  Self-replicating systems based on information-coding polymers are of crucial
importance in biology. They also recently emerged as a paradigm in material
design on nano- and micro-scales. We present a general theoretical and
numerical analysis of the problem of spontaneous emergence of autocatalysis for
heteropolymers capable of template-assisted ligation driven by cyclic changes
in the environment. Our central result is the existence of the first order
transition between the regime dominated by free monomers and that with a
self-sustaining population of sufficiently long chains. We provide a simple,
mathematically tractable model supported by numerical simulations, which
predicts the distribution of chain lengths and the onset of autocatalysis in
terms of the overall monomer concentration and two fundamental rate constants.
Another key result of our study is the emergence of the kinetically-limited
optimal overlap length between a template and each of its two substrates. The
template-assisted ligation allows for heritable transmission of the information
encoded in chain sequences thus opening up the possibility of long-term memory
and evolvability in such systems.
