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PrimAlign: PageRank-inspired Markovian
alignment for large biological networks
Karel Kalecky
1
and Young-Rae Cho
2,
*
1
Institute of Biomedical Studies, Baylor University, Waco, TX 76712, USA and
2
Department of Computer Science,
Baylor University, Waco, TX 76798, USA
*To whom correspondence should be addressed.
Abstract
Motivation: Cross-species analysis of large-scale protein–protein interaction (PPI) networks has
played a significant role in understanding the principles deriving evolution of cellular organizations
and functions. Recently, network alignment algorithms have been proposed to predict conserved
interactions and functions of proteins. These approaches are based on the notion that orthologous
proteins across species are sequentially similar and that topology of PPIs between orthologs is
often conserved. However, high accuracy and scalability of network alignment are still a challenge.
Results: We propose a novel pairwise global network alignment algorithm, called PrimAlign, which
is modeled as a Markov chain and iteratively transited until convergence. The proposed algorithm
also incorporates the principles of PageRank. This approach is evaluated on tasks with human,
yeast and fruit fly PPI networks. The experimental results demonstrate that PrimAlign outperforms
several prevalent methods with statistically significant differences in multiple evaluation measures.
PrimAlign, which is multi-platform, achieves superior performance in runtime with its linear
asymptotic time complexity. Further evaluation is done with synthetic networks and results sug-
gest that popular topological measures do not reflect real precision of alignments.
Availability and implementation: The source code is available at http://web.ecs.baylor.edu/faculty/
cho/PrimAlign.
Contact: young-rae_cho@baylor.edu
Supplementary information: Supplementary data are available at Bioinformatics online.
1 Background
Recent high-throughput techniques have been exploring protein
functions and interactions with other proteins. Apart from experi-
mental studies, computational analyses over existing data are also
performed, as they are considerably faster, less expensive and their
predictions of interactions and functions can substantially expedite
new discoveries. These works have made genome-wide protein–
protein interaction (PPI) data publicly available, collectively referred
to as Interactome. (Koh et al., 2012;Rolland et al., 2014).
From the standpoint of comparative Interactomics, cross-species
comparison of the link patterns in PPI networks have played a sig-
nificant role in this field, since it increases our understanding of
principles deriving evolution of cellular organizations and functions
(Sharan et al., 2005). This type of computational analysis is called
network alignment. It is based on a formal view of PPI networks as
graphs, where proteins are represented as nodes and their interac-
tions as edges. PPI networks of two species are aligned together in
the sense that proteins with the identical function are mapped to
each other. This is possible, since many genes and proteins are
conserved in similar forms across different species; they are called
orthologs. Based on the alignment results, further topological and
functional analyses can be performed. Interactions in one network
can point towards possible interactions in the other network.
Similarly, functions of a protein in one network can predict func-
tions of another protein aligned to it from the other network if they
are true orthologs.
Various network alignment algorithms have been proposed over
the last decade so as to predict conserved interactions and functions
of proteins. Network alignment algorithms are divided into two
groups: global network alignment and local network alignment. The
former deals with aligning entire networks and aims at finding the
maximal set of mapped node pairs. On the other hand, the latter
searches for a set of sub-structures that represent conserved func-
tional components. Sometimes, this distinction is being simplified to
many-to-many mappings for local aligners and one-to-one mappings
with pairing all nodes from the smaller network for global aligners.
We follow the former distinction based on production of conserved
clusters, as we have observed that some global aligners produce
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Bioinformatics, 34, 2018, i537–i546
doi: 10.1093/bioinformatics/bty288
ISMB 2018
many-to-many mappings and do not connect all proteins from the
smaller network. Earlier studies handled local alignment for aligning
small networks whereas most recent studies have proposed global
alignment algorithms for large-scale PPI networks.
One of the earliest local alignment tools was PathBLAST (Kelley
et al., 2004) which discovers conserved pathways by pairing interac-
tions between orthologous proteins. It takes a query pathway and
aligns it to a PPI network, outputting all matching paths from the
network which achieve a certain threshold score. The score of each
path is based on the BLAST (Altschul et al., 1990) e-value of each
aligned protein pair, as well as the ‘probability of a real interaction’
between proteins along the path, defined as the number of studies
which confirm each interaction. NetworkBLAST (Kalaev et al.,
2008), as an upgraded version of PathBLAST, allows for performing
alignment between multiple networks to identify two types of shared
sub-graphs: linear paths of interacting proteins (i.e. signaling path-
ways) and clusters of densely interacting proteins (i.e. protein com-
plexes). It searches for highly similar sub-networks and extends
them in a greedy manner. MaWISh (Koyuturk et al., 2006)isa
graph-theoretic optimization model to solve the maximum weight
induced sub-graph problem. It iteratively searches for a match, mis-
match and duplication of interactions between two PPI networks to
discover highly conserved groups of interactions, inspired by the du-
plication and divergence model for PPI network evolution. Recently,
PINALOG (Phan and Sternberg, 2012), AlignNemo (Ciriello et al.,
2012) and AlignMCL (Mina and Guzzi, 2014) have been introduced
as local aligners.
As for global alignment algorithms, IsoRank (Singh et al., 2008)
is the first algorithm that applies the concept of PageRank to net-
work alignment. It computes a score for each node based on the
principle that neighboring nodes of the nodes aligned to each other
should also be aligned to each other in the other network. It com-
putes a steady-state distribution combined with a personalized
PageRank vector. The upgraded version, IsoRankN (Liao et al.,
2009), uses a spectral clustering to efficiently produce a multiple
alignment, leaving the previous version obsolete. SMETANA
(Sahraeian and Yoon, 2013) and CUFID (Jeong et al., 2016) per-
form a Markov random walk in the joined network to compute a
steady-state distribution. Additional probabilistic consistency trans-
formations are executed on the results. In addition, CUFID calcu-
lates steady-state flow through the edges and applies a bipartite
matching to obtain one-to-one alignment. In contrast, SMETANA
allows for many-to-many alignment where one protein from one
network can be aligned to multiple proteins from the other network.
