Large effects and the infinitesimal model.

The infinitesimal model of quantitative genetics relies on the Central Limit Theorem to stipulate that under additive models of quantitative traits determined by many loci having similar effect size, the difference between an offspring's genetic trait component and the average of their two parents' genetic trait components is Normally distributed and independent of the parents' values. Here, we investigate how the assumption of similar effect sizes affects the model: if, alternatively, the tail of the effect size distribution is polynomial with exponent α<2, then a different Central Limit Theorem implies that sums of effects should be well-approximated by a "stable distribution", for which single large effects are often still important. Empirically, we first find tail exponents between 1 and 2 in effect sizes estimated by genome-wide association studies of many human disease-related traits. We then show that the independence of offspring trait deviations from parental averages in many cases implies the Gaussian aspect of the infinitesimal model, suggesting that non-Gaussian models of trait evolution must explicitly track the underlying genetics, at least for loci of large effect. We also characterize possible limiting trait distributions of the infinitesimal model with infinitely divisible noise distributions, and compare our results to simulations.

Chargaff's second parity rule lies at the origin of additive genetic interactions in quantitative traits to make omnigenic selection possible.

Background: Francis Crick's central dogma provides a residue-by-residue mechanistic explanation of the flow of genetic information in living systems. However, this principle may not be sufficient for explaining how random mutations cause continuous variation of quantitative highly polygenic complex traits. Chargaff's second parity rule (CSPR), also referred to as intrastrand DNA symmetry, defined as near-exact equalities G ≈ C and A ≈ T within a single DNA strand, is a statistical property of cellular genomes. The phenomenon of intrastrand DNA symmetry was discovered more than 50 years ago; at present, it remains unclear what its biological role is, what the mechanisms are that force cellular genomes to comply strictly with CSPR, and why genomes of certain noncellular organisms have broken intrastrand DNA symmetry. The present work is aimed at studying a possible link between intrastrand DNA symmetry and the origin of genetic interactions in quantitative traits. Methods: Computational analysis of single-nucleotide polymorphisms in human and mouse populations and of nucleotide composition biases at different codon positions in bacterial and human proteomes. Results: The analysis of mutation spectra inferred from single-nucleotide polymorphisms observed in murine and human populations revealed near-exact equalities of numbers of reverse complementary mutations, indicating that random genetic variations obey CSPR. Furthermore, nucleotide compositions of coding sequences proved to be statistically interwoven via CSPR because pyrimidine bias at the 3rd codon position compensates purine bias at the 1st and 2nd positions. Conclusions: According to Fisher's infinitesimal model, we propose that accumulation of reverse complementary mutations results in a continuous phenotypic variation due to small additive effects of statistically interwoven genetic variations. Therefore, additive genetic interactions can be inferred as a statistical entanglement of nucleotide compositions of separate genetic loci. CSPR challenges the neutral theory of molecular evolution-because all random mutations participate in variation of a trait-and provides an alternative solution to Haldane's dilemma by making a gene function diffuse. We propose that CSPR is symmetry of Fisher's infinitesimal model and that genetic information can be transferred in an implicit contactless manner.

The infinitesimal model with dominance.

The classical infinitesimal model is a simple and robust model for the inheritance of quantitative traits. In this model, a quantitative trait is expressed as the sum of a genetic and an environmental component, and the genetic component of offspring traits within a family follows a normal distribution around the average of the parents' trait values, and has a variance that is independent of the parental traits. In previous work, we showed that when trait values are determined by the sum of a large number of additive Mendelian factors, each of small effect, one can justify the infinitesimal model as a limit of Mendelian inheritance. In this paper, we show that this result extends to include dominance. We define the model in terms of classical quantities of quantitative genetics, before justifying it as a limit of Mendelian inheritance as the number, M, of underlying loci tends to infinity. As in the additive case, the multivariate normal distribution of trait values across the pedigree can be expressed in terms of variance components in an ancestral population and probabilities of identity by descent determined by the pedigree. Now, with just first-order dominance effects, we require two-, three-, and four-way identities. We also show that, even if we condition on parental trait values, the "shared" and "residual" components of trait values within each family will be asymptotically normally distributed as the number of loci tends to infinity, with an error of order 1/M. We illustrate our results with some numerical examples.

Adaptation of a quantitative trait to a changing environment: New analytical insights on the asexual and infinitesimal sexual models.

