 Research
 Open Access
 Published:
Accurate HLA type inference using a weighted similarity graph
BMC Bioinformaticsvolume 11, Article number: S10 (2010)
Abstract
Background
The human leukocyte antigen system (HLA) contains many highly variable genes. HLA genes play an important role in the human immune system, and HLA gene matching is crucial for the success of human organ transplantations. Numerous studies have demonstrated that variation in HLA genes is associated with many autoimmune, inflammatory and infectious diseases. However, typing HLA genes by serology or PCR is time consuming and expensive, which limits largescale studies involving HLA genes. Since it is much easier and cheaper to obtain single nucleotide polymorphism (SNP) genotype data, accurate computational algorithms to infer HLA gene types from SNP genotype data are in need. To infer HLA types from SNP genotypes, the first step is to infer SNP haplotypes from genotypes. However, for the same SNP genotype data set, the haplotype configurations inferred by different methods are usually inconsistent, and it is often difficult to decide which one is true.
Results
In this paper, we design an accurate HLA gene type inference algorithm by utilizing SNP genotype data from pedigrees, known HLA gene types of some individuals and the relationship between inferred SNP haplotypes and HLA gene types. Given a set of haplotypes inferred from the genotypes of a population consisting of many pedigrees, the algorithm first constructs a weighted similarity graph based on a new haplotype similarity measure and derives constraint edges from known HLA gene types. Based on the principle that different HLA gene alleles should have different background haplotypes, the algorithm searches for an optimal labeling of all the haplotypes with unknown HLA gene types such that the total weight among the same HLA gene types is maximized. To deal with ambiguous haplotype solutions, we use a genetic algorithm to select haplotype configurations that tend to maximize the same optimization criterion. Our experiments on a previously typed subset of the HapMap data show that the algorithm is highly accurate, achieving an accuracy of 96% for gene HLAA, 95% for HLAB, 97% for HLAC, 84% for HLADRB1, 98% for HLADQA1 and 97% for HLADQB1 in a leaveoneout test.
Conclusions
Our algorithm can infer HLA gene types from neighboring SNP genotype data accurately. Compared with a recent approach on the same input data, our algorithm achieved a higher accuracy. The code of our algorithm is available to the public for free upon request to the corresponding authors.
Introduction
In human chromosomal region 6p21, there is a Human Leukocyte Antigen (HLA) superlocus of about 4Mb length with extreme high levels of gene density and variation. The HLA locus contains about 0.5% (> 150) of all known protein coding genes [1] and nearly each HLA gene has more than a dozen different alleles [2]. The HLA genes play important roles in the immune system and encode a group of related proteins known as the HLA complex. The highly polymorphic HLA genes produce hyper variable HLA complex, by which the human immune system differentiates self cells and nonself cells. Mismatches between an organ donor’s HLA genes and a recipient’s HLA genes usually result in rejection reactions and cause a transplantation to fail. The nature of the genetic diversity of the HLA region is complex, and this region has become a research hotspot in human genomics [3]. Recently, numerous researchers have illustrated that different alleles of HLA genes are associated with many autoimmune, inflammatory and infectious diseases [4, 5]. However, direct experimental typing methods of HLA gene alleles such as serology and PCR are laborious, time consuming and expensive, which limit largescale studies concerning HLA genes [6]. So, effective computational techniques are in demand to help determining HLA gene types.
The HLA region is divided into two classical regions called class I and class II, and an intervening region denoted as class III, as illustrated in Figure 1. The classical genes HLAA, HLAB and HLAC are in Class I; and the classical genes HLADP, HLADQ and HLADR are in Class II [3]. There is an elaborate nomenclature for hyper variable HLA gene alleles, as shown in Figure 2. Beginning with HLA and the gene name, each HLA allele name contains up to four sets of digits separated by colons. The first set of digits describes the allele group, which can be determined by serological typing. The next set of digits describes the subtypes and represents different amino acid sequences of the encoded protein. The last two sets of digits denote any synonymous mutations in exons and introns respectively. An additional optional suffix such as ‘L’, ‘S’, ‘C’, ‘A’ or ‘Q’ is used to specify its expression level or other nongenomic data. Although four sets of digits are needed to completely describe an allele, most practical applications usually only require the first set or the first two sets of digits.
