# Coefficient of Inbreeding - implementation issue

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The formula for the coefficient of inbreeding is as follows…

I have a family tree going back 9 generations. Say I find a common ancestor X in the 4th generation on the mothers side and in the 5th generation on the fathers side. I can work out that constituent of F… but what do I do about all the ancestors of X (they will all also be common ancestors). Do I also work out their values and add them to F?… or do I ignore them? The descriptions of the COI I have seen do not make this clear.

The \$F_i\$ term is the term that accounts for inbreeding of your common ancestor X. If that common ancestor is not inbred then the term is 0 and the calculation is a little easier. Using the example you have given:

``X ─┬─ Y │ ┌──┴──┐ GGGF GGM │ │ GGF │ │ GM GF │ │ │ F ─┬─ M │ Mick``

In this case, the common ancestors are not inbred so the \$F_i\$ terms go away (\$1+F_i = 1\$). There are two paths of common ancestry (via X and Y), therefore the calculation of the two paths from the father (F) and mother (M) to X and Y is:

\$F_{Mick} = frac12^{n_{F,X}+n_{M,X}+1}(1+F_X)+frac12^{n_{F,Y}+n_{M,Y}+1}(1+F_Y) \$

\$F_{Mick} = frac12^{4+3+1}(1+0)+frac12^{4+3+1}(1+0) \$

\$F_{Mick} = frac{1}{256} + frac{1}{256} = 0.0078 = 0.78\% \$

In a more complex case, such as where ancestor X was inbred, \$F_i\$ cannot be ignored. In the following example, X's parents were first cousins:

``U ─┬─ V │ ┌──┴──┐ XGF XGM │ | XF─┬─XM │ X ─┬─ Y │ ┌──┴──┐ GGGF GGM │ │ GGF │ │ GM GF │ │ │ F ─┬─ M │ Mick``

For this, we must first calculate \$F_X\$ (i.e. how inbred X is):

\$F_X = frac12^{n_{XF,U}+n_{XM,U}+1}(1+F_U)+frac12^{n_{XF,V}+n_{XM,V}+1}(1+F_V) \$

\$F_X = frac12^{2+2+1}(1+0)+frac12^{2+2+1}(1+0)\$

\$F_X = frac{1}{32} + frac{1}{32} = 0.0625 = 6.25\% \$

Then we can input this \$F_X\$ into the same calculation as done before for the simple case:

\$F_{Mick} = frac12^{n_{F,X}+n_{M,X}+1}(1+F_X)+frac12^{n_{F,Y}+n_{M,Y}+1}(1+F_Y) \$

\$F_{Mick} = frac12^{4+3+1}(1+0.0625)+frac12^{4+3+1}(1+0)\$

\$F_{Mick} = 0.00415 + frac{1}{256} = 0.0081 = 0.81\% \$

A good explanation of these calculations with further examples can be found here: The Coefficient of Inbreeding (F) and its applications. The calculations can get quite complicated even with relatively simple inbreeding.

## Coefficient of Inbreeding - implementation issue - Biology

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Evolution of Inbreeding Coefficients and Effective Size in the Population of Saguenay Lac-St.-Jean (Québec)

Soufia Mourali-Chebil, * Evelyne Heyer *

* Unité Eco-Anthropologie MNHN/CNRPS/P7 UMR5145, Musée de l'Homme, Paris, France.

## Genetics of consanguinity and inbreeding in health and disease

Context: Inbreeding increases the level of homozygotes for autosomal recessive disorders and is the major objective in clinical studies. The prevalence of consanguinity and the degree of inbreeding vary from one population to another depending on ethnicity, religion, culture and geography. Global epidemiological studies have revealed that consanguineous unions have been significantly associated with increased susceptibility to various forms of inherited diseases.

Objective: The study aimed to determine the role of consanguinity in human health and to highlight the associated risks for various diseases or disorders.

Methods: PubMed and Google Scholar search engines were used to explore the published literature on consanguinity and its associated risks using the key words “consanguinity”, “prevalence”, “inbreeding depression”, “coefficient of inbreeding”, “child health”, “mortality”, “human health”, “homozygosity” and “complex diseases” in different combinations. The studies were screened for eligibility on the basis of their epidemiological relevance.

Results: This comprehensive assessment highlights the deleterious consequences in populations with a higher prevalence of consanguinity among different countries worldwide.

