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The fitness landscape encodes the mapping of genotypes to fitness and provides a succinct representation of possible trajectories followed by an evolving population. Evolutionary accessibility is quantified by the existence of…
Consider a graph in which each site is endowed with a value called \emph{fitness}. A path in the graph is said to be "open" or "accessible" if the fitness values along that path is strictly increasing. We say that there is accessibility…
Biological evolution can be conceptualized as a search process in the space of gene sequences guided by the fitness landscape, a mapping that assigns a measure of reproductive value to each genotype. Here we discuss probabilistic models of…
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…
Functional effects of different mutations are known to combine to the total effect in highly nontrivial ways. For the trait under evolutionary selection (`fitness'), measured values over all possible combinations of a set of mutations yield…
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…
Accessibility percolation is a new type of percolation problem inspired by evolutionary biology: a random number, called its fitness, is assigned to each vertex of a graph, then a path in the graph is accessible if fitnesses are strictly…
Inspired by biological evolution, we consider the following so-called accessibility percolation problem: The vertices of the unoriented $n$-dimensional binary hypercube are assigned independent $U(0, 1)$ weights, referred to as fitnesses. A…
The adaptive evolution of a population under the influence of mutation and selection is strongly influenced by the structure of the underlying fitness landscape, which encodes the interactions between mutations at different genetic loci.…
Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in…
We study how correlations in the random fitness assignment may affect the structure of fitness landscapes. We consider three classes of fitness models. The first is a continuous phenotype space in which individuals are characterized by a…
All possible phenotypes are not equally accessible to evolving populations. In fact, only phenotypes of large size, i.e. those resulting from many different genotypes, are found in populations of sequences, presumably because they are…
We consider a random fitness landscape on the space of haploid diallelic genotypes with n genetic loci, where each genotype is considered either inviable or viable depending on whether or not there are any incompatibilities among its allele…
Consider an infinite, rooted, connected graph where each vertex is labelled with an independent and identically distributed Uniform(0,1) random variable, plus a parameter $\theta$ times its distance from the root $\rho$. That is, we label…
Fitness landscapes have historically been a powerful tool for analyzing the search space explored by evolutionary algorithms. In particular, they facilitate understanding how easily reachable an optimal solution is from a given starting…
Darwinian evolution is driven by random mutations, genetic recombination (gene shuffling) and selection that favors genotypes with high fitness. For systems where each genotype can be represented as a bitstring of length $L$, an overview of…
Genotype-to-phenotype maps and the related fitness landscapes that include epistatic interactions are difficult to measure because of their high dimensional structure. Here we construct such a map using the recently collected corpora of…
The relationship between the shape of a fitness landscape and the underlying gene interactions, or epistasis, has been extensively studied in the two-locus case. Gene interactions among multiple loci are usually reduced to two-way…
Fitness landscapes are mappings between genotypes, phenotypes, and fitness that shape evolution. In recent years, empirical work and theoretical models have greatly advanced our understanding of how populations navigate rugged fitness…
We introduce a new model of evolution on a fitness landscape possessing a tunable degree of neutrality. The model allows us to study the general properties of molecular species undergoing neutral evolution. We find that a number of…