Emergence of Bow-tie Architecture in Evolving Feedforward Networks

Abstract

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 strength 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 facilitates the emergence of bow-tie architecture even when starting from strong network links. Our data suggest that bow-tie architecture emerges as a side effect of evolution rather than as a result of evolutionary adaptation. Author SummaryMany biological networks including gene regulatory networks, metabolic networks, and signaling networks, show a characteristic hierarchical structure known as the bow-tie architecture. This architecture consists of a narrow middle layer with multiple inputs and outputs. Understanding why bow-tie architecture has universally appeared through evolution may provide insight into the design principle of a network within a cell. The universality of the bow-tie structure has so far been explained by the adaptive advantages such as high evolvability and capability to classify the inputs. However, our computer simulation demonstrates that the bow-tie structure inevitably emerges even without functional advantages when the molecular interactions within the network are weak at the initial condition of the evolution. We also demonstrate that an increase in the number of inputs (i.e., receptors) and outputs (i.e., downstream genes) leads to the emergence of bow-tie architecture, even when evolution starts from a strong molecular interaction. Although many previous studies have discussed the adaptive properties of bow-tie architecture, our findings suggest that bow-tie architecture is a byproduct of evolution rather than a result of evolutionary adaptation.