A Unified Model for Inclusive Inheritance in Livestock Species.

For years, animal selection in livestock species has been performed by selecting animals based on genetic inheritance. However, evolutionary studies have reported that nongenetic information that drives natural selection can also be inherited across generations (epigenetic, microbiota, environmental inheritance). In response to this finding, the concept of inclusive heritability, which combines all sources of information inherited across generations, was developed. To better predict the transmissible potential of each animal by taking into account these diverse sources of inheritance and improve selection in livestock species, we propose the "transmissibility model." Similarly to the animal model, this model uses pedigree and phenotypic information to estimate variance components and predict the transmissible potential of an individual, but differs by estimating the path coefficients of inherited information from parent to offspring instead of using a set value of 0.5 for both the sire and the dam (additive genetic relationship matrix). We demonstrated the structural identifiability of the transmissibility model, and performed a practical identifiability and power study of the model. We also performed simulations to compare the performances of the animal and transmissibility models for estimating the covariances between relatives and predicting the transmissible potential under different combinations of sources of inheritance. The transmissibility model provided similar results to the animal model when inheritance was of genetic origin only, but outperformed the animal model for estimating the covariances between relatives and predicting the transmissible potential when the proportion of inheritance of nongenetic origin was high or when the sire and dam path coefficients were very different.

FOR years, animal selection in livestock species has been performed by selecting animals to be the parents of the next generation based on their breeding values for the trait of interest (Fisher 1918). The theory behind this method is that, in addition to the common environmental effects shared by relatives, phenotypic resemblance between relatives is of genetic origin. Selection is therefore efficient because additive genetically caused phenotypic variation is intergenerationally stable (Mameli 2004). Indeed, DNA sequence is transmitted from parent to offspring (i.e., inherited), and, because genes play an important role in the development of all phenotypes, the transmission of this genetic information contributes to the phenotypic similarity between parents and offspring. However, it is very simplistic to consider that only the DNA sequence is transmitted from one generation to the next. Evolutionary studies have reported that nongenetic information that drives evolution, and natural selection can also be inherited across generations [see Jablonka and Lamb (2008), Bonduriansky , and O’Dea for examples describing how nongenetic inheritance could be a key transient mechanism by which organisms adapt to rapid environmental change]. In response to this finding, evolutionary biologists have developed the concept of inclusive or general heritability (Mameli 2004; Danchin ), which combines all sources of information inherited across generations. The existence of nongenetic inherited effects is one of the many explanations given to the missing heritability problem that is reported mainly in humans (Manolio ), i.e., genetic variants in genome-wide association studies (GWAS) cannot completely explain the heritability of traits. In addition to the possible complex genetic architecture of traits (epistasis, imprinting, etc.), it has been postulated that mechanisms unrelated to the DNA sequence may explain the differences between heritability measurements using pedigrees or SNPs (Eichler ; Sandoval-Motta ). Many additional inheritance mechanisms have been described, and sometimes it is difficult to distinguish one from another (Rossiter 1996). For simplicity, we propose to classify these mechanisms into two main categories based on the medium of transmission: (1) the inherited information is transmitted across generations via a physical transmission support as is the case for epigenetic marks (epigenetic inheritance), metabolites, and symbionts (microbiota inheritance); and (2) the information is not transmitted via a physical transmission support (environmental inheritance). Numerous studies have emphasized the importance of epigenetics in mammal and plant traits (Quadrana and Colot 2016; Charlesworth ). In addition, there is now evidence that some epigenetic marks (histone modifications, mRNA and DNA methylation, etc.) that modulate gene expression are transmitted transgenerationally (Miska and Ferguson-Smith 2016; van Otterdijk and Michels 2016). Among the other information that is physically transmitted across generations, one of the most important is probably the microbiota. The microbiota consists of the symbiotic microbial cells (bacteria, archaea, viruses, and eukaryotic microbes) that reside in, and on, the bodies of animals and plants (Turner ; Shreiner ). These microbes have tremendous potential to impact physiology, in both health and disease, of their host (Sommer and Bäckhed 2013; Marchesi ). The vertical transmission of the microbiota has been described in different species (Cankar ; Ley ; Abecia ; Sylvain and Derome 2017), confirming that the microbiome is part of the information transmitted across generations (Sandoval-Motta ). The term environmental inheritance that we propose here refers to other sources of information transmitted across generations that do not rely on a physical transmission medium, that is, information that passes from one generation to another via learning mechanisms [behavioral (Jablonka and Lamb 2014), cultural inheritance (Feldman and Cavalli-Sforza 1975; Danchin )] or transmission of environmental conditions (ecological inheritance). All these phenomena have been explored in evolutionary studies to understand the mechanisms that drive natural selection (Mameli 2004). Although discussed (Goddard and Whitelaw 2014), they are not yet taken into account in the context of selection in livestock species. Nonetheless, if, as stated in evolution studies, nongenetic inheritance is much more important than genetic inheritance (Mameli 2004), then not considering it for selection could mean that the maximal progress for the selected population may never be reached, or, even worse, that the transmissible potential of an individual be misevaluated because the model (animal model) used to study the covariances between relatives is inappropriate.

The aim of this study was to present a model for estimating the inclusive heritability of traits in livestock species and predicting the transmissible potential of individuals based on genealogical and phenotypic information. The first part of the paper gives a brief overview of the mathematical models proposed in the literature for modeling the different sources of nongenetic inheritance and describes our transmissibility model for inclusive heritability. The second part reviews the properties of the transmissibility model: demonstration of the structural identifiability of the model, comparison of the practical identifiability of the transmissibility model with that of a model aiming to disentangle genetic and nongenetic sources of inheritance, and power analysis of the transmissibility model. The third part compares, using simulation studies, the performances of the animal model and the transmissibility model for selection (i.e., for estimating covariances between relatives and predicting the transmissible potential of individuals).

Materials and Methods

Overview of the mathematical models for different sources of inclusive inheritance in a livestock context

The path coefficient diagram describing the transmission of the different inherited factors in livestock species is provided in Figure 1. Let us consider an offspring born from sire and dam .

Figure 1

Path coefficient diagram describing the transmission of the different inherited factors in livestock species. a, genetic effects; epi, epigenetic effect; mic, microbiota effect; cult, cultural effect; y, phenotype; e, residual. Indices s, d, o and i refer to sire, dam, offspring, and comate i, respectively.

Genetic inherited effects:

An animal receives half of its DNA from its father and half from its mother. Following Mendelian theory, the additive genetic value (or breeding value) of animal is then , where is the Mendelian sampling [i.e., the deviation of the breeding value of animal i from the average breeding value for both parents (Falconer 1974)], considering no inbreeding, , with the additive genetic variance. Thus with the numerator relationship matrix.

Epigenetic inherited effects:

Tal proposed a mathematical model for the transmission of epigenetic marks used later by Varona and Paiva in the context of animal breeding. The model is built on the idea that, during the vertical transmission of epigenetic marks from mother to offspring and from father to offspring, an unknown proportion of them is lost (thus not transmitted to the next generation). This unknown probability of changing the epigenetic state during gametogenesis and/or early development is called the reset coefficient and can be different for the sire and the dam ( and , respectively). Consequently, the model proposed for the epigenetic value of an individual is: with and (under the assumption that the variance of transgenerational epigenetic effects is constant across generations) . Thus with the matrix of epigenetic relationships between individuals.

