Errors in genetic theory equations

Summary This study addresses the consequences of eliminating terms such as x² and x³ from genetic equations when the variable x is known to be small. This paper indicates logically that to assign such terms a value of 0.0 requires knowing the magnitude of the coefficients for each of these terms as well as the magnitude of all other terms in a given expression. Since most genetic expressions of interest involve several unknowns, the elimination of these terms appears difficult to justify in most situations. The effects of the elimination of a single term from an expression in a classical plant breeding paper were investigated as a simple exemplifying case. In the example, the simplified equation for change in population mean with selection sometimes greatly overestimated the response to selection and in some cases also altered conclusions as to best procedure. Though simplified equations are usually much more tractable and interpretable, the bias which is introduced into the research results and the potential for propagation of such biases in subsequent studies indicates that no term can be uncritically ignored in a genetic equation. The obvious alternatives are (1) do not simplify by eliminating terms, (2) perform a complete error analysis, or (3) restrict the range of values for variables so that terms can be justifiably eliminated in the error analysis.Cooperative investigations of the Alfalfa Production Research Unit, United States Department of Agriculture, Agricultural Research Service, and the Nevada Agricultural Experiment Station, Reno, NV     

Cover distributions of vascular plants in relation to soil chemistry and soil depth in a granite rock ecosystem

The variability in the cover distribution of vascular plants, accounted for by soil chemical properties and soil depth, on a granite slope with shallow autochtonous soil in southeast Sweden was evaluated using multivariate statistical regression and graphical methods. Soil acidity and soil depth were, to an often high degree, able to account for the variability in the distributions of the ca. 30 most frequent species, including Rumex acetosella, Vincetoxicum hirundinaria, Filipendula vulgaris, Satureja acinos, Geranium columbinum, Silene rupestris, and Arenaria serpyllifolia. The best expression of soil acidity was pH-KCl, though exchangeable Ca and Al were also important measures of the soil-plant relationships. The Ca:Al ratio was inferior in this respect. Also exchangeable or acid soluble phosphate was significantly related to the distribution of several species, whereas soil organic matter content was almost unrelated.     

Genetic variance and drift in selfed and intermated populations derived from backcrossing

Summary The genetic variance among random-mated lines derived from backcrossing (BC_gS₁ lines) depends upon the backcross generation (g) and the number (n) of BC_gF₁ plants crossed in generations 1 through g. There is little effect of n on genetic variance for n > 6. The genetic variance among BC_gF₂-derived lines is greater than that among BC_gS₁ lines for all g. If either BC_gF₂-derived or BC_gS₁ lines are used as a base population for recurrent selection, 8, 16, 32, and 64 BC₁F₁, BC₂F₁, BC₃F₁, and BC₄F₁ plants, respectively, should be used to avoid loss of donor alleles to drift.Joint contribution of USDA-ARS and Journal Paper No. J-11224 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 2471Formerly Research Geneticist, USDA-ARS, Ames, Iowa, USA     

Including genetic relationships in selection decisions: alternative methodologies

Investigations are made of variations in an iterative methodology previously introduced for reducing inbreeding by including genetic relationships in selection decisions, using adjusted estimated breeding values (EBV). An alternative computing strategy for maximising the value of the population selection criterion is shown to involve less computation, which results in function values as great or greater than the original method. Alteration of weights for different types of relationships in the adjusted EBV has no detectable effect on genetic gain at a given level of inbreeding. Selection using the adjusted EBV method in one sex and truncation on EBV in the other sex results in less genetic gain at a given level of inbreeding than using adjusted EBV in both sexes, but results in more gain at a given level of inbreeding than three selection strategies that do not include genetic relationships in selection decisions. The advantage of the adjusted EBV method over these three methods is retained when selection is for a sexlimited trait.     

