Page 269 - Applied Probability

12
Models of Recombination
12.1 Introduction

At meiosis, each member of a pair of homologous chromosomes replicates
to form two sister chromosomes known as chromatids. The maternally
and paternally derived sister pairs then perfectly align to form a bundle
of four chromatids. Crossing-over occurs at points along the bundle known
as chiasmata. At each chiasma, one sister chromatid from each pair is
randomly selected and cut at the crossover point. The cell then rejoins the
partial paternal chromatid above the cut to the partial maternal chromatid
below the cut, and vice versa, to form two hybrid maternal–paternal chro-
matids. The preponderance of evidence suggests that the two chromatids
participating in a chiasma are chosen nearly independently from chiasma to
chiasma [31]. This independence property is termed lack of chromatid in-
terference. After crossing-over has occurred, the recombined chromatids
of a bundle are coordinately separated by two cell divisions so that each of
the four resulting gametes receives exactly one chromatid.
The number and positions of the chiasmata along a chromatid bundle
provide an example of a stochastic point process [6]. Most probabilists
are familiar with point processes such as Poisson processes and renewal
processes. This chapter considers point process models for the formation of
chiasmata. Each such chiasma process induces correlated and identically
distributed crossover processes on the four gametes created from a chro-
matid bundle. We can conceive of both chiasma and crossover processes as
occurring on a fixed interval of the real line. When one makes, as we do
in this chapter, the assumption of no chromatid interference, then each of
the four crossover processes is created from the chiasma process by ran-
dom thinning of chiasmata. If we characterize a gamete by the origin
of one of its telomeres (chromosome ends), then it participates in half
of the crossovers on average. Random thinning amounts to independently
choosing for each chiasma whether the gamete does or does not participate
in the underlying crossover event. Because of the symmetry of the model,
these two choices are equally likely.
For any well-behaved set A on the chromatid bundle, the random variable
N A counts the number of random points in A. The chiasma process is de-
termined by these random variables. Two important functions of A are the
intensity measure E(N A ) and the avoidance probability Pr(N A = 0).
It is natural to assume that E(N {a} ) = 0 for every fixed point a and that
random points never coincide. Beyond these assumptions, we seek models