Microbiota inherited effects:

To our knowledge, no mathematical model for the transmission of the microbiota has been proposed in the literature. However, there is evidence that part of the microbiota received by an offspring is vertically transmitted and another part is horizontally transmitted from environmental sources including comates (Bright and Bulgheresi 2010). The vertical transmission reflects the inheritance of symbionts from the mother, and, rarely, from both parents. This is particularly true in livestock species, where generally no direct contact between sires and offspring is possible. Consequently the model for the transmission of the “microbiota value” of an individual following these rules could be: , where is the proportion of microbiota inherited from the mother, is the proportion of microbiota transmitted by individual sharing the same environment as the target individual , and . Then , where is the microbial relationship matrix.

Cultural and behavioral inherited effects:

Culture is a system of inheritance via social learning; it has been described to be transmitted through vertical, horizontal, and oblique pathways. Nearly 50 years ago, Feldman and Cavalli-Sforza (1975) proposed a model for cultural inheritance. In their model, the cultural value of an individual is defined as the sum of a fraction of the cultural values of its parents and of animals having an influence on the target animal (transmission through nonparental elders and animals of the same age group). Due to the rearing conditions used in livestock species, most often among the two parents only the dam can have a parental cultural influence on her offspring (with several exceptions in meat sheep, meat cattle species). Thus, the model can be written as: which is a model equivalent to that proposed for microbiota inheritance. Consequently, it will be difficult to distinguish cultural and microbiota values in practice without a specific experimental design or without additional information on the microbiota and/or the behavior of animals. We thus propose, in the context of selection in livestock species, to combine them into a single term: the single-parent transmitted value . In addition, for simplicity, we will consider that individuals sharing the same environment as the target individual throughout its life are sampled randomly in the population. The model describing the vertical transmission of the single parent transmitted value can therefore be simplified to: . It should be noted that this model is an oversimplification of the transmission of culture/behavior/microbiota, the purpose of which is solely to underline the difficulty of separating genetic from nongenetic effects, even in a simple case.

Ecological inherited effects:

In livestock species, unlike wild species, animals do not choose their environment; it depends entirely on the farming system used. Consequently, we consider that there is no ecological inheritance in livestock species.

Therefore, the different sources of inheritance in livestock species can be modeled using three different models: a model in which both parents influence the inherited value with a known coefficient of transmission (genetic effects); one with a unknown coefficient of transmission (epigenetic effects); and a third model, in which, of the two parents, only the dam vertically transmits the inherited information (microbiota, cultural, and behavioral inheritance).

Model proposed for inclusive inheritance in livestock species

The phenotype of an individual is the result of the combination of all these inherited factors and other environmental effects. To our knowledge, no studies have taken into account all of these sources of variation in a single model in order to explain phenotypic variability. However, several have investigated two inherited factors at a time (Tal ; Danchin ; Varona ) by considering additivity of the effects of the different inherited factors. Under this assumption, the phenotype of an individual considering all the inherited factors is:where is the vector of fixed effects, and the vector that links the fixed effect to the observation of animal ; are the additive genetic, epigenetic and single-parent transmitted values of animal, respectively, that follow the aforementioned models; is the residual with . Given the laws of transmission of the different inherited factors; an animal will transmit to its offspring if is a male (=sire transmissible potential of individual ), while it will transmit to its offspring if is a female (= dam transmissible potential of individual ). In this model, the dam transmissibility (the dam-offspring regression) is and the sire transmissibility (the sire-offspring regression) is . These transmissibilities are useful to compute the response to selection: where and are the female and male selection differentials, respectively. Theoretically, the parameters of Equation 1 are identifiable (see Appendix A for the demonstration). However, a very large number of observations and a particular population structure would be necessary to estimate them in practice. For these reasons, we propose to combine the different transmissible potentials of an individual into a single “transmissible value” with the following model of transmission:where are the path coefficients of transmission from the sire and the dam, respectively, that follow the following constraints: Thus, in this model, the two path coefficients of transmission can be 0.5 as in the traditional animal model, < 0.5 in agreement with the epigenetic model or one coefficient can be > 0.5 in agreement with single parent inheritance, which is of particular interest for the dam side. The corresponding model explaining phenotypic variation is:with and where is the matrix of transmission between individuals (transmission relationship matrix) and the identity matrix. We call this model the “transmissibility model”. The transmission relationship matrix is a symmetric matrix with 1’s on the diagonal and as off-diagonal entries. For the case of two animals with common ancestor with , = , where are the number of sire and dam transmissions between ancestor l and animal , respectively. The transmission relationship matrix can be computed through its inverse, which can be easily obtained by the following decomposition, where is a diagonal matrix with variances of relative to as components (i.e., if both parents are known, for animals of unknown sire, for animals of unknown dam, and 1 for animals for which both parents are unknown), is a lower triangular matrix with 1’s on the diagonal and the negatives of the sire and dam coefficients of transmission as off-diagonal entries.

In the transmissibility model, the proportion of transmitted variance is , the sire-offspring regression and the dam-offspring regression (sire and dam transmissibilities) are , respectively. The response to selection is thus where and are the female and male selection differentials, respectively.

The parameters of the transmissibility model can be estimated with the restricted maximum likelihood method (REML) using ASReml, and the OWN Fortran program that we have developed. This OWN Fortran program is freely available on the zenodo website: https://doi.org/10.5281/zenodo.1487869, it can be used to fit a transmissibility model with different or equal path coefficients of transmission for the sire and the dam.

In the transmissibility model, variance of the transmissible potential, sire, and dam path coefficients of transmission estimates depend on the relative proportion of variance and path coefficients of each inherited factor. Let us describe different schematic cases that help to understand the behavior of the transmissibility model. For simplicity, we consider that phenotypes are derived from Equation 1 with sire and dam epigenetic path coefficients of transmission equal to 0.25 and 0.40, respectively, single parent path coefficient of transmission equals to 0.9, total variance equals to 1, and in all examples.

If the inheritance is mainly of genetic origin , the sire and dam path coefficient of transmission estimated with the transmissibility model will be close to 0.5, but slightly higher for the dam side and which is close but < 0.6;

if the three inherited factors equally participate in the inheritance the dam path coefficient of transmission obtained with the transmissibility model is higher than that of the sire (0.68 vs. 0.26), and decreases to 0.53. Finally,

if the inheritance is mainly of single-parent origin then the disequilibrium between sire and dam path coefficient of transmission is high and is close to 0.6 (0.57). Additional expected estimates in the transmissibility model compared to parameters of the true model of inheritance are provided in Supplemental Material, Figures S1, S2, and S3.

Structural, practical identifiability, and power analysis

The parameters of the transmissibility model are structurally identifiable (see Appendix B for the demonstration). Their practical identifiability was assessed by using methods proposed by Visscher and Goddard (2015), Raffa and Thompson (2016), and Raue . A detailed description of this practical identifiability study is provided in Appendix C. Briefly, the practical identifiability study is as follows: the expected likelihood ratio test (ELRT) comparing the true to the estimation model is literally expressed as a function of the parameters and the structure of the population (i.e., number and size of families with determined relationships). To do so, a simplified population structure with homogeneous families is considered. Then, the profile likelihood (PL), which reflects the minimal value of ELRT, is computed for each parameter to estimate. A close inspection of changes in PL with the value of the parameter being estimated enables the detection of directions where the likelihood flattens out. In such situations, the parameter is considered as practically nonidentifiable. In addition, the power for testing a given hypothesis can be derived from the ELRT as described by Raffa and Thompson (2016).

