Dawson K J
IACR Long Ashton Research Station, University of Bristol, Bristol, Long Ashton, BS41 9AF, United Kingdom.
Theor Popul Biol. 2000 Aug;58(1):1-20. doi: 10.1006/tpbi.2000.1471.
How rapidly does an arbitrary pattern of statistical association among a set of loci decay under meiosis and random union of gametes? This problem is non-trivial, even in the case of an infinitely large population where selection and other forces are absent. J. H. Bennett (1954, Ann. Hum. Genet. 18, 311-317) found that, for an arbitrary number of loci with an arbitrary linkage map, it is possible to define measures of linkage disequilibrium that decay geometrically with time. He found a recursive method for deriving expressions for these variables in terms of "allelic moments" (the factorial moments about the origin of the "allelic indicators"), and expressions for the allelic moments in terms of his new variables. However, Bennett no where stated his recursive algorithm explicitly, nor did he give a general formula for his measures of linkage disequilibrium, for an arbitrary number of loci. Recursive definitions of Bennett's variables were obtained by Lyubich. However, the expressions generated by these recursions are not the same as those found by Bennett. (They do not express Bennett's variables as functions of the allelic moments.) Lyubich's derivations employ genetic algebras. Here, I present a method for obtaining explicit expressions for Bennett's variables in terms of the allelic moments. I show that the transformation from the allelic moments to Bennett's variables and the inverse transformation always have the form that Bennett claimed. (This transformation and its inverse have essentially the same form.) I present general recursions for calculating the coefficients in the forward transformation and the coefficients in the inverse transformation. My derivations involve combinatorial arguments and ordinary algebra only. The special case of unlinked loci is briefly discussed.
在减数分裂和配子随机结合的情况下，一组基因座之间任意的统计关联模式衰减得有多快？即使在没有选择和其他因素的无限大种群的情况下，这个问题也并非易事。J. H. 贝内特（1954年，《人类遗传学杂志》18卷，311 - 317页）发现，对于任意数量具有任意连锁图谱的基因座，可以定义连锁不平衡的度量，这些度量会随时间呈几何级数衰减。他找到了一种递归方法，用于根据“等位基因矩”（关于“等位基因指标”原点的阶乘矩）推导这些变量的表达式，以及根据他的新变量推导等位基因矩的表达式。然而，贝内特从未明确阐述过他的递归算法，也没有给出任意数量基因座的连锁不平衡度量的通用公式。柳比奇得到了贝内特变量的递归定义。然而，这些递归产生的表达式与贝内特所发现的并不相同。（它们没有将贝内特的变量表示为等位基因矩的函数。）柳比奇的推导使用了遗传代数。在此，我提出一种方法，用于根据等位基因矩获得贝内特变量的显式表达式。我表明从等位基因矩到贝内特变量的变换以及逆变换总是具有贝内特所声称的形式。（这种变换及其逆变换本质上具有相同的形式。）我给出了用于计算正向变换系数和逆变换系数的通用递归式。我的推导仅涉及组合论证和普通代数。文中简要讨论了基因座不连锁的特殊情况。
