Abstract
Background
Detailed study of genetic variation at the population level in humans and other species is now possible due to the availability of large sets of single nucleotide polymorphism data. Alleles at two or more loci are said to be in linkage disequilibrium (LD) when they are correlated or statistically dependent. Current efforts to understand the genetic basis of complex phenotypes are based on the existence of such associations, making study of the extent and distribution of linkage disequilibrium central to this endeavour. The objective of this paper is to develop methods to study finescale patterns of allelic association using probabilistic graphical models.
Results
An efficient, lineartime forwardbackward algorithm is developed to estimate chromosomewide LD models by optimizing a penalized likelihood criterion, and a convenient way to display these models is described. To illustrate the methods they are applied to data obtained by genotyping 8341 pigs. It is found that roughly 20% of the porcine genome exhibits complex LD patterns, forming islands of relatively high genetic diversity.
Conclusions
The proposed algorithm is efficient and makes it feasible to estimate and visualize chromosomewide LD models on a routine basis.
Background
Alleles at two loci are said to be in linkage disequilibrium (LD) when they are correlated or statistically dependent. The term refers to the idea that in a large homogeneous population subject to random mating, recombination between two loci will cause any initial association between them to vanish over time. In observed data, however, nonzero allelic associations are pervasive, particularly at short distances, but also at long distances and even between chromosomes. These associations arise in a complex interplay between processes such as mutation, selection, genetic drift and population admixture, and are broken down by recombination. The patterns of association are of interest, partly because they underpin the relation of genotype to phenotype at the population level, and partly because they reflect population history.
Patterns of LD may be represented in different ways. A common method is to display pairwise measures of LD as triangular heatmaps [1,2]: in these displays, LD blocks (genomic intervals within which all loci are in high LD) stand out clearly. Early work in the HapMap project led researchers to hypothesize that the human genome consists of a series of disjoint blocks, within which there is high LD, low haplotype diversity and little recombination, and that are punctuated by short regions with high recombination (recombination hotspots) [36]. Subsequently various authors [7,8] reported that genetic variation follows more complex patterns, for which richer models are required.
Discrete graphical models [9] (also known as discrete Markov networks) provide a rich family of statistical models
to describe the distribution of multivariate discrete data. They may be represented
as undirected graphs in which the nodes represent variables (here, SNPs) and absent
edges represent conditional independence relations, in the sense that two variables
that are not connected by an edge are conditionally independent given some other variables.
To motivate this focus on conditional rather than marginal associations, consider
three loci
Other authors have used graphical models for the joint distributions of allele frequencies. Usually, in highdimensional applications, attention is restricted to a tractable subclass, the decomposable graphical models [10]. In the first use of decomposable models in this context [11], models were selected using a greedy algorithm based on significance tests. In [8,12] methods and programs for selecting decomposable graphical models using Monte Carlo Markov Chain (MCMC) sampling were described. These methods are computationally feasible for modest numbers of markers (say, several hundreds), but not for modern SNP arrays with hundreds of thousands of SNPs per chromosome. To improve efficiency, the search space may be restricted to graphical models whose dependence graphs are interval graphs[13,14]. These are graphs for which each vertex may be associated with an interval of the real line such that two vertices are connected by an edge if and only if their intervals overlap. In this context the ordering of SNPs along the real line is their physical ordering along the chromosome. MCMC sampling from this model class may be performed more efficiently [13,14]. This work was extended in [15] to a more general subclass of decomposable models, namely those in which distant marker pairs (i.e., with more than a given number of intervening markers) are conditionally independent given the intervening markers.
In an alternative approach [1618] latent mixtures of forests have been applied, in order to accommodate short, medium and longrange LD patterns. Also directed graphs (Bayesian networks) have been applied, selecting edges and their directions using causal discovery algorithms [19]. There are close links between decomposable models and Bayesian networks ([10], Sect. 4.5.1).
A rather different approach to modelling the joint distribution of allele frequencies [20,21] is implemented in the software package BEAGLE [22], which is widely used to process data from SNP arrays. The approach is based on a class of models arising in the machine learning literature called acyclic probabilistic finite automata (APFA) [23]. These are related to timevariant variable length Markov chains. For phase estimation and imputation BEAGLE uses an iterative scheme analogous to the EM algorithm, alternating between sampling from a haplotypelevel model given the observed genotype data (the Estep) and selecting a haplotypelevel model given the samples (the Mstep). A similar computational scheme for decomposable graphical models has been described and implemented in the FitGMLD program [15].