We investigated the practical identifiability of the transmissibility model, and of a model that aims to distinguish genetic from nongenetic transmitted effects. For simplicity, we considered, for this practical identifiability study, that nongenetic effects were of epigenetic origin only, i.e., the true model is and the two models of estimation are (mod1) and . To ensure structural identifiability in mod1, six different types of family relationship are sufficient: sire – offspring, dam – offspring, paternal half-sibs, maternal half-sibs, uncle-nephew and aunt-nephew. Those relationships are also sufficient to ensure the structural identifiability of the transmissibility model. We thus investigated the practical identifiability of both mod1 and the transmissibility model for three different pedigree sizes (small, medium, and large), which differed according to the number of family relationships as described in Table 1. Practical identifiability can be evaluated for an infinite number of combinations of the true genetic, epigenetic variances, and reset coefficients. We considered, as an illustration, the case of a trait of moderate heritability (0.2), equal genetic and epigenetic variances, and six different values of the sire and dam epigenetic coefficient of transmission combination (: 0.2 0.2, 0.2 0.3, 0.2 0.4, 0.3 0.3, 0.3 0.4, and 0.4 0.4).

Table 1

Pedigree structure of the three populations considered for the practical identifiability study

	Small	Medium	Large
Number of paternal half-sibs	25 * 30 half-sibs	250 * 30 half-sibs	500 * 100 half-sibs
Number of maternal half-sibs	25 * 30 half-sibs	250 * 30 half-sibs	500 * 100 half-sibs
Number of sire-offspring couples	1000	5000	25,000
Number of dam-offspring couples	1000	5000	25,000
Number of uncle-nephew couples	1000	5000	25,000
Number of aunt-nephew couples	1000	5000	25,000

We then computed the power of the transmissibility model to detect nongenetic inheritance (i.e., at least one of the two path coefficients of transmission differs from 0.5), while the true model of transmission was using the three aforementioned pedigree structures, and considering different combinations of the genetic, epigenetic variances (i.e., or 0.6; or 0.7) in the case of equal sire and dam path coefficients of transmission. Finally, we also computed, for medium pedigree sizes, the power to detect nongenetic inheritance, when the sire and dam have different epigenetic path coefficients of transmission in the true model and different path coefficients of transmission in the transmissibility model used for the estimation.

Validation of the transmissibility model using simulations

Phenotype simulation:

We performed a simulation study to evaluate the ability of the transmissibility model in complex pedigrees to: (1) detect nongenetic inheritance, (2) estimate regression coefficients for different types of relatives, and (3) predict the transmissible values of individuals. In the latter two cases, the performances of the transmissibility model were compared with those of the animal model.

Datasets consisted of a four-generation population. The first generation comprised 25 sires and 100 dams that were randomly mated to give birth to 800 offspring (eight offspring per dam, sex ratio = 1/2). Among the progeny, 25 males and 100 females were sampled randomly to be the parents of the next generation. The same process was repeated for each generation. The final population comprised 2525 individuals.

The genetic, epigenetic, and single-parent transmitted values for each animal in the first generation were sampled from the following distributions: . Then, the different inherited values of animal born to sire and dam in subsequent generations were obtained by sampling from: Phenotypes were then generated for all individuals of the first three generations, and for 1/4 of the animals of the last generation (randomly sampled) by:

where = [10 20 30]T is the vector of the effect of one factor with three levels. The level of the factor is sampled randomly for each individual from a uniform distribution. For the remaining 3/4 of the animals of the last generation (600), no phenotypes were generated. These individuals will be used to evaluate the ability of the models to predict the transmissible potential of an individual based on information from relatives only (no phenotypic information). In addition, among these individuals, 400 were considered as having only one known parent; only the sire was known for 200 and only the dam was known for 200, randomly sampled. To do so, we modified the pedigree to change the ID of the sire or the dam to unknown.

We simulated the four datasets described in Table 2. The sum of the sire and dam transmissibility was the same in all datasets, but datasets differed in the ratio of heritable variance to phenotypic variance , in the proportion of genetic origin for the heritable variance, and in the disequilibrium between sire and dam transmissibility. In dataset 1, the heritable variance was of genetic origin only: sire and dam transmissibilities were equal . In datasets 2 and 3, sire and dam transmissibilities were different. The ratio of heritable variance to phenotypic variance and the ratio of sire transmissibility to dam transmissibility was the same for the two datasets but the proportion of genetic origin for the heritable variance differed (63% in dataset 2, and 10% in dataset 3). Finally, the last dataset (dataset 4) had, as in dataset 3, a low proportion of genetic origin for the transmitted variance (8%) associated with a high disequilibrium between sire and dam transmissibility .

Table 2

Parameters of the simulations

	Dataset 1	Dataset 2	Dataset 3	Dataset 4
σe2	75	75	75	75
σa2	34.357	33	5	5
σepi2	0	17	45	45
12(1−v_s)	0	0.059	0.333	0.200
12(1−v_d)	0	0.265	0.411	0.475
σsp2	0	2	2	10.099
δ	0	0.7	0.7	0.7
σa2+σepi2+σsp2σa2+σepi2+σsp2+σe2	0.314	0.409	0.409	0.445
σa2σa2+σepi2+σsp2	1.000	0.635	0.096	0.083
Sire transmissibility b_OS	0.157	0.138	0.138	0.085
Dam transmissibility b_OD	0.157	0.176	0.176	0.229

Phenotypes were simulated under the following model: .

Parameter estimation

Two models were used to analyze the phenotypes: the animal model and the transmissibility model .

For each dataset type, 100 independent replicates were simulated.

Evaluation of the performance of the transmissibility model

The performance of the transmissibility model was assessed by:

Its ability to detect nongenetic inheritance. The animal model is a special case of the transmissibility model for which sire and dam coefficients of transmission are fixed to 0.5; it is thus nested in the transmissibility model. After model convergence, the null hypothesis H0 “sire and dam coefficients of transmission are equal to 0.5” was tested against the H1 hypothesis “almost one of the coefficients of transmission (sire or dam) differs from 0.5” by performing a likelihood ratio test of size 5% comparing the transmissibility model with the animal model (mixture ). Rejecting the null hypothesis H0 permits the conclusion that the underlying model is not purely additive genetic. The realized type I error is the number of replicates over 100, for which H0 was rejected when analyzing dataset 1, the power is the number of replicates over 100 for which H0 was rejected when analyzing datasets 2–4.

Its ability to estimate the regression coefficient between different types of relatives. The regression coefficients obtained with the two models of estimation were compared with the expected values assuming the hypothesis of the simulation model. Special attention was paid to the sire-offspring and dam-offspring regression coefficients (sire and dam transmissibilities). In addition, sire and dam path coefficients of transmission and the ratio of transmitted variance to phenotypic variance obtained with the genetic and transmissibility models were examined in detail to better comprehend how these models behave with data simulated according to a different model (Equation 1).