Founder effect and genetic disease in Sottunga, Finland

Pedigree data are analyzed in order to determine the factors responsible for the high frequencies of certain genetic disorders in an isolated Swedish-speaking population of Finland's A land archipelago. The founders of Sottunga are identified, and the genetic contributions of each founder to descending birth cohorts are estimated. Founders born before 1700 have far more descendants in the contemporary gene pool than do more recent founders. However, because of migration and depopulation since 1900, the expected genetic contributions of the early founders to the present-day population are similar to those of later founders. A descendant in the contemporary population has a 2% chance of having inherited a particular gene from the founder who makes the largest single contribution to the gene pool. This corresponds approximately to a 2% probability of inheriting an autosomal dominant disease gene from this founder. Given an average inbreeding coefficient of 0.0016, the probability of inheriting two recessive disease genes from this founder is 0.000032. The incidence of autosomal dominant von Willebrand disease in Sottunga is greater than 10% while that of autosomal recessive tapetoretinal disease is 1.5%. We conclude, therefore, that the high frequencies of these diseases are not due to the disproportionate genetic contribution of one or a few particular founders. It is more likely that these disease genes occurred in high frequency in the initial population or were introduced repeatedly through time.     

Delaying copulation in the amphipod crustacean Gammarus palustris: Effects on female fecundity and consequences for the frequency of amplexus

Both physiological and environmental constraints can be important selective factors on behaviors. The results of the present study showed that fecundity is significantly reduced if copulation is delayed for as short a time as 6 ± 6h in the amphipod crustacean Gammarus palustris (Bousfield, 1969). This led to the prediction that selection would favor behaviors which minimize the likelihood of delays. In amphipods of this genus, amplexus occurs before the female molts and copulation occurs within minutes afterwards. The results of field observations support the prediction; it is estimated that 99% of the females are in amplexus when they molt. However, while the physiological constraints that were demonstrated show the advantages of pairing before copulation, it is speculated that amplexus as the specific pairing behavior evolved in response to the selective pressures of the free‐roaming habits of G. palustris.     

Balancing selection response and rate of inbreeding by including genetic relationships in selection decisions

An iterative selection strategy, based on estimated breeding values (EBV) and average relationship among selected individuals, is proposed to optimise the balance between genetic response and inbreeding. Stochastic simulation was used to compare rates of inbreeding and genetic gain with those of other strategies. For a range of heritabilities, population sizes and mating ratios, the iterative strategy, denoted ADJEBV, outperforms other strategies, giving the greatest genetic gain at a given rate of inbreeding and the least breeding at a given genetic gain. Where selection is currently by truncation on the EBV, with a restriction on the number of full-sibs selected, it should be possible to maintain similar levels of genetic gain and inbreeding with a reduction in population size of 10–30%, by changing to the iterative strategy. If performance is measured by the reduction in cumulative inbreeding without losing more than a given amount of genetic gain relative to results obtained under truncation selection on the EBV, then with the EBV based on a family index, the performance of ADJEBV is greater at low heritability, and is generally greater than where EBV are based on individual records. When comparisons of genetic response and inbreeding are made for alternative breeding scheme designs, schemes which give higher genetic gain within acceptable inbreeding levels would usually be favoured. If comparisons are made on this basis, then the selection method used should be ADJEBV, which maximises the genetic gain for a given level of inbreeding. The results indicated that all selection strategies used to reduce inbreeding had very small effects on the variance of gain, and so differences in this respect are unlikely to affect choices among selection strategies. Selection criteria are recommended based on maximising a selection objective which specifies the desired balance between genetic gain and inbreeding.     

Local effects of inbreeding on embryo number and consequences for genetic diversity in Kerguelen mouflon

A classical paradigm in population genetics is that homozygosity or inbreeding affects individual fitness through increased disease susceptibility and mortality, and diminished breeding success. Using data from an insular population of mouflon (Ovis aries) founded by a single pair of individuals, we compare embryo number of ewes with different levels of inbreeding. Contrary to expectations, ewes with the highest levels of homozygosity showed the largest number of embryos. Using two different statistical approaches, we showed that this relationship is probably caused by heterozygosity at specific genes. The genetics of embryo number coupled with cyclic dynamics could play a central role in promoting genetic variation in this population.     

Estimation of heritability and genetic trend in populations at a physiological limit

Summary Limits on physiological processes, though perhaps unknown, must exist. The reported simulations evaluate the effect of a physiological limit on the estimation of genetic parameters and genetic progress. Simulation experiments reveal no change in the estimate of heritability, even for limits as restrictive as 1.5 phenotypic standard deviations above the population mean. However, estimates of additive genetic and environmental variance shrink as limits on performance increase in severity. Simulated physiological limits do not affect the rate of genetic progress. However, absolute measures of estimated genetic change are less than those predicted by response equations.     

