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Figure 5. Topology of the shortest pathways: A) The metabolic network of a eukaryote E. niduLtns of length 26. B) The Internet at AS level of length 10. C) The metabolic network ofan archae Methanococcusjannaschii of length 20. D) WWW of www.nd.edu with length 20. In (A) and (C), circles denote substrates and rectangles denote intermediate states. From Goh KI et al, Proc Nad Acad Sci USA 99:12583-8, ©2002 National Academy of Sciences, USA, with permission.40

Figure 6. Global snapshot of the metabolic network of E. nidulans. The metabolites are shown in blue and the enzymes in light blue. Highlighted in orange (metabolites) and yellow (enzymes) are the shortest pathways of longest length, d = 26, whose starting and end points are indicated in green. A color version of this figure is available online at www.Eurekah.com.

a substrate and an enzyme is taken as the unit of length. From Figure 5 A, one can see that there exists a blob structure inside which vertices are multiply connected, while vertices outside are singly connected. The characteristic of the class I is that the blob is localized in a small region. To give a visual image of the existence of the localized blob, we show the global snapshot of the shortest pathways in the EN metabolic network in Figure 6.

For the class II, the mass depends on distance linearly, M(d) - Ad for large d (Fig. 4C,D). Despite the linear dependence, the shortest pathway topology for the case of A > 1 is more complicated than that of the simple tree structure where A = 1. Therefore, the SF networks in the class II are subdivided into two types, called the class Ila and lib, respectively. For the class Ila, A > 1 and the topology of the shortest pathways includes multiply connected vertices (Fig. 5B and C), while for the class lib, A = 1 and the shortest pathway is almost singly connected (Fig. 5D). Examples in real world networks in the class Ila are the Internet at the AS level (A - 4.5) and the metabolic network for archaea (A-2.0), while that in the class lib is the WWW (A - 1.0).

The WWW is an example belonging to the class lib. For this network, the mass-distance relation exhibits M(d) - 1.0d-, suggesting that the topology of the shortest pathway is almost singly connected, which is confirmed in Figure 5D. When a SF network is of tree structure, one can solve the distribution of load running through each edge analytically, and obtain the load exponent to be 8 = 2.

Eukaryoîes

Eukaryoîes x

Figure 7. Mass-distance relations for the metabolic networks of the three domains of life: 6 archaea, 32 bacteria, and 5 eukaryotes, respectively, are plotted. In the bottom-right panel, M(d) averaged over all species in each domain are compared. + stands for the data for archaea, * for bacteria, and ° for eukayotes. The straight lines have slopes 1.25 (black) and 0.5 (orange), respectively, drawn for the eye. Note that since we count only the metabolites in M(d), M(d) = 0.5^for singly-connected shortest pathways. From Goh KI et al, Proc Nad Acad Sci USA 99:12583-8, ©2002 National Academy of Sciences, USA, with permission.40 A color version of this figure is available online at www.Eurekah.com.

In biological perspectives, the power of the shortest pathway analysis and the resulting classification is exemplified by the success in the categorizing the domains of life. In Figure 7, we show the mass-distance relations of the metabolic networks of all 43 species that we considered, grouped by the domains. Evidendy, M(d) for archaea behave differently from that for bacteria and eukaryotes. The eukaryotes have the class I-type metabolic networks and the archaea have the class II-type ones. The existence of the blob in eukaryotes and lack thereof in archaea implies the formation of such architecture might be driven by evolutionary pressure. One advantage of having the class I-type topology is that it is more resilient to the targeted attack on highly connected vertices.40 It would be interesting to extend such idea to a more realistic situation for the metabolic stability.

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