Theoretical models of genetic map functions
type: Item
instance of: scholarly article
title: Theoretical models of genetic map functions
zbMATH Open document ID: 0539.92013
DOI: 10.1016/0040-5809(84)90013-3
author: Uri Liberman
author: Samuel Karlin
published in: Theoretical Population Biology
publication date: 1984-01-01T00:00:00Z
review text: The map distance \(x_{a,b}\) between two markers at points a and b on the genome is the expected number of crossovers between a and b. If the recombination frequency \(r_{a,b}\) associated with a pair of markers a and b depends only on their map distance \(r_{a,b}=M(x_{a,b})\) then the function M is a genetic map function. The authors compare four different methods for constructing map functions; three of the class of crossover formation methods and one of the class of differential equation methods. In the latter, one assumes that \(r(=r_{a,b})\) satisfies the differential equation \(dr/dx=1-2C(x)r\) for some empirical coincidence function C(x). Haldane's (1919) choice, \(C=1\), leads to the map function \(r=(1-e^{-2x})/2\). In the general crossover formation process, one assumes that the lengths of the intervals between successive crossovers are random variables governed by a distribution function \(F(\cdot)\). The renewal function associated with F is \(R(t)=F(t)+\int^{t}_{0}F(t- \xi)dR(\xi)\) and then \(x_{a,b}=R(b)-R(a)\). If F is taken as a Poisson distribution then \(r=(1-e^{-2x})/2\) again results. The count location method was introduced by the authors in Adv. Appl. Probab. 11, 479-501 (1979; Zbl 0411.60041). It is characterized by a probability sequence \((c_ 0,c_ 1,c_ 2,...)\) and a distribution function F(x) such that along the chromosomal interval [0,l], the probability of k crossovers is \(c_ k\) and conditioned that k crossovers do occur, their locations are determined as realizations of F. Here \(x_{a,b}=(m/2)(F(b)-F(a))\) where m is the expected number of crossovers. The authors prove four theorems one of which is theorem 1: For a crossover formation model of the form of a renewal process with distribution F such that \((d^ nF/dt^ n)(0)\neq 0\) for some n, a map function exists if and only if F is the exponential distribution, i.e., the renewal process is Poisson. The other theorems relate to the multilocus feasibility of map functions. For example, …
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 60K20
zbMATH DE Number: 3858902
zbMATH Keywords: recombination frequency
zbMATH Keywords: genetic map function
zbMATH Keywords: crossover formation methods
zbMATH Keywords: count location method
zbMATH Keywords: exponential distribution
zbMATH Keywords: multilocus feasibility
zbMATH Keywords: differential equations method
zbMATH Keywords: Poisson renewal process
MaRDI profile type: Publication
full work available at URL: 0040-5809(84)90013-3
OpenAlex ID: W2000498059
Wikidata QID: Q52693272
cites work: Q5724331
cites work: Q5512461
cites work: On the Probability Theory of Linkage in Mendelian Heredity
cites work: Classifications and comparisons of multilocus recombination distributions
cites work: A natural class of multilocus recombination processes and related measures of crossover interference
description: scientific article; zbMATH DE number 3858902