Bifurcations and phase transitions in spatially extended two-member hypercycles

Abstract

Mounting theoretical and experimental evidence indicates that the success of molecular replicators is strongly tied to the local nature of their interactions. Local dispersal in a given spatial domain, particularly on surfaces, might strongly enhance the growth and selection of fit molecules and their resistance to parasites. In this work the spatial dynamics of a simple hypercycle model consisting of two molecular species is analysed. In order to characterize it, both mean field models and stochastic, spatially explicit approaches are considered. The mean field approach predicts the presence of a saddle-node bifurcation separating a phase involving stable hypercycles from extinction, consistently with spatially explicit models, where an absorbing first-order phase transition is shown to exist and diffusion is explicitly introduced. The saddle-node bifurcation is shown to leave a ghost in the phase plane. A metapopulation-based model is also developed in order to account for the observed phases when both diffusion and reaction are considered. The role of information and diffusion as well as the relevance of these phases and the underlying spatial structures are discussed, and their potential implications for the evolution of early replicators are outlined. r

Bifurcations analysis of oscillating hypercycles

“…Here, all hypercycle members oscillate for a long period of time before collapsing, in what is known as a delayed transition. Such a dynamics, which arises near saddle-node bifurcations, have been characterized in hypercycles [6,7,9,10,15,19], in autocatalytic replicators [10,11], as well as in singlespecies dynamics under Allee effects [20]. Interestingly, the same dynamics is found below the saddle-node bifurcation of fixed points and below the saddle-node of periodic orbits (Fig.…”

“…Eigen and Schuster proposed the hypercycle theory as a way to ensure the coexistence of distinct replicators able to store larger information allowing to surpass such an information constraint. It is known that hypercycles achieve coexistence of all replicators by means of bistability [1,[6][7][8][9]. Under bistability, large enough initial conditions of hypercycle elements allow the survival and persistence of all the catalytic species.…”

“…On the contrary, low initial conditions involve the asymptotic extinction of the hyperycle since the dynamics are captured by a stable attractor found in the origin of the phase space. The dynamics of autocatalytic replicators [10,11] and two-member hypercycles [7,10] have been widely characterized. 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].…”

“…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

“…As mentioned in the Introduction, our system might be equivalent to a hypercycle system with replicase-mediated replication. The consideration of replicases, as we have shown, might involve the presence of a smoothed phase transition, which is in contrast with the governing bifurcations of ribozyme-like hypercycles, in which the extinction of the replicons is governed by saddle-node bifurcations [45,47] and by absorbing first-order phase transitions [46]. Focusing in our first studied model, we have compared the critical mutation rate under complementation with the critical mutation obtained from a simple standard quasispecies model, also under the Swetina-Schuster fitness landscape.…”