With the advance of high throughput SNP genotyping technologies, it is relatively easy to obtain genomewide SNP genotype data with low costs, and SNP data of many individuals has already been available. Recently, some researchers have studied the problem of HLA gene type inference based on SNP genotype data. The first type of approaches are rooted in the concept of tag SNPs. Based on the linkage disequilibrium between multiallelic HLA genes and their neighboring biallelic SNPs, de Bakker et al.[4] selected up to three tag SNPs as predictors of HLA alleles. Although tag SNP based methods in general can infer some common HLA alleles [7–9], they usually select different sets of tag SNPs for different alleles of the same HLA gene. Moreover, HLA genes are highly polymorphic and the majority alleles are rare. It is known that they generally cannot be distinguished by using combinations of up to three tag SNPs [6]. By extending the above tag SNPs based methods, Leslie et al.[6] selected dozens of SNPs around the HLA genes and proposed a statistical method to infer HLA alleles at class I and class II loci. The method is based on the assumption that a chromosome carrying an HLA allele is an imperfect mosaic of those chromosomes with the same HLA allele. Given a training data of SNP haplotypes and the corresponding phased HLA gene types, a hidden Markov model is used to calculate the posterior probability of a chromosome taking an HLA allele with a particular SNP haplotype. The model requires a fine genetic map of the region [6], and uses a training set of SNP haplotypes with known, phased HLA gene types [2]. Based on the identity by descent (IBD) information between pairs of individuals, Setty et al.[2] proposed an iterative approach for HLA type imputation. At first, a program (GERMLINE [10]) is called to obtain the IBD segments between each pair individuals and an IBDGraph is built for the individuals with known or unknown HLA types. Then the unknown HLA types of some individuals are imputed from the individuals with known HLA types that are involved in the same triplets of the IBDGraph. At last the IBDGraph is updated and a new iteration begins until no more HLA types can be imputed. Though the IBDGraph based method does not require SNP haplotypes as a part of the input, the program GERMLINE [10] needs SNP haplotype information to determine the IBD status between each pair of individuals.
The accuracy of all above methods critically depends on the accuracy of the haplotypes of each individual, which are usually inferred from genotypes based on some computational models. Though the haplotype inference problem from unrelated individuals has been extensively studied recently [11], the accuracy of inferred haplotypes, especially for large chromosome regions (> 100 kb), is not yet satisfactory [12]. Additional information from family data could greatly improve the accuracy of haplotype inference for long chromosomal regions [13, 14]. However, even with pedigree information, there are still potentially many haplotype solutions for a pedigree that satisfy the Mendelian law and have the smallest number of recombination events [13, 14].
In this paper, we jointly model HLA gene type inference and SNP haplotype inference/selection within one unified framework, by utilizing the relationship among individuals (i.e., pedigree information), known HLA gene types of some individuals and the relationship between SNP haplotypes and HLA gene types. We first propose a new haplotype similarity measure and construct a weighted haplotype similarity graph. Known HLA gene types are used to derive additional constraints on edges connecting two haplotypes from the same individual. Based on the principle that different HLA gene types should have different background SNP haplotypes, the algorithm searches for an optimal labeling of all the haplotypes with HLA gene types such that the total weight among the same HLA gene types is maximized. To obtain haplotypes from genotypes, we first utilize the program recently developed in [14] to construct a solution space with the minimum number of recombinants from each pedigree. To deal with ambiguous haplotype solutions for each pedigree, an enumerating procedure is used to select a haplotype configuration that tends to maximize the total similarity among the same HLA gene types. When there are too many solutions for the enumerating procedure to work efficiently, a genetic algorithm is adopted instead. Compared with the existing methods, our algorithm achieves a higher accuracy on a previously studied HapMap dataset.