Selection in favor of nucleotides G and C diversifies evolution rates and levels of polymorphism at mammalian synonymous sites

The impact of synonymous nucleotide substitutions on fitness in mammals remains controversial. Despite some indications of selective constraint, synonymous sites are often assumed to be neutral, and the rate of their evolution is used as a proxy for mutation rate. We subdivide all sites into four classes in terms of the mutable CpG context, nonCpG, postC, preG, and postCpreG, and compare four-fold synonymous sites and intron sites residing outside transposable elements. The distribution of the rate of evolution across all synonymous sites is trimodal. Rate of evolution at nonCpG synonymous sites, not preceded by C and not followed by G, is approximately 10% below that at such intron sites. In contrast, rate of evolution at postCpreG synonymous sites is approximately 30% above that at such intron sites. Finally, synonymous and intron postC and preG sites evolve at similar rates. The relationship between the levels of polymorphism at the corresponding synonymous and intron sites is very similar to that between their rates of evolution. Within every class, synonymous sites are occupied by G or C much more often than intron sites, whose nucleotide composition is consistent with neutral mutation-drift equilibrium. These patterns suggest that synonymous sites are under weak selection in favor of G and C, with the average coefficient s approximately 0.25/Ne approximately 10(-5), where Ne is the effective population size. Such selection decelerates evolution and reduces variability at sites with symmetric mutation, but has the opposite effects at sites where the favored nucleotides are more mutable. The amino-acid composition of proteins dictates that many synonymous sites are CpGprone, which causes them, on average, to evolve faster and to be more polymorphic than intron sites. An average genotype carries approximately 10(7) suboptimal nucleotides at synonymous sites, implying synergistic epistasis in selection against them.     

Effects of population size and linkage on optimal selection intensity

Following Robertson (1970a) it is generally considered that for mass selection the selected proportion that maximises ultimate response is 0.5. This prediction has been partly tested by different authors. Here we explicitly address the question using computer simulations of selection in finite populations with linkage. The results show that the response achieved is always lower than the one predicted by standard methods, and that optimum selection intensity may be much lower than predicted unless population size is small.     

Density dependence hypotheses and the distribution of fecundity

1. Beja & Palma (2008, Journal of Animal Ecology, 77, doi: 10.1111/j.1365-2656.2007.01312.x) attempt to provide a critical analysis of the effectiveness and limitations of a previously published method (Ferrer et al. 2006, Journal of Animal Ecology, 75, 111-117.) to discriminate between Habitat Heterogeneity Hypothesis and the Individual Adjustment Hypothesis using real data from a Bonelli's eagle Hieraaetus fasciatus population. 2. They conclude that significant and strong correlations between mean and CV or skewness are expected under a biologically plausible assumption about brood size distribution, and that the two hypotheses cannot therefore be distinguished. 3. A major concern we have with their paper centres on this biologically plausible brood-size distribution. They used the same quasi-Poisson distribution of brood sizes (typical for a saturate population under Habitat Heterogeneity Hypothesis) for both families of simulations. So, is not surprising that both groups gave similar results. 4. They argued that this approach was 'empirical', free of theoretical assumptions. But in testing between hypotheses, what we are looking for is precisely the differences among theoretical brood-size distributions predicted under the two hypotheses. 5. Summarizing, with the same mean fecundity at high densities, both hypotheses must have different brood-size distributions. So the use of a single left-skewed distribution, typical of a real saturated population (most likely under Habitat Heterogeneity Hypothesis) in attempts to distinguish between the two hypotheses by re-sampling several times on the same left-skewed distribution, as done by Beja & Palma, is clearly inappropriate.     

Expected early genetic gain from selection for milk yield in dairy cattle

Summary A matrix program to predict short term genetic gain from single trait selection for milk yield was developed. Rate of genetic gain was calculated as the annual change in the mean breeding value of all producing females. Several parameters sets representing various selection policies were used to examine situations pertinent to dairy populations of the United States. Approach to the asymptotic rates of genetic gain within the model varied with the choice of parameters, but even with consistent selection policies, predicted total genetic gain in the first 10 years was only half of the expected from classical theory. Considerable year to year variation in the rate of gain occurred. Early gains were more dependent on female selection decisions than gains during the steady state. In a two-phase model, the approach to the linear rate of gain in the second phase was accelerated by starting with an ongoing improvement program, but considerable delays still existed. Selection for sex- limited traits such as milk yield, which require pedigree selection and a waiting time for progeny test results reached asymptotic rates more slowly than previously assumed.     