Predicting the adaptation of populations to a changing environment is crucial to assess the impact of human activities on biodiversity. Many theoretical studies have tackled this issue by modeling the evolution of quantitative traits subject to stabilizing selection around an optimal phenotype, whose value is shifted continuously through time. In this context, the population fate results from the equilibrium distribution of the trait, relative to the moving optimum. Such a distribution may vary with the shape of selection, the system of reproduction, the number of loci, the mutation kernel or their interactions. Here, we develop a methodology that provides quantitative measures of population maladaptation and potential of survival directly from the entire profile of the phenotypic distribution, without any a priori on its shape. We investigate two different systems of reproduction (asexual and infinitesimal sexual models of inheritance), with various forms of selection. In particular, we recover that fitness functions such that selection weakens away from the optimum lead to evolutionary tipping points, with an abrupt collapse of the population when the speed of environmental change is too high. Our unified framework allows deciphering the mechanisms that lead to this phenomenon. More generally, it allows discussing similarities and discrepancies between the two systems of reproduction, which are ultimately explained by different constraints on the evolution of the phenotypic variance. We demonstrate that the mean fitness in the population crucially depends on the shape of the selection function in the infinitesimal sexual model, in contrast with the asexual model. In the asexual model, we also investigate the effect of the mutation kernel and we show that kernels with higher kurtosis tend to reduce maladaptation and improve fitness, especially in fast changing environments.

The "New Synthesis".

When Mendel's work was rediscovered in 1900, and extended to establish classical genetics, it was initially seen in opposition to Darwin's theory of evolution by natural selection on continuous variation, as represented by the biometric research program that was the foundation of quantitative genetics. As Fisher, Haldane, and Wright established a century ago, Mendelian inheritance is exactly what is needed for natural selection to work efficiently. Yet, the synthesis remains unfinished. We do not understand why sexual reproduction and a fair meiosis predominate in eukaryotes, or how far these are responsible for their diversity and complexity. Moreover, although quantitative geneticists have long known that adaptive variation is highly polygenic, and that this is essential for efficient selection, this is only now becoming appreciated by molecular biologists-and we still do not have a good framework for understanding polygenic variation or diffuse function.

Evolutionary dynamics of complex traits in sexual populations in a heterogeneous environment: how normal?

When studying the dynamics of trait distribution of populations in a heterogeneous environment, classical models from quantitative genetics choose to look at its system of moments, specifically the first two ones. Additionally, in order to close the resulting system of equations, they often assume the local trait distributions are Gaussian [see for instance Ronce and Kirkpatrick (Evolution 55(8):1520-1531, 2001. https://doi.org/10.1111/j.0014-3820.2001.tb00672.x.37 )]. The aim of this paper is to introduce a mathematical framework that follows the whole trait distribution (without prior assumption) to study evolutionary dynamics of sexually reproducing populations. Specifically, it focuses on complex traits, whose inheritance can be encoded by the infinitesimal model of segregation (Fisher in Trans R Soc Edinb 52(2):399-433, 1919. https://doi.org/10.1017/S0080456800012163 ). We show that it allows us to derive a regime in which our model gives the same dynamics as when assuming Gaussian local trait distributions. To support that, we compare the stationary problems of the system of moments derived from our model with the one given in Ronce and Kirkpatrick (Evolution 55(8):1520-1531, 2001. https://doi.org/10.1111/j.0014-3820.2001.tb00672.x.37 ) and show that they are equivalent under this regime and do not need to be otherwise. Moreover, under this regime of equivalence, we show that a separation bewteen ecological and evolutionary time scales arises. A fast relaxation toward monomorphism allows us to reduce the complexity of the system of moments, using a slow-fast analysis. This reduction leads us to complete, still in this regime, the analytical description of the bistable asymmetrical equilibria numerically found in Ronce and Kirkpatrick (Evolution 55(8):1520-1531, 2001. https://doi.org/10.1111/j.0014-3820.2001.tb00672.x.37 ). More globally, we provide explicit modelling hypotheses that allow for such local adaptation patterns to occur.

Canalization and Robustness in Human Genetics and Disease.

Canalization refers to the evolution of populations such that the number of individuals who deviate from the optimum trait, or experience disease, is minimized. In the presence of rapid cultural, environmental, or genetic change, the reverse process of decanalization may contribute to observed increases in disease prevalence. This review starts by defining relevant concepts, drawing distinctions between the canalization of populations and robustness of individuals. It then considers evidence pertaining to three continuous traits and six domains of disease. In each case, existing genetic evidence for genotype-by-environment interactions is insufficient to support a strong inference of decanalization, but we argue that the advent of genome-wide polygenic risk assessment now makes an empirical evaluation of the role of canalization in preventing disease possible. Finally, the contributions of both rare and common variants to congenital abnormality and adult onset disease are considered in light of a new kerplunk model of genetic effects.

