Revisiting the evolution of bow-tie architecture in signaling networks.

Bow-tie architecture is a layered network structure that has a narrow middle layer with multiple inputs and outputs. Such structures are widely seen in the molecular networks in cells, suggesting that a universal evolutionary mechanism underlies the emergence of bow-tie architecture. The previous theoretical studies have implemented evolutionary simulations of the feedforward network to satisfy a given input-output goal and proposed that the bow-tie architecture emerges when the ideal input-output relation is given as a rank-deficient matrix with mutations in network link intensities in a multiplicative manner. Here, we report that the bow-tie network inevitably appears when the link intensities representing molecular interactions are small at the initial condition of the evolutionary simulation, regardless of the rank of the goal matrix. Our dynamical system analysis clarifies the mechanisms underlying the emergence of the bow-tie structure. Further, we demonstrate that the increase in the input-output matrix reduces the width of the middle layer, resulting in the emergence of bow-tie architecture, even when evolution starts from large link intensities. Our data suggest that bow-tie architecture emerges as a side effect of evolution rather than as a result of evolutionary adaptation.