Characterization of genetic variation at the population level is of fundamental importance to understanding how phenotypes relate to genotypes. Some specific uses to which joint models for allele frequencies have been put include
1. Insight into the population history of different genomic intervals. Under simplifying assumptions, the ancestral history of a short genomic interval can be reconstructed from a decomposable graphical model for the SNPs in the interval [8].
2. Quality control of genome assembly, in that some motifs may suggest errors in SNP positioning.
3. Phase estimation and imputation as described above [15,21].
4. Derivation of more informative covariates involving multiple loci to plug into genomic prediction models [20].
5. Use in pedigree simulation to model LD in founders, for example in connection with gene drop simulation [14] and assessment of SNP streak statistics [24].
In this paper decomposable graphical models are used to model finescale, local LD patterns. By local is meant that the proposed methods are designed to capture shortrange associations between loci, but not long range ones. The same is true of other approaches [1315] mentioned above. Here, an efficient, lineartime algorithm is developed to select a model using a penalized likelihood criterion, and it is shown how such a model may conveniently be displayed, allowing finescale LD structure to be visualized.
Methods
Graphs and graphical models
The following notation and terminology is mainly based on [9]. A graph is defined as a pair
A path (of length n) between vertices α and β in an undirected graph is a set of vertices
A subset D⊂V in an undirected graph is said to separateA⊂V from B⊂V if every path between a vertex in A and a vertex in B contains a vertex from D. The graph
The boundary bd(A) of a vertex set A⊆V is the set of vertices adjacent to a vertex in A but not in A, that is ,.
Let
for some functions
Selecting graphical models for chromosomewide LD
Suppose that N observations of p SNPs from the same chromosome are sampled from some population. The variable set
is written
where
A technical but nonetheless important point is that calculation of the correct model dimension is not straightforward for highdimensional models, since not all parameters may be estimable. A simple expression exists however for the difference in model dimension between two decomposable models that differ by a single edge ([10], pp. 3740). This is useful when the search space is restricted to decomposable models and the search algorithm only involves comparison of models differing by single edges (as here).
A forwardbackward approach to estimate the graphical model for V is used. The first (forward) step is based on a greedy forward search algorithm.
To take account of the physical ordering of the SNPs, the algorithm starts from a
model
There are several advantages to initially constraining the search space to include the skeleton. An unconstrained search starting off from the null model (with no edges) would not take the physical ordering into account. Since the graphs considered are consistent with this ordering, they are conveniently displayed as LD maps, as illustrated below. Because decomposable models contain no chordless cycles of length ≥4, two distal loci cannot be linked by an edge unless there are sufficiently many intermediate (chordal) edges to prevent the occurrence of such cycles. Thus the algorithm proceeds by filling out around the skeleton by incrementally adding edges. The effect of this is that only local LD is modelled. Since both association and (implicitly) proximity are required, the edges included are more likely to be for real (that is, less likely to be due to chance), so in this sense the model selection process is more reliable. In addition, restricting the search space in this way improves computational efficiency. The lineartime algorithm described in the following section also builds on the physical vertex ordering.
In the second (backward) step the graph found in the first step is pruned, again using a greedy algorithm that seeks to minimize the BIC (AIC) criterion by removing edges, without requiring that the skeletal edges are retained. Keeping these in the model would be appropriate if adjacent SNPs are in high LD, but this is not always the case. For example, there may be recombination hotspots, or genome assembly errors may have led to errors in positioning the SNPs. The graphs may be used to examine whether this has occurred.
To display the resulting graph, a suitable graph layout is needed. After some experimentation, one such was obtained by modifying the horizontal dot layout in Rgraphviz [26], by exponentially smoothing the ycoordinates and replacing the xcoordinates with the SNPnumber. In this way the ycoordinates are chosen so as to clarify the graphical structure: a consequence of this is that nodes with large absolute ycoordinates tend to signal genomic regions of high structural complexity.
