Additional file 1. The appendix contains the proofs omitted from the main text, the ILP formulation for the Copy-Number Triplet Problem, the complete ILP formulation for the Copy-Number Tree Problem, and additional details about the results

Benjamin Raphael (4030346)

Meirav Zehavi (4030337)

Mohammed El-Kebir (4030343)

Roded Sharan (23003)

Ron Shamir (11621)

Ron Zeira (4030340)

Simone Zaccaria (4030334)

Algorithms for Molecular Biology

Mohammed El-Kebir

Benjamin J. Raphael

Ron Shamir

Roded Sharan

Simone Zaccaria

Meirav Zehavi

Ron Zeira

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Complexity and algorithms for copy-number evolution problems

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Abstract Background 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 is 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. Results We model the evolution of a tumor by segmental deletions and amplifications, and gauge distance from profile $$\mathbf {a}$$ a to $$\mathbf {b}$$ b by the minimum number of events needed to transform $$\mathbf {a}$$ a into $$\mathbf {b}$$ 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. Conclusions 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 that scales to practical problem instance sizes. We assess the efficiency and quality of our algorithms on simulated instances. Availability https://github.com/raphael-group/CNT-IL

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BACKGROUND: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 is 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. RESULTS:We model the evolution of a tumor by segmental deletions and amplifications, and gauge distance from profile [Formula: see text] to [Formula: see text] by the minimum number of events needed to transform [Formula: see text] into [Formula: see text]. 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. CONCLUSIONS: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 that scales to practical problem instance sizes. We assess the efficiency and quality of our algorithms on simulated instances. AVAILABILITY:https://github.com/raphael-group/CNT-ILP

El-Kebir, Mohammed

Raphael, Benjamin J

Shamir, Ron

Sharan, Roded

Zaccaria, Simone

Zehavi, Meirav

Zeira, Ron

Princeton University Open Access Repository

Complexity and algorithms for copy-number evolution problems.

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MOESM1 of Complexity and algorithms for copy-number evolution problems

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