Abstract
Implications of genotype × environment interaction resulting from site differences in expression of genetic variation (LoE interaction) were explored for some plausible scenarios for breeding radiata pine. Expected genetic gains were modelled in a Smith-Hazel selection index. Two sites were modelled, addressing two sets of three traits at each site, to create 6 × 6 genetic and phenotypic covariance matrices based on typical heritabilities and between-trait correlations as well as rank-change (RC) interaction. Two of the traits, which behaved like stem volume production and disease resistance respectively, featured in all scenarios, with disease being expressed and influencing volume at only one site. Two alternatives for the third trait behaved like wood density and stem straightness respectively, two levels of genetic trade-off between volume and density being addressed. The impact of LoE interaction was modelled by assigning either zero or non-zero economic weight to a trait at one site. Comparisons of expected genetic gains indicated how substantial economic gains can depend on appropriate recognition, in selection for deployment, of LoE interaction. Expected economic gains from selection for both sites jointly were generally little less than those from selection for individual sites.
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Introduction
Genotype-by-environment interaction (‘G × E’) is a pervasive issue for forest tree breeding. It can involve either or both of changes in the ranking of genotypes’ performance among environments (‘RC’ interaction) and differences among environments in level of expression of genotypic differences (‘LoE’ interaction).Footnote 1 Where present, G × E will tend to erode the achievement and capture of genetic gain, unless the breeder adopts one or both of (1) more complicated and expensive breeding-programme structures, and (2) more closely targeted deployment of genetic material to sites and growing regimes.
The concepts involved in addressing G × E in forest tree breeding are reviewed by White et al. (2007, pp 134−139) and Li et al. (2017). Historically, the focus has tended to be on RC interaction, as reflected in departures from perfect genetic correlation between performance in different environments (type B genetic correlations), and its implications (e.g. Burdon 1977). While RC interaction is of prime significance for evaluation and selection in the breeding operations aimed at continuing genetic gain, LoE interaction is seen as being often more relevant to the economically efficient capture of genetic gain through appropriate deployment of genotypes (Li et al. 2017). Some of the main implications of RC interaction are straightforward, being simple functions of type B genetic correlations (Burdon 1977). The implications of LoE interactions, however, are generally less straightforward mathematically. While some of the implications are intuitively evident, and may be widely acted on, the implications are generally not quantitatively documented. Here, we report a simulation study, based on existing genetic parameter information and some plausible selection scenarios for radiata pine (Pinus radiata D. Don), illustrating the potential economic impacts of LoE interaction for various selection options for genotypic deployment.
First, however, we give a brief conceptual review of LoE interaction, and outline some key implications.
We then outline some theory and concepts for multi-trait selection, in the framework of a Smith-Hazel selection index (e.g. Cotterill and Dean 1990), which is a linear function of the phenotypic values of some traits that the breeder can observe.
The theory is then applied to an extremely simple illustrative (if hypothetical) example. Following that, it is applied to a general scenario involving two sites with three traits, with substantial LoE interaction in addition to some RC interaction, assuming alternative sets of genetic parameters that are seen as realistic for radiata pine in New Zealand. Selection scenarios involve alternative sets of economic weights for the different traits, as a basis for calculating how expected profitability can be affected by certain LoE interaction levels.
An incidental objective was comparing expected multi-trait selection gains, for selecting genotypes for single sites versus gains from selecting individuals jointly for the two sites. Preliminary study was also made of whether a particular trait related to growth performance needed to be assigned an actual economic weight.
LoE interaction concepts and some implications
A classical model for the composition of phenotypic variance (σ2P) is
where σ2G, σ2GE and σ2E are genotypic, genotype × environment interaction and environmental variances respectively.
In turn, σ2GE has two components (Dickerson 1962; Bowman 1972), as
where σ2GE′ reflects departure from perfect linear genetic correlation between genotypes’ performance in different environments, which in effect represents RC interaction variance; and V(σj) reflects variance among the environments (1,…j,…n) in the expression of genetic variation, which represents LoE interaction variance. While an alternative partitioning of σ2GE has been presented by Muir et al. (1992), that is of no interest here.
