Streblospio benedicti, females make either many small eggs that develop into complex larvae that feed in the plankton or few large eggs that develop into benthic juveniles without having to feed as larvae. By isolating the contributions of maternal, paternal, and zygotic genotype to larval traits, we determined that larval anatomical structures are governed by the offspring genotype at a small number of large‐effect loci. Larval size is not shaped by the larva's own genotype but instead depends on loci that act in the mother, and at two genomic locations, by loci that act in the father. The overall phenotype of each larva thus depends on three separate genomes, and a population's response to selection on larval traits will reflect the interactions among them.The evolutionary potential of a population is shaped by the genetic architecture of its life‐history traits. Early‐life phenotypes are influenced by both maternal and offspring genotype, and efforts to understand life‐history evolution therefore require consideration of the interactions between these separate but correlated genomes. We used a four‐generation experimental pedigree to estimate the genetic architecture of early‐life phenotypes in a species with dramatic variation in larval size and morphology. In the polychaete annelid  Trade‐offs between the number and quality of offspring sit at the heart of life‐history theory, dictated by finite maternal resources that can be allocated in different ways. Mothers can produce small numbers of large, well‐provisioned offspring, or large numbers of small, poorly provisioned offspring. Ecological factors govern the optimal position of a population along this trade‐off front Vance .Heliocidaris sea urchins  eggs that develop into characteristic planktonic larvae, which feed on unicellular algae for a period of days or weeks before undergoing a metamorphosis and settling into the benthic mud. Other females produce a much smaller number of much larger eggs . These eggs develop into large larvae that do not need to feed and can immediately metamorphose and settle. In both modes of development, embryogenesis and larval morphogenesis occur in brood pouches on the female's back, and offspring are released into the water as fully formed planktonic larvae that have to feed (planktotrophs) or metamorphosis‐competent larvae that do not require planktonic food, provisioned by yolk (lecithotrophs).To understand how genetic architecture shapes the evolution of larval life‐history traits, in the context of qualitative transitions in larval form, we have developed an experimental model in the estuarine polychaete ng Levin . Some S.S. benedicti larvae differ substantially in form , made from crosses among 115 G2 females and 53 G2 males. Individual G2 males sired an average of 2 G3 families , 2 (n = 12), 3 (n = 13), 4 (n = 8)). The G2s used as parents are a subset of the 183 females and 58 males described in our earlier study , using the sim.geno function in rqtl , number of defensive chaetae, length of the longest chaeta, and presence or absence of anal cirri.To phenotype G3s to genotypes from the G2s, we employed mixed‐effect models that account for family structure and accommodate parental genetic effects. We performed a series of genome‐wide linkage scans to detect QTL. We subsequently tested the significant QTL for their transmission mode , dominance, and epistasis.To map quantitative‐trait loci (QTL) underlying trait variation in the G3 phenotype vector y asTo search the genome for QTL, we step through each genetic marker position, modeling the GQM vector holds the count of lecithotroph alleles at the marker in the G2 mother (0 to 2), the QP vector holds the count of lecithotroph alleles at the marker in the G2 father, and betas are the additive effects of maternal and paternal alleles at the marker. We model the family structure with random effects for the mother and father. Incidence matrices M and P connect each larva with the appropriate entry in the γs, vectors of random effects associated with each parent. Here the random effect of mother accounts for diverse sources of similarity among full sibs; these include background maternal and zygotic genetic effects, maternal environmental effects, and effects due to early development in a common environment. The paternal effect accounts for similarity among half sibs, which should only be present if there are zygotic or paternal genetic effects. ε is the residual error.Here, μ is the global mean vector, the We compare the fit of this model to a no‐QTL model,This model is nested within the previous model and has two fewer parameters. We use twice the difference between the models in log likelihood as a test statistic. This is the canonical likelihood ratio test statistic, asymptotically distributed as a chi‐square under the null hypothesis. For genome scans, we use permutations, described below, to generate genome‐wide null statistics.3 larva,T subscript indicates total allele count (0 to 4). We again test this model against (2), the no‐QTL null model.For purely zygotic‐effect inheritance, we can achieve higher power by using a single beta, as the effects of maternal and parental alleles should be identical. Additive zygotic effects can therefore be modeled as the effect of the sum of alleles across the two parents of each GTo directly test for parental effects at QTL detected by the tests above, we compare models 1 and 3 to one another, testing whether the model fit is significantly improved by allowing allelic effects to differ according to the parental sex. Rejection of model 3 is evidence of maternal effects, paternal effects, both kinds of effects, or a combination of parental and zygotic effects. To test a purely maternal‐effect model at such loci, we compare model 1 to a reduced model with no effect of paternal genotype at the locus:Rejection of model 4 indicates that the paternal genotype at the marker matters, rejecting a purely maternal‐effect model. Similarly, a paternal‐genotype‐only model tests whether the maternal genotype matters:2 male and G2 female genotypes independently, preserving the underlying full‐ and half‐sib relatedness among G3s. The permutations are also stratified by G2 family , as in Zakas et al.  then test for additional Qβ terms. Note that we use permutations to define genome‐wide significance for linkage scans (step 1 in the testing framework), and for subsequent tests involving the loci detected in step 1 use uncorrected p = 0.05 threshold from likelihood ratio tests.When the initial genome scan identified a significant locus , we then incorporated the genotype at that locus into a second genome scan, using the genetic model for the locus identified by comparisons among models 1, 3, and 4. This forward search continued until no additional loci exceeded a genome‐wide permutation‐based threshold of 2 male hemizygosity, unknown sexes among the G3 larvae, and unbalanced G2 genotypes due to the direction of the pedigree‐founding cross. As the sole F1 male in the pedigree carried the X chromosome from its