Offspring Distribution Research Articles

Site recurrence for coalescing random walk

Begin continuous time random walks from every vertex of a graph and have particles coalesce when they collide. We use a duality relation with the voter model to prove the process is site recurrent on bounded degree graphs, and for Galton-Watson trees whose offspring distribution has exponential tail. We prove bounds on the occupation probability of a site, as well as a general 0-1 law. Similar conclusions hold for a coalescing process on trees where particles do not backtrack.

Conditioned multi-type Galton−Watson trees

We consider multi-type Galton Watson trees, and find the distribution of these trees when conditioning on very general types of recursive events. It turns out that the conditioned tree is again a multi-type Galton Watson tree, possibly with more types and with offspring distributions depending on the type of the father node and on the height of the father node. These distributions are given explicitly. We give some interesting examples for the kind of conditioning we can handle, showing that our methods have a wide range of applications.

A note on the maximal outdegrees of Galton–Watson trees

In this note we consider both the local maximal outdegrees and the global maximal outdegree of Galton–Watson trees. In particular, we show that the tail of any local maximal outdegree and that of the offspring distribution are asymptotically equivalent up to a certain factor. However for the global maximal outdegree, this is only true in the subcritical case.

Survival of branching Brownian motion in a uniform trap field

We study a branching Brownian motion Z evolving in Rd, where a uniform field of Poissonian traps are present. We consider a general offspring distribution for Z and find the asymptotic decay rate of the annealed survival probability, conditioned on non-extinction. The method of proof is to use a skeleton decomposition for the Galton–Watson process underlying Z and to show that the particles of finite line of descent do not contribute to the survival asymptotics. This work is a follow-up to Öz and Çağlar (2013) and solves the problem considered therein completely.

Principled Design and Runtime Analysis of AbstractConvexEvolutionarySearch.

Geometric crossover is a formal class of crossovers that includes many well-known recombination operators across representations. In previous work, it was shown that all evolutionary algorithms with geometric crossover (but no mutation) do the same form of convex search regardless of the underlying representation, the specific selection mechanism, offspring distribution, search space, and problem at hand. Furthermore, it was suggested that the generalised convex search could perform well on generalised forms of concave and approximately concave fitness landscapes regardless of the underlying space and representation. In this article, we deepen this line of enquiry and study the runtime of generalised convex search on concave fitness landscapes. This is a first step toward linking a geometric theory of representations and runtime analysis in the attempt to (1) set the basis for a more general, unified approach for the runtime analysis of evolutionary algorithms across representations, and (2) identify the essential matching features of evolutionary search behaviour and landscape topography that cause polynomial performance. We present a general runtime result that can be systematically instantiated to specific search spaces and representations and present its specifications to three search spaces. As a corollary, we obtain that the convex search algorithm optimises LeadingOnes in [Formula: see text] fitness evaluations, which is faster than all unbiased unary black box algorithms.

Hybrid-Lambda: simulation of multiple merger and Kingman gene genealogies in species networks and species trees

BackgroundThere has been increasing interest in coalescent models which admit multiple mergers of ancestral lineages; and to model hybridization and coalescence simultaneously.Results Hybrid-Lambda is a software package that simulates gene genealogies under multiple merger and Kingman’s coalescent processes within species networks or species trees. Hybrid-Lambda allows different coalescent processes to be specified for different populations, and allows for time to be converted between generations and coalescent units, by specifying a population size for each population. In addition, Hybrid-Lambda can generate simulated datasets, assuming the infinitely many sites mutation model, and compute the F ST statistic. As an illustration, we apply Hybrid-Lambda to infer the time of subdivision of certain marine invertebrates under different coalescent processes.Conclusions Hybrid-Lambda makes it possible to investigate biogeographic concordance among high fecundity species exhibiting skewed offspring distribution.

