Two-locus identity probabilities and identity disequilibrium in a partially selfing subdivided population

Measures of association of genes at different loci (linkage disequilibrium) are widely used to determine whether the structure of natural populations is clonal or not, to map genes from population data, or to test for the homogeneity of response of molecular markers to background selection, for example. However, the usual definitions of parameters for gametic associations may not be suitable for all these purposes. In this paper, we derive the recursion equations for one- and two-locus identity probabilities in an infinite island model. We study the role of drift, gene flow, partial selfing and mutation model on the expected association of genes across loci. We define the 'within-subpopulation identity disequilibrium' as the difference between the joint two-locus probability of identity in state and the expected product of one-locus identity probabilities. We evaluate this parameter as a function of recombination rate, effective size, gene flow and selfing rate. Within-subpopulation identity disequilibrium attains maximum values for intermediate immigration rates, whatever the selfing rate. Moreover, identity disequilibrium may be very small, even for high selfing rates. We discuss the implications of these findings for the analysis of data from natural populations.     

Prediction of identity by descent probabilities from marker-haplotypes

The prediction of identity by descent (IBD) probabilities is essential for all methods that map quantitative trait loci (QTL). The IBD probabilities may be predicted from marker genotypes and/or pedigree information. Here, a method is presented that predicts IBD probabilities at a given chromosomal location given data on a haplotype of markers spanning that position. The method is based on a simplification of the coalescence process, and assumes that the number of generations since the base population and effective population size is known, although effective size may be estimated from the data. The probability that two gametes are IBD at a particular locus increases as the number of markers surrounding the locus with identical alleles increases. This effect is more pronounced when effective population size is high. Hence as effective population size increases, the IBD probabilities become more sensitive to the marker data which should favour finer scale mapping of the QTL. The IBD probability prediction method was developed for the situation where the pedigree of the animals was unknown (i.e. all information came from the marker genotypes), and the situation where, say T, generations of unknown pedigree are followed by some generations where pedigree and marker genotypes are known.     

The shapes of neutral gene genealogies in two species: probabilities of monophyly,paraphyly, and polyphyly in a coalescent model

Abstract.— The genealogies of samples of orthologous regions from multiple species can be classified by their shapes. Using a neutral coalescent model of two species, I give exact probabilities of each of four possible genealogical shapes: reciprocal monophyly, two types of paraphyly, and polyphyly. After the divergence that forms two species, each of which has population size N , polyphyly is the most likely genealogical shape for the lineages of the two species. At ∼ 1.300 N generations after divergence, paraphyly becomes most likely, and reciprocal monophyly becomes most likely at ∼1.665 N generations. For a given species, the time at which 99% of its loci acquire monophyletic genealogies is ∼5.298 N generations, assuming all loci in its sister species are monophyletic. The probability that all lineages of two species are reciprocally monophyletic given that a sample from the two species has a reciprocally monophyletic genealogy increases rapidly with sample size, as does the probability that the most recent common ancestor (MRCA) for a sample is also the MRCA for all lineages from the two species. The results have potential applications for the testing of evolutionary hypotheses.     

On the probabilities of identity states in permutable populations.

Génin and Clerget-Darpoux recently discussed the derivation of the probabilities of identity states for populations in which there was some degree of kinship, primarily to allow the extension of the classical affected-sib-pair method to such populations. It is argued here that their derivation makes certain assumptions that are valid only for some very restricted population models and that are not needed for an appropriate treatment. Here the probabilities of the identity states of two individuals with a given genealogical relationship are specified in terms of the kinship parameters of the underlying population, from which the founders of the individuals'' genealogy have been randomly selected. It is argued that an appropriate representation for a permutable population, one in which gene identity does not depend on the pattern of genes across individuals, requires three parameters. This representation is related to that of Génin and Clerget-Darpoux and to that of Weir.     

The expected probabilities of identity for the circular stepping-stone model

The expected values of the probabilities of identity by descent are derived for the circular stepping-stone model. The results are more easily interpreted than those derived previously.     

Simulation of selected genealogies

Algorithms for generating genealogies with selection conditional on the sample configuration of n genes in one-locus, two-allele haploid and diploid models are presented. Enhanced integro-recursions using the ancestral selection graph, introduced by S. M. Krone and C. Neuhauser (1997, Theor. Popul. Biol. 51, 210-237), which is the non-neutral analogue of the coalescent, enables accessible simulation of the embedded genealogy. A Monte Carlo simulation scheme based on that of R. C. Griffiths and S. Tavaré (1996, Math. Comput. Modelling 23, 141-158), is adopted to consider the estimation of ancestral times under selection. Simulations show that selection alters the expected depth of the conditional ancestral trees, depending on a mutation-selection balance. As a consequence, branch lengths are shown to be an ineffective criterion for detecting the presence of selection. Several examples are given which quantify the effects of selection on the conditional expected time to the most recent common ancestor.     

Germs, genomes and genealogies

Genetic diversity in pathogen species contains information about evolutionary and epidemiological processes, including the origins and history of disease, the nature of the selective forces acting on pathogen genes and the role of recombination in generating genetic novelty. Here, we review recent developments in these fields and compare the use of population genetic, or population-model based, approaches to phylogenetic, or population-model free, methodologies. We show how simple epidemiological models can be related to the ancestral, or coalescent, process underlying samples from pathogen species, enabling detailed inference about pathogen biology from patterns of molecular variation.     

