The IBD process along four chromosomes

Abstract

In this note, we present the continuous-time Markov rate matrix that models identity by descent (ibd) patterns among four chromosomes in a population. The equilibrium distribution of this Markov process along a chromosome is the set of 4-gene state probabilities given by the Ewens sampling formula. This model will facilitate inference of identity by descent among the four chromosomes of a pair of individuals, using data at dense SNP loci among which there may be linkage disequilibrium.

1 Introduction

All methods of gene mapping rely, at some level, on the detection of genome shared ibd by individuals who are similar with regard to some trait whose genetic determinants are to be found. Thus some prior model for ibd is required, based either on known or hypothesized pedigree information, or on a model for coancestry in a population.

Traditionally, from Thompson (1975) to Anderson and Weir (2007), pairwise relationships between individuals have been estimated on the basis of data at unlinked loci. Even methods of ibd detection for linkage inferences have used widely spaced markers, such as those of a typical microsatellite genome scan at an average spacing of 5 centiMorgans or 5 × 10⁶bp. (Weeks and Lange, 1988; Thompson, 2000). The increasing availability of dense SNP data, at an average spacing of 0.01cM or 10,000 bp (300K over the genome), provides potential for much more accurate identification of small segments of genome shared ibd between a pair, or among a set, of haplotypes.

At the pedigree level, in the absence of genetic interference, the sequence of inheritance vectors (Lander and Green, 1987) at locations of markers along a chromosome can be modeled as a first-order Markov chain. Where more remote relationships are not accurately known, a simple Markov model for the ibd between a pair of chromosomes can be used to analyze marker data and obtain probabilities conditional on these data (Leutenegger et al., 2003). Whereas the pointwise probability of ibd decreases as 2^−m, where m is the number of meioses separating a pair of individuals, the length of shared segments decreases only as m⁻¹. Use of sets of remotely related individuals, such as distant cousins, decreases the confounding effects of genetic heterogeneity of the trait relative to a population study, and decreases the confounding effects of shared environment relative to studies of close relatives.

The model of Leutenegger et al. (2003) is a continuous-time Markov process for ibd between two chromosomes in a population, where “time” is genetic distance along the chromosomes. This model can be used to analyze, for example, data on potentially inbred individuals, but analysis of sharing between affected relatives requires at least consideration of the four chromosomes of a pair of individuals. Although the marginal (single-locus) equilibrium probabilities of patterns of identity by descent among multiple chromosomes are well known (Balding and Nichols, 1994; Weir, 2001), an appropriate model for the changes in ibd patterns along a chromosome has proved elusive for more than a chromosome pair (Leutenegger, 2003).

Here we present the infinitesimal-rate matrix (Karlin and Taylor, 1975) for changes in ibd pattern among four chromosomes that has the correct population equilibrium probabilities of all 4-gene patterns of ibd. Thus this matrix serves as a prior Markov model for ibd along a chromosome for a pair of remote relatives from a population. Combined with hidden Markov models for population haplotypes such as that of Scheet and Stephens (2006) or of Browning (2006), it can serve as a basis for inferring ibd segments shared by individuals on the basis of dense SNP marker data among which there may be linkage disequilibrium (LD).

2 Identity along four chromosomes

2.1 The case of two chromosomes

As background, we first review briefly the model of Leutenegger et al. (2003) for ibd between a pair of chromosomes. In effect, that model is a continuous-time Markov process, with rate-matrix between the two states of no-ibd (0) and ibd (1)

$$\left(\begin{array}{cc}\hfill -af\hfill & \hfill af\hfill \\ \hfill a(1-f)\hfill & \hfill -a(1-f)\hfill \end{array}\right).$$

In this model, the equilibrium marginal probability of ibd is f and the second parameter a determines the rate of change between the two states in units of genetic distance. The ibd segments are exponential with expected length (a(1 − f))⁻¹, and the relative rate of gain vs loss of ibd is f/(1 − f). In fitting this model using marker data, Leutenegger et al. (2003) allowed for typing error, so that heterozygosity at a marker did not absolutely preclude ibd. With this proviso and with sufficiently dense polymorphic markers, it was found that ibd segments could be well identified, and that posterior probabilities of ibd were not very sensitive to the precise values of f and a. While these parameters could be estimated from the data, any reasonable choice gave similar results for purposes of identifying ibd segments. Although in reality a complex relationship between the parents of the individual result in segments of ibd of a variety of expected lengths, the single parameter a provides a sufficiently flexible model that ibd segments can be accurately imputed. For example, if relatively close relationships are suspected, a choice of f = 0.1 and a = 0.1 per centiMorgan (cM) is appropriate Leutenegger (2003).

