This Article 
 Bibliographic References 
 Add to: 
Computational Problems in Perfect Phylogeny Haplotyping: Typing without Calling the Allele
January-March 2008 (vol. 5 no. 1)
pp. 101-109
A haplotype is an m-long binary vector. The xor-genotype of two haplotypes is the m-vector of their coordinate-wise xor. We study the following problem: Given a set of xor-genotypes, reconstruct their haplotypes so that the set of resulting haplotypes can be mapped onto a perfect phylogeny tree. The question is motivated by studying population evolution in human genetics, and is a variant of the perfect phylogeny haplotyping problem that has received intensive attention recently. Unlike the latter problem, in which the input is "full" genotypes, here we assume less informative input, and so may be more economical to obtain experimentally.Building on ideas of Gusfield, we show how to solve the problem in polynomial time, by a reduction to the graph realization problem. The actual haplotypes are not uniquely determined by that tree they map onto, and the tree itself may or may not be unique. We show that tree uniqueness implies uniquely determined haplotypes, up to inherent degrees of freedom, and give a sufficient condition for the uniqueness. To actually determine the haplotypes given the tree, additional information is necessary. We show that two or three full genotypes suffice to reconstruct all the haplotypes, and present a linear algorithm for identifying those genotypes.

[1] E.M. Arkin and R. Hassin, “Multiple-Choice Minimum Diameter Problems,” unpublished manuscript, 1992.
[2] V. Bafna, D. Gusfield, G. Lancia, and S. Yooseph, “Haplotyping as Perfect Phylogeny: A Direct Approach,” Technical Report UC Davis CSE-2002-21, 2002.
[3] T. Barzuza, J.S. Beckmann, R. Shamir, and I. Pe'er, “Computational Problems in Perfect Phylogeny Haplotyping: XOR-Genotypes and Tag SNPs,” Proc. Ann. Symp. Combinatorial Pattern Matching (CPM '04), pp. 14-31, 2004.
[4] T. Barzuza, J.S. Beckmann, R. Shamir, and I. Pe'er, “Typing without Calling the Allele: A Strategy for Inferring SNP Haplotypes,” The European J. Human Genetics, vol. 13, no. 8, pp.898-901, 2005.
[5] R.E. Bixby and D.K. Wagner, “An Almost Linear-Time Algorithm for Graph Realization,” Math. Operations Research, vol. 13, pp. 99-123, 1988.
[6] A.G. Clark, “Inference of Haplotypes from PCR-Amplified Samples of Diploid Populations,” Molecular Biology and Evolution, vol. 7, no. 2, pp. 111-122, 1990.
[7] M.J. Daly, J.D. Rioux, S.F. Schaffner, T.J. Hudson, and E.S. Lander, “High Resolution Haplotype Structure in the Human Genome,” Nature Genetics, vol. 29, no. 2, pp. 229-232, 2001.
[8] Z. Ding, V. Filkov, and D. Gusfield, “A Linear-Time Algorithm for Perfect Phylogeny Haplotyping,” Proc. Ninth Ann. Int'l Conf. Research in Computational Molecular Biology (RECOMB '05), 2005.
[9] G. Eastbrook, C. Johnson, and F. McMorris, “An Idealized Concept of the True Cladistic Character,” Math. Biosciences, vol. 23, pp. 263-272, 1975.
[10] E. Eskin, E. Halperin, and R.M. Karp, “Efficient Reconstruction of Haplotype Structure via Perfect Phylogeny,” J. Bioinformatics and Computational Biology, vol. 1, no. 1, Apr. 2003.
[11] L. Excoffier and M. Slatkin, “Maximum-Likelihood Estimation of Molecular Haplotype Frequencies in a Diploid Population,” Molecular Biology and Evolution, vol. 12, no. 5, pp. 921-927, 1995.
[12] S.B. Gabriel et al., “The Structure of Haplotype Blocks in Human Genome,” Science, vol. 296, pp. 2225-2229, 2002.
[13] F. Gavril and R. Tamari, “An Algorithm for Constructing Edge-Trees from Hypergraphs,” Networks, vol. 13, pp. 377-388, 1983.
[14] D. Gusfield, Algorithms on Strings, Trees, and Sequences—Computer Science and Computational Biology. Cambridge Univ. Press, 1997.
[15] D. Gusfield, “Haplotyping as Perfect Phylogeny: Conceptual Framework and Efficient Solutions,” Proc. Sixth Ann. Int'l Conf. Research in Computational Molecular Biology (RECOMB '02), pp. 166-175, 2002.
[16] J.E. Hopcroft and R.E. Tarjan, “Dividing a Graph into y Triconnected Components,” SIAM J. Computing, vol. 2, pp. 135-157, 1973.
[17] A.J. Jeffreys, L. Kauppi, and R. Neumann, “Intensely Punctate Meiotic Recombination in the Class II Region of the Major Histocompatibility Complex,” Nature Genetics, vol. 29, pp. 217-222, 2001.
[18] P.Y. Kwok, “Genetic Assoc. by Whole-Genome Analysis,” Science, vol. 294, no. 5547, pp. 1669-1670, 2001.
[19] S. Myers, L. Bottolo, C. Freeman, G. McVean, and P. Donnelly, “A Fine-Scale Map of Recombination Rates and Hotspots Across the Human Genome,” Science, vol. 310, no. 5746, pp. 321-324, 2005.
[20] M.W. Nachman and S.L. Crowell, “Estimate of the Mutation Rate Per Nucleotide in Humans,” Genetics, vol. 156, pp. 297-304, 2000.
[21] N. Patil et al., “Blocks of Limited Haplotype Diversity Revealed by High Resolution Scanning of Human Chromosome 21,” Science, vol. 294, no. 5547, pp. 1719-1723, 2001.
[22] I. Pe'er and J.S. Beckmann, “Resolution of Haplotypes and Haplotype Frequencies from SNP Genotypes of Pooled Samples,” Proc. Seventh Ann. Int'l Conf. Research in Computational Molecular Biology (RECOMB '03), pp. 237-246, 2003.
[23] R. Sachidanandam et al., “A Map of Human Genome Sequence Variation Containing 1.42 Million Single Nucleotide Polymorphisms,” Nature, vol. 409, no. 6822, pp. 928-933, 2001.
[24] W.T. Tutte, “An Algorithm for Determining whether a Given Binary Matroid Is Graphic,” Proc. Am. Math. Soc., vol. 11, pp. 905-917, 1960.
[25] W.T. Whitney, “2-Isomorphic Graphs,” Am. J. Math., vol. 55, pp.245-254, 1933.
[26] W. Xiao and P.J. Oefner, “Denaturing High-Performance Liquid Chromatography: A Review,” Human Mutation, vol. 17, no. 6, pp.439-474, 2001.

Index Terms:
XOR-genotypes, Haplotypes, Perfect Phylogeny, Graph Realization
Tamar Barzuza, Jacques S. Beckmann, Ron Shamir, Itsik Pe'er, "Computational Problems in Perfect Phylogeny Haplotyping: Typing without Calling the Allele," IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 5, no. 1, pp. 101-109, Jan.-March 2008, doi:10.1109/TCBB.2007.1063
Usage of this product signifies your acceptance of the Terms of Use.