Conclusions: To avoid the inbreeding load there is the need to improve socioeconomic and educational status and to increase public awareness of reproductive health and anticipated deleterious effects. Pre-marital and pre-conception counselling of consanguineous populations should be an integral part of health policy to train people and make people aware of its harmful consequences. Furthermore, runs of homozygosity (ROH) and whole-exome sequencing (WES) are useful tools in exploring new genomic signatures for the cause of inbreeding depression.

## How did we get here?

By selective breeding (the mating of two individuals with high genetic merit), we aim to increase the homozygosity of the “good” genes. This has worked well for us. Since the introduction of A.I., and especially genomic selection, genetic progress in dairy cattle has rapidly increased. With the introduction of genomic selection, it was predicted genetic progress would increase due to the considerable reduction of the generation interval as well as the increase of accuracy.

And this prediction certainly came true. It was also predicted, however, that inbreeding would decrease. After all, bull studs could now obtain DNA profiles from every dairy calf they’d want, giving them the opportunity to seek out animals with high genetic merit yet low relatedness. This would increase the chance of proliferating the good genes while avoiding the bad. That prediction wasn’t as accurate. What happened?

What researchers had not taken into account was the incredible rise in economic value of young, high-merit genomic animals. Genomic selection worked so well for dairy cattle the dollar value of elite animals rose to levels previously unimaginable.

Quickly, the race for the highest GTPI animals led to a higher degree of mating close relatives. This wasn’t just a matter of economic value outweighing the risk of inbreeding as much as it was the consequence of a highly competitive market.

Interbreeding family members became a fast track to high-merit animals, as accepting a genetic lag with the competitor wasn’t an option. The current factual situation is: Where genetic theory recommends an inbreeding level of 6 percent, we are more than double that in young genomic bulls and have just passed 7 percent in the national U.S. cow population.

By comparison, the current average level in the Dutch cow population, with a much less competitive market, is only 4.5 percent.

## Abstract

Data obtained from the French Kennel Club and the Fichier National Canin were used to estimate the effect of inbreeding on average litter size and survival in seven French breeds of dog. Depending on the breed, litter sizes were 3.5–6.3 puppies and longevities were 7.7–12.2 years. Estimated heritabilities were 6.0–10.9% for litter size and 6.1–10.1% for survival at 2 years of age. Regression coefficients indicated a negative effect of inbreeding on both individual survival and litter size. Although the impact of baseline inbreeding within breeds appears to be limited, the improper mating of close relatives will reduce biological fitness through significant reduction of litter size and longevity.

## Inbreeding coefficients and coalescence times

This paper describes the relationship between probabilities of identity by descent and the distribution of coalescence times. By using the relationship between coalescence times and identity probabilities, it is possible to extend existing results for inbreeding coefficients in regular systems of mating to find the distribution of coalescence times and the mean coalescence times. It is also possible to express Sewall Wright's F ST as the ratio of average coalescence times of different pairs of genes. That simplifies the analysis of models of subdivided populations because the average coalescence time can be found by computing separately the time it takes for two genes to enter a single subpopulation and time it takes for two genes in the same subpopulation to coalesce. The first time depends only on the migration matrix and the second time depends only on the total number of individuals in the population. This approach is used to find F ST in the finite island model and in one- and two-dimensional stepping-stone models. It is also used to find the rate of approach of F ST to its equilibrium value. These results are discussed in terms of different measures of genetic distance. It is proposed that, for the purposes of describing the amount of gene flow among local populations, the effective migration rate between pairs of local populations, M^ , which is the migration rate that would be estimated for those two populations if they were actually in an island model, provides a simple and useful measure of genetic similarity that can be defined for either allozyme or DNA sequence data.

