Iterative kriging for removing spatial autocorrelation in analysis of forest genetic trials

Conventional analysis of spatially correlated data in inadequately blocked field genetic trials may give erroneous results that would seriously affect breeding decisions. Forest genetic trials are commonly very large and strongly heterogeneous, so adjustments for micro-environmental heterogeneity become indispensable. This study explores the use of geostatistics to account for the spatial autocorrelation in four Pinus pinaster Ait. progeny trials established on hilly and irregular terrains with a randomized complete block design and large blocks. Data of five different traits assessed at age 8 were adjusted using an iterative method based on semivariograms and kriging, and the effects on estimates of variance components, heritability, and family effects were evaluated in relation to conventional analysis. Almost all studied traits showed nonrandom spatial structures. Therefore, after the adjustments for spatial autocorrelation, the block and family × block variance components, which were extremely high in the conventional analysis, almost disappeared. The reduction of the interaction variance was recovered by the family variance component, resulting in higher heritability estimates. The removal of the spatial autocorrelation also affected the estimation of family effects, resulting in important changes in family ranks after the spatial adjustments. Comparison among families was also greatly improved due to higher accuracy of the family effect estimations. The analysis improvement was larger for growth traits, which showed the strongest spatial heterogeneity, but was also evident for other traits such as straightness or number of whorls. The present paper demonstrates how spatial autocorrelation can drastically affect the analysis of forest genetic trials with large blocks. The iterative kriging procedure presented in this paper is a promising tool to account for this spatial heterogeneity.     

Mixed spatial models for data analysis of yield on large grapevine selection field trials

In large field trials, it may be desirable to adjust for spatial correlation due to variation in soil fertility and in other environmental factors. Spatial correlation within a field trial can mask differences in the genotypic values of clones, consequently reducing the possibility of identifying superior genotypes. This paper describes a strategy to improve the precision of statistical data analysis of grapevine selection trials through the use of mixed spatial models. The efficiency of mixed spatial models was compared with that of a classical randomized complete block model (with independent and identically distributed errors). The comparisons were based on yield data from three large experimental populations of clones of the Arinto, Aragonez (Tempranillo) and Viosinho grapevine varieties. The fit of the spatial mixed models applied to yield data was significantly better than that of the classical approach, resulting in a positive impact on selection decisions and increasing the accuracy of genetic gain prediction.     

Effect of migration and environmental heterogeneity on the maintenance of quantitative genetic variation: a simulation study

The paradox of high genetic variation observed in traits under stabilizing selection is a long‐standing problem in evolutionary theory, as mutation rates appear too low to explain observed levels of standing genetic variation under classic models of mutation–selection balance. Spatially or temporally heterogeneous environments can maintain more standing genetic variation within populations than homogeneous environments, but it is unclear whether such conditions can resolve the above discrepancy between theory and observation. Here, we use individual‐based simulations to explore the effect of various types of environmental heterogeneity on the maintenance of genetic variation (V_A) for a quantitative trait under stabilizing selection. We find that V_A is maximized at intermediate migration rates in spatially heterogeneous environments and that the observed patterns are robust to changes in population size. Spatial environmental heterogeneity increased variation by as much as 10‐fold over mutation–selection balance alone, whereas pure temporal environmental heterogeneity increased variance by only 45% at max. Our results show that some combinations of spatial heterogeneity and migration can maintain considerably more variation than mutation–selection balance, potentially reconciling the discrepancy between theoretical predictions and empirical observations. However, given the narrow regions of parameter space required for this effect, this is unlikely to provide a general explanation for the maintenance of variation. Nonetheless, our results suggest that habitat fragmentation may affect the maintenance of V_A and thereby reduce the adaptive capacity of populations.     