Its ability to predict the transmissible potential of individuals. The sire and dam “transmissible values” obtained with the transmissibility model ( for sires, for dams) and the additive genetic values predicted by the animal model (0.5 for sires and dams) were compared with the simulated transmissible values (weighted sum of the different sources of inheritance, i.e., for sires, for dams) for four different groups of animals: all animals with a phenotype, progeny without a phenotype and both parents known, progeny without a phenotype and unknown dam, and progeny without a phenotype and unknown sire. Comparison was based on the correlation between the estimated and simulated transmissible value, and the bias was quantified using the coefficient of regression of the simulated to the estimated transmissible value.

Data availability

The authors state that all information necessary for performing the simulation study presented in the manuscript are represented fully within the manuscript. Code source and tutorial to apply the transmissibility model are available on the zenodo website https://doi.org/10.5281/zenodo.1487869. Additional figures describing expected transmissibility model estimates in comparison with the true model of inheritance are available on Figshare. Figure S1 provides the expected estimates in the transmissibility model, while the true model is . Figure S2 provides the expected estimates in the transmissibility model while the true model is . Figure S3 compares the expected error term variance in the transmissibility model to the Mendelian sampling variance. Supplemental material available at FigShare: https://doi.org/10.25386/genetics.8275307.

Results

Practical identifiability and power analysis

The profile likelihoods of the transmissibility model and the model designed to dissociate genetic from nongenetic transmission (i.e., ) are shown in Figure 2 for six true values of the sire/dam reset coefficient. The shape of the curve for each parameter to estimate illustrates its practical identifiability. Structural nonidentifiable parameters are characterized by a flat profile likelihood that can have a minimum, but do not exceed a threshold for increasing and/or decreasing values of the parameter. On one hand, the profile likelihoods of parameters of the model that aims at dissociating genetic from epigenetic effects were, in the case of equal sire and dam epigenetic path coefficient of transmission , flat for increasing values of the relative importance of the transmitted variance () and decreasing values of the relative importance of the genetic variance , and for both increasing and decreasing values of the epigenetic path coefficient of transmission whatever the pedigree size. Difference in the sire and dam epigenetic path coefficient of transmission helps the practical identifiability of the parameters. However, even for large pedigree sizes, the profile likelihood of the genetic variance had difficulties exceeding the threshold value. On the other hand, even for small pedigree sizes, the profile likelihood of the proportion of transmitted variance and path coefficients of the transmissibility model increased substantially in both upwards and downward directions. These results indicate that all parameters of the transmissibility model are practically identifiable even for small pedigree size, while those of the model designed to dissociate genetic from nongenetic transmission effects are not.

Figure 2

Profile likelihood vs. parameters to estimate in a model designed to disentangle genetic and epigenetic effects and in the transmissibility model. The true model is . The values of the true model are , r = , and six different combinations for the epigenetic path coefficient of transmission are considered. The two models of estimation are (mod1) and the transmissibility model . First four left panels: profile likelihoods for the parameters , and for mod1. Three last panels: profile likelihoods for and . Top to bottom: small, medium, and large pedigree sizes; colors of lines: different values of the combination . Dotted horizontal line: the threshold value.

The power of the transmissibility model to detect nongenetic inheritance, under the assumption that sire and dam path coefficients of transmission are equal, as a function of the epigenetic path coefficient of transmission is provided in Figure 3 for different combinations of the true genetic and epigenetic variances. Its power to detect nongenetic inheritance increased with pedigree size, proportion of transmitted variance and relative importance of epigenetic variance. For small pedigree sizes, the nongenetic inheritance could only be detected in favorable cases: epigenetic path coefficient of transmission above a certain threshold but significantly <0.5 , proportion of transmitted variance >0.4 and relative importance of epigenetic variance >0.5. For medium pedigree sizes, the model’s power to detect nongenetic inheritance was >80% for many combinations of the true parameters. For large pedigree sizes, the power was close to 100% for all combinations. However, if the true epigenetic path coefficient of transmission was close to 0.5, the model was unable to detect nongenetic inheritance, regardless of pedigree size, because epigenetic transmission was confounded with genetic inheritance (same mode of transmission). The power of the transmissibility model to detect nongenetic inheritance considering different path coefficients of transmission for the sire and the dam is provided in Figure 4. Once again, its power to detect nongenetic inheritance increased with the pedigree size, proportion of transmitted variance, relative importance of epigenetic effects, and the distance of the path coefficients to 0.5. In addition, the power to detect nongenetic effects increased with the disequilibrium between sire and dam transmissibility.

Figure 3

Power to detect nongenetic inheritance with a transmissibility model considering equal path coefficients of transmission for the sire and the dam. The true model is considering same sire and dam epigenetic path coefficients of transmission . Values of the true model are or 0.6, r = or 0.7 and varying values for the path coefficient of transmission . The model of estimation is the transmissibility model which considers the same path coefficients of transmission for the sire and the dam . The null hypothesis H0 is , the alternative hypothesis is H1 .

Figure 4

Power to detect nongenetic inheritance with the transmissibility model considering different path coefficients of transmission for the sire and the dam. The true model is . Values of the true model are or 0.6, r = or 0.7 and varying values for the sire and dam epigenetic path coefficient of transmission The model of estimation is the transmissibility model . The null hypothesis H0 is , the alternative hypothesis is H1 or

Simulation

The power of the transmissibility model to detect nongenetic inheritance was 12, 61, and 96% for datasets 2, 3, and 4, respectively. The realized type I error was 4%.

The regression coefficients for different relatives were calculated by using the variance component and path coefficients of transmission used for the simulation or estimated using the transmissibility and animal models as described in Appendix D. The sire and dam transmissibilities (, ) obtained with the animal and transmissibility models are shown in Figure 5. When phenotypes were generated according to the animal model (dataset 1), the sire and dam transmissibilities obtained with the transmissibility and animal models were similar, and in line with the simulated transmissibilities. The SD of the transmissibilities were slightly larger for the transmissibility model, in comparison with the animal model (0.024 vs. 0.021 for sire transmissibility and 0.023 vs. 0.021 for dam transmissibility). The transmissibility model provided a good estimation of the dam and sire transmissibilities for datasets 2–4 (on average 0.17, 0.17, and 0.22 for the dam, and 0.15, 0.14, and 0.09 for the sire for datasets 2–4). For datasets 2–4, the animal model tended to underestimate the dam transmissibility on the one hand, and overestimate the sire transmissibility on the other hand. These under- and over-estimations increased with the proportion of nongenetic heritable variance, and with the disequilibrium between sire and dam transmissibilities. The ratio of the regression coefficient between other relatives obtained with the animal and transmissibility models to the expected regression coefficient under the simulated model are presented in Figure 6. Both models provided good estimations for all the regression coefficients for dataset 1. As for the sire and dam transmissibilities, the discrepancy between the regression coefficients estimated with the animal model and the values expected using the simulated model increased with the proportion of nongenetic heritable variance and the disequilibrium between sire and dam transmissibilities (datasets 2–4). The greatest difference between estimated and simulated regression coefficients was for paternal half-sibs, paternal half-sibs of the dam-offspring (four times higher than the simulated value), and between paternal half-sibs of the sire and offspring (six times higher than the simulated value) in dataset 4. The regression coefficients obtained with the transmissibility model were still well estimated when the proportion of nongenetic heritable variance increased (datasets 2 and 3). However, with the transmissibility model, an increase in the proportion of heritable variance from a single parent origin resulted in a bias in the estimation of the regression coefficients between the offspring and its paternal uncle (the average ratios with simulated values were 0.71 and 1.40 for paternal and maternal half sibs of the dam, respectively).