Gene extinction and allelic origins in complex genealogies

With the increasing emphasis on data analysis in mathematical genetics, problems of parametrizing genealogical structure become of practical importance. A complete specification of the genetic effects of genealogical structure is provided by the probabilities of genetically distinct states of gene identity by descent. Although this provides a direct parametrization for the joint distribution of traits on a set of related individuals, it is an unwieldy tool in the analysis of large and complex genealogies. Probabilities of joint descent of founder genes and likely ancestries of alleles provide alternative characterizations of relationship and have direct application in practical problems. Joint extinction probabilities of founder genes can also be derived as ancestral likelihoods: evolutionarily, the most significant characteristic of a genealogical structure must be its effect on the survival and extinction of genes.     

Intraspecific gene genealogies: trees grafting into networks

Intraspecific gene evolution cannot always be represented by a bifurcating tree. Rather, population genealogies are often multifurcated, descendant genes coexist with persistent ancestors and recombination events produce reticulate relationships. Whereas traditional phylogenetic methods assume bifurcating trees, several networking approaches have recently been developed to estimate intraspecific genealogies that take into account these population-level phenomena.     

Fixation probabilities for the Moran process with three or more strategies: general and coupling results

We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrary to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth–death processes with only two strategies. We also apply our bounds to the problem of evolution of cooperation in a population with three types of individuals already studied in a deterministic setting by Núñez Rodríguez and Neves (J Math Biol 73:1665–1690, 2016). We argue that cooperators will be fixated in the population with probability arbitrarily close to 1 for a large region of initial conditions and large enough population sizes.

     

Risks and probabilities of breast cancer: short-term versus lifetime probabilities.

OBJECTIVE: To calculate age-specific short-term and lifetime probabilities of breast cancer among a cohort of Canadian women. DESIGN: Double decrement life table. SETTING: Alberta. SUBJECTS: Women with first invasive breast cancers registered with the Alberta Cancer Registry between 1985 and 1987. MAIN OUTCOME MEASURES: Lifetime probability of breast cancer from birth and for women at various ages; short-term (up to 10 years) probability of breast cancer for women at various ages. RESULTS: The lifetime probability of breast cancer is 10.17% at birth and peaks at 10.34% at age 25 years, after which it decreases owing to a decline in the number of years over which breast cancer risk will be experienced. However, the probability of manifesting breast cancer in the next year increases steadily from the age of 30 onward, reaching 0.36% at 85 years. The probability of manifesting the disease within the next 10 years peaks at 2.97% at age 70 and decreases thereafter, again owing to declining probabilities of surviving the interval. CONCLUSIONS: Given that the incidence of breast cancer among Albertan women during the study period was similar to the national average, we conclude that currently more than 1 in 10 women in Canada can expect to have breast cancer at some point during their life. However, risk varies considerably over a woman''s lifetime, with most risk concentrated after age 49. On the basis of the shorter-term age-specific risks that we present, the clinician can put breast cancer risk into perspective for younger women and heighten awareness among women aged 50 years or more.     

On the distribution of the general irreversiblen-compartmental model having time-dependent transition probabilities

The cumulant generating function and first two moments are derived for the stochastic distribution of units in a general irreversiblen-compartment model with time-dependent transition probabilities. In this model, a unit in the first compartment can transfer to any one of the remainingn−1 compartments and a unit in the second compartment can transfer to any of the remainingn−2 compartments and so on. In addition, a unit can enter or leave the system through any compartment. The work is related to previous research and a numerical example is given.     

A general method for biological inference: illustrated by the estimation of gene nucleotide transition probabilities

The method of maximum entropy inference developed by Jaynes can be a particularly useful method for obtaining unbiased estimates of biological parameters when the experimental knowledge about a system can be explicitly formulated. Base transition probabilities between genes, though central to evolutionary theory and understanding, present a difficult estimation problem because the ancestral genes are not experimentally accessible. The necessary estimates must therefore be made on the basis of experimental knowledge other than a direct frequency count of base replacements (A leads to C, for example) between contemporary genes. It is shown how maximum entropy inference together with the experimentally observed fact of compositional fidelity in a given gene family can be used to obtain meaningful gene base transition probabilities at each of the three nucleotide positions within codons. Both symmetric and asymmetric transition probabilities are considered. Tables of these probabilities are given for each codon position for the alpha-hemoglobin, beta-hemoglobin, myoglobin, cytochrome c, and the parvalbumin group genes. Tabular values of the average amino acid composition of these five protein families and the average nucleotide composition of their coding genes at varied codon loci are given. It is thus no longer necessary to assume in theories of evolutionary divergence equimolar base ratios A:C:G:U::1:1:1:1 or that each base has an equal chance of mutating to and being fixed as any one of the other three bases.     

Gene genealogies in geographically structured populations.

Population genetics theory has dealt only with the spatial or geographic pattern of degrees of relatedness or genetic similarity separately for each point in time. However, a frequent goal of experimental studies is to infer migration patterns that occurred in the past or over extended periods of time. To fully understand how a present geographic pattern of genetic variation reflects one in the past, it is necessary to build genealogy models that directly relate the two. For the first time, space-time probabilities of identity by descent and coalescence probabilities are formulated and characterized in this article. Formulations for general migration processes are developed and applied to specific types of systems. The results can be used to determine the level of certainty that genes found in present populations are descended from ancient genes in the same population or nearby populations vs. geographically distant populations. Some parameter combinations result in past populations that are quite distant geographically being essentially as likely to contain ancestors of genes at a given population as the past population located at the same place. This has implications for the geographic point of origin of ancestral, "Eve," genes. The results also form the first model for emerging "space-time" molecular genetic data.     

Human hair genealogies and stem cell latency

Background  

Stem cells divide to reproduce themselves and produce differentiated progeny. A fundamental problem in human biology has been the inability to measure how often stem cells divide. Although it is impossible to observe every division directly, one method for counting divisions is to count replication errors; the greater the number of divisions, the greater the numbers of errors. Stem cells with more divisions should produce progeny with more replication errors.