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Does covariance and phenotypic variance have to be for the same trait to calculate narrow-sense heritability?


I'm working on a problem where I'm given that the phenotypic variance of the parent, $p$, for trait $A$ is $175.2$ and the phenotypic variance of the offspring, $o$, for trait $A$ is $146.1$. However, I'm also given that the covariance between parent and offspring of a different trait, $B$, is $14.6$.

Can I still calculate the narrow-sense heritability of $A$ using $$egin{eqnarray} frac{h^2(A)}{2}&=&frac{ ext{Cov}(p,o)}{ ext{Var}(p)}=frac{14.6}{175.2}approx0.0833 h^2(A)&=&2 imes0.0833approx0.167 end{eqnarray} $$ even though they're different traits?

I'm also told that any resemblance between parents and offspring is completely due to genetics, ie. without environmental effects. I've read that if this is the case, then $h^2$ is $0.50$, but I'm a little confused with it.


Does covariance and phenotypic variance have to be for the same trait to calculate narrow-sense heritability?

Yes. See below for more information.

Can I still calculate the narrow-sense heritability of AA [… ] even though they're different traits?

No, you can't. A measure of heritability makes sense only for

  • a given trait
  • a given population
  • at a given time
  • in a given environment

[I'm also told that if any resemblance between parents and offspring is completely due to genetics, ie. without environmental effects, then $h^2$ is 0.5, but I'm a little confused with it.]

The above is not a exact quote from you but the original sentence made no sense. I rephrase what I suppose you tried to say.

The formulation is inaccurate and could therefore be misleading. It is wrong to say that any covariance between parent and offspring is only due to genetics. In reality, environments between parent and offspring often covary. There might also have covariance due to epigenetic reasons and potentially many other reasons.

You cannot make any claim about what value should $h^2$ takes for the special case where covariance between parent and offspring is solely due to genetic variance (or any other special case considering only genetic variance). You still need to know the relative importance of the genetic variance to the total phenotypic variance. If any of this is unclear, then you should keep on with the recommended reading below.

Reading Recommendation

I recommend that you have a look at Why is a heritability coefficient not an index of “how genetic” something is? for an overview of the concept of heritability and common misconceptions.


A function‐valued trait approach to estimating the genetic basis of size at age and its potential role in fisheries‐induced evolution

Natural selection is inherently a multivariate phenomenon. The selection pressure on size (natural and artificial) and the age at which selection occurs is likely to induce evolutionary changes in growth rates across the entire life history. However, the covariance structure that will determine the path of evolution for size at age has been studied in only a few fish species. We therefore estimated the genetic covariance function for size throughout ontogeny using Atlantic silversides (Menidia menidia) as the model system. Over a 3‐year period, a total of 542 families were used to estimate the genetic covariance in length at age from hatch through maturity. The function‐valued trait approach was employed to estimate the genetic covariance functions. A Bayesian hierarchical model was used to account for the unbalanced design, unequal measurement intervals, unequal sample sizes, and family𠄊ggregated data. To improve mixing, we developed a two‐stage sampler using a Gibbs sampler to generate the posterior of a well‐mixing approximate model followed by an importance sampler to draw samples from posterior of the completely specified model. We found that heritability of length is age‐specific and there are strong genetic correlations in length across ages that last 30ꃚys or more. We used these estimates in a hypothetical model predicting the evolutionary response to harvesting following a single generation of selection under both sigmoidal and unimodal patterns of gear selectivity to illustrate the potential outcomes of ignoring the genetic correlations. In these scenarios, genetic correlations were found to have a strong effect on both the direction and magnitude of the response to harvest selection.


Summary

Inbreeding generates covariances between additive and dominance effects (breeding values and dominance deviations). In this work, we developed and applied models for estimation of dominance and additive genetic variances and their covariance, a model that we call “full dominance,” from pedigree and phenotypic data. Estimates with this model such as presented here are very scarce both in livestock and in wild genetics. First, we estimated pedigree-based condensed probabilities of identity using recursion. Second, we developed an equivalent linear model in which variance components can be estimated using closed-form algorithms such as REML or Gibbs sampling and existing software. Third, we present a new method to refer the estimated variance components to meaningful parameters in a particular population, i.e., final partially inbred generations as opposed to outbred base populations. We applied these developments to three closed rabbit lines (A, V and H) selected for number of weaned at the Polytechnic University of Valencia. Pedigree and phenotypes are complete and span 43, 39 and 14 generations, respectively. Estimates of broad-sense heritability are 0.07, 0.07 and 0.05 at the base versus 0.07, 0.07 and 0.09 in the final generations. Narrow-sense heritability estimates are 0.06, 0.06 and 0.02 at the base versus 0.04, 0.04 and 0.01 at the final generations. There is also a reduction in the genotypic variance due to the negative additive–dominance correlation. Thus, the contribution of dominance variation is fairly large and increases with inbreeding and (over)compensates for the loss in additive variation. In addition, estimates of the additive–dominance correlation are −0.37, −0.31 and 0.00, in agreement with the few published estimates and theoretical considerations.


Results

Within a large multigenerational breeding design we measured the fitness of 2883 sons from 666 families (∼4 sons/family) and 5040 wings (∼8 sons/family) from their brothers. We subsequently carried out a genetic analysis using a suite of eight wing phenotypes (interlandmark distance traits: aLM2-4, aLM2-5, aLM2-8, aLM3-7, aLM3-9, aLM4-5, aLM5-6, aLM5-8) and fitness. Consistent with the levels of additive variance as a proportion of total phenotypic variance typically found for quantitative traits ( Lynch and Walsh 1998), estimates of additive genetic variance (VA) for the eight wing traits analyzed here ranged from 0.29 to 0.56 ( Figure 1A and Table 2). Fitness, however, was strikingly different, with a VA estimate of 0.008, 2 orders of magnitude less than wing traits ( Figure 1A and Table 2). Statistical support for the presence of additive genetic variance in all eight wing traits was provided by univariate likelihood-ratio tests, which tested whether excluding VA significantly worsened the fit of the model, with degrees of freedom equal to one (aLM2-4, χ 2 = 75.7, P < 0.001 aLM2-5, χ 2 = 50.4, P < 0.001 aLM2-8, χ 2 = 101.4, P < 0.001 aLM3-7, χ 2 = 79.0, P < 0.001 aLM3-9, χ 2 = 33.0, P < 0.001 aLM4-5, χ 2 = 68.0, P < 0.001 aLM5-6, χ 2 = 63.8, P < 0.001 aLM5-8, χ 2 = 50.4, P < 0.001). This statistical support was retained for all eight traits when the highly conservative sequential Bonferroni correction of the nominated significance level of 0.05 was applied (α = 0.05, c = 8). For fitness, the univariate test for the presence of VA was not statistically significant (χ 2 = 0 d.f. = 1, P = 1) ( Table 2).

(A) The additive (solid dots) and dominance (open dots) genetic variance for fitness and single wing shape traits, corresponding to the diagonal elements of the additive and dominance covariance matrices, respectively. (B) The proportion of additive (solid dots) and dominance (open dots) genetic variance accounted for by the eigenvectors of the additive genetic and dominance genetic covariance matrices for fitness and wing shape traits, obtained by diagonalizing the respective matrices.

(A) The additive (solid dots) and dominance (open dots) genetic variance for fitness and single wing shape traits, corresponding to the diagonal elements of the additive and dominance covariance matrices, respectively. (B) The proportion of additive (solid dots) and dominance (open dots) genetic variance accounted for by the eigenvectors of the additive genetic and dominance genetic covariance matrices for fitness and wing shape traits, obtained by diagonalizing the respective matrices.

The additive genetic covariance matrix for the eight wing traits and fitness

. Fitness . aLM2-4 . aLM2-5 . aLM2-8 . aLM3-7 . aLM3-9 . aLM4-5 . aLM5-6 . aLM5-8 .
Fitness 0.008 0.5990.344−0.3690.0120.5930.473−0.382−0.292
aLM2-4 0.040 0.546 *0.286−0.7930.2240.1950.336−0.097−0.158
aLM2-5 0.020 0.134 0.405 *0.115−0.120−0.4240.508−0.410−0.359
aLM2-8 −0.025 −0.434 0.054 0.548 *−0.296−0.173−0.3790.0690.176
aLM3-7 0.001 0.112 −0.052 −0.149 0.461 *0.3500.1680.561−0.226
aLM3-9 0.029 0.077 −0.144 −0.068 0.127 0.286 *-0.0640.2120.027
aLM4-5 0.030 0.170 0.222 −0.193 0.078 −0.023 0.473 *−0.468 −0.688
aLM5-6 −0.024 −0.049 −0.179 0.035 0.262 0.078 −0.221 0.471 *0.573
aLM5-8 −0.018 −0.079 −0.154 0.088 −0.104 0.010 −0.319 0.265 0.456 *
. Fitness . aLM2-4 . aLM2-5 . aLM2-8 . aLM3-7 . aLM3-9 . aLM4-5 . aLM5-6 . aLM5-8 .
Fitness 0.008 0.5990.344−0.3690.0120.5930.473−0.382−0.292
aLM2-4 0.040 0.546 *0.286−0.7930.2240.1950.336−0.097−0.158
aLM2-5 0.020 0.134 0.405 *0.115−0.120−0.4240.508−0.410−0.359
aLM2-8 −0.025 −0.434 0.054 0.548 *−0.296−0.173−0.3790.0690.176
aLM3-7 0.001 0.112 −0.052 −0.149 0.461 *0.3500.1680.561−0.226
aLM3-9 0.029 0.077 −0.144 −0.068 0.127 0.286 *-0.0640.2120.027
aLM4-5 0.030 0.170 0.222 −0.193 0.078 −0.023 0.473 *−0.468 −0.688
aLM5-6 −0.024 −0.049 −0.179 0.035 0.262 0.078 −0.221 0.471 *0.573
aLM5-8 −0.018 −0.079 −0.154 0.088 −0.104 0.010 −0.319 0.265 0.456 *

Additive genetic variances are along the diagonal, underlined, with covariances below and correlations above (in italics).