Figure 5

Sire and dam transmissibilities estimated using the animal and transmissibility models for the different datasets. Dotted lines show the simulated values.

Figure 6

Ratio of regression coefficient between relatives to simulated regression coefficient estimated by the animal and transmissibility models for the different datasets. The types of relatives are: 1, paternal grand dam-offspring; 2, maternal grand dam-offspring; 3, paternal half-sib of the sire (paternal uncle 1)-offspring; 4, maternal half-sib of the sire (paternal uncle 2); offspring; 5, paternal half-sib of the dam (maternal uncle 1)-offspring; 6, maternal half-sib of the dam (maternal uncle 2)-offspring; 7, paternal half-sibs; 8, full-sibs.

The proportion of transmitted variance and sire and dam path coefficients of transmission estimated with the transmissibility model, and heritability obtained with the animal model, are presented in Table 3 for all datasets. The heritabilities estimated with the animal model were similar for datasets 1–3 (0.31–0.32) and slightly (but not significantly) higher for dataset 4 (0.36). The general trend was an increase of the proportion of transmitted variance estimated with the transmissibility model from dataset 1–4. When data were simulated using the animal model (dataset1), the coefficients of transmission obtained with the transmissibility model were on average slightly lower than 0.5 (0.46 and 0.47 for the sire and the dam path coefficient, respectively) and the proportion of transmitted variance was slightly larger than the simulated heritability (0.35 vs. 0.31). In datasets 2–4, the estimated sire coefficient of transmission was lower (significantly in dataset 4) than the dam coefficient of transmission, in accordance with the lower sire than dam simulated transmissibility.

Table 3

Proportion of transmitted variance and coefficients of transmission estimated with the transmissibility model and heritability estimated with the animal model for each dataset

		Dataset 1	Dataset 2	Dataset 3	Dataset 4
Animal model	h^2	0.32 ± 0.04	0.31 ± 0.05	0.31 ± 0.05	0.36 ± 0.05
Transmissibility model	τ^2	0.35 ± 0.06	0.36 ± 0.06	0.40 ± 0.09	0.42 ± 0.09
	Sire coefficient of transmission	0.46 ± 0.06	0.42 ± 0.06	0.35 ± 0.06	0.22 ± 0.06
	Dam coefficient of transmission	0.47 ± 0.07	0.47 ± 0.07	0.44 ± 0.09	0.55 ± 0.10

The correlations between the true and predicted transmissible potentials obtained with the transmissibility and animal models for the different datasets are presented in Table 4. When the transmissible potential was mainly of genetic origin (datasets 1 and 2), the correlations between true and predicted sire or dam transmissible potentials were the same for the transmissibility and animal models for all types of animals (with or without phenotypes). Within the subgroup of animals without phenotypes, the correlation between simulated and predicted transmissible potentials was highest when both parents were known. When the transmissible potential was mainly not of genetic origin (datasets 3 and 4), the correlations between simulated and predicted dam transmissible potentials were slightly (but not significantly) higher for the transmissibility model in comparison with the animal model, except for animals without phenotypes and unknown dam. The biases in the estimation of the transmissible potential obtained with both models are presented in Table 5. When the transmissible potential was mainly of genetic origin, there was no important bias in the prediction, whatever the model or animal subtype. When the transmissible potential was mainly not of genetic origin (datasets 3 and 4), its prediction tended to be less biased with the transmissibility model than with the animal model, the bias being higher for the sire transmissible potential than for the dam transmissible potential (on average 0.64 and 0.96 for sires with phenotype; 1.13 and 1.00 for dams with phenotype, 0.52 and 0.97 for sires without phenotype, 0.91 and 1.02 for dams without phenotypes for the animal and transmissibility model, respectively). Within the group of animals without phenotypes, the biases obtained with the animal model were higher for animals (sires or dams) with an unknown dam (ranging from 0.32 to 0.75), whereas the biases obtained with the transmissibility model were relevant for the sire transmissible potential of animals without phenotypes and one unknown parent only (ranging from 0.87 to 1.28) but the SD was significant.

Table 4

Correlation (mean ± SD) between predicted and true transmissible potential for the different datasets

	Model		Dataset 1	Dataset 2	Dataset 3	Dataset 4
Animals with phenotype	Animal	Sire	0.71 ± 0.03	0.68 ± 0.03	0.67 ± 0.03	0.61 ± 0.03
		Dam	0.71 ± 0.03	0.71 ± 0.02	0.68 ± 0.03	0.70 ± 0.02
	Transmissibility	Sire	0.71 ± 0.03	0.67 ± 0.03	0.68 ± 0.03	0.60 ± 0.03
		Dam	0.71 ± 0.03	0.71 ± 0.02	0.69 ± 0.02	0.71 ± 0.02
All animals without phenotypes	Animal	Sire	0.42 ± 0.05	0.40 ± 0.06	0.32 ± 0.05	0.28 ± 0.05
		Dam	0.42 ± 0.05	0.39 ± 0.05	0.32 ± 0.05	0.31 ± 0.05
	Transmissibility	Sire	0.42 ± 0.05	0.39 ± 0.06	0.32 ± 0.05	0.28 ± 0.05
		Dam	0.42 ± 0.05	0.38 ± 0.05	0.33 ± 0.05	0.34 ± 0.05
Animals without phenotypes, both parents known	Animal	Sire	0.52 ± 0.07	0.49 ± 0.08	0.39 ± 0.08	0.34 ± 0.08
		Dam	0.52 ± 0.07	0.47 ± 0.07	0.39 ± 0.07	0.38 ± 0.07
	Transmissibility	Sire	0.51 ± 0.07	0.48 ± 0.08	0.39 ± 0.08	0.34 ± 0.09
		Dam	0.51 ± 0.07	0.47 ± 0.07	0.40 ± 0.08	0.42 ± 0.08
Animals without phenotypes, dam known	Animal	Sire	0.37 ± 0.07	0.34 ± 0.07	0.29 ± 0.07	0.30 ± 0.07
		Dam	0.37 ± 0.07	0.35 ± 0.07	0.31 ± 0.08	0.39 ± 0.07
	Transmissibility	Sire	0.37 ± 0.07	0.34 ± 0.07	0.30 ± 0.08	0.30 ± 0.07
		Dam	0.37 ± 0.07	0.35 ± 0.07	0.32 ± 0.07	0.40 ± 0.07
Animals without phenotypes, sire known	Animal	Sire	0.38 ± 0.09	0.36 ± 0.09	0.26 ± 0.09	0.19 ± 0.09
		Dam	0.38 ± 0.09	0.33 ± 0.09	0.25 ± 0.09	0.14 ± 0.08
	Transmissibility	Sire	0.38 ± 0.09	0.35 ± 0.09	0.26 ± 0.09	0.18 ± 0.09
		Dam	0.38 ± 0.09	0.32 ± 0.09	0.25 ± 0.09	0.14 ± 0.07

Table 5

Bias (mean ± SD) in the estimation of the transmissible potential for the different datasets

