Inferences involving individual coefficients of relatedness and inbreeding in natural populations of Abies

Forest Ecology and Management 197 (2004) 171–180

Review

Inferences involving individual coefficients of relatedness and inbreeding in natural populations of Abies Kermit Ritland*, Steve Travis1 Department of Forest Sciences, University of British Columbia, 2424 Main Mall, Vancouver, BC, Canada V6T1Z4

Abstract Individual coefficients of relatedness and inbreeding, as estimated from genetic markers, are increasingly used for fine-scale studies of natural populations. We describe three approaches based upon these coefficients, two of which involve novel statistics, and illustrate their use with a study of four populations of two Abies species (A. amabilis and A. lasiocarpa) in British Columbia, Canada. Prior to their use, we recommend that an estimator be chosen via an evaluation of their properties via Monte-Carlo simulation. First, we examined isolation-by-distance, via both pairwise relatedness and actual variance of relatedness. Significant declines of relatedness with distance were found in two of four populations, and actual variance of relatedness increased with distance. As an alternative to regression, a new statistic, the ‘‘spatial variance of gene identity’’ is introduced, of utility for among-population comparisons, but it displays a high estimation variance. Second, we estimated of ‘‘heritability in the field’’ for relative growth rate. While estimates were significantly positive for A. amabilis (the first demonstration of heritable variation in the field for a conifer), estimates of the actual variance of relatedness were too uncertain to give absolute levels of heritability. Third, the relative depression of fitness due to inbreeding, characterized with a new measure termed the ‘‘inbreeding genetic load’’, was estimated with individual inbreeding coefficients. Levels of load for relative growth rates were positive but non-significant in all four populations. The latter two methods are particularly for long-lived species such as forest trees, where experimental manipulation is difficult. The relevance of these inferences to breeding and gene conservation in natural forest populations is also discussed. # 2004 Elsevier B.V. All rights reserved. Keywords: Relatedness; Inbreeding; Quantitative traits; Abies

1. Introduction Genetic studies of forest tree populations have traditionally documented average levels of diversity, inbreeding and genetic differentiation as observed at genetic markers (Hamrick et al., 1992). Few studies have used genetic markers to examine population *

Corresponding author. Tel.: þ1-604-822-8101; fax: þ1-604-822-9102. E-mail address: [email protected] (K. Ritland). 1 Present address: USGS National Wetlands Research Center, 700 Cajundome Blvd., Lafayette, LA 70506, USA.

structure at the level of individuals (Sweigart et al., 1999). Furthermore, markers and phenotypes can be jointly analyzed at the individual level to make genetic inferences about quantitative traits, such as levels of heritability (Ritland, 2000). An appealing aspect of these inferences is that traits are studied ‘‘in situ’’, in the natural population, without the artifacts introduced by experimentation. They are particularly applicable to organisms with long generation times, such as forest trees, where experimental studies cannot be conducted in a suitable timeframe (Cruzan, 1998). While most forest trees show low levels of amongpopulation differentiation, any limited dispersal of

