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Protein-protein interaction (PPI) networks provide valuable insights into the function of biological systems, and aligning multiple PPI networks can reveal important functional relationships between different species. However, assessing the quality of multiple network alignments is a challenging problem. In this paper, we propose two new measures, the Squared GO Score (SGS) and the Exposed G Score, to evaluate the quality of multiple network alignments while using functional information from Gene Ontology (GO) terms. We also introduce a $p$-value measure, the Statistical Exposed G Score, to compute the exact significance of a multiple network alignment based on its revealed GO terms. We also show that our measures are highly correlated with the recovered Ortholog count, providing further evidence for their effectiveness. Our work contributes to the development of more reliable and accurate measures for evaluating multiple network alignments and has potential applications in predicting gene function and identifying evolutionary relationships between different species using multiple network alignment.

BLAST is a standard tool in bioinformatics for creating local sequence alignments using a "seed-and-extend" approach. Here we introduce an analogous seed-and-extend algorithm that produces local network alignments: BLANT, for Basic Local Alignment of Network Topology. This paper introduces BLANT-seed: given an input graph, BLANT-seed uses network topology alone to create a limited, high-specificity index of k-node induced subgraphs called k-graphlets (analogous to BLASTS's k-mers). The index is constructed so that, if significant common network topology exists between two graphs, their indexes are likely to overlap. BLANT-seed then queries the indexes of two networks to generate a list of common k-graphlets which, when paired, form a seed pair. Our companion paper (submitted elsewhere) describes BLANT-extend, which "grows" these seeds to larger local alignments, again using only topological information., Comment: 13 pages, 12 Figures, 2 Tables

BLAST is a standard tool in bioinformatics for creating local sequence alignments using a "seed-and-extend" approach. Here we introduce an analogous seed-and-extend algorithm that produces local network alignments: BLANT (Basic Local Alignment of Network Topology). In Part 1, we introduced BLANT-seed, which generates graphlet-based seeds using only topological information. Here, in Part 2, we describe BLANT-extend, which "grows" seeds to larger local alignments using only topological information. We allow the user to specify bounds on several measures an alignment must satisfy, including the edge density, edge commonality (i.e., aligned edges), and node-pair similarity if such a measure is used; the latter allows the inclusion of sequence-based similarity, if desired, as well as local topological constraints. BLANT-extend is able to enumerate all possible alignments satisfying the bounds that can be grown from each seed, within a specified CPU time or number of generated alignments. While topology-driven local network alignment has a wide variety of potential applications outside bioinformatics, here we focus on the alignment of Protein-Protein Interaction (PPI) networks. We show that BLANT is capable of finding large, high-quality local alignments when the networks are known to have high topological similarity -- for example recovering hundreds of orthologs between networks of the recent Integrated Interaction Database (IID). Predictably, however, it performs less well when true topological similarity is absent, as is the case in most current experimental PPI networks that are noisy and have wide disparity in edge density which results in low common coverage., Comment: 8 pages, 5 Figures, 3 Tables

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., Comment: 31 pages; 9 Figures; 8 Tables

The function of a protein is defined by its interaction partners. Thus, topology-driven network alignment of the protein-protein interaction (PPI) networks of two species should uncover similar interaction patterns and allow identification of functionally similar proteins. Howver, few of the fifty or more algorithms for PPI network alignment have demonstrated a significant link between network topology and functional similarity, and none have recovered orthologs using network topology alone. We find that the major contributing factors to this failure are: (i) edge densities in current PPI networks are too low to expect topological network alignment to succeed; (ii) when 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 fail to optimize 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 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 1e-300. We predict that topological network alignment has a bright future as edge densities increase towards 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 towards high-throughput functional prediction based on topology-driven network alignment., Comment: 21 pages + 7-page Supplementary; 6 tables, 11 Figures

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. We use combinatorics to precisely count the number of possible network alignments in which $k$ proteins share a particular GO term. When divided by the number of all possible network alignments, this provides an explicit, exact $p$-value for a network alignment with respect to a particular GO term. Just as with BLAST's p-values and bit-scores, this method is designed not to guide the formation of any particular alignment, but instead to provide an after-the-fact evaluation of a fixed, given alignment., Comment: 22 pages, 3 figures, 4 tables

Sequence alignment has had an enormous impact on our understanding of biology, evolution, and disease. The alignment of biological {\em 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 {\it 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 2 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. Availability: https://github.com/waynebhayes/SANA

A human watching a video of closely-packed cells can generally identify every individual cell, regardless of density and noise, but most currently-available cell-tracking software cannot. This is because the human brain automatically builds a physical model of the scene as it progresses, allowing it to readily distinguish cells from noise and not be unduly confused by overlapping cells. Here we introduce software that uses physical rules to create a simulation of the activity in a cell video, synchronizing itself with the video as the activity progresses. Because our simulation includes every individual cell, we are trivially able to track all cell movement, growth, and divisions. Our method is also particularly robust to noise without requiring any substantial image processing. We demonstrate the effectiveness of this method by tracking the motion and lineage tree of a densely-packed colony of cells that grows from 4 to more than 200 individuals., Comment: 21 pages, 14 Figures, 1 Table

