New GO-based measures in multiple network alignment.

MOTIVATION: Protein-protein interaction (PPI) networks provide valuable insights into the function of biological systems. Aligning multiple PPI networks may expose relationships beyond those observable by pairwise comparisons. However, assessing the biological quality of multiple network alignments is a challenging problem. RESULTS: We propose two new measures to evaluate the quality of multiple network alignments using functional information from Gene Ontology (GO) terms. When aligning multiple real PPI networks across species, we observe that both measures are highly correlated with objective quality indicators, such as common orthologs. Additionally, our measures strongly correlate with an alignment's ability to predict novel GO annotations, which is a unique advantage over existing GO-based measures. AVAILABILITY AND IMPLEMENTATION: The scripts and the links to the raw and alignment data can be accessed at https://github.com/kimiayazdani/GO_Measures.git.

Exact p-values for global network alignments via combinatorial analysis of shared GO terms : REFANGO: Rigorous Evaluation of Functional Alignments of Networks using Gene Ontology.

Network alignment aims to uncover topologically similar regions in the protein-protein interaction (PPI) networks of two or more species under the assumption that topologically similar regions tend to perform similar functions. Although there exist a plethora of both network alignment algorithms and measures of topological similarity, currently no "gold standard" exists for evaluating how well either is able to uncover functionally similar regions. Here we propose a formal, mathematically and statistically rigorous method for evaluating the statistical significance of shared GO terms in a global, 1-to-1 alignment between two PPI networks. Given an alignment in which k aligned protein pairs share a particular GO term g, we use a combinatorial argument to precisely quantify the p-value of that alignment with respect to g compared to a random alignment. The p-value of the alignment with respect to all GO terms, including their inter-relationships, is approximated using the Empirical Brown's Method. We note that, just as with BLAST's p-values, this method is not designed to guide an alignment algorithm towards a solution; instead, just as with BLAST, an alignment is guided by a scoring matrix or function; the p-values herein are computed after the fact, providing independent feedback to the user on the biological quality of the alignment that was generated by optimizing the scoring function. Importantly, we demonstrate that among all GO-based measures of network alignments, ours is the only one that correlates with the precision of GO annotation predictions, paving the way for network alignment-based protein function prediction.

BLANT-fast graphlet sampling tool.

SUMMARY: BLAST creates local sequence alignments by first building a database of small k-letter sub-sequences called k-mers. Identical k-mers from different regions provide 'seeds' for longer local alignments. This seed-and-extend heuristic makes BLAST extremely fast and has led to its almost exclusive use despite the existence of more accurate, but slower, algorithms. In this paper, we introduce the Basic Local Alignment for Networks Tool (BLANT). BLANT is the analog of BLAST, but for networks: given an input graph, it samples small, induced, k-node sub-graphs called k-graphlets. Graphlets have been used to classify networks, quantify structure, align networks both locally and globally, identify topology-function relationships and build taxonomic trees without the use of sequences. Given an input network, BLANT produces millions of graphlet samples in seconds-orders of magnitude faster than existing methods. BLANT offers sampled graphlets in various forms: distributions of graphlets or their orbits; graphlet degree or graphlet orbit degree vectors, the latter being compatible with ORCA; or an index to be used as the basis for seed-and-extend local alignments. We demonstrate BLANT's usefelness by using its indexing mode to find functional similarity between yeast and human PPI networks. AVAILABILITY AND IMPLEMENTATION: BLANT is written in C and is available at https://github.com/waynebhayes/BLANT/releases. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

SANA NetGO: a combinatorial approach to using Gene Ontology (GO) terms to score network alignments.

