Copy-Number Evolution Problems: Complexity and Algorithms

  • Mohammed El-Kebir
  • Benjamin J. Raphael
  • Ron Shamir
  • Roded Sharan
  • Simone Zaccaria
  • Meirav Zehavi
  • Ron Zeira
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9838)


Cancer is an evolutionary process characterized by the accumulation of somatic mutations in a population of cells that form a tumor. One frequent type of mutations are copy number aberrations, which alter the number of copies of genomic regions. The number of copies of each position along a chromosome constitutes the chromosome’s copy-number profile. Understanding how such profiles evolve in cancer can assist in both diagnosis and prognosis. We model the evolution of a tumor by segmental deletions and amplifications, and gauge distance from profile \(\mathbf {a}\) to \(\mathbf {b}\) by the minimum number of events needed to transform \(\mathbf {a}\) into \(\mathbf {b}\). Given two profiles, our first problem aims to find a parental profile that minimizes the sum of distances to its children. Given k profiles, the second, more general problem, seeks a phylogenetic tree, whose k leaves are labeled by the k given profiles and whose internal vertices are labeled by ancestral profiles such that the sum of edge distances is minimum. For the former problem we give a pseudo-polynomial dynamic programming algorithm that is linear in the profile length, and an integer linear program formulation. For the latter problem we show it is NP-hard and give an integer linear program formulation. We assess the efficiency and quality of our algorithms on simulated instances.


Span Tree Integer Linear Program Copy Number Aberration Edge Cost Integer Linear Program Formulation 



B.J.R. is supported by a National Science Foundation CAREER Award CCF-1053753, NIH RO1HG005690 a Career Award at the Scientific Interface from the Burroughs Wellcome Fund, and an Alfred P Sloan Research Fellowship. R. Shamir is supported by the Israeli Science Foundation (grant 317/13) and the Dotan Hemato-Oncology Research Center at Tel Aviv University. R.Z. is supported by fellowships from the Edmond J. Safra Center for Bioinformatics at Tel Aviv University and from the Israeli Center of Research Excellence (I-CORE) Gene Regulation in Complex Human Disease (Center No. 41/11). M.Z. is supported by a fellowship from the I-CORE in Algorithms and the Simons Institute for the Theory of Computing in Berkeley and by the Postdoctoral Fellowship for Women of Israel’s Council for Higher Education. Part of this work was done while M.E-K., B.J.R., R. Shamir, R. Sharan and M.Z. were visiting the Simons Institute for the Theory of Computing.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Mohammed El-Kebir
    • 1
  • Benjamin J. Raphael
    • 1
  • Ron Shamir
    • 2
  • Roded Sharan
    • 2
  • Simone Zaccaria
    • 1
    • 3
  • Meirav Zehavi
    • 2
  • Ron Zeira
    • 2
  1. 1.Department of Computer Science, Center for Computational Molecular BiologyBrown UniversityProvidenceUSA
  2. 2.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  3. 3.Dipartimento di Informatica Sistemistica E Comunicazione (DISCo)Univ. Degli Studi di Milano-BicoccaMilanItaly

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