Modeling Biological Networks

• IV.6.i Subproject 1 Topological Analysis of Selected Metabolic and Regulatory Networks for Large-scale Structure
  • IV.6.i.a Datasources
  • IV.6.i.b Methodology
  • IV.6.i.c Large-Scale Statistical Characterization of Pathways
  • IV.6.i.d Clustering
    • IV.6.i.d.1 Clustering algorithms
      • IV.6.i.d.1.i Spatial clustering algorithms
      • IV.6.i.d.1.ii Clustering by thermal annealing
      • IV.6.i.d.1.iii Network clustering
• [ Other Subprojects ]

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This research will employ ideas developed in the last few years to analyze other complex networks, including the world-wide web and citation networks (Barabasi and Albert, 1999; Albert, Jeong and Barabasi, 2000; Strogatz, 2001). We plan to use several additional quantities, ranging from network diameter to generic robustness measures (Albert et al., 2000), to describe the large-scale structure in detail.

While organizing the metabolism of E. coli and other organisms into pathways is convenient, such organization is somewhat arbitrary. Therefore, we need to augment historical pathway classifications with automatic procedures to identify independent pathways starting from the stoichiometric matrix. We define an "independent pathway" as the smallest set of reactions connecting a single network output to the necessary network inputs, in a way that permits the levels of internal species to reach a stationary state (Klapa et al., 1998; Park et al., 1998; Simpson, et al., 1999). Our goal is to use existing methods to identify independent pathways (Schuster et al., 2000). The detailed picture of the pathway structure will serve as input, augmented by insights offered by our large-scale network statistical analysis.

Some important quantities and methods for characterizing the topology of metabolic and genetic networks (Jeong et al., 2000; Jeong et al., 2001) include the shortest path , giving the set of links that offer the shortest distance between two nodes; the diameter <>, which is the average of over all pairs of nodes; and the path length distribution P().

Fig. IV.5. Mapping between a given network topology and the corresponding connection matrix. (a) For binary network, two nodes either connect (1) or not (0). The connectivity matrix contains full topological information on the network: for example, the first row shows that no direct 1 1, 1 2, 1 4 links exist, since in the position 1, 2, 4 we have a 0 in the matrix. However, a 1 3 link exists, denoted by a 1 in position 3. (b) A small network illustrating the concept of the shortest path. To get from node 1 to node 6, we can use 3 alternative routes: 126, 1326, and 1456. Of these, the shortest one is the 126 path. Also, while compared to each other, the paths 126 and 1456 are absolutely independent because teir only common points are their starting and ending nodes, the paths 126 and 1 326 are only marginally independent since they share the 26 segment.

The set of independent paths that connect two randomly selected components is an important quantity characterizing the robustness of a given metabolic network. We call all paths that connect i and j but do not have common edges (links) and/or nodes, completely independent (Figure IV.5), while paths connecting i and j which differ in at least one link are marginally independent (Figure IV.5). The existence of many independent paths increases the robustness of a metabolic network; if only one path links two components (nodes), then damaging or removing any link or node on this path prevents the two components from communicating. Thus, the path structure determines the degree of error tolerance of a metabolic network. We plan to investigate the structure and scaling properties of these paths, and their dependence on other network characteristics. We also plan to correlate the path characteristics with other measures of network topology.

Independent pathways describe functionally relevant subsets of independent paths connecting not randomly chosen but biologically relevant input and output metabolites. We will use the method of Simpson et al. (1999) to identify these for E. coli and other organisms. In addition, after identifying the full ensemble of independent pathways, we will attempt a large-scale statistical analysis to determine how the number of independent pathways scales with the number of participating metabolites and other network characteristics. This analysis will include a comparative study of several prokaryotic organisms.

At a higher level of organization, the structure of the independent paths and pathways relates closely to the concept of elementary modes in metabolic networks (Schuster et al., 2000). An elementary mode represents the minimal set of enzymes that could operate in a steady state, with the relative flux they need to carry for the mode to function weighting the enzymes. An advantage of this approach is that we can express any steady state flux pattern as a non-negative linear combination of the various elementary modes. Using a physical analogy, elementary modes allow us to decompose the full metabolism into a mixture of pure, isolated states. The number of elementary modes is an important quantitative characterization of metabolism. However, the number of elementary modes depends on the size of the metabolic subset considered. For example, monosaccharide metabolism supports seven modes, but we expect to find many more if we investigate a larger superset module. The dependence of the number of modes on the size of the module investigated gives important insight into the interdependence of the various metabolic pathways. Such results help identify modules.