The Ising Model in Physics and Statistical Genetics

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The Ising Model inPhysics and Statistical
Genetics
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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.

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No allele is IBD. One allele is IBS.
Exactly one allele IBD.
At least one allele IBD. Expected 1.5 alleles
IBD.
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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.

Magnetism ASP
Spin up/down Shared/unshared
Interaction between adjacent dipoles Linkage between adjacent genetic markers
External field disease
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Methods

• The Ising model in Physics

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Methods

• Adaptation to Genetic Data
• Data nm 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
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Methods

• Parameter Estimation
• Maximum likelihood-Powells Method
• Z-recursive calculation

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Methods

• Significance Testing