Let be a graph representing a biological network. We assume that has a set of nodes, representing the entities in the network, and a set of edges, where represents the interaction (binding or regulation) between and . When the network is undirected, we assume that both and are present in the set for simplicity. For example, when represents a PPI network, corresponds to a protein, and the edge between and indicates that these proteins can bind to each other. For a pair of interacting nodes such that , we define their interaction reliability as . Similarly, let be another graph with nodes and edges, representing a different biological network. We denote the interaction reliability between two nodes and in the graph as . Finally, we denote the similarity between two nodes and in the respective networks as , which may be derived using the sequence similarity between two biological entities represented by two nodes as in our experiments.

Our goal is to find the best matching pair of paths () and () in the respective networks that maximizes a predefined pathway alignment score . In order to obtain meaningful results, the alignment score should sensibly integrate the similarity score between aligned nodes and (), the interaction reliability scores between and () and between and (), and the penalty for any gaps in the alignment.

Figure 1C illustrates an example of an alignment between two similar paths and , where belongs to and belongs to as shown in Fig. 1A. The dashed lines in Fig. 1A that connect two nodes and indicate that there exist significant similarities between the connected nodes. In the example shown in Figure 1C, the optimal alignment that maximizes the alignment score has two gaps at and . Note that “insertions” and “deletions” are relative terms, and an insertion in (e.g., ) can be viewed as a deletion in the aligned path , and similarly, an insertion in (e.g., ) can be viewed as a deletion in .

To define the alignment score , we adopt the hidden Markov model (HMM) formalism. We begin by constructing two HMMs based on the network graphs and . Let us first focus on the construction of HMM for . Each node in corresponds to a hidden state in the HMM. For convenience, we represent this hidden state using the same notation . For each edge in the graph , we add an edge from state to state in the HMM. The resulting HMM has an identical structure as the network graph . The HMM for can be constructed in a similar way. Figure 2A illustrates the HMMs that correspond to the network graphs shown in Fig. 1A. In order to find the best matching pairs of paths in the given networks, we define the concept of a “virtual” path that contains nodes, as shown in Fig. 1B. A node in the virtual path can be viewed as a symbol that is emitted by a pair of hidden states and in the respective HMMs. From this point of view, the two HMMs can be regarded as generative models that jointly produce (or “emit”) the virtual path , and the underlying state sequence for will be a pair of state sequences and in the respective HMMs. Therefore, the concept of a virtual path can naturally couple a path in with another in , providing a convenient framework for identifying conserved pathways in the original biological networks.

The described HMM-based network representation allows us to naturally integrate the interaction reliability scores and the node similarity scores into an effective probabilistic framework. We first define two mappings and , which convert the interaction reliability scores and between two nodes in and to the following transition probabilities(1)(2)between the corresponding hidden states in the constructed HMMs. The mapping is defined so that (i) for , (ii) for all , and (iii) for . Similarly, the mapping follows the same constraints: (i) for , (ii) for all , and (iii) for . To specify the emission probability of a virtual symbol at a pair of hidden states in the two HMMs, we define another mapping that converts the node similarity score to the following “pairing” probability(3)where is the pair of underlying hidden states for . The mapping is defined so that (i) for all possible pairs of , and (ii) for .

Based on the HMM framework, the problem of finding the best matching pair of paths is transformed into the problem of finding the optimal pair of state sequences in the two HMMs that jointly maximize the observation probability of the virtual path . In an ungapped pathway alignment, the underlying state pair of a virtual symbol directly corresponds to a pair of aligned nodes in the original networks. We can find the optimal pair of paths in polynomial time by using a dynamic programming algorithm defined in the following, which is conceptually similar to the Viterbi algorithm. We first define as the log-probability of the most probable pair of paths for a subsequence of length , where the underlying states for the virtual symbol are and . The log-probability can be recursively computed as follows:(4)

We repeat the above iterations until . At the end of the iterations, the maximum log-probability of the virtual path is given by:(5)where is the optimal pair of state sequences that correspond to the best matching paths in the original biological networks. Once we have computed , it is straightforward to find by tracing the recursive equations that led to the maximum log-probability . Although the above algorithm only finds the top-scoring pair of paths, we can easily extend it to find the top pairs simply by replacing the operator by an operator that finds the largest scores.

The computational complexity of the above algorithm is for finding the top pairs of matching paths, where is the length of the aligned paths that we want to find, is the number of edges in , and is the number of edges in . Note that the complexity is linear with respect to all the parameters , , , and .

