Modeling Methods 21 Theory to Application

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The concept of modeling the properties of a system to enable prediction of its behavior is well accepted in many fields of science (Casti 1997, Kauffman 1993,

Wolfram 2002) and business (Axelrod and Cohen 1999, Bass 1999, Schrage 2000). In association with the evolution of faster computer hardware and software there have been advances in the theory and the range of modeling methods that can be applied to practical problems in genetics and plant breeding (Fraser and Burnell 1970, Kempthorne 1988, Lynch and Walsh 1998, Podlich and Cooper 1998, Cooper et al. 2002). The application of statistical modeling methods in population and quantitative genetics has a long history (Fisher 1918, Wright 1932, Crow and Kimura 1970, Kempthorne 1988, Falconer and Mackay 1996, Lynch and Walsh 1998). The range of statistical methods available to the geneticist has progressed; expanding from the familiar least squares methods applied to linear models to the uses of maximum likelihood and Bayesian approaches within various linear and non-linear modeling frameworks. With advances in technologies for high throughput genomic, environmental and trait phenotype measurements the availability of large multidimensional data sets has broadened the range of experimental investigations that can be conducted to study genetic variation for traits. As the genetic questions have become more comprehensive and complex the computational requirements have become more demanding and necessitated further advances in computing infrastructure and algorithms. With the availability of the required computing infrastructure, fast and comprehensive simulation capabilities have been developed to assist investigation of many of the challenging questions that are relevant to genetics and plant breeding (Podlich and Cooper 1998, 1999). These enhanced computing tools have opened up many new opportunities to study the properties of "real-world" complex systems (Casti 1997, Kauffman 1993, Williams 1997, Podlich and Cooper 1998, 1999, Micallef et al. 2001, Wolfram 2002, Chapman et al. 2003, Crutchfield and Schuster 2003, Wagner 2005, Newman et al. 2006). The main objective of this paper is to review and discuss some potential applications of simulation modeling methodologies as they have and can be used to identify and understand QTL effects of traits and to evaluate the potential for augmenting breeding by MAS. A secondary objective is to relate the status of the empirical trait mapping results that have accumulated to date to some of the expectations that emerge from modeling the properties of genotype-environment systems whenever interactions among the genetic and environmental components are present; the interactions emphasized here are inspired by the genetic concepts of epistasis, gene-by-environment interactions and pleiotropy. Our intent with this second objective is not to make firm conclusions, these fields of investigation are still in their early stages, but to encourage careful scientific consideration of the different mapping results that have accumulated to date.

2.2. Mapping Traits

Here we define a QTL to be any region of the genome that is associated with the standing variation for a trait phenotype in a relevant reference population that can be identified by one or more sequence-based DNA markers when they are applied in combination with a suitable experimental design and statistical analysis method. Similarly MAS is defined to be any application of the associated DNA markers to select for combinations of QTL alleles to create and test genotypes with a predicted trait phenotype in a target genotype-environment system. It is assumed that the markers can be arranged in the form of a genetic map that represents their linear order on chromosomes. It is also assumed that within the region identified as the QTL there is a functional polymorphism in the DNA sequence that influences the differential realization of the trait phenotype for different genotypes of the QTL. Thus, it is expected that mapping traits to identify QTL reveals information about the genomic positions of important functional polymorphisms in the DNA sequence of the organism that are present within the reference mapping population. The genetic and functional bases of QTL variation for traits and their influences on trait phenotypes are considered further below and elsewhere in this book. Typically MAS strategies used in applied plant breeding will involve some combined index utilizing QTL marker and trait phenotypic information.

The mapping resolution that can be achieved in defining the relevant regions of the genome by QTL analysis methods depends in part on the extent of linkage disequilibrium in the reference mapping population. In the perfect situation the marker sequence polymorphism and the functional DNA sequence polymorphism contributing to the trait phenotypic variation would be the same. This may result when a QTL has been previously cloned or when a candidate gene is used to identify markers and a polymorphism at the gene contributes to the standing quantitative trait variation. However, more typically the QTL are identified by a genome scan using many markers selected to cover the genome. In these cases the marker polymorphisms themselves are likely to be neutral and are associated with the functional polymorphism only by the linkage disequilibrium that exists for the DNA molecule within the reference population of genotypes. When such a genome scan is used to identify the QTL that are to be targets for MAS in applied breeding it is important to understand the extent of linkage disequilibrium in the reference population used for identification of QTL and that which exists in the elite breeding populations targeted for application of a MAS strategy. Linkage disequilibrium between marker and functional polymorphisms in the DNA sequence is necessary within a reference population to identify an association between a marker and a QTL. The extent of linkage disequilibrium in mapping populations can be manipulated by controlling the structure and number of cycles of inter-mating of individuals (e.g., Winkler et al. 2003). With relatively sparse genetic maps it will be necessary for the linkage disequilibrium to extend for large segments of the chromosomes to detect the associations. Alternatively with dense genetic maps it is an advantage to have less extensive linkage disequilibrium to enable finer mapping of the QTL. In parallel with the mapping studies that are used to identify QTL, the individuals within the reference population of a breeding program are continually inter-mated in designed crossing schemes and subjected to selection. Thus, in these breeding crosses recombination can continually operate to both break up current and create new physical linkage associations between alleles of different

QTL on the same chromosome and between markers and the functional polymorphisms of the QTL. Therefore, care must be taken to ensure that the linkage phases in the mapping study are relevant to those in the breeding reference population that will be the target for MAS. The two simulation experiments considered below are designed to take into consideration these influences of linkage disequilibrium (Figure 1).

