# 1 Introduction

The R package decompTumor2Sig has been developed to decompose individual tumor genomes (i.e., the lists of somatic single nucleotide variants identified in individual tumors) according to a set of given mutational signatures—a problem termed signature refitting—using a quadratic-programming approach.

Mutational signatures can be either of the form initially proposed by Alexandrov et al. (Cell Rep. 3:246–259, 2013 and Nature 500:415–421, 2013)—in the following called “Alexandrov signatures”—or of the simplified form proposed by Shiraishi et al. (PLoS Genet. 11:e1005657, 2015)—in the following called “Shiraishi signatures”.

For each of the given mutational signatures, decompTumor2Sig determines their contribution to the load of somatic mutations observed in a tumor.

Note: for all functions provided by decompTumor2Sig, please see the inline R help page for further details and explanations.

Note: here and in the following, when naming “mutations” we intend single nucleotide variants (SNVs).

## 1.1 Papers / how to cite

Krüger S, Piro RM (2018) decompTumor2Sig: Identification of mutational signatures active in individual tumors (submitted).

Krüger S, Piro RM (2017) Identification of mutational signatures active in individual tumors. PeerJ Preprints 5:e3257v1 (Proceedings of the NETTAB 2017 Workshop on Methods, Tools & Platforms for Personalized Medicine in the Big Data Era, October 16-18, 2017 in Palermo, Italy).

BibTeX:

@UNPUBLISHED(krueger-decompTumor2Sig-paper,
author = "Kr{\"u}ger, Sandra and Piro, Rosario Michael",
title = "decompTumor2Sig: Identification of mutational signatures active in individual tumors",
note = "(unpublished)",
year = 2018
);

@ARTICLE(krueger-decompTumor2Sig-nettab,
author = "Kr{\"u}ger, Sandra and Piro, Rosario Michael",
title = "Identification of mutational signatures active in individual tumors",
journal = "PeerJ Preprints",
volume = "5",
number = "e3257v1",
year = 2017
);

# 2 Installation

decompTumor2Sig requires several CRAN and Bioconductor R packages to be installed. Dependencies are usually handled automatically, when installing the package using the following commands:

[[[ IMPORTANT NOTE: decompTumor2Sig has been submitted to Bionconductor but not yet included; please follow the manual installation procedure! ]]]

source("http://bioconductor.org/biocLite.R")
biocLite("decompTumor2Sig")

In the unlikely case that a manual installation is required, e.g., if you do not install decompTumor2Sig via Bioconductor (which is highly recommended), before installing decompTumor2Sig make sure the following packages are installed:

CRAN packages:

Bioconductor packages:

For some additional functionalities, you may also want to install the R package pmsignature which is neither part of CRAN nor of Bioconductor and must be downloaded and installed manually (available at: https://github.com/friend1ws/pmsignature).

CRAN packages can be installed from R using the following command:

install.packages("<package_name>")

Bioconductor packages can be installed from R using the following command:

source("http://bioconductor.org/biocLite.R")
biocLite("<package_name>")

Sometimes, it may also be useful to upgrade Bioconductor:

source("http://bioconductor.org/biocLite.R")
biocLite("BiocUpgrade")

# 3 Input data

decompTumor2Sig works with two kinds of input data:

1. a set of given mutational signatures, and
2. a set of somatic mutations (single nucleotide variants) observed in a tumor genome.

Additionally, decompTumor2Sig requires the genomic sequence (in form of a BSgenome object) to determine neighboring nucleotides of the mutated bases. It may also require transcript annotations (in form of a TxDb object) in case the given mutational signatures take information on the transcription direction into account.

## 3.1 Mutational signatures

Mutational signatures can be loaded in two different formats: Alexandrov-type signatures and Shiraishi-type signatures.

### 3.1.1 Alexandrov signatures

Alexandrov signatures are specified in the format used at the COSMIC Mutational Signatures website (http://cancer.sanger.ac.uk/cosmic/signatures -> Download signatures).

