Mutability as an altruistic trait in finite asexual populations

Evolution of bow-tie architectures in biology

Bow-tie or hourglass structure is a common architectural feature found in many biological systems. 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 the information in the evolutionary goal can be compressed. Mathematically speaking, bow-ties evolve when the rank of the input-output matrix describing the evolutionary goal is deficient. The maximal compression possible (the rank of the goal) determines the size of the narrowest part of the network—that is the bow-tie. A further requirement is that a process is active to reduce the number of links in the network, such as product-rule mutations, otherwise a non-bow-tie solution is found in the evolutionary simulations. This offers a mechanism to understand a common architectural principle of biological systems, and a way to quantitate the effective rank of the goals under which they evolved.

Many biological systems show bow-tie (also called hourglass) architecture. A bow-tie means that a large number of inputs are converted to a small number of intermediates, which then fan out to generate a large number of outputs. For example, cells use a wide variety of nutrients; process them into 12 metabolic precursors, which are then used to make all of the cells biomass. Similar principles exist in biological signaling and in the information processing in the visual system. Despite the ubiquity of bow-tie structures in biology, there is no explanation of how they evolved. Here, we find that bow-ties spontaneously evolve when the information in the evolutionary goal they evolved to satisfy can be compressed. Mathematically, this means that the matrix representing the goal has deficient rank. The maximal compression possible determines the width of the bow-tie—the narrowest part in the network (equal to the rank of the goal matrix). This offers a mechanism to understand a common architectural principle of biological systems, and a way to quantitate the rank of the goals under which they evolved.

Mutation rules and the evolution of sparseness and modularity in biological systems

Biological systems exhibit two structural features on many levels of organization: sparseness, in which only a small fraction of possible interactions between components actually occur; and modularity – the near decomposability of the system into modules with distinct functionality. Recent work suggests that modularity can evolve in a variety of circumstances, including goals that vary in time such that they share the same subgoals (modularly varying goals), or when connections are costly. Here, we studied the origin of modularity and sparseness focusing on the nature of the mutation process, rather than on connection cost or variations in the goal. We use simulations of evolution with different mutation rules. We found that commonly used sum-rule mutations, in which interactions are mutated by adding random numbers, do not lead to modularity or sparseness except for in special situations. In contrast, product-rule mutations in which interactions are mutated by multiplying by random numbers – a better model for the effects of biological mutations – led to sparseness naturally. When the goals of evolution are modular, in the sense that specific groups of inputs affect specific groups of outputs, product-rule mutations also lead to modular structure; sum-rule mutations do not. Product-rule mutations generate sparseness and modularity because they tend to reduce interactions, and to keep small interaction terms small.