-
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
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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
27
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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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