Dealing with stochastic environmental variation in space and time: bet hedging by generalist, specialist, and diversified strategies

Building on previous work, we derive an optimization model for a two-state stochastic environment and evaluate the fitnesses of five reproductive strategies across generations. To do this, we characterize spatiotemporal variation and define grain (=patch) size as the scale of fitness autocorrelation. Fitness functions of environmental condition are Gaussian. The strategies include two specialists on each of the environmental conditions; two generalists that each fare equally well under both conditions, but one (a conservative bet hedger) optimizes the shape of the fitness function; and a diversified bet hedger producing an optimal mix of the two specialists within individual broods. When the environment is primarily in one of the two states, the specialist on that state achieves the highest fitness. In the more interesting situation where the two environments are equally prevalent in the long term, with low-moderate environmental variation, a generalist strategy (that copes with both states well) does best. Higher variation favors diversified bet hedgers, or surprisingly, specialists, depending mainly on whether spatial or temporal variation predominates. These strategies reduce variance in fitness and optimize the distribution of offspring among patches differently: specialists by spreading offspring among many independently varying patches, while diversified bet hedgers put all offspring into a few patches or a single patch. We distinguish features consistent with strategies like diversified bet hedgers that spread risk in time from features linked to strategies like specialists that spread risk in space. Finally, we present testable hypotheses arising from this study and suggest directions for future work.

Attractors in Sequence Space: Peptide Morphing by Directed Simulated Evolution.

Certain antimicrobial peptides (AMPs) and mitochondrial targeting peptides (mTPs) share common features such as a positive net charge, helical propensity, and the ability to interact with lipid membranes[1]. While a primary function of AMPs is to disrupt membrane integrity, many mTPs interact with the mitochondrial membranes and the translocator complex, leaving the membranes intact when proteolytically degraded by matrix-located protease(s)[2]. We present a computational approach that may help rationalize the delicate balance of these effects and other sequence-activity relationships. We implemented and applied a directed evolutionary strategy for stepwise peptide morphing of one peptide into another (MoPED, Morphing of Peptides by Evolutionary Design). For the prospective application, we chose cationic peptides as representatives of both peptide classes and converted an AMP (Protonectin, start sequence)[3] into an mTP (target or “attractor” sequence). Novel peptides were generated based on a chemical similarity index (Grantham substitution matrix).[4] The target sequence served as an attractor point for evolutionary exploration of sequence space. We synthesized and tested the individual peptides that were generated during the morphing process. The designed peptides showed systematic loss of membranolytic potential with increasing distance from the start sequence. The results of biophysical and bacterial growth experiments confirmed the applicability of MoPED to designing chemically motivated peptide derivatives. Receiver operating characteristic (ROC) analysis advocates the inducible peptide α-helicity as a semi-quantitative indicator of membranolytic antimicrobial activity.

The vertex-cut-tree of Galton–Watson trees converging to a stable tree

We consider a fragmentation of discrete trees where the internal vertices are deleted independently at a rate proportional to their degree. Informally, the associated cut-tree represents the genealogy of the nested connected components created by this process. We essentially work in the setting of Galton-Watson trees with offspring distribution belonging to the domain of attraction of a stable law of index $\alpha\in(1,2)$. Our main result is that, for a sequence of such trees $\mathcal{T}_n$ conditioned to have size $n$, the corresponding rescaled cut-trees converge in distribution to the stable tree of index $\alpha$, in the sense induced by the Gromov-Prokhorov topology. This gives an analogue of a result obtained by Bertoin and Miermont in the case of Galton-Watson trees with finite variance.