We now review some examples of LoE interaction, and some implications (cf Burdon et al. 2017; Li et al. 2017). A very simple, clear-cut example is with disease resistance/susceptibility. It can only be expressed at all in environments where the disease hazard is high enough for the disease to occur, and will often be negligible even if present under low disease hazard. Where resistance/susceptibility is not expressed appreciably, it will have no economic significance. Where disease is prevalent, however, resistance may assume direct economic significance. In addition, through disease affecting growth rate, resistance may serve as an indirect selection criterion for growth rate through the resulting genetic correlation with growth rate. Another example is extreme genetic variability in radiata pine for growth rate on a P-deficient site resulting from genetic variation in tolerance of the deficiency (Burdon 1971). Other radiata pine examples (albeit often putative) include accentuated expression of genetic variation in tree-form traits under high soil fertility and/or low stockings (Burdon et al. 2017). Under low fertility, tree form may be uniformly so good that the phenotypic variation for the form traits may be of no economic account, in contrast to the case with high fertility. Note, however, that in tree-form traits that must be visually scored, scores made relative to the general mean may show good heritabilities even where tree form is almost uniformly good (Shelbourne and Low 1980; Li et al. 2017). Thus, the resolution of genetic differences may represent useful information for helping selection for environments where such traits are economically important.
Overall, LoE interaction is almost pervasively related to economic-worth functions for the traits concerned. Provided the metric used for a trait represents absolute values (unlike some visual scoring scales), the economic weight for a trait in a given environment will be proportional to σGj (Eq. 2). Thus, economic worth for genetic gain in the trait will depend strongly on the width of the domain in which genetic variation is expressed in the particular environment.
An associated effect, which we will address along with true LoE interaction, is when genetic variation is expressed over a domain that represents a ‘plateau’ region within a globally curvilinear economic function, as when the trait concerned consistently excels a threshold of acceptability. That is economically equivalent to the variation not being expressed.
Largely through its implications for economic-worth functions, LoE interaction can be a more complex issue than RC interaction by itself. There are other considerations, which do represent complications, but need to be addressed for achieving realistic simulation studies. These are between-trait genetic correlations, which can vary among environments, and co-occurrence of RC interaction along with LoE interaction. The traits in our simulation actually represent a spread among the sorts of cases that we have reviewed above.
The selection index
This index (I) is of the form
where b1 is the weight given to the phenotype for trait 1, namely X1, and so on, noting that for realistic solutions the number of traits, n, generally needs to be small. Alternatively, I can be written
where b′ is a row vector of weights given to the selection traits and x is a column vector denoting the phenotypic values for the respective traits.
The theoretical optimum weighting is given by
where b is a column vector (transpose of b′) for the index weights, P is the phenotypic variance-covariance matrix for the selection traits, G is the genetic covariance matrix between the selection traits and the various breeding-goal traits and a is a column vector of economic weights for the breeding-goal traits.
One can then predict genetic and economic gains for various possible combinations of genetic parameters and assumed economic weights. The expected gains for the breeding-goal traits are given by the response column vector R and a given selection intensity (standardised selection differential) i
i being available for given ratios of selections to candidates in truncation selection from look-up tables (White et al. 2007, p. 341), and noting that the b vector need not necessarily be the optimum represented in Eq. 3.
The above equations serve for exploring expected genetic gains under alternative G × E scenarios. Since impacts of RC G × E on genetic gains are well understood, the emphasis was on the impacts of LoE G × E on genetic gains under alternative sets of genetic parameters (for G and P) and alternative sets of economic weights (a).
Expected total economic gain per unit i is given by R´a, R being the row vector transpose of R.
Illustrative example
This involves disease resistance in a very simple case of deployment of select material on a disease-free site. The one other trait considered, growth potential, assumes all the economic significance, although it is uncorrelated genetically with disease resistance. Assume, too, unit genetic variance and heritability of 0.25 for both traits, with zero phenotypic covariance between them. Thus, P and G are 2 × 2 diagonal matrices, with diagonal variances of 4 and 1, respectively. Alternatives for a' are 1, 0, and 1, 1 for the cases of selecting purely for growth rate and coequally for both traits, respectively, b being calculated in each case using Eq. 5. Expected genetic gains are calculated according to Eq. 6. Expected gains per unit i in growth potential were 0.5 and 0.35 for a' = 1, 0, and 1, 1, respectively, the expectations for disease resistance being 0 and 0.35 in the respective cases. This means a 30% sacrifice in potential genetic gain for growth potential from disregarding the non-expression of disease resistance on the site in question.
