Demographic influences on mitochondrial DNA lineage survivorship in animal populations

Summary Probability models of branching processes and computer simulations of these models are used to examine stochastic survivorship of female lineages under a variety of demographic scenarios. A parameter II, defined as the probability of survival of two or more independent lineages over G generations, is monitored as a function of founding size of a population, population size at carrying capacity, and the frequency distributions of surviving progeny.Stochastic lineage extinction can be very rapid under certain biologically plausible demographic conditions. For stable-sized populations initiated by n females and/or regulated about carrying capacity k=n, it is highly probable that within about 4n generations all descendants will trace their ancestries to a single founder female. For a given mean family size, increased variance decreases lineage survivorship. In expanding populations, however, lineage extinction is dramatically slowed, and the final k value is a far more important determinant of II than is the size of the population at founding. The results are discussed in the context of recent empirical observations of low mitochondrial DNA (mtDNA) sequence heterogeneity in humans and expected distributions of asexually transmitted traits among sexually reproducing species.     

Current versus historical population sizes in vertebrate species with high gene flow: a comparison based on mitochondrial DNA lineages and inbreeding theory for neutral mutations

Using inbreeding theory as applied to neutral alleles inherited maternally, we generate expected probability distributions of times to identity by descent for random pairs of mitochondrial genotypes within a population or within an entire species characterized by high gene flow. For comparisons with these expectations, empirical distributions of times to most recent common ancestry were calculated (by conventional mtDNA clock calibrations) from mtDNA haplotype distances observed within each of three vertebrate species--American eels, hardhead catfish, and redwinged blackbirds. These species were chosen for analysis because census population size in each is currently large and because both genetic and life-history data are consistent with the postulate that historical gene flow within these species has been high. The observed molecular distances among mtDNA lineages were two to three orders of magnitude lower than predicted from census sizes of breeding females, suggesting that rate of mtDNA evolution is decelerated in these species and/or that long-term effective population size is vastly smaller than present-day population size. Several considerations point to the latter possibility as most likely. The genetic structure of any species is greatly influenced by historical demography; even for species that are currently abundant, mtDNA gene lineages appear to have been channeled through fairly small numbers of ancestors.      

Probability of successful larval dispersal declines fivefold over 1 km in a coral reef fish

A central question of marine ecology is, how far do larvae disperse? Coupled biophysical models predict that the probability of successful dispersal declines as a function of distance between populations. Estimates of genetic isolation-by-distance and self-recruitment provide indirect support for this prediction. Here, we conduct the first direct test of this prediction, using data from the well-studied system of clown anemonefish (Amphiprion percula) at Kimbe Island, in Papua New Guinea. Amphiprion percula live in small breeding groups that inhabit sea anemones. These groups can be thought of as populations within a metapopulation. We use the x- and y-coordinates of each anemone to determine the expected distribution of dispersal distances (the distribution of distances between each and every population in the metapopulation). We use parentage analyses to trace recruits back to parents and determine the observed distribution of dispersal distances. Then, we employ a logistic model to (i) compare the observed and expected dispersal distance distributions and (ii) determine the relationship between the probability of successful dispersal and the distance between populations. The observed and expected dispersal distance distributions are significantly different (p < 0.0001). Remarkably, the probability of successful dispersal between populations decreases fivefold over 1 km. This study provides a framework for quantitative investigations of larval dispersal that can be applied to other species. Further, the approach facilitates testing biological and physical hypotheses for the factors influencing larval dispersal in unison, which will advance our understanding of marine population connectivity.     

Maximizing phylogenetic diversity in biodiversity conservation: Greedy solutions to the Noah's Ark problem

The Noah's Ark Problem (NAP) is a comprehensive cost-effectiveness methodology for biodiversity conservation that was introduced by Weitzman (1998) and utilizes the phylogenetic tree containing the taxa of interest to assess biodiversity. Given a set of taxa, each of which has a particular survival probability that can be increased at some cost, the NAP seeks to allocate limited funds to conserving these taxa so that the future expected biodiversity is maximized. Finding optimal solutions using this framework is a computationally difficult problem to which a simple and efficient "greedy" algorithm has been proposed in the literature and applied to conservation problems. We show that, although algorithms of this type cannot produce optimal solutions for the general NAP, there are two restricted scenarios of the NAP for which a greedy algorithm is guaranteed to produce optimal solutions. The first scenario requires the taxa to have equal conservation cost; the second scenario requires an ultrametric tree. The NAP assumes a linear relationship between the funding allocated to conservation of a taxon and the increased survival probability of that taxon. This relationship is briefly investigated and one variation is suggested that can also be solved using a greedy algorithm.     

