Reconstruction of clone- and haplotype-specific cancer genome karyotypes from bulk tumor samples

Abstract

Many cancer genomes are extensively rearranged with highly aberrant chromosomal karyotypes. These genome rearrangements, or structural variants, can be detected in tumor DNA sequencing data by abnormal mapping of se-quence reads to the reference genome. However, nearly all cancer sequencing to date is of bulk tumor samples which consist of a heterogeneous mixture of normal cells and subpopulations of cancers cells, or clones, that harbor distinct somatic structural variants. We introduce a novel algorithm, Reconstructing Cancer Karyotypes (RCK), to reconstruct haplotype-specific karyotypes of one or more rearranged cancer genomes, or clones, that best explain the read alignments from a bulk tumor sample. RCK leverages specific evolutionary constraints on the somatic mutation process in cancer to reduce ambiguity in the deconvolution of admixed DNA sequence data into multiple haplotype-specific cancer karyotypes. In particular, RCK relies on generalizations of the infinite sites assumption that a genome re-arrangement is highly unlikely to occur at the same nucleotide position more than once during somatic evolution. RCK’s comprehensive model allows us to incorporate information both from short and long-read sequencing technologies and is applicable to bulk tumor samples containing a mixture of an arbitrary number of derived genomes. We compared RCK to the state-of-the-art method ReMixT on a dataset of 17 primary and metastatic prostate cancer samples. We demonstrate that ReMixT’s limited support for heterogeneity and lack of evolutionary constrains leads to reconstruction of implausible karyotypes. In contrast, RCK’s infers cancer karyotypes that better explain read alignments from bulk tumor samples and are consistent with a reasonable evolutionary model. RCK’s reconstructions of clone- and haplotype-specific karyotypes will aid further studies of the role of intra-tumor heterogeneity in cancer development and response to treatment. RCK is available at https://github.com/raphael-group/RCK.

1 Introduction

The somatic mutations that drive cancer development range across all genomic scales, from single nucleotide mutations through large-scale genome rearrangements [54, 20, 58, 47]. Whole-genome sequencing of tumor samples has enabled the detection of all classes of somatic mutations; however, specialized algorithms are required to identify each class of mutations from the short DNA sequence reads obtained by current technologies [31, 37, 28, 49, 30, 60]. In addition, nearly all cancer sequencing to date has been of bulk tumor tissue, which is generally a mixture of normal (noncancerous) cells and (sub)populations of cancerous cells, or clones, that often are not genetically identical. Quantifying this intra-tumor heterogeneity is essential for understanding the processes that drive cancer development and also helps inform treatment strategies [2, 44, 34].

Here we consider the problem of describing the large-scale organization of one or more cancer genomes that are derived form a normal human reference genome via large-scale rearrangements. The large-scale organization of a cancer genome is described by two features. First, is the number of copies of each segment of the genome. Many methods (e.g. [57, 9, 7, 40, 24, 19, 43, 63]) have been developed to identify copy number values for heterogeneous, bulk tumor samples. Second, are genome rearrangements (e.g. chromosomal inversions and translocations) that link together distant segments of the normal genome. Many methods have been developed to predict such novel adjacencies (e.g. [51, 48, 30, 10, 60, 49, 16, 52, 66, 50, 27, 17]. However, these methods do not distinguish between adjacencies from different homologous chromosomes or from different cancer clones within a bulk sample; i.e. they assume the human genome is haploid reference and that the tumor is homogeneous.

A more challenging problem is to combine and reconcile the information about segment copy numbers and novel adjacencies into genome karyotypes, or the alignment of cancer genome and the healthy genome that depicts the number of occurrences of every segment in the cancer genome, and the adjacencies between these segments on the cancer genome. Multiple methods have been developed to solve some variations of this cancer genomes karyotype reconstruction problem including [42, 32, 46, 35, 14, 11, 15]. However, each of these methods rely on simplifying as-sumptions that do not adequately address the challenges in real cancer sequencing data. For example, SVclone [11] focuses solely on inferring genome-specific copy numbers for novel adjacencies, without attempting to reconstruct complete karyotypes of the derived genomes. PREGO [42] and Karyotype Reconstruction [15] assume that the human reference genome is haploid, thus losing important information about alleles involved in rearrangements. Weaver [32, 46] assumes that the cancer sample contains only a single derived genome (with a possible admixture of the reference genome), and lacks a proper support of reciprocal novel adjacencies, which can emerge both from copy number neutral somatic rearrangements (e.g., inversions, balanced translocations, etc), as well as from more complex “catastrophic rearrangements” such as chromoplexy and chromothripsis [53, 6, 26, 4, 61, 41]. ReMixT [35] allows for tumor heterogeneity, but fixes the number of derived genomes in the observed cancer sample to 2. Moreover, while ReMixT aims to infer genome- and allele-specific segment copy numbers for a 2-genome sample (with a possible admixture of the reference genome), the genome-specific copy numbers for novel adjacencies that are inferred by ReMixT lack information about which homologous copies of the segments are actually involved in observed novel adjacencies. Lastly, Weaver, and ReMixT produce karyotypes with biologically unlikely scenarios where rearrangements occur repeatedly at the same homologous loci in different cancer clones. We summarize these limitations of existing methods in Table S1.

Here we propose a novel algorithm, Reconstructing Cancer Karyotypes (RCK), for deriving the karyotypes of cancer genomes in a heterogeneous tumor sample from next-generation (and 3rd-generation, when available) sequencing data. RCK distinguishes itself from existing methods by several features including: (i) support for diploid reference genome distinguishing between alleles of segment copy numbers and novel adjacencies (ii) joint inference of both segment and adjacency copy numbers in both clone- and haplotype-specific fashion; (iii) comprehensive support for sample heterogeneity ranging from homogeneous samples with a single derived genome to heterogeneous samples with an arbitrary number of clones; (iv) enforcement of somatic evolutionary constraints on all genomes within a sample; (v) unique ability to incorporate groups of novel adjacencies from 3rd-generation sequencing technologies into the inference model. We demonstrate the advantages of RCK by comparing its performance to ReMixT on a dataset of 17 primary and metastatic prostate cancer samples. We find that RCK infers more plausible karyotypes that conform to an evolutionary model and have allele-specific segment copy numbers that agree with leading copy number inference algorithms.

2 Results

2.1 RCK algorithm

We introduce Reconstructing Cancer Karyotypes (RCK), an algorithm to construct the large-scale organization of one or more cancer genomes present in a bulk tumor sample. Each cancer genome in the sample arises from a sequence of somatic genome rearrangements and copy number number aberrations that transform a healthy normal genome into a cancer genome. As a result of these somatic mutations, each cancer genome can be represented as a karyotype graph – or more briefly a karyotype. A karyotype graph includes: (1) a collection of contiguous segments from the human reference genome, each segment with a label (A or B) distinguishing the two homologous chromosomes; an integer copy number for each segment; (3) a collection of adjacencies that join the ends of segments; (4) an integer copy number for each adjacency. The karyotype graph describes an alignment between the cancer genome and healthy genome (analogous to the breakpoint graph [1, 3] in genome rearrangement studies). The karyotype graph also represents the information about the cancer genome sequence that can be inferred from DNA sequencing technologies whose reads lengths are shorter than the length of genome rearrangements.

