Coalescent processes and population models

PI研究了与聚结过程和种群模型有关的几个问题。凝聚过程是模拟粒子系统的随机过程，粒子系统开始时是分离的，随着时间的推移合并成簇。这些过程可以用来描述一个群体的家谱，因为如果从一个群体中提取样本，并沿着祖先的谱系向后追溯，那么祖先的谱系将会合并。当一个有益的突变在一个种群中发生并迅速传播时，许多祖系将几乎同时合并，因为它们都将追溯到具有有益突变的个体。一个目标是利用多合并合并过程理论的结果，进一步深入了解用于检测有益突变的测试。第二个项目是确定种群中一个个体经历k个突变所需的时间分布。PI还将研究一个系统总质量随时间增加的聚结模型。以前曾研究过一种聚结模型，在这种模型中，新粒子的出现率为1，而星团的合并率与质量的乘积成比例，但人们推测，在另一种模型中，星团的合并率与质量的总和成比例，会出现一种性质不同的相变。聚并的随机模型在生物学、物理化学和天文学等其他科学领域有着广泛的应用。对理解进化感兴趣的生物学家关注的是一个种群中一个样本祖先系的合并。应该有可能利用聚结的数学理论来进一步了解有益突变如何影响这一过程。确定人群中一个个体经历几次突变所需时间的项目是由简单的癌症模型推动的，在这些模型中，假设一个细胞只有在发生了几次有害的突变之后才会癌变。对系统质量随时间增加的凝聚过程的研究是由最近对随机增长网络的兴趣所激发的。

The PI studies several problems related to coalescentprocesses and population models. Coalescent processesare stochastic processes that model a system of particleswhich start out separated and merge into clusters as timegoes forward. These processes can be used to describethe genealogy of a population because if one takes asample from a population and follows the ancestral linesbackwards in time, the ancestral lines will coalesce.When a beneficial mutation occurs in a population andspreads rapidly, many ancestral lines will merge atalmost the same time, as they will all be traced back tothe individual that had the beneficial mutation. Onegoal is to use results from the theory of coalescentprocesses with multiple mergers to get further insightinto tests that are used to detect beneficial mutations.A second project is to determine the distribution of thetime that it takes for one individual in a population toexperience k mutations. The PI will also study a modelof coalescence in which the total mass of the systemincreases over time. A coalescent model in which newparticles appear at rate one and clusters merge at a rateproportional to the product of the masses has previouslybeen studied, but a qualitatively different phasetransition is conjectured to arise in an alternativemodel in which clusters merge at a rate proportional tothe sum of the masses.Stochastic models of coalescence have a wide range ofapplications in other fields of science such as biology,physical chemistry, and astronomy. Biologists interestedin understanding evolution are concerned with the mergingof the ancestral lines of a sample from a population. Itshould be possible to use the mathematical theory ofcoalescence to gain further insight into how beneficialmutations impact this process. The project of determiningthe amount of time for one individual in a population toexperience several mutations is motivated by simple modelsof cancer, in which it is assumed that a cell becomescancerous only after several harmful mutations take place.The study of coalescent processes in which the mass of thesystem increases over time is motivated by recent interestin randomly growing networks.