98-04-007
Replication and Mutation on Neutral Networks
Christian Reidys, Christian V. Forst, and Peter Schuster
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: (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.
Keywords:
Error threshold - neutral evolution -
neutral network - random graph - RNA secondary structures
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