RCK solves the following Cancer Karyotype Reconstruction Problem: given allele-specific segment copy numbers and a list of novel adjacencies (i.e. pairs of genomic loci that are measured as adjacent in the cancer genome, but distant in the normal reference) from a bulk tumor sample, derive karyotype graph(s) for the cancer genome(s) present in the tumor sample. Several challenges emerge in the development of an algorithm to solve this problem. The first challenge is that the many methods for inferring allele-specific copy numbers from bulk tumor sequencing data (e.g. [57, 9, 7, 40, 24, 19, 35, 63]) do not preserve the allelic information across multiple adjacent segments. Specifically, these methods output a pair of copy number vectors, ĉ = [ĉ₁, ĉ₂, …,ĉ_m] and č = [č₁, č₂, …, č_m], where the pair (ĉ_j, č_j) ∈ ℕ² indicates the number of copies of each of the two homologous copies of segment j from the reference genome that are present in the cancer genome. However, each of these pairs are unordered: it is not known whether ĉ_j is the number of segment from the maternal chromosome or the paternal chromosome; moreover, the identification of ĉ_j as maternal or paternal is independent for each j.

The second challenge results is that the many methods for inferring novel adjacencies from bulk tumor sequencing data [51, 48, 30, 10, 60, 49, 16, 52, 66, 50, 27, 17] generally do not include two important attributes in their output: (i) the alleles (maternal or paternal) that are joined by the adjacency; (ii) the copy number(s) of the adjacency in each genome in the sample. Because of this incomplete information in the allele-specific copy numbers and novel adjacencies, cancer genome karyotypes are not directly available.

RCK derives optimal cancer genome karyotype(s) from allele-specific copy numbers and novel adjacencies by solving an optimization problem on a graph, called the Diploid Interval Adjacency Graph (DIAG) (Figure 1). The vertices of the DIAG are extremities, or the positions in the human reference genome of the endpoints of the segments that are rearranged to form the cancer genomes present in the sample. Specifically, we enumerate the segments of the reference genome 1, …, m. Each segment j has the form , where and are extremities. The label t indicates that the extremity is the tail, or starting coordinate of the segment in the reference genome, while the label h indicates the head, or ending coordinate in the reference genome. A haplotype label H ∈ {A, B} indicates which copy of the two homologous chromosomes in the reference (A or B) is the source of the segment. Adjacent extremities of consecutive segments that follow each other along the chromosome in the genome constitute an adjacency. We distinguish between two types of adjacencies: reference adjacencies that are present in the reference genome, and novel adjacencies that are not present in the reference genome. Thus, the DIAG has three types of edges: (1) segment edges join extremities from a segment; (2) reference adjacency edges join extremities of adjacent segments on the reference genome; (3) novel adjacency edges , where H, H′ ∈{A,B}, and, σ, σ ′ ∈ {t, h}. Importantly, since a measured novel adjacency does not generally include allelic information, we add all 4 possible labeled versions of the adjacency, and to the DIAG.

Figure 1: Overview of the RCK algorithm.

Read alignments from bulk tumor sample are input to existing algorithms to identify clone- and allele-specific segment copy numbers (left) and novel adjacencies (right). The RCK algorithm (blue shaded elements) builds a diploid interval adjacency graph integrating copy number and novel adjacency infor an mixed-integer linear program (MILP) that finds an optimal assignment of copy numbers and novel adjacencies to alleles and clones, subject to copy number balance on segment ends and satisfying evolutionary constraints from a generalized infinite sites model. Constraints on groups of novel adjacencies from the 3rd generation sequencing technologies may optionally be included. The output of RCK are clone- and haplotype-specific cancer genome karyotypes.

A chromosome in the cancer genome corresponds to a walk in the DIAG that alternates between segment edges and reference/novel adjacency edges, and where the number of times every segment/adjacency edge is visited encodes the respective segment/adjacency copy number (see Methods 4.2). Thus, all vertices (except telomere vertices) should satisfy the copy number balance condition: the copy number of the incident segment edge equals the sum of the copy numbers of the incident reference edge and novel adjacency edge(s).

The Cancer Karyotype Reconstruction Problem thus can be formulated as the problem of finding an edge multiplicity µ_G(e) for each edge e and each cancer genome G such that: (i) each extremity (vertex v) satisfies the copy number balancing conditions (Equations (9), (10) in Methods); (ii) the copy numbers µ_G(j_A) and µ_G(j_B) of homologous segments j_A and j_B are approximately equal to the allele-specific copy numbers (ĉ_j and č_j); (iii) most of the novel adjacencies are present in at least one genome (i.e. µ_G (e) ≥ 0 for novel adjacency edge e in at least one genome G).

A major difficulty with the above formulation of the Cancer Genome Karyotype Reconstruction Problem is that there are often numerous solutions, many of which are biologically implausible. Considerable ambiguity arises from the lack of A /B labels on the measured novel adjacencies. The lack of allelic label means that each measured novel adjacency corresponds to 4 edges in the DIAG. However, selecting one of these four possible allele-specific novel adjacencies independently for each measured novel adjacency is unwise. Rather, the somatic evolutionary process imposes several constraints on the possible structures of inferred karyotypes. In particular, we derive conditions on allowed novel adjacencies from the infinite sites (IS) assumption commonly used in evolutionary studies. The infinite sites assumption is that a mutation does not occur at the same locus more than once during the course of evolution. The locus of a single-nucleotide mutation is readily defined as a genomic position. However, the locus for a large-scale genome rearrangement is not apparent, and could be defined as either (or both) of the genomic positions of the extremities in the adjacency as well as adjacent genomic positions of “reciprocal” extremities. We define multiple constraints on the extremities that may be involved in novel adjacencies (Figure 1). These constraints generalize the infinite sites assumption to the case of multiple genomes that are derived from a diploid reference genome by a sequence of large-scale genome rearrangements. First, extremity-exclusivity is the constraint that an extremity is involved in at most one novel adjacency. Second, homologous-extremity-exclusivity is the constraint that an extremity and its homolog cannot both be involved in a novel adjacency. Third, homologous-reciprocal-extremity-exclusivity is the constraint that an extremity and its reciprocal mate of the homologous chromosome cannot both be involved in a novel adjacency. All of these constraints are natural generalizations of the infinite sites assumption; however, they have not been distinguished consistently in previous publications (See Methods). As a result, previous methods can yield implausible genome reconstructions, as we will demonstrate below.

RCK solves the optimization problem of finding edge multiplicities µ(e) satisfying conditions (i), (ii), and (iii) above and also where the novel adjacencies inferred to be present (µ_G(e) > 0) satisfy the generalized infinite sites constraints jointly across all clones. We solve this problem using a mixed-integer linear program (see Supplement S2.2). RCK also allows for grouping of novel adjacencies that are measured to be present on the same cell or longer read when such information is available from 3rd generation sequencing technologies (e.g. single cell sequencing, linked read sequencing [66, 52, 16], or long read sequencing [49, 17, 50, 27]). See Methods 4.4 for further details.

2.2 Evaluation and comparison of RCK

We compare RCK to ReMixT, the only other existing method which both derives multiple tumor clones from bulk sequencing data and distinguishes between homologous chromosomes. ReMixT takes read alignments and novel adjacencies as input and infers clone- and allele-specific copy numbers for segments, as well as clone-specific copy numbers for novel adjacencies. Importantly, ReMixT does not infer haplotype A /B labels for the extremities that are involved in each novel adjacency. We will show below that this lack of assignment of each novel adjacency to a homologous chromosome leads to unusual genome reconstructions in many cases.