. Fitness . aLM2-4 . aLM2-5 . aLM2-8 . aLM3-7 . aLM3-9 . aLM4-5 . aLM5-6 . aLM5-8 .
Fitness 0.008 0.5990.344−0.3690.0120.5930.473−0.382−0.292
aLM2-4 0.040 0.546 *0.286−0.7930.2240.1950.336−0.097−0.158
aLM2-5 0.020 0.134 0.405 *0.115−0.120−0.4240.508−0.410−0.359
aLM2-8 −0.025 −0.434 0.054 0.548 *−0.296−0.173−0.3790.0690.176
aLM3-7 0.001 0.112 −0.052 −0.149 0.461 *0.3500.1680.561−0.226
aLM3-9 0.029 0.077 −0.144 −0.068 0.127 0.286 *-0.0640.2120.027
aLM4-5 0.030 0.170 0.222 −0.193 0.078 −0.023 0.473 *−0.468 −0.688
aLM5-6 −0.024 −0.049 −0.179 0.035 0.262 0.078 −0.221 0.471 *0.573
aLM5-8 −0.018 −0.079 −0.154 0.088 −0.104 0.010 −0.319 0.265 0.456 *
. Fitness . aLM2-4 . aLM2-5 . aLM2-8 . aLM3-7 . aLM3-9 . aLM4-5 . aLM5-6 . aLM5-8 .
Fitness 0.008 0.5990.344−0.3690.0120.5930.473−0.382−0.292
aLM2-4 0.040 0.546 *0.286−0.7930.2240.1950.336−0.097−0.158
aLM2-5 0.020 0.134 0.405 *0.115−0.120−0.4240.508−0.410−0.359
aLM2-8 −0.025 −0.434 0.054 0.548 *−0.296−0.173−0.3790.0690.176
aLM3-7 0.001 0.112 −0.052 −0.149 0.461 *0.3500.1680.561−0.226
aLM3-9 0.029 0.077 −0.144 −0.068 0.127 0.286 *-0.0640.2120.027
aLM4-5 0.030 0.170 0.222 −0.193 0.078 −0.023 0.473 *−0.468 −0.688
aLM5-6 −0.024 −0.049 −0.179 0.035 0.262 0.078 −0.221 0.471 *0.573
aLM5-8 −0.018 −0.079 −0.154 0.088 −0.104 0.010 −0.319 0.265 0.456 *

Additive genetic variances are along the diagonal, underlined, with covariances below and correlations above (in italics).

An eigenanalysis of the wing trait A revealed that the first seven (of eight) eigenvectors accounted for 99.7% of VA in these traits. Despite the small proportion of variance (0.3%) accounted for by the last eigenvector of the additive matrix, genetic principal component modeling revealed statistical support for all eight genetic dimensions underlying A (reducing from eight to seven dimensions significantly worsened the fit of the model: χ 2 = 14.00 d.f. = 1, P < 0.001). Although uncovering a full-rank genetic covariance matrix is uncommon, there is also evidence in D. melanogaster that the additive matrix for wing shape is full rank ( Mezey and Houle 2005). Consistent with the lack of VA for fitness found in the univariate test, statistical support was found for eight of nine genetic dimensions of the wing trait plus fitness A (reducing from eight to seven dimensions significantly worsened the fit of the model: χ 2 = 13.83 d.f. = 1, P < 0.001).

For wing traits, estimates of dominance variance (VD) were lower, on average, than additive estimates, ranging from 0.029 to 0.317 ( Figure 1A and Table 3). Univariate likelihood-ratio tests, testing whether excluding VD significantly worsened the fit of the model (again, with degrees of freedom equal to one) provided statistical support for dominance variance in three of the eight wing traits (aLM2-5, χ 2 = 4.6, P = 0.033 aLM3-7, χ 2 = 4.8, P = 0.029 aLM3-9, χ 2 = 6.5, P = 0.011). Applying the sequential Bonferroni correction (α = 0.05, c = 8) reduced statistical support to zero of eight traits however, with only eight tests, this likely represents the conservativeness of the test rather than a true lack of significance, with the probability of obtaining a type I error for this family of tests equal to 0.34. Although estimates of VD were, in general, only moderately lower than VA ( Figure 1A), due to the nature of our breeding design, statistical tests of dominance components suffer from reduced power. Here, all information on dominance comes from the 275 double-first-cousin pairs created in the second generation of breeding, 60% less sires than are used for additive genetic estimates. Despite the exceedingly low VA for fitness, its dominance estimate of 0.23 was within the range of the majority of VD estimates found here for wing traits and was statistically supported in a univariate likelihood ratio test (χ 2 = 7.12 d.f. = 1, P < 0.01).

The dominance genetic covariance matrix for the eight wing traits and fitness

. Fitness . aLM2-4 . aLM2-5 . aLM2-8 . aLM3-7 . aLM3-9 . aLM4-5 . aLM5-6 . aLM5-8 .
Fitness 0.253 *−0.170−0.1140.0970.0440.045−0.1310.5320.226
aLM2-4 −0.020 0.055 0.066−0.8730.377−0.0590.687−0.473−0.769
aLM2-5 −0.032 0.009 0.323 *0.1060.089−0.9250.4220.319−0.504
aLM2-8 0.023 −0.095 0.028 0.216 −0.596−0.036−0.6780.2030.747
aLM3-7 0.013 0.050 0.029 −0.157 0.322 *−0.2660.8150.351−0.718
aLM3-9 0.013 −0.008 −0.304 −0.010 −0.087 0.335 *−0.453−0.4730.543
aLM4-5 −0.034 0.083 0.124 −0.163 0.239 −0.135 0.267 0.080 −0.952
aLM5-6 0.100 −0.041 0.068 0.035 0.074 −0.102 0.016 0.140 −0.007
aLM5-8 0.065 −0.102 −0.163 0.197 −0.231 0.178 −0.279 −0.002 0.322
. Fitness . aLM2-4 . aLM2-5 . aLM2-8 . aLM3-7 . aLM3-9 . aLM4-5 . aLM5-6 . aLM5-8 .
Fitness 0.253 *−0.170−0.1140.0970.0440.045−0.1310.5320.226
aLM2-4 −0.020 0.055 0.066−0.8730.377−0.0590.687−0.473−0.769
aLM2-5 −0.032 0.009 0.323 *0.1060.089−0.9250.4220.319−0.504
aLM2-8 0.023 −0.095 0.028 0.216 −0.596−0.036−0.6780.2030.747
aLM3-7 0.013 0.050 0.029 −0.157 0.322 *−0.2660.8150.351−0.718
aLM3-9 0.013 −0.008 −0.304 −0.010 −0.087 0.335 *−0.453−0.4730.543
aLM4-5 −0.034 0.083 0.124 −0.163 0.239 −0.135 0.267 0.080 −0.952
aLM5-6 0.100 −0.041 0.068 0.035 0.074 −0.102 0.016 0.140 −0.007
aLM5-8 0.065 −0.102 −0.163 0.197 −0.231 0.178 −0.279 −0.002 0.322

Dominance genetic variances are along the diagonal, underlined, with covariances below and correlations above (in italics).

. Fitness . aLM2-4 . aLM2-5 . aLM2-8 . aLM3-7 . aLM3-9 . aLM4-5 . aLM5-6 . aLM5-8 .
Fitness 0.253 *−0.170−0.1140.0970.0440.045−0.1310.5320.226
aLM2-4 −0.020 0.055 0.066−0.8730.377−0.0590.687−0.473−0.769
aLM2-5 −0.032 0.009 0.323 *0.1060.089−0.9250.4220.319−0.504
aLM2-8 0.023 −0.095 0.028 0.216 −0.596−0.036−0.6780.2030.747
aLM3-7 0.013 0.050 0.029 −0.157 0.322 *−0.2660.8150.351−0.718
aLM3-9 0.013 −0.008 −0.304 −0.010 −0.087 0.335 *−0.453−0.4730.543
aLM4-5 −0.034 0.083 0.124 −0.163 0.239 −0.135 0.267 0.080 −0.952
aLM5-6 0.100 −0.041 0.068 0.035 0.074 −0.102 0.016 0.140 −0.007
aLM5-8 0.065 −0.102 −0.163 0.197 −0.231 0.178 −0.279 −0.002 0.322
. Fitness . aLM2-4 . aLM2-5 . aLM2-8 . aLM3-7 . aLM3-9 . aLM4-5 . aLM5-6 . aLM5-8 .
Fitness 0.253 *−0.170−0.1140.0970.0440.045−0.1310.5320.226
aLM2-4 −0.020 0.055 0.066−0.8730.377−0.0590.687−0.473−0.769
aLM2-5 −0.032 0.009 0.323 *0.1060.089−0.9250.4220.319−0.504
aLM2-8 0.023 −0.095 0.028 0.216 −0.596−0.036−0.6780.2030.747
aLM3-7 0.013 0.050 0.029 −0.157 0.322 *−0.2660.8150.351−0.718
aLM3-9 0.013 −0.008 −0.304 −0.010 −0.087 0.335 *−0.453−0.4730.543
aLM4-5 −0.034 0.083 0.124 −0.163 0.239 −0.135 0.267 0.080 −0.952
aLM5-6 0.100 −0.041 0.068 0.035 0.074 −0.102 0.016 0.140 −0.007
aLM5-8 0.065 −0.102 −0.163 0.197 −0.231 0.178 −0.279 −0.002 0.322

Dominance genetic variances are along the diagonal, underlined, with covariances below and correlations above (in italics).