A mobility-based cluster formation algorithm for wireless mobile ad-hoc networks

In the last decade, numerous efforts have been devoted to design efficient algorithms for clustering the wireless mobile ad-hoc networks (MANET) considering the network mobility characteristics. However, in existing algorithms, it is assumed that the mobility parameters of the networks are fixed, while they are stochastic and vary with time indeed. Therefore, the proposed clustering algorithms do not scale well in realistic MANETs, where the mobility parameters of the hosts freely and randomly change at any time. Finding the optimal solution to the cluster formation problem is incredibly difficult, if we assume that the movement direction and mobility speed of the hosts are random variables. This becomes harder when the probability distribution function of these random variables is assumed to be unknown. In this paper, we propose a learning automata-based weighted cluster formation algorithm called MCFA in which the mobility parameters of the hosts are assumed to be random variables with unknown distributions. In the proposed clustering algorithm, the expected relative mobility of each host with respect to all its neighbors is estimated by sampling its mobility parameters in various epochs. MCFA is a fully distributed algorithm in which each mobile independently chooses the neighboring host with the minimum expected relative mobility as its cluster-head. This is done based solely on the local information each host receives from its neighbors and the hosts need not to be synchronized. The experimental results show the superiority of MCFA over the best existing mobility-based clustering algorithms in terms of the number of clusters, cluster lifetime, reaffiliation rate, and control message overhead.     

SIR dynamics in random networks with heterogeneous connectivity

Random networks with specified degree distributions have been proposed as realistic models of population structure, yet the problem of dynamically modeling SIR-type epidemics in random networks remains complex. I resolve this dilemma by showing how the SIR dynamics can be modeled with a system of three nonlinear ODE’s. The method makes use of the probability generating function (PGF) formalism for representing the degree distribution of a random network and makes use of network-centric quantities such as the number of edges in a well-defined category rather than node-centric quantities such as the number of infecteds or susceptibles. The PGF provides a simple means of translating between network and node-centric variables and determining the epidemic incidence at any time. The theory also provides a simple means of tracking the evolution of the degree distribution among susceptibles or infecteds. The equations are used to demonstrate the dramatic effects that the degree distribution plays on the final size of an epidemic as well as the speed with which it spreads through the population. Power law degree distributions are observed to generate an almost immediate expansion phase yet have a smaller final size compared to homogeneous degree distributions such as the Poisson. The equations are compared to stochastic simulations, which show good agreement with the theory. Finally, the dynamic equations provide an alternative way of determining the epidemic threshold where large-scale epidemics are expected to occur, and below which epidemic behavior is limited to finite-sized outbreaks.      

The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.     

Logistic growth in the presence of non-white environmental noise

A method is given for studying realistic random fluctuations in the carrying capacity of the logistic population growth model. This method is then applied using an environmental noise based on a Poisson process, and the time-dependent moments of the population probability density calculated. These moments are expressed in terms of a parameter obtained by dividing the correlation time of the environmental fluctuations by the characteristic response time of the population. When this quotient is large (very slow fluctuations tracked by the population) or small (very rapid fluctuations which are averaged), exact solutions are obtained for the probability density itself. It is also shown that at equilibrium, the average population sizes given by these two exact solutions bound all other cases.Numerical simulations confirm these developments and point to a trade-off between population stability and average population size. Additional simulations show that the probability of becoming extinct in a given time is greatest for populations intermediate between tracking and averaging the carrying capacity fluctuations. In addition to specifying when environmental noise can be ignored, these results indicate the direction in which growth parameters evolve in a fluctuating environment.     

Measures of success in a class of evolutionary models with fixed population size and structure

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.     

Estimation of Mitotic Stability in Conidial Fungi: A Theoretical Framework

Mitotic stability refers to the probability that genetic elements are transmitted to both daughters during mitosis. This is of practical importance in molecular genetics because autonomous cloning vectors should be transmitted at high frequency during mitosis. In filamentous coencytic fungi it is difficult to quantify mitotic stability because a fluctuation test is not feasible. We show how to get around this problem by formulating a general model of the transmission of nuclear genetic elements through the course of conidiogenesis. We derive formulas by two different methods for the expected proportion of conidiospores that retain the element as a function of its mitotic stability and the number of generations of spore production. An important by-product yields the exact probability distributions for the number of conidiospores retaining elements at each stage of conidiophore development. We outline, and illustrate through specific numerical examples, how to use these formulas to estimate mitotic stability. Although we use Aspergillus nidulans as our biological paradigm, the same general framework can be extended to other fungal species, and possibly to less closely related systems as well.     

