Heredity (2006) 97, 346–354 & 2006 Nature Publishing Group All rights reserved 0018-067X/06 $30.00 www.nature.com/hdy The genetic analysis of family structured inbreeding depression studies JK Kelly and MK Tourtellot Department of Ecology & Evolutionary Biology, University of Kansas, 1200 Sunnyside Ave, Lawrence, KS 66045-7534, USA An important question emerging from theoretical studies of explicit genetic model. These yield parameter estimates mating system evolution is whether the fitness of a randomly and provide the likelihoods necessary to test hypotheses, for extracted, fully inbred genotype will exceed the mean of example, whether population-level ID is nonzero. Finally, we outbred individuals. We introduce two statistics (I1 and I2) describe a new publicly available computer program titled related to the probability of extracting a high line. I1 and I2 can ‘IDG’ (Inbreeding Depression Genetics) to execute these be estimated from the family structured experimental designs procedures. typically used to estimate inbreeding depression (ID). Heredity (2006) 97, 346–354. doi:10.1038/sj.hdy.6800879; Maximum likelihood procedures are developed from an published online 9 August 2006 Keywords: Collinsia; inbreeding; inbreeding depression; mating systems; Mimulus; self-fertilization Introduction in that selfed progeny are genetically identical (or nearly so) to their parents. If the constellation of alleles fixed Inbreeding depression (ID) is the decline of fitness- within the lineage is favorable, it may displace the related traits that frequently occurs with inbreeding. The background population of sexual genotypes (Lande and magnitude of ID and its genetic basis are critical factors Schemske, 1985). Of course, the spread of the selfing in the evolution of plant mating systems and also have genotype is greatly facilitated if it can also distribute important implications for agriculture and conservation pollen to other plants while self-fertilizing its own ovules (Darwin, 1876; Charlesworth and Charlesworth, 1987; thus exploiting the ‘cost of sex’ (Maynard Smith, 1978). Lynch et al, 1995; Keller and Waller, 2002). Historically, ID The preceding description, in which a novel mutation has been measured at the population level by comparing induces complete selfing, is idealized. Genetic modifiers the mean phenotypes of inbred and outbred individuals. that cause incremental increases in selfing rate will not More recently, interest has focused on ID at the level of become isolated within self-perpetuating lineages and individual families. ID is estimated from the difference in the evolutionary dynamics are a great deal more mean fitness between inbred and outbred individuals complicated (see Uyenoyama and Waller, 1991; Uye- within the same family, oftentimes the progeny of a noyama et al, 1993). However, the example does illustrate single maternal plant. Variation in ID among families has why interest has at least partially shifted from popula- been demonstrated within a variety of natural plant tion to family level. It also points to a conceptual populations (eg, Agren and Schemske, 1993; Carr et al, difficulty with measuring ID as a difference between 1997; Mutikainen and Delph, 1998; Chang and Rausher, inbred and outbred individuals within the same family. 1999; Vogler et al, 1999; Takebayashi and Delph, 2000; Over the long run, a selfing lineage will be competing Fishman, 2001; Rao et al, 2002; Stone and Motten, 2002). with the entire background population and not just the Why is family-level ID interesting? Some theoretical outbred progeny of that family. Low or even negative studies suggest that it may be a more important family-level ID can result because the mean fitness of determinant of mating system evolution than popula- outbred progeny from that family is unusually low. This tion-level ID (Campbell, 1986; Holsinger, 1988; Uyenoya- would not necessarily bode well for a selfing mutation ma et al, 1993; but also see Charlesworth et al, 1990; that happens to occur within that family. Shultz and Willis, 1995). Consider a plant population that From an empirical point of view, we would like to is self-compatible but predominantly outcrossing. A determine the probability that a randomly extracted, mutation occurs that induces complete self-fertilization. fully inbred genotype will have a mean fitness that If this mutation escapes immediate loss, it will become exceeds the mean of outbred genotypes within the fixed within an ‘inbred lineage’, a genotype that is fully population. Here, ‘random extraction’ implies that alleles homozygous for alleles residing in the ancestral outbred fix randomly within a lineage over successive genera- genotype. This genotype is essentially self-perpetuating tions of selfing and that the lineage is founded by a random outbred genotype. In fact, populations of inbred lines have been extracted from several model species and Correspondence: JK Kelly, Department of Ecology & Evolutionary Biology, measurements from these populations can directly University of Kansas, 1200 