An eigenanalysis of the pooled D for wing traits revealed that the first five (of eight) eigenvectors accounted for 99.5% of the dominance variance, with the last eigenvector accounting for only 5 × 10 −5% of the variance in these traits. Although pooling bivariate analyses to generate the “full” dominance covariance matrix precludes us from testing the significant dimensions underlying it, the observation of such a low eigenvalue suggests that a null or nearly-null space may exist within the dominance matrix that does not exist within the additive matrix. Although fitness did have significant VD in a univariate test, the amount of variance captured by the first five (of nine) eigenvectors of D that include wing traits and fitness was equal to the amount described by the first five (of eight) for D that includes only wing traits.

In total there was 47% less dominance than additive genetic variance for wing traits (as given by a trace of 3.66 vs. 1.94 for additive and dominance covariance matrices, respectively). When fitness was included, this difference in variance decreased to 41% less dominance than additive variance. Matrix projection of the nine eigenvectors of D through the full-rank A indicated that there was substantial additive genetic variance in each of these multivariate trait combinations ( Figure 2). In fact, the proportion of additive genetic variance contained in the eigenvectors of D, and corresponding eigenvectors of A, were remarkably similar over most of the space. To determine the proportion of additive and dominance genetic variance in the multivariate trait combination under strongest directional selection we examined the genetic selection gradient, given by the eight-element vector of the additive genetic covariance between each wing trait and fitness ( Table 2) ( Stinchcombe et al. 2013). The projection of this normalized vector through A and D uncovers the proportion of additive and dominance genetic variance, respectively, in this multivariate trait combination. In contrast to patterns of VA and VD for the univariate trait under strongest directional selection, fitness, the proportion of VA in this multivariate trait combination was 0.31, double the proportion of VD, which was equal to 0.16.

The proportion of additive genetic variance accounted for by the eigenvectors of A and D. The solid dots depict the proportion of additive genetic variance accounted for by the eigenvectors of A, calculated by the eigenvalue of the respective vector divided by the trace of A. The open dots depict the proportion of additive genetic variance contained in the eigenvectors of D, calculated by projecting the eigenvectors of D through A and scaling by the trace of A.

The proportion of additive genetic variance accounted for by the eigenvectors of A and D. The solid dots depict the proportion of additive genetic variance accounted for by the eigenvectors of A, calculated by the eigenvalue of the respective vector divided by the trace of A. The open dots depict the proportion of additive genetic variance contained in the eigenvectors of D, calculated by projecting the eigenvectors of D through A and scaling by the trace of A.

The Krzanowski method of subspace comparison identified a 69% similarity in orientation of the subspaces defined by amax to a4 and dmax to d4, indicating that the multivariate traits with the most additive genetic variance also have the most dominance genetic variance. This was demonstrated by a value of 2.76 for the sum of the eigenvalues of S, which range here from zero to four, indicating orthogonal and coincident subspaces, respectively. Although the eigenvalues of A decline less rapidly than those of D, with the first four accounting for only 89% of additive variance but 97% of dominance variance in these traits, Krzanowski’s method is limited to comparing subspaces of dimension kn/2 therefore, a formal comparison of a larger subspace was not possible within this framework.

Despite the similarity between A and D, they are not identical, with notable differences in the distribution of variance over their respective eigenvectors shown by the more rapid decline of variance for the eigenvectors of D compared to A ( Figure 1B). To determine which traits contribute most strongly to this difference we examined the leading eigenvector [(AD)max] of the difference matrix (AD) ( Table 4). The trait (AD)max accounted for the majority of the difference between A and D (0.62 of the total difference in genetic variance of 1.42), with the elements, or trait loadings, of this vector indicating the relative contribution of each trait to this difference. Surprisingly, fitness was one of the weakest contributors to this trait combination, despite being the only trait to have significant dominance variance, and essentially no additive variance. The strongest contributor to (AD)max was aLM5-6, with a trait loading of −0.605. This trait had a significant, moderate, estimate of VA with a relatively low, nonsignificant estimate of VD. Overall the trait loadings across (AD)max appear fairly even, with positive and negative values indicating that the major difference between A and D is primarily driven by contrasting contributions of anterior to posterior shape variation in the proximal region of the wing, to proximal to distal shape variation across the wing.

The multivariate trait combinations describing the first eigenvectors of the additive genetic covariance matrix A (amax), dominance genetic covariance matrix D (dmax), and the difference matrix A–D [(A–D)max]

. . amax . dmax . (A–D)max .
Trait . % variance: . 37 . 50 . 44 .
Fitness 0.048 −0.070 0.157
aLM2-4 0.459 0.132 0.349
aLM2-5 0.278 0.338 0.331
aLM2-8 −0.428 −0.263 −0.191
aLM3-7 0.112 0.406 −0.300
aLM3-9 0.000 −0.381 0.017
aLM4-5 0.494 0.459 0.422
aLM5-6 −0.315 0.080 −0.605
aLM5-8 −0.413 −0.518 −0.271
. . amax . dmax . (A–D)max .
Trait . % variance: . 37 . 50 . 44 .
Fitness 0.048 −0.070 0.157
aLM2-4 0.459 0.132 0.349
aLM2-5 0.278 0.338 0.331
aLM2-8 −0.428 −0.263 −0.191
aLM3-7 0.112 0.406 −0.300
aLM3-9 0.000 −0.381 0.017
aLM4-5 0.494 0.459 0.422
aLM5-6 −0.315 0.080 −0.605
aLM5-8 −0.413 −0.518 −0.271

The % variance indicates the proportion of variance each eigenvector accounts for, determined by the eigenvalue divided by the trace for the corresponding vector and matrix, respectively.

. . amax . dmax . (A–D)max .
Trait . % variance: . 37 . 50 . 44 .
Fitness 0.048 −0.070 0.157
aLM2-4 0.459 0.132 0.349
aLM2-5 0.278 0.338 0.331
aLM2-8 −0.428 −0.263 −0.191
aLM3-7 0.112 0.406 −0.300
aLM3-9 0.000 −0.381 0.017
aLM4-5 0.494 0.459 0.422
aLM5-6 −0.315 0.080 −0.605
aLM5-8 −0.413 −0.518 −0.271
. . amax . dmax . (A–D)max .
Trait . % variance: . 37 . 50 . 44 .
Fitness 0.048 −0.070 0.157
aLM2-4 0.459 0.132 0.349
aLM2-5 0.278 0.338 0.331
aLM2-8 −0.428 −0.263 −0.191
aLM3-7 0.112 0.406 −0.300
aLM3-9 0.000 −0.381 0.017
aLM4-5 0.494 0.459 0.422
aLM5-6 −0.315 0.080 −0.605
aLM5-8 −0.413 −0.518 −0.271

The % variance indicates the proportion of variance each eigenvector accounts for, determined by the eigenvalue divided by the trace for the corresponding vector and matrix, respectively.


Does covariance and phenotypic variance have to be for the same trait to calculate narrow-sense heritability? - Biology

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The Implications of Inheritance for Clinical Management

From the Institute of Cardiovascular Science, University College London, London, United Kingdom (S.S-C., W.J.M.) Department of Epidemiology, Imperial College, London, London, United Kingdom (S.S-C.) Division of Cardiology, Yale School of Medicine, New Haven, CT (D.J., W.J.M.).

From the Institute of Cardiovascular Science, University College London, London, United Kingdom (S.S-C., W.J.M.) Department of Epidemiology, Imperial College, London, London, United Kingdom (S.S-C.) Division of Cardiology, Yale School of Medicine, New Haven, CT (D.J., W.J.M.).

From the Institute of Cardiovascular Science, University College London, London, United Kingdom (S.S-C., W.J.M.) Department of Epidemiology, Imperial College, London, London, United Kingdom (S.S-C.) Division of Cardiology, Yale School of Medicine, New Haven, CT (D.J., W.J.M.).

