Combinatorial Algorithms for Haplotype Inference Pure Parsimony Dan Gusfield

SNP Data A SNP is a Single Nucleotide Polymorphism - a site in the genome where two different nucleotides appear with sufficient frequency in the population (say each with 5% frequency or more). SNP maps have been compiled with a density of about 1 site per SNP data is what is mostly collected in populations - it is much cheaper to collect than full sequence data, and focuses on variation in the population, which is what is of interest.

Genotypes and Haplotypes Each individual has two “copies” of each chromosome. At each site, each chromosome has one of two alleles (states) denoted by 0 and 1 (motivated by SNPs) Two haplotypes per individual Genotype for the individual Merge the haplotypes

Haplotype Map Project: HAPMAP NIH lead project ($100M) to find common haplotypes in the Human population. Used to try to associate genetic-influenced diseases with specific haplotypes, to either find causal haplotypes, or to find the region near causal mutations. Haplotyping individuals is expensive.

Haplotyping Problem Biological Problem: For disease association studies, haplotype data is more valuable than genotype data, but haplotype data is hard to collect. Genotype data is easy to collect. Computational Problem: Given a set of n genotypes, determine the original set of n haplotype pairs that generated the n genotypes. This is hopeless without a genetic model.

The Pure Parsimony Objective For a set of genotypes, find a Smallest set H of haplotypes, such that each genotype can be explained by a pair of haplotypes in H. For each genotype G in the input, assign a pair of haplotypes in H to explain G. The Pure Parsimony Objective reflects simple genetic models of how haplotypes evolve in a population.

Pure Parsimony Explain the N genotypes using the fewest number of distinct haplotypes. Why? Empirically few haplotypes are seen in populations; Coalescent theory; analogy to other parsimony criteria; common attempts to interpret Clark’s haplotype inferral method as a parsimony method; many methods are to a large extent hidden parsimony methods - PHASE

Example of Parsimony distinct haplotypes set S has size 3

Pure Parsimony is NP-hard Earl Hubbel (Affymetrix) showed that Pure Parsimony is NP-hard. However, for a range of parameters of current interest (50 sites and 50 genotypes) a True Parsimony solution can be computed efficiently, using Integer Linear Programming, and two speed-up tricks. For larger parameters (100 sites and 50 genotypes) A near-parsimony solution can be found efficiently.

Why I did this work I wanted to answer two questions: First, can a pure parsimony solution be computed efficiently for a range of problem sizes of current interest in biology? Second, how accurate is the pure parsimony solution, compared to the correct solution (in simulations and in the available real data), and compared to solutions given by other existing computational methods such as PHASE. Accuracy is measured by the number of genotypes whose originating pair of haplotypes are returned in the solution.

The Conceptual Integer Programming Formulation For each genotype (individual) j, create one integer programming variable Yij for each pair of haplotypes whose merge creates genotype j. If j has k 2’s, then This creates 2^(k-1) Y variables. Create one integer programming variable Xq for Each distinct haplotype q that appears in one of the pairs for a Y variable.

Conceptual IP For each genotype, create an equality that says that exactly one of its Y variables must be set to 1. For each variable Yij, whose two haplotypes are given variables Xq and Xq’, include an inequality that says that if variable Yij is set to 1, then both variables Xq and Xq’ must be set to 1. Then the objective function is to Minimize the sum of the X variables.

Example Creates a Y variable Y1 for pair X X2 and a Y variable Y2 for pair X X4 Y1 + Y2 = 1 Y1 - X1 <= 0 Y1 - X2 <= 0 Y2 - X3 <= 0 Y2 - X4 <= 0 Include the following (in)equalities into the IP The objective function will include the subexpression X1 + X2 + X3 + X4 But any X variable is included exactly once no matter how many Y variables it is associated with.

Efficiency Tricks Ignore any Y variable and its two X variables if those X variables are associated with no other Y variable. The Resulting IP is much smaller, and can be used to find the optimal to the conceptual IP. Also, we need not enumerate all X pairs for a given genotype, but can efficiently recognize the pairs we need.

Avoiding Enumeration of unneeded haplotypes For each pair of genotypes, G1, G2 it is easy to find all the haplotypes that appear in an explanation for G1 and in an explanation for G2. Example: V 0 2 V and then generate all combinations of 0,1’s over the V sites. So the time is O(m x # haps in both explanation sets)

The APOE Data: A case where the haplotypes were molecularly determined There are 17 distinct haplotypes in the real data. The IP finds a True Parsimony Solution with 15 distinct haplotypes. PHASE and HAPLOTYPER each use 15 haplotypes also. Over 10,000 executions of Clarks method, the fewest haplotypes it used in any solutions was 20. This data has 9 sites, and 47 genotypes, each with at least two ambiguous sites.

Recombination Recombination is a process whereby a prefix of one sequence is concatenated to a suffix of another sequence to create a third sequence. Ex. ABCDEFG and TUVWXYZ could recombine to create ABCWXYZ DNA sequences evolve by mutations of different types, but also by recombinations.

Recombination Helps Efficiency As the level of recombination increases, the efficiency of the IP increases, because the variable elimination trick becomes more effective, reducing the size of the IP. The reason is that recombination makes the underlying haplotypes in the population more varied, and also increases the number of haplotypes in the population. Hence, each haplotype is less likely to be part of a potential explanation of any given genotype.

Recombination Hurts Accuracy For almost the same reason as recombination helps efficiency, it hurts accuracy. As recombination increases, the number of haplotypes that can be part of the explanation of more than one genotype in the data decreases. That helps efficiency, but it reduces the level of structure and dependency among the potential explanations, and hence the parsimony criteria is less effective.

How Fast? How Good? Depends on the level of recombination in the underlying data. Pure Parsimony can be computed in seconds to minutes for most cases with 50 genotypes and up to 60 sites, faster as the level of recombination increases. As the level of recombination increases, the accuracy of the Pure Parsimony Solution falls, but remains within 5% of the quality of PHASE (for comparison).

Accuracy For 10 sites and moderate recombination, the Pure Parsimony solutions have the same accuracy as PHASE and HAPLOTYPER solutions. As the number of sites and the level of recombination increases, PHASE and HAPLOTYPER tend to be more accurate than the Pure Parsimony solution, but the gap is moderate.

A Hybrid Approach for Large Data Sets We are interested in handling 100 genotypes and 150 sites. This is too large for the IP approach, but we can use a hybrid approach based on Clarks Method and an IP version of it.

Generic Clark Method Given a known haplotype H (original homozygote or single-site heterozygote, or previously inferred), and an unresolved genotype G, if G can be explained by H and another vector H’, then call H’ a known haplotype, available for additional inferrals. example: H G G is “resolved” by H and H’ H’ Clark (1990) Randomize choices, and do the computations many times to find an execution (run) that explains the most genotypes. In a single run, repeat the basic step until stuck - resolve as many genotypes as possible in the data. Basic Step:

Many variations of Clark Variations based on which parts are randomized. We closely examine eight variations on a real data set. Variation 1 randomizes every decision - probably more than Clark originally intended. Truth in advertising - we implemented our own Clark versions - did not actually use Clark’s software.

Clark/Parsimony Hybrid For low recombination, but many sites (say > 100), the pure IP approach blows up. But by connecting to Clark’s approach, using the Digraph view of Clark (Gusfield ISMB 2000, JCB 2001), the size of the IP is dramatically smaller, and can run on a very large number of sites. On datasets where the pure IP can be run, the hybrid method is only a bit inferior to the pure IP method.

Clark/Parsimony Hybrid Find an execution of Clark’s method that a)maximizes the number of genotypes resolved b) minimizes the number of distinct haplotypes used We can do this by mixing the Digraph View of Clark’s method (Gusfield 2001) with the parsimony criteria, and truly find an execution of Clark’s method that minimizes the number of distinct haplotypes used. On datasets where we can compute True Parsimony, this hybrid does only a bit worse than True Parsimony. For low recombination, large (>60) sites

Other uses of IP On datasets where we know the solution, find the best that a Clark method can ever do. IP can find the best possible execution. On the APOE data, Clark’s method can get all get 47 correct! In fact in a huge number of ways. (But the best we found by actually running Clark’s method was 42 correct). This kind of test is not possible for Statistical methods.