins based on the DIP database. Among the top 25% proteins predicted by the CFMM method, there are 668 proteins correctly detected, which indicates that the CFMM method has superior advantages over other methods. From Figure 5B, we can see that CFMM can reach the accuracy of 94.59, 75.54, 70.03, 65.34, 60.08, and 54.68% in the top 1, 5, 10, 15, 20, and 25%, which are superior to all 14 advanced methods, except that in the top 10% CFMM-predicted 257 proteins, they are a little lower than NPRI. Therefore, we can make a conclusion that CFMM always obtains the better prediction accuracy from the top 1% to the top 25%.

Due to the jackknife methodology (Holman et al., 2009) that can evaluate the advantages and disadvantages of the prediction model, in this section, we will apply the jackknife method to assess the predictive effect of our proposed mode CFMM. Figures 6, 7 show the experimental comparisons between CFMM and 14 advanced competitive methods based on the first 1,000 candidate proteins. By observing Figure 6A, it is obvious that CFMM can achieve better performance than the seven network topology-based methods including DC, SC, BC, EC, IC, CC, and NC. What is more, Figure 6B shows that the performance of CFMM is better than the other seven methods that are based on the combination of biological information of proteins and PPI networks including Pec, CoEWC, POEM, ION, TEGS, CVIM, and NPRI. From Figure 7A, we can easily conclude that the CFMM is advanced than these centrality-based methods including DC, IC, EC, BC, CC, SC, and NC. Although the performance curves of CFFM and NPRI overlap partially, as the number of candidate proteins increases to 450, the predictive performance of CFMM will be significantly higher than that of NPRI. Therefore, based on the above description, we can make a conclusion that the performance of CFMM is not only superior to the first category of methods, such as DC, SC, BC, EC, IC, CC, and NC, but also better than these multiple biological data methods including Pec, CoEWC, POEM, ION, TEGS, CVIM, and NPRI.

In order to further prove the accuracy of the CFMM model, we will analyze the differences between CFMM and other models based on the top 100 predicted proteins under the DIP database and the Krogan database separately, and comparison results are shown in Tables 2, 3, respectively. Here ME denotes one of the 14 competitive methods. | CFMM ∩ ME | represents the number of essential proteins predicted by both CFMM and ME. | CFMM − ME | denotes the number of essential proteins recognized by the CFMM but not by ME, and |ME−CFMM| means the number of key proteins predicted by ME but ignored by CFMM. In addition, { CFMM − ME } represents the set of key proteins recognized by CFMM but not by ME. { ME − CFMM } means the set of essential proteins predicted by ME but not by CFMM. Hence, Tables 2, 3 show the difference between the 14 competitive methods and CFMM under the DIP and Krogan datasets separately. Figure 8 indicates that CFMM can achieve much better predictive performance than all these competing methods as a whole.

The receiver operating characteristic (ROC) curve and precision recall curve (PR) are used to scientifically prove the performance of the prediction model. The area under the curve (AUC) is used to evaluate the performance of the prediction method. The closer the AUC value is to 1, the better the prediction performance of the method. The curve can be plotted by the ratio of true positive rate (TPR) to false positive rate (FPR) according to different thresholds (Peng et al., 2020). Hence, we will further utilize the ROC curves to compare CFMM with other advanced models. Figures 9, 10 indicate that the ROC curves and PR curves of CFMM and other competitive models are based on the DIP and Krogan databases separately. It is obvious that CFMM has a higher AUC curve than other competitive models. Although we can see that the ROC curve of CFMM and the NPRI ROC curves overlap slightly, the AUC value of CFMM is higher than NPRI. Finally, in order to prove the applicability of CFMM, we will further test it in the Gavin database and compare with other methods. The experimental results are shown in Tables 4, 5.

In this section, we discuss the effect of the two self-defined parameters α and λ on the prediction results of CFMM. We set the parameter α to vary from 0.1 to 0.9, then the CFMM algorithm is ran nine times from α = 0.1 to α = 0.9 separately. Finally, the number of true essential proteins identified by CFMM based on the DIP and Krogan databases are shown in Tables 6, 7 separately. Here we select from the top 1% to the top 25% of the proteins identified by CFMM. The prediction accuracy is based on the number of essential proteins that are truly identified. It is obvious that the closer α value is to 1, the higher the prediction accuracy CFMM can achieve. So, we consider that the parameter α on all the databases is 0.9, which can achieve the best performance. When α is set to 0.9, and λ is set to 0.65, the amount of true essential protein is closest to its average level. Therefore, as a result, we will set α and λ on the DIP and Krogan databases to 0.9 and 0.65 separately, while for the Gavin database, the optimum parameters α and λ will be set to 0.9 and 0.8, respectively.