Evolutionary origin of power-laws in a biochemical reaction network: Embedding the distribution of abundance into topology

The evolutionary origin of general statistics in a biochemical reaction network is studied here to explain the power-law distribution of reaction links and the power-law distribution of chemical abundance. Using cell models with catalytic reaction networks, we have confirmed that the power-law distribution for the abundance of chemicals emerges by the selection of cells with higher growth rates, as suggested in our previous study [Phys. Rev. Lett. 90, 088102 (2003)]. Through further evolution, this inhomogeneity in chemical abundance is shown to be embedded in the distribution of links, leading to the power-law distribution. We analyze the mechanism of this embedding and discuss the generality of the results.