The size distribution of conspecific populations: the peoples of New Guinea

The size distribution of the language populations in New Guinea, which represent over 15% of the world's languages, is analysed using models analogous to the resource division models of species abundance distribution in ecological communities. A model distribution of resource segments reflecting population size is created by repeated selection of an existing resource segment and its division into two. We found that any dependency of the selection probability on the size of the segment generated negatively skewed abundance distributions after log transformation. Asymmetric segment division further exacerbated the negative skewness. Size-independent selection produced lognormal abundance distributions, irrespective of the segment division method. Size-dependent selection and asymmetric division were deemed reasonable assumptions since large language populations are more likely to generate isolates, which develop into new populations, than small ones, and these isolates are likely to be small relative to the progenitor population. A negatively skewed distribution of the log-transformed population sizes was therefore expected. However, the observed distributions were lognormal, scale invariant for areas containing between 100 and over 1000 language populations. The dynamics of language differentiation, as reflected by the models, may therefore be unimportant relative to the effect of variable growth rates among populations. All lognormal distributions from resource division models had a higher variance than the observed one, where half of the 1053 populations had between 350 and 3000 individuals. The possible mechanisms maintaining such a low variance around a modal population size of 1000 are discussed.     

A multinomial model for estimating the size of a whale population from incomplete census data

This paper adapts the removal method of population size estimation to the problem of estimating the size of the western Arctic stock of bowhead whales. The whales are counted during their spring migration as they pass two census camps located near Point Barrow, Alaska. Whales seen at the first camp are "removed" from the population of concern to the second camp, where only whales missed by the first camp are counted. If both camps were in operation throughout the migration and if the probability of missing a whale were constant, the removal method would provide a population size estimate based on a trinomial model in which the size of the population would be the number of trials, whales counted by each camp would provide the observed cell totals, and whales missed by both camps would represent an unobserved cell total. Since the probability of missing a whale depends on visibility, we model the population size as the sum of the number of trials of several independent trinomial distributions, each of which represents a particular visibility condition occurring during the census. To account for the fact that watch cannot be maintained at both camps throughout the migration, we derive a confidence interval estimate of the number of trials under a more general model allowing for incomplete observation of totals within particular cells as well as for completely unobserved cells.     

The analysis of space use patterns.

In this paper we describe and test a new method for characterizing the space use patterns of individual animals on the basis of successive locations of marked individuals. Existing methods either do not describe space use in probabilistic terms, e.g. the maximum distance between locations or the area of the convex hull of all locations, or they assume a priori knowledge of the probabilistic shape of each individual's use pattern, e.g. bivariate or circular normal distributions. We develop a method for calculating a probability of location distribution for an average individual member of a population that requires no assumptions about the shape of the distribution (we call this distribution the population utilization distribution or PUD). Using nine different sets of location data, we demonstrate that these distributions accurately characterize the space use patterns of the populations from which they were derived. The assumption of normality is found to result in a consistent and significant overestimate of the area of use. We then describe a function which relates probability of location to area (termed the MAP index) which has a number of advantages over existing size indices. Finally, we show how any quantities such as the MAP index derived from our average distributions can be subjected to standard statistical tests of significance.     

Population persistence time under intermittent control in stochastic environments

If there exists a critical population size above which environmental degradation becomes serious, the population should be suppressed or reduced upon reaching that level. Since population size control is accompanied by costs, a reduction in control frequency may be preferable from an economic viewpoint. Although this can be realized by decreasing the population size drastically in each control, such management may result in increased population extinction probability according to environmental stochasticity. The effects of population management on both mean population persistence time and management cost were analyzed theoretically using a diffusion process. The model showed the functional forms of both mean persistence time and control frequency explicitly; these decreased with an increasing number of individuals removed from the population in each control operation. Based on the analysis, indices representing management costs are proposed. Mean persistence time is generally an increasing function of the cost indices. Nevertheless, if the cost of each control increases with the number of individuals removed, even the most conservative management practice (continuous control) may not be overly expensive.     