The effect of inbreeding constraints and offspring distribution on time to the most recent common ancestor

The expected time to the most recent common ancestor (MRCA) of two alleles in a diploid individual is 4N+2 under random mating with a Poisson progeny distribution, but 8N−2 under maximum avoidance of inbreeding, which entails two progeny per mating pair. (N is the number of mating pairs, hence 2N is the number of individuals, hence 4N is the number of alleles.) The interrelationship of inbreeding constraints and offspring distribution is investigated by varying the level of sib mating: prohibiting sib mating increases the time to MRCA by four generations and decreases the variance of the offspring distribution by 2/N. With two progeny per mating pair, the expected time to the MRCA is 8N−2 under both random mating and sib mating prohibited, as well as under maximum avoidance of inbreeding, but this result does not hold for all mating structures with two progeny per mating pair.

A note on the speed of a random walk on Galton-Watson trees

Consider a λ-biased random walk on Galton-Watson trees. It is proved that the speed exists and is bounded above by ( m -λ)=( m +λ), where m is the mean of offsprings. One may further explore the relation between the speed and the offspring distribution. All examples show that the speed is a monotone function of the variance. We confirm this belief by verifying that the recurrent probability, a quantity related to the speed, is a monotone function of the variance of the offspring distribution in some sense, for the fixed m . Some observations are made and some questions are raised.

Applications of the EM Algorithm in Multitype Branching Processes with Power Series Offspring Distributions

We consider the class of multitype branching stochastic processes with power series offspring distributions. In real situations it is difficult to observe the entire family tree up to the current generation. Often one can only register the progenitors and the total number of individuals in each generation. Hence one faces the need of statistical procedures dealing with unobservable data. In our work we use the EM algorithm to obtain the maximum likelihood estimators (MLE) of the offspring distribution parameters for the incomplete-data problem. Firstly, assuming that the entire family tree is observed, we obtain the maximum likelihood estimators of the parameters. Thereafter in the case of data based on generation sizes we propose an R implementation of the EM algorithm. The methods are illustrated via simulations and computational results.

Predicting extinction or explosion in a Galton–Watson branching process with power series offspring distribution

Extinction is certain in a Galton–Watson (GW) branching process if the offspring mean μ≤1, whereas explosion is possible but not certain if μ>1. Discriminating between these two possibilities is a well-studied hypothesis-testing problem. However, deciding whether extinction or explosion will occur for the current realization of the process is a prediction problem. This can be formulated as a different testing problem by considering the conditional distributions of the process given extinction and explosion respectively. For power series offspring distributions, fixed-sample and sequential parametric tests are presented for the prediction problem and illustrated with data on the spread of epidemics and the populations of endangered species.

Estimates of the Offspring Probabilities in a Class of Multitype Branching Processes with Binary Family Trees with Partial Observation

Multitype branching processes (MTBP) traditionally serve as models in biology in studies of cell proliferation. A sampling scheme that appears frequently is observing the cell count in several independent colonies at discrete time points, sometimes only one. The process is not observable in the sense of the whole tree, but only as the generation at given moment in time, which consists of the number of cells of every type. The EM algorithms are widely used to obtain a maximum likelihood (ML) estimation with incomplete observation. An EM estimation of the offspring distribution is presented in the class of Markov branching processes with binary family trees. In this special case it leads to the same estimates, as using the whole tree observation. This result implies that there is no need to observe the whole process in laboratory experiments to obtain the estimates in such models. Acknowledgments: The research was partially supported by the National Fund for Scientific Research at the Ministry of Education and Science of Bulgaria, grant № DFNI-I02/17.

Estimation in second order branching processes with application to swine flu data

This paper discusses inferential issues related to estimation of offspring mean and variance in a second order branching process, when both the offspring distributions are assumed to have identical mean and variance. Estimating equation approach is used to find the estimator of the offspring mean and the fact that a second order branching process model can be modeled as an autoregressive process is utilized to obtain the estimator of the offspring variance. Both the estimators are shown to be consistent and asymptotically normal. The second order branching process model is applied to H1N1 data for Pune, India, and Mexico and is found to be a suitable model. The estimates obtained from this model are used to compute the proportion of vaccination required for elimination of the disease.

