The Ising Model in Physics and Statistical Genetics

1. The Ising Model inPhysics and Statistical Genetics

2. Introduction • ASP (affcted-sib-pair) analysis • Definitions: Any two identical copies of an allele l are said to be identical by state (IBS). If these alleles are inherited from the same individual then they are also identical by descent (IBD). Clearly, IBD implies IBS but not vice versa. • Main idea: if affected siblings share more IBD alleles at some marker locus than randomly expected among siblings, then that locus might be near a locus of a predisposing gene. The idea is that any two siblings are expected to have one allele IBD by chance. When a deviation of this pattern is detected, by examining many sib-pairs, linkage is established between a disease gene and the marker location.

3. 1/1 1/2 1/1 1/1 1/2 1/2 1/3 1/3 1/1 1/3 1/2 1/2 No allele is IBD. One allele is IBS. Exactly one allele IBD. At least one allele IBD. Expected 1.5 alleles IBD.

4. Introduction • Analog • Advantage • Multipoint analyses have the advantage of extracting maximum IBD-sharing information and a lower overall genomewide type I error. • It can be easily adapted to the analysis of complex genetic models with several genetic effects and with interaction, or epistasis, between genes.

5. Methods • The Ising model in Physics

6. Methods • Adaptation to Genetic Data • Data: n*m matrix • Distribution: j(t): if a marker is shared IBD the neighboring markers have an increased chance to be also shared h(t): a genetic effect causing an increase in IBD shagring at the t locus (there will be an apparent strong field close to the disease gene.) There are 2m-1 parameters. 1 1 1 -1 -1 0 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 1 0 1 0 1 1 -1 -1 -1 -1 -1 0 -1

7. Methods • Parameter Estimation • Maximum likelihood-Powell’s Method • Z-recursive calculation

8. Methods • Significance Testing