Following the arguments given above, here we consider mapping traits within the context of an ongoing breeding program (e.g., Figures 1 and 2). Thus, any genetic gain from MAS will need to build on the progress that has been achieved by conventional breeding strategies. The realized genetic gain for quantitative traits that has been achieved by breeding can be understood as a long-term outcome from the application of open recurrent selection strategies that are designed to manage genetic diversity and manipulate multiple traits over multiple cycles of selection to improve and stabilize the yield and quality traits for the sets of genotypes grown by farmers (e.g., Figure 2; Rajaram and van Ginkel 2001, Duvick et al. 2004, Barker et al. 2004). Much of the breeding progress to date for complex traits such as yield has been achieved by pedigree and recurrent selection strategies (e.g., Hallauer and Miranda 1988, Comstock 1996, Duvick et al. 2004) applied to select for the desired trait phenotypes within relevant pools of genetic diversity, rather than by molecular enhanced approaches such as MAS. However, as with previous changes in core breeding methodology in the 20th Century there will be an exploratory transition phase and we can expect this situation to change as the 21st Century unfolds. The availability of large numbers of polymorphic markers that can be assayed rapidly and economically by high throughput technologies in large populations of genotypes has created interest and provides opportunity for augmenting the breeding process by MAS for the QTL polymorphisms at specific regions of the genome that are indicated as being responsible for the standing trait genetic variation (Cahill and Schmidt 2004, Niebur et al. 2004, Crosbie et al. 2006). This is a proven method for traits under the control of a few major genes or major QTL (e.g., Cahill and Schmidt 2004, Crosbie et al. 2006). The extension of this methodology to traits that are genetically more complex is feasible but requires consideration of the relative importance of the additive and non-additive components of genetic variation for the traits within the elite populations used for breeding and importantly an understanding of the genetic bases of these sources of variation and the potential influences of different trait genetics on the outcomes of the chosen MAS strategy (Cooper and Podlich 2002, Niebur et al. 2004, Podlich et al. 2004, Walsh 2005, van Eeuwijk et al. 2005, Cooper et al. 2005, Welch et al. 2005, Tardieu et al. 2005, Hammer et al. 2005). Walsh (2005) reminds us that to exploit the additive effects of alleles requires only identification of the desirable allele of a QTL and selection of that allele, regardless of the allele combinations at other loci. However, to exploit non-additive effects requires methods for identification of desirable combinations of alleles (for dominance, allele combinations at a single locus; and for epistasis, allele combinations at multiple loci) and selection of these allele combinations and their consistent transmission to subsequent generations. This effort becomes even more challenging in the presence of QTL with pleiotropic effects and QTL-by-Environment interactions (QEI). The empirical challenge that these complexities present in applied breeding is the need to conduct more comprehensive mapping studies to both discover the QTL and the desirable allele combinations and to evaluate their practical selection by MAS.

While we emphasize modeling methods in this paper, empirical evaluations of MAS strategies will always be necessary (e.g. Bouchez et al. 2002, Moreau et al. 2004a, Crosbie et al. 2006). However, there are many potential implementations of MAS and many details of the genetic architecture of traits to consider, making experimental evaluation of all possibilities impractical. Therefore, a combined empirical modeling evaluation of the potential of MAS strategies by focusing on some of the common genetic issues is likely to be a more feasible approach (Wang et al. 2003, 2004, Chapman et al. 2003, Cooper et al. 2005, Hammer et al. 2005). A common feature of the many alternatives that have been proposed is that marker alleles associated with favorable QTL alleles by coupling phase linkage are used to manipulate trait phenotypes by selecting for designated favorable combinations of the QTL alleles at one or more QTL. The trait phenotypes for the target QTL genotypes can be predicted based on experimentally determined effects of the QTL alleles. By defining and constructing some of the different target genotypes based on predictions from the multi-QTL models, validation experiments can be conducted to compare the predicted and realized phenotypes and to estimate the realized genetic gain from MAS. Even for restricted cases such comparisons of MAS with other breeding strategies is costly, they take considerable time and it is questionable whether the design of such experiments with adequate power is feasible for complex traits that are typically improved over multiple cycles of selection within a breeding program; the breeding program does not wait for the results of such studies. Given this non-stationary situation within applied breeding programs there is merit in modeling QTL detection methods and MAS strategies as part of a comprehensive research program organized to design, refine and optimize a breeding program if the results are to positively impact the outcomes of the breeding program.

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