Fitness spectrum among random mutants on Mt. Fuji-type fitness landscape

Abstract

Statistical properties of a Mt. Fuji-type fitness landscape on a multi-valued sequence space were analysed. We constructed the model landscape based on additivity of the free energy contributed by each residue on a biopolymer, introducing "tolerance functions" that describe tolerance to residue substitution at each site. The fitness spectrum among a random mutant population around a wild-type sequence was theoretically obtained as the probability density distribution function of fitness. As the Hamming distance from the wild-type to the mutants increases, the mean fitness of the mutant population gradually decreases, and the variance of the fitness increases. These features are originated from the anisotropy of the landscape. On the assumption that the free energy is statistically additive around a wild-type in a sequence space of a real biopolymer, one can estimate the Hamming distance from the wild-type to the optimal biopolymer and the fitness of the optimum. Two sets of experimental data were analysed: (1) a promoter strength spectrum of a mutant population produced by the random mutagenesis of a wild-type lac promoter; (2) four stepwise optimization processes of different peptide mixtures evaluated with ligand binding affinity. Analysis of both experiments showed the compatibility with the hypothesis that local fitness landscapes around contemporary biopolymers are near Mt. Fuji-type. The mean slope of each of the four affinity landscapes for (2) was estimated as delta In K(d)/delta d = 1.3 approximately 2.3, where d denotes the Hamming distance from the optimum and K(d) represents the mean dissociation constant of sequences located at the Hamming distance of d. Mt. Fuji-type landscape can be regarded as a zero-th order approximation to the real local landscape just like an "ideal gas". We showed a method to gauge statistically the shape of a near Mt. Fuji-type landscape by measuring mutant fitness spectra.