Results and discussion
The performance of our algorithm (denoted as WSGHI) is evaluated using the dataset from [2], which can be downloaded from http://www.inflammgen.org/inflammgen/files/data/. The data consists of 180 Utah residents of European ancestry (27 extended families with an average family size of 6.6) from the CEPH collection (CEU) of the HapMap project, which has been described in [4]. 8562 nonredundant variants in the 7.5Mb extended HLA region were genotyped [4], of which 6300 SNPs passed QC [6]. The HLA typing for three class I genes (HLAA, HLAB, and HLAC) and three class II genes (HLADRB1, HLADQA1, and HLADQB1) was carried out with PCRSSOP protocols. Both the SNP data and HLA gene data used in our experiments are unphased and were obtained from [4].
Two important measures coverage and accuracy are used to analyze and compare the performance of different algorithms, which have also been used in [2]. They are defined as follows
where N_{ analyzed } denotes the number of chromosomes analyzed by each algorithm, N_{ called } denotes the number of chromosomes whose HLA genes have been inferred, and N_{ correct } denotes the number of chromosomes whose HLA genes have been correctly inferred.
Sketch of the algorithm and experimental design
A brief sketch of WSGHI is illustrated in Figure 3. The details of WSGHI are described in Section Methods. The input data are the unphased SNP genotype data of a population P and the known unphased HLA gene types of the individuals in the subset R. The algorithm outputs the inferred HLA gene types for the individuals in the subset U=P–R.
We will use two test strategies: leaveoneout and leaveonepedigreeout. In the leaveoneout test, for each individual in the CEU data, its HLA gene types are removed and WSGHI is used to infer the HLA gene types of this member. In the leaveonepedigreeout test, for each pedigree in the population, the HLA gene types of all members of the pedigree are removed and the algorithm is used to infer the HLA gene types of every member of the pedigree.
Results of the leaveoneout test
We download the same data set used in Setty et al.[2], and compare the performance of our algorithm (WSGHI) and theirs (denoted as IBDHI). To evaluate the accuracy, both algorithms use the leaveoneout test method as described in [2]. For each HLA gene, WSGHI takes the genotype data from the region of 200kb centered around the HLA gene under consideration (i.e., the region spanning 100kb upstream and downstream of the gene) as the input. The results at two different HLA gene allele resolution levels (4digit and 2digit) are illustrated in Figure 4. HLA gene types that are not resolved to the required resolution level or occur just once in the data are excluded from the analysis. Since the coverage of WSGHI for all HLA genes is 100% in the experiments and Setty et al. did not provide the coverage of their algorithm in [2], we only compare the accuracy of both algorithms. As shown in Figure 4, both algorithms perform similarly for HLAA and HLAB. However, for the other four genes, WSGHI is more accurate than IBDHI at both 4digit and 2digit resolutions.
We also investigate how the size of the region around an HLA locus, where the SNP genotype information is used by WSGHI, affects the HLA gene type inference accuracy. Figure 5 illustrates that when the size of SNP genotype region changes from 250kb to 100kb, the accuracy of WSGHI varies slightly for almost all the genes, with the only exception of the HLADRB1 gene. For HLADRB1, WSGHI achieves the highest accuracy of 91.3% when using genotype data from the 250kb region surrounding the gene, and its performance deteriorates dramatically when the size of the region gets smaller. This is mainly caused by the fact that the number of heterozygous SNPs near the center of the HLADRB1 gene region is very small (see Additional file 1). Therefore, when using the region of 100kb, there are not enough SNP haplotypes to distinguish all different HLADRB1 gene types.
