Replication and Mutation on Neutral Networks

Folding of RNA sequences into secondary structures is viewed as a map that assigns a uniquely defined base pairing pattern to every sequence. This mapping is non-invertible since many sequences fold into the same (secondary) structure or shape. The preimages of the map, called neutral networks, are uniquely associated with the shapes and vice versa. Random graph theory is used to construct networks in sequence space which are appropriate models for neutral networks as they share most of the evolutionarily relevant properties with them. The theory of molecular quasispecies (in its simplest version) has been applied to replication and mutation on single-peak fitness landscapes. This concept is extended here by considering evolution on degenerate multi-peak landscapes which originate from neutral networks by assuming that one particular shape is fitter than all others. On such a single-shape landscape the superior fitness value is assigned to all sequences belonging to the master shape (whose associated neutral network comprises all vertices of the corresponding graph). All other shapes are lumped together and their fitness values are averaged in a way that is reminiscent of mean field theory. Replication and mutation on neutral networks are modeled by reformulated phenomenological rate equations as well as by a stochastic birth-and-death model. In analogy to the evolution of quasispecies in sequence space a phenotypic error threshold is observed which separates two scenarios:\enskip (i) a stationary (fittest) master shape surrounded by closely related shapes and (ii) populations drifting through shape space in a diffusion-like process. The error classes of the quasispecies model are replaced by distance classes between the master shape and the other structures. Analytical results are derived for single-shape landscapes, in particular, simple expressions are obtained for the mean fraction of master shapes in the population and for phenotypic error thresholds. The analytical results are complemented by data obtained from computer simulation of the underlying stochastic processes. The predictions of the phenomenological approach on the single-shape landscape are very well reproduced by replication and mutation kinetics of tRNAphe. Simulation of the stochastic process resolved in distance classes produces data which are in excellent agreement with the results obtained by the birth-and-death model.