Heredity in Messy Chemistries

For natural selection to progress, there must be a sufficiently large evolutionary space to explore. In systems with template-based replication, this space is combinatorially large in the length of the information-carrying molecules. Previous work has shown that it is also possible for heredity to occur in much less structured chemistries; this opens the question of how the structure of a reaction network relates to the number of heritable states it can support, and in particular, how the number of heritable states scales with system size for a given network topology. Answering this question would allow us to map out the space of possible chemical mechanisms for heredity, and to identify places where they might be found in the space of organic chemistries that might have been found on the early Earth. We show that by linearising around a fixed point in a chemical reaction network and solving the corresponding eigenvalue problem, it is possible to detect the set of independent autocatalytic subnetworks that can operate in the vicinity of that point. We investigate an upper bound on the scaling of the number of such “autocatalytic cores” with the number of distinct chemical species, and show that the number of cores scales at best as log N in the case of unstructured networks, but that adding a strong energy constraint on the network topology allows it to scale linearly, which is the best possible case.