92-01-004

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Mutation in Autocatalytic Reaction Networks:

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An Analysis based on Perturbation Theory

## Peter F. Stadler and Peter Schuster

A class of kinetic equations describing catalyzed and
template induced replication, and mutation is introduced. This ODE in
its most general form is split into two vector fields, a replication
and a mutation field. The mutation field is considered as a
perturbation of the replicator equation. The perturbation expansion is
a Taylor series in a mutation parameter $\lambda$. First second and
higher order contributions are computed by means of the conventional
Raleigh-Schr"odiger approach. Qualitative shifts in the positions of
rest points and limit cycles on the boundary of the physically
meaningful part of concentration space are predicted from flow
topologies. The results of the topological analysis are summarized in
two theorems which turned out to be useful in applications: the
restpoint migration theorem (RPM) and the limit cycle migration
theorem (LCM). Quantitative expressions for the shifts of rest
points are computed directly from the perturbation expansion. The
concept is applied to a collection of selected examples from
biophysical chemistry and biology.

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