General scenario
The general scenario, and its variations, are hypothetical but are seen as representative of situations facing tree breeders, especially for radiata pine in New Zealand, in respect of the nature of the traits and their genetic parameters within and between sites.
It involves three traits, 1, 2 and 3, which are considered on each of two sites, A and B (between which individual traits differ in expression status as well as genotypic rankings), giving six site/trait combinations, which are treated as six separate ‘traits’ (cf Burdon 1979), each of which can provide an index term in Eq. 3.
Nature of candidate material
Individual trees of a large base (candidate) population are present on two sites. For this study, no family structure is assumed, but genetic parameters are treated as being known. For simplicity of analysis, it is assumed that each individual genotype is replicated by one ramet per site, so there are phenotypic values for each individual for all the six ‘traits’. Zero between-sites environmental covariances are also assumed.
The traits
Trait 1 is a growth trait, lowish heritability (0.2), of unconditional economic importance, e.g. stem diameter or volume. Experience for radiata pine has been (unpublished details of Burdon et al. 1992) that these two traits did not differ materially in heritability or genetic correlations with other traits. Henceforth, this trait is denoted VOL.
Trait 2 as a quality-related trait, of higher heritability (0.3–0.5), but of economic importance on some sites but not others. It can be economically unimportant because its genetic variation, despite being expressed, is in a range that is not economically important. Two cases are considered:
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Case 1: Trait 2 having a high heritability (0.5), and being adversely correlated with VOL, as with wood density (or wood stiffness), and is henceforth denoted DEN. Two alternative levels of adverse genetic correlation are assumed: case 1a approximating to the means of the values tabulated by Wu et al. (2008), and which may also be realistic at the whole-crop rather than the individual-tree level because of the nature of competitive interactions between genotypes (cf. Burdon 2010), and case 1b approximating more to ‘consensus’ values (Evison and Apiolaza 2015, Table 2).
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Case 2: Trait 2 having a heritability of 0.3, and being uncorrelated genetically with VOL, as with stem straightness (although zero genetic correlation with VOL may be too low). It is henceforth denoted STR.
Trait 3 is a disease-resistance trait (alternatively pest resistance), of intermediate heritability (0.33) on site A, but much less heritable on sites of minimal hazard (as on site B), and is positively associated with VOL where disease is prevalent. It is henceforth denoted RES.
Although Eq. 5 provides for a distinction between selection traits as and when they are assessed for selection purposes, and breeding-goal traits as expressed at harvest age (Evison and Apiolaza 2015), we deemed it unnecessary for our purposes to draw that distinction
The sites
The scenario roughly models the contrast between two New Zealand site categories: A—volcanic plateau and B—coastal sand dunes (cf Shelbourne and Low 1980). On site A, all three traits are well expressed, in terms of economic worth as well as being heritable. On site B, VOL alone is clearly expressed, in the sense of genetic variation being economically important, although there are the two cases (1a and 1b) to consider for DEN, assuming different levels of genetic correlation with VOL.
Genetic correlation between sites (type B genetic correlation—Burdon 1977), reflecting RC interaction, is varyingly imperfect among traits. It is lower for both VOL and RES which is very poorly expressed on site B, than for the quality-related traits (DEN or STR as the case may be).
Assumed genetic parameters
The assumed genetic parameters are shown, with some variants, in the matrices presented in Table 1. Polygenic inheritance, multivariate normal distributions and linearity were implicit throughout. Table 1 shows assumed heritabilities and phenotypic and genetic correlations, while Supplementary Tables 1 and 2 show the parameters in terms of G and P matrices respectively. In these Tables, G × E is incorporated primarily in terms of RC interaction for the individual traits, as departures of between-sites genetic correlations for those traits (Table 1) from +1. However, there will also be G × E in terms of differences between sites in between-trait genetic correlations; this can to some degree reflect LoE interaction which, as explained later, is modelled in terms of economic weights assigned to trait values at individual sites.
