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Issue No.06 - Nov.-Dec. (2013 vol.10)
pp: 1391-1402
Daniele Catanzaro , Comput. Sci. Dept., Univ. Libre de Bruxelles (ULB), Brussels, Belgium
Martine Labbe , Comput. Sci. Dept., Univ. Libre de Bruxelles (ULB), Brussels, Belgium
Bjarni V. Halldorsson , Dept. of Biomed. Eng., Reykjavk Univ., Reykjavik, Iceland
ABSTRACT
A loss of heterozygosity (LOH) event occurs when, by the laws of Mendelian inheritance, an individual should be heterozygote at a given site but, due to a deletion polymorphism, is not. Deletions play an important role in human disease and their detection could provide fundamental insights for the development of new diagnostics and treatments. In this paper, we investigate the parsimonious loss of heterozygosity problem (PLOHP), i.e., the problem of partitioning suspected polymorphisms from a set of individuals into a minimum number of deletion areas. Specifically, we generalize Halldórsson et al.'s work by providing a more general formulation of the PLOHP and by showing how one can incorporate different recombination rates and prior knowledge about the locations of deletions. Moreover, we show that the PLOHP can be formulated as a specific version of the clique partition problem in a particular class of graphs called undirected catch-point interval graphs and we prove its general NP-hardness. Finally, we provide a state-of-the-art integer programming (IP) formulation and strengthening valid inequalities to exactly solve real instances of the PLOHP containing up to 9,000 individuals and 3,000 SNPs. Our results give perspectives on the mathematics of the PLOHP and suggest new directions on the development of future efficient exact solution approaches.
INDEX TERMS
Bioinformatics, Genomics, DNA, Human factors, Computational biology, Linear programming,single nucleotide polymorphism, Clique partitioning, submodular functions, polymatroid rank functions, undirected catch-point interval graph, combinatorial optimization, mixed integer programming, computational biology, loss of heterozygosity, genome-wide association studies
CITATION
Daniele Catanzaro, Martine Labbe, Bjarni V. Halldorsson, "An Integer Programming Formulation of the Parsimonious Loss of Heterozygosity Problem", IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.10, no. 6, pp. 1391-1402, Nov.-Dec. 2013, doi:10.1109/TCBB.2012.138
REFERENCES
[1] International HapMap Consortium, "A Second Generation Human Haplotype Map of over 3.1 Million SNPs," Nature, vol. 449, no. 18, pp. 851-861, 2007.
[2] W.H. Li and L.A. Sadler, "Low Nucleotide Diversity in Man," Genetics, vol. 129, pp. 513-523, 1991.
[3] D.G. Wang, J.-B. Fan, C.-J. Siao, A. Berno, P. Young, R. Sapolsky, G. Ghandour, N. Perkins, E. Winchester, J. Spencer, L. Kruglyak, L. Stein, L. Hsie, T. Topaloglou, E. Hubbell, E. Robinson, M. Mittmann, M.S. Morris, N. Shen, D. Kilburn, J. Rioux, C. Nusbaum, S. Rozen, T.J. Hudson, R. Lipshutz, M. Chee, and E.S. Lander, "Large-Scale Identification, Mapping, and Genotyping of Single-Nucleotide Polymorphisms in the Human Genome," Science, vol. 280, no. 5366, pp. 1077-1082, 1998.
[4] M. Cargill et al., "Characterization of Single-Nucleotide Polymorphisms in Coding Regions of Human Genes," Nature Genetics, vol. 22, pp. 231-238, 1999.
[5] M. Halushka, J.B. Fan, K. Bentley, L. Hsie, N. Shen, A. Weder, R. Cooper, R. Lipshutz, and A. Chakravarti, "Patterns of Single Nucleotide Polymorphisms in Candidate Genes of Blood Pressure Homeostasis," Nature Genetics, vol. 22, pp. 239-247, 1999.
[6] J. Terwilliger and K. Weiss, "Linkage Disquilibrium Mapping of Complex Disease: Fantasy and Reality?" Current Opinions in Biotechnology, vol. 9, pp. 579-594, 1998.
[7] M. Hoehe, K. Kopke, B. Wendel, K. Rohde, C. Flachmeier, K. Kidd, W. Berrettini, and G. Church, "Sequence Variability and Candidate Gene Analysis in Complex Disease: Association of $\mu$ Opioid Receptor Gene Variation with Substance Dependence," Human Molecular Genetics, vol. 9, pp. 2895-2908, 2000.
[8] D. Catanzaro, M. Andrien, M. Labbé, and M. Toungouz-Nevessignsky, "Computer-Aided Human Leukocyte Antigen Association Studies: A Case Study for Psoriasis and Severe Alopecia Areata," Human Immunology, vol. 71, no. 8, pp. 783-788, 2010.