The feasibility of producing inbred rather than F1 hybrid cultivars in Brussels sprouts (Brassica oleracea var. gemmifera): an assessment of recombinant inbred lines

Results are presented for the performance of improved inbred lines of Brussels sprouts grown in replicated and fully guarded plots. Some lines were identified which out-performed the reference F₁ hybrid, Gower, for the yield of marketable sprouts and for sprout quality. No lines were found which were superior for both of these traits. These results support earlier contentions that inbred line performance in Brussels sprouts could be improved to levels comparable with those of commercial F₁ hybrids. The genetic gains required to achieve commercial parity with hybrids for all agronomically important traits continue to be large. Therefore, the use of inbred lines as commercial cultivars can only be viewed as a long term objective. Previous studies have identified additive x additive epistasis and the segregation of many loci as important factors limiting the genetic gains to be expected from a single cycle of crossing and inbreeding. In addition to these factors the current study identifies areas of difficulty encountered when attempting to screen and select large numbers of inbred lines, produced either by single seed descent or by anther culture, in a single season. Evidence is presented which suggests that imperfect visual selection and/or genotype × seasonal interactions may substantially reduce the efficiency of selections based upon a single trial of very many unreplicated lines.     

FINE-GRAINED SPATIAL AND TEMPORAL VARIATION IN SELECTION DOES NOT MAINTAIN GENETIC VARIATION IN ERIGERON ANNUUS

Because interactions among plants are spatially local, the scale of environmental heterogeneity can have large effects on evolutionary dynamics. However, very little is known about the spatial patterns of variation in fitness and the relative magnitude of spatial and temporal variation in selection. Replicates of 12 genotypes of Erigeron annuus (Asteraceae) were planted in 288 locations within a field, separated by distances of 0.1 to 30.0 m, and replicated in two years. In a given year, most spatial variation in relative fitness (genotype-environment [G × E] interactions for fitness) occurred over distances of only 50 cm. Year effects were as large or larger than the spatial variation in fitness; in particular there was a large, three-way, genotype-year-environment interaction at the smallest spatial scale. The genetic correlation of fitness across years at a given location was near zero, 0.03. Thus, the relative fitness of genotypes is spatially unpredictable and a map of the selective environment has constantly shifting locations of peaks and valleys. Including measurements of soil nutrients as covariates in the analysis removed most of the spatial G × E interaction. Vegetation and microtopography had no effect on the G × E terms, suggesting that differential response to soil nutrients is the cause of spatial variation in fitness. However, the slope of response to NH₄ and P0₄ was negative; therefore the soil nutrients are probably just indicators of other, unknown, environmental factors. We explored via simulation the evolutionary consequences of spatial and temporal variation in fitness and showed that, for this system, the spatial scale of variation was too fine grained (by a factor of 3 to 5) to be a powerful force maintaining genetic variation in the population. The inclusion of both spatial and temporal variation in fitness actually reduced the coexistence of genotypes compared to pure spatial models. Thus the presence of spatial or temporal variation in selection does not guarantee that it is an effective evolutionary force maintaining diversity. Instead the pattern of selection favors generalist genotypes.     

Modelling spatial trends in sorghum breeding field trials using a two-dimensional P-spline mixed model

Key message

A flexible and user-friendly spatial method called SpATS performed comparably to more elaborate and trial-specific spatial models in a series of sorghum breeding trials.

Abstract

Adjustment for spatial trends in plant breeding field trials is essential for efficient evaluation and selection of genotypes. Current mixed model methods of spatial analysis are based on a multi-step modelling process where global and local trends are fitted after trying several candidate spatial models. This paper reports the application of a novel spatial method that accounts for all types of continuous field variation in a single modelling step by fitting a smooth surface. The method uses two-dimensional P-splines with anisotropic smoothing formulated in the mixed model framework, referred to as SpATS model. We applied this methodology to a series of large and partially replicated sorghum breeding trials. The new model was assessed in comparison with the more elaborate standard spatial models that use autoregressive correlation of residuals. The improvements in precision and the predictions of genotypic values produced by the SpATS model were equivalent to those obtained using the best fitting standard spatial models for each trial. One advantage of the approach with SpATS is that all patterns of spatial trend and genetic effects were modelled simultaneously by fitting a single model. Furthermore, we used a flexible model to adequately adjust for field trends. This strategy reduces potential parameter identification problems and simplifies the model selection process. Therefore, the new method should be considered as an efficient and easy-to-use alternative for routine analyses of plant breeding trials.