Accounting for heteroscedasticity and censoring in chromosome partitioning analyses.

A fundamental assumption in quantitative genetics is that traits are controlled by many loci of small effect. Using genomic data, this assumption can be tested using chromosome partitioning analyses, where the proportion of genetic variance for a trait explained by each chromosome (h2 c ), is regressed on its size. However, as h2 c -estimates are necessarily positive (censoring) and the variance increases with chromosome size (heteroscedasticity), two fundamental assumptions of ordinary least squares (OLS) regression are violated. Using simulated and empirical data we demonstrate that these violations lead to incorrect inference of genetic architecture. The degree of bias depends mainly on the number of chromosomes and their size distribution and is therefore specific to the species; using published data across many different species we estimate that not accounting for this effect overall resulted in 28% false positives. We introduce a new and computationally efficient resampling method that corrects for inflation caused by heteroscedasticity and censoring and that works under a large range of dataset sizes and genetic architectures in empirical datasets. Our new method substantially improves the robustness of inferences from chromosome partitioning analyses.

Introgression of a Block of Genome Under Infinitesimal Selection.

Adaptive introgression is common in nature and can be driven by selection acting on multiple, linked genes. We explore the effects of polygenic selection on introgression under the infinitesimal model with linkage. This model assumes that the introgressing block has an effectively infinite number of loci, each with an infinitesimal effect on the trait under selection. The block is assumed to introgress under directional selection within a native population that is genetically homogeneous. We use individual-based simulations and a branching process framework to compute various statistics of the introgressing block, and explore how these depend on parameters such as the map length and initial trait value associated with the introgressing block, the genetic variability along the block, and the strength of selection. Our results show that the introgression dynamics of a block under infinitesimal selection are qualitatively different from the dynamics of neutral introgression. We also find that, in the long run, surviving descendant blocks are likely to have intermediate lengths, and clarify how their length is shaped by the interplay between linkage and infinitesimal selection. Our results suggest that it may be difficult to distinguish the long-term introgression of a block of genome with a single, strongly selected, locus from the introgression of a block with multiple, tightly linked and weakly selected loci.

Establishment in a new habitat by polygenic adaptation.

Maladapted individuals can only colonise a new habitat if they can evolve a positive growth rate fast enough to avoid extinction, a process known as evolutionary rescue. We treat log fitness at low density in the new habitat as a single polygenic trait and use the infinitesimal model to follow the evolution of the growth rate; this assumes that the trait values of offspring of a sexual union are normally distributed around the mean of the parents' trait values, with variance that depends only on the parents' relatedness. The probability that a single migrant can establish depends on just two parameters: the mean and genetic variance of the trait in the source population. The chance of success becomes small if migrants come from a population with mean growth rate in the new habitat more than a few standard deviations below zero; this chance depends roughly equally on the probability that the initial founder is unusually fit, and on the subsequent increase in growth rate of its offspring as a result of selection. The loss of genetic variation during the founding event is substantial, but highly variable. With continued migration at rate M, establishment is inevitable; when migration is rare, the expected time to establishment decreases inversely with M. However, above a threshold migration rate, the population may be trapped in a 'sink' state, in which adaptation is held back by gene flow; above this threshold, the expected time to establishment increases exponentially with M. This threshold behaviour is captured by a deterministic approximation, which assumes a Gaussian distribution of the trait in the founder population with mean and variance evolving deterministically. By assuming a constant genetic variance, we also develop a diffusion approximation for the joint distribution of population size and trait mean, which extends to include stabilising selection and density regulation. Divergence of the population from its ancestors causes partial reproductive isolation, which we measure through the reproductive value of migrants into the newly established population.

The infinitesimal model: Definition, derivation, and implications.