	Model		Dataset 1	Dataset 2	Dataset 3	Dataset 4
Animals with phenotype	Animal	Sire	1.01 ± 0.07	0.89 ± 0.08	0.81 ± 0.07	0.46 ± 0.04
		Dam	1.01 ± 0.07	1.06 ± 0.09	1.03 ± 0.09	1.23 ± 0.10
	Transmissibility	Sire	1.04 ± 0.13	0.98 ± 0.13	0.98 ± 0.15	0.94 ± 0.24
		Dam	1.03 ± 0.13	1.04 ± 0.13	1.00 ± 0.12	0.99 ± 0.10
All animals without phenotypes	Animal	Sire	0.99 ± 0.14	0.89 ± 0.14	0.65 ± 0.12	0.38 ± 0.08
		Dam	0.99 ± 0.14	0.97 ± 0.15	0.83 ± 0.15	0.98 ± 0.18
	Transmissibility	Sire	1.10 ± 0.23	1.10 ± 0.26	1.02 ± 0.31	0.91 ± 0.34
		Dam	1.08 ± 0.24	1.07 ± 0.23	1.04 ± 0.30	1.00 ± 0.27
Animals without phenotypes, both parents known	Animal	Sire	0.98 ± 0.15	0.89 ± 0.16	0.65 ± 0.15	0.38 ± 0.10
		Dam	0.98 ± 0.15	0.96 ± 0.17	0.83 ± 0.18	0.98 ± 0.22
	Transmissibility	Sire	1.09 ± 0.24	1.10 ± 0.28	1.01 ± 0.33	0.91 ± 0.37
		Dam	1.08 ± 0.24	1.07 ± 0.25	1.03 ± 0.33	1.00 ± 0.29
Animals without phenotypes, dam known	Animal	Sire	1.02 ± 0.22	0.88 ± 0.20	0.70 ± 0.20	0.44 ± 0.12
		Dam	1.02 ± 0.22	1.03 ± 0.22	0.93 ± 0.25	1.33 ± 0.28
	Transmissibility	Sire	1.14 ± 0.31	1.05 ± 0.29	1.01 ± 0.35	0.87 ± 0.36
		Dam	1.15 ± 0.44	1.11 ± 0.35	1.07 ± 0.38	1.03 ± 0.32
Animals without phenotypes, sire known	Animal	Sire	1.01 ± 0.27	0.92 ± 0.25	0.63 ± 0.24	0.32 ± 0.16
		Dam	1.01 ± 0.27	0.95 ± 0.27	0.75 ± 0.28	0.55 ± 0.31
	Transmissibility	Sire	1.16 ± 0.40	1.25 ± 0.51	1.12 ± 0.59	1.28 ± 0.86
		Dam	1.13 ± 0.34	1.11 ± 0.37	1.05 ± 0.50	1.01 ± 0.55

Discussion

Analyzing phenotypes with a model designed to disentangle the different sources of inheritance from each other is challenging. This difficulty is well illustrated by the profile likelihoods obtained with a model aimed at dissociating genetic and nongenetic inheritance, even if the parameters of such a model are theoretically identifiable. For equal sire and dam epigenetic path coefficient of transmission, on the one hand, all values of resulted in similar profile likelihoods. On the other hand, the profile likelihood was flat for all estimated values of < 0.5 and of > 0.4. These results suggest a certain amount of mutual dependency between the estimates for the variance components, probably due to the excessive flexibility of the model via that compensates the changes in the value of the different variance components in both upwards and downward directions. If the estimate is close to 0.5 then the epigenetic effect will mimic genetic effects and compensate a decrease in its variance (i.e., different genetic and epigenetic variances combinations lead to the same likelihood), and if the estimate is close to 0, then the epigenetic effect will mimic residual effects (different values of lead to the same likelihood). These findings are in line with the conclusions on this model reported by Varona . Identifiability is improved for and when the sire and dam path epigenetic coefficients of transmission differ significantly. However, difficulties to separate genetic and epigenetic variances remain. As detailed in Appendix A, when only pedigree information is available, the possibility to separate the different components in Equation 1 arises from the comparison between the covariance between different categories of relatives. Depending on the values of the parameters in Equation 1, these covariances can be very similar, and a very large amount of specific structured data required to estimate the parameters for the different sources of inheritance. The same difficulties have been encountered when estimating the variance components in models aimed at separating different kinds of genetic effects [maternal and social effects (Gerstmayr 1992; Cantet and Cappa 2008)]. Because the data structure often does not meet the requirements needed to disentangle the different sources of inheritance [also suggested by Varona when applying a model aimed at separating genetic and nongenetic heritance with the same sire and dam path coefficients of transmission in a large dataset of 78,209 birth weight records], and because this distinction is not required for selection, we proposed the transmissibility model. This model combines the different inherited factors into a single transmissible value that could prove useful for selection. The objective of this model is not to quantify the different sources of inheritance but to take them into account for selection. Less information is needed to estimate the variance components of this model as illustrated by its profile likelihood that shows that, even for small pedigree sizes, the parameters of the transmissibility model were practically identifiable, and justifies its application to real data.

If the objective is to disentangle the different sources of inheritance, then additional information is required to ensure the practical identifiability of the variance components of the different inherited factors. To study epigenetic inheritance, high-throughput technologies that quantify DNA methylation are now well established (Couldrey and Cave 2014). DNA methylation information obtained at the individual level can be used to build the epigenetic relationship matrix between individuals that will be integrated into the model to explain phenotypic variability. However, it should be noted that if stable epigenetic mutations are in high linkage disequilibrium with the single nucleotide polymorphisms (SNPs) available on the SNP chip for the species studied, this method will not bring anything new to genomic selection (Goddard and Whitelaw 2014). DNA methylation information from relatives could also be used to estimate the probability of epigenetic marks being erased during vertical transmission, and provide an estimation of the path coefficients of transmission for this inherited factor that could be included in the model as a known parameter. Accounting for epigenetic effects in the model could also be performed by adding the level of methylation of each individual (Cortijo ) as a covariate explaining phenotype. For microbial inheritance, 16S rDNA amplicon sequencing is now used routinely for microbial community profiling. Information on operational taxonomic units (OTUs) can then be used to evaluate the “distance” between the microbiota of two individuals to compute the microbial relationship matrix (Lozupone ; Camarinha-Silva ; Xia and Sun 2017). The matrix is then considered as known in the model and facilitates the estimation of the different variance components. However, recent studies aimed at estimating microbiota variance using this approach reported that almost all the genetic variance passes into microbiota variance (Gilbert ). These results suggest that there is still some confusion about the two factors, which can be explained by the genetic origin of the variability of the microbiota (Davenport 2016; Camarinha-Silva ). Analyzing the two phenotypes (trait of interest, microbiota) with a recursive model (Gianola and Sorensen 2004) would probably help understanding this point. However, further research is needed to implement this kind of model on such a complex trait (beta diversity). To decipher genetic from cultural effects, an experimental design based on partial cross-fostering has been proposed by Danchin . However, it is not clear how this procedure avoids confusion between the microbiota and the culture as their modes of transmission are so close. Extracting the cultural relationship between individuals by social network analysis (Scott 2017) would probably be more effective. Nonetheless, this requires direct or indirect recording of animal behavior that can be time-consuming and/or expensive and/or difficult to set up for a large number of animals. At present, it seems unrealistic to try to record all this additional information in order to disentangle the different sources of inheritance in the context of genetic evaluation. The transmissibility model overcomes these difficulties by combining all these inherited factors into a single value: the transmissible potential .