Results of the leaveonepedigreeout test
Because the test data consists of many pedigrees, it is conceptually easier to infer the HLA gene types of a tested member in the above leaveoneout strategy. The known HLA gene types of other members in the same pedigree may provide much needed information to correctly infer HLA gene types of the tested member. Therefore, we further test the performance of our algorithm using the leaveonepedigreeout strategy, by deleting the HLA gene type information of a whole pedigree. The experimental results are shown in Table 1. To our surprise, the results are almost as good as those from the leaveoneout strategy. This is probably because the haplotype similarity information at the population level has already provided sufficient information to correctly infer HLA gene types. One does not gain much from additional information provided by family members, especially in our experimental settings when such relationship is not explicitly explored.
Conclusions
HLA genes have important functions in the human immune system, and their variations are associated with many complex diseases. However, directly typing of HLA genes is time consuming and expensive. Accurate and efficient computational algorithms to infer HLA gene types from SNP genotype data are a good alternative. We have designed an accurate HLA gene type inference algorithm (WSGHI). The algorithm takes SNP genotypes of all individuals and HLA gene types of a subset of individuals as input, and infers the HLA gene types of the remaining individuals based on their genotype data. Extensive experimental results on a previously typed dataset have illustrated that WSGHI can infer the HLA gene types from their neighboring SNP genotypes accurately. Compared with a previous approach based on IBD, our algorithm achieves the same accuracy for HLAA and HLAB genes, but is much more accurate for HLAC, HLADRB1, HLADQA1 and HLADQB genes.
Methods
Preliminaries
A single nucleotide polymorphism (SNP) is the change of a single nucleotide at a position of the genome sequence and is the major form of human genome variation. It is believed that most SNPs are biallelic. Therefore, an SNP can usually be represented as a 0 or 1, where ‘0’ denotes the major allele at the SNP locus and ‘1’ denotes the minor allele. When the allele of an SNP is unknown, ‘–’ is used. A sequence of SNP alleles on one of a pair of homologous chromosomes is called a haplotype, and can be denoted by a string over {0, 1, –}. Two conflated (unordered pair of) SNP alleles at each SNP locus of a pair of homologous chromosomes is called a genotype.
Recently, extensive studies have revealed that biallelic SNPs in the HLA region are in strong linkage disequilibrium with multiallelic HLA genes [4], which implies that similar haplotypes will harbor similar HLA gene alleles. However, determining haplotypes using biological techniques is as costly and time consuming as direct HLA gene typing. Because it is relatively inexpensive to genotype largescale SNPs, inferring HLA gene alleles from their neighboring SNPs genotypes offers an attractive alternative to conventional HLA typing. For an HLA gene, given the genotype data around the HLA gene of a population and the HLA gene alleles of some individuals of the population, the HLA gene type inference problem aims to determine the HLA gene alleles (or types) for the individuals whose HLA gene alleles are unknown. Please see Figure 6 for an example. The genotype data of m SNPs of n individuals can be represented by an n × m matrix (called the genotype matrix) with each element representing an unordered SNPs pair, and the HLA gene types of the population can be denoted as an n × 2 matrix (called the HLA matrix) where an unknown gene type is considered as an empty element and denoted by ‘–’. In Figure 6 (right), the HLAA gene types of the second and the last individuals are inferred using a computational approach. Although there are many different methods to infer haplotypes from genotype data, the accuracy of haplotype inference from a population made up of unrelated individuals is unsatisfactory. On the other hand, with additional family information, long range haplotypes can be inferred more accurately [11]. This is why Leslie et al.[6] and Setty et al.[2] chose pedigrees as their test data. However, even if a pedigree is provided, it may be difficult to uniquely determine its haplotype configuration [14]. In Li et al.[14], the authors designed a program DSS that could establish the solution space of the haplotype configurations of a pedigree under the Mendelian and zerorecombinant constraints in almost linear time. DSS uses a disjointset structure D to represent a general solution. The number of total specific solutions (N_{ s }) has a simple relationship with the number of free variables f: N_{ s } = 2^{f}. To deal with recombination, DSS was extended as follows. At first, the whole region is partitioned into some maximal zerorecombinant segments. Then the solution spaces for these segments are combined into a whole solution space for the region with of the goal of minimizing the number of recombinations between neighboring segments. We adopt the DSS algorithm in this paper due to its efficiency and accuracy.