0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2004.05.012

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seed or pollen will result in fine-scale local genetic structure. Fine-scale genetic structuring affects the evolutionary dynamics of populations by influencing effective population size, patterns of viability selection, and rates of consanguineous matings (Schnabel et al., 1998). The major component of such structure is the identity-by-descent of genes among lineal relatives, which in outbreeding forest trees, would consist of primarily full-sibs, half-sibs and parent–offspring relationships. Over the past few years, a wide variety of approaches to inferring genealogical relationships in natural populations from genetic markers have been developed (Blouin, 2003). It would be ideal if we could reconstruct the entire pedigree network of a population from genetic data, and some progress has been made in this area (Smith et al., 2001; Thomas and Hill, 2002). However, the required statistical methods are very complex and the resulting pedigrees are difficult to relate to geographic and phenotypic data. As a first approximation to this complexity, we can employ pairwise comparisons, involving either individual inbreeding coefficients (pair of alleles within individuals), or relatedness between individuals (pairs of alleles between individuals). The pairwise approach neglects the higher-level patterns of joint relatedness present in family-structured populations, and in the case of inbreeding, the correlation of homozygosity within individuals due to mixed mating (where some individuals are produced by selfing). The problem is somewhat analogous to using pairwise differences in a coalescent phylogeny: the history of coalescent events in sample of genes imposes a substantial correlation on pairwise differences (Slakin and Hudson, 1991). This correlation presents problems for the determination of statistical significance and for the optimal use of statistical information, but we expect this has little effect on the expected value (or bias) of the estimates. Polymorphic genetic markers allow the estimation of relatedness and/or inbreeding coefficients without any prior pedigree information (Blouin, 2003). Much literature has accumulated on the estimation of the population inbreeding coefficient or coefficient of relatedness, but the methodology for using markers to estimate relatedness or inbreeding coefficients for individuals has received attention only recently (Lynch, 1988; Queller and Goodnight, 1989; Ritland, 1996a; Lynch and Ritland, 1999; Wang, 2002). While

isozymes are universally employed and quite easy to assay (and used in this study), microsatellites are clearly the marker-of-choice. We note that isozymes have the advantage of low cost of assay, and in fact, when jointly analyzing phenotypes and markers, of more importance is the number of phenotypes and not the precision of relatedness estimation (Ritland, 1996b). Also, amplified fragment length polymophisms (AFLPs) are of increasing use, but due to their dominance, they cannot be used to infer inbreeding coefficients, and also complicate the estimation of relatedness, although Lynch and Milligan (1994) and Hardy (2003) have braved these conditions. Besides providing basic descriptive information about local genetic structure, estimates of these coefficients can be used as covariates in the analysis of geography or phenotype. Here, we describe three such approaches: the analysis of isolation-by-distance, estimation of the heritability for a quantitative trait, and characterization of adult inbreeding depression. We illustrate them with isozyme studies of two British Columbia Abies species, Abies amabilis and A. lasiocarpa, and discuss the relevance of our results to breeding, silviculture and gene conservation.

2. Materials and methods 2.1. Sampling of Abies populations in British Columbia In the summer and fall of 1997, two populations of Pacific silver fir (A. Amabilis) and two populations of subalpine fir (A. Lasiocarpa), both in southwestern British Columbia, were sampled for genotyping. The silver fir populations were generally on mild slopes and showed highly clumped distributions of individuals within mixed stands of other tree species: ‘‘Cheam’’, on the south-facing slope of Cheam Peak bordering the Chilliwack River drainage at an elevation of 1400 m, and ‘‘Callaghan’’, on the west bank of Callaghan Cr. 10–15 km upstream from its confluence with the Cheakamus River. The subalpine fir populations were located on moderate to steep slopes near treeline: ‘‘Blackcomb’’, on a south-facing slope of Blackcomb Mountain, adjacent to Whistler, BC, and ‘‘Botanie’’, on the eastern slope of Botanie Mountain, north of Lytton, BC.

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We collected needles from the terminal 10–20 cm of the lowest-hanging branches of each tree, without regard for the presence or absence of buds. Sample sizes were as follows: Cheam Peak, n ¼ 281; Callaghan Creek, n ¼ 400; Blackcomb Mountain, n ¼ 310; Botanie Mountain, n ¼ 400. Intensive sampling procedures varied by population in accordance with the scale upon which genetic structure was observed to occur, as determined by sampling 100 trees the prior year. Because we observed that mean genetic relatedness remained positive for trees separated by distances of up to 100 m for both the Cheam Peak and Callaghan Creek populations of Pacific silver fir, we ran several parallel sampling transects 75–150 m apart. In contrast, because most structure was apparent within the first 25 m of spatial separation for trees representing the Blackcomb and Botanie Mountain populations of subalpine fir, these transects were separated by only 10–50 m. The number of transects sampled for each population ranged from 2 to 4. To correlate genotypes with phenotypes in these four intensive transects, ages were estimated from tree cores or basal wafers (‘‘cookies’’), depending on whether trees were greater than or less than 1.3 m in height, respectively. After accounting for biases in smaller trees (<1.3 m), growth rates (mm/year) were estimated using diameter at breast height (DBH) or at ground level for smaller trees. In three of four stands, older trees showed higher growth rates on average, probably because these stands were composed of characteristically young trees 200–400 years ago, whereas now the stands are slow-growing oldgrowth. The exception was Botanie Mountain, which had a cohort of 100-year-old trees with fast growth. Thirteen enzyme systems were screened for variability: phosphoglucose isomerase (PGI), phosphoglucomutase (PGM), leucine aminotransferase (LAP), malic enzyme (ME), isocitrate dehydrogenase (IDH), shikimate dehydrogenase (SKD), malate dehydrogenase (MDH), fructose diphosphatase (FDP), 6phosphogluconate dehydrogenase (6PGD), aconitase (ACO), aspartate aminotransferase (AAT), glutamate dehydrogenase (GDH), and uridine diphosphoglucose pyrophosphorylase (UDP). Many of these loci were not sufficiently polymorphic for estimation purposes. The gene frequencies of loci showing sufficient polymorphisms are given in Table 1.

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Table 1 Gene frequencies loci that were sufficiently polymorphic for use in analyses Locus

Number of alleles

Gene frequencies

Pacific silver fir—Cheam PGI 2 PGM-1 2 PGM-2 2

0.655, 0.345 0.823, 0.177 0.956, 0.044

Pacific silver fir—Callaghan PGI 2 PGM 2 UDP 2

0.618, 0.382 0.673, 0.327 0.915, 0.085

Subalpine fir—Blackcomb PGI 3 PGM 2 IDH 2 6PGD 2 UDP 2 AAT 2

0.165, 0.718, 0.936, 0.041, 0.053, 0.042,

Subalpine fir—Botanie PGI 4 IDH 4 UDP 2

0.119, 0.563, 0.242, 0.075 0.860, 0.118, 0.015, 0.006 0.155, 0.845

0.560, 0.276 0.282 0.064 0.959 0.053, 0.894 0.958

2.2. Estimation of pairwise relatedness and individual inbreeding coefficients In marker-based studies of populations, studies of relatedness require informed choice of estimators, explicitly based upon a model of gene identity. While there is just one gene identity based formula for the probability of a pair of relatives between codominant marker loci and hence one likelihood estimator (Ritland, 2000; Blouin, 2003), there are several alternative, moment-based estimators for relatedness, each derived from this probability model. When estimating relatedness or inbreeding for individuals, maximum likelihood estimators can give large statistical bias (Ritland, 1996a), as the sample size is inherently small—the number of marker loci assayed. Hence a number of ‘‘moment’’ estimators have been devised, which suffer less from such bias. To describe these, define the genotype of individual 1 as AiAj and AkAl for individual 2. Efficient descriptions can be obtained by ‘‘indicator variables’’, dij. These are defined such that if Ai ¼ Aj (the same allele in state), then dij ¼ 1; otherwise dij ¼ 0. Between two individuals (four alleles), there are six d’s, one for each

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pairwise comparison of alleles, both within and between individuals. The estimator of pairwise relatedness of Ritland (1996a) is ^r ¼

½ðdik þ dil Þ=pi  þ ½ðdjk þ djl Þ=pj   1 4ðn  1Þ

(1a)

where n is the number of alleles at the locus, and pi the frequency of allele i in the population (estimated from the population sample). The multilocus estimate is the sum of locus-specific estimates, each weighted in proportion by (n  1). Lynch and Ritland’s (1999) estimator of pairwise relatedness is ^r ¼

pi ðdjk þ djl Þ þ pj ðdik þ dil Þ  4pi pj ð1 þ dij Þðpi þ pj Þ  4pi pj

(1b)

where the first genotype, AiAj, is the ‘‘reference’’ genotype. For multilocus estimates, each locus is weighted proportionally by the inverse of 2pi pj = ½ð1 þ dij Þðpi þ pj Þ  4pi pj . This estimator is an asymmetrical measure of relatedness, as it is based upon the expected genotype of the second individual, given the first (as opposed to the Ritland, 1996a estimator, which is based upon the joint probability of both genotypes). It is best to randomly choose the reference genotype (averaging both ways results in bias of estimates). Queller and Goodnight’s (1989) estimator is ^r ¼

0:5ðdjk þ djl þ dik þ dil Þ  pi  pj 1 þ dij  pi  pj

(1c)

Their estimator is not defined when the reference genotype is heterozygous at a diallelic locus (the denominator is zero). For multilocus estimates, the numerator and denominator terms are summed separately across loci. Recently, Wang (2002) described another estimator whose formula is quite complex and cannot be presented here. For the inbreeding coefficient, the Ritland (1996a) estimator is ^f ¼ dij =pi  1 n1

(2a)

while the Lynch and Ritland (1999) approach gives the estimator ^f ¼ dij  pi 1  pi

(2b)

where allele i is arbitrarily chosen as the reference allele. This has variance pi =ð1  pi Þ, which is used to weight estimates among loci.

Inferences using the above coefficients often require estimation of the actual variance of these coefficients (variance above the statistical, e.g., that due to real pedigree variation). As discussed in Ritland (1996b), we can use the fact that estimates are independent across loci, and perform a weighted ANOVA to estimate actual variances. In terms of bias and variance, Lynch and Ritland (1999) found that Eq. (1a) (Ritland, 1996a) is more appropriate for loci with fewer (<6) alleles while their estimator (Eq. (1b)) behaves better for highly polymorphic loci. The Queller and Goodnight estimator clearly has a larger variance; Wang (2002) concludes his estimator is more appropriate in most situations. Van de Casteele et al. (2001) reviewed the properties of most current estimators, and concluded that the Ritland (1996a) estimator most often showed the best properties. However, since the properties of each estimator (bias, variance) depends upon the distribution of gene frequencies and the patterns of relatedness, they proposed that Monte-Carlo simulations be employed using the actual data of the study, to determine the best estimator. 2.3. Analysis of marker data We followed the proposal of Van de Casteele et al. (2001), and tested the alternative estimators via Monte-Carlo simulation using the estimated gene frequencies of each Abies population, generating data in an equal mixture of two levels of relationship: r ¼ 0:15 and r ¼ 0:05 (this generates an actual variance of relatedness, also of interest), with 10,000 replications. While many possible scenarios of relationship are possible, this seems most reasonable given the results that follow. The results (Table 2) show the Ritland (1996a) estimator recovers the least biased estimate of relatedness, with the lowest error, and also recovers the best estimate of the actual variance of relatedness. Simulations at much higherlevels of relatedness indicate the Lynch and Ritland estimator, and the Wang estimator, outperform the Ritland estimator. In the following, we used the Ritland (1996a) estimator for both relatedness and inbreeding. The procedures used in this paper are implemented in the program MARK, written by KR in FORTRAN95 for Windows, available at http:// www.genetics.forestry.ubc.ca/ritland/programs.html.

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Table 2 Results of Monte-Carlo simulations of the properties four measures of pairwise relatedness (r) and actual variance of relatedness (Vr) Population

Estimator Ritland r

Lynch–Ritland

Queller–Goodnight

Wang

Vr

r

Vr

r

Vr

r

Vr

Pacific silver fir—Cheam Mean 0.105 Variance 0.161

0.0072 0.0689

0.083 0.191

0.0068 0.1853

0.021 0.559

0.0142 0.6713

0.057 0.232

0.0058 0.1474

Pacific silver fir—Callaghan Mean 0.100 Variance 0.121

0.0103 0.0439

0.095 0.183

0.0134 0.2020

0.054 0.367

0.0488 0.1454

0.066 0.150

0.0099 0.0751

Subalpine fir—Blackcomb Mean 0.103 Variance 0.091

0.0081 0.0344

0.120 0.095

0.0142 0.0545

0.108 0.172

0.0055 0.1744

0.091 0.143

0.0109 0.0641

Subalpine fir—Botanie Mean 0.098 Variance 0.070

0.0109 0.0217

0.125 0.075

0.0250 0.0315

0.188 0.124

0.0333 0.0912

0.143 0.112

0.0220 0.0455

Input parameters were r ¼ Vr ¼ 0:10.

3. Individual inferences in the field with genetic markers

Pacific silver fir 0.2

0.0 -0.1 -0.2 0.2 Callaghan

0.1 0.0

Relatedness

A popular way to characterize fine-scale local genetic structure is to plot pairwise genetic relatedness as a function of physical distance. If pollen or seed flow is limited, the expected relationship is negative. Fig. 1 shows the relationship between pairwise genetic relatedness and pairwise physical distance, for each of the four stands of two Abies species. The solid lines represent the original data, and the dashed lines represent the 95% confidence levels. To find these levels, we conducted bootstrap analyses, with individual trees along each transect as the unit of resampling. The middle dashed line gives the mean bootstrap estimate and the upper and lower dashed lines giving the 95% confidence intervals for relatedness, across all possible values of distance (in m). All stands show a clear pattern of decline, with mean relatedness declining to near zero by 20–60 m (Fig. 1). Three of four populations show sharp drops of relatedness within 5–10 m, while the fourth (Cheam) shows a more gradual drop. There is no speciesspecific pattern. A relatedness value of 0.125 corresponds to half-sibs, and first-cousins half this; thus the relationship between proximate trees is between first-

Cheam

0.1

3.1. Inferences about isolation-by-distance

-0.1 -0.2

Subalpine fir 0.2

Blackcomb

0.1 0.0 -0.1 -0.2 0.2

Botanie

0.1 0.0 -0.1 -0.2 0

50

100

150

200

Physical distance (m)

Fig. 1. Relationships between genetic distance in physical distance in populations of pacific silver fir and subalpine fir.

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3.2. The spatial variance of gene identity Theoretical treatments of isolation-by-distance models have led to predictions about the pattern of localized genetic structure under isolation-by-distance. Most notably, in two-dimensional space, the natural logarithm of relatedness declines linearly with distance (Barton et al., 2002). Because this decline is inversely proportional to the dispersal variance, one can derive a statistic that can be compared among studies. Here, we propose another statistic for comparing fine-scale structure: the ‘‘spatial variance of gene identity’’, which is the variance of physical distance of all descendents of an ancestral gene about their mean location, averaged over all ancestors. The distribution of copies of a given ancestral gene would follow some type of distribution, such as the normal depicted in Fig. 2a. While the distribution would theoretically have infinite tails, its variance is finite, and in fact, relatively small. In a population at equilibrium, where only 50% of the genes are passed on to the next generation, it is governed by just the first few generations of dispersal. For example, with one-

Table 3 Estimates of isolation-by-distance parameters for all pairs within 50 m, with standard errors brd Pacific silver fir Cheam 0.0045 S.E. 0.0054 Callaghan 0.0166 S.E. 0.0079 Subalpine fir Blackcomb S.E. Botanie S.E. Grand mean S.E.

0.0256 0.0062 0.0058 0.0034 0.0131 0.0030

br2d

s2I

r

153.7 161.8 71.2 108.4

337.2 1761.4 6.8 1804

0.010 0.006 0.005 0.006

46.7 61.1 1078.5 408.3 337.5 114.1

73.5 134.1 3036.9 18220 863.6 4598.5

0.014 0.004 0.000 0.002 0.007 0.002

Pairwise relatedness (r) and actual variance of relatedness (Vr) were regressed on the natural logarithm of distance, giving brd and br2d respectively. The spatial variance of gene identity, s21 , was also determined for all pairs within 50 m, as was mean relatedness, r .

dimensional dispersal, dispersal variance s2D each generation, and 50% survival, the equilibrium varP1 2 i 2 iance is i¼1 isD =2 ¼ 2sD , merely twice the pergeneration dispersal variance. In the population, there are many overlapping distributions of identical genes. One can imagine a 1 (a)

Frequency

cousin and half-sibs, and this declines to half (second cousin—first-cousin) in roughly 30 m. Regressions of relatedness versus distance were also performed on log-transformed data, for all pairs within 50 m of each other (bootstrapped datasets were again generated by resampling individuals). Declines were again found in all four populations, but declines were statistically significant in just two of four populations (Callaghan, Blackcomb). We also examined the relationship between actual variance of relatedness and distance. This quantity, which measures the heterogeneity of relationships, has not been examined before in relation to distance. While the variance of relatedness is a critical quantity for estimation of heritability (Ritland, 1996b), its statistical error can be large. We regressed this quantity on the logarithm of distance and the results (Table 3) unexpectedly show a trend towards increases in actual variance of relatedness with distance (while no single estimate is statistically significant, the grand means is highly significant). Theoretical work is needed to identify the geographic patterns expected for actual variance of relatedness, and why this variance increases with distance.

0 1 (b)

Frequency

176

0

Distance Fig. 2. Idealized patch structure of genes with short-term identityby-descent. (a) Frequency distribution along a transect of a hypothetical patch of a gene descended from a single ancestral copy. (b) Frequency distributions of all patches of genes identicalby-descent at a locus, superimposed on each other.

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transect through a population as slicing through stacked distributions (Fig. 2b). R Now, for a distribution centered about zero, s2I ¼ x2 f ðxÞ dx (where x is distance), but since one can identify identical genes only by pairwise comparisons, we use pairwise relatedness, divided by mean relatedness, in place of the distribution function f(x). The average spatial variance of gene identity is thus estimated via a calculation involving spatial covariance: P r i di2 pairs i^ 2 ^I ¼ P s (3) 2 pairs i^r i where ri is the estimated relatedness and di the physical distance between pair i. The right side is halved because if x and y are the locations of two identical copies, then EðdÞ2 ¼ Eðx  yÞ2 ¼ EðxÞ2 þ EðyÞ2 ¼ 2s2I . The right side is normalized by mean relatedness because EðXjYÞ ¼ EðXÞ=EðYÞ, where E(X|Y) is the expected squared distance given genes are identical, E(X) the expected product of squared distance with identity and E(Y) the probability that genes are identical. One difficulty in implementing Eq. (3) is that, because relatedness is relative, the expected mean relatedness is zero (and actually negative if the sampling variance of gene frequency is not taken into account, see Ritland, 1996a). One must determine a distance ‘‘cutoff’’ beyond which relatedness is zero (otherwise it becomes slightly negative, causing zero overall relatedness). In the analysis of our data, this distance cutoff was determined to be 50 m, but further work on this issue is needed. In the four transects, estimates of the spatial variance of gene identity were not statistically significant (Table 3).pThe ffiffiffiffiffiffiffiffi mean of 863 indicates that a circle of radius 2 863 ¼ 60 m captures 95% of the genes in a typical patch of identical alleles. This seems reasonable but clearly we need better types of marker data. Comparing these estimates with Fig. 1, it is clear that the slope and intercept of the isolationby-distance relationship combine to determine the spatial variance of identity. Similar values can be caused by either high intercept and steep slope, or low intercept and moderate slope. Models for the evolution of the spatial variance of gene identity, and further examination of its estimation properties, are needed.

