93-11-069

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Abstract:

Landscapes Complex Optimization Problems and Biopolymer Structures

## Peter Schuster and Peter F. Stadler

The evolution of RNA molecules in replication assays, viroids and RNA viruses
can be viewed as an adaptation process on a ``fitness'' landscape. The dynamics
of evolution is hence tightly linked to the structure of the underlying
landscape. Global features of landscapes can be described by statistical
measures like number of optima, lengths of walks, and correlation functions.
The evolution of a quasispecies on such landscapes exhibits three dynamical
regimes depending on the replication fidelity: Above the``localization
threshold'' the population is centered around a (local) optimuim. Between
localization and ``dispersion threshold'' the population is
still centered around a consensus sequence, which, however, changes in time.
For very large mutation rates the population spreads in squence space like a
gas. The critical mutation rates separating the three domains depend strongly
on characteristics properties of the fitness landscapes. Statistical
characteristics of RNA landscapes are accesssibly by mathematical analysis and
computer calculations on the level of secondary structures: these RNA
landscapes belong to the same class as well known optimization problems and
simple spin glass models. The notion of a landscapes is extended to
combinatory maps, thereby allowing for a direct statistical investigation of
the sequence structure relationships of RNA at the level of secondary
structures. Frequencies of structures are highly non-uniform: we find
relatively few common and many rare ones, as expressed by a generalized form of
Zipf's law. Using an algorithm for inverse folding we show that
sequences sharing the same structure are distributed randomly over
sequence space. Together with calculations of structure correlations and
a survey of neutral mutations this provides convincing evidence that RNA
landscapes are as simple as they could possibly be for evolutionary
adaptation: Any desired secondary structure can be found close to an arbitrary
intitial sequence and at the same time almost all bases can be substituted
sequentially without ever changing the shape of the molecule. Consequences of
these results for the evolutionary optimization, the early stages of life, and
molecular biotechnology are discussed.

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