For some purposes it is helpful to use more objective measures of complexity, and
two such measures are used here. Consider a chromosome with p SNPs. For each
As a measure of haplotype diversity, the entropy[27] is used here. Suppose that in some genomic interval there are k distinct haplotypes with relative frequencies
A useful way to display marker data over short genomic intervals is the sample tree[23]. This summarizes a set of discrete longitudinal data of the form
A fast selection algorithm
For highdimensional models, the first step of the algorithm described above can be
computationally demanding. To address this a much faster algorithm for the same task
is now described. This involves combining a series of overlapping marginal models
of lower dimension. Call the block length (model dimension) L and the overlap K. Thus the first block is
Suppose that the true model for
A way to construct a model
The rationale for this is that marginal models may include spurious associations on
the boundaries. For example, let
The algorithm proceeds in the obvious fashion combining
Assuming that the time for deriving the estimates
Implementation
The methods are implemented in a set of R functions which are provided as Additional files 1 and 2 to the paper. The selection algorithms build on the forward search algorithm implemented in the stepw function in the gRapHD package [30]. Currently this function requires that there are no missing data. The backward search algorithm was developed by the author in the course of preparing this paper, and has subsequently been implemented in the gRapHD package. The functions to perform the selection algorithms, and others to work with and display the graphs, build on the following packages: igraph [31], graph [32], Rgraphviz [26] and gRapHD. The triangular heatmaps were produced using the LDheatmap package [33]. Simulation from a fitted decomposable model was carried out using the gRain package [34]. The package rJPSGCS provides an interface between R and the FitGMLD program which was used to perform the algorithm of [15].
Additional file 1. LDgraph. The R functions implementing the methods of the paper.
Format: R Size: 6KB Download file
Additional file 2. exampleScript. R code illustrating how the R functions may be used.
Format: R Size: 1KB Download file
Results
To illustrate the methods described above, they were applied to SNP data obtained from three commercial pig breeds. In all 4239, 1979 and 2123 pigs of the Duroc, Landrace and Yorkshire breeds were genotyped using the Illumina Porcine SNP60 BeadChip [35]. After preprocessing on the basis of call rate, minimal allele frequency and other quality criteria, missing values were imputed using BEAGLE [22]. Using the methods described above, decomposable graphical models were selected for each chromosome and breed, using data at the genotype level. The BIC penalizing constant α= log(N) was used throughout.
Figure 1 compares the running times of the two algorithms, and confirms that the fast algorithm
is approximately linear in the number of SNPs. Table 1 gives more detailed information. It is seen that the estimate
Figure 1. Running times of the two algorithms. The running times (in minutes) of the forward step of the standard and fast algorithms for the different chromosomes and breeds are shown in (a) and (b), respectively.
Table 1. The performance of the two algorithms
Figure 2 shows the LD graph for one chromosome (Duroc chromosome 1): the same graph is obtained with both algorithms. It is a long thin graph with 2863 nodes and 4340 edges. It has 32 connected components, of which 20 are isolated vertices — suggesting SNP positioning errors — and four are long intervals with 997, 569, 786 and 452 SNPs. Curiously, the subgraph induced by the last 35 SNPs contains 7 connected components, which suggests positioning errors in this region. The most striking feature of the graph is that for about 80% of the chromosome, a simple serial or nearserial association structure is found, but for the remaining 20% more complex patterns of LD are observed. Similar results are found for all chromosomes and breeds.
Figure 2. The LD graph for Duroc chromosome 1. The estimated LD graph for chromosome, constructed using the unphased genotypes, is shown.
Genomic intervals with simple association structure tend to be associated with low haplotype diversity, and intervals with complex structure with high diversity (Figure 3). To further compare the low and high complexity regions, two representative intervals of the same length were selected: SNPs numbered 18001825 (low complexity), and SNPs numbered 24702495 (high complexity). These have sample entropies of 0.79 and 4.05, respectively. Their subgraphs are shown in Figure 4, and their sample trees in Figure 5. These show the low diversity of the low complexity interval, and the relatively high diversity of the high complexity interval. Corresponding triangular heatmaps are shown in Figures 6 and 7. The low complexity interval has high LD whereas the high complexity interval shows a more mosaic structure.
Figure 3. Entropy versus height and width for Duroc chromosome 1. In (a) a plot of median entropy, a measure of haplotype diversity, versus width for Duroc chromosome 1 is shown. In (b), a corresponding plot of median entropy versus height is shown. The entropy was calculated for each interval of 7 adjacent SNPs using the phased data.
Figure 4. The subgraphs of typical low and high complexity intervals. The subgraphs of a typical low complexity interval (SNPs 18001825) and high complexity interval (SNPs numbered 24702495) are shown in (a) and (b), respectively.
Figure 5. The sample tree for the low and high complexity intervals. Sample trees for the low and high complexity intervals are shown in (a) and (b), respectively. The trees are based on the phased haplotype data. The root node is drawn on the left. Edge widths are proportional to the numbers of observations passing through the edges. The colour of the edges signifies the allele: red is 1 and blue 2. It is seen that the interval shown in (a) has low haplotype diversity, dominated by a single common haplotype, whereas that shown in (b) has high haplotype diversity, with one or two relatively common haplotypes.
