The growth of many artificial replicators is approximately parabolic (sub-exponential) in solution, due to the self-inhibition through duplex formation by the association of single-stranded molecules. This type of growth implies “survival of everybody” under a selection constraint. Parabolic growth requires high enough concentration so that the single strands can find one another. The selective outcome is more complicated when spontaneous decay of molecules is also taken into account. When double strands decompose at a slower rate than single strands, coexistence or survival of the fittest becomes a quantitative issue. Here we investigate the evolution of parabolic replicators by the methods of adaptive dynamics. Directional selection for higher replication rate in general results in a “parabolic quasi-species”, due to the fact that the fittest template is followed by a moving shadow of inferior templates that owe their presence to parabolic growth. Under the assumption of cross-hybridisation between non-identical templates molecular coexistence disappears when such pairing is sufficiently non-selective, because replicators do not inhibit themselves more than they limit the others. At intermediate specificity of pairing adaptive branching of the population becomes feasible, due to the fact that distant enough sequences are able to escape from cross-limitation by other sub-populations.
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