Algorithm
Given the SNP genotypes of some pedigree data and the known HLA gene type information of some individuals, our algorithm jointly models the HLA gene type inference problem and the optimal haplotype selection problem within one framework. In brief, we first define a new haplotype similarity measure. Given a set of haplotypes, we construct a complete weighted graph with each node representing one haplotype. Each edge, connecting a pair of haplotypes, is given a weight using the value of the similarity between the two haplotypes. In addition, for each individual with known HLA types, a constraint edge is added between its two haplotypes and is labeled using its HLA gene types. Given such a graph, our goal is to search for an optimal assignment of each node using HLA gene types that satisfies all the edge constraints and maximizes the similarity measures between the haplotypes with the same HLA gene types. Because there are potentially multiple haplotype solutions for each pedigree, our algorithm will pick solutions that tend to maximize the overall similarities of haplotypes with the same HLA gene types. More specifically, the algorithm WSGHI mainly consists of two steps. The first step is to search for an optimal relationship between HLA gene types and their background haplotypes, and at the same time, select a unique haplotype configuration for each pedigree from its multiple solutions. The second step is to compute an optimal assignment of HLA gene types for the individuals whose HLA gene types are unknown. For simplicity, the relationship between HLA gene types and their background haplotypes is called the HapHLA relation.
In the first step, the extended DSS [14] is applied to establish the solution space of the haplotype configurations of each pedigree in a population P. In many cases, the whole solution space of P may be too large to directly enumerate. For example, though most families of the CEU population have no more than 1024 different haplotype solutions, some pedigree have more than 2^{60} solutions (see Additional file 2). Therefore, an incremental method is adopted to construct the similarity graph. We first pick those pedigrees whose solution spaces contain only a unique solution to form a haplotype subpopulation P^{′}, and a partial HapHLA relation is established based on the graph built from P^{′}. Then the remaining pedigrees are sorted in the ascending order according to the size of their solution space and are precessed sequentially in the following manner. When the solution space of a pedigree is small, every haplotype solution for the pedigree is enumerated and combined with the current similarity graph to generate a new partial HapHLA relation. The optimization criterion is evaluated and the solution with the highest value is selected as the solution of this pedigree. When the number of haplotype solutions of a pedigree is too large to enumerate, a genetic algorithm is adopted. The process continues until all pedigrees have been added. Finally, in step two, unknown HLA allele types can be inferred based on the complete similarity graph by using the HapHLA relation established in the first step. The details of the algorithm are described in the remainder of this subsection.
Similarity between two haplotypes
Let h_{ i } = (s_{ i }_{1}, s_{ i }_{2}, …, s_{ in }) and h_{ j } = (s_{ j }_{1}, s_{ j }_{2}, …, s_{ jn }) be two haplotypes of length n, where s is an SNP. If s_{ il }, s_{ jl } ≠ ‘–’ and s_{ il } ≠ s_{ jl }, then h_{ i } and h_{ j } mismatch at locus l. If s_{ il },s_{ jl } ≠ ‘’ and s_{ il } = s_{ jl }, then h_{ i } and h_{ j } match at locus l. Given a threshold T_{ mis }, [p, q] is a maximum region of nearly identical matching of h_{ i } and h_{ j } if p and q satisfy the following conditions:

(1)
1 ≤ p <q ≤ n;

(2)
h_{ i } and h_{ j } match at loci p and q;

(3)
there are no more than T_{ mis } continuous mismatches of h_{ i } and h_{ j } between p and q; and

(4)
the number of matching loci of h_{ i } and h_{ j } in the region [p, q] is maximized.
