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Hegarty, P. och Martinsson, A. (2014) *On the existence of accessible paths in various models of fitness landscapes*.

** BibTeX **

@article{

Hegarty2014,

author={Hegarty, Peter and Martinsson, Anders},

title={On the existence of accessible paths in various models of fitness landscapes},

journal={The Annals of Applied Probability},

issn={1050-5164},

volume={24},

issue={4},

pages={1375-1395},

abstract={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 is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node v^0 to the all-ones node v^1 tends respectively to 0, 1 and 1, as n tends to infinity.
A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of v^0 is set to some α = α_n ∈ [0, 1]. We prove that there is a very sharp threshold at α_n = (ln n)/n for the existence of accessible paths from v^0 to v^1 . As a corollary we prove significant concentration, for α below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from v^0 to v^1 existing tends to 1 provided the drift parameter θ = θ_n satisfies n(θ_n) → ∞, and for any fitness distribution which is continuous on its support and whose support is connected.},

year={2014},

note={14},

}

** RefWorks **

RT Journal Article

SR Electronic

ID 165031

A1 Hegarty, Peter

A1 Martinsson, Anders

T1 On the existence of accessible paths in various models of fitness landscapes

YR 2014

JF The Annals of Applied Probability

SN 1050-5164

VO 24

IS 4

SP 1375

OP 1395

AB 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 is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node v^0 to the all-ones node v^1 tends respectively to 0, 1 and 1, as n tends to infinity.
A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of v^0 is set to some α = α_n ∈ [0, 1]. We prove that there is a very sharp threshold at α_n = (ln n)/n for the existence of accessible paths from v^0 to v^1 . As a corollary we prove significant concentration, for α below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from v^0 to v^1 existing tends to 1 provided the drift parameter θ = θ_n satisfies n(θ_n) → ∞, and for any fitness distribution which is continuous on its support and whose support is connected.

LA eng

DO 10.1214/13-aap949

LK http://publications.lib.chalmers.se/records/fulltext/165031/local_165031.pdf

LK http://dx.doi.org/10.1214/13-aap949

OL 30