Most of the global alignment algorithms maximize alignment
score which is computed by a combination of sequence similarity
and topological similarity for protein pairs from two (or more) PPI
networks. Recently proposed global aligners include MI-GRAAL
(Kuchaiev and Pr
zulj, 2011), L-GRAAL (Malod-Dognin and Pr
zulj,
2015), SPINAL (Alada
g and Erten, 2013), NETAL (Neyshabur
et al., 2013), NetCoffee (Hu et al., 2014), HubAlign (Hashemifar
and Xu, 2014), MAGNA (Saraph and Milenkovi
c, 2014),
MAGNAþþ (Vijayan et al., 2015), WAVE (Sun et al., 2015) and
SANA (Mamano and Hayes, 2017).
In this paper, we propose a new global network alignment method
called PrimAlign––PageRank-Inspired Markovian Alignment. This al-
gorithm is built upon the idea of modeling the networks as a Markov
chain that is iteratively transited until convergence, combined with
the basic principles of the original PageRank algorithm and sparse
computations. The multi-platform source code in C# is provided. The
method is compared with several previous network alignment
algorithms introduced above while performing pairwise alignment of
human, yeast and fruit fly networks. The proposed method performs
superiorly to the other algorithms with respect to the alignment qual-
ity as well as computation time. Additional evaluation is performed
with 30 synthetic networks from the popular set, NAPAbench
(Sahraeian and Yoon, 2012).
2 Materials and methods
2.1 Markov chains
In computational modeling, Markov chains are well-established
models describing discrete sequences of stochastic processes, in
which the following state depends only on the current state. Markov
chains consist of possible states, in which the system under consider-
ation can be found and probabilities of transition from each state to
each other state when performing a step in the process sequence. As
the model is stochastic, the overall state of the system at each step is
described as a probability distribution over all possible states,
expressing the probability of the system being currently in each par-
ticular state.
Since the model topology is fixed and transition probabilities are
constant, a Markov chain can be defined by a constant transition
matrix T, in which the value at row iand column jrepresents the
probability of transition from state Sito state Sjat step twhen being
in state Si:
Ti:j½¼PS
tþ1ðÞ
jjStðÞ
i
The state distribution at step tcan be represented as a vector p, where
the value at index irepresents the probability of being in state Si:
pðtÞ½i¼PS
tðÞ
i
This formula allows for straightforward computation of subsequent
states:
pðtþ1Þ¼pðtÞT
pðtþ2Þ¼pðtþ1ÞT¼pðtÞTT
In Markov chain models, the usual task is to analyze the long-term
state distribution p, which can be stationary:
p¼lim
t!1 pðtÞ
p¼pT
However, reaching the stationary distribution is not guaranteed un-
less the chain is ergodic, which requires the transition matrix to be
primitive.
2.2 PageRank
The original PageRank algorithm (Langville and Meyer, 2006)is
based on representation of web pages as a Markov chain. The pages
themselves are states while the links connecting them are possible
transitions with non-zero probabilities simulating random browsing.
The task is to find the stationary distribution. Due to the model top-
ology, web pages with multiple incoming links or with links coming
from important pages will be also important, i.e. they achieve a rela-
tively high probability in the stationary distribution. To overcome
the convergence problems, two modifications are incorporated into
the transition matrix: First, states with no transition (pages with no
links) are assigned with transitions to all other states with uniform
i538 K.Kalecky and Y.-R.Cho
probability. This step turns the transition matrix into a true right
stochastic matrix. Second, a damping factor ais introduced to simu-
late a chance of sudden ‘teleportation’ to a random state, which can
occur with the probability of (1–a). This change converts the transi-
tion matrix into a primitive stochastic matrix and its convergence
during the iterative traversal is guaranteed. The modifications are
not performed over the transition matrix directly, but rather the
transition step is adjusted to:
ptemp ¼apðtÞTþqe
n
þ1aðÞ
eTe
n
pðtþ1Þ¼ptemp
kptempk1
where: a¼damping factor
q¼column vector (length n), for each row of T:
1 if the row is all-zero, 0 otherwise.
e¼row vector of 1 s (length n).
T¼transition matrix (size nn).
n¼number of states.
pðtÞ¼state probability distribution vector at step t(length n).
While qe=nrepresents the first modification, (1 aÞeTe=nintrodu-
ces the teleportation effect. The temporary variable ptemp is normal-
ized by its L1-norm, so that the result is a valid probability
distribution vector.
The smaller the damping factor is and the larger the probability
of the random teleportation is, the more affected the results are and
the probability distribution is smoothened although convergence is
faster. Nevertheless, every damping factors smaller than 1 guarantee
the convergence.
2.3 PrimAlign
Our new method searches for stationary-distributed transition probabil-
ities between two joined PPI networks forming a Markov chain with
PageRank-inspired modifications. While the idea of Markov random
walk was previously used in CUFID and a personalized PageRank vec-
tor was used in IsoRank, the proposed PrimAlign algorithm is built
upon both the Markovian representation and PageRank modifications
with sparse computations, nicely integrating similarity weights with the
network topology and guaranteeing the convergence and achieving lin-
ear time complexity (linear with respect to the number of edges on in-
put), which is the theoretical minimum for this task. Supplementary
Figure S1 shows individual steps in the data flow of PrimAlign.
Input: Three files are expected on input. The first two represent
PPI networks which are to be aligned. Any edges can be weighted to
specify the interaction confidence. The third file lists sequence simi-
larities of inter-network protein pairs, all of which are treated as
candidate orthologs. The recommended sequence similarity score is
either BLAST bit-score or -log of BLAST e-value. The fourth param-
eter is the output file path. Optionally, a threshold for selecting
orthologs can be provided, otherwise the program uses a default
threshold (0.75). The exact specification of input and all file formats
are detailed in Supplementary Text S1.
