Theoretical models of genetic map functions
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, theorem 4: Let \(r=M(x)\) be a map function that is multilocus feasible. If all derivatives \(M^{(n)}(x)\) exist, then \((-1)^ nM^{(n)}(0)\leq 0\), \(n=1,2,..\).. On the basis of these theorems the authors conclude that Haldane's choice in the differential equations method is multilocus feasible but several other choices derived by that method are not. Only the Poisson renewal process leads to a multilocus feasibility map function. And every count- location process leads to a feasible multilocus map function.
scientific article; zbMATH DE number 3858902