On the existence of accessible paths in various models of fitness landscapes

  title={On the existence of accessible paths in various models of fitness landscapes},
  author={Peter V. Hegarty and Anders Martinsson},
  journal={Annals of Applied Probability},
We present rigorous mathematical analyses of a number of well-known mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the n-dimensional binary hypercube, for some n, a mutation involves the flipping of a single bit, and each vertex is assigned a real number, called its fitness, according to some rules. Our main concernis with the issue of existence of (selectively) accessible paths; that is, monotonic paths in the hypercube along which fitness… 

Evolutionary Accessibility of Modular Fitness Landscapes

The block model can be viewed as a special case of Kauffman’s NK-model, and it is shown 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 paths statistics.

Accessibility Percolation on Cartesian Power Graphs

This work derives a lower bound on $\beta^\ast$ for general $A$ and conjecture that this bound is tight for a large class of allele graphs, and compares favorably to published numerical results for multiallelic Hamming graphs.

Properties of Random Fitness Landscapes and Their Influence on Evolutionary Dynamics. A Journey through the Hypercube

The individual-based Wright-Fisher model is used to study recombination of genotypes, interactions between individuals and the influence of the underlying fitness landscape on these mechanisms.

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

Accessibility of the global fitness maximum increases with K and can be much higher than for binary sequences, suggesting that evolution can follow many different trajectories on such landscapes and the reconstruction of evolutionary pathways from experimental data might be an extremely difficult task.

Universality Classes of Interaction Structures for NK Fitness Landscapes

A unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of L is developed, and two different universality classes of interaction structures that display different asymptotics of this quantity for large k are identified.

Evolutionary constraints in fitness landscapes

This paper presents some measures of evolutionary constraints based on the similarity between accessible paths and the abundance and characteristics of “chains” of obligatory mutations, that are paths going through genotypes with a single fitter neighbor, and shows how they shed light on evolutionary constraints and predictability in experimentally resolved landscapes.

Accessibility percolation with backsteps

It is shown that, in the large $L$ limit, the probability that an accessible path exists from an arbitrary starting point to the (random) fittest site is no more than 1-\frac12\sinh^{-1}(2) =0.27818\ldots and conjecture that this probability does converge to $x^*_{1/2}$.

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

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 is proposed and compared to the known behavior in the MLM model.

Fitness Landscapes, Adaptation and Sex on the Hypercube

In this thesis, several models of fitness landscapes are analyzed with different analytical and numerical methods to identify characteristics in order to compare the model landscapes to experimental measurements.

Greedy adaptive walks on a correlated fitness landscape.



Evolutionary Accessibility in Tunably Rugged Fitness Landscapes

Some measures of accessibility behave non-monotonically as a function of K, indicating a special role of the most sparsely connected, non-trivial cases K=1 and 2, and the relation between models for fitness landscapes and spin glasses is addressed.

Evolutionary Accessibility of Mutational Pathways

Access to the globally optimal configuration should be accessible to genome wide evolution, but the repeatability of evolutionary trajectories is limited owing to the presence of a large number of alternative mutational pathways.

Towards a general theory of adaptive walks on rugged landscapes.

Analysis of a local fitness landscape with a model of the rough Mt. Fuji-type landscape: application to prolyl endopeptidase and thermolysin.

This model may provide a good approximation of real sections of local landscapes for current biopolymers phenomenologically and theoretically explained discrepancies between the fitnesses of multiple mutants and those predicted based on strict additivity of the component mutations by using a model of the rough Mt Fuji-type landscape.

Adaptive landscapes and protein evolution

It is found that mutant sites in real proteins show significantly more additivity than those obtained from random simulations and is reflected in a summary statistic for adaptive landscapes known as the “roughness,” which for the actual proteins so far examined lies in the smallest 0.5% tail of random landscapes.

A simple model for the balance between selection and mutation

  • J. Kingman
  • Biology
    Journal of Applied Probability
  • 1978
A model for the variation in time of the fitness distribution in a large haploid population is shown to have simple limiting properties which can be elucidated in fairly explicit terms. The novel


The theoretical and empirical considerations imply that strong genetic constraint on the selective accessibility of trajectories to high fitness genotypes may exist and suggest specific areas of investigation for future research.

Darwinian Evolution Can Follow Only Very Few Mutational Paths to Fitter Proteins

It is demonstrated that 102 mutational trajectories linking β-lactamase alleles are inaccessible to Darwinian selection and that many of the remaining trajectories have negligible probabilities of realization, which implies that the protein tape of life may be largely reproducible and even predictable.

The Probabilistic Method

A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.

Advanced Combinatorics: The Art of Finite and Infinite Expansions

I. Vocabulary of Combinatorial Analysis.- 1.1. Subsets of a Set Operations.- 1.2. Product Sets.- 1.3. Maps.- 1.4. Arrangements, Permutations.- 1.5. Combinations (without repetitions) or Blocks.- 1.6.