Quantifying time-inhomogeneous stochastic introgression processes with hazard rates

Introgression is the permanent incorporation of genes from one population into another through hybridization and backcrossing. It is currently of particular concern as a possible mechanism for the spread of modi ed crop genes to wild populations. The hazard rate is the probability per time unit that such an escape takes place, given that it has not happened before. It is a quantitative measure of introgression risk that takes the stochastic elements inherent in introgression processes into account. We present a methodology to calculate the hazard rate for situations with time-varying gene ow from a crop to a large recipient wild population. As an illustration, several types of time-inhomogeneity are examined, including deterministic periodicity as well as random variation. Furthermore, we examine the e ects of an extended tness bottleneck of hybrids and backcrosses in combination with time-varying gene ow. It is found that bottlenecks decrease the hazard rate, but also slow down and delay its changes in reaction to changes in gene ow. Furthermore, we nd that random variation in gene ow generates a lower hazard rate than analogous deterministic variation. We discuss the implications of our ndings for crop management and introgression risk assessment.This research was funded through the research program 'Ecology Regarding Genetically modified Organisms (ERGO)', commissioned by four Dutch ministries. This funding program is managed by the Earth and Life Sciences Council (ALW) of the Netherlands Organisation for Scientific Research (NWO). P. Haccou's research is additionally supported by the NDNS (Nonlinear Dynamics of Natural Systems) program of NWO. M.C. Serra would like to thank the Fundacao para a Ciencia e Tecnologia for financial support through the scholarship SFRH/BPD/47615/2008. We thank Marije Stoops and Prof. Baorong Lu for discussions and comments on a previous version

The likelihood ratio test for the change point problem for exponentially distributed random variables

AbstractLet x1,…, xn+1 be independent exponentially distributed random variables with intensity λ1 for i ⩽ τ and λ2 for i > τ, where τ as well as λ1 and λ2 are unknown. By application of theorems concerning the normed uniform quantile process it is proved that the asymptotic null-distribution of the likelihood ratio statistic for testing λ1 = λ2 (or, equivalently, τ = 0 or n + 1) is an extreme value distribution.Change point problems occur in a variety of experimental sciences and therefore have considerabla attention of applied statisticians. The problems are non-standard since the usual regularity conditions are not satisfied. Explicit asymptotic distributions of likelihood ratio tests have until now only been derived for a few cases. The method of proof used in this paper is based on the ‘strong invariance principle’.Furthermore it is shown that the test is optimal in the sense of Bahadur, although the Pitman efficiency is zero. However, simulation results indicate a good power for values of n that are relevant for most applications.The likelihood ratio test is compared with another test which has the same asymptotic null-distribution. This test has Bahadur efficiency zero. The simulation results confirm that the likelihood ratio test is superior to the latter test

Prospects & Overviews Bet hedging or not? A guide to proper classification of microbial survival strategies

Bacteria have developed an impressive ability to survive and propagate in highly diverse and changing environments by evolving phenotypic heterogeneity. Phenotypic heterogeneity ensures that a subpopulation is well prepared for environmental changes. The expression bet hedging is commonly (but often incorrectly) used by molecular biologists to describe any observed phenotypic heterogeneity. In evolutionary biology, however, bet hedging denotes a risk-spreading strategy displayed by isogenic populations that evolved in unpredictably changing environments. Opposed to other survival strategies, bet hedging evolves because the selection environment changes and favours different phenotypes at different times. Consequently, in bet hedging populations all phenotypes perform differently well at any time, depending on the selection pressures present. Moreover, bet hedging is the only strategy in which temporal variance of offspring numbers per individual is minimized. Our paper aims to provide a guide for the correct use of the term bet hedging in molecular biology

Stochasticity in the adaptive dynamics of evolution: The bare bones

First a population model with one single type of individuals is considered. Individuals reproduce asexually by splitting into two, with a population-size-dependent probability. Population extinction, growth and persistence are studied. Subsequently the results are extended to such a population with two competing morphs and are applied to a simple model, where morphs arise through mutation. The movement in the trait space of a monomorphic population and its possible branching into polymorphism are discussed. This is a first report. It purports to display the basic conceptual structure of a simple exact probabilistic formulation of adaptive dynamics

Dynamics of escape mutants

We use multi-type Galton-Watson branching processes to model the evolution of populations that, due to a small reproductive ratio of the individuals, are doomed to extinction. Yet, mutations occurring during the reproduction process, may lead to the appearance of new types of individuals that are able to escape extinction. We provide examples of such populations in medical, biological and environmental contexts and give results on (i) the probability of escape/extinction, (ii) the distribution of the waiting time to produce the first individual whose lineage does not get extinct and (iii) the distribution of the time it takes for the number of mutants to reach a high level. Special attention is dedicated to the case where the probability of mutation is very small and approximations for (i)-(iii) are derived

High-Resolution Gene Flow Model for Assessing Environmental Impacts of Transgene Escape Based on Biological Parameters and Wind Speed.

