遺傳標幟（如DNA分子標幟）常被遺傳學與育種學家用來代表某特定基因型（包括品種或品系），這些標幟多散佈於整個基因組裡，它們在不同個體上的多型性以及在族群裡的分離情形透過基因型鑑定所觀察。當人們擁有夠多數量具有多型性的標幟，就可以輕易地辨識出一個個體或一組相似的基因型。標幟與基因間在族群中的不獨立分離讓某些標幟上的基因型可代表一個或數個相似表現型個體，若樣本族群中的表現型有所差異，（數量）性狀基因座的定位便可能造就。本研究旨在推導在單交與雙交雜種後代自、異交族群中的多基因座（連鎖與不連鎖）基因型頻度。在單交雜種自交族群裡，我們給連結基因型與其頻度的互換分數（recombination score）提供了證明。在Hospital等人所給予互換分數的定義下，具有相同互換分數的基因型在任意世代中的出現有理論上相同的機率。這樣的「對稱性」在其他雜交族群中亦有類似的變形。在雙交的異交族群裡，我們使用了三階層的互換分數來歸類任意世代中擁有相同頻度的配子型。由於基因型頻度理論值的數目少於基因型的數目，我們只要利用較少維度的轉移矩陣作乘法運算，便可得到任意世代所有的基因型頻度。最後，我們提供了一組模擬單交雜種自交F6族群的資料，作為多基因座頻度應用於多區間定位的範例。

Genetic markers such as DNA have long been used to represent the genotype of an individual (precisely, a lineage) by geneticists and breeders. These markers are developed by some means throughout the genome of the particular organism and being genotyped. Polymorphism of each marker characterizes different individuals. The characterization would be much more specific with the amount of polymorphic genetic markers we recognized. The genotypes of these markers are associated with the phenotypic values in the mapping of quantitative trait loci (QTL). In this study, we derived the multilocus genotypic frequencies for recombinant inbred and advanced intercrossed populations from 2- and 4-way crosses of inbred lines. We provide the mathematical proof for the relationship between the theoretical genotypic frequencies and the recombination scores of individual in the selfed populations derived from biparental cross of inbred lines. It is showed that genotypes with the same recombination score would have the equal probability to show up in any generation beyond the F2. This arisen symmetry also has its similar variants in 2-way random mating as well as 4-way selfing and random mating populations. Multi-level recombination score is proposed to identify the gametes with the same theoretical frequency among the random-mated 4-way cross derivatives. By using these symmetries, we reduced the dimensions of frequencies-transition matrix for each population. The reduction of matrix size lightens the computation effort in the multiplications for obtaining the advanced generation genotypic frequencies. At the end of this study, we provide a simple simulated case studying involving a biparental selfed F6 population and its multiple interval QTL mapping.

口試委員會審定書 i
致謝 ii
中文摘要 iv
Abstract v
Contents vi
List of Figures vii
List of Tables viii
1 Introduction 1
2 Theory and Algorithm 5
2.1 Self-fertilization 5
2.1.1 Biparental cross of inbred lines (2-way cross) 6
2.1.2 4-way cross 15
2.2 Random mating 18
2.2.1 2-way cross 18
2.2.2 4-way cross 19
3 Case Studying 22
3.1 Map construction 22
3.2 Genome scanning 25
4 Discussion 30
Appendix: Proofs of Property, Lemma and Theorem 35
A.1 Number of distinct genotypes in a 2-way-cross-derived population 35
A.2 Reason for some properties in recombination scores 35
A.3 Lemmas prior to Theorem 2 37
A.4 Number of distinct frequencies in a 2-way-cross-derived selfed population 41
A.5 Multi-level recombination scores for 4-way cross random-mated population 43
Literature Cited 44
List of Figures
2.1 A selfed population derived from the biparenatal cross of inbred lines
in terms of a diploid chromosome 5
2.2 Possible parents for a particular genotype 7
2.3 Ties of frequency among genotypes with recombination scores different
only by signs and contain at least one “1” in the F2 population from
biparental cross of inbred lines 12
2.4 Parents for genotypes i and j, denoted k and l (k′ and l′), constructed
in the same way 14
2.5 A 4-way cross 16
3.1 The LOD scores in the mapping of QTLs 28
List of Tables
2.1 The numbers of distinct genotypes and of distinct frequencies in the
advanced population derived from 2-way cross 9
3.1 The likely models for QTL and their corresponding statistics 29

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