     

Genetic differentiation of neutral markers and quantitative traits in predominantly selfing metapopulations: confronting theory and experiments with Arabidopsis thaliana

The comparison of the genetic differentiation of quantitative traits (QST) and molecular markers (FST) can inform on the strength and spatial heterogeneity of selection in natural populations, provided that markers behave neutrally. However, selection may influence the behaviour of markers in selfing species with strong linkage disequilibria among loci, therefore invalidating this test of detection of selection. We address this issue by monitoring the genetic differentiation of five microsatellite loci (FST) and nine quantitative traits (QST) in experimental metapopulations of the predominantly selfing species Arabidopsis thaliana, that evolved during eight generations. Metapopulations differed with respect to population size and selection heterogeneity. In large populations, the genetic differentiation of neutral microsatellites was much larger under heterogeneous selection than under uniform selection. Using simulations, we show that this influence of selection heterogeneity on FST can be attributable to initial linkage disequilibria among loci, creating stronger genetic differentiation of QTL than expected under a simple additive model with no initial linkage. We found no significant differences between FST and QST regardless of selection heterogeneity, despite a demonstrated effect of selection on QST values. Additional data are required to validate the role of mating system and linkage disequilibria in the joint evolution of neutral and selected genetic differentiation, but our results suggest that FST/QST comparisons can be conservative tests to detect selection in selfing species.     

Doubled haploid versus S1 family recurrent selection for testcross performance in a maize population

Theoretically, in a recurrent selection program, the use of doubled haploids (DH) can increase genetic advance per unit of time. To evaluate the efficiency expected from the use of DH for the improvement of grain yield in a maize (Zea mays L.) population, two recurrent selection programs for testcross performance were initiated using testcross progenies from DH lines and S₁ families. In 4 years one selection cycle using DH and two selection cycles using S₁ families were carried out with the same selection intensity for both methods. As expected, testcross genetic variance was twice as high among DH lines as among S₁ families. The predicted genetic gain was 8.2% for the DH selection cycle, and 10.6% for the two S₁ selection cycles, giving a per year advantage of 29% for the S₁ family method over the DH method with a cycle of 4 years. With a 3-year cycle for the DH method, both methods were expected to be equivalent. Using a tester related to the one used for selection, the genetic gains obtained were equivalent for both methods: 6.6% for the DH cycle and 7.0% for the two S₁ cycles. With a 3-year cycle for the DH method, the advantage would have been in favor of DH method. Furthermore, the DH method has the advantage of simultaneously producing lines that are directly usable as parents of a hybrid. Thus, if the genetic advance per unit of time is evaluated at the level of developed varieties even with the same or with a lower genetic advance in population improvement, the DH method appears to be the most efficient.     

Evaluation of experimental designs and spatial analyses in wheat breeding trials

Thirty-three wheat breeding trials were conducted from 1994 to 1996 in the Northern Grains Region (QLD and Northern NSW) of Australia to evaluate the influence of experimental designs and spatial analyses on the estimation of genotype effects for yield and their impact on selection decisions. The relative efficiency of the alternative designs and analyses was best measured by the average standard error of difference between line means. Both more effective designs and spatial analyses significantly improved the efficiency relative to the randomised complete block model, with the preferred model (which combined the design information and spatial trends) giving an average relative efficiency of 138% over all 33 trials. When the Czekanowski similarity coefficient was used, none of the studied models were in full agreement with the randomised complete block model in the selection of the top lines. The agreement was influenced by selection proportions. Hence, the use of these methodologies can impact on the selection decisions in plant breeding. Received: 17 December 1998 / Accepted: 29 July 1999     