Edge reweighting: The sequence similarity scores are cubed.
Thanks to the normalization in the next phase, this exponentiation
amplifies the ratios between individual scores (large scores become
even larger and small scores become even smaller), while they still
sum up to 1. Different exponents were explored, and the effect is vis-
ible as long as any exponentiation occurs, so the exact exponent
could have been set differently with a little change in results
(Supplementary Fig. S2). The motivation behind this step is the
assumption that differences in sequence similarity are more relevant
than differences in network topology and protein interaction
weights, so the differences in sequence similarity are amplified.
Building transition matrix: The transition matrix is first built as
a symmetric square matrix with one row and one column for each
protein detected across the input files. The number at row rand col-
umn crepresents a similarity weight between r-th and c-th protein
or zero if the protein pair is not defined in any input file. This weight
is either PPI similarity if the proteins come from the same network
(or 1, respectively, if PPI weights are not specified in the input file),
or reweighted sequence similarity if the proteins come from different
networks. Then, the matrix is normalized to form a valid transition
matrix by scaling each row to sum up to 1. If a protein can transit
both within its network and into the other network, its weights are
scaled so that the total probability of transiting within its network is
the same as the total probability of transiting into the other network.
Alternatively put, we can imagine the whole matrix as a compos-
ition of four partial transition matrices depending on whether the
source and destination of the transition is network Aor B:
T¼TA!ATA!B
TB!ATB!B
"#
where: TA!A;TB!B¼partial matrices with same-network transi-
tions built from PPI network weights.
TA!B;TB!A¼partial matrices with inter-network transi-
tions built from sequence similarity weights.
In this view, the normalization can be done as row normalization of
the partial matrices with subsequent row normalization of the whole
matrix.
PageRank-inspired stationary distribution computation: Starting
with a uniformly-distributed probability vector over the proteins, the
transition matrix is repeatedly traversed and the probability vector is
updated. The iteration ends when a stationary distribution is reached
(1e–5 in L2-norm change of the vector p) or after a maximum number
of iterations (200, but in our experiments it typically converges within
20 iterations). The algorithm is inspired with PageRank by including
its two principles mentioned earlier: Proteins with no transitions
(which could occur when a protein is listed on input, but all its weights
and scores are 0) simulates transitions to all other proteins with uni-
form probability, and the damping factor ais included to simulate ran-
dom ‘teleportation’ from each protein. This change converts the
transition matrix into a primitive stochastic matrix and its convergence
during the iterative traversal is guaranteed. awas set to 0.85 as in the
original PageRank algorithm (Langville and Meyer, 2006). This par-
ameter seems to be robust as well (Supplementary Fig. S2).
As the number of proteins in both species can be relatively large,
resulting in matrix Mwith around 1 000 000 000 elements, and be-
cause of repeated vector-matrix multiplications, it is crucial to repre-
sent the matrices as sparse objects. Otherwise, the computation load
would be unfeasible for computation with real-world PPI networks
in a genomic scale. The formula stated above is in a form allowing
simple reading of the incorporated PageRank modifications, but
operates with full non-sparse matrices and the performance would
be degraded greatly. Therefore, the real equation used is based on a
form converted by regrouping the algebraic elements, so that the
sparse computations may be preserved:
ptemp ¼apðtÞTapðtÞqþ1a
e
n
Inter-network traversal probabilities extraction: Traversal probabil-
ity is computed for each inter-network protein pair (candidate
PrimAlign: PageRank-inspired Markovian alignment for large biological networks i539
ortholog) as the stationary probability for one protein multiplied by
the probability of selecting its paired protein as a transition target
out of other possible inter-network transitions (within-network
transitions are of no interest now), and summed symmetrically with
the probability computed in the same manner for the other protein
in the pair. As the probabilities rather dissolve with the increasing
number of proteins, the final score results from normalization of the
traversal probabilities by multiplication with the number of unique
proteins. This ensures that the values are comparable across align-
ments of networks of varied sizes. Formally, for a-th protein from
network Aand b-th protein from network B:
score a;b
ðÞ
¼pa
½ TA!Ba;b½
kTA!Ba;1:nB
½k
1
þpnAþb
½
TB!Ab;a½
kTB!Ab;1:nA
½k
1
n
where: n;nB¼number of proteins in network A;B.
Thresholding: The cut-off threshold to apply to the final score to
select high-quality ortholog candidates is 0.75 by default and can be
user-defined on input. We recommend using a number between 0.5
[more permissive, more false positives (FPs)—likely to select pro-
teins that are not real orthologs] and 1 [more strict, more false nega-
tives (FNs)—likely to miss some real orthologs]. The effect of these
threshold values is shown in Supplementary Figure S3. Setting the
threshold to 0 outputs all ortholog candidates.
Output: Ortholog candidates with the scores greater or equal to
the threshold are written to a file specified on program input to-
gether with their scores in a tab-delimited format (Supplementary
Text S1).
3 Experiment design
3.1 Data acquisition
In this study, the PPI networks of human (Homo sapiens), yeast
(Saccharomyces cerevisiae) and fruit fly (Drosophila melanogaster)
were selected, as they are well-explored and of a reasonable size. All
the networks were acquired from BioGRID (Chatr-aryamontri et al.,
2017) and filtered for physical interactions. The interacting proteins
were paired with genes that they are produced by, and maintained
and treated as gene-to-gene interactions. This approach helps with
connecting information from multiple databases together.
Semantic information was provided through ontology annota-
tions. The annotation files were downloaded from GO (The Gene
Ontology Consortium, 2015) containing over 400 000 annotations
with GO terms for human, and over 100 000 GO annotations for
yeast. Only pairs, in which the genes have at least one GO annota-
tion, were considered. GO itself was used in the basic version to
guarantee safe propagation of annotations within the ontology hier-
archy and all types of GO relationships were treated equally for the
purpose of creating a parent-child hierarchy and subsequent annota-
tion propagation. Annotations marked as IEA (Inferred from
Electronic Annotation) were excluded.