planktotroph mother, all G2s females carry at least one planktotroph X chromosome. Denoting the lecithotroph allele L and planktotroph allele P, G2 females can never be LL homozygotes at X‐linked loci. We can nevertheless test for linkage to X‐chromosome markers by comparing the full model to the no‐QTL model (1 vs. 2), and we can test for parental effects by comparing the full model to the one‐parent‐only model (1 vs. 4).The analyses described above were applied to autosomal markers. The models are not precisely suited to the transmission genetics of the X chromosome, because of G2 population, there are four types of crosses  with X‐linked allelic affects weighted according to the sex and cross. For X‐linked markers in our Gses Fig. . Under t2 phenotypes found no evidence for sex‐linkage or for differences between sexes in any larval phenotypes , number of larval chaetae (on one side of the larva), length of the longest larval chaeta, and presence or absence of anal cirri. Number of chaetae is a mixture of a point mass at zero (24.8% of larvae) and distribution with a mode at 4 chaetae and a long right tail . We therefore modeled number of chaetae as two separate traits: presence/absence of chaetae as a binary trait, fit with a generalized linear mixed model approach , and Box‐Cox‐transformed chaetae number conditional on their presence, using the standard approach. These two approaches to the data yielded extremely similar results, and so we proceeded by combining the two models . The result is a mixed‐model analog of the two‐part model of Broman . The larTo test for directional polygenic effects, under the assumption that loci that differ between lecithotrophs and plankotrophs will tend to affect larval traits in a predictable direction, we used expected autosomal ancestry  as a predictor in a mixed effect model that also includes the genome‐wide significant loci. We consider both biparental ancestry  and uniparental ancestry . We estimate ancestry as the fraction of lecithotroph alleles at autosomal markers, excluding the chromosomes that carry genome‐wide significant loci for the trait . This exclusion step is necessary because genotypes across the chromosomes with major‐effect loci are correlated by linkage with the major‐effect locus genotype, and so the contribution of that chromosome to polygenic effects is confounded with its major‐locus effect.3 larvae. Instead, we tested for dominance by asking whether the fit of the additive zygotic model was significantly improved by addition of a coefficient for the expected heterozygosity of the larva .For loci inferred to act via parental genetic effect, we tested for dominance effects by incorporating a term for parental heterozygosity. For loci inferred to act via zygotic effects, we cannot use this direct approach because we lack individual‐level genotypes for the G3 phenotypes, we integrated G2 and G3 phenotype data into a single genome scan for zygotic effects by summing the G2 and G3 log likelihoods at each marker under an additive zygotic model and testing against the summed log likelihoods of the relevant no‐QTL models. We also performed this genome scan after incorporating each of the established QTLs into the model, with dominance where appropriate, as covariates. Likelihoods for the G2s were estimated under simple fixed effect models. Note that this approach allows for the allelic effects of a marker to differ between the generations. Significance thresholds were estimated by structured permutation of the male and female G2 genotypes, preserving relatedness between each G2 animal and its progeny.To improve mapping resolution beyond that achieved with Gh2) using mixed‐effect parent‐offspring regressions for traits with continuous distributions . For additive zygotic‐effect inheritance, the slope of the regression of offspring phenotypes on the average phenotype of the parents, accounting for family structure with random effects, is an estimator of h2. We tested whether the heritabilities were different from zero by comparing the regressions to a null model with no effect of parental phenotype, and we tested for differences between maternal and paternal estimates by including the two parental phenotypes as separate fixed effects. Note that our heritability estimates apply only to transmission of traits from G2 to G3 in this specific cross. To estimate whether the detected QTLs account for the estimated heritability, we incorporated the QTLs as fixed effects in the regressions and compared these to no‐QTL regressions. We tested whether heritabilities were different from zero using likelihood ratio tests.We estimated narrow‐sense heritability  are genetically heterogeneous and the phenotypes of their G3 offspring therefore permit us to detect loci acting both parentally in the G2s and zygotically in the G3s.We studied the great‐grandchildren from a cross of a planktotrophic female and a lecithotrophic male. The grandchildren . These results imply that we have detected the loci responsible for the bulk of the additive genetic basis for these traits.We used mixed‐model parent‐offspring regression to estimate the heritability of chaetae length and number Table . We esti2 and G3 generations, we can achieve higher power to detect and localize zygotic‐effect quantitative trait loci. We found that the genetic signals added and background noise largely cancelled, leaving the few significant QTL defined more clearly , consistent with paternal effects. Both loci also nominally reject the uniparental paternal‐effect model , suggesting maternal effects. The direction of the inferred effects depends on the parent; lecithotroph alleles carried by mothers are associated with larger larvae, and the same alleles carried by fathers are associated with smaller progeny . However, this complication should not result in a parent‐dependent change in the sign of an allelic effect, such as we observe on the X‐linked locus affecting larval size.3 data do not allow us to simultaneously estimate maternal, paternal, and zygotic effects, but we can test for zygotic effects in a setting that controls for maternal and paternal effects. Specifically, the G2 generation varies zygotically but all individuals share heterozygous F1 parents and so parental‐effect variation is absent. We found no evidence for direct effects of the loci on X (p = 0.88) or chromosome 9 (p = 0.21) in an analysis of 240 G2 larvae , the fraction of remaining maternal genome derived from the lecithotrophic founder explains additional variation in larval size . Wild S. benedicti: its multilocus basis guarantees that parental types will be exceedingly rare among segregating cross progeny, and most advanced‐generation hybrids will be phenotypically intermediate. The genetic architecture therefore implies that the absence of intermediates in nature reflects strong selection against advanced‐generation hybrids.This result may in part be down to numbers. Although our study includes thousands of individuals, they come from only 115 full‐sib families. Particular homozygous genotype