Effects of local mutations in quadratic iterations

Abstract We introduce mutations in replication systems in which the intact copying mechanism is performed by discrete iterations of a complex quadratic map in the family fc ( z ) = z 2 + c . More specifically, we consider a “correct” function fc 1 acting on the complex plane (representing the DNA to be copied). A “mutation” fc 0 is a different (“erroneous”) map acting on a locus of given radius r around a mutation focal point ξ ∗. The effect of the mutation is interpolated radially to eventually recover the original map fc 1 when reaching an outer radius R . We call the resulting map a “mutated” map. In the theoretical framework of mutated iterations, we study how a mutation (replication error) affects the temporal evolution of the system, in the context of cell differentiation and tumor formation. We use the prisoner set of the system to quantify simultaneously the long-term behavior of the entire space under mutated maps. We analyze how the position, timing and size of the mutation can alter the system's long-term evolution (as encoded in the topology of its prisoner set), its progression into disease and its ability to recover or heal. In the context of genetics, mutated iterations may shed some light on aspects such as the importance of location, size and type of mutation when evaluating a system’s prognosis, and suggest ways to customize intervention.