A network approach to modeling population aggregation and genetic control of pest insects

Historically, models of the invasion and biological control of insect pests have omitted heterogeneities in the spatial structure of the targeted populations. In this study, we use stochastic network simulations to examine explicitly population heterogeneity as a function of landscape structure and insect behavior. We show that when insects are distributed non-randomly across a heterogeneous landscape, control can be significantly hindered. However, when insect populations are clustered as a result of limited dispersal, genetic control efficiency can be enhanced. In developing the model, we relax a key assumption of previous theoretical studies of genetic control: that released genetic control insects remain homogenously distributed irrespective of the spatial structure of the wild type populations. Here, this behavior (termed the ‘coverage proportion’) is parameterized and its properties are explored. We show that landscape heterogeneity and limited dispersal have little effect on the critical coverage proportion necessary for control.     

The geometry of coexistence

Understanding the processes that maintain diversity has been the focus of extensive study, yet there is much that has not been integrated into a cohesive framework. First, there is a separation of perspective. Ecological and evolutionary approaches to diversity have progressed in largely parallel directions. Second, there is a separation of emphasis. In both ecology and population genetics, classical theories favour local explanations with emphasis on population dynamics and selection within populations, while contemporary theories favour spatial explanations, with emphasis on population structure and interactions among populations. What is lacking is a comparative approach that evaluates the relative importance of local and spatial processes in maintaining genetic and ecological diversity. I present a framework for diversity maintenance that emphasizes the comparative approach. I use a well-known but little-used mathematical approach, the perturbation theorem for dynamical systems, to identify key points of contact between ecological and population genetic theories of coexistence. These connections provide for a synthesis of several important concepts: population structure (source-sink versus extinction-colonization), spatial heterogeneity (intrinsic versus extrinsic) in fitness and competitive ability, and temporal scales over which local and spatial processes influence diversity. This framework ties together a large and diverse body of theory and data from ecology and population genetics. It yields comparative predictions that can serve as guidelines in biodiversity management.     

Comparison of bread wheat lines selected by doubled haploid, single-seed descent and pedigree selection methods

 Yield performance of each group of ten spring bread wheat lines selected by doubled haploid (DH), single-seed descent (SSD) and pedigree selection (PS) methods from three F₁ crosses was compared with the aim of evaluating the DH method in breeding programs. Populations of 65–97 DH lines and 110 SSD lines per cross were used for selection. PS lines were developed by repeated selections from 1500 F₂ plants. Yield evaluation was performed at the F₆ generation of SSD and PS lines along with DH lines in a 2-year field experiment. It took only 2 years from the planting of wheat materials for DH production to the planting of selected DH lines for yield evaluation. There was no significant difference in grain yield between DH lines and PS lines selected from an F₁ cross whose parental varieties were closely related in their pedigrees. In two crosses with low coefficients of parentage and a large variation in their progenies, grain yield of selected DH lines was significantly lower than those of selected SSD and PS lines. These results confirm that the DH method can save time in obtaining recombinant inbred lines ready for yield evaluation. However, a larger DH population is required to achieve the same level of genetic advance with the PS method in crosses containing greater genetic variation. Received: 23 December 1997 / Accepted: 12 March 1998     

Spatial genetic structure of a small rodent in a heterogeneous landscape

Gene flow in natural populations may be strongly influenced by landscape features. The integration of landscape characteristics in population genetic studies may thus improve our understanding of population functioning. In this study, we investigated the population genetic structure and gene flow pattern for the common vole, Microtus arvalis, in a heterogeneous landscape characterised by strong spatial and temporal variation. The studied area is an intensive agricultural zone of approximately 500 km² crossed by a motorway. We used individual-based Bayesian methods to define the number of population units and their spatial borders without prior delimitation of such units. Unexpectedly, we determined a single genetic unit that covered the entire area studied. In particular, the motorway considered as a likely barrier to dispersal was not associated with any spatial genetic discontinuity. Using computer simulations, we demonstrated that recent anthropogenic barriers to effective dispersal are difficult to detect through analysis of genetic variation for species with large effective population sizes. We observed a slight, but significant, pattern of isolation by distance over the whole study site. Spatial autocorrelation analyses detected genetic structuring on a local scale, most probably due to the social organisation of the study species. Overall, our analysis suggests intense small-scale dispersal associated with a large effective population size. High dispersal rates may be imposed by the strong spatio-temporal heterogeneity of habitat quality, which characterises intensive agroecosystems.     