Using published estimates to arrive at the alternative parameter values assumed within and across sites entailed some judgement calls. This was because most genetic parameters can vary among environments (especially for forest trees), different parameter values were based on different subsets of field trials, and the various point estimates were of varying precision, creating problems of achieving appropriate weightings and adjusting for missing estimates. Except for RES at site B, genetic variances are standardised to a value of 1, so variances and between-sites covariances were assigned in Supplementary Tables 1 and 2 to fit that condition and the specified heritabilities and correlations. With the one exception, phenotypic variances thus correspond to the inverses of their heritabilities (1/h2), and genetic covariances equal genetic correlations. Because of zero environmental covariance between sites, the between-site phenotypic covariances equal the corresponding genetic covariances, and the between-sites phenotypic correlations are set to fit the heritabilities and genetic correlations. The phenotypic variance of RES at site B is arbitrarily set to its value for site A, with its genetic variance and its covariances adjusted from there according to its assumed heritability and between-trait correlations (Supplementary Tables 1 and 2). The exact value assigned to this phenotypic variance is deemed to be immaterial. The only appreciably non-zero covariance shown by RES at site B is between its values at the two sites.
A partial check on the validity of the assumed genetic parameters is whether the P and G matrices are clearly positive definite, an indication of internal consistency and thus inherent biological feasibility. If both types of matrix are positive definite, and heritabilities are within bounds, the P−1G matrix product, which subsumes all heritabilities and genetic and phenotypic correlations, must also be positive definite.
Assignment of economic weights and modelling of LoE interaction
Alternative sets of assumed economic weights (a′) (1 or 0) are shown in Table 2, with brief explanations of the selection scenarios. Assigning values of 1 to genetic variances automatically scales the economic weights to a common basis, with economic weights of 1 representing equal weights for the traits concerned. Modelling the impact of LoE interaction was done by comparing expected genetic gains between cases assigning different alternative economic weights to the expression of a trait in different environments (as proposed by Li et al. 2017), in this study either 1 or 0 for a trait at a site. Assigning a zero value implies that expressed variation in the trait is of no economic importance, and that any genetic shift in the trait can be ignored for the given case and selection scenario.
Simulation runs
For all cases, positive definite status for all G and P matrices (Supplementary Tables 1 and 2 respectively) was confirmed, meaning substantial internal consistency even if they are not optimally conditioned.
Then, for each of the various cases and selection scenarios (Table 4), the following were calculated:
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1.
Index weights (b′) for the six ‘traits’, applying Eq. 5 to the various G and P matrices (Supplementary Tables 1 and 2 respectively) and alternative sets of economic weights (Table 2).
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2.
Expected genetic gains (Eq. 6), using alternative G and P matrices and the calculated b vectors (Eq. 5), for the six different ‘traits’ at the respective sites.
Results
Index weights
Calculated index weights (b′) are shown in Table 3, for the different cases and sets of economic weights (or selection scenarios), the relativities among weights assigned to traits being important rather than the absolute figures. No such weights are exactly zero, because all phenotypic information is being used for any value it has for direct or indirect selection. Some weights, though, vary only trivially from zero. Index weights corresponding to non-zero economic weights are shown in boldface. Assigning a positive index weight but zero economic worth to a trait means that it is used entirely as an indirect selection criterion.
Trait 1 (VOL) assumed appreciable or strong index weights throughout, although those weights could be markedly affected by variation in economic weights assigned to other traits,
Trait 2, which for both cases (DEN and STR) was strongly correlated genetically between sites, assumed appreciable index weights at site B, but much more when assigned an economic weight there (scenarios 1 vs 2, 4 vs 5, and 6 vs 7). Where VOL was assigned zero economic weight at site B (scenarios 3 and 8, selecting for site A only), its index weight there was still appreciable but markedly less than when it was assigned an economic weight (scenarios 2 and 7 respectively).