2.2.1 Data processing

We analyze a cancer sequencing dataset from Gundem et al. [23], which consists of whole-genome sequencing data from 49 samples from 10 metastatic prostate cancer patients. Segment copy numbers inferred by Battenberg [40] were obtained from the publication [23] and read alignments for every sample were obtained from the authors. For each sample, Battenberg output includes: (i) the number of clones; (ii) allele-specific copy numbers for each genomic segment in each clone; (iii) the occurrence of a whole genome duplication (WGD) when reported tumor ploidy > 3. We also used HATCHet [63], a recently developed algorithm that infers allele-specific copy numbers for one or more cancer clones as well as the presence of WGD by joint analysis of multiple sequenced samples from the same patient. We considered the 17 samples where both Battenberg and HATCHet agreed on the number of clones present.

For novel adjacencies, we used the predictions from brass2 (https://github.com/cancerit/BRASS), which we obtained from the the original publication [23]. brass2, like most methods that identify novel adjacencies from aligned DNA sequence reads, has some uncertainty in the exact genomic coordinate involved in a novel adjacency. This uncertainty can be an issue when determining whether an adjacency is part of a reciprocal event (e.g. inversion or reciprocal translocation). Thus, we adjusted the coordinates of extremities to obtain refined coordinates for loci involved in reciprocal novel adjacencies. For RCK, we also aligned the positions of extremities of segments from Battenberg or HATCHet to the positions of extremities from novel adjacencies determined by brass2. See Methods 4.6 for further details.

We divided the cancer samples into two groups according to the number of tumor clones predicted by both Battenberg and HATCHet : homogeneous samples containing only one tumor clone (samples A21g, A21h, A24c, A24d, A24e, A34a, A34c, A34d); and heterogeneous samples containing two tumor clones (samples A10c, A12c, A12d, A17d, A31a, A31d, A31e, A31f, A32e). Notably, there was only one sample (A12c) where Battenberg and HATCHet disagreed on the presence of a WGD.

For each sample, we ran RCK requiring that: (1) the only telomeres in the inferred cancer genomes are telomeres from the reference genome (i.e. extremities that are not the endpoints of reference chromosomes have copy number balance); (2) at least a fraction P of the input novel adjacencies are present in at least one of the derived genomes in a sample, for P = 1.0, 0.9, 0.75, 0.5. ReMixT does not allow control over telomeres or the fraction of novel adjacencies, and thus we ran ReMixT using default parameters.

2.2.2 Heterogeneous tumor samples

We first compared the allele-specific segment copy numbers inferred by ReMixTand the haplotype-specific segment copy numbers inferred by RCKto the allele-specific copy numbers from HATCHetand Battenberg, using a length-weighted segment copy number distance (equation (15) in Methods). We found that in all but two cases (samples A10cand A12cwith a NAs utilization parameter P= 1.0), the segment copy numbers inferred by RCK are more similar to the copy numbers from HATCHet(Figure 2A) and Battenberg(Figure S2A). We also observed that RCK’s ability to control the fraction of input novel adjacencies that are required to be utilized in the inferred karyotype (Methods 4.3, 4.5) allows for more plausible reconstructions: the distance between input copy numbers is largest when we require RCKto use all novel adjacencies, but the distance decreases and stabilizes when a small fraction of novel adjacencies are excluded (P0.9). We note that the largest distances between ReMixTand HATCHet(or Battenberg) inferred copy numbers are on four samples (A31a, A31d, A31e, and A31f) where both Battenbergand HATCHetinferred a WGD. In these four samples, the high segment copy number values output by ReMixTalso suggest many copy number changes; however, the large distances indicate that these inferred copy numbers may not align well with copy numbers expected from a WGD.

Figure 2:

A) Length-weighted segment copy number distances (eq. (15)) between segment copy numbers from HATCHetand segment copy numbers output by ReMixTand RCK. B) Fractions of novel adjacencies (NAs) from input that are inferred to be present by ReMixTor RCKfor each sample in the heterogeneous group. RCKused segment copy numbers from HATCHetin input and novel adjacency utilization parameter P= 1.0, 0.9, 0.75, 0.5.

We next compared the fraction of input novel adjacencies that were contained in the genomes reconstructed by ReMixTand RCK. This value ranged from 0.75 to 0.92 for ReMixT(Figure 2B). In contrast, for RCKfraction of utilized input novel adjacencies ranged from 0.5 to 1.0 with its lower bound explicitly controlled via the Pparameter. We observe that RCKfrequently utilizes more novel adjacencies than the minimum required (value of P). This occurs on 6/9 cancer samples (A10c, A31a, A31d, A31e, A31f, A32e) with HATCHetcopy numbers and P= 0.75, P= 0.5, and 5/9 samples with Battenberginput. RCK’s incorporation of novel adjacencies at a higher proportion than the minimum required fraction Psuggests that RCKis selectively including those novel adjacencies required to achieve copy number balance.

Next, we analyzed the structure of karyotypes inferred by each method. Since ReMixTdoes not output A/Blabels for extremities involved in novel adjacencies, we investigated whether it was possible to derive A/Blabels on ReMixTadjacencies to produce reasonable cancer genomes that would allow for a copy number balance/excess on the extremities of segments and comply with generalized IS constraints. We first observed that the karyotypes reconstructed by ReMixThad a large number of extremities that are not telomeres in the reference and have copy number excess (ranging from 41 to 133 per genome), corresponding to a large number of novel telomeres (Figure S1). Such karyotypes correspond to unlikely cancer genomes having dozens or even hundreds of linear chromosomes with novel telomeres, in addition to ∼46 (∼92 in WGD samples) derived linear chromosomes with reference telomeres. In contrast, the RCKresults reported here use only reference telomeres and thus the karyotypes have at most 48 (89 in WGD samples) linear chromosomes in total.

We examined the frequency of violations of the generalized IS constraints. By construction, RCKkaryotypes have no such violations. In contrast, we identified three types of violations of generalized IS conditions in the ReMixTkaryotypes. The first is an intra-genome violation of the homologous-extremity-exclusivity constraint. This violation occurs when the inferred segment copy numbers require that a novel adjacency a be assigned both a label Aand a label Bin order to achieve copy number balance (Figure 3A). This situation requires that at least two large-scale somatic rearrangements occur independently at the same genomic position on both homologous chromosomes, which is highly unlikely. We find that karyotypes reconstructed by ReMixTcontain such violations in 6/9 samples, ranging from 1 to 8 violations per genome, and from 1 to 12 violations per sample (Figure 3B).

Figure 3:

ReMixT karyotypes from the heterogeneous group of metastatic prostate cancer samples have numerous violations of the generalized IS constraints. In graph panels A), C), and E) solid edges represent segment edges, black-dashed edges represent reference adjacency edges, and red-dashed edges represent novel adjacency edges. Integer values indicate copy numbers of corresponding segment and adjacency edges. A) Example of a violation of the homologous-extremity-exclusivity constraint. To obtain copy number balance, both homologous vertices and must be involved in novel adjacencies. B) Number of novel adjacencies (NAs) in each cancer karyotype inferred by ReMixT in each sample that violate the homologous-extremity-exclusivity constraint. C) Example of a violation of the inter-genome homologous-extremity-exclusivity constraint. To obtain copy number balance, both homologous vertices and (in different genomes) must be involved in novel adjacencies. D) Fractions ^x/_y of the number (x) of novel adjacencies (NAs) violating the inter-genome homologous-extremity-exclusivity constraint (on at least one of the extremities involved in every novel adjacency) in ReMixT karyotypes, over the total number (y) of novel adjacencies reported by ReMixT as being present in both genomes in the every sample. E) Example of a violation of the intra-genome homologous-reciprocal-extremity-exclusivity constraint. To obtain copy number balance, both homologous-reciprocal vertices and must be involved in novel adjacencies. We note that in addition to the violation of the intra-genome homologous-reciprocal-extremity-exclusivity) constraint, violations of the inter-genome or both version(s) of this constraint are also possible (See Figure S7). F) Fractions ^x/_y of the number (x) of reciprocal locations with violations of either intra- or inter-genome (or both) homologous-reciprocal-extremity-exclusivity constraint in ReMixT karyotypes, over the total number (y) of reciprocal locations with both involved NAs reported as being present by ReMixT.