To allow for genotyping errors, T_{ mis } should an integer greater than 0. When T_{ mis } is small, our experimental results (Table 2) show that our algorithm is robust with respect to the parameter. In our experiments, the default value of T_{ mis } is set to 2. Let the set of the maximum regions of nearly identical matching of h_{ i } and h_{ j } be S_{ r }(h_{ i },h_{ j }), and the number of matching loci of h_{ i } and h_{ j } in a region [p, q] be N_{ pq }(h_{ i },h_{ j }). The similarity measure of h_{ i } and h_{ j } is defined as
Intuitively, Similarity(h_{ i }, h_{ j }) reflects the likelihood that the two haplotypes h_{ i } and h_{ j } harbor a same HLA gene type. When two haplotypes h_{ i } and h_{ j } are identical, Similarity(h_{ i }, h_{ j }) reaches the maximum value 1, and the HLA gene types reside in these two haplotypes are equal with a high probability.
Weighted similarity graph
Given a haplotype set H of a population inferred from the genotype matrix, a corresponding weighted similarity graph G_{ H } can be constructed as follows. For an individual with a pair of haplotypes {h_{ i }, h_{ j }} and a pair of HLA gene alleles {α,β}, there are two vertices i and j in G_{ H } and a constraint edge c_{ ij } between i and j. The edge c_{ ij } takes {α, β} as a constraint. If α ≠ β, c_{ ij } is a heterozygous constraint edge. If α = β, c_{ ij } is a homozygous constraint edge. Between any two vertices p and q, there is a similarity edge e_{ pq } taking a weight w_{ pq } = Simlarity(h_{ p }, h_{ q }). The set of vertices of G_{ H } is denoted as V(G_{ H }) and the set of HLA gene types in all constraints is denoted as C(G_{ H }). An example is given in Figure 7(a), where two individuals I_{1} and I_{2} are shown with their haplotypes {h_{1}, h_{2}} and {h_{3}, h_{4}}, as well as their HLAA gene types {0201, 0101} and {0201, 2401}, respectively.
Optimal labeling of a weighted similarity graph
A labeling function l : l(i) = α defined on a similarity graph G_{ H } is a mapping from V(G_{ H }) to C(G_{ H }), where i ∈ V(G_{ H }) and α ∈ C(G_{ H }), which represents the assignment of an HLA gene type α to a vertex i (i.e., its corresponding haplotype h_{ i }). For a constraint edge c_{ ij } with constraint {α, β}, if l(i) ∪ l(j) = {α, β}, the constraint edge is satisfied. When all constraints can be satisfied, the labeling function l is feasible and describes a HapHLA relation. A feasible form of G_{ H } is constructed by removing all the constraint edges (see Figure 7(b) for an example). A graph G_{ H } may have many different feasible forms that represent different HapHLA relations. To select an optimal one, a measure Con is introduced:
where denotes the set of vertices of . Given a weighted similarity graph G_{ H }, the optimal labeling problem is to find a feasible labeling l for G_{ H } such that is maximized.
A naive bruteforce search will take O(2^{N}_{h}) time for a graph G_{H} with N_{h} heterozygous constraint edges, which is impractical when N_{ h } is large. We develop a heuristic procedure HeuLabel (Additional file 3)to solve the problem. In HeuLabel, the vertices adjacent to homozygous constraint edges are labeled unambiguously in Step 2. In Step 3, we use a threshold T_{ s } to remove the similarity edges with small weights in G_{ H } and obtain a sparse graph G. We observed that when T_{ s } varies from 0.55 to 0.90, the performance of WSGHI changed only slightly in our experiments (see Table 3). In the experimental tests, the default value of T_{ s } is set to 0.65. In the following steps, vertices are labeled in such a way that most vertices in a connected component of G are labeled by the same HLA gene type. The details are illustrated in Additional file 3. For briefness, the feasible form of the weighted similarity graph G_{ H } obtained by the procedure HeuLabel is denoted by G(H).