Motivation: Gene Ontology (GO) terms are frequently used to score alignments between protein-protein interaction (PPI) networks. Methods exist to measure GO similarity between proteins in isolation, but proteins in a network alignment are not isolated: each pairing is dependent on every other via the alignment itself. Existing measures fail to take into account the frequency of GO terms across networks, instead imposing arbitrary rules on when to allow GO terms. Results: Here we develop NetGO, a new measure that naturally weighs infrequent, informative GO terms more heavily than frequent, less informative GO terms, without arbitrary cutoffs, instead downweighting GO terms according to their frequency in the networks being aligned. This is a global measure applicable only to alignments, independent of pairwise GO measures, in the same sense that the edge-based EC or S3 scores are global measures of topological similarity independent of pairwise topological similarities. We demonstrate the superiority of NetGO in alignments of predetermined quality and show that NetGO correlates with alignment quality better than any existing GO-based alignment measures. We also demonstrate that NetGO provides a measure of taxonomic similarity between species, consistent with existing taxonomic measuresa feature not shared with existing GObased network alignment measures. Finally, we re-score alignments produced by almost a dozen aligners from a previous study and show that NetGO does a better job at separating good alignments from bad ones. Availability and implementation: Available as part of SANA. Contact: whayes@uci.edu. Supplementary information: Supplementary data are available at Bioinformatics online.

SANA: simulated annealing far outperforms many other search algorithms for biological network alignment.

SUMMARY: Every alignment algorithm consists of two orthogonal components: an objective function M measuring the quality of an alignment, and a search algorithm that explores the space of alignments looking for ones scoring well according to M . We introduce a new search algorithm called SANA (Simulated Annealing Network Aligner) and apply it to protein-protein interaction networks using S 3 as the topological measure. Compared against 12 recent algorithms, SANA produces 5-10 times as many correct node pairings as the others when the correct answer is known. We expose an anti-correlation in many existing aligners between their ability to produce good topological vs. functional similarity scores, whereas SANA usually outscores other methods in both measures. If given the perfect objective function encoding the identity mapping, SANA quickly converges to the perfect solution while many other algorithms falter. We observe that when aligning networks with a known mapping and optimizing only S 3 , SANA creates alignments that are not perfect and yet whose S 3 scores match that of the perfect alignment. We call this phenomenon saturation of the topological score . Saturation implies that a measure's correlation with alignment correctness falters before the perfect alignment is reached. This, combined with SANA's ability to produce the perfect alignment if given the perfect objective function, suggests that better objective functions may lead to dramatically better alignments. We conclude that future work should focus on finding better objective functions, and offer SANA as the search algorithm of choice. AVAILABILITY AND IMPLEMENTATION: Software available at http://sana.ics.uci.edu . CONTACT: whayes@uci.edu. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

A BMP-FGF morphogen toggle switch drives the ultrasensitive expression of multiple genes in the developing forebrain.

Borders are important as they demarcate developing tissue into distinct functional units. A key challenge is the discovery of mechanisms that can convert morphogen gradients into tissue borders. While mechanisms that produce ultrasensitive cellular responses provide a solution, how extracellular morphogens drive such mechanisms remains poorly understood. Here, we show how Bone Morphogenetic Protein (BMP) and Fibroblast Growth Factor (FGF) pathways interact to generate ultrasensitivity and borders in the dorsal telencephalon. BMP and FGF signaling manipulations in explants produced border defects suggestive of cross inhibition within single cells, which was confirmed in dissociated cultures. Using mathematical modeling, we designed experiments that ruled out alternative cross inhibition mechanisms and identified a cross-inhibitory positive feedback (CIPF) mechanism, or "toggle switch", which acts upstream of transcriptional targets in dorsal telencephalic cells. CIPF explained several cellular phenomena important for border formation such as threshold tuning, ultrasensitivity, and hysteresis. CIPF explicitly links graded morphogen signaling in the telencephalon to switch-like cellular responses and has the ability to form multiple borders and scale pattern to size. These benefits may apply to other developmental systems.

SANA: cross-species prediction of Gene Ontology GO annotations via topological network alignment.