The second type of violation is an inter-genome violation of the homologous-extremity-exclusivity constraint. This violation occurs when a novel adjacency a is reported as being present in more than one genome in the sample, but a label A must be assigned to at least one a’s extremities in in one genome, and a label B must be assigned to at least one a’s extremities in another genome. This situation requires at least two large-scale somatic rearrangements occur independently at the same homologous genomic location in two different tumor clones, which is highly unlikely. We found that the karyotypes produced by ReMixT had such violations in all samples, with a substantial fraction (ranging from 0.09 to 0.28) of novel adjacencies containing such violations (Figure 3D).

The third type of violation concerns pairs of reciprocal novel adjacencies. For a pair a = {x, uh}, b = {(u + 1)t, y} of reciprocal novel adjacencies that involve reference adjacent extremities uh, (u+1)t possible violations of generalized IS include intra/inter-genome violation of the homologous-extremity-exclusivity or intra/inter-genome violation of the homologous-reciprocal-extremity-exclusivity constraints, or both. Any such violation requires that at least two large-scale somatic rearrangements occur independently on the same or homologous genomic location both producing pairs of reciprocal novel adjacencies, a situation which is highly unlikely. We found that karyotypes produced by ReMixT had such violations in all samples; furthermore in 6/9 samples more than half of reciprocal novel adjacencies had such violations (Figure 3F).

2.2.3 Homogeneous tumor samples

We ran RCK and ReMixT on cancer samples from the homogeneous group and analyzed the karyotypes output by both methods, following the procedures described above for the heterogeneous samples. Since ReMixT assumes that an input sample contains exactly two cancer clones, ReMixT’s results disagree with both Battenberg’s and HATCHet’s predictions of one cancer clone in these samples. To obtain a partial comparison of the segment copy number profiles inferred by ReMixT with the profiles inferred by Battenberg and HATCHet in each sample, we used ReMixT’s clone with the highest cellular frequency. Overall, our analysis of inferred cancer genomes karyotypes in the homogeneous group aligned with the findings for the heterogeneous group. In particular, we found that on every sample in the homogeneous group, the segment copy numbers inferred by RCK (with P ≤ 0 9) are more similar to the copy numbers from Battenberg (Figure S6A) and HATCHet (Figure S6B) compared to the segment copy numbers inferred by ReMixT. We also found that the fraction of input novel adjacency that were present in inferred karyotypes ranged from 0.82 to 0.94 in ReMixT results and from 0.5 to 1.0 in RCK results (Figure S5). As in the case of heterogeneous samples, we observed that segment copy number distances are largest for RCK when we require RCK to use all novel adjacencies (a larger proportion than used in ReMixT), but the distances decrease and stabilize when some novel adjacencies are excluded (P ≤ 0 9).

Similar to the heterogeneous samples, we observed that karyotypes inferred by ReMixT had implausible features including a large number (and multiplicity) of novel telomeres (Figure S3) and violations of the generalized infinite sites constraints (Figures S4). In contrast, karyotypes inferred by RCK had no such issues. Overall, our analysis of inferred cancer genomes karyotypes in the homogeneous group aligned with the findings for the heterogeneous group.

3 Discussion

We presented RCK, a novel algorithm for reconstructing clone- and haplotype-specific cancer genomes karyotypes from bulk tumor samples. RCK accounts for heterogeneity in the observed tumor sample, correctly models the diploid reference genome, and enforces biologically reasonable evolutionary constraints that generalize the infinite sites constraints to somatic large-scale rearrangements. RCK is, to the best of our knowledge, the only algorithm with these features and also the only algorithm that can combine both next- and 3rd-generation sequencing data into the reconstruction process, leveraging the long-range adjacency information from 3rd-generation sequencing technologies.

On real cancer sequencing data, we found that RCK infers cancer karyotypes which inferred segment copy numbers are closer to those produced by state-of-the-art copy number inference tools (HATCHet and Battenberg), and which novel adjacencies conform with constraints from an infinite sites evolutionary model. In contrast, ReMixT’s approach of using novel adjacencies to “adjust” copy numbers generally led to allele-specific segment copy numbers that were different from those of HATCHet and Battenberg. Moreover, the novel adjacencies that are present in ReMixT inferred karyotypes often require biologically implausible rearrangements. These results demonstrate that “linking” of copy numbers via novel adjacencies without considering the underlying somatic evolutionary process is not advisable.

While the proposed RCK method uses a very comprehensive somatic evolutionary model and addresses several shortcomings of the previous approaches, there are limitations and avenues for future improvements. First, in the RCK results presented here, we assume that no new telomeres are introduced in the cancer genomes, i.e. all telomeres are telomeres of the reference genome. RCK allows for non-reference telomeres to be specified; however, we have not incorporated telomere selection into the objective function of the optimization. Such novel telomeres can correspond to real telomeres, but in many cases are likely due to missing novel adjacencies in the input data. Second, we can further generalize RCK to simultaneously analyze multiple samples from the same individual, perhaps including a phylogenetic [62] or longitudinal constraints [38]. Simultaneously analysis of multiple samples has proved useful in copy number inference [63]. Third, it would be helpful to model a patient-specific germline genome that includes germline structural variations, long repetitive segments, etc. Finally, one could further leverage information in 3rd-generation sequencing data by including haplotype-specific labeling of extremities involved in groups of novel adjacencies.

RCK’s inference of clone- and haplotype-specific cancer karyotypes enables further studies of the somatic mutational processes that produce highly rearranged cancer genomes, as well as improved characterization of specific functional changes (e.g., loss of heterozygosity, novel haplotype-specific fusion genes, etc). Higher-resolution reconstructions of cancer karyotypes can also help researchers illuminate differences/similarities between different types of cancer in general and lead to a more targeted and personalized medical treatments in specific patients.

4 Methods

We start by considering the case of “perfect” input data and the problem of reconstructing karyotype of a single mutated genome in sections 4.1 – 4.2. Then we extend to the case of a heterogeneous cancer sample (section 4.3). and describe how our model can incorporate (when available) information from 3rd-generation sequencing technologies (section 4.4). In section 4.5 we describe a more general case where there is uncertainty in input segment copy number values.

4.1 Single derived genome

We view cancer as a process propagated by a sequential application of somatic large-scale rearrangements starting with a diploid reference genome R and ending with a derived genome G. Every chromosome in a diploid reference genome R is present in two homologous copies, which we label by A and B respectively. A segment is a contiguous part of a reference chromosome labeled A; its endpoints and are called extremities. We label segments 1 through m in a multichromosomal diploid reference genome R. In a mutated genome that is derived from the reference via large-scale rearrangements, segments can be absent, present more than once, and appear both in forward and reverse orientation. We denote by a reversed instance of segment j_A.

Extremities that demarcate the beginning and the end of a chromosome are called telomeres and we define by τ (G) the set of telomeres in genome G.

A pair (j_A, k_B) of consecutive segments on a chromosome determines an adjacency (i.e., a pair of extrem-ities that are adjacent on a chromosome). A genome G determines a set(G) of adjacencies present in it.

