Coalescent processes and population models

PI研究与联合中心过程和种群模型相关的几个问题。聚结过程是一种随机过程，它模拟了一个粒子系统，随着时间的推移，这些粒子开始是分开的，然后合并成簇。这些过程可以用来描述一个群体的谱系，因为如果一个人从一个群体中抽取样本，并在时间上追溯到祖先的血统，祖先的血统就会合并。当一个有益的突变在一个人口中发生并迅速传播时，许多祖先的血统将几乎同时合并，因为它们都将被追溯到发生有益突变的个人。OneGoal是使用多个合并的联合中心过程理论的结果来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个突变所需时间的分布。PI还将研究一个聚合模型，在该模型中，系统的总质量会随着时间的推移而增加。以前已经研究过一种聚结模型，其中新粒子以1的速率出现，团簇以与质量的乘积成正比的速度合并，但人们推测，在另一种模型中，团簇以与质量的总和成比例的速度合并，会出现性质不同的相变。聚结的随机模型在其他科学领域有广泛的应用，如生物学、物理化学和天文学。对了解进化感兴趣的生物学家关心的是从种群中融合样本的祖先系。应该有可能利用结合的数学理论来进一步洞察有利的突变如何影响这一过程。确定一个群体中的一个个体经历几个突变的时间量的项目是由简单的癌症模型推动的，在这个模型中，假设一个细胞只有在发生几个有害的突变后才变得癌细胞。对系统质量随时间增加的合并过程的研究受到最近对随机增长网络的兴趣的推动。

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.