Topological network alignment aims to align two networks node-wise in order to maximize the observed common connection (edge) topology between them. The topological alignment of two protein-protein interaction (PPI) networks should thus expose protein pairs with similar interaction partners allowing, for example, the prediction of common Gene Ontology (GO) terms. Unfortunately, no network alignment algorithm based on topology alone has been able to achieve this aim, though those that include sequence similarity have seen some success. We argue that this failure of topology alone is due to the sparsity and incompleteness of the PPI network data of almost all species, which provides the network topology with a small signal-to-noise ratio that is effectively swamped when sequence information is added to the mix. Here we show that the weak signal can be detected using multiple stochastic samples of "good" topological network alignments, which allows us to observe regions of the two networks that are robustly aligned across multiple samples. The resulting network alignment frequency (NAF) strongly correlates with GO-based Resnik semantic similarity and enables the first successful cross-species predictions of GO terms based on topology-only network alignments. Our best predictions have an AUPR of about 0.4, which is competitive with state-of-the-art algorithms, even when there is no observable sequence similarity and no known homology relationship. While our results provide only a "proof of concept" on existing network data, we hypothesize that predicting GO terms from topology-only network alignments will become increasingly practical as the volume and quality of PPI network data increase.

On the current failure-but bright future-of topology-driven biological network alignment.

Since the function of a protein is defined by its interaction partners, and since we expect similar interaction patterns across species, the alignment of protein-protein interaction (PPI) networks between species, based on network topology alone, should uncover functionally related proteins across species. Surprisingly, despite the publication of more than fifty algorithms aimed at performing PPI network alignment, few have demonstrated a statistically significant link between network topology and functional similarity, and none have demonstrated that orthologs can be recovered using network topology alone. We find that the major contributing factors to this surprising failure are: (i) edge densities in most currently available experimental PPI networks are demonstrably too low to expect topological network alignment to succeed; (ii) in the few cases where the edge densities are high enough, some measures of topological similarity easily uncover functionally similar proteins while others do not; and (iii) most network alignment algorithms to date perform poorly at optimizing even their own topological objective functions, hampering their ability to use topology effectively. We demonstrate that SANA-the Simulated Annealing Network Aligner-significantly outperforms existing aligners at optimizing their own objective functions, even achieving near-optimal solutions when the optimal solution is known. We offer the first demonstration of global network alignments based on topology alone that align functionally similar proteins with p-values in some cases below 10-300. We predict that topological network alignment has a bright future as edge densities increase toward the value where good alignments become possible. We demonstrate that when enough common topology is present at high enough edge densities-for example in the recent, partly synthetic networks of the Integrated Interaction Database-topological network alignment easily recovers most orthologs, paving the way toward high-throughput functional prediction based on topology-driven network alignment.

An Introductory Guide to Aligning Networks Using SANA, the Simulated Annealing Network Aligner.

Sequence alignment has had an enormous impact on our understanding of biology, evolution, and disease. The alignment of biological networks holds similar promise. Biological networks generally model interactions between biomolecules such as proteins, genes, metabolites, or mRNAs. There is strong evidence that the network topology-the "structure" of the network-is correlated with the functions performed, so that network topology can be used to help predict or understand function. However, unlike sequence comparison and alignment-which is an essentially solved problem-network comparison and alignment is an NP-complete problem for which heuristic algorithms must be used.Here we introduce SANA, the Simulated Annealing Network Aligner. SANA is one of many algorithms proposed for the arena of biological network alignment. In the context of global network alignment, SANA stands out for its speed, memory efficiency, ease-of-use, and flexibility in the arena of producing alignments between two or more networks. SANA produces better alignments in minutes on a laptop than most other algorithms can produce in hours or days of CPU time on large server-class machines. We walk the user through how to use SANA for several types of biomolecular networks.

Shadowing high-dimensional hamiltonian systems: the gravitational N-body problem.

A shadow is an exact solution to a chaotic system of equations that remains close to a numerically computed solution for a long time. Using a variable-order, variable-time-step integrator, we numerically compute solutions to a gravitational N-body problem in which many particles move and interact in a fixed potential. We then search for shadows of these solutions with the longest possible duration. We find that in "softened" potentials, shadow durations are sufficiently long for significant evolution to occur. However, in unsoftened potentials, shadow durations are typically very short.