To complete the practical identifiability study and to better understand the behavior of the transmissibility model and the animal model in regard to genetic and nongenetic inheritance, we evaluated by simulation the performance of the transmissibility model, and compared the results with the animal model. We deliberately used a different model to simulate the data from the models used for the estimations (except for dataset 1 which coincide with an animal model). Indeed, our purpose was to not favor one of the models but to evaluate their performances in the “real” world. Consequently, except for dataset 1, the variance components obtained with the two models are not comparable to those used for the simulation and the performance of the models were assessed as their ability to match as closely as possible the expected covariances between relatives and to predict the transmissible potential of individuals.

The covariances between different kinds of relatives are used to estimate the parameters of the transmissibility and animal models. On the one hand, the animal model has to fit these different covariances with the major restriction of using equal to 0.5 sire and dam path coefficients of transmission. On the other hand, the transmissibility model offers more flexibility with sire and dam path coefficients of transmission than can vary from 0 to 1, although their sum must always be < 1. This higher flexibility resulted in a closer fit of the transmissibility model with the simulated model compared with the animal model in different situations as shown in Figure 5 and Figure 6. When data were simulated under the assumption that the covariance between relatives is of genetic origin only, the path transmission coefficients (sire and dam) estimated with the transmissibility model were close but < 0.5. This apparent underestimation is the consequence of the bounded parameter space for the sum of the two path coefficients of transmission. Indeed, pure genetic inheritance correspond to a transmissibility model with parameters at the boundary of the parameter space (sum of the sire and dam path coefficients of transmission = 1). The distribution of the estimated sum of these two path coefficients over the simulation is thus truncated to the right with a point mass at 1. This problem of parameter estimation at the boundary of parameter space has been reported previously (Kopylev 2012). It is thus not possible to obtain an average estimation of the two parameters at 0.5. However, the transmissibility model was able to correctly estimate the covariance between relatives and correctly assess that the inheritance was of genetic origin only, as confirmed by the realized type I error obtained, which was close to 5%. The predicted transmissible potentials were similar to the breeding values obtained with the animal model. The transmissibility model is thus able to model appropriately the genetic transmission of phenotypes. When data were simulated under the assumption that the covariance between relatives is of both genetic and nongenetic origin, the transmissibility model was able to correctly estimate sire and dam transmissibilities in all situations, which was not the case of the animal model, especially when the transmissibilities were very different (dataset 4). On the one hand, the flexibility of the transmissibility model that results from the possibility of using different values for sire and dam path coefficients of transmission enables it to adjust the estimated regression between relatives to the simulated values, even if the simulation model is different. Of course, all the regression coefficients could not be estimated exactly, especially when the sources of inheritance were diverse and of non-negligible importance (i.e., the simulated model differed greatly from the estimation model; dataset 4). In such cases, the limitations of the transmissibility model that arise from the use of equal transmitted variance for both sexes become apparent. However, the bias in the estimation of part of the regression coefficients is low, and involves coefficients of low values. On the other hand, to fit as close as possible to the simulated regression coefficients with its intrinsic constraints, the animal model estimated a genetic variance that resulted in an estimated equal sire and dam transmissibility between the sire- and dam-simulated transmissibilities. This phenomenon is particularly apparent when the disequilibrium between sire and dam transmissibilities is important. As a consequence, the estimations of the different regressions between other relatives obtained with this approach are biased. Thus, the transmissibility model better estimate covariances between relatives than the animal model in the case of nongenetic inheritance.

Nevertheless, the greater flexibility of the transmissibility model compared with the animal model has certain disadvantages. Indeed, the estimation of the path coefficients of transmission in the transmissibility model is more time-consuming than parameter estimation in the animal model. We evaluated that the CPU time for one iteration of convergence on a linux system and intelXeonE5-2698v3 processor is proportional to the cube of the number of individuals in the pedigree . To overcome this computation time problem, it is preferable to estimate path coefficients on a representative subset of the data, and to fix them to these estimated values when applying the transmissibility model to a large dataset.

Correlations between true and predicted transmissible values were similar for the animal model and the transmissibility model. The incorrect estimation of some regressions between relatives with the animal model did not influence these correlations. This is because these correlations were mainly dependent on the information used to build the evaluations and not on the regression coefficient and they were the same for both models. The high correlations (1, 0.98, 0.96, and 0.96 for datasets 1–4) between EBV obtained with the animal model and predicted transmissible values obtained with the transmissibility model confirm that, opposite to expectation, the animal model is unable to isolate the genetic component of inherited factors; genetic and nongenetic sources of inheritance being integrated in the EBV. Thus, when there are nongenetic sources of inheritance, selection based on EBV obtained with the animal model does not correspond to a pure genetic selection, but rather to selection on transmissible values. In this context, the limits of the animal model were observed on the bias. This bias was important and leads to an overestimation or an underestimation of the transmissible values depending on the source of information available: the performance of the paternal or maternal half-siblings, the sire’s or the dam’s own performance. Therefore, in practice, if the animal model performs well when very standardized selection schemes are used, with very homogeneous selection information between candidates, and in the absence of overlapping generations, it would not in all other conditions. This was probably the reason for the slight discrepancy between correlations for the two models for dataset 4 when all nonphenotyped animals are combined in a single set even though the information available for each individual is different; whereas the correlations are similar when the same kind of information is available. In the presence of overlapping generations, the biased estimations obtained do not ensure a fair comparison between candidate animals. In this case, which is common in real-life situations, the transmissibility model will be more effective than the animal model for selecting animals on their transmissible potential to improve the trait of interest in the selected population, provided that the nongenetic inheritance of this trait is not negligible. It should be noted that, when selection is relaxed, unlike with pure genetic selection, part of the benefit on the transmissible potential achieved by previous selection will theoretically gradually disappear, and only genetic progress will be maintained. This proportion depends on the relative importance of the nongenetic effect in the inherited factors. Indeed, nongenetic effects are diluted in future generations (Tal ), resulting in a decline to 0 of their average effects when there is no selection. The long-term benefit of selection will thus depend mainly on the genetic benefit, which will be conserved even when selection is relaxed. Additional calculations on the simulated data showed that the correlation between the true breeding values and the sire- or dam-estimated transmissible potentials, and their bias, were in the same range for the transmissible and the animal models (results not shown), indicating that, even if selection is relaxed, the long-term benefit of the transmissibility model will be higher than that obtained with the animal model. It should be noted that these results were obtained for a restricted set of simulated data (datasets 1–4); additional investigations are necessary to ensure that the same conclusion can be drawn in all situations.

Evidence of the vertical transmission of nongenetic effects has been reported widely in livestock (Abecia ; Sanga 2010; Braunschweig ; Feeney ). However, the quantification of nongenetic inheritance for traits under selection is rare. Paiva reported an epigenetic heritability of 0.10 for body weight in meat quails, while Difford estimated simultaneously that significant parts of the variance for CH4 emission in cattle were explained by genetics and the microbiota (0.21 and 0.13, respectively). To our knowledge, the part of variance explained by culture/behavioral inheritance has never been reported in the literature for livestock species. The limited amount of data on the quantification of nongenetic inheritance in livestock is probably due to the novelty of the subject for these species and the lack of easy tools for estimating it. The transmissibility model will help overcome this obstacle by estimating a deviation from the pure genetic inheritance using routinely recorded field data (phenotype and pedigree). Analysis of the power of the transmissibility model to detect nongenetic inheritance demonstrated that its power depends on pedigree size, the relative importance of the nongenetic inheritance and also the difference between the sire and dam path coefficients of transmission. There is nothing we can do for the two last components, but we suggest using populations with complex pedigree structures, because more information will be provided for the same computing time.