Introduction

Since the advent of genotyping, recognition of heritable disease has been perceived as an opportunity for genetic diagnosis or new gene identification studies to advance understanding of pathogenesis. Until recently, however, clinical application of DNA-based testing was confined largely to Mendelian disorders. Even within this remit, predictive testing of relatives is cost-effective only in diseases in which the majority of families harbor mutations in known causal genes, such as adult polycystic kidney disease and hypertrophic cardiomyopathy, but not dilated cardiomyopathy. Confirmatory genetic testing of index cases with borderline clinical features may be economic in the still smaller subset of diseases with limited locus heterogeneity, such as Marfan syndrome. Furthermore, Mendelian diseases account for ≈5% of total disease burden. 1 Genome-wide association studies have made headway in elucidating the genetic contribution to the more common, complex diseases, and high throughput techniques promise to facilitate integration of genetic analysis into clinical practice. Nevertheless, many genes remain to be identified and implementation of genomic profiling as a population screening tool would not be cost-effective at present. The implications of heredity, however, extend beyond serving as a platform for genetic analysis, influencing diagnosis, prognostication, and treatment of both index cases and relatives, and enabling rational targeting of genotyping resources. This review covers acquisition of a family history, evaluation of heritability and inheritance patterns, and the impact of inheritance on subsequent components of the clinical pathway.

Family History

Eliciting a family history is the first step to determining whether a known diagnosis is heritable or symptoms of unknown etiology have a hereditary basis. Both narrative and diagrammatic approaches are integral to data collection, the former including questioning for diseases that recur within the family and the latter involving construction of a pedigree or family tree. Incorporation of psychosocial and interactional data, such as emotional relationships (harmony, apathy, hostility, etc) upgrades the pictoral representation into a genogram. 2

As a result of media coverage, internet access, and health promotion campaigns, many patients nowadays are medically literate and well aware of the impact of genetics on health. If the patient perceives this aspect of the interview as intrusive, however, it may be necessary to explain its relevance (eg, “Sometimes the health of a person’s family members may affect one’s own health”) and emphasize confidentiality.

For the narrative portion of the family history, various questioning styles may be appropriate, from an open-ended starter (“Do you know of any conditions that run in the family?”) to a checklist ranging from prostate cancer to osteoporosis. Just as medical students are encouraged to take a comprehensive screening history while experienced diagnosticians adopt a more tailored approach, the specialist tends to acquire a directed family history. Cardiologists will often concentrate on ischemic heart disease, the heritable risk factors thereof (viz., diabetes, hypertension, hyperlipidemia), cerebro- and peripheral vascular disease, and premature sudden cardiac death (SCD) shared environmental factors, such as passive smoking, also may be addressed. Extraneous details are thereby avoided, at the cost of potentially missing clues to multisystemic disorders, such as Anderson-Fabry disease, and historic misdiagnoses, such as apparent epilepsy or drowning in a family with long QT syndrome. Construction of a comprehensive pedigree should, however, rectify most of the omissions from directed family history acquisition, and has the added benefit of jogging the patient’s memory through the focus on specific relatives.

The Pedigree

At minimum, a pedigree covers 3 generations (typically subject, parents, and grandparents) and first- and second-degree relatives at each level on both sides of the family. The children of the subject’s generation are included from puberty onwards, although younger children and infants are also important if juvenile onset diseases are under survey. Ideally, the pedigree is expanded to include the details of as many generations and distant relatives as the subject is able to give larger is better for diseases with low penetrance, for scrutiny of inheritance patterns, or as a guide for subsequent cosegregation studies. Standardized notation for pedigrees is summarized in Figure 1 examples from families with Mendelian and complex diseases are shown in Figure 2 and online-only Data Supplement Figure III.

Figure 1. Genogram symbols. By convention, squares and circles on a pedigree are the symbols for males and females, respectively, labeled with names and current ages. Deceased individuals are depicted by striking through the symbol with a forward slash, or placing an X inside the symbol, although the latter precludes use of shading to highlight clinically affected individuals. Age and cause of death are recorded. A single horizontal line between a man and woman indicates union, a double line is drawn for consanguineous mating, and cross hatches along the line indicate dissolution of the union. A vertical line descends from the union line and then connects to another horizontal line, the sibship line. For each child resulting from the union, a vertical line of descent drops from the sibship line. There are 3 notable exceptions: a dashed line of descent, with square brackets around the symbol, denotes a child adopted into the family an inverted V descending from the sibship line represents fraternal twins and identical twins are shown by drawing a short vertical line that subsequently bifurcates. Alternatively, twins of both types may be depicted by the short vertical line and bifurcation, with monozygotic status distinguished by an additional horizontal line connecting the branching diagonals or the symbols themselves. Variations also exist in the conventions for presenting pregnancy loss or intrauterine death, potentially relevant in congenital onset disorders. One common approach is a triangle to depict pregnancy, with a diagonal slash or X for a miscarriage and an additional horizontal strikethrough for a termination a stillbirth is denoted by a smaller square/circle, also struck through (not shown). Standardized notation notwithstanding, these variations underscore the importance of providing annotations, footnotes, or a key on the pedigree to facilitate communication within a multidisciplinary healthcare team. All individuals within a single generation are shown at the same level, adjacent to each other, with the first-born farthest to the left. Preceding generations are above and younger generations are below. Each generational level is labeled with a Roman numeral, beginning at the top of the pedigree individuals within each generation are assigned consecutive Arabic numbers from left to right. If a specific disease is of interest, then the squares/circles are shaded for clinically affected individuals, while a dot in the center of the symbol depicts an unaffected mutation carrier. The index case, defined as the individual through whom the family is ascertained, is denoted by an arrow.

Figure 2. Anderson-Fabry pedigree. The index case (arrow) presented in her early 60s with atypical chest pain and was found to have hypertension with mild left ventricular hypertrophy, for which she was prescribed a β-blocker. Ten years later, she learnt that her sister (II.7) had been diagnosed with hypertrophic cardiomyopathy, prompting her to seek a second opinion. During her subsequent evaluation, a detailed pedigree was compiled. Her two nieces (III.5 and III.6) were noted to have sensorineural deafness, while a nephew (III.8) had renal failure. The spectrum of clinical abnormalities within the family raised suspicion of Anderson-Fabry disease, which was confirmed on genetic testing.

There are 2 common strategies for acquisition of the pedigree. The first is to compile it in the presence of the patient, which has the advantage of allowing immediate discussion of pertinent findings. Completion of the task at the initial consultation is often not feasible, because a single subject may not be familiar with the medical history of all maternal and paternal relatives. This possible drawback is offset in consultations attended by several relatives together, as is practice (with mutual consent) at some centers specializing in inherited disease. The alternative approach is to mail questionnaires to patients well in advance of the consultation, to enable them to research their family history. A draft pedigree is constructed from the responses and subsequently modified through clarification and further exploration during the interview.

The Specialist Family History

The specialist family history usually entails deliberate omission of general screening questions in favor of detailed, tailored questioning. Rheumatological and renal disorders, for example, arguably influence cardiovascular health, but questioning for a family history thereof is not requisite, although a personal history is noteworthy. On the other hand, it is not sufficient for the tailored family history to be limited to cardiovascular disease, risk factors, and instances of SCD. Deaths from progressive heart failure should be distinguished from SCD. Coronary intervention and bypass surgery are worth specific queries, as are device implants, radiofrequency ablations, and transplantation if the focus is a cardiomyopathy or arrhythmic disorder. Undiagnosed symptoms, such as palpitation and syncope, also may be relevant, particularly in relatives who subsequently suffered SCD. The age of the relative at symptom onset and at the time of clinical diagnosis, intervention, or events merits documentation.

The circumstances of SCD also warrant exploration and may stratify the differential diagnosis in cases where autopsy was not performed or inconclusive for example, SCD during sleep in a Southeast Asian man with noncontributory postmortem raises suspicion of Brugada syndrome. For the same reason, it may be worth exploring the background to apparently accidental deaths, to determine whether loss of consciousness might have preceded and precipitated trauma, as in fatal road traffic accidents and drowning. If a specific inherited disorder is under investigation, the clinician will often enquire about extracardiac symptoms that point to a syndromic form, such as congenital deafness in a family with suspected long QT (autosomal recessive Jervell-Lange-Nielsen syndrome) or kidney disease and neuropathy in the setting of hypertrophic cardiomyopathy (X-linked recessive Anderson-Fabry disease).

