Bow-tie or hourglass structure is a common architectural feature found in biological and technological networks. A bow-tie in a multi-layered structure occurs when intermediate layers have much fewer components than the input and output layers. Examples include metabolism where a handful of building blocks mediate between multiple input nutrients and multiple output biomass components, and signaling networks where information from numerous receptor types passes through a small set of signaling pathways to regulate multiple output genes. Little is known, however, about how bow-tie architectures evolve. Here, we address the evolution of bow-tie architectures using simulations of multi-layered systems evolving to fulfill a given input-output goal. We find that bow-ties spontaneously evolve when two conditions are met: (i) the evolutionary goal is rank deficient, where the rank corresponds to the minimal number of input features on which the outputs depend, and (ii) The effects of mutations on interaction intensities between components are described by product rule -namely the mutated element is multiplied by a random number. Product-rule mutations are more biologically realistic than the commonly used sum-rule mutations that add a random number to the mutated element. These conditions robustly lead to bow-tie structures. The minimal width of the intermediate network layers (the waist or knot of the bow-tie) equals the rank of the evolutionary goal. These findings can help explain the presence of bow-ties in diverse biological systems, and can also be relevant for machine learning applications that employ multi-layered networks. Many biological systems show bow-tie architecture: a huge number of inputs are converted to a small number of intermediates, which then fan out to generate a huge number of outputs. Examples are metabolic and signaling networks. However, there is no explanation of how bowties evolve in biology. Here, we find that bow-ties spontaneously evolve when the information in the evolutionary goal can be compressed, known in mathematics as a deficient goal rank. This rank is a number that determines the size of the narrowest part of the bow-tie in the network. This offers a first mechanism to understand a common architectural principle of biological systems, and a way to quantitate the rank of the goals under which they evolved. 3
The third study REF showed that a layered and directed network can evolve to a bow-tie structure if the relation between inputs and outputs can be represented with a rank-deficient matrix, and if the mutations in the intensity (weights) of module interactions (network edges) can be modeled as products by a random number (rather than sums).
Evolution of bow-tie architectures in biology
PLoS Comput Biol 11(3): e1004055
Biology Medicine Computer Science