A Linear-Time Algorithm for the Copy Number Transformation Problem

Zeira, Ron; Zehavi, Meirav; Shamir, Ron

doi:10.1089/cmb.2017.0060

Cited by 13 publications

(34 citation statements)

References 18 publications

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“…position (we note that in [24], null positions make the problem more complex, but not here). In particular, the hardness also holds if only deletions are allowed.…”

Section: Strong Np-hardnessmentioning

confidence: 93%

“…We consider amplification and deletion events, which respectively have the effect of increasing and decreasing the number of copies in a chromosome. As in [22,24], we assume that events affect a set of positions that are contiguous in the reference chromosome.…”

Section: Preliminary Notionsmentioning

confidence: 99%

“…The problem of computing the minimum number of subinterval alterations to transform one CNP into another was solved in exponential-time in [22] by modeling CNP events with a finite-state transducer. Zeira et al [24] gave a linear time algorithm, using a clever trick for computing each row of a quadratic-size dynamic programming table in constant time (similar to the techniques used in [25]). In [11], the large phylogeny problem under this model is shown NP-hard, though solvable with an ILP.…”

Section: Introductionmentioning

confidence: 99%

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Comparing copy-number profiles under multi-copy amplifications and deletions

Cordonnier

Lafond

2020

BMC Genomics

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Background: during cancer progression, malignant cells accumulate somatic mutations that can lead to genetic aberrations. In particular, evolutionary events akin to segmental duplications or deletions can alter the copy-number profile (CNP) of a set of genes in a genome. Our aim is to compute the evolutionary distance between two cells for which only CNPs are known. This asks for the minimum number of segmental amplifications and deletions to turn one CNP into another. This was recently formalized into a model where each event is assumed to alter a copy-number by 1 or −1, even though these events can affect large portions of a chromosome.Results: we propose a general cost framework where an event can modify the copy-number of a gene by larger amounts. We show that any cost scheme that allows segmental deletions of arbitrary length makes computing the distance strongly NP-hard. We then devise a factor 2 approximation algorithm for the problem when copy-numbers are non-zero and provide an implementation called cnp2cnp. We evaluate our approach experimentally by reconstructing simulated cancer phylogenies from the pairwise distances inferred by cnp2cnp and compare it against two other alternatives, namely the MEDICC distance and the Euclidean distance.Conclusions: the experimental results show that our distance yields more accurate phylogenies on average than these alternatives if the given CNPs are error-free, but that the MEDICC distance is slightly more robust against error in the data. In all cases, our experiments show that either our approach or the MEDICC approach should preferred over the Euclidean distance.

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“…position (we note that in [24], null positions make the problem more complex, but not here). In particular, the hardness also holds if only deletions are allowed.…”

Section: Strong Np-hardnessmentioning

confidence: 93%

Section: Preliminary Notionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Comparing copy-number profiles under multi-copy amplifications and deletions

Cordonnier

Lafond

2020

BMC Genomics

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“…We propose a greedy algorithm ( Table S1 ) which guarantees to find an optimal solution within a runtime of O ( m ) ( Appendix ), where m is the size of the array 55 . We add an additional restriction that MED equals to infinity, if the copy number at any site is going from 0 to any other number.…”

