Clustering tree based on the adjusted Rand index values for the investigated CoCo clusterings.

<p>The clusterings from single networks are based on the greedy heuristic for approximating the MODULARITY problem. All other clusterings are based on the greedy heuristic for COCONETS. The tree is derived by agglomerative clustering with a distance matrix derived from the adjusted Rand index values for all pairwise comparisons of the obtained CoCo clusterings. The stress conditions are denoted as follows: cold (c), heat (h), lactose diauxie (ld), oxidative (o), and stationary phase (s); their pairwise combinations are marked with ‘/’, and the clustering over all five stresses, by ‘all’. The number of clusters in each CoCo clustering is included next to the abbreviations for the stresses.</p

Condition-specific network properties and pairwise network similarities for eight environmental conditions in <i>Arabidopsis thaliana</i>.

<p>The upper part of the table includes seven seminal network properties together with the thresholds used to establish the edges in the correlation networks of metabolites under eight investigated stresses: 4 C and darkness (4-D), 21 C and darkness (21-D), 32 C and darkness (32-D), 4 C and light (4-L), 21 C and low-light (21-LL), 21 C and high light (21-HL), and 32 C and light (32-L). The lower part of the table includes the Jaccard similarity between the edge-sets of the condition-specific networks.</p

Illustration of COCONETS.

<p>Given are three networks in the top row. The clusters in the optimal clustering for each network are marked in different colors (red and blue). The optimal concurrent conditional clustering of the two networks given in the middle row is of value 0 and is suboptimal for the network to the left. The optimal clusterings for any of the two networks shown in the bottom row are suboptimal concurrent conditional clustering; the shown clustering yields a value of −0.24, while the optimal clustering is of value 0, whereby all nodes form a single cluster (not shown).</p

Illustration of network comparison based on community structure.

<p>Shown are three networks, , , and . Nodes belonging to the same community in each network are marked by the same color. Networks and differ in 11 edges, while networks and do not share 4 edges. Nevertheless, the community structures between and are equivalent, while this is not the case for the community structures in and .</p

Concurrent Conditional Clustering of Multiple Networks: COCONETS

<div><p>The accumulation of high-throughput data from different experiments has facilitated the extraction of condition-specific networks over the same set of biological entities. Comparing and contrasting of such multiple biological networks is in the center of differential network biology, aiming at determining general and condition-specific responses captured in the network structure (<i>i.e.</i>, included associations between the network components). We provide a novel way for comparison of multiple networks based on determining network clustering (<i>i.e.</i>, partition into communities) which is optimal across the set of networks with respect to a given cluster quality measure. To this end, we formulate the optimization-based problem of concurrent conditional clustering of multiple networks, termed COCONETS, based on the modularity. The solution to this problem is a clustering which depends on all considered networks and pinpoints their preserved substructures. We present theoretical results for special classes of networks to demonstrate the implications of conditionality captured by the COCONETS formulation. As the problem can be shown to be intractable, we extend an existing efficient greedy heuristic and applied it to determine concurrent conditional clusters on coexpression networks extracted from publically available time-resolved transcriptomics data of <i>Escherichia coli</i> under five stresses as well as on metabolite correlation networks from metabolomics data set from <i>Arabidopsis thaliana</i> exposed to eight environmental conditions. We demonstrate that the investigation of the differences between the clustering based on all networks with that obtained from a subset of networks can be used to quantify the specificity of biological responses. While a comparison of the <i>Escherichia coli</i> coexpression networks based on seminal properties does not pinpoint biologically relevant differences, the common network substructures extracted by COCONETS are supported by existing experimental evidence. Therefore, the comparison of multiple networks based on concurrent conditional clustering offers a novel venue for detection and investigation of preserved network substructures.</p></div

Segmentation for yeast’s metabolic cycle.

<p>The partitions found by applying our method are highlighted in light grey. The phases of the yeast metabolic cycle are indicated with colored rectangles above each panel following Tu <i>et al. </i><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062974#pone.0062974-Tu1" target="_blank">[36]</a>. R/C stands for reductive charging, OX oxidative metabolism, and R/B, reductive metabolism. (a) shows the segmentations caught by relative density as global property; (b) illustrates the segmentations based on degree; (c) and (d) demonstrate segmentations with local-global properties, betweenness and closeness, respectively. The segmentations in panel (a) performs particularly well due to the global changes in the form of global cycles in the data set from yeast.</p

Directed acyclic graph (DAG) used as input in Algorithm 1 (Fig. 3).

<p>The DAG for time points is depicted. It contains nodes, including the special nodes and . The label of each node corresponds to the time points , , and .</p

Optimal segmentation for synthetic data.

<p>The upper part of the table shows the result of the optimal segmentation for synthetic data based on dynamic programming, while the lower part contains the result based on the method of Ramakrishnan <i>et al.</i><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0062974#pone.0062974-Ramakrishnan1" target="_blank">[15]</a>. In the upper table, the first and second columns show the name and the type of network properties used to determine the distances: G stands for global, L for local, and LG for local-global. The third column includes the number of segments that maximize the objective with the dynamic programming approach. The resulting segments are given in the forth column, while the fifth and sixth columns contain the corresponding values of lower () and upper () bound of the tuning parameter . The lower part also includes minimum and maximum length of the segments, i.e., and , as parameters of the contending method.</p

Network-Based Segmentation of Biological Multivariate Time Series

<div><p>Molecular phenotyping technologies (<i>e.g.</i>, transcriptomics, proteomics, and metabolomics) offer the possibility to simultaneously obtain multivariate time series (MTS) data from different levels of information processing and metabolic conversions in biological systems. As a result, MTS data capture the dynamics of biochemical processes and components whose couplings may involve different scales and exhibit temporal changes. Therefore, it is important to develop methods for determining the time segments in MTS data, which may correspond to critical biochemical events reflected in the coupling of the system’s components. Here we provide a novel network-based formalization of the MTS segmentation problem based on temporal dependencies and the covariance structure of the data. We demonstrate that the problem of partitioning MTS data into segments to maximize a distance function, operating on polynomially computable network properties, often used in analysis of biological network, can be efficiently solved. To enable biological interpretation, we also propose a breakpoint-penalty (BP-penalty) formulation for determining MTS segmentation which combines a distance function with the number/length of segments. Our empirical analyses of synthetic benchmark data as well as time-resolved transcriptomics data from the metabolic and cell cycles of <i>Saccharomyces cerevisiae</i> demonstrate that the proposed method accurately infers the phases in the temporal compartmentalization of biological processes. In addition, through comparison on the same data sets, we show that the results from the proposed formalization of the MTS segmentation problem match biological knowledge and provide more rigorous statistical support in comparison to the contending state-of-the-art methods.</p></div

Illustration of the segmentation for synthetic data with relative density as network property.

<p>The resulting partitions are highlighted in light grey and the simulated segmentation points are marked with red bars.</p