Complete Numerical Solution of the Diffusion Equation of Random Genetic Drift

A numerical method is presented to solve the diffusion equation for the random genetic drift that occurs at a single unlinked locus with two alleles. The method was designed to conserve probability, and the resulting numerical solution represents a probability distribution whose total probability is unity. We describe solutions of the diffusion equation whose total probability is unity as complete. Thus the numerical method introduced in this work produces complete solutions, and such solutions have the property that whenever fixation and loss can occur, they are automatically included within the solution. This feature demonstrates that the diffusion approximation can describe not only internal allele frequencies, but also the boundary frequencies zero and one. The numerical approach presented here constitutes a single inclusive framework from which to perform calculations for random genetic drift. It has a straightforward implementation, allowing it to be applied to a wide variety of problems, including those with time-dependent parameters, such as changing population sizes. As tests and illustrations of the numerical method, it is used to determine: (i) the probability density and time-dependent probability of fixation for a neutral locus in a population of constant size; (ii) the probability of fixation in the presence of selection; and (iii) the probability of fixation in the presence of selection and demographic change, the latter in the form of a changing population size.     

Numerical Equilibrium Analysis for Structured Consumer Resource Models

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for “Daphnia consuming algae” models in C-code. The results obtained by way of this implementation are shown in the form of graphs.     

A DNA solution of SAT problem by a modified sticker model

Among various DNA computing algorithms, it is very common to create an initial data pool that covers correct and incorrect answers at first place followed by a series of selection process to destroy the incorrect ones. The surviving DNA sequences are read as the solutions to the problem. However, algorithms based on such a brute force search will be limited to the problem size. That is, as the number of parameters in the studied problem grows, eventually the algorithm becomes impossible owing to the tremendous initial data pool size. In this theoretical work, we modify a well-known sticker model to design an algorithm that does not require an initial data pool for SAT problem. We propose to build solution sequences in parts to satisfy one clause in a step, and eventually solve the whole Boolean formula after a number of steps. Accordingly, the size of data pool grows from one sort of molecule to the number of solution assignments. The proposed algorithm is expected to provide a solution to SAT problem and become practical as the problem size scales up.     

Population growth makes waves in the distribution of pairwise genetic differences.

Episodes of population growth and decline leave characteristic signatures in the distribution of nucleotide (or restriction) site differences between pairs of individuals. These signatures appear in histograms showing the relative frequencies of pairs of individuals who differ by i sites, where i = 0, 1, .... In this distribution an episode of growth generates a wave that travels to the right, traversing 1 unit of the horizontal axis in each 1/2u generations, where u is the mutation rate. The smaller the initial population, the steeper will be the leading face of the wave. The larger the increase in population size, the smaller will be the distribution's vertical intercept. The implications of continued exponential growth are indistinguishable from those of a sudden burst of population growth Bottlenecks in population size also generate waves similar to those produced by a sudden expansion, but with elevated uppertail probabilities. Reductions in population size initially generate L-shaped distributions with high probability of identity, but these converge rapidly to a new equilibrium. In equilibrium populations the theoretical curves are free of waves. However, computer simulations of such populations generate empirical distributions with many peaks and little resemblance to the theory. On the other hand, agreement is better in the transient (nonequilibrium) case, where simulated empirical distributions typically exhibit waves very similar to those predicted by theory. Thus, waves in empirical distributions may be rich in information about the history of population dynamics.     

Reducing liposome size with ultrasound: bimodal size distributions

Sonication is a simple method for reducing the size of liposomes. We report the size distributions of liposomes as a function of sonication time using three different techniques. Liposomes, mildly sonicated for just 30 sec, had bimodal distributions when surface-weighted with modes at about 140 and 750 nm. With extended sonication, the size distribution remains bimodal but the average diameter of each population decreases and the smaller population becomes more numerous. Independent measurements of liposome size using Dynamic Light Scattering (DLS), transmission electron microscopy (TEM), and the nystatin/ergosterol fusion assay all gave consistent results. The bimodal distribution (even when number-weighted) differs from the Weibull distribution commonly observed for liposomes sonicated at high powers over long periods of time and suggests that a different mechanism may be involved in mild sonication. The observations are consistent with the following mechanism for decreasing liposome size. During ultrasonic irradiation, cavitation, caused by oscillating microbubbles, produces shear fields. Large liposomes that enter these fields form long tube-like appendages that can pinch-off into smaller liposomes. This proposed mechanism is consistent with colloidal theory and the observed behavior of liposomes in shear fields.