Biological network alignment is currently in a state of disarray, with more than two dozen network alignment tools having been introduced in the past decade, with no clear winner, and other new tools being published almost quarterly. Part of the problem is that almost every new tool proposes both a new objective function and a new search algorithm to optimize said objective. These two aspects of alignment are orthogonal, and confounding them makes it difficult to evaluate them separately. A more systematic approach is needed. To this end, we bring these two orthogonal issues into sharp focus in two companion papers. In Part 1 (this paper) we show that simulated annealing, as implemented by SANA, far outperforms all other existing search algorithms across a wide range of objectives. Part 2 (our companion paper) then uses SANA to compare over a dozen objectives in terms of the biology they recover, demonstrating that some objective functions recover useful biological information while others do not. We propose that further work should focus on improving objective functions, with SANA the obvious choice as the search algorithm., Comment: 8 pages main text, 2 pages Supplementary

In this paper we study the morphological properties of spiral galaxies, including measurements of spiral arm number and pitch angle. Using Galaxy Zoo 2, a stellar mass-complete sample of 6,222 SDSS spiral galaxies is selected. We use the machine vision algorithm SpArcFiRe to identify spiral arm features and measure their associated geometries. A support vector machine classifier is employed to identify reliable spiral features, with which we are able to estimate pitch angles for half of our sample. We use these machine measurements to calibrate visual estimates of arm tightness, and hence estimate pitch angles for our entire sample. The properties of spiral arms are compared with respect to various galaxy properties. The star formation properties of galaxies vary significantly with arm number, but not pitch angle. We find that galaxies hosting strong bars have spiral arms substantially ($4-6^\mathrm{o}$) looser than unbarred galaxies. Accounting for this, spiral arms associated with many-arm structures are looser (by 2$^\mathrm{o}$) than those in two-arm galaxies. In contrast to this average trend, galaxies with greater bulge-to-total stellar mass ratios display both fewer and looser spiral arms. This effect is primarily driven by the galaxy disc, such that galaxies with more massive discs contain more spiral arms with tighter pitch angles. This implies that galaxy central mass concentration is not the dominant cause of pitch angle and arm number variations between galaxies, which in turn suggests that not all spiral arms are governed by classical density waves or modal theories., Comment: 18 pages, 13 figures, accepted for publication in MNRAS

Graphlets are small connected induced subgraphs of a larger graph $G$. Graphlets are now commonly used to quantify local and global topology of networks in the field. Methods exist to exhaustively enumerate all graphlets (and their orbits) in large networks as efficiently as possible using orbit counting equations. However, the number of graphlets in $G$ is exponential in both the number of nodes and edges in $G$. Enumerating them all is already unacceptably expensive on existing large networks, and the problem will only get worse as networks continue to grow in size and density. Here we introduce an efficient method designed to aid statistical sampling of graphlets up to size $k=8$ from a large network. We define graphettes as the generalization of graphlets allowing for disconnected graphlets. Given a particular (undirected) graphette $g$, we introduce the idea of the canonical graphette $\mathcal K(g)$ as a representative member of the isomorphism group $Iso(g)$ of $g$. We compute the mapping $\mathcal K$, in the form of a lookup table, from all $2^{k(k-1)/2}$ undirected graphettes $g$ of size $k\le 8$ to their canonical representatives $\mathcal K(g)$, as well as the permutation that transforms $g$ to $\mathcal K(g)$. We also compute all automorphism orbits for each canonical graphette. Thus, given any $k\le 8$ nodes in a graph $G$, we can in constant time infer which graphette it is, as well as which orbit each of the $k$ nodes belongs to. Sampling a large number $N$ of such $k$-sets of nodes provides an approximation of both the distribution of graphlets and orbits across $G$, and the orbit degree vector at each node., Comment: 13 pages, 4 figures, 2 tables. Accepted to PLOS ONE

