  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.