Optimal haplotype configurations and the HapHLA relation
In this subsection, we discuss the details of the incremental step of adding pedigrees with multiple haplotype solutions. Assume that an optimal haplotype configuration H′ and a HapHLA relation l′ of a subpopulation P′ have already been determined, which are encoded in the graph . When a new pedigree P″ is added to P′, the following procedure is used to search for an optimal haplotype assignment H and an optimal HapHLA relation l for P′ ∪ P″.
For the new pedigree P″, the extended DSS program uses a disjointset D to describe the solution space of haplotype assignments of P″. It can output a particular solution after assigning a particular value to a vector S = (v_{1}, …,v_{f}) of binary variables, where f is the number of free variables of the solution space. When the total number of different haplotype solutions of P″ is not larger than 2^{T}_{c} (T_{c} = 10 in our experiments), a simple exhaustive enumeration procedure EnumAlg illustrated in Additional file 4 is used to list all solutions one by one. For each haplotype solution H″ of P″, the procedure EnumAlg adds new vertices and new edges to and obtains an updated graph for P′ ∪ P″. By applying the procedure HeuLabel to , we can obtain a new labeling l for the graph. Among all the solutions, EnumAlg selects the solution that gives the maximum as the assignment of pedigree P″ and also obtain an optimal HapHLA relation of P′ ∪ P″.
When the number of solutions is too large (i.e., f >T_{ c }), we use a genetic algorithm denoted as GeneticAlg to search for the optimal haplotype configuration and HapHLA relation. GeneticAlg directly uses the solution variable S from the DSS algorithm to express an individual code, which uniquely represents a haplotype solution of P″, denoted as H(S). The fitness function of S is Con(G(H′ ∪ H(S))), where H′ is the haplotype configuration of P′ and H″ ∪ H(S) is the haplotype configuration of the population P′ ∪ P″. The hypothesis space is H = {(v_{1},v_{2},…,v_{ f })v_{ i } ∈ {0,1}, i = 1, 2,…, f}. Both tournament selection and roulette wheel selection [15] are adopted as genetic selection operators. To produce new individuals, singlepoint mutation and singlepoint crossover [15] are adopted. Please see Additional file 5 for details. In our experiments, the parameters of GeneticAlg are set as follows: population size p_{ s } = 400, the maximum number of population generation g_{ m } = 50, crossover rate r_{c} = 0.8, and mutation rate r_{ m } = 0.2.
HLA gene type inference
Let R be the set of the individuals whose HLA gene types are known and U the set of other individuals.
Let H(I) denote the haplotype pair of an individual I, H_{1}(I) and H_{2}(I) the haplotypes in H(I), and V_{1}(I) and V_{2}(I) the vertices in G_{ H } corresponding to H_{1}(I) and H_{2}(I) respectively. Let w_{ m }(i) denote the maximum weight of the similarity edges adjacent to vertex i. For a pair of HLA gene types {g_{1}, g_{2}}, let w_{ m }(I;g_{1}, g_{2}) denote
If after the procedure HeuLabel, the set U is still not empty, we use the following procedure to label vertices corresponding to an individual I ∈ U, based on the principle that similar haplotypes harbor similar HLA allele types. First, if w_{ m }(V_{1}(I)) or w_{ m }(V_{2}(I)) is smaller than the threshold T_{ s } (set to 0.65 in the experimental tests), the HLA gene types of I cannot be inferred. Otherwise, the HLA gene types of I can be inferred as follows. Let L(V_{1}(I)) and L(V_{2}(I)) be the set of labels of the vertices adjacent to V_{1}(I) and V_{2}(I) by the similarity edges with the maximum weights respectively, i.e.,
L(V_{1}(I)) = {l(p)  w_{pV}_{1}(I) = w_{m}(V_{1}(I)) ∧ l(i) ≠ ‘–’};
L(V_{2}(I)) = {l(p)  w_{pV}_{2}(I) = w_{m}(V_{2}(I)) ∧ l(i) ≠ ‘–’}.