For remote relationships, or for detecting ibd segments among chromosomes in a population, the overall level of ibd is at least an order of magnitude lower, and ibd segments are smaller. However, if the marginal prior probability of ibd, f, is chosen too small, ibd can be hard to detect: f = 0.01 provides a compromise. Even where ibd is a priori improbable, ibd segments are few rather than small. Even at 100 meioses (50 generations) separation, a shared segment is of expected length 1 cM, corresponding closely to a value a = 1 per cM. With dense marker data, segments of ibd much larger than the expected length are easily detected despite long lengths being a priori improbable. Further, due to the a priori exponential distribution of segment lengths, segments of less than the expected length are a priori quite probable and can also be detected given dense data. Thus the values f = 0.01 and a = 1 per cM are expected to work well for analysis of remote relationships at the population level.

2.2 The fifteen states for four chromosomes

Second, we review patterns of gene identity among four ordered genes. There are 15 patterns of gene identity, labeled by four digits (Thompson, 1974). Genes with the same label are ibd and those with different digit labels are not. For example, the state 1223 indicates that the second and third genes are ibd. If the four genes are the maternal and paternal genes of two individuals, then for the purposes of single-locus genotypes on the pair, the 15 states reduce to 9 state classes. These state classes are characterized by whether the 2 genes of the first individual are ibd (1 if so, 0 otherwise), similarly for the second individual, and by the number of distinct genes shared between the individuals. For example, the state class 101 consists of the two states 1112 and 1121. We will use this labeling of state classes in order to keep clear the distinction between the 15- and the 9-state representations. Although, for many purposes, the nine states are sufficient, if LD along a chromosome is to be modeled, it will be necessary to distinguish the two chromosomes within an individual. The 9 state-classes and corresponding 15 states are shown in Table 1.

Table 1.

The state classes among 4 genes

State class	ibd states	class characterization	equilibrium state probability	equilibrium class probability
1	1111	1 1 1	6β2η/(1 − β)	–
2	1122	1 1 0	βη	–
3	1112 and 1121	1 0 1	2βη	4βη
4	1123	1 0 0	η(1 − β)	–
5	1211 and 1222	0 1 1	2βη	4βη
6	1233	0 1 0	η(1 − β)	–
7	1212 and 1221	0 0 2	βη	2βη
8	1213, 1231, 1223 and 1232	0 0 1	η(1 − β)	4η(1 − β)
9	1234	0 0 0	η(1 − β)2/β	–

2.3 Gene identity in an equilibrium population

Third, we review the single-locus multigene identity probabilities at equilibrium under an infinite alleles model (Ewens, 1972). In a sample of size n, let a_i be the number of allelic types present in i copies. Then the number of observed alleles is k = Σ a_i, n = Σ ia_i and

$$P_n(a_1,\dots ,a_n)=\frac{n!}{\theta (\theta +1)\dots (\theta +n-1)}{\displaystyle \prod _{j=1}^n{\left(\frac{\theta }{j}\right)}^{a_j}\frac{1}{a_j!}}$$ (1)

where θ is the population mutation rate parameter. Note that we are not here assuming the infinite alleles model for our observed marker data. We use this model only for the latent ibd among sampled genes. Under an infinite-alleles model, genes deriving from each mutation event are identical by descent. Under the model (1) the coancestry coefficient β is given by

$$\beta =P_2(a_2=1)=\frac{2}{\theta (\theta +1)}\frac{\theta }{2}=\frac{1}{1+\theta }$$

Ewens’ sampling formula (1) may be rewritten in terms of coancestry β as

$$P_n(a_1,\dots ,a_n)=\frac{n!{\beta }^{n-k}{(1-\beta )}^{k-1}}{(1+\beta )(1+2\beta )\dots .(1+(n-2)\beta )}{\displaystyle \prod _{j=1}^n{(j^{a_j}{a}_j!)}^{-1}}$$ (2)

We follow Balding and Nichols (1994) in expressing higher order gene identities in terms of β by considering equilibrium under an infinite alleles model using a coalescent framework. However, rather than the parametrization of Balding and Nichols (1994) we use equation (2) directly to obtain the probabilities of the 15 states. For convenience, write η = β(1 − β)/(1 + β)(1 + 2β).

Then

$$\begin{array}{cc}\hfill P_4(a_4=1)=6{\beta }^2\eta /(1-\beta );\hfill & \hfill P_4(a_2=2)=3\beta \eta \hfill \\ P_4(a_3=a_1=1)=8\beta \eta ;\hfill & P_4(a_1=2,a_2=1)=6\eta (1-\beta )\hfill \\ \text{and}{P}_4(a_1=4)=\eta {(1-\beta )}^2/\beta .\hfill & \hfill \end{array}$$

Since all configurations with the same values of a_j have the same probability, the probabilities in the last two columns of Table 1 immediately follow.