Trait 3 (RES) only assumed substantial positive index weights when it was given an economic weight in its own right (scenarios 1–3), despite a quite strong genetic correlation with VOL. The index weights assigned to RES at site B under selection scenarios 1–3 are immaterial because of its very weak expression there.
Between cases 1a and 1b, calculated index weights did not differ substantially.
Expected genetic gains in relation to LoE interaction simulated by economic weights
Expected genetic gains are shown in Table 4. Gains for which economic weights are non-zero are shown in boldface. Remaining gains (or genetic shifts) associated with zero economic weights are deemed to be economically null. Thus, the sums presented for individual sites and both sites amount to economic gains. As with index weights, expected gains did not differ substantially between cases 1a and 1b.
Key findings are shown in Table 5, involving losses in expected gain for VOL incurred if an economic weight is wrongly assigned to DEN or STR at site B. For cases 1a and 1b, with DEN adversely correlated with VOL, placing a pseudo-economic weight on DEN [(aDEN = 1)] at site B more than halved the expected genetic gain for VOL there compared with assigning zero economic weight (aDEN = 0) there (scenarios 1 vs 2, 4 vs 5 and 6 vs 7). Also, it substantially reduced expected gain for VOL in selection for both sites jointly. Thus, ignoring a lack of economically important expression of genetic variation in DEN at site B can sacrifice important economic gain there. However, pursuit of all the potential gains for VOL at site B would incur a substantial negative shift (deemed effectively null) in DEN there. For case 2, with STR neutrally correlated genetically with VOL, the corresponding proportional reductions of expected genetic gain in VOL were much less, but far from negligible.
Impact of selection for specific sites versus selection for both sites jointly
Expected effects of selection for specific sites compared with selection for both sites jointly are illustrated in Table 6, noting that gains from scenarios 2, 3, 5, 7 and 8 involve gains in DEN or STR at site B that are deemed to be effectively null. An expected economic advantage of single-site selection was generally evident, but was variable and for scenario 2 was absent in case 1a and only very minor for the other two cases.
Impact of assigning zero economic weight to trait 3
Where zero economic weights were assigned to RES some genetic gain was expected for that trait (scenarios 6–8), reflecting its substantial genetic correlation with VOL, despite RES being assigned no appreciable index weight. Where selection was entirely for site B (scenarios 4 and 5), expected gains for RES were mostly small or even weakly negative.
Discussion
Underpinning models
Likely departures from the polygenic (‘infinitesimal’) model seem to be too small, for radiata pine and forest trees in general (Isik 2014), to disrupt the predictive power of the model which has proved very robust with respect to detailed phenomena of gene action (Huang and Mackay 2016).
Multivariate normality, while not explicitly invoked, is a likely precondition for linearity. While correlations are not necessarily linear, reports of non-linearity and its impacts are generally lacking; moreover, non-linearities may need to be severe to be of practical importance. Linear selection weights, while implying linear economic-worth functions, may very often be satisfactory approximations despite certain non-linearities of economic worth (Burdon 1990), especially with selection of parents for producing genetically segregating offspring.
Not addressed are possible differences in costs or logistical difficulties between using single-site and joint-sites selection.
Nature of simulation
The specific situation modelled was artificial in several respects:
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replication of candidate genotypes between sites but not within sites;
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the population being large, without family structure;
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nature of heritability unspecified;
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no differentiation between selection- and breeding-goal traits;
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gain expectations being deterministic;
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all economic weights being 1 or 0.
The pattern of replication was convenient for the simulations, because it allowed the use of single 6 × 6 P and G matrices for each genetic-parameter scenario, and facilitated the joint accommodation of both LoE and RC interaction which will typically co-occur. Also, when clonal selection and deployment are envisaged, between-sites replication of genotypes is advantageous for coping with RC interaction. Within-site replication has, among its advantages and costs, the practical advantage of some protection against data loss through mortality, which we could ignore in a theoretical study. As such replication increases, index weights will approach economic weights more closely, and benefits of selection for single sites will increase.
