A logistic branching process for population genetics

A logistic (regulated population size) branching process population genetic model is presented. It is a modification of both the Wright-Fisher and (unconstrained) branching process models, and shares several properties including the coalescent time and shape, and structure of the coalescent process with those models. An important feature of the model is that population size fluctuation and regulation are intrinsic to the model rather than externally imposed. A consequence of this model is that the fluctuation in population size enhances the prospects for fixation of a beneficial mutation with constant relative viability, which is contrary to a result for the Wright-Fisher model with fluctuating population size. Explanation of this result follows from distinguishing between expected and realized viabilities, in addition to the contrast between absolute and relative viabilities.     

A bisexual multitype branching process with applications in population genetics

A bisexual multiple branching process is studied. Consider a population with respect to three genotypes in both the female and male populations and let $$X(n) = \left\langle {X_1 (n), X_2 (n), X_3 (n)} \right\rangle and Y(n) = \left\langle {Y_1 (n), Y_2 (n), Y_3 (n)} \right\rangle$$ be random vectors giving the number of females and males (respectively) of each genotype in generationn. The mating of females and males is accommodated in the model withZ _ij (n) representing the number of females of theith genotype mated with a male of thejth genotype in generationn. The mating system is such that a female may be mated to only one male but a male may be mated with more than one female. By arranging the nine random variablesZ _ij (n),i, j=1, 2, 3, in a 1×9, vectorZ(n) it is shown that under certain conditions there is a positive constant ? such that when ?>1 the vectorsZ _n /ρⁿ,X _n /ρⁿ andY _n /ρⁿ converge almost surely asn→∞ to random vectors with fixed directions. The paper is divided into four sections. In section 1 the model is described in detail and its potential applications to population genetics are discussed. In section 2, the generating function of the transition probabilities of theZ-process are derived. Section3 is devoted to the study of the limiting behavior of the first and second moments of theZ-process, and in section4 the results of section3 are utilized to study the behavior of the random vectorsZ(n),X(n) andY(n) asn→∞.     

The influence of genes on the aging process of mice: a statistical assessment of the genetics of aging

Genetic interventions that accelerate or retard aging in mice are crucial in advancing our knowledge over mammalian aging. Yet determining if a given intervention affects the aging process is not straightforward since, for instance, many disease-causing mutations may decrease life span without affecting aging. In this work, we employed the Gompertz model to determine whether several published interventions previously claimed to affect aging in mice do indeed alter the aging process. First, we constructed age-specific mortality tables for a number of mouse cohorts used in longevity experiments and calculated the rate at which mortality increases with age. Estimates of age-independent mortality were also calculated. We found no statistical evidence that GHRHR, IGF1R, INSR, PROP1, or TRX delay or that ATM + TERC, BubR1, klotho, LMNA, PRDX1, p53, WRN + TERC, or TOP3B accelerate mouse aging. Often, changes in the expression of these genes affected age-independent mortality and so they may prove useful to other aspects of medicine. We found statistical evidence that C/EBP, MSRA, SHC1, growth hormone, GHR, PIT1, and PolgA may influence aging in mice. These results were interpreted together with age-related physiological and pathological changes and provide novel insights regarding the role of several genes in the mammalian aging process.     

An application of kinship process to the gene frequencies: linkage disequilibrium due to random drift in mendelian genetics with reversible mutation and in molecular genetics.

Selection at the colony level in social Hymenoptera with colonies containing single, once-mated queens is examined under a simple two-allele model. The condition for balanced polymorphism is $\text{X>}\text{2V}{}^2\text{(V + 1)}$, where V is the fitness of colonies with all workers homozygous and X that of colonies with both heterozygous and homozygous workers, relative to the fitness of colonies with all workers heterozygous. For certain fitness combinations satisfying the above relationship and characterized by values of V and X much lower than one, iteration reveals the development of stable limit cycles of allele frequencies rather than convergence to an equilibrium point. Addition of a third allele, or overlap between generations, eliminates these cycles. Queen-level overdominance is sufficient but not necessary for balanced polymorphism when V < 1, is both sufficient and necessary when V = 1, and is necessary but not sufficient when V > 1. Colony-level selection is a potentially powerful force maintaining genetic variation in populations of social insects, but does not imply correspondence between queen and worker genotype frequencies.     

Exploring the genetics of aging in a wild passerine bird

Senescence is the decline in survival and reproduction as an organism ages and is known to occur in collared flycatchers Ficedula albicollis. We consider annual fitness (the estimated genetic contribution that an individual makes to next year's gene pool) as a measure of age-specific fitness. We apply a restricted maximum likelihood linear mixed-model approach on 25 years of data on 3,844 male and 4,992 female collared flycatchers. Annual fitness had a significant additive genetic component (h2 of about 4%). Annual fitness declined at later ages in both sexes. Using a random regression animal model, we show that the observed age-related phenotypic changes in annual fitness were not present on the additive genetic level, contrary to predictions of genetic hypotheses of senescence. Our study suggests that patterns of aging in the wild need to be interpreted with caution in terms of underlying genetics because they may be largely determined by environmental processes.     

The Galton-Watson branching process as a quantitative tool in parasitology.

