Topological peculiarities of mammalian networks with different functionalities: transcription, signal transduction and metabolic networks

We have comparatively investigated three different mammalian networks - on transcription, signal transduction and metabolic processes - with respect to their common and individual topological traits. The networks have been constructed based on genome- wide data collected from human, mouse and rat. None of these three networks exhibits a pure power-law degree distribution and, therefore, could be considered scalefree. Rather, the degree distributions of all three networks were best fitted by mixed models of a power law with an exponential tail. The networks differ from one another in the quantitative parameters of the models. Moreover, the transcription network can also be very well approximated by an exponential law. The connectivity within each network is rather robust, as is seen when removing individual nodes and computing the values of their pairwise disconnectivity index (PDI), which characterizes the topological significance of each node v by the number of direct or indirect connections in the network that critically depend on the presence of v. The results evidence that the networks are not centralized: none of nodes globally controls the integrity of each network. Just a few vertices appeared to strongly affect the coherence of the networks. These nodes are characterized by a broad range of degrees, thereby indicating that the degree alone is not the decisive criteria of a node's importance. The networks reveal distinct architectures: The transcriptional network exhibits a hierarchical modularity, whereas the signaling network is mainly comprised of semi-autonomous modules. The metabolic network seems to be made by a more complex mixture of substructures. Thus, despite being encoded by the same genomes, the networks significantly differ from one another in their general architectural design. Altogether, our results indicate that the subsets of genes and relationships that constitute these networks have co-evolved very differently and through multiple mechanisms

Decidability of membership problems for flat rational subsets of GL(2, Q) and singular matrices

This work relates numerical problems on matrices over the rationals to symbolic algorithms on words and finite automata. Using exact algebraic algorithms and symbolic computation, we prove new decidability results for 2 × 2 matrices over Q. Namely, we introduce a notion of flat rational sets: if M is a monoid and N ≤ M is its submonoid, then flat rational sets of M relative to N are finite unions of the form L0g1 L1 ··· gtLt where all Lis are rational subsets of N and gi ∈ M. We give quite general sufficient conditions under which flat rational sets form an effective relative Boolean algebra. As a corollary, we obtain that the emptiness problem for Boolean combinations of flat rational subsets of GL(2, Q) over GL(2, Z) is decidable. We also show a dichotomy for nontrivial group extension of GL(2, Z) in GL(2, Q): if G is a f.g. group such that GL(2, Z) < G ≤ GL(2, Q), then either G ≅ GL(2, Z) × Zk, for some k ≥ 1, or G contains an extension of the Baumslag-Solitar group BS(1, q), with q ≥ 2, of infinite index. It turns out that in the first case the membership problem for G is decidable but the equality problem for rational subsets of G is undecidable. In the second case, decidability of the membership problem is open for every such G. In the last section we prove new decidability results for flat rational sets that contain singular matrices. In particular, we show that the membership problem is decidable for flat rational subsets of M(2, Q) relative to the submonoid that is generated by the matrices from M(2, Z) with determinants 0, ± 1 and the central rational matrices

The pairwise disconnectivity index as a new metric for the topological analysis of regulatory networks

<p>Abstract</p> <p>Background</p> <p>Currently, there is a gap between purely theoretical studies of the topology of large bioregulatory networks and the practical traditions and interests of experimentalists. While the theoretical approaches emphasize the global characterization of regulatory systems, the practical approaches focus on the role of distinct molecules and genes in regulation. To bridge the gap between these opposite approaches, one needs to combine 'general' with 'particular' properties and translate abstract topological features of large systems into testable functional characteristics of individual components. Here, we propose a new topological parameter – the pairwise disconnectivity index of a network's element – that is capable of such bridging.</p> <p>Results</p> <p>The pairwise disconnectivity index quantifies how crucial an individual element is for sustaining the communication ability between connected pairs of vertices in a network that is displayed as a directed graph. Such an element might be a vertex (i.e., molecules, genes), an edge (i.e., reactions, interactions), as well as a group of vertices and/or edges. The index can be viewed as a measure of topological redundancy of regulatory paths which connect different parts of a given network and as a measure of sensitivity (robustness) of this network to the presence (absence) of each individual element. Accordingly, we introduce the notion of a path-degree of a vertex in terms of its corresponding incoming, outgoing and mediated paths, respectively. The pairwise disconnectivity index has been applied to the analysis of several regulatory networks from various organisms. The importance of an individual vertex or edge for the coherence of the network is determined by the particular position of the given element in the whole network.</p> <p>Conclusion</p> <p>Our approach enables to evaluate the effect of removing each element (i.e., vertex, edge, or their combinations) from a network. The greatest potential value of this approach is its ability to systematically analyze the role of every element, as well as groups of elements, in a regulatory network.</p

270 Genome Informatics 16(2): 270–278 (2005) Topology of Mammalian Transcription Networks

We present a first attempt to evaluate the generic topological principles underlying the mammalian transcriptional regulatory networks. Transcription networks, TN, studied here are represented as graphs where vertices are genes coding for transcription factors and edges are causal links between the genes, each edge combining both gene expression and trans-regulation events. Two transcription networks were retrieved from the TRANSPATH r ○ database: The first one, TN RN, is a ‘complete ’ transcription network referred to as a reference network. The second one, TN p53, displays a particular transcriptional sub-network centered at p53 gene. We found these networks to be fundamentally non-random and inhomogeneous. Their topology follows a power-law degree distribution and is best described by the scale-free model. Shortest-path-length distribution and the average clustering coefficient indicate a small-world feature of these networks. The networks show the dependence of the clustering coefficient on the degree of a vertex, thereby indicating the presence of hierarchical modularity. Clear positive correlation between the values of betweenness and the degree of vertices has been observed in both networks. The top list of genes displaying high degree and high betweennes, such as p53, c-fos, c-jun and c-myc, is enriched with genes that are known as having tumor-suppressor or proto-oncogene properties, which supports the biological significance of the identified key topological elements