The log-probability can serve as a good alignment score for the paths and that effectively combines node similarity and interaction reliability. In principle, we can also use non-stochastic emission (pairing) scores and transition scores and in the recursive equation (4), in place of the log-probabilities , , and , respectively. This will yield a non-stochastic pathway alignment score instead of an observation probability.

As we can see, the concept of the “virtual” path provides an intuitive way of coupling states in two different HMMs. In fact, by taking a closer look at the recursive equation (4), the proposed alignment algorithm can also be viewed as a Markovian walk on a product graph, whose nodes consist of all possible pairs of hidden states in the respective HMMs and the edges between these nodes are determined by the connectivity (or transition probability) between the corresponding states in the HMMs. The algorithm searches for the optimal path (or the top- paths) in the product graph that yields the highest score based on the parameters of the given HMMs.

To accommodate gaps in the aligned paths and , we modify the previous HMMs as follows. First, we add an accompanying state for every state in , and similarly, we add an accompanying state for every state in . Next, we add an outgoing edge from each state to the corresponding accompanying state. In addition to this, we also add outgoing edges from the accompanying state to all the neighboring states of the original state. To be more precise, will have an outgoing edge to every , and will have an outgoing edge to every . By varying the transition probabilities and , we can control the probabilities of having insertions and/or deletions, and thereby control the “gap penalties” in a pathway alignment. We adjust the outgoing transition probability from so that ; and for the outgoing transition probability from so that . We can also control the probabilities of having consecutive insertions or deletions by adjusting the probabilities and for making self-transitions at either or . The outgoing transition probabilities from an accompanying state are chosen so that they are proportional to and satisfy . The transition probabilities in can be chosen in a similar manner. The structures of the modified HMMs are depicted in Fig. 2B. Note that, in a gapped alignment, the matching paths (or state sequences) and will still contain nodes each, and the only difference from an ungapped alignment is that the paths may now contain one or more accompanying nodes which represent gaps. The proposed framework does not impose any restriction on the number of gaps and their locations in the pathway alignment.

In order to find the optimal pair of paths (and their alignment) that maximize the pathway alignment score, we can apply the same dynamic programming algorithm described in the previous section. The retrieved paths can contain any of the hidden states and in the modified HMMs, where we define and for notational convenience. The optimal paths is the best matching pair of paths from two networks, and they may now contain insertions and/or deletions. As before, if we want to find the top pairs instead of a single top-scoring pair, we can simply replace the operator by an operator that finds the largest scores. Note that the computational complexity of the algorithm is , which is still linear with respect to all the parameters.

It is straightforward to extend the described pairwise network alignment algorithm for aligning multiple networks. Without loss of generality, we only consider the extension to the alignment of three networks. Given three network graphs , , and , we construct the corresponding HMMs based on their structures. We again use the concept of virtual paths, and now we assume that a virtual path is jointly emitted by these three HMMs. The emission of a virtual symbol is now governed by a pairing probability of three hidden states , , and that belong to the HMMs that correspond to , , and , respectively. We can find the best matching paths based on the following recursive equation:(6)where is assumed for simplicity. We repeat the above iterations until we reach and compute the maximum log-probability as follows:(7)where corresponds to the set of best matching paths in the three networks.

It should be noted that although we fix the length of the virtual path to , we can in fact find any top-scoring alignment with a shorter length , since we store all the alignment scores for shorter alignments while running the dynamic programming algorithm. The recursive equations in (4) and (6) do not restrict multiple occurrence of the same node in the final pathway alignment. However, when it is desirable to avoid such multiple occurrence, we can easily incorporate a “look-back” step into each iteration in order to prevent adding a node that is already included in the (intermediate) alignment. As this requires tracing the intermediate optimal (or top ) alignment, the computational complexity of the recursive equations (4) and (6) with a “look-back” step will be increased in proportion to the length of the intermediate alignment.

In order to obtain more general subnetwork alignments, not just alignments of linear paths, we can combine the overlapping paths among the top retrieved pairs of paths. The edges that are already contained in the constructed subnetwork alignment (which correspond to the conserved molecular interactions in the biological networks) are then removed from the HMMs, and we run the dynamic programming algorithm again to find another subnetwork alignment that does not overlap with the retrieved subnetworks. By repeating this “search and peel-off” process, we can effectively find diverse subnetwork regions that are conserved in the given networks.

The memory complexity of the proposed algorithm is for finding the top pathway alignments for networks. Although the required amount of memory increases only linearly with respect to each parameter, it can still make the algorithm infeasible when we want to align multiple number of large networks. To overcome this problem, we may assign non-zero pairing probabilities to a set of nodes (in the respective networks) only if every pair in this set has considerable node similarity that exceeds a certain threshold. Assuming that there are sets of nodes that satisfy this condition, we only need to consider these possible node alignments, in which case the overall memory complexity reduces to . Since is often much smaller than , this scheme can save significant amount of memory, thereby making the algorithm feasible.

