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Selection, the tendency of some traits to become more frequent than others in a population under the influence of some (natural or artificial) agency, is a key component of Darwinian evolution and countless other natural and social phenomena. Yet a general theory of selection, analogous to the Fisher-Tippett-Gnedenko theory of extreme events, is lacking. Here we introduce a probabilistic definition of selection and show that selected values are attracted to a universal family of limiting distributions. The universality classes and scaling exponents are determined by the tail thickness of the random variable under selection. Our results are supported by data from molecular biology, agriculture and sport.
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We study metapopulation models for the spread of epidemics in which different subpopulations (cities) are connected by fluxes of individuals (travelers). This framework allows to describe the spread of a disease on a large scale and we focus here on the computation of the arrival time of a disease as a function of the properties of the seed of the epidemics and of the characteristics of the network connecting the various subpopulations. Using analytical and numerical arguments, we introduce an easily computable quantity which approximates this average arrival time. We show on the example of a disease spread on the world-wide airport network that this quantity predicts with a good accuracy the order of arrival of the disease in the various subpopulations in each realization of epidemic scenario, and not only for an average over realizations. Finally, this quantity might be useful in the identification of the dominant paths of the disease spread.
Cancer cell population dynamics often exhibit remarkably replicable, universal laws despite their underlying heterogeneity. Mechanistic explanations of universal cell population growth remain partly unresolved to this day, whereby population feedback between the microscopic and mesoscopic configurations can lead to macroscopic growth laws. We here present a unification under density-dependent birth events via contact inhibition. We consider five classical tumor growth laws: exponential, generalized logistic, Gompertz, radial growth, and fractal growth, which can be seen as manifestations of a single microscopic model. Our theory is substantiated by agent based simulations and can explain growth curve differences in experimental data from in vitro cancer cell population growth. Thus, our framework offers a possible explanation for the large number of mean-field laws that can adequately capture seemingly unrelated cancer or microbial growth dynamics.
In exponentially proliferating populations of microbes, the population typically doubles at a rate less than the average doubling time of a single-cell due to variability at the single-cell level. It is known that the distribution of generation times obtained from a single lineage is, in general, insufficient to determine a populations growth rate. Is there an explicit relationship between observables obtained from a single lineage and the population growth rate? We show that a populations growth rate can be represented in terms of averages over isolated lineages. This lineage representation is related to a large deviation principle that is a generic feature of exponentially proliferating populations. Due to the large deviation structure of growing populations, the number of lineages needed to obtain an accurate estimate of the growth rate depends exponentially on the duration of the lineages, leading to a non-monotonic convergence of the estimate, which we verify in both synthetic and experimental data sets.
Cells forming various epithelial tissues have a strikingly universal distribution for the number of their edges. It is generally assumed that this topological feature is predefined by the statistics of individual cell divisions in growing tissue but existing theoretical models are unable to predict the observed distribution. Here we show experimentally, as well as in simulations, that the probability of cellular division increases exponentially with the number of edges of the dividing cell and show analytically that this is responsible for the observed shape of cell-edge distribution.
Statistics of Poincare recurrences is studied for the base-pair breathing dynamics of an all-atom DNA molecule in realistic aqueous environment with thousands of degrees of freedom. It is found that at least over five decades in time the decay of recurrences is described by an algebraic law with the Poincare exponent close to $beta=1.2$. This value is directly related to the correlation decay exponent $ u = beta -1$, which is close to $ uapprox 0.15$ observed in the time resolved Stokes shift experiments. By applying the virial theorem we analyse the chaotic dynamics in polynomial potentials and demonstrate analytically that exponent $beta=1.2$ is obtained assuming the dominance of dipole-dipole interactions in the relevant DNA dynamics. Molecular dynamics simulations also reveal the presence of strong low frequency noise with the exponent $eta=1.6$. We trace parallels with the chaotic dynamics of symplectic maps with a few degrees of freedom characterized by the Poincare exponent $beta sim 1.5$.
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