论文标题
求解选择重组方程：选择和重组的祖先线
Solving the selection-recombination equation: Ancestral lines under selection and recombination
论文作者
论文摘要
确定性的选择重组方程式描述了在大量法律中选择和重组的种群遗传类型组成的演变。到目前为止，明确的解决方案似乎已经遥不可及。只有在三个站点的特殊情况下，选择在其中一个地点才能找到一个近似的解决方案，但没有明显的泛化途径。我们对\ emph {任意}链接到一个选定位点的中性位点数量的情况使用分析和概率，家谱方法。这导致溶液的递归积分表示。从祖先选择重组图的变体开始，我们开发了一个有效的家谱结构，该结构可以等效地表示为加权分区过程，具有启动和重置的Yule过程家族以及一个起始过程。我们证明它们是对差分方程在时间前进的解的双重，因此可以获得确定性解的随机表示，以及Markov Semigroup以封闭形式获得的。
The deterministic selection-recombination equation describes the evolution of the genetic type composition of a population under selection and recombination in a law of large numbers regime. So far, an explicit solution has seemed out of reach; only in the special case of three sites with selection acting on one of them has an approximate solution been found, but without an obvious path to generalisation. We use both an analytical and a probabilistic, genealogical approach for the case of an \emph{arbitrary} number of neutral sites linked to one selected site. This leads to a recursive integral representation of the solution. Starting from a variant of the ancestral selection-recombination graph, we develop an efficient genealogical structure, which may, equivalently, be represented as a weighted partitioning process, a family of Yule processes with initiation and resetting, and a family of initiation processes. We prove them to be dual to the solution of the differential equation forward in time and thus obtain a stochastic representation of the deterministic solution, along with the Markov semigroup in closed form.
