Evolutionary Dynamics in Finite Populations Mix Rapidly

In this paper we prove that the mixing time of a broad class of evolutionary dynamics in finite, unstructured populations is roughly logarithmic in the size of the state space. An important special case of such a stochastic process is the Wright-Fisher model from evolutionary biology (with selection and mutation) on a population of size N over m genotypes. Our main result implies that the mixing time of this process is O(log N) for all mutation rates and fitness landscapes, and solves the main open problem from [4]. In particular, it significantly extends the main result in [18] who proved this for m = 2. Biologically, such models have been used to study the evolution of viral populations with applications to drug design strategies countering them. Here the time it takes for the population to reach a steady state is important both for the estimation of the steady-state structure of the population as well in the modeling of the treatment strength and duration. Our result, that such populations exhibit rapid mixing, makes both of these approaches sound. Technically, we make a novel connection between Markov chains arising in evolutionary dynamics and dynamical systems on the probability simplex. This allows us to use the local and global stability properties of the fixed points of such dynamical systems to construct a contractive coupling in a fairly general setting. We expect that our mixing time result would be useful beyond the evolutionary biology setting, and the techniques used here would find applications in bounding the mixing times of Markov chains which have a natural underlying dynamical system

Human mobility networks and persistence of rapidly mutating pathogens

Rapidly mutating pathogens may be able to persist in the population and reach an endemic equilibrium by escaping hosts' acquired immunity. For such diseases, multiple biological, environmental and population-level mechanisms determine the dynamics of the outbreak, including pathogen's epidemiological traits (e.g. transmissibility, infectious period and duration of immunity), seasonality, interaction with other circulating strains and hosts' mixing and spatial fragmentation. Here, we study a susceptible-infected-recovered-susceptible model on a metapopulation where individuals are distributed in subpopulations connected via a network of mobility flows. Through extensive numerical simulations, we explore the phase space of pathogen's persistence and map the dynamical regimes of the pathogen following emergence. Our results show that spatial fragmentation and mobility play a key role in the persistence of the disease whose maximum is reached at intermediate mobility values. We describe the occurrence of different phenomena including local extinction and emergence of epidemic waves, and assess the conditions for large scale spreading. Findings are highlighted in reference to previous works and to real scenarios. Our work uncovers the crucial role of hosts' mobility on the ecological dynamics of rapidly mutating pathogens, opening the path for further studies on disease ecology in the presence of a complex and heterogeneous environment.Comment: 29 pages, 7 figures. Submitted for publicatio

Culture and Cancer

Genetic mechanisms, since they broadly involve information transmission, should be translatable into information dynamics formalism. From this perspective we reconsider the adaptive mutator, one possible means of 'second order selection' by which a highly structured 'language' of environment and development writes itself onto the variation upon which evolutionary selection and tumorigenesis operate. Our approach uses recent results in the spirit of the Large Deviations Program of applied probability that permit transfer of phase transition approaches from statistical mechanics to information theory, generating evolutionary and developmental punctuation in what we claim to be a highly natural manner

Counterintuitive properties of the fixation time in network-structured populations

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth-death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations

Extraordinary Sex Ratios: Cultural Effects on Ecological Consequences

We model sex-structured population dynamics to analyze pairwise competition between groups differing both genetically and culturally. A sex-ratio allele is expressed in the heterogametic sex only, so that assumptions of Fisher's analysis do not apply. Sex-ratio evolution drives cultural evolution of a group-associated trait governing mortality in the homogametic sex. The two-sex dynamics under resource limitation induces a strong Allee effect that depends on both sex ratio and cultural trait values. We describe the resulting threshold, separating extinction from positive growth, as a function of female and male densities. When initial conditions avoid extinction due to the Allee effect, different sex ratios cannot coexist; in our model, greater female allocation always invades and excludes a lesser allocation. But the culturally transmitted trait interacts with the sex ratio to determine the ecological consequences of successful invasion. The invading female allocation may permit population persistence at self-regulated equilibrium. For this case, the resident culture may be excluded, or may coexist with the invader culture. That is, a single sex-ratio allele in females and a cultural dimorphism in male mortality can persist; a low-mortality resident trait is maintained by father-to-son cultural transmission. Otherwise, the successfully invading female allocation excludes the resident allele and culture, and then drives the population to extinction via a shortage of males. Finally, we show that the results obtained under homogeneous mixing hold, with caveats, in a spatially explicit model with local mating and diffusive dispersal in both sexes.Comment: final version, reflecting changes in response to referees' comment

Heterogeneous individual motility biases group composition in a model of aggregating cells

Aggregative life cycles are characterized by alternating phases of unicellular growth and multicellular development. Their multiple, independent evolutionary emergence suggests that they may have coopted pervasive properties of single-celled ancestors. Primitive multicellular aggregates, where coordination mechanisms were less efficient than in extant aggregative microbes, must have faced high levels of conflict between different co-aggregating populations. Such conflicts within a multicellular body manifest in the differential reproductive output of cells of different types. Here, we study how heterogeneity in cell motility affects the aggregation process and creates a mismatch between the composition of the population and that of self-organized groups of active adhesive particles. We model cells as self-propelled particles and describe aggregation in a plane starting from a dispersed configuration. Inspired by the life cycle of aggregative model organisms such as Dictyostelium discoideum or Myxococcus xanthus, whose cells interact for a fixed duration before the onset of chimeric multicellular development, we study finite-time configurations for identical particles and in binary mixes. We show that co-aggregation results in three different types of frequency-dependent biases, one of which is associated to evolutionarily stable coexistence of particles with different motility. We propose a heuristic explanation of such observations, based on the competition between delayed aggregation of slower particles and detachment of faster particles. Unexpectedly, despite the complexity and non-linearity of the system, biases can be largely predicted from the behavior of the two corresponding homogenous populations. This model points to differential motility as a possibly important factor in driving the evolutionary emergence of facultatively multicellular life-cycles