For a diploid reference genome R with k chromosomes we define a set of reference telomeres and note that|𝒯 (R) |= 4k. We further note that a multichromosomal diploid reference R determines a set𝒜 (R) of reference adjacencies as follows: A derived genome G corresponds to a collection of concatenation of segments (i.e., derived chromosomes), where segments in each novel concatenation can originate from any homologous copy of any of the chromosomes in the diploid reference R. Each derived chromosome thus corresponds to a word from the following alphabet: Adjacencies that are present in a mutated genome G but are not present in the reference are called novel and we denote by 𝒜N (G) a set of novel adjacencies in genome G. We note that since there are no novel adjacencies in the reference we have 𝒜 N (R) = Ø. We say that a set 𝒜 of adjacencies satisfies infinite sites if no two adjacencies in 𝒜 involve the same extremity. For a reference adjacency we call extremities and reciprocal.

We assume that the large-scale somatic rearrangements that “break” and “reglue” chromosomes do not affect the same genomic locations (on either of the or A/B copies) more than once during the entire somatic evolutionary process (i.e., generalized infinite sites). We note that under generalized IS only reference adjacencies can participate in breaks, however we also note that novel adjacencies produced by rearrangements can further be amplified/deleted via other rearrangements. For a break r of an adjacency involving extremities and under the generalized IS at every point before and after r in the somatic evolutionary process none of the reference/novel adjacencies involving either or can be involved in any other rearrangement(s) (breaks), where H, H ′ ∈ {A, B}, Â = B, and . Examples of rearrangements that violate the generalized IS and consecutive implications for novel adjacencies in the derived genomes are shown in supplementary Figure S9.

With the generalized IS assumption for somatic evolution propagated by large-scale rearrangements we naturally obtain several constraints for the derived genome which we list below:

1. extremity-exclusivity: every extremity is involved in at most one novel adjacency from 𝒜_N (G). This con-straint is based on the fact that for a novel adjacency a to involve an extremity there must have been a large-scale rearrangement breaking a reference adjacency involving in the first place (and possible several other reference adjacencies). Having more than 1 novel adjacency involving would correspond to the scenario where some other rearrangement must have broken some adjacency involving , which is prohibited under generalized IS.

2. homologous-extremity-exclusivity: if an extremity is involved in a novel adjacency from 𝒜 _N (G), then the homologous extremity is not involved in any novel adjacency from 𝒜 _N (G). This constraint follows the logic outlined in extremity-exclusivity, but considers A/B labeled homologous extremities and : for both and extremities to be involved in novel adjacencies there must have been at least two large-scale rearrangements breaking homologous reference adjacencies involving both extremities and , which is prohibited under the generalized IS.

3. homologous-reciprocal-extremity-exclusivity: if an extremity from the reference adjacency is involved in a novel adjacency from 𝒜 _N (G), then the homologous extremity , is not involved in any novel adjacency from 𝒜 _N (G). This constraint follows the justification provided in homologous-extremity-exclusivity for both extremities and to be involved in novel adjacencies there must have been two large-scale rearrangements breaking both homologous reference adjacencies and which is prohibited under the generalized IS.

We call a genome G proper, if the above three conditions are met.

A genome G determines a diploid segment copy number profile C_G = (a = [a₁, a₂, …, a_m], b = [b₁, b₂, …, b_m]), where values (a_j, b_j) ∈ ℕ² indicate the number of copies of segments j_A and j_B in G. We note that in a diploid reference R we have a_j = b_j =1 for every segment j. An example of a diploid segment copy number profiles C_G and C_R for a derived genome G and a reference R are shown in Figure 4.

Figure 4:

A) Example of a diploid reference genome R containing two pairs of homologous chromosomes (chromosomes labeled by A are shown in dark blue/green, and homologous copies labeled by B are shown in light blue/green) that are “partitioned” into 12 consecutive segments labeled 1 through 12. B) Reference genome R is shown as a collection of concatenations of segments, with segments located on chromosomes labeled A shown in dark blue/green and segments located on chromosomes labeled B shown in light blue/green. The “pointy” end of each segment j correspond to extremity jh, while the “flat” end corresponds to extremity jt. Dashed lines determine adjacencies between segments’ extremities. A set corresponds to telomeres in the shown diploid reference genome R. A diploid segment copy number profile C_R = (a, b) is shown for the genome R with colors (dark/light blue/green) corresponding to A/B labeled segments. C) A derived genome G obtained via multiple large-scale rearrangements from the reference genome R. Red dashed lines correspond to novel adjacencies (e.g., ). A diploid segment copy number profile C_G = (a, b) is shown for the genome G with colors (dark/light blue/green) corresponding to A/B labeled segments. A set 𝒯 (G) = 𝒯 (R) of telomeres in the derived genome G equals to that in the original reference genome R.

When a mutated genome G is derived from a diploid reference R current technologies do not allow us to measure its diploid segment copy number profile C_G directly. Rather there exist several methods [57, 9, 7, 40, 24, 19, 35, 63] that are capable of measuring a pair ,of vectors, where for every segment j an unlabeled (allele-specific) pair represents copy numbers of segments j_A and j_B in G, but without A/B labels explicitly associated with the measured values. In other words, we know that , but it is unclear whether or (example shown in Figure 6).

Figure 5:

constructed on a set {1, 2, …, 12} of segments, and a set of adjacencies, where a set 𝒜(R) corresponds to reference adjacencies in a diploid reference R shown in Figure 4B, and a set represents unlabeled novel adjacencies that were measured from a derived genome G shown in Figure 4C. Telomere vertices 𝒯 (G) =𝒯(R)⊆V are shown as squares, and non-telomere vertices are shown as circles. Solid edges correspond to segment edges in E_S, with dark blue/green edges corresponding to segments labeled A, and light blue/green edges corresponding to segments labeled B. Reference adjacency edges E_R are shown as black-dashed edges, and novel adjacency edges E_N are shown as red-dotted edges.

Figure 6: Description of errors in noise-free segment copy number (SCN) inference for a heterogeneous (i.e., 2 genomes) and haplotype-specific (i.e., A/B labeled segments) sample S = (G₁, G₂) under different limiting assumption about the sample’s structure.

A) A 2-genome proper sample S = (G₁, G₂) with every genome G_i ∈ S depicted both as collections of adjacent blocks as well as corresponding sequences of signed block. B) SCN inference under the assumption that the sample in question is homogeneous (i.e., comprised of a single derived genome) and with no consideration given to the fact that every segment has two distinct A/B instances of it (haploid-reference). In a vector c = [c₁, c₂, c₃, c₄] for a segment j a value c_j corresponds to an average over sums of diploid/allele-specific SCNs across genomes G_i ∈ S. C) SCN inference under the assumption that the sample is homogeneous, but distinguishing between A/B labeled copies of every segment, though not preserving the alleles labels mapping to true A/B labels across segments. Colors encode true labeling (dark blue – A, light blue – B), flipped alleles are shown for segments 2 and 4). In vectors and for a segment j values correspond to averages and of genome- and allele-specific copy number values. D) SCN inference under the assumption that the sample is heterogeneous, but with a haploid-reference assumption. In vectors c₁ = [c_1,1, c_1,2, c_1,3, c_1,4] and c₂ = [c_2,1, c_2,2, c_2,3, c_2,4] for a segment j and genome G_i the value c_i,j equals to the sum of allele-specific copy number values in a genome G_i. E) Allele- and genome specific SCN inference. Colors encode true labeling (dark blue – A, light blue – B), flipped alleles are shown for segment 2 and 4 (i.e., and ).