The genes were further annotated with KEGG Orthology (KO)
functional annotations (Kanehisa et al., 2017), which describe mo-
lecular functions of genes and proteins and can be used for orthol-
ogy matching. Furthermore, two lists of high-confident orthologous
genes were obtained. The first was exported through ENSEMBL
BioMart (Kinsella et al., 2011) and the other was downloaded from
InParanoid (Sonnhammer and O
¨stlund, 2015). While all of these
three sources partially rely on sequence similarity when finding
matching pairs, each method approaches the task differently. While
InParanoid employs a pairwise BLAST-based approach, ENSEMBL
uses a phylogeny-based method and KO functional annotations
undergo a comprehensive check against pathways of multiple spe-
cies in KEGG database and are partially manually reviewed. We as-
sume these sets contain true orthologs even though they may be
highly incomplete (there may be many orthologs undetected by these
methods).
To map individual datasets together, various types of identifiers
were connected via UniProt mapping service (The UniProt
Consortium, 2017). These include: UniProt KB protein IDs, UniProt
KB gene IDs, gene names and their synonyms, BioGRID protein IDs,
Saccharomyces Genome Database gene IDs, ENSEMBL gene IDs.
Synonyms were considered as long as they did not collide with other
names.
Overall, the transformations and filtering resulted in the net-
works of almost 270 000 interactions of 16 000 unique genes for
human, almost 90 000 interactions of 6000 unique genes for yeast
and more than 41000 interactions of 14 000 unique genes for fruit
fly. KO annotations were available for over 12 000 human genes,
3000 yeast genes and almost 6000 fruitfly genes. ENSEMBL and
InParanoid lists contain almost 6000 and 2000 human-yeast ortho-
logs, 13 500 and 4500 human-fruit fly orthologs and 5000 and 2000
yeast-fruit fly orthologs.
3.2 Alignment task
All three pairwise combinations of the three networks were aligned
with PrimAlign as well as the following algorithms for comparison:
AlignMCL,AlignNemo,CUFID,HubAlign,IsoRankN,
MAGNAþþ,MI-GRAAL,NETAL,NetCoffee,NetworkBLAST,
PINALOG,SANA,SMETANA and WAVE . Thus, we do not avoid
comparison with both global and local aligners, since both catego-
ries attempt to discover correct functional orthologs at first, regard-
less of whether they additionally group them into conserved clusters
or not and whether they assume one-to-one mappings or not. We
compare them directly at the same size of output to avoid a possible
bias caused by isolation of high confidence pairs comparing to more
complete mappings as explained in Section 3.3.
On input, the algorithms accept a network file for each species
with a list of interacting pairs, either unweighted or weighted
according to the interaction strength. Another input is sequence
similarity of pairs between the networks as produced by BLAST ana-
lysis, or within each network as well for some algorithms. We com-
puted the PPI weights in two forms: (i) as semantic similarity scores,
called sim
GIC
(Pesquita et al., 2008), and (ii) all set to 1, making
them effectively unweighted. This enables us assessing the effect of
weighting using semantic similarity.
Sequence similarity on input is an essential part of biological net-
work alignment. Without it, an algorithm could just optimize the
topological similarity, which is prone to overfitting and missing the
true functional matches, especially in incomplete networks. The in-
put BLAST information is either BLAST bit-score or BLAST e-value.
The main difference between them is that the e-value is adjusted
according to lengths of protein sequences, so the e-value and the bit-
score are not directly convertible without knowing the protein
lengths and the total size of the BLAST library. Nevertheless, the
algorithms accept a form of both except NetworkBLAST which is
strict in demanding e-values. Therefore, the e-value measure was
used for all the algorithms. The datasets bundled with NetCoffee
have been processed for this purpose. As a result, the BLAST files
contain the scores of over 60 000 human-yeast pairs, 40 000 human-
fruit fly pairs, 9000 yeast-fruit fly pairs, 200 000 human-human
pairs, 30 000 yeast-yeast pairs and 8000 fruit fly-fruit fly pairs.
i540 K.Kalecky and Y.-R.Cho
IsoRankN is applicable with weighted networks, but in fact, its
authors do not recommend inserting weights due to insufficient test-
ing of the algorithm for weighted data. Therefore, only unweighted
networks have been used for IsoRankN. Binaries available for
NetCoffee are not up-to-date and are marked as obsolete although
new source files are available. Therefore, the latest code from their
Git repository was compiled and used for comparison. It was discov-
ered that CUFID and SMETANA share a bug of rewriting network
weights: Although both algorithms read and store the weight values,
they are later ignored and replaced by 1s. As a result, the outputs
were identical when using either of the weighting methods. Be it a
bug or a feature, we have fixed the code to process the weights.
Additionally, we have restricted PrimAlign in two forms to pro-
cess either only sequence similarities without any network files on
input (PrimAlign-Seq), or only topology of unweighted networks
without sequence similarities (PrimAlign-Topo). These versions
serve for comparison of contribution of topology information and
sequence similarities to the final power of PrimAlign.
All the algorithms and their inputs are summarized in Table 1.
The parameters were selected according to recommendations from
provided manuals with respect to the size of the tasks or default
values were used. Processing of sequence similarities as well as multi-
threading was always enabled (if applicable). The log transformations
of e-values equal to 0 have been substituted by log of the smallest
positive value that a number of double precision could represent.
Computation times of the algorithms were also compared. To
guarantee a fair comparison, all algorithms were run on the same
machine, with 8 GB RAM and quad-core 3 GHz CPU; either in
Windows 10 environment (CUFID,SMETANA,PrimAlign)or
Ubuntu Server 16.10 (others) and without active usage of the ma-
chine for other tasks (after reboot). Both environments enable multi-
core processing. The time of start and end of runs was monitored
programmatically and rounded to seconds.