combinations are rare in this collection. For example, only three of the families involve crosses of a male and a female that are both homozygous LL at the chromosome 5 locus affecting cirri. For traits affected by multiple loci, particular parental genotype combinations may be altogether absent. This reinforces an important feature of the genetic architecture of larval life‐history in 1s, whose autosomal genomes should be equivalent: F1s with lecithotroph mothers often lacked chaetae entirely, while those with planktotroph mothers almost always had chaetae. The findings from reciprocal F1s suggest that we may be missing maternal effect loci.In addition, our findings are partly inconsistent with previous work that studied inheritance of larval traits using classical line‐cross methods, comparing phenotype distributions in reciprocal crosses and backcrosses rather than tracking individual genotypes . To test the hypothesis of a polygenic epistatic effect that manifests only at the extremes, we tested the G3 data for quadratic and cubic effects of maternal genetic background and found no significant effects. Thus, if present, maternal‐effect epistasis is likely to involve higher order interactions that are absent or rare in the G3 population. In nature, however, most hybrids will be F1s, with purely lecithotroph or planktotroph maternal backgrounds, and these animals will most often backcross to the locally common form rather than intercross with other hybrids. We therefore anticipate that higher‐order epistasis, manifest only in “pure” genetic backgrounds, could play an important role in inter‐population gene flow in wild S. benedicti.The second class of explanation for missing maternal effects invokes epistasis, specifically genes that only show their effects when a large fraction of the maternal genome comes from a single background. In reciprocal FStreblospio benedicti permits several possible mechanisms of paternal effect. Because our larval size trait is measured after embryogenesis, it may reflect both prezygotic variation, such as egg size, and postzygotic variation, such as differential allocation of yolk resources to growth and metabolism.One surprising result of our analysis is the discovery of paternal‐effect loci acting on larval size. This finding is consistent with our earlier line‐cross study of larval size (Zakas and Rockman S. benedicti males and females do not have to interact to mate; males release spermatophores onto the benthic mud and females retrieve them and store sperm for subsequent fertilizations. Plausible mechanisms for paternal effects in this case include effects of spermatophore components on oogenesis. During laboratory matings, males and females are paired in small arenas for days or weeks, during which time water‐borne chemical signaling between male and female is also possible. A suggestion of signaling between adults comes from an account of experimental crosses between genetically divergent Streblospio populations and species: incubated with distantly related females, S. benedicti males stopped producing spermatophores, even though they readily produce spermatophores when housed by themselves (Schulze et al. Although egg‐size is superficially a purely maternal trait, females may alter oogenesis in response to genetically variable signals from males. Streblospio is genetic variation in imprinting, or parent‐of‐origin‐dependent gene expression. Imprinting is fundamentally different from ordinary parental effects; the former involves gene expression in the offspring and the latter gene expression in the parents. Nevertheless, genotype‐dependent imprinting generates a genotype‐by‐parent‐of‐origin interaction whose phenotypic effects can mimic antagonistic maternal and paternal effects. Alleles at the two paternal‐effect loci may themselves be differentially imprinted, or they may act in trans to differentially regulate imprinting of other loci. To test imprinting models with transmission genetics, we would require additional data on the genotypes of individual G3s and their parental origins (Hager et al. Alternatively, paternal genetic variation may influence larval size through postzygotic effects on growth and metabolism. Although such paternal effects could be mediated directly by sperm‐borne cytoplasmic components (Seidel et al. The larvae of marine invertebrates provide a rich testing ground for models of life‐history evolution, with major transitions between planktotrophy and lecithotrophy a frequent motif Thorson  and poweBoth authors designed and conducted the experiments, performed the analyses, and wrote the manuscript.All data discussed in the manuscript are provided to readers in Supplementary File 1, along with all scripts required to reproduce the results and figures.We have no conflicts of interest to disclose.Associate Editor: T. FlattHandling Editor: T. ChapmanSupplementary MaterialClick here for additional data file.Supplementary MaterialClick here for additional data file.Table S1. G3 linkage scan results.Table S2. Heritability analyses from parent‐offspring regressions.Table S3. Joint G2‐G3 additive zygotic linkage scan results.Click here for additional data file.


We consider an entropic distance analog quantity based on the density of the Gini index in the Lorenz map, i.e., gintropy. Such a quantity might be used for pairwise mapping and ranking between various countries and regions based on income and wealth inequality. Its generalization to f-gintropy, using a function of the income or wealth value, distinguishes between regional inequalities more sensitively than the original construction. The study of social and economic inequality seems to belong to social sciences and humanities. However, based mostly on economic and econophysics studies, analyses of income and wealth data in terms of mathematical quantities such as the Lorenz curve  and the Given that most interesting distributions have infinite support, the norms on their space are not equivalent. That is why there are numerous measures of distance between distributions. Clearly, the Kullback–Leibler divergence is the most popular one. It is well known that to produce a good estimate of the KL divergence using a limited sample is not easy. That is one of the reasons why several generalization and particular alternatives have been developed. Some are for a specific family of distributions ,11; otheCertainly, the Gini index is only one among many possible approaches and does not replace the study of the distributions themselves . In thisIn an earlier work we introduced this quantity called gintropy  and demox, with its monotonic function, f-gintropy concepts, e.g., using x, describe in a later section.In this paper we attempt to generalize this concept to a gintropic distance measure between two PDFs, making it possible to draw a distance graph between countries with varying inequality. By doing so we also investigate the generalization potential in replacing the obvious variable, the income value This article is constructed as follows: First, we review the