Integration Into Clinical Practice

Acquisition of an accurate family history depends on the reliability of both the patient and the clinician. The validity of patient reporting was assessed in a sample of 10 diabetic cases and 10 controls. The 20 index cases indicated the diabetic status of over 200 of their primary relatives, the majority of whom were subsequently interviewed. There were no discrepancies between the family histories provided by the index cases and the information obtained from their relatives. Although the sample number was small, these results suggest that patient reporting of family history is generally reliable. 3

The extent to which healthcare workers request and record these details is also pertinent. In a study of family medicine residents interviewing model patients, the average time to obtain and document a family history was 16 (range 9–30) minutes, with ≈83% of available information typically elicited. 4 Perhaps owing to time constraints, audits of clinical practice among primary care physicians suggest that the family history is of low priority. Research nurses observing 4454 patient visits to 138 family practitioners reported that family history was discussed during 51% of new patient interviews and 22% of visits by established patients, with the average duration of the discussion being under 2 1/2 minutes a pedigree was present in 11% of patient records. 5 The quality of the information obtained also may be limited. A separate investigation reviewed 53 charts from 15 randomly selected primary care physicians, collected from 223 consultations (mean 4.2 per patient) over a 5-year period. Although 39 patients self-reported a family history of at least 1 common disease, the case records reflected this in only 14 (36%). Study patients identified 115 first-degree and 213 second-degree relatives with 1 of the diseases under survey the physicians documented this family history in only 23 (20%) and 4 (2%) of cases, respectively. 6

The task of obtaining an adequate family history therefore may fall to specialists. At centers with a major interest in hereditary diseases, acquisition of a detailed pedigree often is integral to the work-up, with sharing of responsibility between physicians, nurses, and genetic counselors. As a consequence of variable and age-related expression, affected relatives of index cases with a genetically determined disease often have mild clinical manifestations and few, if any, symptoms. Unfortunately, limited disease expression may belie significant risk of events in some genetic diseases, including arrhythmogenic cardiomyopathy and the inherited arrhythmia syndromes, such as long QT and Brugada syndrome. 7,8 Prospective clinical evaluation of relatives, sometimes on a periodic basis, is indicated in many scenarios. Compilation of a pedigree, therefore, is best viewed as a dynamic process, with periodic need for revision of earlier drafts to reflect the clinical status of family members as they age and undergo repeated examination. Further information also may become available on deceased relatives in the field of inherited cardiovascular disease, obtaining postmortem reports and, if available, retained blocks for re-examination, frequently is invaluable. Electronic storage of family data facilitates updating dedicated software packages are now available for pedigree construction and analysis.

Implications of Family History

Recurrence of a trait within a family may be a corollary of genetic transmission, shared environment (“household effects”), or common behaviors. Evidence of genetic contribution is sought at the family level through examination of the pedigree for inheritance patterns and at the population level through heritability studies.

Inheritance Patterns

If the prevalent mode of inheritance for a trait is well- established, the clinician may merely seek assurance that the pedigree is consistent with it. The downside of this common, time-saving approach may be a tendency to overlook more unusual inheritance patterns. Like most structural cardiovascular disorders, dilated cardiomyopathy is transmitted predominantly as autosomal dominant, and in this form accounted for 56% of a series of 39 affected families. 9 A number of other subtypes, however, also were identified, including an autosomal recessive form in 16%, characterized by adverse prognosis, and X-linked recessive in 10%, associated with mutations in the dystrophin gene. Without meticulous scrutiny of the pedigree, it is possible to overlook departures from autosomal dominant transmission, such as skipping of generations and male-only disease small, nuclear families further obscure inheritance patterns, underscoring the importance of recruiting and evaluating distant relatives. Pointers to different modes of Mendelian inheritance, summarized in Table 1, also are potentially explicable by incomplete penetrance or coincidence, further compounding the difficulties.

Simple Versus Complex Traits

The incomplete penetrance and variable expressivity that accompany many autosomal dominant traits implies the influence of genetic background and environmental factors on expression of the causal mutation. Recognition of the role of modifiers challenges the traditional dichotomy between “simple” Mendelian and “complex” multifactorial traits as artificial: a representation of perception rather than biological reality. Based on the multi-hit premise, the genetic contribution to complex diseases is attributable to interaction of 2 or more independently inherited alleles the primacy of any individual gene is not discernible, but that is the sole distinction from a Mendelian model. It can therefore be argued that heritable traits represent a continuum from a discernible primary (causal) gene interacting with modifiers to increasingly shared influence by multiple genes and environmental effects. 10 Awareness of this concept is important because of the practical constraints it imposes on genetic diagnosis. If penetrance and expression within a family are variable to the point of complexity, a single genetic variant identified in an index case cannot be presumed a primary mutation. In isolation, the variant may not be sufficient for clinical expression, limiting its predictive capacity among proven carriers. At the same time, the variant may not be necessary for clinical expression, precluding reassurance of relatives who do not carry it. Arrhythmogenic cardiomyopathy serves as a real-world example of this scenario. Autosomal dominant transmission predominates and penetrance is near complete in some kindreds. Conversely, in other families penetrance may be as low as 20% and the pedigrees reminiscent of multifactorial traits. A number of studies have now demonstrated that variants previously presumed to be independently pathogenic occur at low frequency in healthy control subjects, and that 2 or more variants frequently are necessary for clinical disease expression. 11–13 The emerging genetic complexity of arrhythmogenic cardiomyopathy poses an obstacle to commercialization of predictive testing.

Non-Mendelian Inheritance Patterns

Allele-allele interactions, allele-dose effects, and environmental triggers are but 3 of the factors that may contribute to variable penetrance and expressivity. Variations may arise at every stage at which gene products undergo regulation: transcription, splicing, translation, protein folding, oligomerization, translocation, compartmentalization within the cell or export from it, and turnover. 14 Molecular chaperones, which double as heat shock proteins, assist in the correct assembly, folding, and localization of proteins. Noncoding RNA derived from both introns and extrons is also processed into micro RNA, small nuclear RNA, and other small regulatory RNAs. The resulting RNA regulatory networks control multiple facets of gene expression, including chromatin architecture, transcription, RNA splicing, editing, translation, and turnover. 15 An array of mechanisms may therefore give rise to phenotypic diversity and hence to segregation pattern variability. The key non-Mendelian inheritance patterns are summarized in Table 2. 14

Worthy of further discussion are epigenetic marks, defined as heritable alterations of genome function that are extrinsic to the primary DNA nucleotide sequence. DNA methylation, small regulatory RNAs, covalent modification of histone proteins, and chromatin conformation all fall under the umbrella of epigenetics. Normal epigenetic patterns are essential for growth and development. For example, the modification of histone acetylation and methylation, which is controlled by families of histone acetylases/deacetylases and methyltransferases/demethylases, regulates stem cell maintenance, differentiation, and function. 16,17

Genomic imprinting (Table 2) refers to an epigenetic mark that is specific for the parent-of-origin and results in preferential expression of only 1 of the 2 parental alleles, while the other is “switched off”. Placental mammals may have evolved imprinting to fine tune the growth and development of the fetus. In general, paternally inherited genes are associated with enhanced fetal growth through increased demands on the mother, while maternally inherited genes favor conservation of the mother’s resources for her future offspring as well as her own survival. Imbalances in imprinted gene expression appear to underlie key complications of pregnancy, such as gestational trophoblastic disease, and various congenital syndromes. 18–21 As an example, hypomethylation of the imprinting control region 1 at 11p15 and maternal duplication of 11p15 are associated with the intrauterine and postnatal growth retardation of the Silver-Russell syndrome. Conversely, hypermethylation of the same region and paternal uniparental disomy of 11p15 result in the overgrowth and organomegaly of Beckwith-Wiedemann syndrome. 22,23 Imprinting anomalies usually arise sporadically. When familial clustering is observed, however, the inheritance pattern is not Mendelian heritable genetic defects reveal effects only when inherited from the appropriate parent. 14

Heritability

A phenotypic trait is considered heritable, in common parlance, when at least 1 of its determinants is transmissible between generations. Heritable need not mean inherited. Bilateral anophthalmia, for example, is caused by de novo loss of function mutations in the SOX2 gene in a significant proportion of cases. 24 Index cases have unaffected parents because the mutations arise sporadically at germ-line level, but are capable of passing the defect on to their offspring. Furthermore, describing a trait as heritable gives no indication of the mechanism or pattern of intergenerational transmission, or the extent to which genetic and epigenetic factors contribute to the phenotype. If the scope of the term heritable appears restrictive, then the definition of heritability is still more specific. Heritability is the proportion of total phenotypic variance (σp) in a given population that is due to variation in genetic factors (σG). 25 Estimation of heritability is discussed further in the online-only Data Supplement. 26,27

Limitations of Heritability

Estimating heritability has long been an integral first step to elucidating the etiology of traits with evidence of familial recurrence but unknown or indistinct inheritance pattern. More recently, it has been argued that the heritability is anachronistic, an oversimplification of intricate biological systems, limited in scope and hence limited in use. 28 The opposing school of thought holds that heritability retains its relevance in the genomics era, but the limitations of the concept must be understood to enable profitable application. 25

First, the partitioning of phenotypic variance that forms the basis of heritability calculations assumes the absence of genetic–environmental covariance. This is not always a safe assumption. Dairy cattle, for example, may be fed according to the milk production capacity of their particular lines, leading to positive covariance. 25

Second, heritability is a measure of the genetic and environmental contributions not to the phenotype itself, but to its variance around the mean for a given population. Thus, a low heritability implies that only a small proportion of the total phenotypic variation is due to genetic variation, not that the additive genetic variance itself is trivial. Nor does high heritability necessarily indicate predominant genetic determination. If a trait is highly heritable, then the phenotype of an individual in the current status quo should be a good predictor of genotype. Knowledge of genotype does not, however, predict the absolute phenotype, which may be influenced by changing environmental factors. The 150-year trend toward increasing height in successive generations in most European countries is probably attributable to improved nutrition, which in no way contravenes the reported heritability of

Third, heritability estimates provide no insight into the cause of differences between populations. In the mid-19th century, Caucasian men in the United States were, on average, 9 cm taller than their Dutch counterparts, but by the end of the 20th century, the Dutch had overtaken them by

5 cm. In spite of the high heritability of adult height, this reversal is most likely environmental rather than genetic in etiology. 25,29