2.4 State-changes along a continuous genome

We can now present the continuous-chromosome rate-matrix, Q, that has these required equilibrium probabilities. From each of the 15 states, we consider the ways in which ibd can be gained (rate g) or lost (rate h). The resulting matrix is shown in Table 2. Consider, for example, the state 1213. The pair of ibd genes can lose identity at rate h, leading to state 1234. On the other hand ibd may be gained in several ways: the gene labeled 2 may become ibd to either of the two labeled 1, giving state 1112 (rate 2g); the gene labeled 3 may become ibd to either of the two labeled 1, giving state 1211 (rate 2g); the genes labeled 2 and 3 may become ibd giving state 1212 (rate g). Transition rates from other states with three distinct genes may be derived similarly. As another example consider the state 1111. Any one of the four genes may lose ibd with the remainder, leading to rate h changes to each of states 1112, 1121, 1211, and 1222.

Table 2.

Transitions among the fifteen ibd states between four chromosomes.

ibd	One or both individuals inbred								non-inbred states
state	111	110	101		100	011		101	002		001				000
1111	–	0	h	h	0	h	h	0	0	0	0	0	0	0	0
1122	0	–	0	0	h	0	0	h	0	0	0	0	0	0	0
1112	3g	0	–	0	h	0	0	0	0	0	h	0	h	0	0
1121	3g	0	0	–	h	0	0	0	0	0	0	h	0	h	0
1123	0	g	2g	2g	–	0	0	0	0	0	0	0	0	0	h
1211	3g	0	0	0	0	–	0	h	0	0	h	h	0	0	0
1222	3g	0	0	0	0	0	–	h	0	0	0	0	h	h	0
1233	0	g	0	0	0	2g	2g	–	0	0	0	0	0	0	h
1212	0	0	0	0	0	0	0	0	–	0	h	0	0	h	0
1221	0	0	0	0	0	0	0	0	0	–	0	h	h	0	0
1213	0	0	2g	0	0	2g	0	0	g	0	–	0	0	0	h
1231	0	0	0	2g	0	2g	0	0	0	g	0	–	0	0	h
1223	0	0	2g	0	0	0	2g	0	0	g	0	0	–	0	h
1232	0	0	0	2g	0	0	2g	0	g	0	0	0	0	–	h
1234	0	0	0	0	g	0	0	g	0	0	g	g	g	g	–

For reasons of space, the diagonal terms are omitted. They are such that each row sums to 0.

Because we are considering change rates over infinitesimal chromosome lengths, only single events need to be considered. For example, there is 0 rate of switching from 1122 to 1234, since this would require two separate losses of ibd. Additionally, since in the infinite alleles model new alleles arise a singletons, here only singletons can “coalesce” to become ibd either with each other or with a larger group. Hence the 3 states with a₂ = 2 cannot instantaneously gain ibd. A transition must be made via one of the states with a₁ = 2, a₂ = 1. On the other hand, the other states with two distinct genes (a₁ = a₃ = 1) can gain ibd, moving at rate 3g to the state a₄ = 1, since the singleton gene can, without constraint, become ibd to any of the other three.

In Table 2 the detailed 15 states label the rows, while for clarity the nine state classes are used to label groups of corresponding columns. For reasons of space the diagonal terms are omitted. Since each row sum of any rate-matrix is zero (Karlin and Taylor, 1975), they are easily determined as −4h for state 1111; −2h for states 1122, 1212 and 1221, −3h −3g for states 1112, 1121, 1211 and 1222; −h −5g for states 1123, 1233, 1213, 1231, 1223 and 1232; and −6g for state 1234.

The equilibrium probabilities of this rate-matrix must satisfy πQ = 0 (Karlin and Taylor, 1975), and are a function of g/h = α only. Further, the equilibrium probability for states with k distinct genes have relative probabilities of order α^−k. It is easily shown that this rate-matrix has equilibrium probabilities

$$\begin{array}{c}\pi (1122)=\pi (1212)=\pi (1221)\hfill \\ \pi (1112)=\pi (1211)=\pi (1222)=\pi (1121)\hfill \\ \pi (1123)=\pi (1233)=\pi (1231)=\pi (1213)=\pi (1223)=\pi (1232)\hfill \end{array}$$

and further that

$$\begin{array}{cc}\hfill \pi (1111)& =3\alpha \pi (1112)\hfill \\ \hfill \pi (1112)& =2\pi (1122)\hfill \\ \hfill \pi (1122)& =\alpha \pi (1123)\hfill \\ \hfill \text{and}\pi (1123)& =\alpha \pi (1234)\hfill \end{array}$$

in agreement with the equilibrium probabilities of Table 1 with g/h = α = β/(1 − β).