Analysis of interspecific competition in perennial plants using Life Table Response Experiments

The impact of interspecific competition is usually measured by its effect upon plant growth, neglecting impacts upon other stages of the life cycle such as fecundity which have a direct influence upon individual fitness and the asymptotic population growth rate of a population (λ). We used parameterized matrix models for three perennial plant species grown with and without interspecific competition to illustrate how the methodology of Life Table Response Experiments (LTRE) can be used to link any change in population dynamics to changes in any part of the life cycle. Plants were herbaceous grassland species grown for two years in a field experiment at Rothamsted Experimental Station, England. Interspecific competition reduced λ by over 90% in all species. Survival and growth were slightly affected by competition whereas plant fecundity was greatly reduced. Nearly all of the observed difference in λ between the competition treatments was explained by the fecundity terms, and more precisely by a large difference in the number of seeds, and a high sensitivity of λ to the germination rate. Whereas most competition studies focus on the measurement of change in individual fitness, our study illustrates how informative it is to take account not only of the effect of competition upon vital rates but also of how different vital rates affect population growth rate.     

Quantitative genetic variation and selection of enzyme activities in Drosophila melanogaster

The nature of the genetic variation for the activity of three enzymes (α-GPD, ME, and SOD) was studied by means of analyses of variance among full-sib and half-sib families. The results presented here indicate that the genetic variation of activity of these enzymes consist primarily of non-additive genetic variance. A moderate level of additive genetic variation was found only for α-GPD activity. We also examined the question whether an association exists between enzyme activities and selection for preadult developmental time. Using the method developed by Lande and Arnold (1983), significant directional selection was observed for α-GPD activity.     

Evolution of late-life fecundity in Drosophila melanogaster

Late-life fecundity has been shown to plateau at late ages in Drosophila analogously to late-life mortality rates. In this study, we test an evolutionary theory of late life based on the declining force of natural selection that can explain the occurrence of these late-life plateaus in Drosophila. We also examine the viability of eggs laid by late-age females and test a population genetic mechanism that may be involved in the evolution of late-life fecundity: antagonistic pleiotropy. Together these experiments demonstrate that (i) fecundity plateaus at late ages, (ii) plateaus evolve according to the age at which the force of natural selection acting on fecundity reaches zero, (iii) eggs laid by females in late life are viable and (iv) antagonistic pleiotropy is involved in the evolution of late-life fecundity. This study further supports the evolutionary theory of late life based on the age-specific force of natural selection.     

Effect of mass selection on the within-family genetic variance in finite populations

Summary The adequacy of an expression for the withinfamily genetic variance under pure random drift in an additive infinitesimal model was tested via simulation in populations undergoing mass selection. Two hundred or one thousand unlinked loci with two alleles at initial frequencies of 1/2 were considered. The size of the population was 100 (50 males and 50 females). Full-sib matings were carried out for 15 generations with only one male and one female chosen as parents each generation, either randomly or on an individual phenotypic value. In the unselected population, results obtained from 200 replicates were in agreement with predictions. With mass selection, within-family genetic variance was overpredicted by theory from the 12th and 4th generations for the 1,000 and 200 loci cases, respectively. Taking into account the observed change in gene frequencies in the algorithm led to a much better agreement with observed values. Results for the distribution of gene frequencies and the withinlocus genetic covariance are presented. It is concluded that the expression for the within-family genetic variance derived for pure random drift holds well for mass selection within the limits of an additive infinitesimal model.     

Habitat geometry, population viscosity and the rate of genetic drift

In all natural populations, individuals located close to one another tend to interact more than those further apart. The extent of population viscosity can have important implications for ecological and evolutionary processes. Here we develop a spatially explicit population model to examine how the rate of genetic drift depends upon both spatial population structure and habitat geometry. The results show that the time to fixation for a new and selectively neutral mutation is dramatically increased in viscous populations. Furthermore, in viscous populations the time to fixation depends critically on habitat geometry. Fixation time for populations of identical size increases markedly as landscape width decreases and length increases. We suggest that similar effects will also be important in metapopulations, with the spatial arrangement of subpopulations and their connectivity likely to determine the rate of drift. We argue that the recent increases in computer power should facilitate major advances in our understanding of evolutionary landscape ecology over the next few years, and suggest that the time is ripe for a unification of spatial population dynamics theory, landscape ecology and population genetics.