Evolutionary accessibility of modular fitness landscapes

A fitness landscape is a mapping from the space of genetic sequences, which is modeled here as a binary hypercube of dimension $L$, to the real numbers. We consider random models of fitness landscapes, where fitness values are assigned according to some probabilistic rule, and study the statistical properties of pathways to the global fitness maximum along which fitness increases monotonically. Such paths are important for evolution because they are the only ones that are accessible to an adapting population when mutations occur at a low rate. The focus of this work is on the block model introduced by A.S. Perelson and C.A. Macken [Proc. Natl. Acad. Sci. USA 92:9657 (1995)] where the genome is decomposed into disjoint sets of loci (`modules') that contribute independently to fitness, and fitness values within blocks are assigned at random. We show that the number of accessible paths can be written as a product of the path numbers within the blocks, which provides a detailed analytic description of the path statistics. The block model can be viewed as a special case of Kauffman's NK-model, and we compare the analytic results to simulations of the NK-model with different genetic architectures. We find that the mean number of accessible paths in the different versions of the model are quite similar, but the distribution of the path number is qualitatively different in the block model due to its multiplicative structure. A similar statement applies to the number of local fitness maxima in the NK-models, which has been studied extensively in previous works. The overall evolutionary accessibility of the landscape, as quantified by the probability to find at least one accessible path to the global maximum, is dramatically lowered by the modular structure.Comment: 26 pages, 12 figures; final version with some typos correcte

Sign Epistatis Networks

In this master thesis I consider primarily the NK model for fitness landscapes and adaptive walks on such landscapes. Specifically a notation for a low-dimensional representation of (reciprocal) sign epistasis on the space of genetic loci is introduced for these models and used to derive sign-dependent properties, in particular the so called accessibility property of the landscape

Structure and Accessibility of Fitness Landscapes

In this thesis I take a look at stochastic models for fitness landscapes, specifically the House-of-Cards (HoC) model of completely uncorrelated fitness values and the NK-type models which are built through combination of HoC landscapes as building blocks. These models are parameterized by a parameter k, which is considered to determined the "ruggedness" of the fitness landscape, with a maximal value of k corresponding to the HoC model and a value of k=1 corresponding to non-epistatic landscapes. I consider the behavior of two properties related to ruggedness on these landscapes, namely the number of local fitness maxima and the accessibility of genotypes via paths of monotonic fitness increase. Although high ruggedness is connected to a higher number of local maxima and therefore intuitively also a lower probability of distance genotypes being accessible from one another, this turns out to not hold generally. Contrary to assumptions made when the NK model was first introduced\autocite{weinberger1991}, it can be shown that asymptotic different quantitative results for the number of local maxima can be found for different choices of interaction structures between loci of the genotype. These models have analogous interpretations in solid state physics as the random energy model and spin glass models

Accessibility Percolation on Cartesian Power Graphs

A fitness landscape is a mapping from the space of genotypes to the real numbers. A path in a fitness landscape is a sequence of genotypes connected by single mutational steps. Such a path is said to be accessible if the fitness values of the genotypes encountered along the path increase monotonically. We study accessible paths on random fitness landscapes of the House-of-Cards type, which implies that the fitness values are independent, identically and continuously distributed random variables. The genotype space is taken to be a Cartesian power graph $A^L$, where $L$ is the number of genetic loci and the allele graph $A$ encodes the possible allelic states and mutational transitions on one locus. The probability of existence of accessible paths between two genotypes at a distance linear in $L$ displays a sharp transition from 0 to 1 at a threshold value $\beta^\ast$ of the fitness difference between the initial and final genotype. We derive a lower bound on $\beta^\ast$ for general $A$ and conjecture that this bound is tight for a large class of allele graphs. Our results generalize previous results for accessibility percolation on the biallelic hypercube, and compare favorably to published numerical results for multiallelic Hamming graphs. A key tool of our analysis is the concept of quasi-accessibility, which eliminates the need to account for the self-avoidance of accessible paths.Comment: 10 page

Universality classes of interaction structures for NK fitness landscapes

Kauffman's NK-model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems. Genotypes are represented as sequences of $L$ binary loci. The fitness assigned to a genotype is a sum of contributions, each of which is a random function defined on a subset of $k \le L$ loci. These subsets or neighborhoods determine the genetic interactions of the model. Whereas earlier work on the NK model suggested that most of its properties are robust with regard to the choice of neighborhoods, recent work has revealed an important and sometimes counter-intuitive influence of the interaction structure on the properties of NK fitness landscapes. Here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible (that is, fitness-monotonic) mutational pathways. In particular, we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of $L$, and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large $k$. Moreover, we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in $L$ for a large class of interaction structures that we characterize as locally bounded. Finally, we discuss the impact of the NK interaction structures on the dynamics of evolution using adaptive walk models.Comment: 61 pages, 9 figure

Measuring epistasis in fitness landscapes: The correlation of fitness effects of mutations

Genotypic fitness landscapes are constructed by assessing the fitness of all possible combinations of a given number of mutations. In the last years, several experimental fitness landscapes have been completely resolved. As fitness landscapes are high-dimensional, simple measures of their structure are used as statistics in empirical applications. Epistasis is one of the most relevant features of fitness landscapes. Here we propose a new natural measure of the amount of epistasis based on the correlation of fitness effects of mutations. This measure has a natural interpretation, captures well the interaction between mutations and can be obtained analytically for most landscape models. We discuss how this measure is related to previous measures of epistasis (number of peaks, roughness/slope, fraction of sign epistasis, Fourier Walsh spectrum) and how it can be easily extended to landscapes with missing data or with fitness ranks only. Furthermore, the dependence of the correlation of fitness effects on mutational distance contains interesting information about the patterns of epistasis. This dependence can be used to uncover the amount and nature of epistatic interactions in a landscape or to discriminate between different landscape models. (C) 2016 Elsevier Ltd. All rights reserved