The statistical dynamics of epochal evolution
Year: 1999
Type: dissertation
Abstract: In this thesis, a new mathematical formalism for analyzing\nevolutionary dynamics is developed. This formalism combines ideas and\nmethods from statistical mechanics, mathematical population genetics,\nand dynamical systems theory to describe the dynamics of evolving\npopulations. In particular, the work shows how the maximum entropy\nformalism of statistical mechanics can be extended to apply to simple\nevolutionary systems, such that "macroscopic" equations of motion can\nbe constructed from an underlying "microscopic" evolutionary dynamics.\n\nMore specifically, the thesis studies "epochal evolution"; a dynamical\nphenomenon which is frequently observed in evolutionary dynamics. In\nepochal evolution, some macroscopic state variables that describe the\nevolving population exhibit an alternation of periods of stasis\n(epochs) and sudden transitions (innovations). For populations\nevolving in a constant selective environment there are two main\nmechanisms that may bring about epochal evolution. In the first\nmechanism the population is imagined to evolve on a "fitness\nlandscape" that assigns a fitness to each point in a space of\ngenotypes. Metastability then occurs when the population gets stuck\naround a "local optimum" in the fitness landscape. An innovation takes\nplace when a rare sequence of mutations creates a lineage of\nindividuals that crosses a valley of low fitness toward a local\noptimum of higher fitness. In the second mechanism, the genotype space\nis thought to decompose into a relatively small number of "neutral\nsubbasins": large connected sets of equal fitness genotypes. In this\nview, an epoch corresponds to a time period in which the fittest\nmembers of the population diffuse through a neutral subbasin under\nmutation until one of them discovers a (rare) connection to a neutral\nsubbasin of higher fitness. In somewhat different methodology, in the\nfirst mechanism the metastability is caused by a "fitness barrier",\nwhereas in the second mechanism it is caused by an "entropy barrier".\n\nIn this thesis, the evolutionary dynamics in the presence of both\nfitness and entropy barriers is studied, although the focus is on\nfitness functions with entropy barriers. For a large class of simple\nfitness functions we derive, using the maximum entropy methodology,\nequations of motion on the level of fitness distributions from the\nunderlying microscopic dynamics of selection and mutation on\ngenotypes. In the "thermodynamic limit" of infinite population sizes\nthe population follows these equations of motion precisely while for\nfinite populations the population only follows these equations of\nmotion on average at each time step. From this formulation of the\nfinite population dynamics we determine explicitly the locations of\nepochs in the space of fitness distributions, the stochastic dynamics\nwithin and between the epochs, the average durations of epochs, and\nthe stability of epochs.\n\nThe results also bear directly on the dynamics of evolutionary\nsearch algorithms such as genetic algorithms. In two chapters of this\nthesis, our mathematical model is used to derive optimal parameter\nsettings for evolutionary search on a wide class of fitness functions.\nThe analysis suggests that optimal evolutionary search occurs in a\nparameter regime where the highest fitness strings in the population\nare only marginally stable. \n\nWe also studied in detail the way in which an evolving population will\nspread through a neutral subbasin or neutral network. As the theory\nshows, the limit distribution of the population over a neutral network\nis independent of almost all evolutionary parameters and is determined\nsolely by the topology of the neutral network. Additionally, this\ndistribution is concentrated at those areas of the neutral network\nwhere the "neutrality" is largest. This implies that under neutral\nevolution the population evolves to increase its "mutational\nrobustness" and that the mutational robustness that it attains is only\ndependent on the topology of the neutral network on which the\npopulation evolves.\n\nFinally, we studied and compared the barrier crossing times for\nfitness and entropy barriers. The results show that entropy barrier\ncrossing in evolving population takes place on much shorter time\nscales than fitness barrier crossing. This suggests that in evolution,\nthe escape from a metastable state is most likely to occur along\nneutral paths in genotype space.
Author: Erik van Nimwegen
Language: English
Subfield: Genetics
Domain: Life Sciences
Open Access status: green
OpenAlex ID: W1483618554