Environmental impacts caused by transgene flow from genetically engineered (GE) crops to their wild relatives mediated by pollination are longstanding biosafety concerns worldwide. Mathematical modeling provides a useful tool for estimating frequencies of pollen-mediated gene flow (PMGF) that are critical for assessing such environmental impacts. However, most PMGF models are impractical for this purpose because their parameterization requires actual data from field experiments. In addition, most of these models are usually too general and ignored the important biological characteristics of concerned plant species; and therefore cannot provide accurate prediction for PMGF frequencies. It is necessary to develop more accurate PMGF models based on biological and climatic parameters that can be easily measured in situ. Here, we present a quasi-mechanistic PMGF model that only requires the input of biological and wind speed parameters without actual data from field experiments. Validation of the quasi-mechanistic model based on five sets of published data from field experiments showed significant correlations between the model-simulated and field experimental-generated PMGF frequencies. These results suggest accurate prediction for PMGF frequencies using this model, provided that the necessary biological parameters and wind speed data are available. This model can largely facilitate the assessment and management of environmental impacts caused by transgene flow, such as determining transgene flow frequencies at a particular spatial distance, and establishing spatial isolation between a GE crop and its coexisting non-GE counterparts and wild relatives

Quantifying stochastic introgression processes in random environments with hazard rates

Introgression is the permanent incorporation of genes from the genome of one population into another. Previous studies have found that stochasticity in number of offspring, hybridisation, and environment are important aspects of introgression risk, but these factors have been studied separately. In this paper we extend the use of the hazard rate which we previously used to studied effects of demographic stochasticity with repeated invasion attempts, to incorporate temporal environmental stochasticity. We find that introgression risk varies much in time, and in some periods it can be much enhanced in such environments. Furthermore, effects of plant life history parameters, such as flowering and survival probabilities, on hazard rates depend on characteristics of the environmental variation.This research was funded through the research program 'Ecology Regarding Genetically modified Organisms (ERGO)', commissioned by four Dutch ministries. This funding program was managed by the Earth and Life Sciences Council (ALW) of the Netherlands Organisation for Scientific Research (NWO) grant number 838.06.031. P. Haccou's research was additionally supported by the NDNS (Nonlinear Dynamics of Natural Systems) program of NWO. M.C. Serra's research was supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the Fundacao para a Ciencia e a Tecnologia, through the Project PEstOE/MAT/UI0013/2014

The mixing advantage is less than 2

Corresponding to n independent non-negative random variables X_1,...,X_n , are values M_1,...,M_n , where each M_i is the expected value of the maximum of n independent copies of X_i. We obtain an upper bound for the expected value of the maximum of X_1,...,X_n in terms of M_1,...,M_n . This inequality is sharp in the sense that the random variables can be chosen so that the bound is approached arbitrarily closely. We also present related comparison results

Bet hedging or not? A guide to proper classification of microbial survival strategies

Bacteria have developed an impressive ability to survive and propagate in highly diverse and changing environments by evolving phenotypic heterogeneity. Phenotypic heterogeneity ensures that a subpopulation is well prepared for environmental changes. The expression bet hedging is commonly (but often incorrectly) used by molecular biologists to describe any observed phenotypic heterogeneity. In evolutionary biology, however, bet hedging denotes a risk-spreading strategy displayed by isogenic populations that evolved in unpredictably changing environments. Opposed to other survival strategies, bet hedging evolves because the selection environment changes and favours different phenotypes at different times. Consequently, in bet hedging populations all phenotypes perform differently well at any time, depending on the selection pressures present. Moreover, bet hedging is the only strategy in which temporal variance of offspring numbers per individual is minimized. Our paper aims to provide a guide for the correct use of the term bet hedging in molecular biology.