Section: Methodsmentioning

confidence: 99%

Single-cell copy number lineage tracing enabling gene discovery

Wang

Mohanty

et al. 2020

Preprint

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Aneuploidy plays critical roles in genome evolution.Alleles, whose dosages affect the fitness of an ancestor, will have altered frequencies in the descendant populations upon perturbation.Single-cell sequencing enables comprehensive genome-wide copy number profiling of thousands of cells at various evolutionary stage and lineage. That makes it possible to discover dosage effects invisible at tissue level, provided that the cell lineages can be accurately reconstructed.Here, we present a Minimal Event Distance Aneuploidy Lineage Tree (MEDALT) algorithm that infers the evolution history of a cell population based on single-cell copy number (SCCN) profiles. We also present a statistical routine named lineage speciation analysis (LSA), which facilitates discovery of fitness-associated alterations and genes from SCCN lineage trees.We assessed our approaches using a variety of single-cell datasets. Overall, MEDALT appeared more accurate than phylogenetics approaches in reconstructing copy number lineage.From the single-cell DNA-sequencing data of 20 triple-negative breast cancer patients, our approaches effectively prioritized genes that are essential for breast cancer cell fitness and are predictive of patient survival, including those implicating convergent evolution. Similar benefits Recent advances in single-cell DNA sequencing (e.g., tagmentation based approach 15 and single-cell CNV solution by the 10X Genomics) have enabled large-scale acquisition of singlecell copy number (SCCN) profiles in tens of thousands of cells at around 100 kb resolution (~0.1X sequencing coverage per cell) [16][17][18][19] . Other platforms such as single-cell RNA-sequencing 20, 21 and single-cell ATAC-sequencing 22 have also been utilized for SCCN profiling.These SCCN profiles not only present a rich pool of genetic perturbations that are invisible at tissue level, but also potentiate reconstruction of cellular lineage, based on which the impact of an allele on cellular fitness can be measured. Thus, statistical approaches that integrate cellular lineage tracing with population genetic analysis 23 can enable discovery of novel disease genes and mechanisms of disease progression.So far, studies performing retrospective lineage tracing from single-cell data have largely been utilizing phylogenetics approaches designed to model species evolution, which is quite different from cellular evolution in terms of duration, scale, genetics and dynamics 24, 25 . Many existing phylogenetics approaches assume that genomic sites evolve independently and follow the socalled infinite site assumption (ISA) 26 . But in the context of aneuploidy, a genome site can often be altered repeatedly by different CNAs, due partly to constraints on genome and chromatin structures, properties of DNA replication/repairing 27 and functional selection. To apply conventional maximum parsimony approaches on SCCN data, one has to over-segment genomic regions and represent copy numbers as characters in disjoint intervals, which illrepresents the properties of DNAs and dis...

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“…MEDICC ( Schwarz et al , 2014 ), the first algorithm to compute the CND, uses a heuristic to reconstruct a phylogenetic tree from CNPs, and has been used successfully in several cancer studies ( Mangiola et al , 2016 ; Schwarz et al , 2015 ; Sottoriva et al , 2015 ). More recently, Zeira et al (2017) showed that the CND between a pair of profiles can be computed in linear time and El-Kebir et al (2017) gave an integer linear programming formulation for reconstructing a phylogenetic tree between CNPs with the minimum number of events.…”

Section: Introductionmentioning

confidence: 99%

Copy number evolution with weighted aberrations in cancer

Zeira

Raphael

2020

Bioinformatics

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Motivation Copy number aberrations (CNAs), which delete or amplify large contiguous segments of the genome, are a common type of somatic mutation in cancer. Copy number profiles, representing the number of copies of each region of a genome, are readily obtained from whole-genome sequencing or microarrays. However, modeling copy number evolution is a substantial challenge, because different CNAs may overlap with one another on the genome. A recent popular model for copy number evolution is the copy number distance (CND), defined as the length of a shortest sequence of deletions and amplifications of contiguous segments that transforms one profile into the other. In the CND, all events contribute equally; however, it is well known that rates of CNAs vary by length, genomic position and type (amplification versus deletion). Results We introduce a weighted CND that allows events to have varying weights, or probabilities, based on their length, position and type. We derive an efficient algorithm to compute the weighted CND as well as the associated transformation. This algorithm is based on the observation that the constraint matrix of the underlying optimization problem is totally unimodular. We show that the weighted CND improves phylogenetic reconstruction on simulated data where CNAs occur with varying probabilities, aids in the derivation of phylogenies from ultra-low-coverage single-cell DNA sequencing data and helps estimate CNA rates in a large pan-cancer dataset. Availability and implementation Code is available at https://github.com/raphael-group/WCND. Supplementary information Supplementary data are available at Bioinformatics online.

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A Linear-Time Algorithm for the Copy Number Transformation Problem

Cited by 13 publications

References 18 publications

Comparing copy-number profiles under multi-copy amplifications and deletions

Comparing copy-number profiles under multi-copy amplifications and deletions

Single-cell copy number lineage tracing enabling gene discovery

Copy number evolution with weighted aberrations in cancer

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