We drew no distinction between narrow-sense and broad-sense heritabilities. The former are appropriate for selecting candidates as seed parents, the latter for selecting clones for mass vegetative propagation and commercial deployment. Our study is seen as more relevant to clonal selection and deployment, because it is with deployment that LoE is of greater interest. However, with clonal deployment, possible non-linearities in economic-worth functions could well pose appreciable complications (Burdon 1990), but these are beyond the scope of this study.
Modern advanced generation tree breeding typically involves family structures, which are often complex, so the appropriate selection algorithms can represent generalised best linear predictions (BLP) or best linear unbiased predictions (BLUP). Nevertheless the Smith-Hazel selection index is seen as a satisfactory tool for probing the sensitivities of the gain expectations that we have studied. In this connection, our simulations could be extended to accommodate additional variants of input parameters (cf case 1a versus case 1b).
Selection traits in practice typically differ from breeding-goal (or breeding-objective) traits (cf Evison and Apiolaza 2015), in that the former are measures of quite early performance whereas the latter represent harvest-age performances. This is not seen as affecting our main conclusions, although it has likely implications in respect of VOL, which we discuss later.
Although our simulations were essentially deterministic, they could be extended to stochastic simulations which could accommodate sampling distributions about assumed parameter values and could eliminate some potential biases arising in deterministic expectations. However, achieving that with respect to whole genetic parameter matrices could be challenging. Stochastic simulation could be especially advantageous with very finite populations which could have family structures.
As for the simulated economic weights, more nuanced variation in simulated economic weights would be ideal, especially for DEN in the variants of case 1. Small, non-zero economic weights could be specified, but with the information available they would be only guesses. With the quite strong negative genetic correlation between VOL and DEN in case 1a, the pursuit of all potential gain in VOL at site B would require that the breeder accepts quite a strong negative genetic shift in DEN there.
While correlations are not necessarily linear, reports of non-linearity and its impacts are generally lacking; moreover, non-linearities may need to be severe to be of much practical importance. Linear selection weights, while implying linear economic-worth functions, may very often be satisfactory approximations despite certain non-linearities of economic worth (Burdon 1990), especially with selection of parents for producing seedlings.
Overall, the various artificial features of the model are not seen as invalidating our key findings.
The traits, genetic parameters and economic weights
For each genetic-parameter case, we considered subsets of three conventional traits, but treated the expression of a trait at different sites as effectively different traits. There were no inconsistencies in the assumed genetic parameters large enough to mean singular matrices, but the anomalous and inconsistent comparisons in Table 6 may reflect suboptimal conditioning of matrices. Expanding the matrices and economic-weight vectors to accommodate four (or more) conventional traits would in principle be straightforward, although obtaining good information on all the genetic parameters would be more challenging. Ideally, the breeder could use all available phenotypic data for selection traits of worthwhile heritability and genetic correlations with breeding-goal traits, and know the gamut of variances and covariances for all traits of interest, as they are expressed in various site categories. Since genetic parameters are never known without error, there are major challenges in achieving very large parameter matrices without either some internal inconsistencies or biasing important genetic correlations towards zero, and realised multi-trait gains will tend to be less than predicted ones.
The assignment of economic weights to either DEN or STR addresses the well-known phenotypic plasticity of radiata pine (Burdon and Miller 1992; Burdon 2001), whereby means and practical importance of variation in some quality-related traits can vary dramatically among sites. Our site B input parameters were based on familiar tree behaviour and some genetic parameter estimates for radiata pine on a nitrogen-deficient coastal dune site (Shelbourne and Low 1980), one that is broadly representative of a significant if secondary site category for growing radiata pine in New Zealand. There, while resolution of family differences could be achieved in stem straightness, with quite good type B genetic correlations with other sites, stem crookedness was not a problem. Similarly, variation in wood density and especially stiffness there is likely to be of very small economic significance compared with variation in stem volume. In any case, the study can be repeated for any relevant sets of parameters and economic weights. By contrast, the site A input parameters were deemed to be broadly representative for a major portion of the country’s commercial forest estate.
While variations in comparative economic weights for conventional traits were addressed as a consequence of LoE interaction, they can arise for other reasons. Obvious examples include where the harvested wood may be subject to different processes or produce different types of end-product, or where cost structures in the value chain vary according to the situation. For instance, variation in transport costs can differentially affect the net economic worth of different traits (G. Dutkowski pers. comm. 2017), reflecting location of plantations rather than any intrinsic properties of environments or the trees.
