Coalescent processes and population models

University Of California-San Diego, La Jolla CA

Investigators

Abstract

The PI studies several problems related to coalescent processes and population models. Coalescent processes are stochastic processes that model a system of particles which start out separated and merge into clusters as time goes forward. These processes can be used to describe the genealogy of a population because if one takes a sample from a population and follows the ancestral lines backwards in time, the ancestral lines will coalesce. When a beneficial mutation occurs in a population and spreads rapidly, many ancestral lines will merge at almost the same time, as they will all be traced back to the individual that had the beneficial mutation. One goal is to use results from the theory of coalescent processes with multiple mergers to get further insight into tests that are used to detect beneficial mutations. A second project is to determine the distribution of the time that it takes for one individual in a population to experience k mutations. The PI will also study a model of coalescence in which the total mass of the system increases over time. A coalescent model in which new particles appear at rate one and clusters merge at a rate proportional to the product of the masses has previously been studied, but a qualitatively different phase transition is conjectured to arise in an alternative model in which clusters merge at a rate proportional to the sum of the masses. Stochastic models of coalescence have a wide range of applications in other fields of science such as biology, physical chemistry, and astronomy. Biologists interested in understanding evolution are concerned with the merging of the ancestral lines of a sample from a population. It should be possible to use the mathematical theory of coalescence to gain further insight into how beneficial mutations impact this process. The project of determining the amount of time for one individual in a population to experience several mutations is motivated by simple models of cancer, in which it is assumed that a cell becomes cancerous only after several harmful mutations take place. The study of coalescent processes in which the mass of the system increases over time is motivated by recent interest in randomly growing networks.