definitions of gintropy and f-gintropy, controlling their entropy-like properties. Then we construct the f-gintropic divergence measure in analogy to the entropic divergence and study its basic mathematical properties. Finally, based on available data on regional and national income distributions, we develop an inequality distance map. We suggest using these maps as a ”first sight” tool for judging the real effect of inequalities on the population parts near the average.G in terms of cumulative distributions of the population and the wealth or income possessed by them. These cumulatives,The Gini index is a number between zero and one characterizing the inequality inherent in income and wealth distributions. Its definition,The gintropy, As such the The function rated in .x. Moreover, based on this, we construct an entropic divergence measure for the purpose of using it as an ”inequality distance” between PDFs based on their generalized Lorenz curve behavior, comprised in the quantity f-gintropy.In the present paper we attempt to generalize this construction to a possible use of another quantity’s cumulative tail more integral than the original variable, P and do not use any subscript related to it. Later, when considering two different PDFs and constructing their f-gintropic distance measure, we shall retain such indices.Let f be a monotonic, non-negative differentiable function with Let Equation . Invertif-Gini index is constructed based on the geometry of the Lorenz curve. First, one easily proves thatThe Second, the corresponding Gini index is defined as the difference between these two integralsLet us first prove the statement about the sum in Equation . The firx and y among each other,The second integral can be rewritten by exchanging the dummy integration variables Then we interchange the order of integration to arrive atIndeed, the sum of Equations  and 11)11) becomHere, obviously the two integrals inside the square bracket can be unified in a single integral proving thatInspecting a general Lorenz curve drawn in the unit square on the Equation , as the We are interested further in its equivalent forms. We would like to view he forms  and 9)  Gf also Exchanging the integration variables again and reordering the integrations, an equivalent expression emergesNow we fix a few requirements for the function quations  and 15)f(x). To This agrees with the original definition for Based on the above, it is natural to introduce f-gintropy asIt is never negative, It is zero for It is maximal at the x or It has a single maximum only, either as a function of The f-gintropy, This quantity has the following properties, based on the already discussed restrictions on x, when We give a short analysis of the above statements as follows.f for all Again, due to the monotonity of x-derivative,Let us turn now to the discussion of concavity. For that purpose we calculate the first and second derivatives of gintropy w.r.s.p to This quantity changes sign exactly where The f, we have Due to the strict monotonity of Let us recall that the Gini index and its generalization can be written as an integral over the gintropy, f-entropy due to Csiszár replaces the logarithm of the probability with a general function in the entropy formula, we generalize the tail cut integral of the income to a function of it. The latter function will be restricted, as discussed later.We built the f-gintropic divergence measure following the construction pattern of the Kullback–Leibler divergence. It is also in line with the idea how Csiszár’s f-divergence has been introduced ,25. For Instead of using the PDFs, we suggest applying a similar formula to the gintropies. Since the original gintropy function is not normalized, we use its normalized versionThen s instead of the logarithm with It is straightforward to generalize further by using a general convex function Analogous to the definition of mutual information, the mutual gintropy, We will exemplify the use of gintropic distance reconsidering the income distribution data studied in . Data foOne can learn from population biology that in many cases the use of the probability density function is not the best methodology to illustrate the relevant differences in abundance . For hea society ,33.Gintropic distances can be calculated using the well-known Kullback–Leibler divergences for the Approximating the above integral from the interpolated experimental data for One can now go further and test whether the The corresponding normalized gintropy is proportional to the Tsallis q-entropy form :(30)σ^q we have demonstrated that: (i) the gintropy emphasizes data near the average income, and (ii) the use of weight functions rising steeper than


Nkx2-5+/− mice from two inbred strain crosses illustrates the fundamental role of epistasis. Modifier genes raise or lower the risk of specific defects via pairwise (G×GNkx) and higher-order (G×G×GNkx) interactions with Nkx2-5. Their effect sizes correlate with the severity of a defect. The risk loci for mild, atrial septal defects exert predominantly small G×GNkx effects, while the loci for severe, atrioventricular septal defects exert large G×GNkx and G×G×GNkx effects. The loci for moderately severe ventricular septal defects have intermediate effects. Interestingly, G×G×GNkx effects are three times more likely to suppress risk when the genotypes at the first two loci are from the same rather than different parental inbred strains. This suggests the genetic coadaptation of interacting G×G×GNkx loci, a phenomenon that Dobzhansky first described in Drosophila. Thus, epistasis plays dual roles in the pathogenesis of congenital heart disease and the robustness of cardiac development. The empirical results suggest a relationship between the fitness cost and genetic architecture of a disease phenotype and a means for phenotypic robustness to have evolved.In newborns, severe congenital heart defects are rarer than mild ones. This epidemiological relationship between heart defect severity and incidence lacks explanation. Here, an analysis of ~10,000  Congenital heart disease (CHD) is recognized as the most common birth defect, but cardiac malformation actually encompasses anatomically distinct phenotypes in which the severity and incidence of a defect are inversely related. For example, secundum atrial septal defects (ASD), which are compatible with survival into middle age, are three times more common than atrioventricular septal defects (AVSD), which can cause death during infancy a 1,2]. ,2. 