We designed our transmissibility model to take into account nongenetic sources of inheritance in order to estimate the path coefficients of transmission between parents and offspring, and to predict the transmissible potential of individuals. Estimations are based on the covariances between different types of relatives. It should be noted that phenomena other than nongenetic inherited factors reported here, such as the X-chromosomal inheritance, parent-of-origin effects, mitochondrial inheritance, etc., can cause deviation from the law of transmission assumed in the animal model (Hutchison ; Fernando and Grossman 1990; Neugebauer ). Disentangling all these phenomena with no information other than pedigrees and phenotypes is challenging (and certainly unfeasible), and has never been attempted before. Another phenomenon that could create confusion with nongenetic effects is the maternal effect. Indeed, maternal genetic effects induce different covariances between the offspring and the mother, and between the offspring and the father (Table D1 in Appendix D), that can mimic the transmission of nongenetic effects. To illustrate this point, we performed an additional simulation of 100 replicates (results not shown) in which the phenotypes were generated with the same parameters as in dataset 3 plus a maternal effect with a genetic variance =5 (independent from direct genetic effects). The application of the transmissibility model to these simulated data resulted in a higher estimated value for the dam coefficient of transmission than obtained for dataset 3 (0.53 vs. 0.44) while the estimates of the sire coefficient of transmission and remained similar to those obtained for dataset 3. This result illustrates that maternal effects have been wrongly integrated into the dam transmissible potential. To avoid such confusion between maternal genetic effects and nongenetic effects, it is possible to include maternal genetic effects in the transmissibility model as an additional random effect (demonstration of the structural identifiability of the model is provided in Appendix E). The application of the transmissibility model with maternal genetic effects to the aforementioned simulated data (i.e., dataset 3 + maternal genetic effects) resulted in estimates of the sire/dam coefficients of transmission and similar to those obtained with the transmissibility model applied to dataset 3, while the estimates of the maternal genetic variance were in line with those simulated (4.7 on average). This result confirms that it is possible to disentangle maternal genetic effects and nongenetic effects with the transmissibility model.

Table D1 Regression coefficient between relatives depending on the model

Covariance between offspring and	Formula for the regression coefficients
	Simulated model	Simulated value				Transmissibility model	Animal model	Maternal model
	y_i=x_iβ+a_i+epi_i+sp_i+e_i					y_i=x_iβ+t_i+e_i	y_i=x_iβ+u_i+e_i	y_i=x_iβ+u_i+m_d+e_i
	epi_i=12λ_sepi_s+12λ_depi_d+ε_epi,i
	sp_i=δsp_d+ε_sp,i
Sire	12σa2+12λ_sσepi2	0.157	0.138	0.138	0.085	ω_sσt2	12σu2	12σu2+14σ_u,m
Dam	12σa2+12λ_dσepi2+δσsp2	0.157	0.176	0.176	0.229	ω_dσt2	12σu2	12σu2+54σ_u,m+12σm2
Paternal grand dam	14σa2+14λ_sλ_dσepi2	0.079	0.067	0.058	0.041	ω_sω_dσt2	14σu2	14σu2+18σ_u,m
Maternal grand dam	14σa2+14λd2σepi2+δ^2σsp2	0.079	0.082	0.077	0.121	ωd2σt2	14σu2	14σu2+58σ_u,m+14σm2
Paternal half-sib of the sire (paternal uncle 1)	18σa2+18λs3σepi2	0.039	0.033	0.018	0.007	ωs3σt2	18σu2	18σu2
Maternal half-sib of the sire (paternal uncle 2)	18σa2+18λ_sλd2σepi2	0.039	0.033	0.025	0.020	ω_sωd2σt2	18σu2	18σu2+14σ_u,m
Paternal half-sib of the dam (maternal uncle 1)	18σa2+18λs2λ_dσepi2	0.039	0.033	0.021	0.011	ωs2ω_dσt2	18σu2	18σu2+14σ_u,m
Maternal half-sib of the dam (maternal uncle 2)	18σa2+18λd3σepi2+δ^3σsp2	0.039	0.040	0.035	0.066	ωd3σt2	18σu2	18σu2+12σ_u,m+12σm2
Paternal half-sib	14σa2+14λs2σepi2	0.079	0.065	0.049	0.023	ωs2σt2	14σu2	14σu2
Full-sib	12σa2+14(λs2+λd2)σepi2+δ^2σsp2	0.157	0.148	0.127	0.144	(ωs2+ωd2)σt2	12σu2	14σu2+σ_u,m+σm2

It should be noted that our transmissibility model assumes that the transmitted variance is the same for all individuals in the population, and is constant across generations. However, epigenetic, microbiota, and cultural effects are subject to change with environmental conditions (Aguilera ; Spor ). It should be possible to take into account this influence of the environment on part of the heritable factors by considering the transmitted variance as a function of the environment or of time in a manner similar to that used in the structured antedependence model (SAD) (David ) (the transmissibility model is a particular form of the SAD model).

Finally, our model combines the different sources of inheritance into a single transmissible value. However, the OWN Fortran program we propose to run the transmissibility model allows the different sources of inheritance to be separated if required, especially genetic and epigenetic inheritance, as described by Varona or Paiva ,b). Furthermore, unlike previous methods proposed, the program that we propose has the advantage of (1) using a frequentist method for the estimations (faster than the Bayesian approach), (2) estimating the path coefficients (conversely to the grid search proposed by Paiva ), and (3) allowing for different path coefficients for sires and dams, which has never been proposed before.

Table A1 List of covariances between phenotypes of relatives useful for identifiability of the parameters in Equation 1

	Covariance between	Formula for the covariance given Equation 1
COVS-O	Sire -offspring	[12r(1−θ)+λ_s(1−r)(1−θ)]α
COVD-O	Dam-Offspring	[12r(1−θ)+λ_d(1−r)(1−θ)+δθ]α
COVPGS-O	Paternal grand sire-offspring	[14r(1−θ)+λs2(1−r)(1−θ)]α
COVMGS-O	Maternal grand sire-offspring	[14r(1−θ)+λ_sλ_d(1−r)(1−θ)]α
COVMGD-O	Maternal grand dam-offspring	[14r(1−θ)+λd2(1−r)(1−θ)+δ^2θ]α
COVPU-O	paternal half-sib of the sire (paternal uncle 1) - offspring	[18r(1−θ)+λs3(1−r)(1−θ)]α
COVPUGD-O	maternal half-sib of the sire (paternal uncle 2)	[18r(1−θ)+λ_sλd2(1−r)(1−θ)]α

Table E1 List of covariances between phenotypes of relatives useful for parameter identifiability in the transmissibility model with maternal effects

	Covariance between	Formula for the covariance
COVS-O	Sire-offspring	ω_sσt2+ω_sω_dσ_t,m
COVD-O	Dam-offspring	ω_dσt2+(ωd2+1)σ_t,m+0.5σm2
COVPGD-O	Paternal grand dam-offspring	ω_sω_dσt2+ω_sωd2σ_t,m
COVPU-O	Paternal half-sib of the sire (paternal uncle 1) - offspring	ωs3σt2
COVPHS	Paternal half-sibs	ωs2σt2