To illustrate the potential capability of aligning different types of molecular networks, we first tested our algorithm using two small synthetic examples, which include a pair of undirected networks and another pair of directed networks. These examples were obtained from the tutorial files in the PathBLAST plugin of software Cytopscape (version 1.1, http://www.cytoscape.org/plugins1.php) and they were used for the validation of a network alignment algorithm called MNAligner [24].

In order to validate the accuracy of our algorithm for predicting functional modules that are conserved in different organisms, we performed additional experiments using three microbial PPI networks obtained from [9]. In our experiments, we performed a pairwise alignment between the E. coli and the C. crescentus networks as well as a pairwise alignment between the E. coli and the S. typhimurium networks. We assessed the accuracy of our algorithm based on the consistency of the KEGG ortholog (KO) group annotations [29] of the aligned proteins. In order to measure the consistency of KO group annotations, we computed the specificity of the predictions based on a similar methodology that was used in [14]. To compute this measure, we first remove all the aligned protein pairs that do not have complete KO annotations, and then compute the total number of annotated protein pairs. An annotated protein pair is regarded as being correct if both proteins have the same KO group annotations, and incorrect if the annotations do not agree. The specificity is defined as the ratio of the number of “correct” protein pairs among all annotated protein pairs.

For this experiment, the parameters of the HMMs have been chosen as follows. First, the transition scores have been obtained by taking the logarithm of the protein interaction probabilities in the microbial networks, which had been assigned by the SRINI algorithm [30]. The emission scores have been computed based on the sequence similarity between the proteins and , as in the previous section, where the protein similarities have been estimated based on the BLASTP hit scores between protein pairs provided in [9].

Based on the constructed HMMs, we used our algorithm to find the top-scoring pathway alignment with gaps. At each iteration, we looked for the top aligned pair of paths, stored the alignment, and removed the interactions included in the alignment from the respective networks for the next iteration. By repeating this iteration, we found 200 high-scoring path alignments. This experiment has been repeated with varying virtual path length: , , , , and . In all our experiments, we disallowed multiple occurrence of identical protein pairs and set the gap/mismatch penalty to . For each experiment, we computed the cumulative specificity for the top alignments, which is given by(9)where is the total number of correctly aligned protein pairs in the top alignments, and is the total number of annotated protein pairs also in the top alignments. The result from the pairwise alignment of the E. coli and the C. crescentus networks is shown in Fig. 4A, and the result from the alignment of the E. coli and the S. typhimurium networks is shown in Fig. 4B. As we can see in both Fig. 4A and Fig. 4B, the cumulative specificity generally decreases when we increase the alignment length . This is expected since the algorithm tends to recruit more protein pairs in the alignment if we increase . Furthermore, generally decreases if we increase . This is natural, since alignments with lower scores correspond to less conserved pathways with larger variations. Although it is difficult to directly compare our results with those reported in [14], it is still worth to note that the cumulative specificity (for the top 200 alignments) of the proposed HMM-based algorithm is higher than the specificity of the alignment algorithm Græmlin 2.0 [14], for both pairwise network alignments. These results clearly indicate that our HMM-based algorithm can produce accurate network alignments that are biologically meaningful.

Further analysis of the predicted alignments led to a number of interesting observations. For example, the alignment of E. coli and C. crescentus networks and the alignment of E. coli and S. typhimurium networks both detected conserved DNA replication modules. The module contained components of the primosome (dnaA, gyrA, gyrB), subunits of topoisomerase IV (parC, parE), and a subunit of DNA polymerase III (dnaN). These protein families are all known to be involved in DNA replication. We also found other interesting conserved modules, which include both large and small subunits of ribosomal protein complexes (rplA, rplB, rplC, rplE, rplK, rplP; and rpsA, rpsB, rpsC, rpsE, rpsG, rpsK); DNA-directed RNA polymerase complex containing rpoA, rpoB, rpoC, and other subunits; the citrate cycle (TCA cycle) containing 2-oxoglutarate dehydrogenase E1 component (sucA, sucB) and succinyl-CoA synthetase (sucC, sucD); NADH dehydrogenase I (nuoA, nuoB, nuoC, nuoF, nuoH, nuoI, nuoL, nuoM), which is a part of the oxidative phosphorylation pathway; nitrate reductase 1 (with narG, narH, narI, and narJ); and a portion of the bacterial secretion system (with secA, secD, secY).