Stochastic Ordering of Infinite Geometric Galton–Watson Trees

We consider Galton–Watson trees with Geom $$(p)$$ offspring distribution. We let $$T_{\infty }(p)$$ denote such a tree conditioned on being infinite. We prove that for any $$1/2\le p_1 <p_2 \le 1$$ , there exists a coupling between $$T_{\infty }(p_1)$$ and $$T_{\infty }(p_2)$$ such that $${\mathbb {P}}(T_{\infty }(p_1) \subseteq T_{\infty }(p_2))=1$$ .

Nucleotide variation and balancing selection at the Ckma gene in Atlantic cod: analysis with multiple merger coalescent models.

High-fecundity organisms, such as Atlantic cod, can withstand substantial natural selection and the entailing genetic load of replacing alleles at a number of loci due to their excess reproductive capacity. High-fecundity organisms may reproduce by sweepstakes leading to highly skewed heavy-tailed offspring distribution. Under such reproduction the Kingman coalescent of binary mergers breaks down and models of multiple merger coalescent are more appropriate. Here we study nucleotide variation at the Ckma (Creatine Kinase Muscle type A) gene in Atlantic cod. The gene shows extreme differentiation between the North (Canada, Greenland, Iceland, Norway, Barents Sea) and the South (Faroe Islands, North-, Baltic-, Celtic-, and Irish Seas) with FST > 0.8 between regions whereas neutral loci show no differentiation. This is evidence of natural selection. The protein sequence is conserved by purifying selection whereas silent and non-coding sites show extreme differentiation. The unfolded site-frequency spectrum has three modes, a mode at singleton sites and two high frequency modes at opposite frequencies representing divergent branches of the gene genealogy that is evidence for balancing selection. Analysis with multiple-merger coalescent models can account for the high frequency of singleton sites and indicate reproductive sweepstakes. Coalescent time scales vary with population size and with the inverse of variance in offspring number. Parameter estimates using multiple-merger coalescent models show that times scales are faster than under the Kingman coalescent.

Estimation of the variance effective population size in age structured populations

The variance effective population size for age structured populations is generally hard to estimate and the temporal method often gives biased estimates. Here, we give an explicit expression for a correction factor which, combined with estimates from the temporal method, yield approximately unbiased estimates. The calculation of the correction factor requires knowledge of the age specific offspring distribution and survival probabilities as well as possible correlation between survival and reproductive success. In order to relax these requirements, we show that only first order moments of these distributions need to be known if the time between samples is large, or individuals from all age classes which reproduce are sampled. A very explicit approximate expression for the asymptotic coefficient of standard deviation of the estimator is derived, and it can be used to construct confidence intervals and optimal ways of weighting information from different markers. The asymptotic coefficient of standard deviation can also be used to design studies and we show that in order to maximize the precision for a given sample size, individuals from older age classes should be sampled since their expected variance of allele frequency change is higher and easier to estimate. However, for populations with fluctuating age class sizes, the accuracy of the method is reduced when samples are taken from older age classes with high demographic variation. We also present a method for simultaneous estimation of the variance effective and census population size.

An inhomogeneous controlled branching process∗

We consider a discrete-time branching process in which the offspring distribution is generation-dependent and the number of reproductive individuals is controlled by a random mechanism. This model is a Markov chain, but, in general, the transition probabilities are nonstationary. Under not too restrictive hypotheses, this model presents the classical duality of branching processes: it either becomes extinct or grows to infinity. Sufficient conditions for the almost sure extinction and for a positive probability of indefinite growth are given. Finally, the rates of growth of the process are studied, provided that there is no extinction.

On the scaling limits of Galton-Watson processes in varying environments

We establish a general sufficient condition for a sequence of Galton–Watson branching processes in varying environments to converge weakly. This condition extends previ- ous results by allowing offspring distributions to have infinite variance. Our assumptions are stated in terms of pointwise convergence of a triplet of two real- valued functions and a measure. The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes. Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes.