A generalization of Kingman's model of selection and mutation and the Lenski experiment

Kingman's model of selection and mutation studies the limit type value distribution in an asexual population of discrete generations and infinite size undergoing selection and mutation. This paper generalizes the model to analyse the long-term evolution of Escherichia. coli in Lenski experiment. Weak assumptions for fitness functions are proposed and the mutation mechanism is the same as in Kingman's model. General macroscopic epistasis are designable through fitness functions. Convergence to the unique limit type distribution is obtained.