Mathematical Models of Molecular Evolution

This thesis deals with mathematical models of Darwinian evolution in populations of simple living organisms. We study models which describe how the representation of different gene sequences in the population is affected by the fundamental forces of evolutionary processes, such as selection pressure, the mutation rate and the time dynamics of the underlying fitness landscape. The thesis includes an introductory text and five appended research papers. The introductory text is meant to serve both as a background to the papers, but also as a brief review of closely related models in theoretical biology. It is written mainly for people with a background in physics. The introductory text starts with a short review of biochemistry. The concept of fitness landscapes is then introduced. Structural aspects of the landscapes, such as neutrality and ruggedness, are discussed. Eigen\u27s quasispecies model is reviewed in some detail since it gives a background mainly to Paper I, which study sexually replicating quasispecies, but also to Paper II and III, dealing with quasispecies and error thresholds in dynamic environments. Two different models of dynamic fitness landscapes are discussed. In Paper II we study the error thresholds occurring in a population living on a landscape with a single fitness peak, which moves around randomly in sequence space. In Paper III the fitness peak is fluctuating and we study the response in the population dynamics, e.g., the phase shift. In the the last part of the thesis we discuss evolved mutation rates. Two different models, describing how the mutation rates in simple organisms evolve, are presented in Paper IV and V. The first paper concerns organisms living in dynamic environments. The second paper studies evolution on a fitness staircase, where the population evolves form low fitness peaks to higher peaks in the close neighborhood. Both models predict that the mutation rate per genome is approximately independent of genome length, a phenomenon that has been observed for real living organisms