![]() ![]() The result is a new class of probability distributions that are based on a novel use of generalized hyperbolic functions. The crossover process is a then an alternating renewal process that toggles between two states (recombination/no recombination), and its finite dimensional distributions are given in closed form. The interesting mathematical ideas are that one can explicitly write down the finite dimensional distributions of a counting renewal process with Erlang inter-arrival distances. Since we hope that these results will be used by geneticists, we have attempted to make them more accessible by stating them without the standard lemma/theorem format and separating the proofs to Section 6. Section 5 reviews our findings and makes some general comments about the plausibility of renewal models for recombination. The description of the identity by descent process and the effect on genome wide thresholds are contained in the following section. The next section derives closed form expressions for map functions and coincidence functions for Erlang models of recombination, filling in some gaps in the work of Cobbs ( 1978) and Foss et al. We show that the (infinite series) matrix functions they consider can also be used to specify the multilocus probabilities for the crossover process as well, and we give closed form expressions for both the crossover process and the four strand chiasma process. ( 1995), where infinite series expressions for multilocus probabilities are given for the chiasma model on the four strand bundle. These results in Section 2 are based on Zhao et al. The main results of this paper are closed form expressions for multilocus probabilities when the inter-event distribution is Erlang. Expressions for map functions and coincidence functions follow from the multilocus probabilities. The approach here is to specify a model for recombination and derive multilocus probabilities directly from that model. ( 1993) appears to provide more, but neither can fully characterize interference in general. The “adjacent interval” coincidence coefficient (Muller ( 1916) Sturtevant ( 1915)) provides some information, and the “nonadjacent interval” coincidence coefficient of Foss et al. The root of the problem is that a map function can only describe what happens among loci and a multilocus probability requires more information. However, as Zhao and Speed ( 1996) point out, such efforts cannot accurately describe general multilocus probabilities because different models can yield the same map function. Geiringer ( 1944) Karlin and Liberman ( 1994, 1979) Liberman and Karlin ( 1984) Risch and Lange ( 1983) Schnell ( 1961) Weeks et al. Several authors have attempted to express multilocus probabilities in terms of map functions, e. A map function is a relation r= r( d) that expresses the recombination fraction r between two locations on a chromosome to the genetic distance d between them. Various models have been proposed to represent that interference. It is widely accepted that there is positive crossover interference - a crossover at a point apparently inhibits crossovers at nearby points, see Kwiatkowski et al. Its use in genetics was introduced by Haldane ( 1919), who knew it assumes no crossover interference. The standard model is a Poisson process which is used because it was a reasonable first approximation and it is mathematically tractable. ![]() genetic units, not physical units.) Just as important, the multilocus probabilities depend on the model used to describe the way crossovers occur. (Throughout this paper, distances will be expressed in Morgans, e.g. These multilocus probabilities depend on inter-marker distances d 1,…, d n, where d j= distance between markers \(\mathcal \). ![]()
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