Fourth, the heritability estimate is strictly applicable only to the original population under test conditions. Age and sex, for example, are recognized covariates in heritability calculations. Theoretically, the heritability of a trait may vary by population and environment, although in practice it is often similar in other populations of the same species and even across species. 25,28,30

Contemporary Role of Heritability

Couched in the above caveats, the contemporary role of heritability estimation can be revisited. The rationale for genome-wide association studies is the “common disease/common variant” hypothesis: that the genetic contribution to complex traits is due to alleles occurring at high population frequency but exerting modest effects on phenotype in the individual. More than 300 replicated associations now have been reported between common variants and complex traits, ranging from height to type-2 diabetes, obesity, atrial fibrillation, cardiac conduction, and renal function. 31–36 Yet, the variance explained by the validated single nucleotide polymorphisms is usually only a fraction of the narrow-sense heritability. For example, although genome-wide association studies have elicited ≈50 variants associated with adult height, they appear to account for a mere ≈5% of the total phenotypic variance. 31 Among the potential sources of this “missing heritability” are gene–environment interactions, inherited epigenetic factors, copy number variants, such as insertions and deletions, copy neutral variation, such as inversions and translocations, and the “common disease/rare variant” (or “genetic heterogeneity”) hypothesis 37,38 The “missing heritability” of complex traits is discussed in more detail in the online-only Data Supplement. 27,39–42

Application of Heritability to Mendelian Traits

Heritability, therefore, remains a valuable benchmark for monitoring progress in the elucidation of common, complex traits. Both the principles and methods of heritability analysis also are applicable to the phenotypes associated with Mendelian diseases. Nested analysis of variance (ANOVA) further allows evaluation of inter- and intrafamilial differences. The combination of heritability estimation and nested ANOVA enables dissection of the relative contribution of mutational heterogeneity, modifier genes, and environmental factors to continuous phenotypic measures. The original test sample for this analysis comprised >300 relatives from type 1 autosomal dominant polycystic kidney disease, which is characterized by marked variation in the severity and progression of renal and extrarenal phenotypes. The results suggested that inherited modifiers in the genetic background were important contributors to the diversity in traits, including serum creatinine, urinary protein excretion, renal volume, number of liver cysts, and age at diagnosis of hypertension and end-stage renal disease. 43 The approach was subsequently to investigate a number of quantitative traits associated with arrhythmogenic cardiomyopathy, also known for its broad phenotypic spectrum. Heritability estimates ranged from 20% to 77%, being highest for left ventricular ejection fraction and lowest for the ventricular arrhythmia grade, suggesting differing genetic and environmental contributions to these traits. ANOVA models indicated a predominant mutation effect for left ventricular fibrosis, as indicated by late gadolinium enhancement on cardiac magnetic resonance. Conversely, the modifier genetic effect appeared significant for right ventricular end-diastolic volume and ejection fraction, left ventricular ejection fraction, and importantly, for arrhythmic events. 44

Implications of Inheritance

Acquiring a reliable and comprehensive family history is the first step to determining whether an observed trait might have a hereditary basis. If familial clustering is observed, then pedigree analysis of a kindred or heritability estimation of a cohort facilitates confirmation of inheritance. Establishing a trait as heritable has important implications for both clinical practice and public health promotion.

Clinical Practice

An inherited trait influences every component of the clinical pathway, from history taking to physical examination, investigations, diagnosis, and therapy. The forthcoming review series in Circulation: Cardiovascular Genetics focuses on the impact of heredity in specific cardiovascular disorders. The general principles are introduced here with a few key examples.

History and Physical Examination

Revisiting the clinical history is often necessary after compiling a detailed pedigree. Multiple instances of SCD in the family of an index case with apparent dilated cardiomyopathy may instigate reinterrogation for palpitation and symptoms of impaired consciousness, which, if predominant, suggest an alternative diagnosis. The time course also is relevant: presentation with symptoms suggestive of arrhythmia, with eventual progression to left ventricular failure, is more typical of left-dominant or biventricular arrhythmogenic cardiomyopathy in dilated cardiomyopathy heart failure is typically the first manifestation of the disease. 45 The propensity to arrhythmia and SCD is also prominent in familial dilated cardiomyopathy with conduction system disease any history of accompanying skeletal muscle weakness raises suspicion of Emery-Dreifuss muscular dystrophy, which may be transmitted as an autosomal dominant, recessive, or X-linked trait. 46

Recurrent extracardiac abnormalities in relatives may point to an inherited multisystems disorder in spite of apparently isolated cardiovascular disease in the index case confirmation warrants a careful review of systems in all family members to ensure early recognition of complications. The example shown in Figure 2 is that of Anderson-Fabry disease the family history was the first clue to the diagnosis in an index case who presented with apparently uncomplicated left ventricular hypertrophy.

Suspicion of an inherited trait also merits vigilance for specific features on physical examination. Recurrent hyperlipidemia within the family may prompt the clinician to seek abdominal tenderness from pancreatitis, lipaemia retinalis, various types of xanthomata, xanthelasma, and arcus cornealis the presence of stigmata may give clues to the Fredrickson type before lipid subfraction analysis or genotyping can be conducted on all family members. 47 Returning to the example of Anderson-Fabry disease, angiokeratoma may be the most visible early clinical feature, taking the form of a reddish purple maculopapular rash on abdomen, thighs, and hips corneal opacities may also be detected by slit lamp examination. 48

Investigations and Diagnosis

Subsequent investigations will also need tailoring. Prospective assessment for dilated cardiomyopathy usually is confined to a 12-lead ECG and 2D-echocardiogram ambulatory ECG monitoring is performed for risk prediction on diagnostic confirmation. If the family history is suggestive of left-dominant arrhythmogenic cardiomyopathy or dilated cardiomyopathy with conduction system disease, however, ambulatory ECG monitoring and exercise testing should be integral components of the screening work-up. Any hint of Anderson-Fabry disease in a pedigree from a family with apparent hypertrophic cardiomyopathy warrants assay of α-galactosidase A activity in leukocytes, which (if low) is diagnostic in men. Although the disease is inherited as an X-linked recessive trait, women may be affected owing to random X-chromosome inactivation their α-galactosidase A activity may be normal, however, necessitating mutation screening of the GLA gene for confirmation. 48

A critical aspect of the diagnosis of Anderson-Fabry disease is the recognition that a single family member (particularly a woman) may not express the phenotype to an extent sufficient to arouse clinical suspicion. Diagnosis often requires recognition of a pattern within the extended family: the coexistence of apparently disparate abnormalities, such as nephropathy, ischemic cerebrovascular disease, progressive hearing loss and vestibular impairment, and even less well-known complications, such as osteoporosis and chronic cough and wheeze from respiratory involvement (Figure 2). This need to build up a composite familial phenotype is also inherent to the evaluation of the surviving relatives of sudden unexplained death victims. Postmortem examination of the index case has proved noncontributory, and family members often demonstrate ostensibly nonspecific abnormalities. The experienced clinician may, however, be able to discern a pattern from the findings in the extended family for example, right precordial T-wave inversion in a parent, combined with arrhythmia of right ventricular origin in 1 or more siblings, raises suspicion of arrhythmogenic right ventricular cardiomyopathy, a disease commonly missed on autopsy. 7,49

Perhaps the most important impact of proven heredity is in lowering the threshold necessary for diagnosis of the trait in relatives. In the index case, a key challenge in establishing the diagnosis is the exclusion of phenocopies: nonhereditary states that mimic the genetically determined disease. The likelihood of disease in a relative is, however, manifold higher than the baseline prevalence in the general population, reducing, albeit not obviating, the need to exclude phenocopies. Furthermore, as previously discussed, the affected relatives of index cases with Mendelian disorders commonly show incomplete phenotypic expression less stringent diagnostic criteria may be requisite for recognition of familial disease. The increased pretest probability, coupled with variable expressivity, is reflected by the provision of modified diagnostic guidelines for relatives in hypertrophic cardiomyopathy and a number of other inherited diseases. 50,51

Prognostication and Therapy

The role of heredity in prognostication and therapy is currently less well-defined. In hypertrophic cardiomyopathy, follow-up studies repeatedly have confirmed a family history of SCD as risk factor for events in the individual. 52 Experience indicates that the predictive capacity of family history holds true for disease due to mutations in 3 out of 4 of the major genes: MYH7, MYBPC3, and TNNT2. The exception is TNNI3 disease, which is characterized by markedly variable penetrance and expressivity in affected families. 53 Knowledge of the mutation itself does not enhance the prognostic power above that of family history. In contrast, there is no evidence to support the use of family history as a prognostic indicator in arrhythmogenic cardiomyopathy. 7 In dilated cardiomyopathy, its primary importance may be to identify the subset of families with associated conduction system disease and high arrhythmic risk, which can be verified by mutation screening of LMNA, or emerin (EMD) in families with X-linked recessive Emery-Dreifuss muscular dystrophy. 46 A similar scenario arises in long QT syndrome: the family history is useful chiefly as lead-in to genotyping. The trigger for events may provide a clue to the disease subtype (and hence the causal gene). The most well-known precipitants include swimming and diving for LQT1 (KCNQ1), auditory stimuli for LQT2 (KCNH2), and sleep for LQT3 (SCN5A), underscoring the importance of determining the circumstances of any sudden deaths in the family. 8,49 Regardless of whether the family history suggests a particular subtype, however, a definitive clinical diagnosis of long QT syndrome merits mutation screening of the main causal genes to enable predictive testing and guide treatment. Beta-blocker therapy, for example, is particularly efficacious in LQT1 patients, but perhaps less so in LQT2 and LQT3. 8,54

