Error Propagation in the Hypercycle

We study analytically the steady-state regime of a network of $n$ error-prone self-replicating templates forming an asymmetric hypercycle and its error tail. We show that the existence of a master template with a higher noncatalyzed self-replicative productivity, $a$, than the error tail ensures the stability of chains in which $m < n-1$ templates coexist with the master species. The stability of these chains against the error tail is guaranteed for catalytic coupling strengths ($K$) of order of $a$. We find that the hypercycle becomes more stable than the chains only for $K$ of order of $a^2$. Furthermore, we show that the minimal replication accuracy per template needed to maintain the hypercycle, the so-called error threshold, vanishes like $\sqrt(n/K)$ for large $K$ and $n \leq 4$.