Basic Theory of IBD

Author: C. Cannings
Submitted: Friday 6th of July 2012 09:49:56 AM
Submitted by: egf
Language: English
Content type: Learning resource
Educational levels: expert, qc2, qc3

Abstract

Two individuals are (biologically) related if they share a common ancestor. In that case they may have derived regions of their chromosomes from a common ancestral chromosomes and thus they will, in the absence of mutations, have the same genetic material across these regions. We begin by considering a single locus; essentially a point on the chromosomes. For a pair of genes at this locus, either within a single individual or in two separate individuals, the complete information is captured by calculating the probability that those genes are copies of some ancestral gene. If so they are said to be identical by descent. In order to specify the relationships we introduce various coefficients. For two genes in two individuals we consider the coefficient of kinship and for a single individual the coefficient of inbreeding. There are several different coefficients defined in the literature, see Malecot (1948) and Wright(1923), and care must be taken to recognize the subtle differences between them. These ideas will be demonstrated with respect to a number of relationships, e.g. sibs, first cousins. We demonstrate how these coefficients can be used to make calculations of the probability an individual has a particular genotype. In many situations we may wish to deal with many individuals rather than just one or two, and with all the homologous genes rather than just two. This requires the introduction of the notion of an identity state. This will be done here for two individuals only. All of the above has concentrated on a single point (locus) of the genome. Attention now switches to the process of identity along the chromosomes. We make the assumption that the occurrence of recombination along the chromosomes follows a Poisson process (whose properties will be introduced briefly), thus ignoring interference. Using the ideas of Donnelly(1983) we show how the process can be modelled. We demonstrate how as the process moves along the chromosome transitions are made between distinct states (a specification of the pattern of recombinations in the genealogy) and the rates at which these occur. We thus have a Markov Chain and this allows us to derive various results of interest.We return to the notion of identity states, and in particular examine the case of offspring of non-inbred individuals, deriving the probabilities for the identity states of those offspring. The equivalent for inbred individuals will be introduced briefly. Finally (very briefly) we consider the process for identity states along chromosomes (this will be raised again in the final talk of the workshop). References Donelly KP (1983) The probability that related individuals share some section of the genome identical by descent. Theoret. Pop.Biol.,23, 34-64. Malecot G (1948) Les mathematiques de l'heredite. Paris, Masson et cie

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C. Cannings. Basic Theory of IBD. EUROGENE portal. July 2012. online: http://eurogene.open.ac.uk/content/basic-theory-ibd

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