Epistatic pathways in evolvable mechanical networks

An elastic spring network is an example of evolvable matter. It can be pruned to couple separated pairs of nodes so that when a strain is applied to one of them, the other responds either in-phase or out-of-phase. This produces two pruned networks with incompatible functions that are nearly identical but differ from each other by a set of "mutations," each of which removes or adds a single bond in the network. The effect of multiple mutations is epistatic; that is, the effect of a mutation depends on what other mutations have already occurred. We generate ensembles of network pairs that differ by a fixed number, $M$, of discrete mutations and evaluate all $M!$ mutational paths between the in- and out-of phase behaviors up to $M = 14$. With a threshold response for the network to be considered functional, so that non-functional networks are disallowed, only some mutational pathways are viable. We find that there is a surprisingly high critical response threshold above which no evolutionarily viable path exists between the two networks. The few remaining pathways at this critical value dictate much of the behavior along the evolutionary trajectory. In most cases, the mutations break up into two distinct classes. The analysis clarifies how the number of mutations and the position of a mutation along the pathway affect the evolutionary outcome.