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Predictive Model and Software for Inbreeding-Purging Analysis of Pedigreed Populations

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Predictive Model and Software for Inbreeding-Purging Analysis of Pedigreed Populations G3: Genes|Genomes|Genetics Early Online, published on September 7, 2016 as doi:10.1534/g3.116.032425 1 Predictive model and software for inbreeding-purging analysis of pedigreed populations 2 3 AURORA GARCÍA-DORADO*, JINLIANG WANG† and EUGENIO LÓPEZ-CORTEGANO* 4 5 * Departamento de Genética, Universidad Complutense, Madrid 28040, Spain 6 † Institute of Zoology, Zoological Society of London, London NW1 4RY, United Kingdom 7 © The Author(s) 2013. Published by 1the Genetics Society of America. 8 Running title: Purging in pedigreed data 9 10 Key words: 11 - Inbreeding depression 12 - purging coefficient 13 - rate of inbreeding depression 14 - inbreeding load 15 - logarithmic fitness 16 17 Corresponding author: 18 Aurora García-Dorado 19 Departamento de Genética, Facultad de Biología, Universidad Complutense, Ciudad 20 Universitaria, Madrid 28040, Spain 21 Phone number: 0034 3944975 22 Email address: [email protected] 23 24 2 25 Abstract 26 The inbreeding depression of fitness traits can be a major threat for the survival of 27 populations experiencing inbreeding. However, its accurate prediction requires taking into 28 account the genetic purging induced by inbreeding, which can be achieved using a “purged 29 inbreeding coefficient”. We have developed a method to compute purged inbreeding at the 30 individual level in pedigreed populations with overlapping generations. Furthermore, we derive 31 the inbreeding depression slope for individual logarithmic fitness, which is larger than that for 32 the logarithm of the population fitness average. In addition, we provide a new software PURGd 33 based on these theoretical results that allows analyzing pedigree data to detect purging and to 34 estimate the purging coefficient, which is the parameter necessary to predict the joint 35 consequences of inbreeding and purging. The software also calculates the purged inbreeding 36 coefficient for each individual, as well as standard and ancestral inbreeding. Analysis of 37 simulation data show that this software produces reasonably accurate estimates for the 38 inbreeding depression rate and for the purging coefficient that are useful for predictive purposes. 39 40 3 41 Introduction 42 Inbreeding depression for fitness, due to the increase in the frequency of homozygous 43 genotypes for (partially) recessive deleterious alleles under inbreeding, is a major threat for the 44 survival of small populations (Falconer & Mackay 1996, Saccheri et al. 1998, Hedrick & 45 Kalinowski 2000, Frankham 2005). However, as these alleles become more exposed under 46 inbreeding, an increase in the efficiency of natural selection against them is also expected, which 47 is known as genetic purging and tends to reduce the frequency of deleterious alleles and, 48 consequently, the fitness decline induced by inbreeding (Templeton & Read 1984, Hedrick 1994, 49 Ballou 1997, García-Dorado 2012, 2015). 50 The first models developed to detect the consequences of purging on inbreeding depression 51 from pedigree data accounted for purging by using an ancestral purging coefficient Fa that 52 represents the proportion of an individual’s genome that is expected to have been exposed to 53 homozygosis by descent in at least one ancestor (Ballou 1997, Boakes & Wang 2005). The 54 rational is that, due to genetic purging, inbred individuals with inbred ancestors would have 55 fewer deleterious alleles than individuals with the same inbreeding but non-inbred ancestors. 56 More recently, a theoretical Inbreeding-Purging (IP) approach has been developed that 57 predicts the evolution of fitness under inbreeding by taking purging into account by means of a 58 purged inbreeding coefficient g. This IP model considers that purging acts against a purging 59 coefficient (d) that quantifies the component of the deleterious effects that are only expressed 60 under inbreeding (García-Dorado 2012). For a single locus model, d represents the per copy 61 excess of the deleterious effect in the homozygous over that expected on an additive hypothesis, 62 and its value ranges from d=0 (no purging) to d=0.5 (purging against recessive lethal alleles). In 63 practice, as d varies across loci, a single value, known as the effective purging coefficient 64 (denoted by de in García-Dorado 2012; here denoted by d for simplicity), can be used to compute 4 65 approximate predictions for the overall consequences of purging over the whole genome. 66 Estimating this effective d value is of main interest as it will provide a measure of the purging 67 occurred, and will allow us to use the model to predict the expected evolution of fitness. 