Pin-Align: A New Dynamic Programming Approach to Align Protein-Protein Interaction Networks

To date, few tools for aligning protein-protein interaction networks have been suggested. These tools typically find conserved interaction patterns using various local or global alignment algorithms. However, the improvement of the speed, scalability, simplification, and accuracy of network alignment tools is still the target of new researches. In this paper, we introduce Pin-Align, a new tool for local alignment of protein-protein interaction networks. Pin-Align accuracy is tested on protein interaction networks from IntAct, DIP, and the Stanford Network Database and the results are compared with other well-known algorithms. It is shown that Pin-Align has higher sensitivity and specificity in terms of KEGG Ortholog groups.


Introduction
In the last decade, high throughput techniques in experimental biology produced a large amount of biological data which usually can be modeled by large networks such as metabolic networks, gene regulatory networks, and proteinprotein interaction (PPI) networks [1]. Analogous to biological sequence comparison [2][3][4], comparing large biological networks can improve our biological understanding of them. Comparison of PPI networks is a well-studied area due to the important roles of PPI networks. The main idea behind comparison of PPI networks is to find evolutionary conserved interaction modules which describes functional relevance. Exact comparison of large PPI networks is a NP-hard problem and there is no polynomial time algorithm for it. The NPhardness of this problem is based on the fact that this problem can be reduced to the subgraph isomorphism problem. In this way, the exact comparison of PPI networks is computationally infeasible, and PPI networks comparison is often addressed by heuristic methods [5][6][7].
Several algorithms have been proposed for biological network alignment. One of the first proposed network alignment algorithms is Path-Blast [8]. This algorithm finds highscore conserved pathways. After that, Sharan et al. designed Network-Blast [9] in order to identify conserved protein complexes in multiple species. Koyuturk et al. devised an evolution-based scoring scheme to detect conserved clusters, called MaWISh (maximum weight induced subgraph) [10]. Flannick et al. proposed Graemlin as the first method to detect conserved subnetworks of arbitrary structures with progressive alignment method [11]. All these mentioned algorithms are called local alignment algorithms owing to the fact that they started to find subnetworks such as pathways and protein complexes and expanded their search results to obtain the feasible alignment. In local network alignment, each protein may be mapped to several proteins. On the other hand, global network alignment algorithm is defined to detect the best overall mapping across all parts of the input networks. IsoRank is the first global network alignment algorithm aiming to maximize the overall match between two networks [12]. Flannick et al. extended Graemlin to global network alignment by using a training set of known network alignments to learn parameters for the scoring function. This novel algorithm is called Graemlin 2.0 and is claimed to have linear time complexity with the number of PPIs [13]. The next version of IsoRank, IsoRankN [14], uses Nibble-Page-Rank algorithm [15] to align input networks locally and globally. Tian and Samatova introduced a connected-components based algorithm, called HopeMap, for pairwise network alignment with the focus on fast identification of maximal conserved patterns [16]. After that, GRAAL [6] and H-GRAAL [1] are presented as the global network alignment algorithms based on seed-extend approaches and Hungarian algorithm, respectively. Both of GRAAL and H-GRAAL algorithms are capable of aligning any kind of network owing to the fact that they are specifically designed based on network structure and do not use sequence similarity data.
Observing that the size of true orthologs across species is small comparing to the total number of proteins in all species, Tian and Samatova presented a different approach based on a precompiled list of homologs identified by KO terms [17]. In this paper, Pin-Align (protein interaction network alignment) algorithm is introduced as a novel local network alignment algorithm for aligning two protein-protein interaction networks. In Pin-Align, Hub Clustering and hierarchical clustering algorithms are applied in the heuristic and dynamic programming phases in order to reduce time and space complexity of the problem. Two different types of scoring, Node-Scoring and Structural-Scoring, are used to find the best local alignments. Node-Scoring is simply based on BLAST bit score [18] and Structural-Scoring is based on the importance of Bridges (Bridges are considerable edges, which take part in most of the paths and consequently conserved paths in the networks) in biological networks.