177

3.3. Heritability in the field While many studies have measured heritabilities under artificial conditions, its levels in natural populations are largely speculative due to the difference between artificial and natural environments. This difficulty can been overcome via a marker-based method for assessing heritability (Ritland, 1996b). Pairwise relatedness can be used to estimate ‘‘heritability in the field’’ for growth rate in these Abies populations in the following way. Let the value of a quantitative trait Y for two individuals i and j be Yi for the first and Yj for the second. Their shared phenotypes are measured as Zij ¼

ðYi  UÞðYj  UÞ V

where U and V are the sample mean and variance of Y, respectively, in the population. Among all pairs, the average Zij equals the phenotypic correlation. If shared phenotypes are determined by the sharing of both genes and environments, then Zij ¼ 2rij h2 þ re þ eij where rij is the relatedness coefficient, re the correlation due to sharing of environments (assumed constant for all pairs), and eij the random error. This is a linear model, so over several pairs of individuals, we can estimate heritability as the regression ^h2 ¼ CovðZij ; rij Þ 2Varðrij Þ

(4)

 are the means of the Zij and rij, Cov(Zij, where Z and R rij) is the covariance between estimated relatedness and phenotypic similarity, and Var(rij) is the actual variance of relatedness. If actual variance of relatedness is statistically not significant, then at least, the presence of genetic variation can be ascertained by testing for positive Cov(Zij, rij). Fig. 3 shows the regressions of character similarity for growth rates upon estimated relatedness, for each of the four transects. A positive slope indicates heritability. In subalpine fir, there was no evidence for heritability, while in silver fir, both slopes were significantly positive, indicating heritability. However, estimates of the actual variance of relatedness were very uncertain, precluding a determination of the absolute level of heritability.

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10 5

Pacific silver fir Cheam

Pairwise phenotypic similarity for growth rate

0 -5 y=.025+.049 r (p<.05)

-10 10 5

Callaghan

0 -5 y=.018+.043 r (p<.05)

-10

6 3

Subalpine fir Blackcomb

0 -3 y=.028-.050 r (ns)

-6 10 5

Botanie

0

During the past three decades, many studies have attempted to correlate fitness with heterozygosity. A positive association may be due to overdominance at or very near the marker gene loci (Mitton and Grant, 1984) or due to variation of inbreeding among individuals combined with inbreeding depression (Strauss, 1986). Many studies have found that in populations with such positive associations, there are significant heterozygote deficiencies, suggesting the latter hypothesis is most frequently correct (Lynch and Walsh, 1998). If this is so, the strength of association between fitness and heterozygosity can give information about inbreeding depression. Here, we show that estimates of individual inbreeding coefficients (Sweigart et al., 1999), together with fitness (growth rate) measures, can predict levels of adult inbreeding depression in natural populations, and the consequent ‘‘genetic load’’ incurred by inbreeding. If we assume that fitness declines linearly with the inbreeding coefficient f; for the ith individual with coefficient fi, the expected fitness is wi ¼ wo  bfi

-5 -10

3.4. Inbreeding depression

y=-.004+.019 r (ns)

-1 0 1 2 Estimated pairwise relatedness

Fig. 3. Graphical portrayal of the relationship between phenotypic similarity (for grow rate) vs. genetic relationship, estimated from markers. Slopes were significantly positive for silver fir, but not subalpine fir.

The is the first demonstration of heritability in the wild for a conifer, and these results indicate that ‘‘natural’’ levels are probably low, at least compared to the traditional estimates based on controlled crosses grown in the common garden (where heritabilities of 10–30% would be found). The wide scatter of points is due to the estimation error of both relatedness and of the correlation of phenotypes, but is not a substantial problem in this analysis. The main problem here is the lack of actual variance of relatedness, or at least, detectable variance of relatedness. This can be due to either low levels of the variance of actual relatedness, or to markers of low information. This may pose a real limitation for future studies of heritabilities in the field for outbreeding organisms.

where wo is the fitness of an outbred individual (f ¼ 0), and b the regression of fitness on f. Noting that the average fitness is  ¼ wo  bf w the relative reduction of fitness due to inbreeding in the population is  wo  w b ¼ f wo wo

(5)

