Abstract

In this thesis, a new mathematical formalism for analyzing
evolutionary dynamics is developed. This formalism combines ideas and
methods from statistical mechanics, mathematical population genetics,
and dynamical systems theory to describe the dynamics of evolving
populations. In particular, the work shows how the maximum entropy
formalism of statistical mechanics can be extended to apply to
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simple
evolutionary systems, such that "macroscopic" equations of motion can
be constructed from an underlying "microscopic" evolutionary dynamics.
More specifically, the thesis studies "epochal evolution"; a dynamical
phenomenon which is frequently observed in evolutionary dynamics. In
epochal evolution, some macroscopic state variables that describe the
evolving population exhibit an alternation of periods of stasis
(epochs) and sudden transitions (innovations). For populations
evolving in a constant selective environment there are two main
mechanisms that may bring about epochal evolution. In the first
mechanism the population is imagined to evolve on a "fitness
landscape" that assigns a fitness to each point in a space of
genotypes. Metastability then occurs when the population gets stuck
around a "local optimum" in the fitness landscape. An innovation takes
place when a rare sequence of mutations creates a lineage of
individuals that crosses a valley of low fitness toward a local
optimum of higher fitness. In the second mechanism, the genotype space
is thought to decompose into a relatively small number of "neutral
subbasins": large connected sets of equal fitness genotypes. In this
view, an epoch corresponds to a time period in which the fittest
members of the population diffuse through a neutral subbasin under
mutation until one of them discovers a (rare) connection to a neutral
subbasin of higher fitness. In somewhat different methodology, in the
first mechanism the metastability is caused by a "fitness barrier",
whereas in the second mechanism it is caused by an "entropy barrier".
In this thesis, the evolutionary dynamics in the presence of both
fitness and entropy barriers is studied, although the focus is on
fitness functions with entropy barriers. For a large class of simple
fitness functions we derive, using the maximum entropy methodology,
equations of motion on the level of fitness distributions from the
underlying microscopic dynamics of selection and mutation on
genotypes. In the "thermodynamic limit" of infinite population sizes
the population follows these equations of motion precisely while for
finite populations the population only follows these equations of
motion on average at each time step. From this formulation of the
finite population dynamics we determine explicitly the locations of
epochs in the space of fitness distributions, the stochastic dynamics
within and between the epochs, the average durations of epochs, and
the stability of epochs.
The results also bear directly on the dynamics of evolutionary
search algorithms such as genetic algorithms. In two chapters of this
thesis, our mathematical model is used to derive optimal parameter
settings for evolutionary search on a wide class of fitness functions.
The analysis suggests that optimal evolutionary search occurs in a
parameter regime where the highest fitness strings in the population
are only marginally stable.
We also studied in detail the way in which an evolving population will
spread through a neutral subbasin or neutral network. As the theory
shows, the limit distribution of the population over a neutral network
is independent of almost all evolutionary parameters and is determined
solely by the topology of the neutral network. Additionally, this
distribution is concentrated at those areas of the neutral network
where the "neutrality" is largest. This implies that under neutral
evolution the population evolves to increase its "mutational
robustness" and that the mutational robustness that it attains is only
dependent on the topology of the neutral network on which the
population evolves.
Finally, we studied and compared the barrier crossing times for
fitness and entropy barriers. The results show that entropy barrier
crossing in evolving population takes place on much shorter time
scales than fitness barrier crossing. This suggests that in evolution,
the escape from a metastable state is most likely to occur along
neutral paths in genotype space.
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