Materials and Methods
Protein-protein interaction (PPI) network is defined as the set of relationships among proteins. Here, a PPI network is modeled by an undirected and weighted graph = ( , ), where nodes correspond to proteins and each weighted edge specifies the probability that two proteins interact. For two graphs 1 = ( 1 , 1 ) and 2 = ( 2 , 2 ) (such as two sample networks and in Figure 1), the goal of network alignment is to identify possible mappings which map vertices of 1 into vertices of 2 .
In addition, for each mapping, the corresponding set of conserved edges is also identified. Mappings may be partial; that is, they do not need to be defined for all the nodes of the graphs. Each mapping implies a common subgraph between the two graphs. When protein 1 from graph 1 is mapped to protein 2 from graph 2 , then 1 and 2 refer to the same node in the common subgraph; the edges in the common subgraph correspond to the conserved edges [23].
Pin-Align algorithm is a local network alignment algorithm, mainly designed based on the dynamic programming approach. To explain Pin-Align in more detail, first a formal definition of local network alignment is presented. Given two graphs 1 ( 1 , 1 ) and 2 ( 2 , 2 ), local alignment LA 1 , 2 (Sg 1 , Sg 2 , ) is a triplet where Sg 1 and Sg 2 are the subgraphs of 1 and 2 , respectively, and is defined as the mapping which aligns vertices of Sg 1 into vertices of Sg 2 . Two sample local alignments of networks and of Figure 1 are shown in Figure 2(a).
Following, the best local network alignment (BLA) is defined as a function to obtain the best local alignment end for (12) end for (13) end for (14) end for (15)  for each subgraph of 1 . The best local alignment of Sg 1 as a subgraph of 1 with a subgraph of 2 is shown by BLA 1 , 2 (Sg 1 ), which represents a local alignment with the highest score among all possible alignments between Sg 1 and all possible subgraphs of 2 that can be mapped to Sg 1 . Now, a naive dynamic programming algorithm for network alignment is demonstrated in the next subsection, and later Pin-Align algorithm is presented.

Dynamic Programming Algorithm to Align PPI Networks.
In alignment of two weighted graphs 1 and 2 through dynamic programming technique, we generate all LAs of size one (subgraphs with a single vertex) in the first step. For obtaining LAs of larger subgraphs of 1 , we merge all possible pairs of LA of smaller subgraphs. In other words, for each subgraph of 1 such as Sg 1 , there are several pairs of distinct subgraphs (subgraphs with no common vertices) such as Figure 2 demonstrated the merge process of two sample local alignments of networks and of Figure 1. We produce all possible local alignments for Sg 1 such as LA 1 , 2 (Sg 1 , Sg 2 , ) by merging every possible local alignments of 1 such as LA 1 , 2 ( 1 , 1 , ) with every possible local alignment of 2 such as LA 1 , 2 ( 2 , 2 , ). This process is done iteratively, and finally all local alignments of 1 including BLA 1 , 2 ( 1 ) are obtained. Inspired by dynamic programming technique, we propose Simple-Align algorithm in Algorithm 1. This algorithm can find all BLAs; however by considering time and memory complexity, this algorithm is not feasible. To overcome this, we use several heuristics to reduce memory and time complexity and propose a new local alignment algorithm, Pin-Align, in next section.

