1

21 1-14

* * *

Schwarz et al.

MEDICC [2]

Zeira et al. [1]

Zeira et al.

(genome rearrangement)

2006

TCGA(The Cancer Genome Atlas)

*

Tunghai Science Vol. 21: 15−28 15 July, 2019

2

DNA (deletions) (amplifications)

TCGA Research Network, 2011 [3] (copies

or alleles) 2 (copy

number profile) CNP DNA

CNP CNP

CNP G (G-

banding) (Fluorescence in situ hybridization, FISH)

(array CGH) (deep sequencing)

CNP

CNP

Euclidean Schwarz et al.,

2014 [2] Chowdhury et al. FISH CNP (edit distance)

Chowdhury et al.,

2013, 2014, 2015 [4][5][6]

TuMult CNP Letouzé

et al., 2010 [7]

Schwartz et al. Schwarz et al., 2014

[2] CNP 1

MEDICC [2]

Zeira et al.

[1]

16

3

CNP

DNA CNP

1 1

3 CNP S CNP T

Copy Number

Transformation CNT CNP S CNP T CNT

CNT

CNP

CNP CNTP

CNPs

CNT CNT

17

4

Zeira et al. CNTP

3.1 Zeira et al. [1]

1.

CNT

ordered

1.

ordered CNT

2.

1 elongated

2.

ordered CNT elongated

3.

CNT

zero-skipping

3.

CNT zero-skipping ordered

4.

bounded

4.

zero-skipping ordered CNT bounded

deletions amplifications

1 2 3 4 5 6 7 8

18

5

3.2 Zeira et al. CNTP DpCntpAlg

DpCntpAlg NewDpCntpAlg

CNP DpCntpAlg

CNT

ordered elongated zero-skipping bounded

[1]

3

DpCntpAlg

zero-skipping

1

3

j 1 2 3 ~ 9 10 S 5 3 3 6 T 9 6 0 8 = 0 4 4 9 = 1 6 5 10 = 2 8 7 11 = 3 10 7 12 = 4 12 8 13 … =

19

6

DpCntpAlg

NewDpCntpAlg

DpCntpAlg NewDpCntpAlg j

j

zero-skipping

20

7

3.3 Zeira et al. CNTP LinearCntpAlg

LinearCntpAlg DpCntpAlg M piecewise linear

function [1]

GreedyCntpAlg

Excel [ ]

(random value) S T DpCntpAlg (Greedy

algorithm)

CNT

GreedyCntpAlg

DpCntpAlg

21

8

F E

( + , + )

( - , - )

( - , + )

( + , - )

22

9

+K

+2

+K

+0

+2 2+1

+0

+2 22+1

23

10

+2

+2

+2 +1 222

+1

+0

24

11

+2 2222+1 +2

+1

+2 +1

+2 222222+2+2+1

25

12

GreedyCntpAlg Zeira et al.

LinearCntpAlg

1. zero-skipping S T

2. LinearCntpAlg

GreedyCntpAlg

26

13

1. Zeira R., Zehavi M., Shamir R. 2017 A Linear-Time Algorithm for the Copy Number Transformation Problemhttps://www.ncbi.nlm.nih.gov/pubmed/28837352

2. Schwarz R.F., Trinh A., Sipos B., Brenton J.D., Goldman N., Markowetz F. 2014 Phylogenetic quantification of intra-tumour heterogeneityhttps://www.ncbi.nlm.nih.gov/pubmed/24743184

3. TheCancer GenomeAtlas Research Network. 2011 Integrated genomic analyses of ovarian carcinomahttps://www.ncbi.nlm.nih.gov/pubmed/21720365

4. Chowdhury S.A., Shackney S.E., Heselmeyer-Haddad K., Ried T., Schäffer A.A., Schwartz R. 2013 Phylogenetic analysis of multiprobe fluorescence in situ hybridization data from tumor cell populationshttps://www.ncbi.nlm.nih.gov/pubmed/23812984

5. Chowdhury S.A., Shackney S.E., Heselmeyer-Haddad K., Ried T., Schäffer A.A., Schwartz R. 2014 Algorithms to model single gene, single chromosome, and whole genome copy number changes jointly in tumor phylogenetics PLoS Comput. Biolhttps://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1003740

6. Chowdhury S.A., Gertz E.M., Wangsa D., Heselmeyer-Haddad K., Ried T., Schäffer A.A., Schwartz R. 2015 Inferring models of multiscale copy number evolution for single-tumor phylogeneticshttps://www.ncbi.nlm.nih.gov/pubmed/26072490

7. Letouzé E., Allory Y., Bollet M.A., Radvanyi F., Guyon F. 2010 Analysis of the copy number profiles of several tumor samples from the same patient reveals the successive steps in tumorigenesishttps://www.ncbi.nlm.nih.gov/pubmed/20649963

Excel DpCntpAlg GreedyCntpAlg LinearCntpAlg

https://drive.google.com/drive/folders/1b2Mkk1xuff0OX2f92TvyqjdYUKvexrj5?usp=sharing

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14

Correction of the linear-time alogorithm for the Copy Number Transformation Problem

Dun-Siang Yang* Tsung-Yi Yang* Wei-Hua Hsieh*

Abstract

Schwarz et al. set the distance between copy number profiles to explore the gap between normal genome and tumor genome. They proposed the MEDICC algorithm for calculating distance[2], but did not analyze its complexity. However, in some cases, MEDICC would be exponential time. After that, Zeira et al. proposed a linear time algorithm [1], but we found the process that leading out the result has something wrong. So we propose a new algorithm and make some changes to the original method.

Keywords: Cancer, Genome rearrangement, Copy number profile

*Department of Applied Mathematics, Tunghai University, Taichung 40704, Taiwan

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