1,2].NKX2-5 mutation, which causes isolated CHD, is most commonly associated with atrial septal defects, although severe defects such as tetralogy of Fallot and double outlet right ventricle were reported [Questions regarding how the fitness cost of a defect relates to the robustness of the underlying pathways are practical and fundamental. An appreciation of the genetic architecture of a heart defect—i.e., the genes involved, alleles, interactions, and their effects—could inform genetic risk models and strategies for prevention. The genetic architecture encompasses more than the proximate cause of disease. As a rule, the same deleterious mutation in different individuals can manifest as a mild, moderate, or severe defect or no defect at all. The relationship holds in general for causes of syndromic and isolated CHD. For example, despite the well-known associations of trisomy 21 and 22q11.2 deletion with atrioventricular septal defects and tetralogy of Fallot, respectively, their most common cardiac phenotypes are structurally normal heart or an atrial or ventricular septal defect ,4. Likewreported .Modifier genes affect the phenotypic outcome . GeneticNkx2-5+/− mice, a model of non-syndromic human CHD, demonstrate that the incidence of specific defects can vary between genetic backgrounds [Nkx2-5+/− mice in the C57BL/6N background or the second-generation (F2) offspring of backcrosses to the parental strains or intercrosses (F1×F1). The results suggest a role in CHD genetics for heterosis, a well-described phenomenon in agricultural genetics in which heterozygosity across the genome increases the average level of robustness in a population [Nkx2-5 mutation and hence the risks of anatomically distinct malformations [Inbred strain crosses of kgrounds . We prevpulation . Furtherrmations ,10.Drosophila [C. elegans [Epistasis contributes quantifiably to fitness traits in simple organisms, such as yeast , Drosophosophila , and C.  elegans . On the  elegans ,15. Why  elegans . One rea elegans .Nkx2-5+/− mice from two different inbred strain crosses. Each QTL represents a pairwise interaction between a modifier gene and Nkx2-5 (G×GNkx). In addition, we examined the non-linear effects of higher-order interactions between two QTLs and Nkx2-5 (G×G×GNkx). Empirical data are sorely required regarding the nature of genetic risk in CHD. More fundamentally, a quantitative analysis of epistasis in a set of phenotypes with varying fitness costs could inform a longstanding debate regarding the origins of mutational robustness [We reasoned that a comparative analysis of the genetic architecture of four CHD phenotypes caused by the same mutation could illuminate the subject. The four defects—ASD, muscular and membranous ventricular septal defects (VSD), and AVSD—have mild to severe fitness costs, as defined by the likelihood of survival to reproductive age in humans . We detebustness .Nkx2-5+/− mutant line was maintained in the C57BL/6N background [Nkx2-5+/− C57BL/6N males were crossed to wild-type A/J or FVB/N females. The Nkx2-5+/− F1 was then crossed to produce A×B and F×B F2 progeny. We also produced F10 and F14 advanced intercross generations of F×B by random mating from the F2 generation onward. No matings between siblings or first cousins were allowed from the F3 and F4 generations onward [Inbred C57BL/6N (B) and FVB/N (F) mice were purchased from Charles River  and A/J (A) from the Jackson Laboratory . Mice were housed under standard conditions in the same room with access to water and chow ad libitum. The ckground ,19. To ps onward . BreederNkx2-5 genotyping, all Nkx2-5+/− and a subset of wild-type hearts were dissected and embedded in paraffin. Hearts were completely sectioned in the frontal plane at 6 µm thickness. Every section was collected, stained with hematoxylin and eosin, and evaluated by two individuals. Defects were diagnosed by the morphology of the atrial and ventricular septae, semilunar and atrioventricular valves, chambers, and the anatomic relationships between them. Secundum ASDs were distinguished from a patent foramen ovale based on the size of the latter in wild-type hearts.We collected and phenotyped hearts as previously described . BrieflyNkx2-5+/− mice. In the F×B advanced intercross, normal controls were randomly selected from unaffected siblings.Genomic DNA was isolated by phenol–chloroform extraction. SNP genotyping was performed on affected and normal The A×B and F×B F2 intercrosses were genotyped using custom panels of ~120 SNPs on the Sequenom MassARRAY system . The aveQTLs that modify the risk of a heart defect were mapped via case-control association tests. Each type of heart defect was analyzed separately. The same normal controls were used in each analysis.N = 5000) was performed to determine significance thresholds. Significant and suggestive loci had LOD scores >3.3 (α = 0.05) and >2.7 (α = 0.2), respectively.We utilized R/qtl (version 1.40-8) to map QTLs under a binary trait model in the A×B F2 intercross . PermutaP values < 9.2 × 10−4 (0.05/542) and <1.8 × 10−3 (1/542). The combined analysis of the F2 and advanced intercross generations enhances the resolution of mapping but poses two special considerations [We implemented a univariate linear mixed model in GEMMA (version 0.94.1) for a combined analysis of the F×B F2 and advanced intercross generations . A likeleen SNPs ,24. Signerations .First, some SNPs were not genotyped in either the F×B F2 or advanced intercross generations and were imputed. In the F2, 758 SNPs were imputed using R/qtl . In the Second, to minimize type I error, we accounted for unequal relatedness between individuals in the advanced intercross generations. The univariate linear mixed model, Equation (1), incorporates a genetic relatedness matrix (GRM) that captures the phenotypic variance due to the genome-wide similarity between individuals. For each case-control analysis, we used GEMMA to estimate centered GRMs using the SNP genotypes from the combined F×B generations. Because F2 individuals are essentially full siblings, their pairwise relatedness coefficients were set to 0.5 and 1 on the diagonal in the GRM. We then adjusted the entire matrix (F2 + advanced intercross generations) to the nearest positive definite matrix using the nearPD function in the “Matrix” R package (version 1.2-6). The adjustment adds a small amount of noise to the GRM to remove negative eigenvalues, which is necessary for the stability of the mixed model. A Mantel test for the similarity between the pre- and post-adjusted GRMs showed >95% similarity in all cases.l) according to Equation (2):The proportion of the phenotypic variance explained (PVE) by all genotyped SNPs was estimated using restricted maximum likelihood (REML) in the linear mixed model in GEMMA. This method uses the GRM to estimate the genome-wide PVE and its standard error for each type of heart defect in the A×B and F×B intercrosses . PVE esthreshold .SNP, PVESNP, was calculated according to Equation (3):SNP estimates were corrected and transformed to the liability scale in the same manner as the genome-wide PVE estimates.The proportion of the phenotypic variance explained by an Nkx effects using logistic regression models. The odds ratios for each A×B locus were obtained using the base glm function in R. The odds ratios for each F×B locus were obtained using the logistic mixed model implemented in GMMAT [We estimated the odds ratios for the G×Gin GMMAT . The mixNkx effect is determined by the difference between the observed and the expected incidences of a defect at a two-locus genotype under the null model of independently acting loci. The expected incidence is calculated from the unweighted regression of the two-locus genotype incidences onto the two single-locus incidences. To compare G×G×GNkx effects between defects that have different incidences, the effects were divided by the unweighted average incidence of the defect across all nine genotype combinations between two loci. Effects that raise or lower risk are deemed positive or negative, respectively. To assess statistical significance, we arcsine-transformed the effects to obtain unbiased variances [T-tests to identify significant, non-zero effects. The significance threshold was Bonferroni-corrected for the number of two-locus genotypes in a pairwise analysis, i.e., P = 0.05/9x, where x is the number of pairwise combinations of loci for a defect. To compare effects between defects that have different incidences, the significant epistatic effects were divided by the unweighted average incidence of each defect [Nkx modifier locus was identified for ASD and muscular VSD in the F × B population, we assessed G×G×GNkx effects in combination with the next most significant SNP The contribution of higher-order epistatic effects to the risk of a heart defect was calculated according to the physiological epistasis model . The G×Gariances . Using th defect . BecauseNkx effects versus defect severity , we controlled for variation in the observed incidence at two-locus genotypes to avoid bias by an underlying correlation.GEMMA was run on the Ubuntu operating system (version 14.04 LTS). All other statistical analyses were performed in the R Statistical Computing environment (version 3.3.1). All t- and z-tests for comparisons of means and proportions, respectively, were two-sided. We used the R package “ppcor”  to calcuNkx2-5+/− newborns from intercrosses between the inbred strains A/J and C57BL/6N  and FVB/N and C57BL/6N . An ASD is the most common defect associated with human NKX2-5 mutation [Nkx2-5+/− mice are born at the expected Mendelian ratio. In fact, we obtained pups with very severe heart defects, such as double outlet right ventricle and tricuspid atresia, but too few for quantitative genetic analysis. About 70% of Nkx2-5+/− mice from either intercross have a normal heart.We phenotyped ~10,000 mutation . An ASD Nkx interactions in developmental pathways leading to a defect.The total, quantitative effect of genetic modifiers on risk could be a function of the number of genes, their effect sizes, or both. To distinguish these possibilities, we mapped the modifier QTLs in the F2 intercross of A×B and in the combined F2 and advanced intercross generations of F×B . MappingQTLs that overlap between defects may share a gene involved in the development of multiple cardiac structures . OverlapNkx effect sizes, however, vary with the severity of a defect. G×GNkx interactions have a small effect on ASD risk and increasingly larger effects for VSDs and AVSD. The correlation holds whether the effect is calculated as the risk associated with a susceptibility allele or the phenotypic variance explained by a locus . Statistically significant G×G×GNkx effect sizes vary between defects. AVSD effects are larger than ASD and VSD effects. The relationship is not secondary to an underlying correlation, such as between the observed incidence at a two-locus genotype and the G×G×GNkx effect size or the severity of a defect.Similar to G×Ga defect e. We calNkx effects suggest that positive selection promoted the coadaptation of interacting genes, as opposed to the elimination of intrinsically deleterious alleles during the inbreeding of mouse strains. Among significant G×G×GNkx effects, two-locus genotypes that are homozygous at both loci for alleles from the same inbred strain or “syn-homozygous”, e.g., BB at both loci, are more likely to lower risk. Conversely, anti-homozygous G×G×GNkx effects, e.g., homozygous FF at one locus and BB at the other, are more likely to raise the risk. Heterozygosity at either locus is equally likely to raise or lower risk , eight persons carried two or more mutations of the six genes [A complex genetic model plausibly explains the inverse relationship between the severity and incidence of a defect in humans. Whereas a modestly deleterious mutation may be sufficient to cause a mild defect, additional genetic interactions may be necessary to push a mutation carrier past the liability threshold for a severe defect, as observed in the ix genes . Larger The model suggests a corollary: epistasis should matter less for strongly deleterious mutations that place individuals near the liability threshold for a defect. To exert the same phenotypic effect as a weak mutation and its contributing genetic interactions, a strong mutation must either obstruct one critical pathway or perturb several simultaneously. The dramatic effect of de novo mutations of histone-modifying genes, which regulate the expression of multiple developmental genes, is consistent with the latter possibility .Nkx interactions are consistent with genetic coadaptation, a phenomenon first described by Dobzhansky in 1948. In crosses of Drosophila isolated from geographically distant locales, he noted “gene complexes” that conferred greater fitness when their alleles were from the same, rather than different populations [Drosophila [The protective and deleterious effects of syn- and anti-homozygosity in G×G×Gulations ,42. Incrulations , male feosophila , body coosophila , and vitosophila . An elegosophila . A/J andnkx2-5 mutation was probably not the original selection pressure because there are many potential causes of CHD. Nevertheless, outbreeding in the intercrosses produced individual variation in the robustness to the Nkx2-5 mutation. Modifier locus genotypes permit us to observe the analog of pre- and post-selected states in the same population. If humans similarly vary in the robustness of cardiac developmental pathways, the genetic basis of CHD is a function not only of deleterious variants but also of the degree of allelic compatibility between genes in the individual.Finally, the results suggest a rare example of selection for genetic robustness . Whether


