Sorting Cancer Karyotypes Sorting Cancer Karyotypes by Elementary Operationsby Elementary Operations

Michal Ozery-Flato and Ron ShamirSchool of Computer Science,

Tel Aviv University

Outline

• Introduction

• Modeling the evolution of cancer karyotypes

• The karyotype sorting problem

• Combinatorial Analysis

• Results

2

3

Introduction

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http://www.ncbi.nlm.nih.gov/sky/skyweb.cgi

Normal female

karyotype

5

The "Philadelphia chromosome"

6

http://www.ncbi.nlm.nih.gov/sky/skyweb.cgi

Breast cancer

karytype (MCF-7)

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Chromosomal Instability

• A phenotype of most cancer cells.– Losses or gains of chromosomes result from

errors during mitosis

– Chromosome rearrangements are associated with "double strand breaks"

multi-polar mitoses

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Double Strand Breaks

• Constitute the most dangerous type of DNA damage– A successful repair ligates two matching

broken ends– Mis-repair can result in rearrangements (e.g.

translocations) or deletions

M.C. Escher, 1953

Double strand break

The Challenge

Analyze the evolution of aberration events in cancer karyotypes

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The Mitelman Database of Chromosome Aberrations in Cancer• Over 55,000 cancer karyotypes, culled from

over 8000 scientific publications

• Can be parsed automatically (CyDAS parser www.cydas.org)

• The largest current data resource on cancer genomes' organization

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Modeling the Evolution of Cancer Karyotypes

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The Normal Karyotype • Band = basic unit observable in karyotype. A

unique region in the genome, identified by integer

• Normal Chromosome = interval of bands– Two normal chromosomes are either disjoint or

equivalent

• Normal karyotype = a collection of normal chromosomes– Usually contains two copies of each chromosome

(with the possible exception of the sex chromosomes)

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The Cancer karyotype

• Fragment = a sub-interval (>1 bands)

of a normal chromosome

• Chromosome = – One fragment, or a concatenation of several

fragments – Orientation-less: [1,4]::[37,40] [40,37]::[4,1]

• Cancer karyotype = a collection of chromosomes

concatenation (breakpoint)

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Elementary Operations

Breakage

Fusion

du

plicatio

n

deletion

These operations can generate all known chromosomal aberrations!

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The Karyotype Sorting Problem

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The Karyotype Sorting (KS) Problem• Find a shortest sequence of elementary

operations that transforms the normal karyotype into given cancer karyotype

• Find the elementary distance = #operations in such a solution to KS.

???

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The Karyotype Sorting (KS) Problem(inverse formulation)

• Find a shortest sequence of inverse elementary operations that transforms the given cancer karyotype into the normal karyotype

???

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Inverse Elementary Operations

Breakage

Fusion

du

plicatio

n

deletion

c-d

eletion

addition

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Assumptions

• ~95% of the karyotypes in the Mitelman Database have no recurrent breakpoints

• Assumptions:– The cancer karyotype contains no recurrent

breakpoints– Every added chromosome contains no

breakpoints

[20,39]::[12,1]Breakpoint ID={390,120}

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The Reduced Karyotype Sorting (RKS) Problem

• Assumptions reduced problem:– No breakpoints in the cancer karyotype

(i.e every chromosome is an interval)

– No breakpoints created by fusions / additions All the normal chromosomes are identical

1 2 3 4 5 6 7 8 9 10 110 1 2 3 4 5 6 7 8 9 10 110

The normal karyotype The cancer karyotype

breakage, fusion, c-deletion, addition

identical chromosomes

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Combinatorial Analysis

(RKS Problem)

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Extending the karyotypes

1 2 3 4 5 6 7 8 9 10 110

The normal karyotype

1 2 3 4 5 6 7 8 9 10 110

The cancer karyotype

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Parameter 1: f = #disjoint pairs of complementing interval ends

• Observation: f = -1 for fusion; f = 1 for breakage f {0,-1,-2} for c-deletion f {0,1,2} for addition

f =5

1 2 3 4 5 6 7 8 9 10 110

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The histogram

• Parameter 2: w = #bricks

• Observations:

– w is even w = 0 for breakage / fusion w {0,2} for addition / c-deletion

1 2 3 4 5 6 7 8 9 10 110

The cancer karyotype

1 2 3 4 5 6 7 8 9 10 110

The histogram

A wall with 2 bricks

A brick

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Simple Bricks• A brick is simple if

– no lower brick (in the same wall), and– no complementing interval ends

• Parameter 3: s = #simple bricks• Observation:

s {0,-1} for breakage s =0 for constrained-deletion– |s| 2 for addition

Simple bricks

1 2 3 4 5 6 7 8 9 10110

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The Weighted Bipartite Graph of Bricks

• Parameter 4: m = the minimum weight of a perfect matching

weight v-,v+:simple v-,v+:non-simple otherwise

v- < v+ 2 0 1

v+ < v- 0 2 1

1 2 3 4 5 6 7 8 9 10110

Positive bricks

Results

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Main Theorem

• The elementary-distance, d, satisfies:

w/2+f+s+m-2N d 3w/2+f+s+m-2N

N = #intervals in the normal karyotype

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Results (2)

• Used the main theorem to devise a polynomial-time 3-Approximation algorithm

• Combined with a greedy heuristic on real data (95% of Mitelman DB) optimal solutions computed for 100% of karyotypes– 99.99% cases : lower bound is achieved

(hence solution is optimal)– 30 cases: lower-bound+2 but actually optimal

(manual verification)

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Summary

• A new framework for analyzing chromosomal aberrations in cancer

• A 3-approximation algorithm when there are no recurrent breakpoints – 100% success on 57,252 karyotypes (with no

recurrent breakpoints) from the Mitelman DB.

• Future work: handle recurrent breakpoints– Analyze the remaining 5% of the karyotypes in

the Mitelman DB.

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Thank for your attention.

Questions?