Mathematical Sciences: Stochastic Models in Population Genetics

[505129] [Donnelly]摘要提出的研究分为三大类。第一个问题涉及测量值人口模型。用粒子表示法研究了一类“中性”过程（包括Dawson-Watanabe和Fleming-Viot）的性质及其诱导的系谱结构。该提案的一个主要目的是扩展这种结构，最明显的是纳入选择（相当于空间设置中的位置依赖分支率），但也（在遗传学背景下）允许重组和群体亚结构。在解释人类种群遗传数据的问题的激励下，研究者和他的同事研究了一般模型的特定子类，以期了解关于种群动态和可能的人口统计学相关性的各种假设对家谱的影响，从而对遗传数据的影响。该建议的第二个领域涉及地理结构群体遗传学模型的特性。最近关于这些模型中聚结时间的结果允许明确计算与分子DNA数据相关的突变结构（超过无限等位基因）的身份测量矩。相反，对这些模型的理解激发了对分子遗传数据的群体分化（和相关估计器）测量的研究，这些数据概括了赖特的F_st系数。提案的第三个领域涉及将现有的和预期的理论结果应用于遗传数据分析中的统计问题。这些包括广泛的选择性模型的精确似然比（因此有效的推理程序），在测试中立性中使用等位基因年龄顺序信息的潜力，来自连锁基因样本的进化参数估计的相关结构，以及（不）存在来自单个e-locus数据的某些进化参数的一致估计。该项目旨在促进我们对进化过程的理解。数学模型的发展使我们能够对特定进化机制的结果进行检验：有人可能会问：“如果进化是按照以下方式进行的，那么我们在遗传数据中会看到什么样的模式呢？”数学建模是必要的，因为许多进化力量在极长的时间尺度上运作，而直接观察或实验是不可能的。然后，将模型预测与实际遗传数据进行比较，可以深入了解假设的机制起作用的程度。调查人员在一般水平上研究有关感兴趣的人口结构和人口统计的特定假设的后果。该项目的特定方面涉及人类种群进化的模型。在解释遗传数据时出现的各种统计问题也包括在内。

9505129 Donnelly Abstract The proposed research falls in three broad areas. The first of these concerns measure-valued population models. A particle representation is used to study properties of a general class (including the Dawson-Watanabe and Fleming-Viot) of "neutral" processes and their induced genealogical structure. A major aim of the proposal is to extend this construction, most notably to incorporate selection (equivalently location dependent branching rates in the spatial setting), but also (in the genetics context) to allow for recombination and population substructure. Motivated by problems in the interpretation of human population genetic data, the investigator and his colleagues study particular subclasses of the general models with a view to understanding the effects on genealogy, and hence on genetic data, of various assumptions about population dynamics, and possible demographic correlations. A second area of the proposal concerns properties of geographically structured population genetics models. Recent results relating to coalescence times in these models allow explicit calculation of moments of identity measures for mutation structures (beyond infinite alleles) relevant to molecular DNA data. Conversely, an understanding of the models motivates a study of measures of population differentiation (and associated estimators) for molecular genetic data which generalize Wright's F_st coefficient. The third area of the proposal concerns the application of existing and anticipated theoretical results to statistical questions in the analysis of genetic data. These include exact likelihood ratios for a wide class of selective models against neutrality (and hence efficient inference procedures), the potential for use of information on the age order of alleles in testing neutrality, the correlation structure of estimates of evolutionary parameters from samples of linked genes, and the (non-)existence of consistent estimators of certain evolutionary parameters from singl e-locus data. The project is concerned with advancing our understanding of evolutionary processes. The development of mathematical models allows an examination of the consequences of particular evolutionary mechanisms: one may ask "if evolution acted in the following way, what sort of patterns would we expect in genetic data?". Mathematical modeling is necessary because many evolutionary forces operate over extremely long time scales for which direct observation, or experimentation, is impossible. Comparisons of model predictions with actual genetic data then provides insight into the extent to which the posited mechanisms are acting. The investigator studies, at a general level, the consequences of particular assumptions about the structure and demography of populations of interest. Particular aspects of the project relate to models for the evolution of human populations. Various statistical questions which arise in the interpretation of genetic data are also included.