Genetic Map Functions

Abstract

A genetic map function M gives a relation r = M(d) connecting recombination fractions r and genetic map distances d between pairs of loci along a chromosome arm. Recombination fractions and map distances are summary statistics concerning potentially observable characteristics of the single chromosomes (also known as chromatids) that are the products of meiosis, and that go into gametes. The recom-bination fraction between two loci is the proportion of such chromosomes that are recombinant, that is, that have genetic material of differing parental origins , at the two loci (see Linkage Analysis, Model Based). The genetic map distance between two loci is the average number of exchange points that occur along such a chromosome between the loci, where an exchange point, also known as a crossover point, is a point where the parental origin of the genetic material changes. In these definitions, proportions and averages are calculated in the hypothetical infinite population of single chromosomes resulting from meiosis in a given organism, occurring under standard conditions. Variations between organisms within the same species, or of the conditions of meiosis, may lead to small, but observable, differences in these quantities. It should be noted that some authors (e.g. [1] and [9]) use the term map function for the function M −1 in the inverse relation d = M −1 (r) expressing d in terms of r. We follow Karlin [7] and others in calling M a map function, mainly because the theoretical development is slightly simpler for M than for M −1. Map functions have been widely used in genetics because of two facts. The first is that genetic map distances are additive by definition, whereas recom-bination fractions are not. Thus, map distances are preferred for mapping chromosomes. The second is that recombination fractions are much easier to estimate from data, although with human data indirect techniques may need to be used, see [9]. This is because recombination refers only to features of chromosomes at the endpoints of intervals. By contrast, to estimate a map distance information concerning exchanges in the entire interval between two loci is required and, until recently, such information was Reproduced from the Encyclopedia of Biostatistics, 2nd Edition.  John Wiley & Sons, Ltd. ISBN: 0-470-84829-4. rarely, if ever, available. Modern molecular genetic methods now exist permitting the identification of points of exchange along chromosomes, and in the near future it may become much easier to estimate map distances directly (see [8]). The traditional use of map functions has been to take an estimated recombination fractionˆrfractionˆ fractionˆr between two loci and a map function M deemed appropriate for the organism in question, and estimate the map distance between the loci by the quantityˆd quantityˆ quantityˆd = M −1 (ˆ r). Perhaps the simplest case is the map function r = d, with inverse d = r. This is quite satisfactory for small r and d, say, in the interval (0, 0.05), but the relative error increases as the magnitudes of d and r increase. If two loci can be linked by a chain of intermediate loci, each having a recom-bination fraction of no more than 0.05 (say) with its successor, then a quite satisfactory estimate of the map distance between the initial and final locus can be obtained by adding the successive interlocus recombination fractions. The notion of map function is helpful in situations where such intermediate loci are not available. The recombination fraction and map length of an interval will differ when there is a nonzero chance of multiple exchange points occurring in the interval. The chance of this occurring increases as the size of the interval increases. If we denote the distribution of exchange points in a particular interval by (p 0 , p 1 , p 2 , p 3 ,. . .), so that p k is the expected proportion of single chromosomes that have k exchange points in the interval, then the recombination fraction is r = p 1 + p 3 + · · · (1) (i.e. the probability of an odd number of exchange points), while the map length is d = p 1 + 2p 2 + 3p 3 + · · ·. (2) For example, if p k = e −d d k /k!, then the map length is easily seen to be d, while the recombination fraction is r = e −d + e −d d 3 3! + · · · = 1 2 (1 − e −2d). (3) This relation is known as Haldane's map function, and it is widely used today, nearly 80 years after Haldane [6] first described it. Although simple and easy to use, especially for multilocus calculations,