Adaptation in Tunably Rugged Fitness Landscapes: The Rough Mount Fuji Model

Abstract

Much of the current theory of adaptation is based on Gillespie's mutational landscape model (MLM), which assumes that the fitness values of genotypes linked by single mutational steps are independent random variables. On the other hand, a growing body of empirical evidence shows that real fitness landscapes, while possessing a considerable amount of ruggedness, are smoother than predicted by the MLM. In the present article we propose and analyze a simple fitness landscape model with tunable ruggedness based on the rough Mount Fuji (RMF) model originally introduced by Aita et al. in the context of protein evolution. We provide a comprehensive collection of results pertaining to the topographical structure of RMF landscapes, including explicit formulas for the expected number of local fitness maxima, the location of the global peak, and the fitness correlation function. The statistics of single and multiple adaptive steps on the RMF landscape are explored mainly through simulations, and the results are compared to the known behavior in the MLM model. Finally, we show that the RMF model can explain the large number of second-step mutations observed on a highly fit first-step background in a recent evolution experiment with a microvirid bacteriophage.T HE genetic adaptation of an asexual population to a novel environment is governed by the number and fitness effects of available beneficial mutations, their epistatic interactions, and the rate at which they are supplied (Sniegowski and Gerrish 2010). Despite the inherent complexity of this process, recent theoretical work has identified several robust statistical patterns of adaptive evolution (Orr 2005a,b). Most of these predictions were derived in the framework of Gillespie's mutational landscape model (MLM), which is based on three key assumptions (Gillespie 1983(Gillespie , 1984(Gillespie , 1991Orr 2002). First, selection is strong enough to prevent the fixation of deleterious mutations and mutation is sufficiently weak such that mutations emerge and fix one at a time [the strong selection/weak mutation (SSWM) regime]. Second, wild-type fitness is high, which allows one to describe the statistics of beneficial mutations using extreme value theory (EVT). Third, the fitness values of new mutants are uncorrelated with the fitness of the ancestor from which they arise. This last assumption implies that the fitness landscape underlying the adaptive process is maximally rugged with many local maxima and minima (Kauffman and Levin 1987;Kauffman 1993;Jain and Krug 2007), a limiting situation that is often referred to as the house of cards (HoC) landscape (Kingman 1978). Thus, the MLM is concerned with a population evolving in a HoC landscape under SSWM dynamics, starting from an initial state of high fitness.The validity of the SSWM assumption depends primarily on the population size N. Denoting the mutation rate by u and the typical selection strength by s, the criterion for the SSWM regime reads Nu ( 1 ( Ns, which can always be satisfied by a suitable choice o...

On the (un)predictability of a large intragenic fitness landscape

“…Moreover, because multiple alleles at the same site are contained within the landscape, we may study whether changes in the shape of the landscape are site-or amino acid-specific. We computed various landscape statistics (roughness-to-slope ratio, fraction of epistasis, and the recently proposed gamma statistics; SI Appendix, Supporting Information 1: Extended Materials and Methods) (10,11,14) and compared them with expectations from theoretical landscape models [NK (36)(37)(38), RMF (39,40), HoC (41), egg-box landscapes (14); for brief definitions of these terms, see SI Appendix, Supporting Information 2: Overview of Different Fitness Landscape Models Introduced in the Main Text]. Whenever necessary, we provide an analytical extension of the used statistic to the case of multiallelic landscapes (Materials and Methods).…”

δ-exceedance records and random adaptive walks

“…However, since our analysis neglects the effect of fluctuations, the subleading behavior of (60) cannot generally be expected to be exact. The leading order behavior D RAW ≈ (ln L) 1/γ /A with A and γ given in (21) and (22), respectively, is compared to simulations in figure 8, showing excellent agreement.…”

“…To sum up, we found that where γ and A are given in (22) and (21). Hence, the MFA becomes exact as l → ∞.…”

Scaling properties of evolutionary paths in a biophysical model of protein adaptation

“…In most cases the computational and experimental cost of analyzing empirical models has required simplified sequence spaces, especially binary sequences (indicating only the presence or absence of a mutation at each site) [3,5,8,9], genomes or proteins with reduced lengths [14][15][16], and reduced alphabets of amino acids [16,17] or protein structural components [18]. However, it is not clear how properties of landscapes and evolutionary paths change under these implicit coarse-graining schemes.…”