Studies on horse breeding and genetics are often dispersed in many scientific journals, and little attention is generally paid to topics such as horse genetic improvement and horse genomics, despite horses have been selected over time for a number of traits like speed, gaits, jumping performances, strength, or for more conventional traits like morphology or temperament. Worldwide, many horse breeds have been selected for preserving and improving traits of interest for sport performances or work. In addition, most horse breeds are small close populations with high levels of inbreeding and homozygosis, requiring appropriate breeding management. Novel traits such as fertility, longevity, and health have been recently included or should be included in breeding decisions. Furthermore, in spite of the current lack of genomic data, an implementation of genomic selection in equine management could provide substantial benefits, because of the long generation interval typical of horses. Additionally, new genomic features have increased interest in analyzing genetic diversity among horse breeds, and in attaining deep knowledge on the functionality of single or groups of genes involved in the expression of economical important traits. Gene networking or studies on other &ldquoomics-science&rdquo are also hot topics in horse breeding and genetics. We hereby are glad to invite authors to submit original manuscripts that address any aspect related to horse breeding and genetics. Topics of interest include the genetic improvement of novel and traditional traits, genetic correlations, and the response to selection genome-wide association studies, genomic selection, and pathway analyses of traits and genetic diversity, optimal contribution selection, characterization of horse genome variation, and studies on gene functionality.

Dr. Cristina Sartori
Prof. Roberto Mantovani
Guest Editors

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## Genetic Effects of Multiple Generations of Supportive Breeding

Institute of Cell, Animal and Population Biology, University of Edinburgh, West Mains Road, Edinburgh EH9 3JT, United Kingdom

Division of Population Genetics, Stockholm University, S-106 91 Stockholm, Sweden, email [email protected]

Institute of Cell, Animal and Population Biology, University of Edinburgh, West Mains Road, Edinburgh EH9 3JT, United Kingdom

Division of Population Genetics, Stockholm University, S-106 91 Stockholm, Sweden, email [email protected]

### Abstract

Abstract: The practice of supporting weak wild populations by capturing a fraction of the wild individuals, bringing them into captivity for reproduction, and releasing their offspring into the natural habitat to mix with wild ones is called supportive breeding and has been widely applied in the fields of conservation biology and fish and wildlife management. This procedure is intended to increase population size without introducing exogenous genes into the managed population. Previous work examining the genetic effects of a single generation of supportive breeding has shown that although a successful program increases the census population size, it may reduce the genetically effective population size and thereby induce excessive inbreeding and loss of genetic variation. We expand and generalize previous analyses of supportive breeding and consider the effects of multiple generations of supportive breeding on rates of inbreeding and genetic drift. We derived recurrence equations for the inbreeding coefficient and coancestry, and thereby equations for inbreeding and variance effective sizes, under three models for selecting captive breeders: at random, preferentially among those born in captivity, and preferentially among those born in the wild. Numerical examples indicate that supportive breeding, when carried out successfully over multiple generations, may increase not only the census but also the effective size of the supported population as a whole. If supportive breeding does not result in a substantial and continuous increase of the census size of the breeding population, however, it might be genetically harmful because of elevated rates of inbreeding and genetic drift.

### Abstract

Resumen: La práctica de apoyar poblaciones silvestres débiles mediante la captura de una fracción de los individuos silvestres, su cautiverio para la reproducción y la liberación a su descendencia en habitas naturales para que convivan con organismos silvestres se conoce como reproducción de apoyo y se ha empleado ampliamente en la biología de la conservación y en el manejo de pesca y vida silvestre. Este procedimiento tiene la intención de incrementar el tamaño de la población sin introducir genes exógenos en la población bajo manejo. Trabajos previos sobre los efectos genéticos de una sola generación de reproducción de apoyo muestran que, aunque un programa exitoso incrementa el tamaño poblacional, puede reducir la población genéticamente efectivae inducir así un exceso de consanguinidad y pérdida de variación genética. Expandimos y generalizamos análisis previos de la reproducción de apoyo y consideramos los efectos de múltiples generaciones de reproducción de soporte en las tasas de consanguinidad y de deriva génica. Derivamos ecuaciones de recurrencia para el coeficiente de consanguinidad y de coancestría, y por tanto ecuaciones de tamaños efectivos de consanguinidad y de varianza, para tres modelos de selección de reproductores en cautiverio : aleatoria, preferentemente entre los nacidos en cautiverio y preferentemente entre los nacidos en libertad. Los ejemplos numéricos indican que la reproducción de apoyo, cuando es exitosa en múltiples generaciones, puede ser favorable para el incremento no solo del tamaño, sino del tamaño efectivo de la población soportada en su conjunto. Sin embargo, si la reproducción de soporte no resulta en un incremento sustancial y continuo del tamaño de la población, puede ser genéticamente dañina debido a las altas tasas de consanguinidad y de deriva genética.