This quantity is the ‘‘inbreeding genetic load’’, or the proportional reduction in population mean fitness caused by inbreeding and inbreeding depression. Its estimation requires regressing fitness on individual f, and like the estimation of heritability with markers, unbiased estimation requires disentangling statistical variance of f from actual variance of f. Fig. 4 plots relative growth rate against individual inbreeding coefficients, for each of the four stands. All four populations showed a negative association between growth rate and inbreeding coefficient. The estimates of inbreeding genetic load were: Cheam 0.02, Callaghan 0.05, Blackcomb 0.10 and Botanie

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Pacific Silver Fir

179

4. Discussion

1.5

Cheam 1.0 0.5 0.0

y=0.311-.007F (ns)

1.5

Callaghan

Growth rate (mm/year)

1.0 0.5 y=0.277-0.014F (ns)

0.0

1.5

Subalpine Fir Blackcomb

1.0 0.5 0.0

y=0.496-0.025F (ns)

Botanie 1.0

0.0

y=0.535-0.024F (ns)

-2.0

-1.0

0.0

1.0

Estimated individual F Fig. 4. Effect of inbreeding upon growth rate, as measured by estimates of individual F, regressed upon growth rate.

0.05; these were all non-significantly different from zero. However, the grand average of 0.05 was significant. This 5% reduction in fitness is supported by more indirect calculations: given that ‘‘apparent’’ selfing rates among outcrossed progeny are 0.05–0.10 in many conifers, and if this is due to 20–40% half-sib mating, and if mating to half-sibs causes a 25% reduction in fitness, then the predicted inbreeding genetic load is ca. 5–10%. Interestingly, this is similar to the genetic gain realized by first generation phenotypic selection of plus trees from wild populations. More studies on the latent genetic load due to local population structure are needed in tree populations with natural genetic structure, and the potential for the use of the appropriate controlled crossing of nursery seedlings to alleviate this load.

Knowledge of the level of local population structure in natural populations can allow prediction of the genetic effects of various silvicultural practices involving natural regeneration, as they can alter patterns of seed and pollen flow, as well as provide guidelines about seed collections that maximize genetic diversity. Likewise, knowledge of the levels of heritability allows predictions of the effects of silvicultural practices such as selective thinning and highgrading. Traditionally, assessing heritabilities from natural populations of long-lived species has been made difficult by the necessity of conducting controlled breeding experiments. Here, we have demonstrated a genetic component for growth rate in both of two populations of Pacific silver fir, but did not find significant components in two populations of subalpine fir. Levels of inbreeding depression can also be of importance in silvicultural practices and predicting the effects of reforestation. The culling of trees might be based upon their heterozygosity, although the cost of assay and the logistics of sampling might outweigh the benefits of increased productivity. In the arena of mating system analysis, pairwise approaches also provide extensions to more traditional analyses. The ‘‘correlated matings model’’ (Ritland, 1989) estimates the ‘‘correlation of paternity’’, defined as the proportion of full-sibs among all pairwise comparisons of outcrossed siblings. Second, the correlation of selfing between pairs of loci within individuals can provide an alternative measure of biparental inbreeding (Ritland, 2002). Finally, at the individual level, family estimates of outcrossing rate (Cruzen and Arnold, 1994; Ritland, 2002) can be obtained using approaches similar to those for estimating relatedness (above), and these should be of use in fine-scale studies of mating systems in relation to environments or silviculture practice. However, the pairwise approach does not take advantage of the multilocus structure of data. Examples of such structure may occur when (1) several fullsib families of fish are raised in a tank, but parentage of these fish is unknown, (2) progenies from open-pollinated trees are grown in field trials; while maternal parentage is known, families consist of unknown proportions of full- versus half-sibs. As well, most localized genetic structure in tree populations may

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effectively consist of mixtures of half-sib, full-sib and unrelated individuals (cousins and further relatives being not detectable with markers). Models that classify individual by relatedness (e.g., Blouin et al., 1996) might more accurately portray patterns of isolationby-distance in natural populations, and allow stronger inferences about heritabilities in the field.

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