Furthermore, when a genome G derives from a diploid reference R we can not measure a set A (G) of adjacencies in G directly, but rather we can only measure an obfuscated version of the set 𝒜 _N (G). of novel adjacencies in G. That is, for every novel adjacency (G) we can only measure an unlabeled(i.e., with involved extremities missing the A/B labels) adjacency (e.g., for a derived genome G shown in Figure 4 instead of measuring a novel adjacency. we measure an unlabeled novel adjacency {3^h, 7^h}). There exist several methods capable of producing the unlabeled novel adjacencies both from a standard short-read bulk sequencing data [51, 48, 30, 10, 60] as well as from 3rd-generation sequencing technologies [49, 16, 52, 66, 50, 27, 17].

We note that if a set of unlabeled novel adjacencies is measured from a proper derived genome, it satisfies the generalized infinite sites conditions: since in unlabeled novel adjacencies involved extremities lack A/B labels, only the (unlabeled) extremity-exclusivity constraint (i.e., on unlabeled extremities) must be satisfied, which is achieved, because in proper genome conditions extremity-exclusivity and homologous-extremity-exclusivity guarantee that for every pair of homologous extremities at most one of them is involved in any novel adjacency from 𝒜_N (G), and thus the unlabeled extremity j is also involved in at most one measured unlabeled novel adjacency from .

We assume that large-scale rearrangements that generated a mutated genome G from a diploid reference R have not created novel telomeres (i.e., 𝒯 (G) ⊆ 𝒯 (R)), and formulate the following problem of reconstructing mutated genome from measurement data:

Problem 1. Given a diploid reference R, allele-specific copy number measurements for every segment j, and a set of unlabeled novel adjacencies that satisfies (unlabeled) extremity-exclusivity constraint, find a proper genome G satisfying:

1. for every adjacency either or a ∈ A(R);

2. for every adjacency there exist labels H, H′ ∈ {A, B}, such that ;

3. for every (a_j, b_j) ∈ C_G either or

4. 𝒯 (G) ⊆ 𝒯 (R).

Since the measured unlabeled novel adjacencies do not have the A/B labels, we do not know the true underlying novel adjacencies that produced a measurement. For an unlabeled novel adjacency we defined by a set of the four possible novel adjacencies that can be obtained by A/B labeling extremities in a. For a given set 𝒜 of unlabeled novel adjacencies we define a set ℋ (𝒜) of all possible novel adjacencies as follows: We note that when a set of measured unlabeled novel adjacencies comes from a genome G, it follows that . A union of sets 𝒜 (R) and represents all possible adjacencies that can be present in the observed mutated genome G.

4.2 Diploid Interval Adjacency Graph

We reformulate Problem 1 of finding a proper derived genome G from the measurement data as a graph-theoretic problem. First, we define the diploid interval adjacency graph (DIAG), which can be viewed as a generalization of a breakpoint graph used in the area of comparative genomics [1, 65, 3], or graphs used in the area of structural analysis of normal and cancer genomes with haploid reference structure [36, 42, 32, 12, 15, 35. is constructed on a set {1, 2, …, m} of segments, and a set of adjacencies.

The set V of vertices is in one-to-one correspondence with all segments’ extremities. Formally we define V as follows: The set E of edges in a DIAG is comprised of two types of edges: segment edges E_S and adjacency edges E _𝒜. The set E_S of segment edges represents segments as follows: The set E_𝒜 of adjacency edges is in a one-to-one correspondence with a set of adjacencies: i.e., every adjacency is represented by a corresponding adjacency edge (e.g., an example DIAG is shown in Figure 5).

Every adjacency edge e_a ∈ E _𝒜 that corresponds a reference adjacency a ∈𝒜 (R) we call a reference adjacency edge, and we denote by E_R ⊆ E _𝒜 a set of all reference adjacency edges in E _𝒜. We also define a set E_N = E _𝒜 \ E_R of novel adjacency edges, with edges in E_N respectively corresponding to novel adjacencies in . Since adjacency edges and adjacencies are in one-to-one correspondence we allow ourselves to use adjacencies when referring to adjacency edges and vice versa.

Since every vertex is incident to exactly one segment edge , we define e_S (v) ∈ E_S to be a segment edge incident to a vertex v, and define e_s (j_H) ∈ E_S to be a segment edge corresponding to a segment j_H. Every vertex v ∈ V is incident to at most one reference adjacency edge, and we define e_R(v) ∈ E_R to be a reference adjacency edge containing vertex v, if such adjacency exists. Naturally, we define E_N (v) ⊆ E_N to be a set of novel adjacency edges incident to v ∈ V.

Every chromosome in a derived genome G determines a segment-adjacency edge alternating walk in the corresponding DIAG, that starts and ends at telomere vertices in 𝒯 (G) (examples are shown in the supplement Figure S8B). Such an alternating walk spells out a concatenation of segments from the reference genome, corresponding to a derived chromosome in G. Thus, a derived genome G determines a collection of segment-adjacency edge alternating walks. The number of times a segment edge is traversed (in either direction) across all walks determined by G corresponds to the segment copy number (e.g., ). Similarly, the number of times an adjacency edge is traversed (in either direction) across all walks determined by G corresponds to an adjacency copy number (i.e., the number of times an adjacency corresponding to an edge e is present in G). A genome G thus determines an edge multiplicity function μ : E → 𝒩 on both segment and adjacency edges (example is shown in the supplement Figure S8A). We call the corresponding DIAG G(R, 𝒜_N, μ) a weighted DIAG.

We note that DIAG is allowed to have self-loop adjacency edges that correspond to a self-loop novel adjacencies in . Such self-loop novel adjacencies can be produced by breakage-fusion-bridge cycles, inverted tandem duplications, and other more complex large-scale genome rearrangements that have been observed in cancer [22, 64, 33, 25]. We define by l(a) : E_𝒜→{1, 2} an auxiliary function that outputs 2 if a is a self-loop adjacency (edge), and 1 otherwise. We say that a vertex v ∈V exhibits a copy number balance provided: Similarly, we say that a vertex v ∈ V exhibits a copy number excess provided: The following theorem follows directly from previous work [29, 45]:

Theorem 1. A weighted DIAG G = (V, E, μ), can be decomposed into a collection of segment-adjacency edge alternating walks that start and end at a set 𝒯⊆V of telomere vertices, such that every edge e ∈ E is traversed μ(e) times, if:

1. every non-telomere vertex v ∈ V \𝒯 is copy number balanced,

2. and every telomere vertex v ∈ 𝒯 ⊆V has a copy number excess.

When the derived genome is allowed to have circular chromosomes, which have been extensively observed and studied in cancer [8, 21, 59, 18, 56], Theorem 1 provides not only a necessary, but also a sufficient condition for a derived genome to exist. For an extended discussion about DIAG decomposition into segment-adjacency edge alternating walks please refer to supplementary material section S2.1.

For every unlabeled novel adjacency and a we define by h^E (a) ⊆E_N a subset of novel adjacency edges corresponding to adjacencies in h(a). Furthermore, given a weighted , for every unlabeled novel adjacency we define by a subset of adjacency edges with positive multiplicities as follows: Now we readily reformulate the Problem 1, allowing a derived genome to contain circular chromosomes, into a problem of finding edge multiplicities in the associated DIAG as follows:

Problem 2. Given a , where the set of unlabeled novel adjacencies satisfies (unlabeled) extremity-exclusivity constraint, and allele-specific copy number measurements for every segment j of telomere vertices find an edge multiplicity function μ : E → N such that:

1. for every unlabeled adjacency ;

2. for every self-loop unlabeled adjacency ;

3. for every pair of unlabeled novel adjacencies, such that , there exist and, where H,H ′,H ″ ∈ {A, B};

4. for every segment j, either or ;

5. every non-telomere vertex v∈V \𝒯(R) exhibits copy number balance (eq. (6));

6. every telomere vertex v ∈𝒯(R) ⊆V exhibits either copy number balance (eq. (6)) or copy number excess (eq. (7)).