If L(V_{1}(I)) (or L(V_{2}(I))) contains only one element, V_{1}(I) (or V_{2}(I)) is labeled by the element and the HLA gene type of H_{1}(I) (or H_{2}(I)) is determined. If there are more than one elements in L(V_{1} (I)) or L(V_{2}(I)), HLA gene types g_{1} and g_{2} that satisfy the following condition are selected to label V_{1}(I) and V_{2}(I) respectively: g_{1} ∈ L(V_{1}(I)), g_{1} ∈ L(V_{2}(I)) and w_{ m }(I;g_{1},g_{2}) is the maximum (see the procedure HLAtype in Additional file 6 for details).
Algorithm WSGHI
The pseudocode of the algorithm WSGHI is given as follows.
INPUT: genotype matrix M_{ G } and HLA matrix M_{ H } of a population P made up of pedigrees p_{1},…,p_{ k }.
OUTPUT: inferred HLA gene types for the individuals whose HLA gene types are unknown.
STEP 1: (find out an optimal HapHLA relation for P)
Step 1.1: for i = 1,…, k do apply the extended DSS to obtain a disjointset structure D_{ i } and free variables v_{1},…,v_{f}_{i} that describe the solution space of the haplotype configuration of pedigree p_{i};
Step 1.2: sort the pedigrees in P in ascending order according to f_{ i };
Step 1.3: H′ = P′ = Ø; i = 1;
Step 1.4: while f_{ i } = 0 and i ≤ k do {P′ = P′ ∪p_{ i }; H′ = H′∪ the unique haplotype configuration of p_{ i }; i + +;}
Step 1.5: build a weight similarity graph G_{ P }_{′} for P′ using M_{ G } and M_{ H } ;
Step 1.6: apply procedure HeuLabel to G_{ P }_{′}, and obtain G(H″) using M_{ G } and M_{ H };
Step 1.7: while f_{ i } < T_{ c } and i ≤ k do
{ G(H′) =EnumAlg (G(H′),p_{ i }); i + +;}
Step 1.8: while i ≤ k do
{ G(H′) = GeneticAlg(G(H′),p_{ i }); i + +;}
Step 2: (infer the HLA gene types for the individuals whose HLA gene types are unknown)
Step 2.1: scan M_{ H } to calculate the set R of the individuals in P whose HLA gene types are known and the set U of the other individuals;
Step 2.2: HLAtype(G(H′), R,U);
Step 2.3: the HLA gene types of an individual I ∈ U with the haplotype configuration (h_{ p }, h_{ q }) is ( l ( p), l (q));
The running time of WSGHI is mainly determined by the maximum size of the haplotype solution spaces of the pedigrees in the population P. When taking genotype data from the 200kb region as the input and running on a PC with Linux CentOS 5, 1GB memory and 2.4GHz CPU, WSGHI takes 20 minutes to 1 hour to finish the test results for each HLA gene. Most of the time is used in the first step to compute an optimal HapHLA relation, and less than 0.01 seconds are used in Step 2. To check whether the performance is robust and stable, we tested WSGHI five times using the leaveoneout strategy with the default parameter values. For each HLA gene, the tests exhibited very similar accuracy and running time (see Additional file 7 for the details).
Authors contributions
MX designed the algorithm, performed the computational experiments. MX and JL drafted the manuscript. TJ supervised the project and polished the manuscript. All authors read and approved the manuscript.
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Acknowledgements
This research was supported in part by the National Institutes of Health/National Library of Medicine grant 2R01LM008991 and the National Natural Science Foundation of China grant 61070145. The authors would like to thank X. Li for helpful discussions on the DSS program.
This article has been published as part of BMC Bioinformatics Volume 11 Supplement 11, 2010: Proceedings of the 21st International Conference on Genome Informatics (GIW2010). The full contents of the supplement are available online at http://www.biomedcentral.com/14712105/11?issue=S11.
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Keywords
 Human Leukocyte Antigen
 Single Nucleotide Polymorphism Genotype
 Human Leukocyte Antigen Allele
 Single Nucleotide Polymorphism Locus
 Human Leukocyte Antigen Type