With the emphasis on analysing genotype-by-environment interactions within the framework of genomic selection and genome-wide association analysis, there is an increasing demand for reliable tools that can be used to simulate large-scale genomic data in order to assess related approaches.We proposed a theory to simulate large-scale genomic data on genotype-by-environment interactions and added this new function to our developed tool GPOPSIM. Additionally, a simulated threshold trait with large-scale genomic data was also added. The validation of the simulated data indicated that GPOSPIM2.0 is an efficient tool for mimicking the phenotypic data of quantitative traits, threshold traits, and genetically correlated traits with large-scale genomic data while taking genotype-by-environment interactions into account.This tool is useful for assessing genotype-by-environment interactions and threshold traits methods. Access to dense single nucleotide polymorphism (SNP) markers across the genome has created the opportunity for finely identifying quantitative trait loci (QTLs) through genome-wide association studies (GWASs) and accurately predicting genetic values through genomic selection (GS) for economically important traits in animal and plant breeding –3. The rGenotype-by-environment (G-by-E) interactions have long been a topic of research interest. Generally, models applied to genetic evaluations do not consider G-by-E interactions, resulting in reductions in genetic gains. Many studies have reported that models accounting for G-by-E interactions improved the accuracy of estimates of genetic parameters and breeding values for complex traits –6. MeanwMany traits of economic importance, such as litter size of large mammals, degree of calving difficulty and resistance to disease, show a discrete character of phenotypes, and are defined as threshold traits . Due to Previously, we developed the simulation tool GPOPSIM, which can simulate large-scale genomic data including population structure, polymorphic markers and multiple quantitative traits based on the mutation-drift equilibrium model . The objc is the environmental value; α0 and α1 are the zero- and first-order random regression coefficients of the breeding value on c, respectively; and e0 and e1 are the zero- and first-order random regression coefficients of the residual effect on c, respectively.Generally, G-by-E interactions are analysed by a multi-trait model or a reaction norm model . If envic is further divided into two components:The environmental value α0, β and α1 are affected by all QTLs simultaneously, and these three effects of each QTL are drawn from a multivariate normal distribution with a vector of means 0 and the variance-covariance structure pi(1 − pi)mi, where pi is the frequency of one allele of the ith QTL and mi is the effect of the ith QTL for α0, β or α1. Then, the substitution effects are rescaled to ensure total variances α0, β and α1, respectively. e0, e1 and ϵ for each individual are sampled from a multivariate normal distribution with a vector of means 0 and the variance-covariance structure c do not have a genetic relationship. Moreover, we can also generate the phenotype y and the environmental value c through the model y = α0 + α1 ∗ c + e0 without accounting for heterogeneous residual variance.We assume that A parameter file is required to run GPOPSIM2.0 software. We specified various parameters for the simulation in this file. The parameter settings influence the historical population and the population structure, pedigree structure and genome structure of the current populations Fig. . The simα0, β or α1 are its QTL effects multiplied by genotypes, and then the means of the pseudo TBVs are scaled to 0. Finally, the environmental value c of each individual is obtained by adding the cumulative effect across all QTLs for β with the residual ϵ, and then the phenotype y of each individual is generated through the model y = α0 + α1 ∗ c + e0 + e1 ∗ c or the model y = α0 + α1 ∗ c + e0 without accounting for heterogeneous residual variance.For the simulation of a G-by-E interaction, Additionally, threshold traits can be simulated by GPOPSIM2.0, which lies in the idea that discontinuous characters have an underlying continuity liability , and threshold values divide the liability into discontinuity, resulting in some kinds of visible expression . It is ahttps://github.com/SCAU-AnimalGenetics/GPOPSIMv2). GPOPSIM2.0 can now simulate one or more independent/correlated quantitative traits, one or more independent/correlated threshold traits, genetically correlated quantitative-threshold traits and G-by-E interactions. The format of the input and output files is the same as that in GPOPSIM1.0.The GPOPSIM2.0 program is written in Fortran 90, and the source code is available free online. Executable files are currently performed on Windows and Linux platforms. GPOPSIM2.0 is free of charge for all users, and no licence is required , according to a two-sample t test.Additionally, GPOPSIM2.0 can generate good-quality simulated data for threshold traits. The incidences calculated from simulated data were very close to the set incidences 30% or 40%) from Fig. 0% from FAccording to the validation of simulated data, GPOPSIM2.0 has successful new functions for simulating genomic data for G-by-E interactions and threshold traits. GPOPSIM2.0 is a user-friendly tool for simulating large-scale genomic data, and these new functions will aid in the development of new approaches for analysing G-by-E interactions and threshold traits within the framework of GS and GWASs. Nevertheless, there is still room for further improvement of GPOPSIM2.0, such as accommodating QTL epistatic effects and longitudinal genomic data.Project name: GPOPSIM2.0.https://github.com/SCAU-AnimalGenetics/GPOPSIMv2Project home page: Operating system(s): Compiled for Windows and Linux.Programming language: Fortran 90.Other requirements: None.License: None.Any restrictions to use by non-academics: None.


The RSA traits of the stiff stalk subgroup (SS) showed lower values than those of the non-stiff stalk subgroup (NSS) and tropical/subtropical subgroup (TST). Using the RSA traits, the genome-wide association study identified 63 SNPs and 189 candidate genes. Among them, nine candidate genes co-localized between RSA and aboveground architecture traits. A further co-expression analysis identified 88 candidate genes having high confidence levels. Furthermore, we identified four highly reliable RSA candidate genes, GRMZM2G099797, GRMZM2G354338, GRMZM2G085042, and GRMZM5G812926. This research provides theoretical support for the genetic improvement of maize root systems, and it identified candidate genes that may act as genetic resources for breeding.Roots are important plant organs for the absorption of water and nutrients. To date, there have been few genome-wide association studies of maize root system architecture (RSA) in the field. The genetic basis of maize RSA is poorly understood, and the maize RSA-related genes that have been cloned are very limited. Here, 421 maize inbred lines of an association panel were planted to measure the root systems at the maturity stage, and a genome-wide association study was performed. There was a strong correlation among eight RSA traits, and the RSA traits were highly correlated with the aboveground plant architecture traits (e.g., plant height and ear leaf length,  Global agricultural production is facing its greatest challenge owing to climate change, land degradation and population growth ,2. BreedPlant RSA has a strong plasticity, and it is affected by both environmental signals and genotype-associated internal mechanisms, which change in response to the external environment . Root deArabidopsis [Hormones are the main components, on the physiological level, of the internal regulatory mechanism of RSA . Hormonebidopsis . A studybidopsis . Cytokinbidopsis . In addibidopsis .GRMZM2G1537222, which contains nine significant SNPs, most likely related to maize root development [GRMZM2G02110, which is specifically expressed in the root system, was located on chromosome 5 [The RSA is a complex quantitative trait having wide phenotypic variation among different genotypes. Because root sampling is destructive and labor-intensive, there are limited studies on the genetic mechanisms of crop RSA in the field . In receelopment . In addimosome 5 . A previmosome 5 . Moreovemosome 5 . In recemosome 5 ,34,35. FTo elucidate the genetic basis of maize RSA, we measured related traits at maturity under field conditions using an associated population of 421 inbred lines. Combined with the RSA phenotypes, we used high-density genotypes to perform a GWAS to investigate the genetic basis of maize RSA. Additionally, a co-expression analysis was performed using root transcriptome data to identify candidate genes related to maize RSA.−1 in a randomized block design. Each inbred line was planted in a 4 m single row, with a 50 cm row spacing and a 20 cm inter-plant spacing. There was a base application of compound fertilizer . Other field measures were implemented in accordance with conventional management practices.The maize association panel used in this study included 421 maize inbred lines  from ChiAt the mature stage, 3–5 plants of each inbred line with similar aboveground growth were selected and their root systems were excavated to a depth of 30 cm. The excavated roots were gently shaken to remove most of the attached soil and soaked in a water mixed with detergent. A washing apparatus with adjustable water pressure was then used to remove any remaining soil attached to the roots. After being cleaned, the root system was transferred to a studio with stable light-emitting diode (LED) light to collect two-dimensional images of the root systems using a camera . All the root images were stored in JPEG file format. The software DIRT was used for the high-throughput quantitative analysis of two-dimensional root images with a masking threshold = 20 . In totahttps://www.r-project.org/, accessed on 10 August 2021), and the pearson correlation coefficient was calculated using R package “Hmisc”. The principal component analysis was performed using R package “ggbiplot”. On the basis of subgroup classifications, box plots of the eight root traits were constructed and compared using R package “boxplot”.The phenotypic data of aboveground traits used in this study, including plant height, ear position, and leaf traits, were derived from published studies  Tables . A statihttp://www.maizego.org/index.html, accessed on 8 June 2020). The genotypic data were compared with the reference genome B73 RefGen_v2 . In this study, eight root traits and eight aboveground traits were used as phenotypes. The genome-wide association study was performed by TASSEL5.0 [10  = 5.0 as the threshold for determining whether the marker was significantly associated with the root traits. Manhattan plots and a linkage disequilibrium heat map were constructed using R package “CMplot” and “LDheatmap”, respectively. Using B73 RefGen_v2 as the reference genome, root candidate genes were searched for in the regions 50-kb up- and downstream of significant SNPs. MaizeGDB , NCBI , and TAIR  databases were used to annotate candidate genes.The genotypes, population structure, and kinship matrix of the association panel used for the GWAS in this study were downloaded from MaizeGO  was used extract total RNA in accordance with the manufacturer’s instructions. RNA integrity was assessed using the RNA Nano 6000 Assay Kit and the Agilent Bioanalyzer 2100 system . Sequencing libraries were generated using an NEBNext Ultra RNA Library Prep Kit for Illumina . Libraries were sequenced on an Illumina HiSeq 6000 system. HISAT2  was usedr = −0.31–−0.16; p < 0.05; r = 0.26–0.99, p < 0.05; r = 0.93–0.99, p < 0.01; r = 0.13–0.25, p < 0.05, p > 0.05; The eight root traits measured in this study were root top angle (ANG_TOP), root bottom angle (ANG_BTM), root skeleton width (SKL_WIDTH), the maximum width of the root system (WIDTH_MAX), the median width of the root system (WIDTH_MED), projected root area (AREA), average root density (AVG_DEN) and the number of root tip paths (RTP_COUNT) . The detp > 0.05, p < 0.05, p < 0.05, p < 0.05, To investigate the variations in RSAs of different germplasm resources, we analyzed the variances of root traits among different groups. Overall, compared with Non-Stiff Stalk (NSS), Mixed group (Mixed), and Tropical/subtropical (TST) subgroups, the Iowa Stiff Stalk Synthetic (SS) subgroup had lower root trait values a–e. Our GRMZM2G099797, co-localized with RTP_COUNT, WIDTH_MAX, WIDTH_MED, and SKL_WIDTH  were associated with SNP markers (chr6.S_142873590) corresponding to ANG_BTM   , which emeristem . The oveeostasis . Thus, wGRMZM2G354338 (ZmARGOS8) gene identified in this study  encodes a rhamnogalacturonan II xylosyltransferase, and its expression results in significant root defects and partial plant death [GRMZM5G812926, which is significantly associated with RTP_COUNT (Arabidopsis homolog AT1G30440 (NPH3), which encodes a protein having an NPY3 domain [Arabidopsis [GRMZM2G099797, GRMZM2G354338, GRMZM2G085042, and GRMZM5G812926) identified in this study are worthy of attention as RSA-related candidate genes.Furthermore, ethylene plays an important role in regulating plant development . The GRMis study ,encodes ethylene . With thncreased , which mnditions , and it nditions . This su ANG_BTM . The funnt death . FurtherTP_COUNT , may be 3 domain . This dobidopsis . TherefoOverall, owing to the difficulty and inefficiency of root trait identification, breeders tend to ignore the genetic analysis and improvement of root traits and focus instead on aboveground traits. With the continuous development of computer science and image recognition technology, the throughput acquisition of root phenotypes has been significantly increased. Using a high-throughput root phenotypic determination method, our study acquired mature maize root phenotypes, which laid an important foundation for the analysis of maize root genetic mechanisms. To date, we have a very limited understanding of the genes that control maize root development, and no major RSA genes that affect maize breeding have been found . With thIn this study, an association panel consisting of 421 maize inbred lines was used to measure RSA traits at the maturity stage under field conditions. First, we found that, compared with other germplasms, the SS germplasm has a narrower and steeper RSA. We then identified 189 candidate genes using a GWAS. In addition, we found that there were significant genetic differences between the root and the aboveground agronomic traits of maize. Finally, we conducted a co-expression analysis using root transcriptome data and identified 88 high-confidence candidate genes. We then identified four potential RSA-related candidate genes on the basis of gene function annotation. The results of this study provide valuable resources for the future genetic improvement of maize RSA.