  Genetic sequences are known to possess non-trivial composition together with
symmetries in the frequencies of their components. Recently, it has been shown
that symmetry and structure are hierarchically intertwined in DNA, suggesting a
common origin for both features. However, the mechanism leading to this
relationship is unknown. Here we investigate a biologically motivated dynamics
for the evolution of genetic sequences. We show that a metastable (long-lived)
regime emerges in which sequences have symmetry and structure interlaced in a
way that matches that of extant genomes.
