Dualities for branching-coalescing processes in population genetics
Final Report Abstract
In mathematical population genetics, stochastic processes are crucial to understand on a fundamental level the impact of evolutionary forces on populations. There are two dominating approaches. On the one hand, forward in time, allele-frequency processes are studied. A suitable class of stochastic processes are so-called Wright-Fisher processes. On the other hand, processes that evolve backward in time carry important traces of evolution. These backward-intime processes commonly arise, though not exclusively, from genealogical considerations, thus often establishing connections to branching-coalescing processes. One of the contributions of the current project is a wide-ranging analysis of the potential long term behavior of rather general Wright-Fisher processes. In this way, the project contributes to a better understanding of the long term effects of evolution. For Wright-Fisher processes with two allelic types, and under consideration of genetic drift and selection, four qualitatively distinct long-term behaviors and their corresponding parameter regions were identified. In one of these scenarios, genetic variability is maintained with absolute certainty. In the other three instances, where alleles may disappear, bounds for the rate of extinction were established. In models that solely incorporate genetic drift, yet consider a finite number of alleles, explicit expressions for the expected times of allele extinction were derived. This extension broadens previous findings to encompass a wide range of Wright-Fisher processes. Ultimately, this facilitates a more precise prediction of allele extinction times. A pivotal role in deriving these results is played by backward-in-time processes. In the aforementioned cases, these processes are linked to the allele-frequency process through what are known as Siegmund and moment dualities. This work serves as an example of how these corresponding dualities can be leveraged to analyze forward-in-time processes. For some biological systems, Wright-Fisher processes are inadequate; for example, if the population size affects the strength of evolutionary forces. The antibody affinity maturation within the immune system is a situation that requires therefore a new model. This was achieved in this project, where in collaboration with computational biologists, a novel mean-field branching process model was developed to capture the relevant properties of such systems. In addition to population genetics applications, the project was also concerned with a general problem in the theory of Brownian motions. Namely, we described an explicit construction for a coupling of Brownian motions with different drift in a way that maximizes their initial agreement. Also there, a crucial role is played by a backward-in-time process.
Publications
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Absorption and stationary times for the Λ-Wright-Fisher process. arXiv e-prints
A. Blancas; A. Gonzalez Casanova; S. Hummel & S. Palau
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Lines of descent in a Moran model with frequency-dependent selection and mutation. Stochastic Processes and their Applications, 160, 409-457.
Baake, E.; Esercito, L. & Hummel, S.
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Constructing maximal germ couplings of Brownian motions with drift. Electronic Communications in Probability, 29(none).
Hummel, Sebastian & Quinn, Jaffe Adam
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Mean-field interacting multi-type birth–death processes with a view to applications in phylodynamics. Theoretical Population Biology, 159, 1-12.
DeWitt, William S.; Evans, Steven N.; Hiesmayr, Ella & Hummel, Sebastian
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Λ-Wright–Fisher processes with general selection and opposing environmental effects: Fixation and coexistence. The Annals of Applied Probability, 35(1).
Cordero, Fernando; Hummel, Sebastian & Véchambre, Grégoire
