Metabolic networks are almost nonfractal: A comprehensive evaluation

Network self-similarity or fractality are widely accepted as an important topological property of metabolic networks; however, recent studies cast doubt on the reality of self-similarity in the networks. Therefore, we perform a comprehensive evaluation of metabolic network fractality using a box-covering method with an earlier version and the latest version of metabolic networks and demonstrate that the latest metabolic networks are almost self-dissimilar, while the earlier ones are fractal, as reported in a number of previous studies. This result may be because the networks were randomized because of an increase in network density due to database updates, suggesting that the previously observed network fractality was due to a lack of available data on metabolic reactions. This finding may not entirely discount the importance of self-similarity of metabolic networks. Rather, it highlights the need for a more suitable definition of network fractality and a more careful examination of self-similarity of metabolic networks.

Figure 1

Comparison between the coefficients of determination of an exponential fit ($R_{\exp }^2$) and power-law fit ($R_{\mathrm{pow}}^2$). (a) Scatter plot of $R_{\exp }^2$ versus $R_{\mathrm{pow}}^2$. Distributions of $R_{\exp }^2-R_{\mathrm{pow}}^2$ in the earlier version (b) and latest version (c) of metabolic networks.

Figure 2

Scatter plots of the normalized number of boxes [i.e., $P_B\left(l_B\right)=N_B\left(l_B\right)/N$] versus box size $l_B$ in three representative organisms, obtained from the latest version (upper row) and earlier version (lower row) of metabolic networks: E. coli (left column), Saccharomyces cerevisiae (yeast) (middle column), Arabidopsis thaliana (thale cress) (right column). Solid and dashed lines indicate exponential fits and power-law fits, respectively.

Figure 3

Negative correlation between ${\Delta }_{R_{\mathrm{pow}}^2}$ and $R_D$ (Spearman's rank correlation coefficient $r_s=-0.30$ and $p=5.2\times {10}^{-5}$).

Figure 4

Distributions of $R_{\exp }^2-R_{\mathrm{pow}}^2$, obtained from the latest version of metabolic networks, in archaea (a), bacteria (b), and eukaryotes (c).

Figure 5

Scatter plot of $R_{\exp }^2$ versus $R_{\mathrm{pow}}^2$ in the latest version of eukaryotic metabolic networks.