[9] H. Stefansson, D. Rujescu, S. Cichon, O.P.H. Pietiläinen, A. Ingason, S. Steinberg, R. Fossdal, E. Sigurdsson, T. Sigmundsson, J.E. Buizer-Voskamp, T. Hansen, K.D. Jakobsen, P. Muglia, C. Francks, P.M. Matthews, A. Gylfason, B.V. Halldorsson, D. Gudbjartsson, T.E. Thorgeirsson, A. Sigurdsson, A. Jonasdottir, A. Jonasdottir, A. Bjornsson, S. Mattiasdottir, T. Blondal, M. Haraldsson, B.B. Magnusdottir, I. Giegling, H.J. Moeller, A. Hartmann, K.V. Shianna, D. Ge, A.C. Need, C. Crombie, G. Fraser, N. Walker, J. Lonnqvist, J. Suvisaari, A. Tuulio-Henriksson, T. Paunio, T. Toulopoulou, E. Bramon, M.D. Forti, R. Murray, M. Ruggeri, E. Vassos, S. Tosato, M. Walshe, T. Li, C. Vasilescu, T.W. Moehleisen, A.G. Wang, H. Ullum, S. Djurovic, I. Melle, J. Olesen, L.A. Kiemeney, B. Franke, C. Sabatti, N.B. Freimer, J.R. Gulcher, U. Thorsteinsdottir, A. Kong, O.A. Andreassen, R.A. Ophoff, A. Georgi, M. Rietschel, T. Werge, H. Petursson, D.B. Goldstein, M.M. Nöthen, L. Peltonen, D.A. Collier, D.S. Clair, and K. Stefansson, "Large Recurrent Microdeletions Associated with Schizophrenia," Nature, vol. 455, pp. 232-236, 2008.
[10] B. Halldórsson, D. Aguiar, R. Tarpine, and S. Istrail, "The Clark Phase-Able Sample Size Problem: Long-Range Phasing and Loss of Heterozygosity in GWAS," J. Computational Biology, vol. 18, no. 3, pp. 323-333, 2011.
[11] M. Goedert and M.G. Spillantini, "A Century of Alzheimer's Disease," Science, vol. 314, no. 5800, pp. 777-781, 2006.
[12] D.A. Elder, K. Kaiser-Rogers, A.S. Aylsworth, and A.S. Calikoglu, "Type I Diabetes Mellitus in a Patient with Chromosome 22q11.2 Deletion Syndrome," Am. J. Medical Genetics, vol. 101, no. 1, pp. 17-19, 2001.
[13] M. Shinawi, T. Sahoo, B. Maranda, S.A. Skinner, C. Skinner, C. Chinault, R. Zascavage, S.U. Peters, A. Patel, R.E. Stevenson, and A.L. Beaudet, "11p14.1 Microdeletions Associated with ADHD, Autism, Developmental Delay, and Obesity," Am. J. Medical Genetics, vol. 155, no. 6, pp. 1272-1280, 2011.
[14] K. Momma, R. Matsuoka, and A. Takao, "Aortic Arch Anomalies Associated with Chromosome 22q11 Deletion," Pediatric Cardiology, vol. 20, no. 2, pp. 97-102, 1999.
[15] C.M. Ogilvie, J.W. Ahn, K. Mann, R.G. Roberts, and F. Flinter, "A Novel Deletion in Proximal 22q Associated with Cardiac Septal Defects and Microcephaly: A Case Report," Molecular Cytogenetics, vol. 2, no. 9, pp. 1-5, 2009.
[16] S. Puvabanditsin, M.S. Nagar, M. Joshi, G. Lambert, E. Garrow, and E. Brandsma, "Microdeletion of 16p11.2 Associated with Endocardial Fibroelastosis," Am. J. Medical Genetics, vol. 152, no. 9, pp. 2382-2386, 2010.
[17] J. McClellan and M.C. King, "Genetic Heterogeneity in Human Disease," Cell, vol. 141, pp. 210-217, 2010.
[18] D. Catanzaro and M. Labbé, "The Pure Parsimony Haplotyping Problem: Overview and Computational Advances," Int'l Trans. Operations Research, vol. 16, no. 5, pp. 561-584, 2009.
[19] D.F. Conrad, D. Pinto, R. Redon, L. Feuk, O. Gokcumen, Y. Zhang, J. Aerts, T.D. Andrews, C. Barnes, P. Campbell, T. Fitzgerald, M. Hu, C.H. Ihm, K. Kristiansson, D.G. MacArthur, J.R. MacDonald, I. Onyiah, A.W.C. Pang, S. Robson, K. Stirrups, A. Valsesia, K. Walter, J. Wei, T.W.T.C.C. Consortium", C. Tyler-Smith, N.P. Carter, C. Lee, S.W. Scherer, and M.E. Hurles, "Origins and Functional Impact of Copy Number Variation in the Human Genome," Nature, pp. 704-712, 2009.