A network approach to modeling population aggregation and genetic control of pest insects

Historically, models of the invasion and biological control of insect pests have omitted heterogeneities in the spatial structure of the targeted populations. In this study, we use stochastic network simulations to examine explicitly population heterogeneity as a function of landscape structure and insect behavior. We show that when insects are distributed non-randomly across a heterogeneous landscape, control can be significantly hindered. However, when insect populations are clustered as a result of limited dispersal, genetic control efficiency can be enhanced. In developing the model, we relax a key assumption of previous theoretical studies of genetic control: that released genetic control insects remain homogenously distributed irrespective of the spatial structure of the wild type populations. Here, this behavior (termed the ‘coverage proportion’) is parameterized and its properties are explored. We show that landscape heterogeneity and limited dispersal have little effect on the critical coverage proportion necessary for control.     

The stability of the G‐matrix: The role of spatial heterogeneity

The temporal stability of the genetic variance‐covariance matrix ( G ) has been discussed for a long time in the evolutionary literature. A common assumption in all studies, including empirical ones, is that spatial heterogeneity is minor such that the population can be represented by a single mean and variance. We use the well‐established allocation‐acquisition model to analyze the effect of relaxing of this assumption, simulating a case where the population is divided into patches with a variance in quality between patches. This variance can in turn differ between years. We found that changes in spatial variance in patch quality over years can make the G ‐matrix vary substantially over years and that the estimated genetic correlations, evolvability, and response to selection are different dependent on whether spatial heterogeneity is taken into account or not. This will have profound implications for our ability to predict evolutionary change and understanding of the evolutionary process.     

Evolution of fitnesses in structured populations with correlated environments

The outcome of selection in structured populations with spatially varying selection pressures depends on the interaction of two factors: the level of gene flow and the amount of heterogeneity among the demes. Here we investigate the effect of three different levels of spatial heterogeneity on the levels of genetic polymorphisms for different levels of gene flow, using a construction approach in which a population is constantly bombarded with new mutations. We further compare the relative importance of two kinds of balancing selection (heterozygote advantage and selection arising from spatial heterogeneity), the level of adaptation and the stability of the resulting polymorphic equilibria. The different levels of environmental heterogeneity and gene flow have a large influence on the final level of polymorphism. Both factors also influence the relative importance of the two kinds of balancing selection in the maintenance of variation. In particular, selection arising from spatial heterogeneity does not appear to be an important form of balancing selection for the most homogeneous scenario. The level of adaptation is highest for low levels of gene flow and, at those levels, remarkably similar for the different levels of spatial heterogeneity, whereas for higher levels of gene flow the level of adaptation is substantially reduced.     

Analysis of F ST outliers at allozyme loci in Pacific salmon: implications for natural selection

Natural selection has been invoked to explain the observed geographic distribution of allozyme allele frequencies for a number of teleost species. The effects of selection on allozyme loci in three species of Pacific salmon were tested. A simulation-based approach to estimate the null distribution of population differentiation (F _ST) and test for F _ST outliers was used. This approach showed that a majority of allozyme loci conform to neutral expectations predicted by the simulation model, with relatively few F _ST outliers found. No consistent F _ST outlier loci were found across species. Analysis of population sub-groups based on geography and genetic identity reduced the number of outlier loci for some species, indicating that large geographic groups may include genetically divergent populations and/or that there is geographic heterogeneity in selection pressure upon allozyme loci. Two outlier allozyme loci found in this analysis, lactate dehydrogenase-B and malic enzyme, have been shown to be influenced by selection in other teleost species. This approach is also useful in identifying allozyme loci (or other genetic markers) that meet assumptions for population genetic study.     