Although seldom factored into clinical decision making, heredity also influences risk and therapeutic response in common, complex diseases. The Paris Prospective Study included over 7000 men aged 43 to 52 years without a known history of ischemic heart disease, who were followed for an average of 23 years. Parental sudden death was associated with a relative risk of 1.8 for sudden death in the individual, after adjusting for confounders, including family history of myocardial infarction. 55 The case-control AGNES study included individuals with and without ventricular fibrillation during the early phase of a first ST-elevation myocardial infarction familial sudden death occurred significantly more frequently among cases than controls (43.1% and 25.1%, odds ratio 2.72). 56 A replicated association was subsequently found between a common variant at 21q21 and ventricular fibrillation during acute myocardial infarction. 57 Family history of SCD is, therefore, relevant not only in the identification of inherited arrhythmogenic disease, but also in the risk stratification of ischemic heart disease. In the therapeutic arena, hereditary factors in the form of common genetic variants have also been implicated in the risk of overanticoagulation and bleeding events from warfarin, and in the development of myopathy from statins. 58,59

Public Health Promotion

For inheritance to become a tool for preventive medicine at a population level, at least 3 conditions would have to be met. First, heredity would have to be a simple, inexpensive, and reliable risk factor for a substantial proportion of diseases of public health significance. The most obvious solution would be to employ family history as a surrogate for more definitive proofs of heredity, such as inheritance patterns or genotype. Accumulating evidence confirms that family history indicates susceptibility to a majority of common, chronic, or life-threatening diseases, including diabetes, asthma, osteoporosis, breast, prostate, or colorectal cancer, and melanoma. 60 Second, the selected marker of heredity, in this case, family history, should not just identify a minority of high risk subjects, but stratify subjects into high, moderate, and average (general population level) risk categories. The predictive capacity of family history, in isolation, increases with the number of family members affected, with the proximity of the kinship to affected relatives, and by the prematurity of onset of the disease. Various scoring systems and risk classifications have been proposed that take these factors into account the ideal tool would also be robust to family size and inflation by a single individual, and incorporate covariates such as age and sex. 61

Using a standardized quantitative family risk score, the degree of familial aggregation of ischemic heart disease, stroke, hypertension, and diabetes was obtained from >120 000 families, chiefly from Utah. A positive family history of ischemic heart disease was present in only 14% of the general population, but accounted for 72% of premature cases and 48% of cases at all ages. For cerebrovascular disease, 11% of families with a positive risk score accounted for 86% of early (<75 years) and 68% of all strokes. The instrument used to collect family history showed 77% sensitivity and 85% specificity. 60,62 Family history tools appear, therefore, to satisfy the criteria of feasibility, validity, and use.

The third requirement for an effective public health (or clinical) tool is tangible benefit to the subjects. Neither early prediction nor diagnosis is arguably of value unless effective strategies exist for prevention or intervention. Pervasive noncompliance with lifestyle advice, such as smoking cessation, suggests that knowledge of risk is frequently insufficient to modify behavior. The motivation to do so may, however, may be enhanced by the belief that change is both possible and salutary. Participation in screening programs for colorectal and breast cancer, for example, appears to be higher among individuals with a family history of the disease. 60 Public health campaigns have been successful in areas ranging from cot death to skin cancer prevention with increasing media attention and research focus on genetic discoveries, there has arguably never been a better time to highlight the importance of familial disease and its implications to both healthcare providers and consumers. 63 Until comprehensive genomic profiling becomes scientifically achievable, commercially viable, and universally accessible, such time-honored surrogates for inheritance will retain their value in both clinical management and public health promotion.

Acknowledgments

We are grateful to Shaughan Dickie and Dr Sripurna Das for their assistance in pedigree construction and proof reading.

Sources of Funding

The authors were supported by the British Heart Foundation (SSC, WJM), the EU 5th Framework Program Research and Technology Development ( QLG1-CT-2000-01091 ), and the Department of Health’s NIHR Biomedical Research Centres funding scheme.


Limited genetic variability and phenotypic plasticity detected for cavitation resistance in a Mediterranean pine

Given the magnitude of the expected increase in world average temperatures and in the frequency of extreme climatic events (Beniston et al., 2007 Della-Marta & Beniston, 2008 Sterl et al., 2008 van Oldenborgh et al., 2009 Wigley, 2009 ), the rate of adaptation driven by natural selection and migration may no longer keep pace with climate change (Davis & Shaw, 2001 Davis et al., 2005 Corlett & Westcott, 2013 S. Delzon et al., unpublished). This is particularly true for long-lived organisms such as forest tree species forced to cope with these drastic rapid climate changes – possibly within a single generation (Breda et al., 2006 Bréda & Badeau, 2008 Lindner et al., 2010 ). Indeed, several authors have reported that recent forest die-backs could be linked to severe drought events, which are manifestations of climate change (Allen & Breshears, 1998 Breshears et al., 2005 Breda et al., 2006 Granier et al., 2007 Allen, 2009 Allen et al., 2010 ). Whether organisms can pass through such abiotic filters depends on their fitness, that is, the probability to both survive and reproduce. The reproduction component is often considered the main driver of fitness, but in a changing world with stochastic extreme droughts, survival could become a bigger challenge than reproduction. Quantifying the extent and relative amount of genetic and environmental variations in relevant fitness-related traits during extreme drought is therefore a prerequisite to understanding the evolutionary processes that lead these organisms to cope with such climatic events and predicting their adaptive potential in response to climate change (Lindner et al., 2008 , 2010 ).

One of the most relevant traits for tracking tree survival during extreme droughts is cavitation resistance, that is, the ability to conduct water though the xylem even during drought events (Cochard et al., 2008 ). This is grounded by several lines of evidence. Meta-analyses have shown that, on average, species from drier climates are more cavitation-resistant than species from wetter climates (Maherali et al., 2004 Choat et al., 2012 ), while experimental drought and recovery monitoring experiments have established a causal link between resistance to cavitation and lethal water potential (Brodribb & Cochard, 2009 Brodribb et al., 2010 ). For instance, in conifers, a 50% loss of hydraulic conductance in the seedling stem leads to death by dehydration, showing that more cavitation-resistant species survive stronger drought (Brodribb & Cochard, 2009 Brodribb et al., 2010 ). Overall, resistance to cavitation (estimated by the pressure corresponding to 50% loss of hydraulic conductance (P50)) varies widely among species (Maherali et al., 2004 ), especially conifers (P50 ranged between −2 and −16 MPa see Delzon et al., 2010 ). A recent study showed that most of the variability in this trait was attributable to genera within a botanical family, whereas species explained < 10% of the variance (S. Delzon et al., unpublished).

To cope with increasingly severe drought events, the adaptation of sessile organisms will rely on the level of standing genetic variation and phenotypic plasticity (Aitken et al., 2008 ). Genetic diversity can be seen as a pool of variants among which natural selection, at a given time and in a given point in space, keeps the fittest. This diversity is naturally renewed over time by the interplay of evolutionary forces (migration, genetic drift, natural selection, recombination and mutation). Only three recent studies have assessed genetic variation in resistance to cavitation (Corcuera et al., 2011 Lamy et al., 2011 Wortemann et al., 2011 ). It was found that phenotypic variation is low (coefficient of variation < 10%) and that variation between populations is limited, as most genetic variation resides within population. Lamy et al. ( 2011 ) proposed that cavitation resistance is a genetically canalized trait in Pinus pinaster, that is, the average value of this trait is similar for populations of dry and wet origins, making it robust to genetic perturbations (i.e. mutation, recombination, etc). Phenotypic plasticity is a second component of adaptation. It is classically split into two components: a reaction norm corresponding to a (linear, quadratic or sigmoid) function that links phenotypic variation to environmental changes and the genetic variability of the reaction norm, that is, the genotype-by-environment (G × E) interaction (Debat & David, 2001 Pigliucci, 2005 ). G × E is of major concern in the attempt to develop plants adapted to wide geographical ranges. It is of considerable concern for forest trees species, as it provides clues to understand mechanisms that have shaped local adaptation. The term ‘phenotypic plasticity’ can be found in most discussion sections of papers dealing with ecological implications of variation in cavitation resistance (Kolb & Sperry, 1999 Maherali & DeLucia, 2000 Maherali et al., 2002 Jacobsen et al., 2007 Beikircher & Mayr, 2009 Martinez-Vilalta et al., 2009 ), yet it has rarely been quantified by appropriate experiments and robust estimators. This has prompted some authors to postulate that cavitation resistance is a highly plastic trait (Jacobsen et al., 2007 Beikircher & Mayr, 2009 ), whereas other authors claim that cavitation resistance is not a plastic trait at all (Maherali et al., 2002 Martinez-Vilalta et al., 2004 , 2009 ). These conclusions arose from the comparison between phenotypic variation, as assessed in situ, and genetic variation, as measured in provenance trials. However, until recently, working with a large sample size (to properly estimate variance components in well-designed experiments) was inconceivable given the technology available. This phenotyping barrier was removed thanks to technical advances allowing high-throughput phenotyping (Cochard, 2002 Cochard et al., 2005 ). This high-throughput method has reduced the cost of experimentation, allowing researchers on a fixed budget to obtain much more precise estimates of variances and variance components, opening up new perspectives for quantifying the relative amount of genetic variation and phenotypic plasticity in resistance to cavitation.