General implications of level-of-expression interaction
Despite various reservations, the results give a clear indication that assigning economic weights where expressed genetic variation in traits is of no economic consequence can entail major sacrifice of worthwhile economic gains, especially with adverse genetic correlations between economic and non-economic traits. This is doubtless addressed, tacitly or intuitively, in many deployment decisions, but we do not know of published figures on the topic.
Comparison of gains from selecting for specific sites versus jointly for both sites
Perennially debated is whether to select for specific environments, or for performance across environments (in effect for stable performers) (Li et al. 2017). Johnson and Burdon (1990) reported quite modest advantages in expected genetic gain from selecting for specific site types, despite considerable RC interaction. Here, we also observed mostly modest advantages in expected gain from specialised selection for specific sites (Table 6), but the picture was inconsistent. Any such advantages of single-site selection could be much greater if candidate genotypes are not replicated across sites and, of course, if between-sites genetic correlations are very imperfect. The practical importance of our results could depend strongly on whether selection is for advancing a breeding population or for deployment populations. Depending on the propagation system(s) for delivering genetic gain, specialised deployment may be much easier to implement than specialisation in the breeding population, whatever the nature of G × E. If so, quite modest gain advantages may justify specialisation for deployment.
Economic weight for disease resistance?
An incidental finding was that, without assigning an actual economic weight to RES resistance, it assumed essentially zero index weights. What economic weight, if any, should be assigned to RES is problematic. It can take either an economic weight in its own right, or a zero economic weight and be allowed to take an index weight by virtue of its genetic correlation with some truly economic trait(s). Giving RES an actual economic weight would be appropriate where the main impact of RES is through reducing the costs of disease/pest control (as in parasite control in farm animals). With forest trees, however, where exploiting genetic resistance may be the sole (or pre-eminent) tool for disease control, there is a case for assigning no actual economic weight to RES but letting it take an index weight through its genetic correlation(s) with other traits, notably VOL. In this study, RES surprisingly assumed no appreciable non-zero index weight as a purely indirect selection criterion for the growth trait. However, if selection-age and harvest-age traits are differentiated, and the genetic correlations between RES and harvest-age VOL is strong and the age-age correlations for growth variables are very imperfect (cf King and Burdon 1991), the partial genetic correlations between RES and harvest-age VOL could become very strong. RES might then contribute significantly as a purely indirect selection criterion for final VOL yield and piece size.
For specific cases, high-quality information on the genetic correlation of disease resistance with harvest-age productivity and the magnitude of its influence would clearly be important, whether or not resistance is assigned an actual economic weight.
Concluding
The various oversimplifications in our model are not seen as invalidating the basic conclusion, that LoE interaction can seriously erode expected economic gain if it is not addressed by setting appropriate economic weights for the expressions of a trait in different environments. We have also illustrated how much, or how little, expected multi-trait economic gain can be enhanced, in the presence of both RC and LoE interaction, by selection for specific sites. Moreover, we have obtained some insight into the appropriateness of assigning economic weight to disease resistance (RES) when it is a partial driver of growth rate (VOL).
Despite our study being based on some familiar behaviour of radiata pine, its messages will surely be relevant to some other tree-breeding situations.
Notes
For this, the term “scale interaction” is often used, but we reserve that term for cases where data transformation (e.g. log or square root) makes the interaction disappear statistically.
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Acknowledgements
We thank Greg Dutkowski for helpful comments on a draft, and Tim Mullin for helpful comments and suggestions for the presentation. RDB had use of Scion office facilities, and YL’s time was covered by Scion’s Forest Genetics core funding allocation.
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Burdon, R.D., Li, Y. Genotype-environment interaction involving site differences in expression of genetic variation along with genotypic rank changes: simulations of economic significance. Tree Genetics & Genomes 15, 2 (2019). https://doi.org/10.1007/s11295-018-1308-3
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DOI: https://doi.org/10.1007/s11295-018-1308-3