The influence of mutation on autocatalytic reaction networks
A particular class of ordinary differential equations describing catalyzed, template-induced, and erroneous replication is investigated. Apparently different models for the dynamical behaviour of self- replicating macromolecules (e.g. the quasi-species model, the hypercycle model) may be viewed as particular cases of a general replication- mutation process characterized by an error-prone replication mechanism (Section 2). Section 3 presents some general results derived from the perturbation approach. The first-order selection-mutation equation is compared with Eigen's quasi-species model in Section 4. General properties of second-order kinetics are outlined in Section 5, while Section 6 analyzes the second-order model by means of the perturbation theory. Section 7 investigates in detail three special models: Schlögl's model, uniform cooperation model, and the hypercycle. The proposed simplified version of the quasi-species model exhibits transitions between different master species as a function of the mutation rate solely as a consequence of the different influence of replication and degradation rate. This concentration of the error tail increases linearly between the transitions but decreases at the transition points. Although these transitions are reminiscent of similar phenomena in the classical quasi- species model, they do not depend on the fitness of close-by neighbours in the sequence space. A comparison of the proposed mutation description with the models based on Eigen's \(Q\) matrix highlights that dynamics simplify with increasing mutation rate in both models, and the error tail becomes globally stable for sufficiently high mutation rates, while models based on the \(Q\) matrix exhibit a globally stable rest point close to the center of the simplex in this case.
scientific article; zbMATH DE number 665732