Sunnyside Ave, Lawrence, KS 66045-7534, USA. E-mail: [email protected] estimate this probability (Simmons and Crow, 1977; Received 26 December 2005; accepted 23 June 2006; published Takano et al, 1987; Hughes, 1997; Willis, 1999b). As we online 9 August 2006 discuss in greater detail below, most ID studies contain Focus on the family JK Kelly and MK Tourtellot 347 data sufficient for at least an indirect appraisal. The fraction of density in the ‘tail’ of the normal distribution generation of high performing inbred lines is relevant is a simple function of the standard deviation (here, the not only to mating system evolution, but also has direct tail is the collection of inbred lines that exceed the bearing on agriculture (Crow, 1987). outbred mean). Referring to tabulations of the standard We introduce two statistics (I1 and I2) related to the normal (eg, Rice, 1989, p 558), we find that the probability that a random inbred line will exceed the probability that a random line exceeds M is only 0.023 1 outbred mean phenotype in the population. Neither I1 if I2 ¼ 2. This probability increases to 0.159 if I2 ¼ 1 and to nor I2 yields the probability directly, except under 0.309 if I2 ¼ 2. specific assumptions about the fitness distribution across I1 and I2 are dimensionless ratios, a feature that inbred lines. However, each should be positively facilitates comparisons across different traits and studies. correlated with this probability and thus represents a However, there are important statistical issues associated useful abstraction of the genetic information resident with ratios of estimators (Rice, 1989, pp 146–147). In within ID experiments. In the second section of the particular, substantial bias is introduced if the denomi- paper, I1 and I2 are directly estimated from the observed nator has a large standard error (SE). This is unlikely to distribution of inbred line means within a large experi- be a difficulty for I1, because in all but the smallest mental study of Mimulus guttatus (Kelly, 2005a). Esti- experiments, the SE of M should be small relative to its mates from multiple traits support our contention that I1 estimated value. However, this need not be true for and I2 are correlated with the probability that a line will estimates of b. Estimates for I2 are ruled ‘suspect’ by the 1 exceed the outbred mean. The third part of the paper IDG program if the SE of b is greater than 4 its estimated develops methods for estimating I1 and I2 from three magnitude. standard experimental designs. The estimation proce- I1 requires that trait values are positive which should dures are based on an explicit genetic model and provide usually be true of measurement on their original scale. a likelihood ratio test for the presence of population-level However, log-transformation of fractional values, for ID. Finally, we describe a new computer program titled example, proportion surviving, will yield negative ‘IDG’ (Inbreeding Depression Genetics) to execute these values. This can be remedied simply by adding a procedures. constant to all measurements before calculating I1.A second issue concerns the logical derivation of I1.We assumed that inbreeding decreases trait values and that ID statistics high trait values are favorable. However, for characters Let M denote the outbred mean for the trait under such as time to sexual maturity, inbreeding may increase consideration. The mean value of fully inbred genotypes trait values and lower values may be favorable. For such is defined to be MÀb, where b is the ‘inbreeding load’. As situations, I1 can be adapted to predict the emergence of inbreeding typically depresses trait values, we expect low lines simply by reversing the sign of b before that b should usually be positive. Finally, let VGI denote substituting it into Equation (1), thus insuring that the (genetic) variance among fully inbred lines. Given (MÀb)2oM2. these definitions, we will consider the following two aggregate statistics: 2 A direct study of I1 and I2 in M. guttatus ðM À bÞ þ VGI I1 ¼ ð1Þ A large collection of inbred lines have been extracted M2 from a single natural population of M. guttatus (Willis, and 1999a; Kelly and Arathi, 2003). Each line was initiated ffiffiffiffiffiffiffiffi p from a single outbred genotype and synthesized by VGI I2 ¼ ð2Þ successive generations of single seed descent (self- b fertilization with random selection of progeny). These Higher values for I1 and I2 imply an increased lines each had between seven and nine generations of probability that if an outbred genotype produces a fully selfing in their ancestry and line inbreeding coefficients inbred descendant, its phenotypic value (fitness) exceed greater than 0.99 (high homozygosity confirmed with the outbred mean. genetic markers: Willis, 1999a; Liza Holeski, unpublished The first statistic (I1) is derived from Chebyshev’s results). As part of a larger breeding design, the lines inequality, an identity concerned with the probability of were randomly paired and crossed to produce F1 extreme outcomes (Feller, 1968, p 233).