Talk in Population Genetics

Abstract:
A system of Poissonian interacting trajectories (PIT) was recently introduced in our joint paper with Felix Hermann, Adrián González Casanova, Renato Soares dos Santos, and Anton Wakolbinger. As I will mention in the beginning of my talk, such a system of [0,1]-valued piecewise linear trajectories arises as a scaling limit of the system of logarithmic subpopulation sizes in a Moran model with mutation and selection. Here, selection is strong and the rate of beneficial mutations is in the so-called Gerrish-Lenski regime, where the inter-arrival times between consecutive mutations are of the same order as the durations of a mutant invasion (selective sweep). Changes of the resident population yield kinks (slope changes) in the resident population. We showed that the PIT exhibits an almost surely positive asymptotic rate of fitness increase (speed of adaptation), which turns out to be finite if and only if fitness increments have a finite expectation.

In a joint follow-up paper with Katalin Friedl and Viktória Nemkin, we study algorithmic aspects of interacting trajectories, including the case when the mutation times are deterministic. We show that although the interaction of n trajectories may cause Ω(n²) slope changes in total, the resident fitness function can be computed in O(n) time using a deterministic algorithm. This algorithm uses the so-called continued lines representation of the interacting trajectories, which was introduced in our previous paper.

Moreover, we show that the expected number of slope changes of the PIT up to time t is O(t), and thus the aforementioned quadratic number of kinks in the worst case is exceptional in the Poissonian case. The proof of this result is strongly related to the renewal argument that we used in the previous paper to characterize the speed of adaptation. A modification of this renewal argument implies that the time-average of the number of kinks of the PIT converges almost surely to a deterministic limit. This limit turns out to be positive and finite for any fitness increment distribution.

If time permits, I will also mention some interesting (and apparently difficult) open problems related to the speed of adaptation and related topics.