推导IBS评分在全同胞对人群中概率分布的计算公式

Equation Derivation of the Probability Distribution of IBS Score among Full Sibling Pairs

目的推导通过STR等位基因频率计算生物学全同胞对间状态一致性(identity by state,IBS)评分概率分布的通用计算公式。方法依据孟德尔遗传规律和生物学全同胞(full sibling,FS)的父母为无关个体这一假设,推导得到不同基因型组合的无关个体生育两名子代时,子代不同基因型组合的IBS评分及所对应的概率。结果以fi表示某STR基因座第i个等位基因的频率(i=1,2,…,m),则生物学全同胞对在该基因座出现2个相同等位基因的概率(p2FS)的计算公式为:p2FS=1/4×[1+2∑i=1^m fi^2+2(∑i=1^m fi^2)^2-∑i=1^m i^f4];在该基因座出现1个相同等位基因的概率(p1FS)的计算公式为:p1FS:1/2×[1+∑i=1^m fi^2-2(∑i=1^m fi^2)^2-1∑i=1^m fi^3+2∑i=1^m fi^4];在该基因座出现0个相同等位基因的概率(P0FS),的计算公式为:p0FS=1/4×[1-4∑i=1^m fi^2+2(∑i=1^m fi^2)^2+4∑i=1^m fi^3-3∑i=1^m fi^4];p2FS、p1FS、p0FS的和为1。对于包含n个STR基因座的多重分型系统,生物学全同胞间的IBS评分符合二项分布:IBS^B(2n,π1)。其中总体率π1的计算公式为:π1=1/n∑l=1^n p2FSl+1/2n∑l=1^n p1FSl。结论生物学全同胞鉴定中的备择假设为两名被鉴定人为生物学全同胞,对任意STR基因座组合、IBS评分所对应的备择假设概率均可通过本文所推导的公式直接进行计算,计算结果是进行证据解释的基础。

Objective To derive the general equation of the probability distribution of identity by state(IBS)score among biological full sibling pairs by calculating STR allele frequency.Methods Based on the Mendelian genetics law and the hypothesis that parents of biological full siblings(FS)were unrelated individuals,the IBS score and corresponding probability of different genotype combinations in the offspring when unrelated individuals of different genotype combinations give birth to two offsprings were derived.Results Given fi(i=1,2,…,m)as the frequency of the ith allele of a STR locus,the probability of sharing 2 alleles(p2 FS),1 allele(p1 FS)or 0 allele(p0 FS)with biological full sibling pairs on the locus can be respectively expressed as follows:p2FS=1/4×[1+2∑i=1^m fi^2+2(∑i=1^m fi^2)^2-∑i=1^m i^f4],and p0FS=1/4×[1-4∑i=1^m fi^2+2(∑i=1^m fi^2)^2+4∑i=1^m fi^3-3∑i=1^m fi^4].The sum of p2 FS,p1 FS and p0 FS must be 1.As for the multiple genotyping system that contained n STR loci,IBS scores between biological full sibling pairs conform to binomial distribution:IBS^B(2 n,π1).The population rate can be given by the formula:π1=1/n∑l=1^n p2FSl+1/2n∑l=1^n p1FSl.Conelusion The alternative hypothesis in biological full sibling testing is that two appraised individuals are biological full siblings.The probability of the corresponding alternative hypothesis of any STR locus combination or IBS score can be directly calculated by the equations presented in this study,and the calculation results are the basis for explanations of the evidence.