The Galaxy Zoo has provided morphological data on many galaxies. Several biases have been identified in the Galaxy Zoo data. Here we report on a newly discovered selection effect: astronomers interested in studying spiral galaxies may select a set of spiral galaxies based upon a threshold in spirality (the fraction of Galaxy Zoo humans who report seeing spiral structure). SpArcFiRe is an automated tool that decomposes a spiral galaxy into its constituent spiral arms, providing objective, quantitative data on their structure. SpArcFiRe measures the pitch angle of spiral arms. We have observed that when selecting a set of spiral galaxies based on a threshold on spirality, the pitch angle of spiral arms appear increase with redshift. We hypothesize that this is a selection effect: tightly-wound spiral arms become less visible as images degrade with increasing redshift, leading to fewer such galaxies being included in the sample at higher redshifts. We corroborate this hypothesis by artificially degrading images of nearby galaxies, then using a machine learning algorithm trained on Galaxy Zoo data to provide a spirality for each artificially degraded image. It correctly predicts that spirality decreases as image quality degrades. Thus, the mean pitch angle of those galaxies remaining above the spirality threshold is higher than those eliminated by the selection effect. This demonstrates that users who select samples of galaxies using a threshold of Galaxy Zoo votes must carefully consider the possibility of selection effects on morphological measures, even if the measure itself is believed to be objective and unbiased. Finally, we also perform an empirical sensitivity analysis to demonstrate that SpArcFiRe's output changes in a smooth and predictable fashion to changes in its internal algorithmic parameters., Comment: 11 pages, 20 figures

Gene Ontology (GO) terms are frequently used to score alignments between protein-protein interaction (PPI) networks. Methods exist to measure the GO similarity between two proteins in isolation, but pairs of proteins in a network alignment are not isolated: each pairing is implicitly dependent upon every other pairing via the alignment itself. Current methods fail to take into account the frequency of GO terms across the networks, and attempt to account for common GO terms in an ad hoc fashion by imposing arbitrary rules on when to "allow" GO terms based on their location in the GO hierarchy, rather than using readily available frequency information in the PPI networks themselves. Here we develop a new measure, NetGO, that naturally weighs infrequent, informative GO terms more heavily than frequent, less informative GO terms, without requiring arbitrary cutoffs. In particular, NetGO down-weights the score of frequent 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 by creating alignments of predetermined quality based on homologous pairs of nodes and show that NetGO correlates with alignment quality much 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 measures--a feature not shared with existing GO-based 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 than existing measures at separating good alignments from bad ones.

Given an approximately centered image of a spiral galaxy, we describe an entirely automated method that finds, centers, and sizes the galaxy and then automatically extracts structural information about the spiral arms. For each arm segment found, we list the pixels in that segment and perform a least-squares fit of a logarithmic spiral arc to the pixels in the segment. The algorithm takes about 1 minute per galaxy, and can easily be scaled using parallelism. We have run it on all ~644,000 Sloan objects classified as "galaxy" and large enough to observe some structure. Our algorithm is stable in the sense that the statistics across a large sample of galaxies vary smoothly based on algorithmic parameters, although results for individual galaxies can sometimes vary in a non-smooth but easily understood manner. We find a very good correlation between our quantitative description of spiral structure and the qualitative description provided by humans via Galaxy Zoo. In addition, we find that pitch angle often varies significantly segment-to-segment in a single spiral galaxy, making it difficult to define "the" pitch angle for a single galaxy. Finally, we point out how complex arm structure (even of long arms) can lead to ambiguity in defining what an "arm" is, leading us to prefer the term "arm segments"., Comment: 4 pages (twocolumn),5 figures, 2 tables. Submitted to ApJ. Letters

We demonstrate that the system of Jovian planets (Sun+Jupiter+Saturn+Uranus+Neptune), integrated for 200 million years as an isolated 5-body system using many sets of initial conditions all within the uncertainty bounds of their currently known positions, can display both chaos and near-integrability. The conclusion is consistent across four different integrators, including several comparisons against integrations utilizing quadruple precision. We demonstrate that the Wisdom-Holman symplectic map using simple symplectic correctors as implemented in Mercury 6.2 (Chambers 1999) gives a reliable characterization of the existence of chaos for a particular initial condition only with timesteps less than about 10 days, corresponding to about 400 steps per orbit. We also integrate the canonical DE405 initial condition out to 5 Gy, and show that it has a Lyapunov Time of 200--400 My, opening the remote possibility of accurate prediction of the Jovian planetary positions for 5 Gy., Comment: 35 pages, 9 figures, 2 tables. Submitted to AJ

The existence of chaos in the system of Jovian planets has been in question for the past 15 years. Various investigators have found Lyapunov times ranging from about 5 millions years upwards to infinity, with no clear reason for the discrepancy. In this paper, we resolve the issue. The position of the outer planets is known to only a few parts in 10 million. We show that, within that observational uncertainty, there exist Lyapunov timescales in the full range listed above. Thus, the ``true'' Lyapunov timescale of the outer Solar System cannot be resolved using current observations., Comment: 8 pages, 2 figures

A {\it shadow} is an exact solution to a chaotic system of equations that remains close to a numerically computed solution for a long time, ending in a {\it glitch}. We study the distribution of shadow durations at low dimension and how shadow durations scale as dimension increases up to 300 in a slightly simplified gravitational n-body system. We find that ``softened'' systems are shadowable for many tens of crossing times even for large n, while in an ``unsoftened'' system each particle encounters glitches independently as a Poisson process, giving shadow durations that scale as 1/n., Comment: 4 pages, 4 figures, submitted to PRL