2.5 The nine-state ibd model

In the event that paternal and maternal chromosomes need not be distinguished, and in particular if analysis is in the absence of LD, we can reduce for the 15-states to the 9 genotypically distinguishable classes. The relevant matrix is shown in Table 3, and is obtained simply by collapsing the 15-state matrix into the 9 classes. Since within any state class there are no transitions among the states, the rate of leaving the class is the same as that of leaving each state in the class, and transitions occur to the classes corresponding to the states in Table 2. For example, the two states 1212 and 1221 are each left at rate 2h, and so also is the combined state class 002 = {1212, 1221}. As discussed above the only possible transitions from this class is to the class 001, which therefore occurs at rate 2h. Likewise all four states in the class 001 = {1213, 1231, 1223, 1232} are left at rate 5g + h and so also is the class, and the transitions to other classes at rates 2g, 2g, g and h follow: compare rows 11 to 14 of Table 2 with the penultimate row of Table 3.

Table 3.

Transitions among the nine ibd states between four chromosomes

ibd state	inbred-states						non-inbred
111	−4h	0	2h	0	2h	0	0	0	0
110	0	−2h	0	h	0	h	0	0	0
101	3g	0	−3h-3g	h	0	0	0	2h	0
100	0	g	4g	−h-5g	0	0	0	0	h
011	3g	0	0	0	−3h-3g	h	0	2h	0
010	0	g	0	0	4g	−h-5g	0	0	h
002	0	0	0	0	0	0	−2h	2h	0
001	0	0	2g	0	2g	0	g	−h-5g	h
000	0	0	0	g	0	g	0	4g	−6g

Note that only certain transitions are possible in a single step, but since this is a continuous-time process all transitions are possible between any two distinct points of the genome. The equilibrium solution of the process has

$$\begin{array}{ccc}\hfill \pi (101)& =\hfill & \pi (011)=2\pi (002)=4\pi (110),\hfill \\ \hfill \pi (001)& =\hfill & 4\pi (100)=4\pi (010),\hfill \\ \hfill \pi (111)& =\hfill & \frac{3}{4}\alpha (\pi (101)+\pi (011))=6\alpha \pi (110),\hfill \\ \hfill \text{and}\pi (000)& =\hfill & {(6\alpha )}^{-1}(\pi (100)+\pi (010)+\pi (001)={\alpha }^{-1}\pi (100).\hfill \end{array}$$

Indeed, the solution is that of Table 1, again with g/h = α = β/(1 − β). Note α is the analogue of the ratio f/(1 − f) for the two-state model.

3 Discussion

Markov models, and more generally hidden Markov models (HMM), are pervasive in complex stochastic systems, not least because they permit likelihood and probability computations (Baum et al., 1970; Baum, 1972). The continuous chromosome ibd model of section 2.4 can be used as an HMM underlying genetic marker data on pairs of relatives, and allow the inference of small segments of genome shared ibd from data on dense SNP markers at a spacing of a few thousand base pairs (bp). However, at this scale, failure to take linkage disequilibrium (LD) into account in modeling haplotype frequencies would lead to severe biases. Haplotypes that are common in the population would be modeled as rare, through multiplication of allele frequencies over loci, leading to false inference of ibd. Other haplotypes, not existing in the population, would be given positive frequencies, and might be falsely imputed in the case of unphased genotypic data.

For likelihood computations on small pedigrees (Abecasis and Wigginton, 2005), the approach of clustered SNPs has been used to accommodate LD among marker loci. That is, SNPs in high LD are combined into “super-markers” within which no recombination is permitted. Between clusters there is assumed to be no LD. For simple relationships, such as sib pairs, an alternative approach is to incorporate LD in parental haplotypes, using a Markov model or HMM for the allelic types along a chromosome. For example, the cluster-based hidden Markov model for population haplotypes of Scheet and Stephens (2006) may be combined with the hidden Markov model of inheritance vectors (Lander and Green, 1987) to provide a tractable model for lod-score or ibd-based methods of linkage inference in the presence of LD in parental haplotypes (Fu and Thompson, 2007).

While for small pedigrees the dichotomy of no-LD or no-recombination will often suffice, it is less clear that this is so for remote relatives separated by many meioses. A more flexible model allowing both recombination and LD is desirable. Just as for the Markov inheritance vectors, the Markov model of Section 2.4 for latent ibd can be combined with hidden Markov LD-models for population haplotypes (Scheet and Stephens, 2006; Browning, 2006) to provide a framework for inference of ibd segments in a pair of remote relatives, using data at dense SNP markers exhibiting LD.