Metabolic network modularity arising from simple growth processes

Metabolic networks consist of linked functional components, or modules. The mechanism underlying metabolic network modularity is of great interest not only to researchers of basic science but also to those in fields of engineering. Previous studies have suggested a theoretical model, which proposes that a change in the evolutionary goal (system-specific purpose) increases network modularity, and this hypothesis was supported by statistical data analysis. Nevertheless, further investigation has uncovered additional possibilities that might explain the origin of network modularity. In this work we propose an evolving network model without tuning parameters to describe metabolic networks. We demonstrate, quantitatively, that metabolic network modularity can arise from simple growth processes, independent of the change in the evolutionary goal. Our model is applicable to a wide range of organisms and appears to suggest that metabolic network modularity can be more simply determined than previously thought. Nonetheless, our proposition does not serve to contradict the previous model; it strives to provide an insight from a different angle in the ongoing efforts to understand metabolic evolution, with the hope of eventually achieving the synthetic engineering of metabolic networks.

Figure 1

Schematic diagram depicting the model of growing metabolic networks. (a) Event I. Nodes in black and gray represent, respectively, a new node and a randomly selected existing node. (b–d) Event II. Red (gray) lines correspond to new edges. Nodes in red (rightmost gray nodes) are randomly selected existing nodes; those in green (the other gray nodes) are existing nodes selected by a random walk from each red node. The dashed line indicates the reduced path on which nodes may exist. The new edge becomes a shortcut between the red and the green nodes. Note that the shortcut path, bypassing a path of length $l$ selected through the random walk mechanism, is accepted with the probability $q\left(l\right)$.

Figure 2

Probability $q\left(l\right)$ decays exponentially with the length $l$ of bypassed paths. The filled symbols correspond to the empirical cumulative probability distributions $q\left(l\right)$ of (a) Pyrobaculum aerophilum, (b) Methanothermobacter thermautotrophicus, (c) Bacillus subtilis, (d) Escherichia coli, (c) Saccharomyces cerevisiae, and (f) Homo sapiens. Note that a cumulative distribution is defined as $P(X\ge x)$. Probability distributions from the null model (open symbols) are averaged over 300 realizations. The solid line indicates approximations of the empirical probability function $q\left(l\right)$ using the geometric distribution.

Figure 3

Geometric distributions of $P$ values of empirical and predicted models, derived using the Kolmogorov-Smirnov (KS) test. The KS test was performed in 113 organisms; the median $P$ value is approximately 0.8. The inset shows the $P$-value distributions of the empirical data and the null model [i.e., $q\left(l\right)$ from randomized networks] obtained using the KS test. $P$ values are averaged over 300 realizations, and the median is less than ${10}^{-7}$.

Figure 4

Comparison of the network modularity measure (a) and the number of modules (b) between the empirical and model metabolic networks in 113 organisms. The network modularity measure and the number of modules from our model and the null model are averaged over 300 realizations.

Figure 5

Cumulative distributions of module size in six representative organisms: (a) Pyrobaculum aerophilum, (b) Methanothermobacter thermautotrophicus, (c) Bacillus subtilis, (d) Escherichia coli, (e) Saccharomyces cerevisiae, and (f) Homo sapiens. Note that a cumulative distribution is defined as $P(X\ge x)$. The cumulative distributions derived from our model and the null model are averaged over 300 realizations.

Figure 6

$P$-value comparison of empirical distributions and predicted distributions using the Kolmogorov-Smirnov (KS) test. (a) $P_{\mathrm{model}}$ corresponds to the $P$ value when our model is applied, and $P_{\mathrm{null}}$ corresponds to that when the null model is applied. The medians of the $P_{\mathrm{model}}$ (b) and $P_{\mathrm{null}}$ (c) distributions are 0.66 and 0.31, respectively. The KS test was performed in 113 organisms. The $P$ values are averaged over 300 realizations.

Figure 7

Correlation between the normalized network modularity measure and the model parameter. See Appendix for the normalization procedure of network modularity measure.