Error propagation in the hypercycle

Abstract

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 non-catalyzed 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 n/K for large K and n ≤ 4.

Bifurcations analysis of oscillating hypercycles

“…We consider the hypercycle system analyzed in [16] (see also [14]). The system is composed by n functional templates with heterocatalytic connections and a pool of mutant sequences, which do not participate in the catalysis, thus acting as parasites [9,14,17].…”

“…The system is composed by n functional templates with heterocatalytic connections and a pool of mutant sequences, which do not participate in the catalysis, thus acting as parasites [9,14,17]. The model is given by the following set of differential equations:…”

“…Hypercycles with three and four species display the same asymptotic behavior than smaller catalytic networks but the coexistence attractor is approached by means of damped oscillations [1,6]. Hypercycles with n > 4 can undergo periodic oscillations due to the presence of attracting periodic orbits [1,8,[12][13][14].…”

“…Previous research has characterized these bifurcations in small hypercycles with replicators degradation [6,7] and mutation [14,15]. The switch from the bistable to the monostable (i.e., onlyextinction) regimes in small hypercycles is usually governed by saddle-node bifurcations, which have been thoroughly characterized in small catalytic systems [6,7,15].…”

Error threshold in RNA quasispecies models with complementation

“…Hence we study an unstructured model that only gathers the replication kinetics with a constant population, obviating the details of viral intracellular amplification. This is a standard method used in replicator theoretical models (see e.g., [39][40][41]). For mathematical convenience we studied the Swetina-Schuster single peak fitness landscape [42], where mutations are assumed to be largely deleterious and all mutant genotypes have identical low fitness.…”