3.3 Evaluation metrics
The output of each algorithm is a list of putative orthologous pairs.
If an algorithm outputted a different structure, e.g. pairs in clusters
of conserved regions, we always transformed the results into aligned
pairs, the basic unit of alignment. Multiple measures were computed
over the pairs to see basic statistics of alignments and to assess the
alignment quality.
Four groups of evaluation measures are summarized and detailed
in Table 2.Informative measures are included for overall statistics
of the alignment: the size of the output (i.e. the number of aligned
pairs), gene coverage (number of genes that are aligned), the number
of conserved edges (an edge from one network that is aligned to an
edge in the other network) and the size of the largest conserved con-
nected component (LCCC). The first three measures are well-
defined. LCCC represents the lower number of edges in the largest
common connected sub-graph (LCCS) as defined in (Kuchaiev et al.,
2010). To avoid any ambiguity, we define it as the number of edges
in the largest connected component from one network consisting of
conserved edges that map to a connected component in the other
network (and then can be mapped back to itself), for which the net-
work with the smaller projection of the component is selected (due
to many-to-many mapping, the component in each network can
have a different number of edges). LCCS can be characterized by
both the number of edges and nodes. For simplicity, we chose only
the former variant, preferentially because it reflects the density of
LCCS and the number of conserved relationships. These informative
measures are not meant for evaluation of the alignment quality.
Note that here we included several topological measures that are
sometimes used directly for evaluation. However, we consider such
a usage pre-mature. Natural changes in topology (e.g. deletion), es-
pecially in the networks that are still incomplete, can result in func-
tional conservation which is topologically sub-optimal, in which
case topologically optimal alignment could have little biological
meaning. Therefore, topological evaluation measures can be
misleading.
The other three groups are computed to assess the alignment
quality. In all quality measures, the higher value implies the better
alignment. Annotation-based measures use either functional annota-
tions from KO to indicate how many aligned pairs are aligned func-
tionally correctly (share a KO annotation), or GO to evaluate how
Table 1. Summary of algorithms for comparison
Algorithm BLAST data Algorithm parameters Weights One-to-one Enforced
coverage
Type
AlignMCL -log
2
(e-value) inter-network Yes No No Local
AlignNemo e-values inter-network Yes No No Local
CUFID -log
2
(e-value) inter-network Yes Yes No Global
HubAlign -log
2
(e-value) inter-network No Yes Yes Global
IsoRankN -log
2
(e-value) intra & inter-network -K 10 –thresh 1e–5 –alpha
0.7 –maxveclen 2000000
No No No Global
MAGNAþþ -log
2
(e-value) inter-network -p 15000 -n 2000 -a 0.5 -t 4 -m S3 No Yes Yes Global
MI-GRAAL -log
2
(e-value) inter-network -p 19 No Yes Yes Global
NETAL -log
2
(e-value) intra & inter-network -b 0.5 -c 0.5 No Yes No Global
NetCoffee e-values intra & inter-network No Yes No Global
NetworkBLAST e-values inter-network beta 0.9; blast_th 1e–30;
true_factor0 0.5; true_factor1 0.5
Yes No No Local
PINALOG -log
2
(e-value) inter-network No Yes No Local
SANA -log
2
(e-value) inter-network -s3 0.5 -sequence 0.5 No Yes Yes Global
SMETANA -log
2
(e-value) inter-network Yes No No Global
WAVE -log
2
(e-value) inter-network No Yes Yes Global
PrimAlign-Seq e-values inter-network No No No Global
PrimAlign-Topo unweighted inter-network No No No Global
PrimAlign -log
2
(e-value) inter-network Yes No No Global
Note: Enforced coverage means that all proteins from the smaller network need to be aligned.
PrimAlign: PageRank-inspired Markovian alignment for large biological networks i541
big the shared semantic context is (in terms of Jaccard indices of
annotated GO terms). Ground-truth based measures consider the
lists of orthologous genes from ENSEMBL and InParanoid and ex-
press how many of them were identified among the aligned pairs.
Combined topological measures are derived from topological in-
formative measures but combined with the previous two groups:
Only either functionally correct aligned pairs (those sharing a KO
annotation) or ground-truth correct aligned pairs (those among the
lists of orthologs) are considered here. Combining topological meas-
ures with the other measures seems to be a more reasonable ap-
proach than using topological measures alone.
In order to evaluate alignments with various numbers of aligned
pairs, annotation-based and ground-truth-based evaluation meas-
ures are complemented with their cost forms, saying how many
aligned pairs on output are needed on average per one unit of the
given evaluation measure. Formally for each measure X,
cX ¼#aligned pairs
X
The cost measures have the opposite semantic than the other meas-
ures: The higher the cost value is, the lower the performance.
However, these measures serve only as a hint for cross-algorithm
comparison, as the scores are adjusted only in a linear manner with
respect to the number of aligned pairs––and this assumption of lin-
earity is not valid as shown in the Results Section. Instead, algo-
rithms tend to produce more accurate pairs first and less accurate
predictions with the growing output. This is not surprising (it is fully
in the spirit of ROC curve), but it makes the comparison more diffi-
cult; e.g. accuracy of two algorithms with a distinct output size
should not be directly compared.
Therefore, PrimAlign was additionally run to produce the same
number of aligned pairs as the other algorithms for a fair individual
comparison at the same size of output. To achieve this, we set the
optional threshold parameter in PrimAlign to 0 to output all
candidate orthologs and their scores and then we always selected the
desired number of top pairs with the highest score, as if the thresh-
old was set to produce the right number of pairs.