Efficiency of augmented p-rep designs in multi-environmental trials

Key message

The paper shows that unreplicated designs in multi-environmental trials are most efficient. If replication per environment is needed then augmented p-rep designs outperform augmented and replicated designs in triticale and maize.

Abstract

In plant breeding, augmented designs with unreplicated entries are frequently used for early generation testing. With limited amount of seed, this design allows to use a maximum number of environments in multi-environmental trials (METs). Check plots enable the estimation of block effects, error variances and a connection of otherwise unconnected trials in METs. Cullis et al. (J Agri Biol Environ Stat 11:381–393, 2006) propose to replace check plots from a grid-plot design by plots of replicated entries leading to partially replicated (p-rep) designs. Williams et al. (Biom J 53:19–27, 2011) apply this idea to augmented designs (augmented p-rep designs). While p-rep designs are increasingly used in METs, a comparison of the efficiency of augmented p-rep designs and augmented designs in the range between replicated and unreplicated designs in METs is lacking. We simulated genetic effects and allocated them according to these four designs to plot yields of a triticale and a maize uniformity trial. The designs varied in the number of environments, but have a fixed number of entries and total plots. The error model and the assumption of fixed or random entry effects were varied in simulations. We extended our simulation for the triticale data by including correlated entry effects which are common in genomic selection. Results show an advantage of unreplicated and augmented p-rep designs and a preference for using random entry effects, especially in case of correlated effects reflecting relationships among entries. Spatial error models had minor advantages compared to purely randomization-based models.     

Genetic and Economic Analysis of a Targeted Marker-assisted Wheat Breeding Strategy

The advent of molecular markers as a tool to aid selection has provided plant breeders with the opportunity to rapidly deliver superior genetic solutions to problems in agricultural production systems. However, a major constraint to the implementation of marker-assisted selection (MAS) in pragmatic breeding programs in the past has been the perceived high relative cost of MAS compared to conventional phenotypic selection. In this paper, computer simulation was used to design a genetically effective and economically efficient marker-assisted breeding strategy aimed at a specific outcome. Under investigation was a strategy involving the integration of both restricted backcrossing and doubled haploid (DH) technology. The point at which molecular markers are applied in a selection strategy can be critical to the effectiveness and cost efficiency of that strategy. The application of molecular markers was considered at three phases in the strategy: allele enrichment in the BC₁F₁ population, gene selection at the haploid stage and the selection for recurrent parent background of DHs prior to field testing. Overall, incorporating MAS at all three stages was the most effective, in terms of delivering a high frequency of desired outcomes and at combining the selected favourable rust resistance, end use quality and grain yield alleles. However, when costs were included in the model the combination of MAS at the BC₁F₁ and haploid stage was identified as the optimal strategy. A detailed economic analysis showed that incorporation of marker selection at these two stages not only increased genetic gain over the phenotypic alternative but actually reduced the over all cost by 40%.     

Genetic variance in the F2 generation of divergently selected parents

 Either by selective breeding for population divergence or by using natural population differences, F₂ and advanced generation hybrids can be developed with high variances. We relate the size of the genetic variance to the population divergence based on a forward and backward mutation model at a locus with two alleles with additive gene action. The effects of population size and initial gene frequency are also explored. Larger parental population sizes increase the F₂ genetic variance if the initial probability distribution is uniform or U-shaped. However, population size has the opposite effect if the initial distribution of gene frequencies is skewed such as it would be with newly arriving alleles. These alleles contribute to the genetic variance sooner when the selection pressure is higher or when the effective population size is smaller. Received: 5 April 1998 / Accepted: 22 April 1998