68 Until now, the only empirical estimates of the purging coefficient d have been obtained 69 from the evolution of fitness average in Drosophila bottlenecked populations (Bersabé & García- 70 Dorado 2013; López-Cortegano et al. 2016). However, in conservation practice, fitness data are 71 often available for pedigreed populations. Two versions of the IP model were originally 72 proposed, one aimed to predict mean fitness as a function of the number of generations under a 73 reduced effective population size Ne, the other one aimed to predict individual fitness from 74 pedigree information. Nonetheless, the latter version was only developed for data with non- 75 overlapping generations, which imposes serious limitations to its use in experimental and 76 conservation practice. 77 Here we extend the IP model to compute the purged inbreeding coefficient g for 78 individuals in pedigrees with overlapping generations. Furthermore, we derive a new expression 79 that gives the expected individual log-fitness as a function of g and of the initial inbreeding load 80 , deriving the slope of inbreeding depression for individual logarithmic fitness, which is larger 81 than that for the logarithm of average population fitness. In addition, we present the new free 82 software PURGd, based on this IP approach, that is able to use data for fitness traits in pedigreed 83 samples to test for purging and to estimate the corresponding effective purging coefficient d. 84 This software also estimates the inbreeding depression rate for individual fitness, and computes 85 the standard (F), ancestral (Fa) and purged (g) inbreeding coefficients for the pedigreed 86 individuals. 87 88 5 89 THE MODEL 90 The rate of inbreeding depression estimated from individual fitness. 91 In order to analyze and interpret the consequences of inbreeding and purging at an 92 individual level, we must first consider the relationship between individual fitness and 93 inbreeding in a neutral model with no natural selection. 94 Assume a population where a number of deleterious alleles segregate at a low frequency q 95 at different loci acting multiplicatively on fitness. Here onwards we will concentrate just on 96 (partially) recessive deleterious alleles, which are assumed to be responsible for inbreeding 97 depression. Each locus has two alternative alleles, the wild one and the mutant deleterious allele. 98 It has three genotypes, with average fitness 1, 1-hs and 1-s for the wild homozygous genotype, 99 the heterozygous genotype and the deleterious homozygous genotype, respectively. Therefore, 100 the population inbreeding load, which can be measured by the number of lethal equivalents 101 (Morton et al. 1956), is 102 =2dq(1-q), (1) 103 where d=s(1/2-h), and the sum is over all the relevant loci. 104 For simplicity, we will assume that the initial frequency of each deleterious allele is small 105 enough that homozygous genotypes are only produced due to inbreeding. Furthermore, in this 106 section, we will also assume completely recessive gene action (h=0; s=2d). This assumption 107 smooths the explanation below, but is not necessary for the validity of the conclusions. 108 After some inbreeding, the fitness of an individual that is homozygous by descent for 109 deleterious alleles at n loci is 휀 110 푊 = 푊푚푎푥(1 − 휀)(1 − 2푑) , (2) 6 111 where Wmax is the maximum possible fitness value and is the proportional reduction of the 112 fitness of that individual due to all kind of environmental and genetic factors, excluding 113 inbreeding depression. 114 If the inbreeding load is due to many loosely linked deleterious loci and deleterious alleles 115 segregate at low frequency, the number ni of deleterious alleles in homozygosis for an individual 116 i with standard Wright´s inbreeding coefficient Fi should be Poisson distributed. Since the 117 probability of being homozygous for a deleterious allele in non-inbred individuals is assumed to 118 be negligible, the expected value of this number should be is E(ni) = Fi q(1-q) (Falconer & 119 Mackay 1996). Thus, substituting q(1-q) from Equation (1), we obtain that the mean of this 120 Poisson distribution is 121 = E(ni) = Fi / 2d . (3) 122 Therefore, from Equation 2, and assuming that and F are independent, the expected 123 fitness of an individual i that has genealogical inbreeding Fi is −휆 푛 ∞ 푒 휆 푛 124 E(Wi) = E(W0) ∑ (1 − 2푑) 푛=0 푛! 125 where E(W0) = E[Wmax(1-)] is the expected fitness of a non-inbred individual. The equation 126 above can be rewritten as −휆 푛 휆2푑 −휆2푑 ∞ 푒 휆 푒 푛 127 E(Wi) = E(W0) 푒 ∑ (1 − 2푑) , 푛=0 푛! 128 and can be rearranged as −휆(1−2푑) 푛 −휆2푑 ∞ 푒 [휆(1−2푑)] 129 E(Wi) = E(W0) 푒 ∑ . 푛=0 푛! ∞ −휆(1−2푑) 푛 130 Noting that ∑푛=0 푒 [휆(1 − 2푑)] /푛! adds up all the probabilities for a Poisson 7 131 distribution with mean (1-2d) (i.e., it equals 1), and since = Fi / 2d (Equation 3), we obtain 132 the exponential expression −훿퐹 133 E(Wi) = E(W0) 푒 푖 . (4) 134 and, similarly, the average fitness of a population with average inbreeding Ft in generation t , as 135 far as the number of loci homozygous for a deleterious allele per individual can be assumed to be 136 Poisson distributed with mean = Ft / 2d, is −훿퐹 137 E(Wt ) = E(W0) 푒 푡 , (5) 138 In order to estimate from observed inbreeding depression, logarithms are usually taken in 139 Equations 4 or 5 to obtain a linear model of the kind ln(W) = ln(W0) - F.
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