Second-Order Catalytic Quasispecies Yields First-Order Phase Transition

The quasispecies model describes processes related to the origin of life and viral evolutionary dynamics. We discuss how the error catastrophe that reflects the transition from localized to delocalized quasispecies population is affected by catalytic replication of different reaction orders. Specifically, we find that 2 order mechanisms lead to 1 order discontinuous phase transitions in the viable population fraction, and conclude that the "higher" the interaction the "lower" the transition. We discuss potential implications for understanding the replication of highly mutating RNA viruses. *Corresponding authors Emails: [email protected] (ET), [email protected] (GA) The quasispecies model can be formulated as a set of differential equations describing the evolution of self-replicating molecules.[1-2] These equations were originally developed by Eigen and Schuster [3-4] as a way to model molecular evolutionary processes related to the origin of life. In addition to molecular evolution, the quasispecies model has also been extensively applied toward the understanding of viral evolutionary dynamics, in particular that of rapidly mutating singlestranded RNA viruses (e.g. HIV).[5-7] Likewise, catalytic reactions have been shown to be relevant to the origin of life and early molecular evolution by facilitating self-replication and dynamic evolution.[8-12] Such reactions can operate both autocatalytically and cross catalytically, where autocatalysis is a mechanism for selfreplication and cross catalysis may lead to mutation [12]. It has been further shown how interacting catalytic reactions of various reaction order can form catalytic networks of increasing complexity.[13-15] In a recent paper [16] we have shown how certain crucial features of network complexity require at least second order catalysis [3,12,17]. With the goal of developing a more complete understanding of evolutionary dynamics in higher order autocatalytic networks, we have developed a quasispecies approach for analyzing mutation and selection in catalytic reactions of varying order. Typically, quasispecies models are characterized by an upper mutational threshold, beyond which natural selection can no longer localize the population about the fittest sequences.[18-20] Below this error threshold, the population consists of a cloud of related strains termed a quasispecies, while above this error threshold the evolutionary dynamics is characterized by random genetic drift over the sequence space. The transition from a localized population distribution to a delocalized distribution is known as the error catastrophe. In this Letter we show how the error catastrophe is affected when different replication models are considered. To do so, we determine how models of replication dynamics characterized by a catalyst and a template species influence the nature of the transition at the error threshold. Most interestingly, we find that 2 order catalytic mechanisms lead to 1 order phase transitions with respect to the viable population fraction. Understanding the mutation-selection balance and the error threshold in catalytic networks of varying order is both theoretically interesting, and has potentially important implications in understanding cooperative phenomena in the replication mechanisms of highly mutating genetic sequences, such as RNA viruses. We therefore propose a generalized two-stage cooperative catalytic model (Eq. 1), which assumes a population of single-stranded sequences of some biopolymer (e.g. RNA or protein), denoted by σi, where each σi represents a distinct sequence that can both catalyze the replication of other sequences, as well as act as a template for the production of new sequences: