Genetic aspects of genealogy

The supplementary historical discipline genealogy is also a supplementary genetic discipline. In its formation, genetics borrowed from genealogy some methods of pedigree analysis. In the 21th century, it started receiving contribution from computer-aided genealogy and genetic (molecular) genealogy. The former provides novel tools for genetics, while the latter, which employing genetic methods, enriches genetics with new evidence. Genealogists formulated three main laws ofgenealogy: the law of three generations, the law of doubling the ancestry number, and the law of declining ancestry. The significance and meaning of these laws can be fully understood only in light of genetics. For instance, a controversy between the exponential growth of the number of ancestors of an individual, i.e., the law of doubling the ancestry number, and the limited number of the humankind is explained by the presence of weak inbreeding because of sibs' interference; the latter causes the pedigrees' collapse, i.e., explains also the law of diminishing ancestry number. Mathematic modeling of pedigrees' collapse presented in a number of studies showed that the number of ancestors of each individual attains maximum in a particular generation termed ancestry saturated generation. All representatives of this and preceding generation that left progeny are common ancestors of all current members of the population. In subdivided populations, these generations are more ancient than in panmictic ones, whereas in small isolates and social strata with limited numbers of partners, they are younger. The genealogical law of three generations, according to which each hundred years contain on average three generation intervals, holds for generation lengths for Y-chromosomal DNA, typically equal to 31-32 years; for autosomal and mtDNA, this time is somewhat shorter. Moving along ascending lineas, the number of genetically effective ancestors transmitting their DNA fragment to descendants increases far slower than the number of common ancestors, because the time to the nearest common ancestor is proportional to log2N, and the time to genetically effective ancestor, to N, where N is the population size. In relatively young populations, the number of genetically effective ancestors does not exceed the number of recombination hot spots, which is equal to 25000-50000. In ancient African populations with weaker linkage disequilibrium, their number may be higher. In genealogy, the degree of kinship is measured by the number of births separating the individuals under comparison, and in genetics, by Wright's coefficients of relationship (R). Genetic frames of a "large family" are limited by the average genomic differences among the members of the human population, which constitute approximately 0.1%. Conventionally it can be assumed that it is limited by relatives, associated with the members of the given nuclear family by the 7th degree of relatedness (R approximately 0.78%). However, in the course of the HapMap project it was established that 10-30% of pairs of individuals from the same population have at least one common genome region, which they inherited from a recent common ancestor. A nuclear family, if it is not consanguinous, unites two lineages, and indirectly, a multitude of them, constituting a "suprafamily" equivalent to a population. Some problems ofgenealogy and related historical issues can be resolved only with the help of genetics. These problems include identification of "true" and "false" Rurikids and the problem of continuity of the Y-chromosomal lineage of the Romanov dynasty. On the other hand, computer-aided genealogy and molecular genealogy seem to be promising in resolving genetic problems connected to recombination and coalescence ofgenomic regions.     

Mathematical consequences of the genealogical species concept

A genealogical species is defined as a basal group of organisms whose members are all more closely related to each other than they are to any organisms outside the group ("exclusivity"), and which contains no exclusive group within it. In practice, a pair of species is so defined when phylogenies of alleles from a sample of loci shows them to be reciprocally monophyletic at all or some specified fraction of the loci. We investigate the length of time it takes to attain this status when an ancestral population divides into two descendant populations of equal size with no gene exchange, and when genetic drift and mutation are the only evolutionary forces operating. The number of loci used has a substantial effect on the probability of observing reciprocal monophyly at different times after population separation, with very long times needed to observe complete reciprocal monophyly for a large number of loci. In contrast, the number of alleles sampled per locus has a relatively small effect on the probability of reciprocal monophyly. Because a single mitochondrial or chloroplast locus becomes reciprocally monophyletic much faster than does a single nuclear locus, it is not advisable to use mitochondrial and chloroplast DNA to recognize genealogical species for long periods after population divergence. Using a weaker criterion of assigning genealogical species status when more than 50% of sampled nuclear loci show reciprocal monophyly, genealogical species status depends much less on the number of sampled loci, and is attained at roughly 4-7 N generations after populations are isolated, where N is the historically effective population size of each descendant. If genealogical species status is defined as more than 95% of sampled nuclear loci showing reciprocal monophyly, this status is attained after roughly 9-12 N generations.     

Effect of selection on the topology of genealogical trees

Selection is one of the factors that most influence the shape of genealogical trees. Here we report results of simulations of the infinite-sites version of Moran's model of population genetics aiming at quantifying how the presence of selection affects the branching pattern (topology) of binary genealogical trees. In particular, we consider a scenario of purifying or negative selection in which all mutations are deleterious and each new mutation reduces the fitness of the individual by the same fraction. Analysis of five statistical measures of tree balance or symmetry borrowed from taxonomy indicates that the genealogical trees of samples of populations in which selection is actuating are in the average more asymmetric than neutral trees and that this effect is enhanced by increasing the sample size. However, a quantitative evaluation of the power of these balance measures to detect a tree topology significantly distinct from the neutral one indicates that they are not useful as tests of neutrality of mutations.     

The mapping of epistatic effects onto a genealogical tree in haploid populations

In this paper we present a model that maps epistatic effects onto a genealogical tree for a haploid population. Prior work has demonstrated that genealogical structure causes the genotypic values of individuals to covary. Our results indicate that epistasis can reduce genotypic covariance that is caused by genealogical structure. Genotypic effects (both additive and epistatic) occur along the branches of a genealogical tree, from the base of the tree to its tips. Epistasis reduces genotypic covariance because there is a reweighting of the contribution of branches to the states of genotypes compared to the additive case. Branches near the tips of a genealogical tree contribute proportionally more genetic effects with epistasis than without epistasis. Epistatic effects are most numerous at basal positions in a genealogical tree when a population is constant in size and experiencing no selection, optimizing selection, diversifying selection or directional selection, indicating that epistatic effects are typically old. For a population that is growing in size, epistatic effects are most numerous at midpoints in a genealogical tree, indicating epistatic effects are of moderate age. Our results are important in that they suggest epistatic effects may typically explain deep (old) divergences and broad patterns of divergence that exist in populations, except in growing populations. In a growing population, epistatic effects may cause more within group divergence higher up in a tree and less between group divergence that is deep in a tree. The distribution of the number of epistatic effects and the expected variance and covariance in the number of epistatic effects is also provided assuming neutrality.     

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.     

Fruit set in Styrax obassia (Styracaceae): the effect of light availability, display size, and local floral density

Maternal reproductive success was examined in Styrax obassia (Styracaceae), a bumble-bee pollinated mass-flowering tree in a cool-temperate deciduous forest in northern Japan. The effects of flower number on the success of individual flowers at three levels (inflorescence, individual, and population) were considered. During 1995 and 1996, variations in size, light availability to branches, floral display size, and fruit set were monitored in 37 out of 211 individual S. obassia trees in a 4-ha forest plot. In addition, the locations of the 211 trees in this plot were mapped and the number of inflorescences in each tree was counted. A multiple regression analysis showed that flower number per inflorescence and inflorescence number per individual had negative effects on fruit set, and inflorescence number of aggregated clumps of flowering trees, tree size, and light resource had positive effects on fruit set although significant level were marginal. It is concluded that pollinator attraction may occur not at the individual tree level, but at the level of a clump of flowering trees. It is also suggested that geitonogamy increased with inflorescence number of tree and inflorescence size and that resource limitation was related to the light condition and variation of tree size.     

Breeding population size of a fragmented population of a Costa Rican dry forest tree species.

Pollen immigration can offset the effects of genetic drift and inbreeding in small populations. To understand the genetic consequences of forest fragmentation, estimates of pollen flow into remnant fragments are essential. Such estimates are straightforward for plants with singly sired, multiseeded fruits, since the pollen donor genotype for each fruit can be unambiguously reconstructed through full-sib genealogical analyses. Allozyme analyses were used to estimate pollen donor numbers from the progeny of fruits of the tropical dry forest tree Enterolobium cyclocarpum in a small (9.8 ha) fragmented population (N = 11) over three reproductive seasons (1994, 1995, and 1996). These analyses indicate that each tree receives pollen from many pollen donors. When data are pooled for the site, estimated maximum pollen donor pool sizes in all years exceed the number of individuals (56) in the 227 ha study area. Although unidentified pollen donors may be located as close as 250 m to the study trees, the number of unidentified pollen donors indicates that individuals in this forest fragment are part of a large network of reproductively active individuals.     

Genomic runs of homozygosity record population history and consanguinity

The human genome is characterised by many runs of homozygous genotypes, where identical haplotypes were inherited from each parent. The length of each run is determined partly by the number of generations since the common ancestor: offspring of cousin marriages have long runs of homozygosity (ROH), while the numerous shorter tracts relate to shared ancestry tens and hundreds of generations ago. Human populations have experienced a wide range of demographic histories and hold diverse cultural attitudes to consanguinity. In a global population dataset, genome-wide analysis of long and shorter ROH allows categorisation of the mainly indigenous populations sampled here into four major groups in which the majority of the population are inferred to have: (a) recent parental relatedness (south and west Asians); (b) shared parental ancestry arising hundreds to thousands of years ago through long term isolation and restricted effective population size (N(e)), but little recent inbreeding (Oceanians); (c) both ancient and recent parental relatedness (Native Americans); and (d) only the background level of shared ancestry relating to continental N(e) (predominantly urban Europeans and East Asians; lowest of all in sub-Saharan African agriculturalists), and the occasional cryptically inbred individual. Moreover, individuals can be positioned along axes representing this demographic historic space. Long runs of homozygosity are therefore a globally widespread and under-appreciated characteristic of our genomes, which record past consanguinity and population isolation and provide a distinctive record of the demographic history of an individual's ancestors. Individual ROH measures will also allow quantification of the disease risk arising from polygenic recessive effects.     

Genetic aspects of genealogy

The supplementary historical discipline genealogy is also a supplementary genetic discipline. In its formation, genetics borrowed from genealogy some methods of pedigree analysis. In the 21th century, it started receiving contribution from computer-aided genealogy and genetic (molecular) genealogy. The former provides novel tools for genetics, while the latter, which employing genetic methods, enriches genetics with new evidence. Genealogists formulated three main laws of genealogy: the law of three generations, the law of doubling the ancestry number, and the law of declining ancestry. The significance and meaning of these laws can be fully understood only in light of genetics. For instance, a controversy between the exponential growth of the number of ancestors of an individual, i.e., the law of doubling the ancestry number, and the limited number of the humankind is explained by the presence of weak inbreeding because of sibs’ interference; the latter causes the pedigrees’ collapse, i.e., explains also the law of diminishing ancestry number. Mathematic modeling of pedigrees’ collapse presented in a number of studies showed that the number of ancestors of each individual attains maximum in a particular generation termed ancestry saturated generation. All representatives of this and preceding generation that left progeny are common ancestors of all current members of the population. In subdivided populations, these generations are more ancient than in panmictic ones, whereas in small isolates and social strata with limited numbers of partners, they are younger. The genealogical law of three generations, according to which each hundred years contain on average three generation intervals, holds for generation lengths for Y-chromosomal DNA typically equal to 31–32 years; for autosomal and mtDNA, this time is somewhat shorter. Moving along ascending lines, the number of genetically effective ancestors transmitting their DNA fragments to descendants increases far slower than the number of common ancestors, because the time to the nearest common ancestor is proportional to log2N, and the time to genetically effective ancestor, to N, where N is the population size. In relatively young populations, the number of genetically effective ancestors does not exceed the number of recombination hot spots, which is equal to 25 000–50000. In ancient African populations with weaker linkage disequilibrium, their number may be higher. In genealogy, the degree of kinship is measured by the number of births separating the individuals under comparison, and in genetics, by Wright’s coefficients of relationship (R). Genetic frames of a “large family” are limited by the average genomic differences among the members of the human population, which constitute approximately 0.1%. Conventionally it can be assumed that it is limited by relatives, associated with the members of the given nuclear family by the 7th degree of relatedness (R ∼ 0.78%). However, in the course of the HapMap project it was established that 10–30% of pairs of individuals from the same population have at least one common genome region, which they inherited from a recent common ancestor. A nuclear family, if it is not consanguinous, unites two lineages, and indirectly, a multitude of them, constituting a “suprafamily” equivalent to a population. Some problems of genealogy and related historical issues can be resolved only with the help of genetics. These problems include identification of “true” and “false” Rurikids and the problem of continuity of the Y-chromosomal lineage of the Romanov dynasty. On the other hand, computer-aided genealogy and molecular genealogy seem to be promising in resolving genetic problems connected to recombination and coalescence of genomic regions.     

The accumulation of mutations in asexual populations and the structure of genealogical trees in the presence of selection

We study the accumulation of unfavourable mutations in asexual populations by the process of Muller's ratchet, and the consequent inevitable decrease in fitness of the population. Simulations show that it is mutations with only moderate unfavourable effect that lead to the most rapid decrease in fitness. We measure the number of fixations as a function of time and show that the fixation rate must be equal to the ratchet rate once a steady state is reached. Large bursts of fixations are observed to occur simultaneously. We relate this to the structure of the genealogical tree. We derive equations relating the rate of the ratchet to the moments of the distribution of the number of mutations k per individual. These equations interpolate between the deterministic limit (an infinite population with selection present) and the neutral limit (a finite size population with no selection). Both these limits are exactly soluble. In the neutral case, the distribution of k is shown to be non-self-averaging, i.e. the fluctuations remain very large even for very large populations. We also consider the strong-selection limit in which only individuals in the fittest surviving class have offspring. This limit is again exactly soluble. We investigate the structure of the genealogical tree relating individuals in the same population, and consider the probability (T) that two individuals had their latest common ancestor T generations in the past. The function (T) is exactly calculable in the neutral limit and the strong-selection limit, and we obtain an empirical solution for intermediate selection strengths.     

Inbreeding, pedigree size, and the most recent common ancestor of humanity

How many generations ago did the common ancestor of all present-day individuals live, and how does inbreeding affect this estimate? The number of ancestors within family trees determines the timing of the most recent common ancestor of humanity. However, mating is often non-random and inbreeding is ubiquitous in natural populations. Rates of pedigree growth are found for multiple types of inbreeding. This data is then combined with models of global population structure to estimate biparental coalescence times. When pedigrees for regular systems of mating are constructed, the growth rates of inbred populations contain Fibonacci n-step constants. The timing of the most recent common ancestor depends on global population structure, the mean rate of pedigree growth, mean fitness, and current population size. Inbreeding reduces the number of ancestors in a pedigree, pushing back global common ancestry times. These results are consistent with the remarkable findings of previous studies: all humanity shares common ancestry in the recent past.     

Inferring phylogeny despite incomplete lineage sorting

It is now well known that incomplete lineage sorting can cause serious difficulties for phylogenetic inference, but little attention has been paid to methods that attempt to overcome these difficulties by explicitly considering the processes that produce them. Here we explore approaches to phylogenetic inference designed to consider retention and sorting of ancestral polymorphism. We examine how the reconstructability of a species (or population) phylogeny is affected by (a) the number of loci used to estimate the phylogeny and (b) the number of individuals sampled per species. Even in difficult cases with considerable incomplete lineage sorting (times between divergences less than 1 N(e) generations), we found the reconstructed species trees matched the "true" species trees in at least three out of five partitions, as long as a reasonable number of individuals per species were sampled. We also studied the tradeoff between sampling more loci versus more individuals. Although increasing the number of loci gives more accurate trees for a given sampling effort with deeper species trees (e.g., total depth of 10 N(e) generations), sampling more individuals often gives better results than sampling more loci with shallower species trees (e.g., depth = 1 N(e)). Taken together, these results demonstrate that gene sequences retain enough signal to achieve an accurate estimate of phylogeny despite widespread incomplete lineage sorting. Continued improvement in our methods to reconstruct phylogeny near the species level will require a shift to a compound model that considers not only nucleotide or character state substitutions, but also the population genetics processes of lineage sorting. [Coalescence; divergence; population; speciation.].     

Intergenerational and Genealogical Approaches for the Study of Longevity in the Saguenay-Lac-St-Jean Population

The mechanisms of longevity have been the subject of investigations for a number of years. Although the role of genetic factors is generally acknowledged, important questions persist regarding the relative impact of environmental exposures, lifestyle characteristics, and genes. The BALSAC population register offers a unique opportunity to study longevity from an intergenerational and genealogical point of view. Individuals from the Saguenay-Lac-St-Jean population who died at age 90 or older between 1950 and 1974 were selected from this database (n?=?576), along with a control group of individuals born in the same period who died between 50 and 75 years of age. For these subjects and controls, spouses’ ages at death and parental ages at death and at their birth were investigated using regression analysis. Genealogical reconstructions were carried out for each individual, and various analyses were performed on both groups. Both fathers’ and mothers’ mean ages at death were significantly higher among the longer-lived cases than among controls whereas spouses’ ages at death and parental ages at birth had no effect. Regression analysis confirmed the positive effect of both fathers’ and mothers’ age at death. Mean kinship coefficients for the parents’ generations displayed significant differences, indicating that kinship was higher among subjects than controls (this effect was stronger among the oldest 10% of the subjects). Frequencies and genetic contributions of ancestors were very similar for the two groups, and none of these ancestors appeared more likely to have introduced genetic variants involved in longevity patterns in this French Canadian population.     

Species trees from gene trees: reconstructing Bayesian posterior distributions of a species phylogeny using estimated gene tree distributions

The desire to infer the evolutionary history of a group of species should be more viable now that a considerable amount of multilocus molecular data is available. However, the current molecular phylogenetic paradigm still reconstructs gene trees to represent the species tree. Further, commonly used methods of combining data, such as the concatenation method, are known to be inconsistent in some circumstances. In this paper, we propose a Bayesian hierarchical model to estimate the phylogeny of a group of species using multiple estimated gene tree distributions, such as those that arise in a Bayesian analysis of DNA sequence data. Our model employs substitution models used in traditional phylogenetics but also uses coalescent theory to explain genealogical signals from species trees to gene trees and from gene trees to sequence data, thereby forming a complete stochastic model to estimate gene trees, species trees, ancestral population sizes, and species divergence times simultaneously. Our model is founded on the assumption that gene trees, even of unlinked loci, are correlated due to being derived from a single species tree and therefore should be estimated jointly. We apply the method to two multilocus data sets of DNA sequences. The estimates of the species tree topology and divergence times appear to be robust to the prior of the population size, whereas the estimates of effective population sizes are sensitive to the prior used in the analysis. These analyses also suggest that the model is superior to the concatenation method in fitting these data sets and thus provides a more realistic assessment of the variability in the distribution of the species tree that may have produced the molecular information at hand. Future improvements of our model and algorithm should include consideration of other factors that can cause discordance of gene trees and species trees, such as horizontal transfer or gene duplication.     

Gene Genealogy in Three Related Populations: Consistency Probability between Gene and Population Trees

A genealogical relationship among genes at a locus (gene tree) sampled from three related populations was examined with special reference to population relatedness (population tree). A phylogenetically informative event in a gene tree constructed from nucleotide differences consists of interspecific coalescences of genes in each of which two genes sampled from different populations are descended from a common ancestor. The consistency probability between gene and population trees in which they are topologically identical was formulated in terms of interspecific coalescences. It was found that the consistency probability thus derived substantially increases as the sample size of genes increases, unless the divergence time of populations is very long compared to population sizes. Hence, there are cases where large samples at a locus are very useful in inferring a population tree.     

Coalescent theory for a completely random mating monoecious population

Consider a large random mating monoecious diploid population that has N individuals in each generation. Let us assume that at time 0 a random sample of ninfinity. It is then possible to obtain a generalization of coalescent theory for haploid populations if the distribution of G1 has a finite second moment and E[G(1)(3)]/N-->0 as N-->infinity.     

Evolution of genetic variation for selected traits in successive breeding populations of maritime pine

Directional selection impacts a trait distribution by shifting its mean and reducing its variance. The change of variance is of major importance as the response to selection in subsequent generations is highly dependent of the genetic variability available in the population. In this contribution, evolution of genetic variation was investigated through the first breeding populations of the French maritime pine (Pinus pinaster Ait.) breeding program. We considered three populations: P0 (the forest where plus trees were initially selected), G0 (the plus tree population) and G1 (the population composed of trees selected in the progenies of G0). Analyses focused on the following selected traits: total height (H), girth at 1.30 m (D) and stem deviation to verticality (S). More than 150,000 trees from 25 tests of three distinct populations were studied with an individual genetic model. Accurate genetic parameters were obtained by taking all relationships between trees into account. For H and D, we found a strong decrease of the genetic variation from P0 to G0 corresponding to the initial selection of plus trees, which constitutes the base population of the breeding program. Then, despite the second step of selection applied, no appreciable evolution arose from comparisons between G0 and G1 for these traits. For S, the evolution is less significant as phenotypic variation slightly increased, possibly due to changes of silvicultural practices.     

Bayesian Inference of Sampled Ancestor Trees for Epidemiology and Fossil Calibration

Phylogenetic analyses which include fossils or molecular sequences that are sampled through time require models that allow one sample to be a direct ancestor of another sample. As previously available phylogenetic inference tools assume that all samples are tips, they do not allow for this possibility. We have developed and implemented a Bayesian Markov Chain Monte Carlo (MCMC) algorithm to infer what we call sampled ancestor trees, that is, trees in which sampled individuals can be direct ancestors of other sampled individuals. We use a family of birth-death models where individuals may remain in the tree process after sampling, in particular we extend the birth-death skyline model [Stadler et al., 2013] to sampled ancestor trees. This method allows the detection of sampled ancestors as well as estimation of the probability that an individual will be removed from the process when it is sampled. We show that even if sampled ancestors are not of specific interest in an analysis, failing to account for them leads to significant bias in parameter estimates. We also show that sampled ancestor birth-death models where every sample comes from a different time point are non-identifiable and thus require one parameter to be known in order to infer other parameters. We apply our phylogenetic inference accounting for sampled ancestors to epidemiological data, where the possibility of sampled ancestors enables us to identify individuals that infected other individuals after being sampled and to infer fundamental epidemiological parameters. We also apply the method to infer divergence times and diversification rates when fossils are included along with extant species samples, so that fossilisation events are modelled as a part of the tree branching process. Such modelling has many advantages as argued in the literature. The sampler is available as an open-source BEAST2 package (https://github.com/CompEvol/sampled-ancestors).     

Temporal Changes in the Inequality of Early Growth of Cunninghamia lanceolata (Lamb.) Hook.: A Novel Application of the Gini Coefficient and Lorenz Asymmetry

Growth within tree populations varies among individuals due to changes in biotic and abiotic factors. The degree of such variation, defined as growth inequality, serves as a useful indicator of the uniformity of growth within a population in response to the prevalent environmental conditions. By application of the Gini coefficient (G), an index for inequality, we characterized the early growth inequality of ninety crosses of Chinese fir (Cunninghamia lanceolata (Lamb.) Hook.) and their open-pollinated parental lines. Tree cumulative height was measured annually for 8 consecutive years. Both the crosses and parental lines exhibited temporal changes in growth inequality. The inequality of total height among the crosses decreased logarithmically with age by nearly 3-fold after 13 growing seasons, suggesting that tree height became less variable among the crosses as trees grew larger. Interestingly, the Lorenz asymmetry, an index reflecting the shape of the Lorenze curve from which G is derived, revealed that the inequality of annual height increment among the crosses resulted from an alternate contribution of the fast-growing and slow-growing trees. Among parental lines, two provenances with the smallest and the largest overall inequality in total height showed a similar pattern of changes in annual growth inequality, and the provenance differences were consistent over time. Compared to the other provinces, a local provenance exhibited less variation in total height among individual trees as reflected by a smaller value of inequality, and was better adapted to the field conditions. Our results demonstrated the sensitivity and usefulness of the Gini coefficient and Lorenz asymmetry for the analysis of growth inequality in non-natural populations. Growth inequality is a potentially useful evaluation criterion for early selection. Given comparable initial growth, provenances/families with lower growth inequality values would likely outperform those with higher growth inequality, and eventually tree size of the latter would be more variable due to greater variations among individual trees. Assessment of growth inequality at early ages will advance our understanding of variability of tree growth within a population, facilitate forest genetics improvement programs, and enhance the efficiency of tree breeding.     

The probability distribution under a population divergence model of the number of genetic founding lineages of a population or species

The composition of genetic variation in a population or species is shaped by the number of events that led to the founding of the group. We consider a neutral coalescent model of two populations, where a derived population is founded as an offshoot of an ancestral population. For a given locus, using both recursive and nonrecursive approaches, we compute the probability distribution of the number of genetic founding lineages that have given rise to the derived population. This number of genetic founding lineages is defined as the number of ancestral individuals that contributed at the locus to the present-day derived population, and is formulated in terms of interspecific coalescence events. The effects of sample size and divergence time on the probability distribution of the number of founding lineages are studied in detail. For 99.99% of the loci in the derived population to each have one founding lineage, the two populations must be separated for 9.9N generations. However, only approximately 0.87N generations must pass since divergence for 99.99% of the loci to have <6 founding lineages. Our results are useful as a prior expectation on the number of founding lineages in scenarios that involve the evolution of one population from the splitting of an ancestral group, such as in the colonization of islands, the formation of polyploid species, and the domestication of crops and livestock from wild ancestors.