[20] J.A. Corbel, A. Abyzov, X. Mu, N. Carreiro, P. Cayting, Z. Zhang, M. Snyder, and M. Gerstein, "PEMer: A Computational Framework with Simulation-Based Error Models: for Inferring Genome Structaral Variants from Massive Paired-End Sequencing Data," Genome Biology, vol. 38, no. 10, p. R23, 2009.
[21] K. Chen, J. Wallis, M. McLellan, D. Larson, J. Kallick, C. Pohl, S. McGrath, M. Wendl, Q. Zhang, D. Locke, X. Shi, R. Fulton, T. Ley, R. Wilson, L. Ding, and E. Mardis, "BreakDancer: An Algorithm for High Resolution Mapping of Genomic Structural Variation," Nature Methods, vol. 6, pp. 677-681, 2009.
[22] D.F. Conrad, T.D. Andrews, N.P. Carter, M.E. Hurles, and J.K. Pritchard, "A High-Resolution Survey of Deletion Polymorphism in the Human Genome," Nature Genetics, vol. 38, pp. 75-81, 2006.
[23] E. Corona, B. Raphael, and E. Eskin, "Identification of Deletion Polymorphisms from Haplotypes," Proc. 11th Ann. Int'l Conf. Research in Computational Molecular Biology (RECOMB '07), pp. 354-365, 2007.
[24] S.A. McCarroll, F.G. Kuruvilla, J.M. Korn, S. Cawley, J. Nemesh, A. Wysoker, M.H. Shapero, P.I.W. de Bakker, J.B. Maller, A. Kirby, A.L. Elliott, M. Parkin, E. Hubbell, T. Webster, R. Mei, J. Veitch, P.J. Collins, R. Handsaker, S. Lincoln, M. Nizzari, J. Blume, K.W. Jones, R. Rava, M.J. Daly, S.B. Gabriel, and D. Altshuler, "Integrated Detection and Population-Genetic Analysis of SNPS and Copy Number Variation," Nature Genetics, vol. 40, pp. 1166-1174, 2008.
[25] D. Catanzaro, M. Labbé, and L. Porretta, "A Class Representative Model for Pure Parsimony Haplotyping Under Uncertain Data," PLoS one, vol. 6, no. 3, p. e17937, 2011.
[26] T.I.H. Consortium, "The International Hapmap Project," Nature, vol. 426, no. 18, pp. 789-796, 2003.
[27] V.A. Albert, Parsimony, Phylogeny, and Genomics. Oxford Univ. Press, 2005.
[28] D. Catanzaro, "The Minimum Evolution Problem: Overview and Classification," Networks, vol. 53, no. 2, pp. 112-125, 2009.
[29] D. Catanzaro, "Estimating Phylogenies from Molecular Data," Mathematical Approaches to Polymer Sequence Analysis and Related Problems, R. Bruni, ed. Springer, 2011.
[30] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, 2003.
[31] P.C. Fishburn, Interval Orders and Interval Graphs: Study of Partially Ordered Sets. John Wiley and Sons Inc., 1985.
[32] E. Prisner, "A Characterization of Interval Catch Digraphs," Discrete Math., vol. 73, pp. 285-289, 1989.
[33] L. Lovász, "Perfect Graphs," Selected Topics in Graph Theory, L. W. Beineke and R. J. Wilson, eds., vol. 2, pp. 55-87, Academic Press, 1983.
[34] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, "The Strong Perfect Graph Theorem," Annals of Math., vol. 164, no. 1, pp. 51-229, 2006.
[35] A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency. Springer, 2003.
[36] G. Dion, V. Jost, and M. Queyranne, "Clique Partitioning of Interval Graphs with Submodular Costs on the Cliques," RAIRO Operations Research, vol. 41, pp. 275-287, 2007.
[37] M. Grötschel and Y. Wakabayashi, "Facets of the Clique Partitioning Polytope," Math. Programming, vol. 47, pp. 367-387, 1990.
[38] H.J. Bandelt, M. Oosten, J.H.G.C. Rutten, and F.C.R. Spieksma, "Lifting Theorems and Facet Characterization for a Class of Clique Partitioning Inequalities," Operations Research Letters, vol. 24, no. 5, pp. 235-243, 1999.
[39] M. Oosten, J.H.G.C. Rutten, and F.C.R. Spieksma, "The Clique Partitioning Problem: Facets and Patching Facets," Networks, vol. 38, no. 4, pp. 209-226, 2001.
[40] M. Matsumoto and T. Nishimura, "Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator," ACM Trans. Modeling and Computer Simulation, vol. 8, no. 1, pp. 3-30, 1998.
[41] A. Kong, G. Thorleifsson, D.F. Gudbjartsson, G. Masson, A. Sigurdsson, A. Jonasdottir, G.B. Walters, A. Jonasdottir, A. Gylfason, K.T. Kristinsson, S.A. Gudjonsson, M.L. Frigge, A. Helgason, and U.T.K. Stefansson, "Fine-Scale Recombination Rate Differences between Sexes, Populations and Individuals," Nature, vol. 467, pp. 1099-1103, 2010.
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