Sessão Temática 2 Análise Bayesiana Utilizando a abordagem Bayesiana no mapeamento de QTL´s Roseli Aparecida Leandro ESALQ/USP 11 o SEAGRO / 50ª RBRAS Londrina, Paraná 04 a 08 de Julho de 2005

Colaboradores  Prof. Dr. Cláudio Lopes Souza Jr.  Prof. Dr. Antônio Augusto Franco Garcia (Departamento de Genética ESALQ/USP)  Elisabeth Regina de Toledo (PPG Estatística e Experimentação Agronômica, ESALQ/USP)

Qualitative trait Mendelian gene

Quantitative trait

Bayesian mapping of QTL Geneticists are often interested in locating regions in the chromosome contributing to phenotypic variation of a quantitative trait.

Effects : Additive, dominance Location QTL Genetics Markers

Escala dos Valores Genotípicos Se d = 0Se d/a = 1Se d /a < 1Se d/a > 1 Efeito Aditivo Codominância Dominância Completa Dominância ParcialSobredominânci em que: d/a é o grau de dominância

Chromosomal regions of known location Do not have a physiological causal association to the trait under study

Genetics Markers

By studying the joint pattern of inheritance of the markers and trait Inferences can be made about the number, location and effects of the QTL affecting trait.

Experimental Design Offspring data: Divergent inbred lines Backcross ( code 0=aa, 1=Aa ) (Recessive) F2 (code –1=aa, 0=Aa, 1=AA) Reason: maximize linkage desiquilibrium

F2 Design

Data set

QTL phenotype model One QTL

Multiple QTL phenotype Model

Our aim is to make joint inference about the number of QTL, their positions (loci) and the sizes of their effects. Assume that a linkage map has been developed for the genome.

Genetic Map

0 < r = fração de recombinação < 0.5

Classic approach Interval mapping (Lander & Botstein,1989) Least squares method (Haley & Knott,1992) Composite interval mapping (Jansen, 1993; Jansen and Stam, 1994; Zeng 1993, 1994)

Bayesian approach Satagopan et al. (1996) Satagopan & Yandell (1998) Sillanpää & Arjas (1998)

The joint posterior distribution of all unknowns (s, , Q,  ) is proportional to

In practice, we observe the phenotypic trait. and the marker genotypes but NOT the QTL genotypes. For convenience consider only one linkage group with ordered markers {1,2,...,m}. Assume that genotypes: The markers are assumed to be at known distances

The conditional distribution * assuming the loci segregate independently ** under Haldane assumption of independent recombination

The marginal likelihood of the parameters s,  and  for the ith individual may be obtained from the joint distribution of traits and QTL genotypes. by summing over the set of all possible QTL genotypes for the ith individual,

Therefore, When the data Y are n independent observations, the marginal likelihood for the trait data is the product over individuals, a familiar misture model likelihood,

The joint likelihood is a mixture of densities, and hence, is difficult to evaluate when there are multiple QTL. The joint posterior distribution of all unknowns (s, , Q,  ) is proportional to

A Bayesian approach combined with reversible jump MCMC is well suited for QTL studies

Random-sweep Metropolis-Hastings algorithm for general state spaces (Richardson and Green, 1997) Suppose current state of the chain indexed by s.

The chain can (1) move to a “birth” step (number of loci s  s+1 ) (2) move to a “death” step (number of loci s  s-1 ) (3) continue with “current” number (s) of loci

Simulation Simulated F2 intercross n=250 1 cromossome 2 QTL

Referências Satagopan, J. M.; Yandell, B. S. (1998) Bayesian model determination for quantitative trait