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First published online October 27, 2005
doi: 10.1242/10.1242/jcs.02714

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Scale-free networks in cell biology

Department of Physics and Huck Institutes of the Life Sciences, Pennsylvania State University, University Park, PA 16802, USA

View larger version (24K): [in a new window]   Fig. 1. Graph representation and graph analysis reveals regulatory patterns of cellular networks. The number of interactions a component participates in is quantified by its (in/out) degree, for example node O has both in-degree and out-degree 2. The clustering coefficient characterizes the cohesiveness of the neighborhood of a node - for example the clustering coefficient of I is 1, indicating that it is part of a three-node clique. The graph distance between two nodes is defined as the number of edges in the shortest path between them. For example, the distance between nodes P and O is 1, and the distance between nodes O and P is 2 (along the OQP path). The degree distribution P(k) [P(kin) and P(kout) in directed networks] quantifies the fraction of nodes with degree k, while the clustering-degree function C(k) gives the average clustering coefficient of nodes with degree k. (a) A linear pathway can be represented as a succession of directed edges connecting adjacent nodes. Because there are no shortcuts or feedbacks in a linear pathway, the distance between the starting node and end node increases linearly with the number of nodes. The in-degree and out-degree distribution indicates the existence of a source (kin=0) and a sink (kout=0) node. (b) This undirected and disconnected graph is composed of two connected components (EFGH and IJK), has a range of degrees from 1 to 3 and a range of clustering coefficients from 0 (for F) to 1 (for I, J and K). The connected component IJK is also a clique (completely connected subgraph) of three nodes. (c) This directed graph contains a feed-forward loop (MON) and a feedback loop (POQ), which is also the largest strongly connected component of the graph. The in-component of this graph contains L and M, while its out-component consists of the sink nodes N and R. The source node L can reach every other node in the network.

View larger version (17K): [in a new window]   Fig. 2. Comparison between the degree distribution of scale-free networks () and random graphs () having the same number of nodes and edges. For clarity the same two distributions are plotted both on a linear (left) and logarithmic (right) scale. The bell-shaped degree distribution of random graphs peaks at the average degree and decreases fast for both smaller and larger degrees, indicating that these graphs are statistically homogeneous. By contrast, the degree distribution of the scale-free network follows the power law P(k) = Ak-3, which appears as a straight line on a logarithmic plot. The continuously decreasing degree distribution indicates that low-degree nodes have the highest frequencies; however, there is a broad degree range with non-zero abundance of very highly connected nodes (hubs) as well. Note that the nodes in a scale-free network do not fall into two separable classes corresponding to low-degree nodes and hubs, but every degree between these two limits appears with a frequency given by P(k).

View larger version (84K): [in a new window]   Fig. 3. C. elegans protein interaction network. The nodes are colored according to their phylogenic class: ancient, red; multicellular, yellow; and worm, blue. The inset highlights a small part of the network. Figure reproduced with permission from the American Association for the Advancement of Science (Li, S. et al., 2004).

View larger version (13K): [in a new window]   Fig. 4. Topological properties of the yeast protein interaction network constructed from four different databases. (a) Degree distribution. The solid line corresponds to a power law with exponent = 2.5. (b) Clustering coefficient. The solid line corresponds to the function C(k) = B/k2. (c) The size distribution of connected components. All the networks have a giant connected component of >1000 nodes (on the right) and a number of small isolated clusters. Figure reproduced with permission from Wiley-VCH (Yook et al., 2004).

View larger version (19K): [in a new window]   Fig. 5. Three possible representations of a reaction network with three enzyme-catalyzed reactions and four reactants. (a) The most detailed picture includes three types of node - reactants (circles), reactions (ovals) and enzymes (squares) - and two types of edge - mass flow (solid lines) or catalysis (dashed lines). The edges are marked by the stochiometric coefficients of the reactants. (b) In the metabolite network all reactants that participate in the same reaction are connected; thus the network is composed of a set of completely connected subgraphs (triangles in this case). (c) In the reaction network, two reactions are connected if they share a reactant. A similar graph can be constructed for the enzymes as well.

View larger version (19K): [in a new window]   Fig. 6. Rank (cumulative distribution) of metabolite node degree (left panel) and reaction node degree (right panel) for metabolic networks of H. pylori. The straight lines correspond to a power-law degree distribution with exponent = |slope| + 1 = 2.32. The figure illustrates that functionally different metabolites tend to cover different ranges of the degree spectrum. Reproduced with permission from the American Physical Society (Tanaka, 2005).

View larger version (34K): [in a new window]   Fig. 7. Interactions among 52 genes in the transcriptional regulation network of S. cerevisiae. The gene names are arranged in such a way that left to right illustrates causality. The number of non-regulatory genes regulated by each column of regulatory genes is shown above. Bold type indicates self-activation, bold italics indicates self-inhibition, and borders indicate essential genes. Reproduced with permission from the Nature Publishing Group (http://www.nature.com/naturegenetics) (Guelzim et al., 2002).

View larger version (19K): [in a new window]   Fig. 8. Genome-wide distribution of transcriptional regulators in S. cerevisiae. (A) Solid symbols represent the number of transcription factors bound per promoter region (corresponding to the in-degree of the regulated gene). Open symbols represent the in-degree distribution of a comparable randomized network. (B) Distribution of the number of promoter regions bound per regulator (i.e. the out-degree distribution of transcription factors). Figure reproduced with permission from the American Association for the Advancement of Science (Lee et al., 2002).

View larger version (9K): [in a new window]   Fig. 9. Connections between pathway redundancy and synthetic lethal interactions. Consider a hypothetical cellular network module (a) that receives exogeneous signals through node A and whose sink node F determines the response to the signal (or the phenotype). There are two node-independent (redundant) pathways between nodes A and F that can compensate for each other in case of node disruptions. By defining synthetic lethal interactions as pairs of nodes whose loss causes the disconnection of nodes A and F, one would find graph b. The two graphs present complementary and non-overlapping information.

View larger version (42K): [in a new window]   Fig. 10. Network motifs and themes in the integrated S. cerevisiae network. Edges denote transcriptional regulation (R), protein interaction (P), sequence homology (H), correlated expression (X) or synthetic lethal interactions (S). (a) Motifs corresponding to the `feed-forward' theme are based on transcriptional feed-forward loops; (b) motifs in the `co-pointing' theme consist of interacting transcription factors that regulate the same target gene; (c) motifs corresponding to the `regulonic complex' theme include co-regulation of members of a protein complex; (d) motifs in the `protein complex' theme represent interacting and coexpressed protein cliques. For a given motif, Nreal is the number of corresponding subgraphs in the real network, and Nrand is the number of corresponding subgraphs in a randomized network. Figure reproduced with permission from BioMed Central (Zhang et al., 2005).

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