Mapping functions

Abstract

Accurate estimation of map distance between markers is important for the construction of large-scale linkage maps because it provides reliable and useful linkage information of markers on chromosomes. How to improve accuracy of estimating map distances depends on an appropriate mapping function. We used the coefficient of coincidence to integrate the Haldane function, in which crossovers are assumed to be independent and the Morgan function, in which crossovers are assumed to be interfered, and produce a new mapping function. The mapping function based on positive interference is referred to as the positive function and that on negative interference as the negative function. In these two mapping functions, map distances between loci are determined by both recombination frequencies and the coefficient of coincidence. We applied our mapping functions to four examples and show that our map estimates have much higher goodness-of-fit to the observed mapping data than the Haldane and Kosambi functions. Therefore, they can provide much more precise estimates of map distances than the two conventional mapping functions. Furthermore, our mapping functions produced almost linear (additive) map distances.

Appendix

To fit the negative function to observed data, we need to calculate expected frequencies of single crossovers within intervals 1(ts4–lg2), 2 (lg2–T), and 3 (T–al) and double crossovers between intervals 1 and 2, between intervals 2 and 3, and between intervals 1 and 3 from the map distances (Fig. 3) estimated by the negative function. The map distances between intervals 1 and 2, between intervals 2 and 3, and between intervals 1 and 3 were estimated as 27.8618, 20.4349 and 47.5829 cM, respectively. By placing these estimated map distances and observed coefficients of coincidence (λ = 1.0621, 1.7203, and 1.043) in Eqs. 31 and 30, the recombination frequencies between intervals 1 and 2, between intervals 2 and 3, and between intervals 1 and 3 are expected as 0.2212, 0.2695, and 0.3639, respectively. According to Eqs. 18 and 19, expected frequencies of single crossovers within intervals 1 and 2 in triplet ts4–lg2−T is found to be 0.2212 × 27.5049/27.8618 = 0.2183 and 0.2212 − 0.2184 = 0.0028, respectively. Similarly, expected frequencies of single crossovers within intervals 2 and 3 in triplet lg2–T–al are 0.0048 and 0.2647, respectively. Solutions for expected frequencies of double crossovers between two adjacent intervals 1 and 2 and between intervals 2 and 3 from f _E/[(θ₁ + f _E)(θ₂ + f _E)] = 1 are respectively 0.000786 and 0.0017. Frequency of double crossovers under complete negative interference is f _C = 0.0028 + 0.0048 = 0.0076. These estimated parameters are substituted into Eq. 20 and observed frequencies of double crossovers between intervals 1 and 2 and between intervals 2 and 3 under negative interference are expected as f _O12 = 0.0012 and f _O23 = 0.006. Similarly, f _O13 = 0.0631 between intervals 1 and 3. Thus, the numbers of crossovers in various intervals are expected as

• Interval 1: 632 × (θ₁ − f _O13) = 632(0.2184 − 0.0631) = 98.08,

• Interval 2: 632 × (θ₂ − f _O12 − f _O23) = 632(0.0076 − 0.0012 − 0.006) = 0.253,

• Interval 3: 632 × (θ₃ − f _O13) = 632(0.2647 − 0.0631) = 127.474,

• Intervals 1 and 2: 632 × f _O12 = 632 × 0.0012 = 0.764,

• Intervals 2 and 3: 632 × f _O23 = 632 × 0.0048 = 3.792,

• Intervals 1 and 3: 632 × f _O13 = 632 × 0.0631 = 39.879,

• Parental type: 632 − (98.086 + 0.253 + 127.474 + 0.764 + 3.792 + 39.879) = 361.752.