Figure 6. A triangular heatmap for the low complexity interval. A heatmap of the pairwise LD values in the low complexity interval is shown. All associations are high.
Figure 7. A triangular heatmap for the high complexity interval. A heatmap of the pairwise LD values in the high complexity interval is shown. The LD values form a mosaic of mostly low associations.
Thus low complexity regions tend to consist of series of haplotype blocks with high LD and low haplotype complexity, and may be dominated by a few common haplotypes. In contrast, regions of high complexity tend to have high haplotype diversity and little or no haplotype block structure. It must be stressed that the SNPs in such regions are not generally in linkage equilibrium: on the contrary, complex patterns of association, not marginal independences, are observed.
Figure 2 and Table 1 show the results of applying the algorithms to unphased, genotype data, but as mentioned above they may also be applied to phased haplotypelevel data. This implicitly regards the inferred haplotypes as a random sample of size 2N from an underlying population of haplotypes. Analyses based on inferred haplotypes may be subject to a loss of efficiency [36]. For the current data phase imputation was carried out using the BEAGLE software [22]: applying the fast algorithm to the phased data for Duroc chromosome 1 resulted in the graph shown in Figure 8. This is slightly denser than the graph in Figure 2, with 375 more edges. This may be ascribed to the reduced model dimension due to the use of binary rather than trichotomous variables, which leads to a weaker penalization of complex models in (1). Haplotype and genotypelevel graphical models are related but in general distinct: it has been shown that they are identical only when the haplotypelevel graph is acyclic [37].
Figure 8. The LD graph for Duroc chromosome 1 based on phased data. The estimated LD graph for Duroc chromosome 1, constructed using the phased haplotypes, is shown.
To assess the accuracy of the algorithm, twenty simulated data sets each with the
same number of observations as the data sets analyzed above (N=4239) were constructed. The first ten were generated by taking N random samples from the skeleton, that is, the graphical model with edges between
physically adjacent SNPs only, fitted to the Duroc chromosome 1 genotype data. The
second ten were generated by taking N random samples from the model shown in Figure 2, fitted to the same data. The results of applying the fast algorithm to each data
set are summarized in Table 2. For the skeleton, the edgewise false negative rate was 0.029 and the edgewise false
inclusion rate was 0.00024. For the model in Figure 2 the corresponding rates were 0.031 and 0.019. To visually represent the latter results,
Figure 9 shows a graph
Table 2. Results from simulations
Figure 9. The union of LD graphs for Duroc chromosome 1 derived from ten simulated data sets. The figure represents the results of applying the fast algorithm to ten data sets simulated under the LD graph shown in Figure 1. The union of the ten selected graphs is shown. Edges present in all ten graphs are black, and the colour of the others is interpolated between light blue (those found once) and blue (those found nine times).
Finally, for comparison purposes the algorithm of [15] was applied to the Duroc chromosome 1 genotype data. This algorithm automatically imputes phase and missing marker data, cycling between imputation and model selection as sketched in the background section. The selected graph is shown in Figure 10. Like the graphs shown in Figures 2 and 8, it is long and stringy, but rather more dense. This greater density may be explained by the use in [15] of a Metropolis acceptance rule that is based on the penalized likelihood (1), but with a smaller penalizing constant α=(1/8) log(N). A detailed comparison of the two algorithms would be valuable but is not attempted here.
Figure 10. Haplotypelevel graph found using the algorithm of [[15]]. The figure shows the result of applying the algorithm of [15] to the same data set as used with Figure 2. The algorithm estimates a haplotypelevel graph for the data.
Discussion
This paper has introduced an efficient, lineartime forwardbackward algorithm to estimate chromosomewide probabilistic graphical models of finescale linkage disequilibrium, and has described a convenient way to display these models. In illustration, the methods have been applied to data obtained from three commercial breeds of pigs using the Illumina Porcine SNP60 BeadChip.
The forward part of the algorithm proceeds by combining a series of overlapping marginal decomposable models of dimension L and overlap length K. This implicitly assumes that the maximum width of the true graph is at most K. The resulting model is then triangulated and backward selection performed. The search space closely resembles that of [15], which samples from decomposable models with a given maximum width, using sliding windows. The difference in approaches lies primarily in the search method: here greedy search to optimize a penalized likelihood criterion is used, whereas in [15] MCMC sampling methods are applied.