We note, that finding an edge multiplicity function μ in Problem 2 guarantees the existence of a proper derived genome that determines μ, but such derived genome does not necessarily need to be unique. A resulting weighted DIAG G = (V, E, μ) thus determines a haplotype-specific karyotype of the derived genome in question.

4.3 Multiple derived genomes

The sequencing assays in cancer genomics can involve biological samples that can be genetically heterogeneous (i.e., comprised of cells with different derived genomes, also sometimes referred to in the literature as clones). Let us assume that a sample (i.e., set of genomes) S = (G₁, G₂, …, G_n) in question is comprised of n genomes all of which have derived from a diploid reference R via large-scale rearrangements. A sample S = (G₁, G₂, …, G_n) determines a pair C_S = (A = [a₁, a₂, …, a_n]^T, B = [b₁, b₂, …, b_n]^T) of n ×m diploid segment copy number matrices, where genome-specific segment copy number vectors a_i = [a_i,1, a_i,2, …, a_i,m] and b_i = [b_i,1, b_i,2, …, b_i,m] contain integer values a_i,j, b_i,j ∈ℕ that correspond to the number of times segments j_A and j_B appear in genome G_i ∈ S respectively. We denote by A_[j] = [a_1,j, a_2,j, …, a_n,j]^T and by B_[j] = [b_1,j, b_2,j, …, b_n,j]^T vectors of copy number values for segments j_A and j_B across all genomes G_i ∈S.

For a sample s = (G₁, G₂, …, G_n) we do not measure the pair C_S = (A, B) of its n ×m diploid segment copy matrices directly, but rather we measure a pair of n ×m allele-specific segment copy number matrices, such that for every segment j either or . Examples of allele-specific vs diploid alongside other errors in a noise-free segment copy number inferences with different limiting assumptions about the sample’s structure are shown in Figure 6.

For a sample S = (G₁, G₂, …, G_n) we define by a set of all adjacencies and by a set of all novel adjacencies present in any (subset) of the genomes in S.

Similarly to the case of a single derived genome, out ability to measure novel adjacencies from a sample S = (G₁,G₂, …, G₂) is obfuscated. For every novel adjacency we can only measure an unlabeled counterpart and we also loose the information about which genome(s) in sample S the underlying novel adjacency a is actually present in. We define by a set of unlabeled adjacencies measured from a sample S.

We generalize the previously introduced constraints on possible structures of the derived genomes G_i ∈S for the sample S. We call a sample s = (G₁, G₂, …, G_n) proper if the extremity-exclusivity, homologous-extremity-exclusivity, and homologous-reciprocal-extremity-exclusivity assumptions hold, with a set 𝒜_N (G) substituted with a set 𝒜 (S) (i.e., considering a set 𝒜_N(S) of novel adjacencies across all of the genomes in the observed sample S). Substituting 𝒜_N(G) with 𝒜_N (S) allows us to impose the generalized IS constraints for the whole somatic evolutionar process (i.e., take into account rearrangement that occur on all the branches of the somatic phylogenetic tree) that produced the observed sample S. We note that if a set of unlabeled novel adjacencies comes from a roper sample S, then satisfies the generalized IS conditions (by satisfying the (unlabeled) extremity-exclusivity constraint). Moreover, we note that if a sample S = (G₁, G₂, …, G_n) is proper, then any subsample (including individual derived genomes G_i ∈ S) of S is also proper.

A generalized version of Problem 1 for a sample S = (G₁, G₂, …, G_n) is stated below:

Problem 3. Given a diploid reference R, a pair of n × m allele-specific segment copy number matrices, and a set of measured unlabeled novel adjacencies that satisfies (unlabeled) extremity-exclusivity constraint, find a proper sample s = (G₁, G₂, …, G_n) such that:

1. for every adjacency , either or

2. for every adjacency , there exists a unique pair of labels, such that ;

3. for every segment j, either or ;

4. for every genome G_i ∈ s, the telomere set τ (G_i) ⊆ τ (R).

In a sample S = (G₁, G₂, …, G_n) and a set of unlabeled novel adjacencies measured form S, we observe a DIAG . Every genome G_i ∈ S determines a genome-specific edge multiplicity function μ_i : E ⟶ ℕ as was previously described in a case of a single derived genome.

We extend previously introduced copy number balancing conditions (6) and (7) on vertices in V, using genome-specific edge multiplicity functions. For a genome G_i ∈ S, a vertex v ∈ V exhibits copy number balance provided: and a vertex v ∈ V exhibits copy number excess provided: For every unlabeled adjacency and a genome G_i ∈ S we define by a subset of novel adjacency edges in h^E (a) with positive copy number as determined by the genome-specific edge multiplicity function µ_i as follows: and we then naturally generalize the definition of for the sample S = (G₁, G₂, …, G_n) case: For every segment j_H we define by µ_[j,H] = [µ₁(e_S (j_H)), µ₂(e_S (j_H)), …, µ_n(e_S (j_H))]^T a vector of genome-specific edge multiplicity functions’ values on the segment edge e_S (j_H) ∈ E_S.

We now reformulate a general Problem 3 of finding a proper sample S = (G₁, G₂, …, G_n) in terms of finding edge multiplicity functions µ₁, µ₂, …, µ_n : E ⟶ ℕ in the corresponding DIAG as follows:

Problem 4. Given a DIAG G(R, Ã_N) = (V, E), where a set Ã_N of unlabeled novel adjacencies satisfies (unlabeled) extremity-exclusivity constraint, and a pair of n × m allele-specific segment copy number matrices, find edge multiplicity functions µ₁, µ₂, …, µ_n : E ⟶ ℕ such that:

1. for every adjacency ;

2. for every i ∈ [n] and every adjacency ;

3. for every pair a = {u, j^h}, b = {(j + 1)^t, v} ∈ Ã_N of unlabeled novel adjacencies, such that , there exists and , where H, H′, H″ ∈ {A, B};

4. for every segment j, either or ;

5. for every i ∈ [n] and every non-telomere vertex v ∈ V \ 𝒯 (R) the equality (9) holds;

6. for every i ∈ [n] and every telomere vertex v ∈ 𝒯 (R) ⊆ V either the equality (9) or the inequality (10) hold.

4.4 3rd generation sequencing technologies and novel adjacency groups

Besides the cost-efficient next-generation sequencing technologies (i.e., bulk-sequencing with short paired-end reads), there exist other, more expensive, 3rd-generation sequencing technologies (e.g., single-cell, barcoded linked reads, and long-read sequencing) that can provide additional insight about measured unlabeled novel adjacencies [16, 52, 66, 49, 50, 27, 17]. We observe a sample S = (G₁, G₂, …, G_n) and a set Ã_N of unlabeled novel adjacencies coming from S. We define a 3rd-generation sequencing experiment as either all reads obtained in a single-cell sequencing essay, a set of reads annotated with the same barcode in the barcoded sequencing experiment, or a single long read obtained by a long-read sequencing technology. Let us assume that a 3rd-generation sequencing experiment on a S = (G₁, G₂, …, G_n) identifies a group u ⊆ Ã_N of unlabeled novel adjacencies. Since every 3rd-generation sequencing experiment is conducted either on a single cell (e.g., single-cell) and thus produces data from a single derived genome, or on a part of a single derived chromosome (e.g., bar-codded, long-range) present in a single derived genome, the group u of unlabeled adjacencies is guaranteed to originate from a single derived genome G_i ∈ S.