Pin-Align Algorithm.
As explained, Pin-Align uses dynamic programming approach to solve the local network alignment problem and overcome the deficiencies of dynamic programming such as time and space complexity by applying heuristic approach. The steps of the Pin-Align algorithm are summarized as follows.
These clusters are dense subgraphs of 1 . The partitioning is done by using a new clustering algorithm named Hub Clustering. Clusters obtained in this step are named Hub-Clusters.
(a) create local alignments of size one with BLAST bit score greater than 0, between vertices of and vertices of 2 ; local alignments obtained by this approach are named  Flowchart of Pin-Align algorithm is presented in Figure 3 and its details are explained by an example on two networks and given in Figure 1.
In Step 1, in order to reduce the search space of the problem, input graph 1 = ( 1 , 1 ) is clustered into some dense subgraphs and Step 2 of the algorithm runs for each dense subgraph separately. Generally PPI networks are sparse graphs, but they have dense regions containing high degree vertices named hubs. These regions usually contain protein complexes; therefore conserved modules among different PPIs usually exist in these regions.
To detect dense subgraphs in the graph 1 , a novel clustering algorithm, called Hub Clustering, is proposed. The importance of hub nodes and density of subgraphs are two major criteria for this clustering method. First, a portion of highly connected vertices are selected as hub nodes. The vertices degree of these nodes is greater than 95% of other nodes. The hub nodes are considered as the center vertices of dense regions of graph 1 and are supposed to be initial clusters. For other nodes of 1 , each node is joint to the nearest cluster (the cluster that contains its nearest hub). For this aim, the length of each edge is defined as the negative logarithm of its weight (from adjacency matrix), and then Dijkstra shortest path algorithm [24] is used to find the nearest cluster. The obtained clusters are named Hub-Clusters. In Step 2, each of Hub-Clusters of 1 (obtained in Step 1) is aligned to 2 . As mentioned, Step 2(a) creates Candidates of size one based on BLAST bit score of vertices of two graphs 1 and 2 (BLAST bit score is a normalized score which is calculated by basic local alignment search tool, based on sequence similarity [18]). Figure 4(a) shows BLAST bit scores of proteins of networks and .
After that, all Candidates of each vertex of 1 are collected into a Candidate-Collection with size one as a seed in Step 2(b). In other words, Candidate-Collection of each vertex of 1 such as Sg 1 contains all Candidates for Sg 1 from 2 which can be aligned with Sg 1 . For example, Candidates of networks and of Figure 1 are shown in Figure 4(b) and those Candidate-Collections are shown in Figure 4(c). In Step 2(c), two seeds (i.e., Candidate-Collections CC(Sg ) and CC(Sg )) are merged in order to produce a   new seed (new-CC) where its size is sum of sizes of CC(Sg ) and CC(Sg ). Figure 5 illustrates two Candidate-Collections, - The important problem in this step is discovering the feasible combining pattern to find pairs of Candidate-Collections where merging them generates high score Candidate-Collections. In order to find high score Candidates and avoid complex searches, combining pattern for merging Candidate-Collections should produce Candidates with more conserved edges. Hence we use hierarchical clustering as combining patterns [25]. Hierarchical clusters are generally constructed by generating a sequence of partitions where each subcluster belongs to one supercluster in its entity.
In hierarchical clustering we should determine how the distance between two clusters is computed. If = ( , ) and = ( , ) are subgraphs of 1 = ( 1 , 1 ), the distance between two Candidate-Collections CC( ) and CC( ) is computed, using the following formula: In this equation, V, is equal to one if there is an edge between vertices and V; otherwise it is zero, and V, is the negative logarithm of probability of interaction between two proteins V and in the graph 1 . We use the hierarchical clustering obtained based on formula, as the combining Node-Score. Node-score of each Candidate = LA 1 , 2 (Sg 1 , Sg 2 , ) is obtained by using the following equation: In this equation, is an iterator on vertices of Sg 1 and Sg 1 is the th vertex of Sg 1 ; MV(V, ) indicates a vertex of Sg 2 mapped into V ∈ Sg 1 by the mapping .
Structural-Score. Due to the existence of hubs and small words in biological networks, certainly the network contains bridges. Bridges are considerable edges, which take part in most of the paths (specifically conserved paths) in the networks. Because of the importance of bridges, if Sg 1 = ( 1 , 1 ) is a subgraph of 1 = ( 1 , 1 ) and Sg 2 = ( 2 , 2 ) is a subgraph of 2 = ( 2 , 2 ), the structural scoring function for each Candidate = LA 1 , 2 (Sg 1 , Sg 2 , ) is defined as follows: In this equation, ( ) shows the number of conserved edges at vertex in the subgraph Sg 1 ; is a constant value between 0 and 1 and shows the importance of adjacent vertices degrees; ( , ) is the probability of interaction between protein and protein in the network G.
As mentioned before, to control the exponential growth of the cardinality of Candidate-Collections in combining process, we use a multiobjective optimization method on node score and structural score, as optimization objectives. In this way, after creating each Candidate-Collection among all Candidates obtained in that, we select only a limited number of Candidates (based on upon scoring functions) and discard others. After creating all Final-Candidate-Collections, in Step 3 of Pin-Align Algorithm, an Initial Similarity Bipartite Graph ISBG( 1 , 2 ) = ( 1 , 2 , ) is created which consists of vertices of 1 = ( 1 , 1 ) as its first part and vertices of 2 = ( 2 , 2 ) as its second part, and is a set of weighted edges between 1 and 2 . The weight of each edge represents the similarity between its incident vertices. Let (V , ) be a set of all Final-Candidates which maps V into . In ISBG( 1 , 2 ), where Node Sim (V , ) is the similarity between two proteins V and and it is calculated by using the following formula: Neighborhoods Sim (V , ) is the similarity between conserved neighborhoods of V and and is calculated as follows: BLASTBitScore (V , ) .  Figure 1 is shown in Figure 6(a). After creating the Initial Similarity Bipartite Graph (ISBG), we can find just a single mapping for each vertex by finding maximum matching in this graph. The maximum matching can be found by applying the Hungarian algorithm (with polynomial time complexity) on the ISBG [27].
As mentioned, this approach can find just a single mapping for each vertex, although it cannot support some evolutionary functions such as duplication and diversion. To support duplication and diversion, we should consider matching of a single vertex from one part into two vertices of other parts in the ISBG. This type of matching is named DMatch. In other words, DMatchs are paths with length 2 such as (V , V , V ) in the ISBG which aligns V from one graph into both V and V from another graph. Here the problem is to find a set of Matchs and DMatchs from the ISBG where sum of their weights is maximum. This problem is named Maximum Weighted DMatching. To decrease the size of the solution space of this problem, we decrease the size of edge set of the ISBG by deleting some edges. The above procedure is performed by Algorithms 2 and 3.
Algorithm 2 gives the ISBG as input and creates the Final Similarity Bipartite Graph. In Steps 1-4 of Algorithm 2 all vertices of one part of the ISBG are duplicated and each duplicated vertex is connected to all vertices conjunct to the main vertex with the same weight as shown in Figure 6(b); then the Hungarian algorithm is performed on this graph in Step 5 of Algorithm 2. Obtained bipartite graph is named 1 . The result of these steps for ISBG of Figure 6(a) is shown in Figure 6(c). All duplicated vertices of 1 are merged together in Step 6 of Algorithm 2. In Steps 7-12 of Algorithm 2, the same process is implemented on the other part of the obtained ISBG and the matching 2 is obtained. Figures 7(a) and 7(b) illustrate these steps for ISBG of Figure 6