Statistical evaluation of the performance differences between
Prim-Align and the other algorithms has been performed using
two-proportion two-tailed z-test for measures, where each pair or
edge can be marked as either positive or not (i.e. all evaluation
measures except for GO) and using two-tailed Welch’s t-test
for measures, where each edge contributes to a real value (only GO).
P-value 0.05 was considered significant.
3.4 Synthetic networks
Next, we have performed another set of tests with 30 synthetic net-
works NAPAbench, each time aligning networks with 4000 and
3000 proteins. The networks are constructed by applying three dif-
ferent evolutionary models, which incorporate common biological
events such as gene deletions, gene duplications, gene mutations and
new functional specialization, followed by adjusting similarity
scores. Although the models are based on certain assumptions, they
have been shown to reflect multiple statistical properties of real evo-
lution (Sahraeian and Yoon, 2012).
Comparing to real biological networks, synthetic networks have
the advantage that the functional assignments are known, i.e. we
know all true orthologs. Therefore, we can directly evaluate per-
formance of the algorithms in terms of true positives (TP), FPs and
FNs, consolidated as precision TP and recall TP. Thus, we do not
have to rely on estimating the performance based on topology meas-
ures or incomplete sets of orthologs. However, we compute other
measures designed for comparing local and global aligners (Meng
et al., 2016), namely generalized symmetric sub-structure score
(GS
3
) and node coverage combined with GS
3
(NCV-GS
3
).
In this test, we compared PrimAlign with SANA and AlignMCL
at their size of output as well as using the default threshold. SANA
Table 2. Overview and classification of evaluation measures
Abbr. Measure Calculation
Informative measures AP Aligned Pairs # of aligned pairs
Cov Coverage # of unique aligned genes
CE Conserved edges # of edges from one network that are aligned to an edge
in the other network
LCCC Largest conserved connected
component
# of edges in the largest connected component
assembled from conserved edges
Annotation-based measures KO KO functionally correct
alignments
# of aligned pairs where both genes share KO functional
annotation
GO GO semantically correct
alignments
Sum of ratios of GO annotations shared by aligned
pairs (i.e. sum of Jaccard indices)
Ground-truth-based measures EN Discovered ENSEMBL orthologs # of ENSEMBL orthologs found among aligned pairs
IP Discovered InParanoid orthologs # of InParanoid orthologs found among aligned pairs
Combined topological measures CE-F Conserved edges of functionally
correct pairs
# of edges from one network that are aligned to an edge
in the other network with both aligned gene pairs
being functionally correct
CE-O Conserved edges of known
orthologs
# of edges from one network that are aligned to an edge
in the other network with both aligned gene pairs
being among ENSEMBL or InParanoid orthologs
LCCC-F Largest conserved connected
component from functionally
correct pairs
# of edges in the largest connected component
assembled from conserved edges between functional-
ly correctly aligned pairs
LCCC-O Largest conserved connected
component from known
orthologs
# of edges in the largest connected component
assembled from conserved edges between aligned
pairs also present among ENSEMBL or InParanoid
orthologs
i542 K.Kalecky and Y.-R.Cho
and AlignMCL were selected as superior global and local aligners,
respectively, based on the results in the first part of our evaluation.
4 Results
4.1 Overview of alignment results
As an overview of the produced outputs, the informative measures
and informative cost forms of the annotation-based and ground-
truth-based measures are summarized in Table 3. These results are
only descriptive and the highest or lowest values do not imply the
best or worst performance although the cost measures serve as a
hint for comparison.
For human-yeast network alignment, the number of aligned pairs
on output ranges between 2310 (NetCoffee) and 7904
(NetworkBLAST). Using the default threshold, PrimAlign produced
3801 pairs or 3752 pairs when using unweighted or weighted PPI
networks, respectively. The highest coverage of proteins was
achieved with MAGNAþþ,SANA,WAVE and HubAlign, as they
enforce full coverage of the smaller network (HubAlign actually
missed a few proteins lacking interaction). HubAlign also reached
the highest number of conserved edges and LCCC.
The cost measures point towards inefficiencies in MAGNAþþ,
SANA,WAVE, but also NetworkBLAST, which tend to produce
more aligned pairs, seemingly less confident pairs. On the other
hand, AlignMCL also produced a high number of aligned pairs, but
its cost measures are relatively lower, suggesting it performs qualita-
tively better than them. Aligners with smaller output achieved lower
costs except for NETAL.
Similar results were obtained for other alignment tasks and are
shown in Supplementary Information as Tables S1a and S1b.
IsoRankN did not align any proteins in human-fruit fly and yeast-
fruit fly tasks. NetworkBLAST did not align any proteins in the
yeast-fruit fly task. MI-GRAAL repeatedly failed during pre-proc-
essing human network, and for the smallest alignment task, even the
pre-processing time would take around 2weeks (based on progress
after several hours). Therefore, no results were obtained for MI-
GRAAL.
