We introduce mutations in replication systems, in which the intact copying
mechanism is performed by discrete iterations of a complex quadratic map. More
specifically, we consider a "correct" function acting on the complex plane
(representing the space of genes to be copied). A "mutation" is a different
("erroneous") map acting on a complex locus of given radius r around a mutation
focal point. The effect of the mutation is interpolated radially to eventually
recover the original map 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, on both a
local and global scale (from cell diffetentiation to tumor formation). We use
the Julia 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 topology of the Julia set, hence the
system's long-term evolution, its progression into disease, but also its
ability to recover or heal. In the context of genetics, mutated iterations may
help shed some light on aspects such as the importance of location, size and
type of mutation when evaluating a system's prognosis, and of customizing
intervention.Comment: 15 pages, 15 figures, 7 reference