Input: Sim Graph
end if (6) end for (7) ℎ = (8) 1 = first part of (9) 2 = second part of (10) for all vertex V ∈ 1 do (11) 1 = first edge connected to V (12)  (1) If the degree of the vertex V is one and degree of adjacent vertex is also one, these two vertices are assigned to each other.
(2) If the degree of the vertex V is two and V is adjacent to V and V , the path (V , V, V ) is created as a potential Dmatch.
(3) If the degree of the vertex V is three and V is adjacent to V , V , and V , potential Dmatchs (V , V, V ), (V , V, V ), and (V , V, V ) are created.
The weight of a given DMatch can be calculated by the following formula: After creating all potential DMatchs, we sort them based on their weights; then in a successive rounds we choose the highest weight DMatchs (V , V, V ) and delete all other edges connected to V , V, or V . By this way all assignments are found and finally the local alignment of two graphs is obtained in Steps 24-27 of Algorithm 3. Therefore the output of this algorithm is the final alignment between two given graphs 1 and 2 . The final alignment between networks and in Figure 1 is shown in Figure 8.

Results
In this section, we intend to compare our algorithm to other well-known local alignment algorithms (NetworkBLAST [9], MaWISh [10], Graemlin [11], and Graemlin 2.0 [13]), in a way that Graemlin 2.0 compares its results, because it is the only method that compares results of all local alignment tools against KEGG Ortholog (KO) groups [22] on different data sets. The data sets contain the PPI networks data from SNDB [21] for the organism pairs (E. coli, S. typhimurium) and (E. coli, C. crescentus), PPI networks data from DIP [20] for the organism pair (human, mouse), and data from IntAct [19] database for the organism pair (yeast, fly). The comparison is proceeded based on specificity and sensitivity in terms  of KO groups introduced in Graemlin 2.0 [13]. In shortas Graemlin 2.0 defined-to uniquely specify an alignment, the mapping should be transitive; that is, if protein is aligned with proteins and , then protein B must also be aligned with protein . Mathematically, this means that the mapping is an equivalency relation, so groups of aligned proteins are referred to as equivalence classes. An equivalent class is defined as correct if all protein members in the class are in the same KO group. To measure specificity, Graemlin 2.0 computed two metrics: (1) the fraction of equivalence classes that were correct ( eq ), (2) the fraction of nodes that were in correct equivalence classes ( node ), and to measure sensitivity, it computed two metrics: (1) the total number of nodes that were in correct equivalence classes ( or ), (2) the number of equivalence classes that contained 2 species (Tot).
As we described in Step 1 of the Pin-Align algorithm, Pin-Align chooses some vertices as hubs whose degrees are greater than about 95% of total vertices. For networks S. typhimurium, C. crescentus, human, and yeast, Pin-Align chooses vertices as hubs in a way that their degrees are greater than 264, 126, 16, and 23, respectively. The reason why these degrees are greater than 95 percent of total vertices degrees is clear from cumulative frequency of their vertices degrees which is shown in charts of Figure 9. For example, in the PPI network of S. typhimurium the degrees of about 95% of vertices are less than 264, so in this network we choose vertices with degrees greater than 264 as hubs. Table 1 shows Pin-Align results with different hub degrees. Column two of this table (percentage column) contains different percentages which are used for hub clustering step (Step 1).
The results of the algorithms are demonstrated in Tables 2 and 3. In these tables NB stands for NetworkBlast, MW for MaWISh, Gr for Graemlin, and Gr 2.0 for Graemlin 2.0. They show that Pin-Align is the most specific and sensitive aligner in comparison with other alignment tools like NetworkBlast, MaWISh, Graemlin, and Graemlin 2.0. Because Graemlin 2.0 Table 3: PPI networks from DIP and IntAct for the organism pairs (human, mouse) and (yeast, fly).