The R function used for greedy forward search currently requires that the input data contain no missing values, so it was necessary to impute missing values prior to performing the algorithm, and the BEAGLE software [22] was used for this. This raises the possibility of circularity, or more precisely, that the model selection is influenced by constraints or assumptions implicit in the models used by BEAGLE. But given BEAGLE’s high imputation accuracy with such data it seems unlikely that this plays an important role here.
The algorithm was found to have high accuracy when applied to simulated data sets of the dimensions considered here (N∼4000;p∼3000), with edgewise false positive and negative rates of around 3%. That it is good at reconstructing the model generating the data is reassuring. Needless to say, the algorithm does not necessarily identify the “true” model, which may not be in the search space. As noted previously, the approach captures short range associations, but not long range ones. Moreover, higher edgewise false negative rates may occur when observed data are used if there are infinitesimal departures from a model that are not detectable at the given sample size, as these are filtered out when data are simulated under a selected model.
A comparison of triangular heatmaps with LD graphs is instructive. The former are compact graphical representations of all pairwise marginal associations for a set of SNPs. They are particularly wellsuited to identify LD blocks, which stand out as highlighted triangles. LD graphs supplement heatmaps by showing patterns of association but not their strength. Since they are parametric models they can be put to a number of quantitative uses as described in the background section. At one level, heatmaps and LD graphs convey similar information, since intervals with simple dependence structures tend to appear as series of LD blocks in the heatmap, whereas those with complex structures tend to occur in the interblock regions and have a more mosaic appearance. But the graphs provide a more incisive characterisation of genetic variation, building on the concept of conditional rather than marginal dependence. In this regard, it may be helpful to regard conditional independence statements as expressing the notion of irrelevance, in the sense that A⊥ ⊥B  C implies that if we know C, information about A is irrelevant for knowledge of B. Thus the graphs say something about connections between specific SNPs, for example about which SNPs are required to predict a specific SNP.
The graphs may also be useful in fine mapping. Genomewide association studies seek to find the genetic basis of complex traits, typically using single locus methods  that is, by identifying SNPs with strong marginal association with the complex trait. Due to LD, many SNPs in a genomic region may exhibit strong association with the trait, making it hard to identify the causal loci. A way to address this is to assess the effect of a putative causal SNP on the complex trait in a linear model that also includes terms for the two flanking SNPs, in order to adjust for confounding with nearby effects due to LD. This implicitly assumes a simple serial dependence structure between the SNPs, and when the dependence structure is more complex such adjustment might be insufficient, leading to false positives. This may be prevented by including terms for the neighbours of the putative causal SNP, not just for the flanking SNPs. Any SNP is separated from the remaining SNPs by its neighbours, so by the global Markov property it is independent of the remaining SNPs given its neighbours. A similar method has been proposed based on Bayesian networks [38].
Like graphical models, the APFA models that underlie BEAGLE may be represented as graphs that encode a set of conditional independence relations, but in a very different way. In the present context, for example, nodes in APFA graphs represent haplotype clusters rather than SNPs. Where the model classes intersect, APFA graphs are much more complex than the corresponding dependence graphs, and so less amenable to visualization.
A striking feature of the LD graphs for the pig data was that for roughly 80% of the genome, simple serial or nearserial LD patterns were found, but for the remaining 20%, more complex patterns were observed. The regions with the simple serial structure tend to have low haplotype diversity, which is to be expected in livestock breeds with small effective population sizes [39]. Perhaps more unexpected is that roughly 20% of the porcine genome exhibits complex LD patterns, forming islands of relatively high genetic diversity. This information may be useful in an animal breeding context, to identify regions with high genetic variation. It will also be interesting to compare graphs obtained using different SNP densities in a given breed or species to examine whether and how their topologies change with varying marker densities.
Conclusions
The proposed algorithm is efficient and makes it feasible to estimate and visualize chromosomewide LD models on a routine basis.
Competing interests
The author declares he has no competing interests.
Acknowledgements
The data were kindly provided by the Danish Pig Research Centre and in particular Tage Ostersen. The work was performed in a project funded through the Green Development and Demonstration Programme (grant no. 3405110279) by the Danish Ministry of Food, Agriculture and Fisheries, the Pig Research Centre and Aarhus University. Ole F. Christensen, Peter Sørensen, Luc Janss and Lei Wang are acknowledged for helpful discussions.
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