To explore, in concert, the phenotypic variability, phenotypic plasticity and standing genetic variation of this key fitness trait, we carried out a unique case study on maritime pine (Pinus pinaster), a forest tree species with a fragmented distribution in the western Mediterranean region. The scattered distribution of this species may have prevented or limited gene flow between different groups of populations, promoting high genetic divergence between ecotypes as a result of genetic drift (Ribeiro et al., 2002a , b Bucci et al., 2007 ) and/or natural selection (Quezel & Barbero 1998, in Richardson, 1998 ). In this study, we analysed growth and resistance to cavitation data of six populations planted in France (Lamy et al., 2011 ) and in Spain using a provenance-progeny design, and data from individuals of these populations were sampled in situ. We characterized the phenotypic variance in situ, the genotypic variance in two constrasted trials and the phenotypic plasticity. In total, we measured 513 genotypes for resistance to cavitation, which to the best of our knowledge makes the results of this study the largest data set produced for cavitation resistance.

The aims of this study were three-fold: to quantify the magnitude of phenotypic variability in resistance to cavitation in situ to study the degree of environmental or genetic determinism for this trait by quantifying both genetic variation and phenotypic plasticity using common garden experiments and to estimate the correlations between resistance to cavitation and climate variables. This knowledge leads us to elaborate hypotheses on both micro- and macro-evolution of resistance to cavitation in this pine species.


1. Introduction

Variance component models have played an important role in detecting quantitative trait loci (QTL) for the last couple of decades in both animal breeding (Fernando & Grossman, Reference Fernando and Grossman 1989 Goddard, Reference Goddard 1992 Arendonk et al., Reference Arendonk, Tier and Kinghorn 1994 Wang et al., Reference Wang, Fernando, van der Beek, Grossman and van Arendonk 1995 George et al., Reference George, Visscher and Haley 2000) and human genetics (Goldgar, Reference Goldgar 1990 Schork, Reference Schork 1993 Fulker & Cardon, Reference Fulker and Cardon 1994 Olson, Reference Olson 1995 Xu & Atchley, Reference Xu and Atchley 1995 Blangero et al., Reference Blangero, Williams and Almasy 2001). To construct the variance–covariance matrix of the random QTL effect, identity-by-descent (IBD) probabilities are required. The IBD probabilities describe the correlation structure between individuals with respect to the frequency of their shared (common) alleles. The genetic variance component estimates, and the corresponding likelihoods, are usually calculated using an estimated IBD matrix.

The IBD matrix can be estimated from marker information using either deterministic (Wang et al., Reference Wang, Fernando, van der Beek, Grossman and van Arendonk 1995 Pong-Wong et al., Reference Pong-Wong, George, Woolliams and Haley 2001 Besnier & Carlborg, Reference Besnier and Carlborg 2007) or stochastic algorithms (Thompson & Heath, Reference Thompson and Heath 1999 Pérez-Enciso et al., Reference Pérez-Enciso, Varona and Rothschild 2000 Mao & Xu, Reference Mao and Xu 2005). All these methods actually calculate an average IBD matrix, where each entry is the average frequency of shared alleles, based on partially informative markers. Namely, all the IBD values are known in the statistical models. Instead of using the average IBD matrix, which we refer to as the expectation method (Xu, Reference Xu 1996), the uncertainty of the IBD matrix itself may also be included in the likelihood. Such a method accounting for the uncertainty of the IBD matrix is referred to as the distribution method. The likelihood function that the distribution method uses is called the full likelihood function, in contrast to the expectation method that uses an approximated likelihood function.

Comparison of distribution methods with expectation methods has been a thoroughly investigated problem in human genetics, especially for regression models in QTL analysis and genome-wide association studies (GWAS). Using genotype imputation, GWAS can gain power at positions with uncertain genotypes (Marchini & Howie, Reference Marchini and Howie 2010) where the expectation method gives good power and accuracy (Kutalik et al., Reference Kutalik, Johnson, Bochud, Mooser, Vollenweider, Waeber, Waterworth, Beckmann and Bergmann 2011) by using SNP probabilities as covariates. For QTL analyses based on regression models, the QTL effect is treated as fixed, and several studies have applied the idea of a full likelihood function (Elston & Stewart, Reference Elston and Stewart 1971 Morton & Maclean, Reference Morton and Maclean 1974 Lander & Botstein, Reference Lander and Botstein 1989), which is referred to as the maximum likelihood (ML) method in QTL analysis and has been implemented in, for instance, MAPMAKER-QTL (Lander & Botstein, Reference Lander and Botstein 1989). The implementation is based on an Expectation–Maximization (EM) algorithm (Dempster et al., Reference Dempster, Laird and Rubin 1977). However, ML estimates based on a regression model can be approximated very well by the simple Haley–Knott (HK) regression (Haley & Knott, Reference Haley and Knott 1992), which is the corresponding expectation method using line-origin probabilities as covariates.

In random effect models, the QTL effect is regarded as random, and considering a full likelihood method is still important to avoid losing statistical power (Schork, Reference Schork 1993). Replacing the IBD matrix using its expectation can only approximate the ML estimates, and the approximation was shown, by means of simulations, to be non-negligible in the analyses of sib-pairs (Kruglyak & Lander, Reference Kruglyak and Lander 1995). To resolve these problems, a weighted likelihood approach has been implemented in the software package Mx (Eaves et al., Reference Eaves, Neale and Maes 1996) for the analysis of small human pedigrees where the probability of IBD states are used as weights. However, knowing the distribution of the IBD matrix is crucial for deriving the full likelihood function. In human full-sib studies, the closed form of the joint distribution of the additive IBD matrix and the dominance IBD matrix has been derived, but this is feasible only for pedigrees including small full-sib families (Gessler & Xu, Reference Gessler and Xu 1996 Xu, Reference Xu 1996). These earlier studies show that the full likelihood function is statistically more powerful and often gives higher likelihood at the QTL.

Three problems were raised from previous studies. First, for animal pedigrees, deriving the distribution of the IBD matrix is infeasible due to the large size. Therefore, approximating the full likelihood function using a Monte Carlo strategy is a reasonable idea but has not been implemented (Xu, Reference Xu 1996). Second, full-sib studies in humans calculate IBD probabilities for F 1 individuals. For a crossing design in animals, where e.g. F 2 individuals are studied, deriving the IBD distribution is difficult even for small pedigrees. An application of the distribution method to F 2 individuals has therefore not been investigated before, even though the theory was claimed to be able to extend to different kinds of crosses. Third, when there is inbreeding, for instance, in an F 2 intercross, diagonal elements of the IBD matrix need to be adjusted. After adjusting for inbreeding, the full likelihood theory still holds. If the marker density is low or the markers are partially informative, the difference between the distribution method and the expectation method might be substantial for experimental crosses as well.

The aim of our study is to evaluate the performance of the distribution method in animal intercross designs by assessing the magnitude and direction of bias for the expectation method. We try to account for the above problems and investigate the full likelihood function for animal pedigrees. The rest of this paper is arranged as follows. We first describe the statistical model that our study is based on and introduce the theory about the full likelihood. Two illustrative examples of F 2 pedigrees are simulated, where one is used to show the difference from full-sib studies, and the other is used to show the performance of the distribution method in adjusting the bias of heritability estimates. We compare the distribution method with the expectation method for a real experimental dataset, with simulations based on real genotypes for comparing the power of the two methods. The paper is concluded by discussing possible applications and suggesting future developments.


What is the response to selection?

The breeder's equation, which predicts evolutionary change when a phenotypic covariance exists between a heritable trait and fitness, has provided a key conceptual framework for studies of adaptive microevolution in nature.

Additionally, what does a positive selection differential mean? Selection differentials increase with angling intensity and the associated higher annual exploitation rates (Fig. 3A Table 1). Positive selection differentials would cause the mean phenotype to increase, whereas negative selection differentials would cause it to decrease.

Similarly one may ask, what factors influence a population's evolutionary response to selection?

Five different forces have influenced human evolution: natural selection, random genetic drift, mutation, population mating structure, and culture. All evolutionary biologists agree on the first three of these forces, although there have been disputes at times about the relative importance of each force.

How do you determine the strength of a selection?

Selection coefficient is a measure of the relative strength of selection acting against a genotype. Calculate the selection coefficient (s) by subtracting each fitness value from 1.0 (that is, s = 1-w). Interpretation of selection coefficient: sdd = 0.0 means genotype dd is not being selected against.


Affiliations

Department of Biostatistics and Computational Biology, School of Life Sciences, Laboratory of Population & Quantitative Genetics, State Key Laboratory of Genetic Engineering, Fudan University, Shanghai, 200433, China

Shengjie Yang, Ning Jiang & Zewei Luo

School of Biosciences, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

Jing Chen, Lindsey Leach, Zewei Luo & Minghui Wang

Division of Cardiovascular & Diabetes Medicine, University of Dundee, Dundee, DD1 9SY, UK


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