4.2 Evaluation results
PrimAlign was compared with all the other algorithms individually
at their number of aligned node pairs. The results are captured for
each evaluation measure separately in Figure 1 (or Supplementary
Fig. S4a for higher resolution) for human-yeast alignment and
Supplementary Figures S4b and S4c for the other tasks. Notice that:
Table 3. Alignment overview with informative measures
Aligner Weighted AP Cov CE LCCC cKO cGO cEN cIP
AlignMCL No 7711 7064 19 528 7716 4.39 3.34 2.80 6.00
AlignMCL Yes 7711 7064 19 528 7716 4.39 3.34 2.80 6.00
AlignNemo No 4776 2826 9385 3936 4.37 3.24 3.09 6.46
AlignNemo Yes 4081 2515 8127 3425 4.43 3.28 3.21 6.38
CUFID No 5654 11 308 13 892 6664 4.54 4.34 4.03 5.30
CUFID Yes 5250 10 500 14 744 7070 4.14 3.78 3.69 4.85
HubAlign No 5926 11 852 50 016 24 978 5.19 4.36 4.50 5.99
IsoRankN No 2964 5065 8711 3945 2.18 2.80 1.93 2.53
MAGNAþþ No 5933 11 866 3388 1601 11.19 6.10 9.10 13.93
MI-GRAAL No NA NA NA NA NA NA NA NA
NETAL No 3100 6200 456 129 17.74 3100 1
NetCoffee No 2310 4620 3742 1654 2.94 2.85 2.62 3.68
NetworkBLAST No 7904 3185 12 552 5224 6.68 3.47 4.46 9.81
NetworkBLAST Yes 4008 2195 10 404 4432 5.37 3.47 4.06 7.68
PINALOG No 5317 10 634 32 792 16 285 4.19 4.01 3.76 4.70
PrimAlign No 3801 4883 16 518 6971 2.31 2.86 1.71 3.07
PrimAlign Yes 3752 4843 16 408 6934 2.29 2.86 1.70 3.04
SANA No 5933 11 866 43 524 21 738 9.11 5.14 7.43 11.03
SMETANA No 3487 5384 18 770 8981 2.83 2.97 2.31 3.46
SMETANA Yes 3854 5718 14 417 6320 2.86 3.03 2.28 3.55
WAVE No 5933 11 866 32 500 16 138 10.85 5.48 9.90 12.68
Fig. 1. Results of human-yeast alignment. Each chart shows one evaluation
measure (as detailed in Table 2). PrimAlign and its modifications were run to
output the same num-ber of aligned pairs as other algorithms for direct com-
parison. The red cross marks denote the level of aligned pairs for PrimAlign
using the default threshold (dark red––for U; light red––for W; they mostly
overlap). U––unweighted networks, W––weighted networks
PrimAlign: PageRank-inspired Markovian alignment for large biological networks i543
i. For seven out of eight evaluation measures, PrimAlign exhibits
a saturation curve typical for higher accuracy in more confi-
dent pairs for small outputs. This suggests that PrimAlign’s
confidence score correlates with accuracy.
ii. The semantical measure GO as the only one results in a linear
trend and is almost identical to other algorithms except for the
aligners with enforced coverage, which shows lower semanti-
cal content and NETAL. This suggests that more confident
pairs are not semantically more similar comparing to less con-
fident pairs. However, it points towards inefficiencies in align-
ers with enforced coverage and NETAL.
iii. NETAL is an outlier, for which none of its aligned pairs was
found among known orthologs, indicative of biologically in-
correct alignment.
iv. Apart from GO, aligners with enforced coverage achieve lower
scores across other evaluation measures as well. This is also an
indicator of their low biological quality.
v. Results for semantically weighted and unweighted networks
on input are almost identical for all algorithms supporting
weights except for NetworkBLAST, which shows a big differ-
ence in the number of aligned pairs (but similar cost measures).
This might mean that semantical weighting does not provide
valuable information for aligners.
vi. PrimAlign-Seq and PrimAlign-Topo also exhibit a saturation
trend, although none of them reaches the score of PrimAlign.
This suggests that both sequence similarity and network top-
ology contribute towards the final power of PrimAlign.
vii. The default threshold of PrimAlign produces a reasonable
number of aligned pairs across all three alignment tasks.
This suggests that the default threshold is good for alignment
of networks of various sizes and density. More detailed effects
of changing the threshold are shown in Supplementary
Information as Figure S3.
viii. PrimAlign outperforms other algorithms at their size of output
with a statistical significance across all categories of evaluation
measures, as shown in Table 4 for human-yeast alignment and
Supplementary Tables S2a and S2b for the other alignment tasks.
ix. However, for the smallest alignment task, aligners with
enforced coverage shows very high scores in combined topo-
logical measures although this is not accompanied with such
an increase in other measures and not replicated in other align-
ment tasks. This could mean that these algorithms are overfit-
ting with topological optimization and for such small
networks with relatively few known orthologs the combined
scores might be skewed towards the contribution of topologic-
al similarity.
x. Results of KO, IN and EN measures tend to correlate even
though their lists of orthologs differ and overlap only partially.
This is not surprising, as we expect that all of them provide
quality orthologs and that the measures should correlate with
algorithms accuracy regardless of which sub-set of quality
orthologs is used. Results of combined topological measures
also correlate with each other, suggesting that when filtered
for known orthologs, CE and LCCC are of similar evaluation
power.
4.3 Results with synthetic networks
Results of all 30 runs with unweighted synthetic networks in terms
of precision, recall, GS
3
and NCV-GS
3
are summarized in Figure 2.
For PrimAlign and AlignMCL, the results are very similar across the
measures at the same size of aligned pairs. By default, PrimAlign still
outputs more confident pairs with mean precision above 70%,
whereas AlignMCL produces pairs with precision around 55% in
exchange for higher recall. SANA achieves mean precision around
45% even though its recall is also low. On the other hand, SANA
scores well in topological measures GS
3
and NCV-GS
3
. This corre-
sponds with our previous tests and confirms the suspicion that
SANA overfits alignment topology and produces biologically less
precise predictions. Figure 3 shows precision-recall curve for
PrimAlign together with locations of AlignMCL and SANA for the
first synthetic network. Unfortunately, data of neither of these algo-
rithms allow to construct their own precision-recall (p-r) curves,
which would be the ideal case for comparison and statistical evalu-
ation of their performance in terms of auPR (area under p-r curve).