Stochastic growth processes abound in the biology of parasitism, and one mathematical tool that is particularly well suited for describing such phenomena is the Galton-Watson branching process. Introduced more than a century ago to settle a debate over the rate of disappearance of surnames in the British peerage, branching processes are applied today in fields as diverse as quantum physics and theoretical computer science. In this article, Dale Taneyhill, Alison Dunn and Melanie Hatcher provide a simple introduction to branching processes, and demonstrate their uses in quantitative parasitology.     

An age-dependent branching process model for the analysis of CFSE-labeling experiments

Background  

Over the past decade, flow cytometric CFSE-labeling experiments have gained considerable popularity among experimentalists, especially immunologists and hematologists, for studying the processes of cell proliferation and cell death. Several mathematical models have been presented in the literature to describe cell kinetics during these experiments.     

On a modified Markov branching process

We modify a Markov branching process such that each particle produces offspring only if its life length is greater than T. We find exact expressions for the mean number of particles at time t.     

The genetics of aging

Once thought to be an extremely complex conundrum of weak genetic and environmental effects, exceptional longevity is beginning to yield genetic findings. Numerous lower organism and mammalian models demonstrate genetic mutations that increase life-span markedly. These variations, some of them evolutionarily conserved, inform us about biochemical pathways that significantly impact upon longevity. Centenarian studies have also proven useful as they are a cohort that, relative to younger age groups, lacks genotypes linked to age-related lethal diseases and premature mortality. Pedigree studies have demonstrated a significant familial component to the ability to survive to extreme old age and a recent study demonstrates a locus on chromosome 4 linked to exceptional longevity indicating the likely existence of at least one longevity enabling gene in humans. Thus, a number of laboratories are making substantial and exciting strides in the understanding of the genetics of aging and longevity which should lead to the discovery of genes and ultimately drugs that slow down the aging process and facilitate people's ability to delay and perhaps escape age-associated diseases.     

Daltonism and the genetics of aging

In order to test whether mutations giving rise to color vision deficiencies are more frequently inherited from older fathers, an exhaustive screening of births in the Namur region has allowed to isolate a sample of 225 descending sons of maternal grandfathers who were older than 45 years at their daughter's birth. The incidence of color vision defects was compared between this set of cases and three control groups totalling 959 boys from independent families. While these comparisons were not conclusive, we propose new hypotheses concerning the population dynamics of color vision deficiencies. Neomutations in X-linked pigment genes may be a marker of the overall genetic load borne by the X chromosome. Selection against such loaded X chromosomes may occur in the second generation, either in the course of embryogenesis, or during female gametogenesis. The future assessment of these novel hypotheses relies on the arbitration of molecular genetics.     

On a stochastic integral of a branching process

This paper is concerned with the properties of a stochastic integral which arises in the study of a modified Markov branching process. Explicit expressions are found for the mean and the limit distribution of the integral.     

Evolutionary genetics of aging in Daphnia

Evolutionary theory of aging stipulates that aging is inevitable consequence of low effectiveness of natural selection acting on traits expressed late in the life span of the organisms. Two main hypotheses exist: the neutralist mutation-accumulation theory and selectionist antagonistic pleiotropy theory. Both theories predict the increase of genetic variance with age; the antagonistic pleiotropy theory also predicts negative genetic correlation between fitness related traits in the beginning and in the end of the life span. In order to test these predictions we measured life expectancy and age specific mortality in cohorts of 26 c lones of Daphnia magna extracted from a single cyclic parthenogen population. Simultaneously, fecundity and age to maturity were measured in representatives of the same clones. Log mortality increased linearly with age, with little evidence for leveling off, although some replicate cohorts did show a significant leveling off of mortality. Genetic variance of log mortality was significantly higher in the last quarter of the life span than in earlier time intervals. There was a significant positive genetic correlation between early fecundity and early mortality, but not between early fecundity and late mortality. This indicates that, although there is a trade-off between fecundity and survival, this trade-off is not based on pleiotropy across ages and therefore the data does not support the prediction of antagonistic pleiotropy theory.     

The genetics of aging in the yeastSaccharomyces cerevisiae

The yeastSaccharomyces cerevisiae possesses a finite life span similar in many attributes and implications to that of higher eukaryotes. Here, the measure of the life span is the number of generations or divisions the yeast cell has undergone. The yeast cell is the organism, simplifying many aspects of aging research. Most importantly, the genetics of yeast is highly-developed and readily applicable to the dissection of longevity. Two candidate longevity genes have already been identified and are being characterized. Others will follow through the utilization of both the primary phenotype and the secondary phenotypes associated with aging in yeast. An ontogenetic theory of longevity that follows from the evolutionary biology of aging is put forward in this article. This theory has at its foundation the asymmetric reproduction of cells and organisms, and it makes specific predictions regarding the genetics, molecular mechanisms, and phenotypic features of longevity and senescence, including these: GTP-binding proteins will frequently be involved in determining longevity, asymmetric cell division will be often encountered during embryogenesis while binary fission will be more characteristic of somatic cell division, tumor cells of somatic origin will not be totipotent, and organisms that reproduce symmetrically will not have intrinsic limits to their longevity.