Beyond the Hypercube: Evolutionary Accessibility of Fitness Landscapes with Realistic Mutational Networks

Zagórski, Marcin; Burda, Z.; Waclaw, Bartłomiej

doi:10.1371/journal.pcbi.1005218

“…However, the condition for a local fitness maximum now involves mutations to different alleles at the same site, which may lead to a nontrivial dependence on A. On the basis of a recent study of evolutionary accessibility in multiallelic sequence spaces, one may expect the fitness landscapes to become less rugged with increasing A (Zagorski et al 2016), but this conjecture would have to be corroborated by a detailed analysis.…”

Section: Discussionmentioning

confidence: 99%

Genotypic Complexity of Fisher’s Geometric Model

Hwang

¹

,

Park

²

,

Krug

³

2017

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Fisher's geometric model was originally introduced to argue that complex adaptations must occur in small steps because of pleiotropic constraints. When supplemented with the assumption of additivity of mutational effects on phenotypic traits, it provides a simple mechanism for the emergence of genotypic epistasis from the nonlinear mapping of phenotypes to fitness. Of particular interest is the occurrence of reciprocal sign epistasis, which is a necessary condition for multipeaked genotypic fitness landscapes. Here we compute the probability that a pair of randomly chosen mutations interacts sign epistatically, which is found to decrease with increasing phenotypic dimension , and varies nonmonotonically with the distance from the phenotypic optimum. We then derive expressions for the mean number of fitness maxima in genotypic landscapes comprised of all combinations of random mutations. This number increases exponentially with , and the corresponding growth rate is used as a measure of the complexity of the landscape. The dependence of the complexity on the model parameters is found to be surprisingly rich, and three distinct phases characterized by different landscape structures are identified. Our analysis shows that the phenotypic dimension, which is often referred to as phenotypic complexity, does not generally correlate with the complexity of fitness landscapes and that even organisms with a single phenotypic trait can have complex landscapes. Our results further inform the interpretation of experiments where the parameters of Fisher's model have been inferred from data, and help to elucidate which features of empirical fitness landscapes can be described by this model.

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“…However, the condition for a local fitness maximum now involves mutations to different alleles at the same site, which may lead to a non-trivial dependence on A. On the basis of a recent study of evolutionary accessibility in multi-allelic sequence spaces one may expect the fitness landscapes to become less rugged with increasing A (Zagorski et al 2016), but this conjecture would have to be corroborated by a detailed analysis. Finally, whereas the present work focused on the structure of the fitness landscapes induced by FGM, it is of obvious importance to understand how the adaptative process actually proceeds on such a landscape (Orr 2005).…”

Section: Discussionmentioning

confidence: 99%

Genotypic complexity of Fisher’s geometric model

Hwang

¹

,

Park

²

,

Krug

³

2016

Preprint

View full text Add to dashboard Cite

Fisher's geometric model was originally introduced to argue that complex adaptations must occur in small steps because of pleiotropic constraints. When supplemented with the assumption of additivity of mutational effects on phenotypic traits, it provides a simple mechanism for the emergence of genotypic epistasis from the nonlinear mapping of phenotypes to fitness. Of particular interest is the occurrence of reciprocal sign epistasis, which is a necessary condition for multipeaked genotypic fitness landscapes. Here we compute the probability that a pair of randomly chosen mutations interacts sign epistatically, which is found to decrease with increasing phenotypic dimension n, and varies nonmonotonically with the distance from the phenotypic optimum. We then derive expressions for the mean number of fitness maxima in genotypic landscapes comprised of all combinations of L random mutations. This number increases exponentially with L, and the corresponding growth rate is used as a measure of the complexity of the landscape. The dependence of the complexity on the model parameters is found to be surprisingly rich, and three distinct phases characterized by different landscape structures are identified. Our analysis shows that the phenotypic dimension, which is often referred to as phenotypic complexity, does not generally correlate with the complexity of fitness landscapes and that even organisms with a single phenotypic trait can have complex landscapes. Our results further inform the interpretation of experiments where the parameters of Fisher's model have been inferred from data, and help to elucidate which features of empirical fitness landscapes can be described by this model. KEYWORDS fitness landscape; genotype-phenotype map; epistasis; adaptation; fitness peaks A fundamental question in the theory of evolutionary adaptation concerns the distribution of mutational effect sizes and the relative roles of mutations of small vs. large effects in the adaptive process (Orr 2005). In his seminal 1930 monograph, Ronald Fisher devised a simple geometric model of adaptation in which an organism is described by n phenotypic traits and mutations are random displacements in the trait space (Fisher 1930). Each trait has a unique optimal value and the combination of these values defines a single phenotypic fitness optimum that constitutes the target of adaptation. Because random mutations act pleiotropically on multiple traits, the probability that a given mutation brings the phenotype closer to the target decreases with increasing n. Fisher's analysis showed Jibong-ro, Wonmi-gu, Bucheon 14662, Republic of Korea. E-mail: spark0@catholic.ac.kr that, for large n, the mutational step size in units of the distance to the optimum must be smaller than 1/ √ n for the mutation to be beneficial with an appreciable probability. He thus concluded that the evolution of complex adaptations involving a large number of traits must rely on mutations of small effect. This conclusion was subsequently qualified by the realization tha...

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“…The number of allelic states A is largely irrelevant for the analysis of direct paths. In contrast, indirect paths become more complex for A > 2 because of the possibility of distance-neutral mutations that neither increase nor decrease the distance to the target [102]. Here we mostly restrict our analysis to the biallelic case, where the genotype spaces are hybercubes.…”

Section: Definitionsmentioning

confidence: 99%

Universality Classes of Interaction Structures for NK Fitness Landscapes

Hwang¹,

Schmiegelt²,

Ferretti³

et al. 2018

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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 ≤ 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

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Beyond the Hypercube: Evolutionary Accessibility of Fitness Landscapes with Realistic Mutational Networks

Cited by 33 publications

References 67 publications

Genotypic Complexity of Fisher’s Geometric Model

Genotypic Complexity of Fisher’s Geometric Model

Genotypic complexity of Fisher’s geometric model

Universality Classes of Interaction Structures for NK Fitness Landscapes

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