Evolution in the weak-mutation limit: Stasis periods punctuated by fast transitions between saddle points on the fitness landscape.

A mathematical analysis of the evolution of a large population under the weak-mutation limit shows that such a population would spend most of the time in stasis in the vicinity of saddle points on the fitness landscape. The periods of stasis are punctuated by fast transitions, in lnNe/s time (Ne , effective population size; s, selection coefficient of a mutation), when a new beneficial mutation is fixed in the evolving population, which accordingly moves to a different saddle, or on much rarer occasions from a saddle to a local peak. Phenomenologically, this mode of evolution of a large population resembles punctuated equilibrium (PE) whereby phenotypic changes occur in rapid bursts that are separated by much longer intervals of stasis during which mutations accumulate but the phenotype does not change substantially. Theoretically, PE has been linked to self-organized criticality (SOC), a model in which the size of "avalanches" in an evolving system is power-law-distributed, resulting in increasing rarity of major events. Here we show, however, that a PE-like evolutionary regime is the default for a very simple model of an evolving population that does not rely on SOC or any other special conditions.

Universal Statistics of Incubation Periods and Other Detection Times via Diffusion Models.

We suggest an explanation of typical incubation times statistical features based on the universal behavior of exit times for diffusion models. We give a mathematically rigorous proof of the characteristic right skewness of the incubation time distribution for very general one-dimensional diffusion models. Imposing natural simple conditions on the drift coefficient, we also study these diffusion models under the assumption of noise smallness and show that the limiting exit time distributions in the limit of vanishing noise are Gaussian and Gumbel. Thus, they match the existing data as well as the other existing models do. The character of our models, however, allows us to argue that the features of the exit time distributions that we describe are universal and manifest themselves in various other situations where the times involved can be described as detection or halting times, for example response times studied in psychology.

Large deviations for random trees and the branching of RNA secondary structures.

We give a Large Deviation Principle (LDP) with explicit rate function for the distribution of vertex degrees in plane trees, a combinatorial model of RNA secondary structures. We calculate the typical degree distributions based on nearest neighbor free energies, and compare our results with the branching configurations found in two sets of large RNA secondary structures. We find substantial agreement overall, with some interesting deviations which merit further study.

Large Deviations for Random Trees.

We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.