Table 4. Statistical comparison of PrimAlign with the others
Aligner Weighted KO GO EN IP CE-F CE-O LCCC-F LCCC-O
AlignMCL No *** *** ** **
AlignMCL Yes *** *** ** **
AlignNemo No *** *** *** *** *** *** *** ***
AlignNemo Yes *** *** *** *** *** *** *** ***
CUFID No *** *** *** *** *** *** *** ***
CUFID Yes *** *** *** *** *** *** *** ***
HubAlign No *** *** *** *** *** *** *** ***
IsoRankN No *** *** *** *** *** ***
MAGNAþþ No *** *** *** *** *** *** *** ***
MI-GRAAL No NA NA NA NA NA NA NA NA
NETAL No *** *** *** *** *** *** *** ***
NetCoffee No *** *** *** *** *** *** *** ***
NetworkBLAST No *** *** *** *** *** *** *** ***
NetworkBLAST Yes *** *** *** *** *** *** *** ***
PINALOG No *** *** *** *** *** *** *** ***
SANA No *** *** *** *** *** *** *** ***
SMETANA No *** *** *** *** *** *** *** ***
SMETANA Yes *** *** *** *** *** *** *** ***
WAVE No *** *** *** *** *** *** *** ***
Notes: Statistical comparison of differences in evaluation measures between PrimAlign with unweighted networks on input and the other algorithms for
human-yeast alignment. (empty field) P>0.05, *P0.05, **P0.01, ***P0.001. Green ¼improvement.
i544 K.Kalecky and Y.-R.Cho
4.4 Runtime comparison
Runtimes of individual algorithms are listed in Supplementary Table
S5.NetCoffee is the fastest algorithm. However, its performance
was shown to be poor, mostly worse than both restricted versions
PrimAlign-Seq (which actually finishes within 1 second) and
PrimAlign-Topo.PrimAlign was the only other algorithm running
less than 1 minute for the largest alignment task, showing its great
scalability. On the other side of the spectrum with runtime more
than 1 hour are PINALOG,MAGNAþþ (more than 10 hours),
NetworkBLAST (more than 1 day), IsoRankN (more than 4 days)
and MI-GRAAL (estimated 2weeks of pre-processing for the small-
est alignment task). As visible from runtime ratios between the tasks,
some algorithms are very difficult to scale.
5 Conclusion
Biological network aligners pursue the goal of finding pairs of func-
tionally matching proteins between the networks. This is still a task
with incomplete information and needs to be solved heuristically.
Some algorithms adopt the constraints of one-to-one alignment or
complete coverage of proteins from the smaller network. (Neither
one of them is biologically valid due to the gene duplications and
other evolutionary mechanisms.) Other algorithms may use more
subtle assumptions. Some of them perform subsequent grouping of
aligned pairs into highly conserved clusters (local aligners). In each
case, each algorithm has its own assumptions or heuristics to apply
to recognize correct orthologs.
In this paper, we introduced PrimAlign, a new algorithm for
pairwise global alignment of PPI networks, based on the Markovian
representation and PageRank technique. PrimAlign performs global
many-to-many alignment with asymptotic time complexity O(n),
which is the theoretical minimum for this task, guaranteeing high
scalability. The performance was evaluated on alignment tasks
with three model species and compared to 14 prevalent network
alignment algorithms. Various evaluation measures were used:
ground-truth-based measures with data from two different orthol-
ogy databases, annotation-based measures to evaluate functional
and semantic consistency and topology-based measures combined
with the previous measures. Adjusting the output size in terms of the
number of aligned pairs allowed a direct comparison between
PrimAlign and the other algorithms. Additional evaluation was per-
formed with 30 synthetic networks and high performing representa-
tives of global and local aligners.
As a result, the proposed method outperforms the other algo-
rithms on real networks with statistically significant differences
demonstrated for all evaluation measures except for the semantic
similarity measure. This measure exhibits approximately the same
trend across the algorithms, grows linearly with respect to the num-
ber of aligned pairs, and therefore, it seems to be of little value for
evaluation. Furthermore, weighting PPI networks with semantic
similarity seems to be of no benefit. The previous algorithm with
closest results was AlignMCL. In the task with smallest and least
dense networks, four other algorithms (SANA,WAVE,HubAlign
and PINALOG) achieved significantly higher scores in combined
topological evaluation measures. We speculated that this result
sourced from high topological optimization of these algorithms and
might not be biologically superior, indicated by non-superior scores
in other categories and non-replicability in the alignment of larger
networks.
Additional comparison with AlignMCL and SANA on synthetic
networks, where true functional assignments are known, confirmed
the case. Even though SANA achieved very high topological score,
its real performance was poor with a high ratio of FPs as well as
FNs. This result justifies our decision not to include topological
scores among evaluation measures because of the suspicion that
optimizing topology might easily result in overfitting and failing to
correlate with functional conservation. Furthermore, AlignMCL
performs very similarly to PrimAlign. However, AlignMCL pro-
duced relatively many aligned pairs, leading to lower precision in
compromise to achieve higher recall, and there is no parameter to
tune the confidence threshold, in contrary to PrimAlign. This also
means that while PrimAlign can be evaluated with auPR, we cannot
construct precision-recall curve for AlignMCL, even though auPR
would be probably the most reasonable measure to statistically
evaluate performance in synthetic networks.
The computation time of PrimAlign is excellent thanks to sparse
computations. On the test machine, it runs faster comparing to other
algorithms by 1–4 orders of magnitude, which makes it suitable
for alignment of complex biological networks. Only NetCoffee
was faster although it did not produce comparably good results.
Fig. 2. Results of tests with 30 synthetic networks. Comparing precision, recall
and topological measures of PrimAlign with SANA and AlignMCL at their size
of aligned node pairs
Fig. 3. Example of P-R curve. P-R curve of PrimAlign for the 1st synthetic
network compared with SANA and AlignMCL. The result of PrimAlign for the
default threshold is highlighted.
PrimAlign: PageRank-inspired Markovian alignment for large biological networks i545
The source code of the proposed method, written in C# language, is
available at http://web.ecs.baylor.edu/faculty/cho/PrimAlign. With
less than 70 lines of code, it is probably the most compact and light-
weight alignment algorithm available, it is easy to read and easy to
be extended. The raw evaluation results and input files are also
available.
Conflict of Interest: none declared.
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