We note that for every unlabeled novel adjacency measured from a sample S = (G₁, G₂, …, G_n) there exist a unique novel adjacency counterpart , when S is proper. Or, more formally, |h (a) ∩𝒜_N(S) |= 1. Thus, for every group u⊆ Ã_N of unlabeled novel adjacencies measured via a single 3rd-generation sequencing experiment on a proper sample S = (G₁, G₂, …, G_n) for at least one genome G_i ∈ S we have:

4.5 Uncertainty in copy number measurements

As we have stated before, there exist several methods that for a given sample S = (G₁, G₂, …, G_n) aim to infer a pair of n × m allele-specific segment copy number matrices. Loss of explicit information about A/B labels across segments in such inference (while preserving segment’s specific allele separation across all genomes in S) is often not the only limitation of these methods. It is also often the case for sequences of reference-adjacent segments to be grouped together into larger non overlapping fragments, for which the allele-specific copy numbers are inferred.

More formally, we call a sequence (j,j+1,….,j+l) of reference adjacent segments a fragment and denote it by f_[j,l]. We denote by ℱ a collection of non overlapping fragments that cover all of the segments.

When allele-specific copy numbers are inferred on fragments, rather than individual segments, we naturally obtain the same copy number values for all segments within every overarching fragment, which may be incorrect for some or even all segments within the observed fragment. On the other hand, allele-specific nature of the inferred fragments copy numbers preserves allele separation not only across genomes, but also across segments within each fragment. We thus view available allele-specific copy numbers for fragments as an approximation of the true underlying segment copy numbers, and try to infer the true underlying diploid segment copy number values, while leveraging the allele separation preservation across segments within each fragment.

Let us observe a pair of n × m allele-specific segment copy number matrices, a pair C = (A, B) of n × m diploid segment copy number matrices, and a set ℱ of fragments. For every fragment f ∈ ℱ we define a length-weighted copy number distance as follows: where L(j) is the total number of base pairs (i.e., length) of segment j. We further define a copy number distance between pairs C and of diploid and allele-specific segment copy number matrices as follows: We now extend the previous Problem 3 of finding a sample from the measured data to the case when the measured allele-specific segment copy numbers are noisy and (optionally) information from 3rd-generation sequencing experiments is available:

Problem 5. Given a diploid reference R, a pair of n × m allele-specific segment copy number matrices, a set ℱ of fragments, a set Ã_N of measured unlabeled novel adjacencies that satisfies (unlabeled) extremity-exclusivity constraint, and (optionally) a set 𝒰 of groups of unlabeled novel adjacencies, find a proper sample s = (G₁, G₂, …, G_n) such that:

1. for every adjacency either or a ∈ 𝒜(R);

2. for every adjacency there exists a unique pair H, H′ ∈ {A, B} of labels, such that ;

3. for every adjacency group u ∈𝒰, there exists (at least one) genome G_i ∈ s such that |𝒜_N (G_i) ∩ℋ (u) =|u|;

4. for every genome G_i ∈s the 𝒯 (G_i)⊆ 𝒯 (R); and the copy number distance is minimized.

A reformulation of Problem 5 in terms of finding edge multiplicity functions on the edges of the corresponding DIAG is provided below:

Problem 6. Given a , where a set of unlabeled measured novel adjacencies satisfies (unlabeled) extremity-exclusivity constraint, a (optionally) set 𝒰 of groups of unlabeled novel adjacencies, a pair of n × m allele-specific segment copy number matrices, and a set ℱ of fragments, find edge multiplicities functions μ₁, μ₂, …, μ_n : E → ℕ such that:

1. for every adjacency we have ;

2. for every i ∈ [n] and every adjacency ;

3. for every adjacency group u ∈ 𝒰 there exists (at least one) i ∈ [n] such that ;

4. for every pair of unlabeled novel adjacencies, such that , there exists and , where H, H′, H″ ∈ {A, B};

5. for every i ∈ [n] and every non-telomere vertex v ∈ V \ 𝒯 (R) the equality (9) holds;

6. for every i ∈ [n] and every telomere vertex v ∈ 𝒯 (R) ⊆ V either the equality (9) or the inequality (10) hold; and such that for a pair C_μ = (A_μ, B_μ) of diploid segment copy number matrices (determined by values of edge multiplicity functions μ₁, μ₂, …, μ_n on segments edges E_S), the copy number distance is minimized.

In the Supplement, we derive a mixed integer linear program (MILP) optimization problem that solves Problem 6.

4.6 Deriving extremities and novel adjacencies from data

Segment copy number inference methods often define a fixed-size partition of the reference genome into segments and thus constrain the coordinates of segments extremities. Every measured unlabeled novel adjacency determines a pair {(chr₁, coord₁, str₁), (chr₂, coord₂, str₂)}, where chr_idetermines the chromosome of origin the genomic loci i, coord_i determined the coordinate of the genomic loci i on the respective chromosome chr_i, and str_i ∈ {+, -} determined the strand of origin of the genomic loci i.

Extremities of segments that are inferred by methods that measure clone- and allele-specific segment copy numbers and those involved in measured unlabeled novel adjacencies do not always align. Moreover, there is often a small uncertainty in the exact values of the coordinate coord_i of the genomic loci i involved in a novel adjacencies.

We first address the issue of refining the positions of extremities involved in reciprocal novel adjacencies. For every sample S we first observe all unlabeled novel adjacencies Ã_N measured from S and sort the positions involved in adjacencies from Ã_N on every chromosome (in descending order of the coordvalues). Then, using a sliding window approach, we update the coordinates for any consecutive pair p_i, p_j of positions which resembles a reciprocal signature: i.e., if the distance | coord_i - coord_j | was less than 50 base pairs and str_i ≠ str_j, we update the values of the coordinates in positions p_i and p_j so that they have a coordinate distance of 1, with the position having a + strand appearing prior to the position having a - strand (Figure 7A).

Figure 7: Derivation of the input for ReMixTand RCK.

A) An example of derivation of coordinates that resemble a reciprocal signature in measured unlabeled novel adjacencies on a chromosome a. Positions p₁ = (a, 100, +) and p₂ = (a, 107, -) have reciprocal signature (i.e., | coord₁ - coord₂ | = 7 < 50 and str₁ = – ≠ str₂ = +). Updated pair of coordinates constitutes a reciprocal location. B) An example of partitioning of a set ℱ = {f₁, f₂, f₃, f₄} of fragments from allele-specific copy number calls into a set S = {s₁, s₂, s₃, s₄, s₅, s₆, s₇, s₈} of segments. Extremities of segments in S correspond to either preprocessed coordinates of unlabeled novel adjacencies (e.g., ) or to the extremities of fragments in ℱ (e.g. ).

Then, for allele-specific segment copy number input (e.g., from Battenbergand HATCHet) we partition fragments, on which allele-specific copy number values are measured, into smaller segments such that extremities of obtained segments either correspond to the coordinates of extremities involved in the preprocessed novel adjacencies from Ã_N, or to the extremities of the original fragments (Figure 7B). Copy numbers on newly obtained segments are inherited from the values of the “parent” fragments.

Lastly, in order to compute length-weighted segment copy number distances between RCK, ReMixT, Battenberg, and HATCHetinferences on the prostate cancer samples, we refined the fragments/segments on which the copy numbers were inferred as demonstrate in Figure S10).

Code availability

RCK is available on GitHub at https://github.com/raphael-group/RCK.

Acknowledgments

This work is supported by a US National Institutes of Health (NIH) grants R01HG007069 and U24CA211000 and US National Science Foundation (NSF) CAREER Award (CCF-1053753) to BJR.

Footnotes

• ¹ We assume that every segment appears exactly once in a forward orientation the reference genome.