Our focus here is on the infinitesimal model. In this model, one or several quantitative traits are described as the sum of a genetic and a non-genetic component, the first being distributed within families as a normal random variable centred at the average of the parental genetic components, and with a variance independent of the parental traits. Thus, the variance that segregates within families is not perturbed by selection, and can be predicted from the variance components. This does not necessarily imply that the trait distribution across the whole population should be Gaussian, and indeed selection or population structure may have a substantial effect on the overall trait distribution. One of our main aims is to identify some general conditions on the allelic effects for the infinitesimal model to be accurate. We first review the long history of the infinitesimal model in quantitative genetics. Then we formulate the model at the phenotypic level in terms of individual trait values and relationships between individuals, but including different evolutionary processes: genetic drift, recombination, selection, mutation, population structure, …. We give a range of examples of its application to evolutionary questions related to stabilising selection, assortative mating, effective population size and response to selection, habitat preference and speciation. We provide a mathematical justification of the model as the limit as the number M of underlying loci tends to infinity of a model with Mendelian inheritance, mutation and environmental noise, when the genetic component of the trait is purely additive. We also show how the model generalises to include epistatic effects. We prove in particular that, within each family, the genetic components of the individual trait values in the current generation are indeed normally distributed with a variance independent of ancestral traits, up to an error of order 1∕M. Simulations suggest that in some cases the convergence may be as fast as 1∕M.

Genetic mapping and genomic selection using recombination breakpoint data.

The correct models for quantitative trait locus mapping are the ones that simultaneously include all significant genetic effects. Such models are difficult to handle for high marker density. Improving statistical methods for high-dimensional data appears to have reached a plateau. Alternative approaches must be explored to break the bottleneck of genomic data analysis. The fact that all markers are located in a few chromosomes of the genome leads to linkage disequilibrium among markers. This suggests that dimension reduction can also be achieved through data manipulation. High-density markers are used to infer recombination breakpoints, which then facilitate construction of bins. The bins are treated as new synthetic markers. The number of bins is always a manageable number, on the order of a few thousand. Using the bin data of a recombinant inbred line population of rice, we demonstrated genetic mapping, using all bins in a simultaneous manner. To facilitate genomic selection, we developed a method to create user-defined (artificial) bins, in which breakpoints are allowed within bins. Using eight traits of rice, we showed that artificial bin data analysis often improves the predictability compared with natural bin data analysis. Of the eight traits, three showed high predictability, two had intermediate predictability, and two had low predictability. A binary trait with a known gene had predictability near perfect. Genetic mapping using bin data points to a new direction of genomic data analysis.

Modelling wild-domestic interbreeding: how selection on a quantitative trait affects gene flow at a neutral locus.

Interbreeding between domesticated individuals and their wild conspecifics occurs in a range of species. The rate of gene flow into the wild population, estimated using genetic markers, is often smaller than the fraction of immigrants, as immigrants and their descendants generally have lower relative fitness. Here the difference between one-way migration rate and gene flow (effective migration rate) is explored using quantitative genetic simulations. We model a trait undergoing stabilizing selection in the recipient population, influenced by an infinite number of loci, each with small effect. Immigrants have suboptimal trait values, and differ in allele frequency at an unlinked neutral marker locus. We derive an analytical approximation for the effective migration rate, and show that in the limiting case of low migration rates, the ratio between effective and actual migration rate approximately equals the ratio between mean fitness of immigrants and in the admixed population. This ignores indirect selection on the marker locus within the admixed population due to covariance with the trait value, which may be substantial when the genetic difference between the populations is large. For higher migration rates, the gene flow/migration ratio rises with increasing migration rate, inflating the rate of gene flow.

BDNF contributes to the genetic variance of milk fat yield in german holstein cattle.

The gene encoding the brain-derived neurotrophic factor (BDNF) has been repeatedly associated with human obesity. As such, it could also contribute to the regulation of energy partitioning and the amount of secreted milk fat during lactation, which plays an important role in milk production in dairy cattle. Therefore, we performed an association study using estimated breeding values (EBVs) of bulls and yield deviations of German Holstein dairy cattle to test the effect of BDNF on milk fat yield (FY). A highly significant effect (corrected p-value = 3.362 × 10(-4)) was identified for an SNP 168 kb up-stream of the BDNF transcription start. The association tests provided evidence for an additive allele effect of 5.13 kg of fat per lactation on the EBV for milk FY in bulls and 6.80 kg of fat of the own production performance in cows explaining 1.72 and 0.60% of the phenotypic variance in the analyzed populations, respectively. The analyses of bulls and cows consistently showed three haplotype groups that differed significantly from each other, suggesting at least two different mutations in the BDNF region affecting the milk FY. The FY increasing alleles also had low but significant positive effects on protein and total milk yield which suggests a general role of the BDNF region in energy partitioning, rather than a specific regulation of fat synthesis. The results obtained in dairy cattle suggest similar effects of BDNF on milk composition in other species, including man.