Example:

Substitution Type  Trinucleotide  Somatic Mutation Type  Signature 1     Signature 2     ...
C>A                ACA            A[C>A]A                0.011098326166  0.000682708227  ...
C>A                ACC            A[C>A]C                0.009149340734  0.000619107232  ...
C>A                ACG            A[C>A]G                0.001490070468  0.000099278956  ...
C>A                ACT            A[C>A]T                0.006233885236  0.000323891363  ...
...
T>G                TTG            T[T>G]G                0.002031076880  0.000206615168  ...
T>G                TTT            T[T>G]T                0.004030128160  0.000023598204  ...

To load Alexandrov-type signatures, use the command loadAlexandrovSignatures. By the default, the command loads the signatures directly from COSMIC:

signatures <- loadAlexandrovSignatures()

Alternatively, the signatures can be loaded from a file of the format described above:

signatures <- loadAlexandrovSignatures(file="<signature_file>")

Please note: by default, Alexandrov signatures report mutation frequencies for nucleotide triplets where the mutated base is at the center. Also, the basic Alexandrov signatures do not take transcription direction into account when computing mutation frequencies.

### 3.1.2 Shiraishi signatures

Shiraishi signatures are specified as matrices (without headers and row names; as flat files, one file per signature).

Format (see Shiraishi et al. PLoS Genetics 11(12):e1005657, 2015):

• The first row: Frequencies of the six possible base changes C>A, C>G, C>T, T>A, T>C, and T>G. Please note that due to the complementarity of base pairing these six base changes already include A>? and G>?.

• The following 2k rows (for k up- and downstream flanking bases): Frequencies of the bases A, C, G, and T, followed by two 0 values.

• The optional last row (only if transcription direction is considered): Frequencies of occurrences on the transcription strand, and on the opposite strand, followed by four 0 values.

Example for sequence patterns of 5 bases (mutated base plus two up- and downstream bases) with transcriptional direction:

1.8874e-14   0.10974   0.045918   0.11308   0.07429   0.65697
3.8079e-01   0.12215   0.191456   0.30561   0.00000   0.00000
1.5311e-01   0.34214   0.179774   0.32497   0.00000   0.00000
1.2378e-01   0.10243   0.163461   0.61032   0.00000   0.00000
3.4891e-01   0.15346   0.156687   0.34094   0.00000   0.00000
5.6435e-01   0.43565   0.000000   0.00000   0.00000   0.00000

Among its examples, the decompTumor2Sig package provides a small set of four signatures in flat files. These signatures were obtained from 21 breast cancer genomes (Nik-Zainal et al. Cell 149:979–993, 2012) using the R pmsignature package (Shiraishi et al. PLoS Genet. 11:e1005657, 2015).

To load these flat files as signatures, you can use the following example:

# take the signature flat files provided with decompTumor2Sig
sigfiles <- system.file("extdata",
paste0("Nik-Zainal_PMID_22608084-pmsignature-sig",1:4,".tsv"),
package="decompTumor2Sig")

# load the signature flat files
signatures <- loadShiraishiSignatures(files=sigfiles)

### 3.1.3 Get signatures from the package pmsignature

The third possibility is to convert Shiraishi-type signatures directly from the package that computes them (pmsignature; Shiraishi et al. PLoS Genet. 11:e1005657, 2015).

Example workflow:

library(pmsignature)

# use pmsignature to determine signatures from the set of tumor genomes
numsig = 4
G <- readMPFile("myMutDB.MPF.txt.gz", numBases = 5, trDir = TRUE)
Param <- getPMSignature(G, K=numsig)

# extract the signatures from the parameters estimated by pmsignature
signatures <- getSignatureListFromEstimatedParameters(Param)

Please note that pmsignature is neither part of CRAN nor of Bioconductor and must be downloaded and installed manually (available at: https://github.com/friend1ws/pmsignature). To load mutational signatures without pmsignature being installed, see the previous sections.

### 3.1.4 Conversion of Alexandrov signatures to Shiraishi signatures

An Alexandrov-type signature can be converted into a Shiraishi-type signature (but not vice versa due to the loss of information). Consider the following example:

sign_a <- loadAlexandrovSignatures()

sign_s <- convertAlexandrov2Shiraishi(sign_a)

Note: since the standard Alexandrov-type signatures regard nucleotide triplets and do not take transcription direction into account, the obtained Shiraishi-type signatures will also be limited to triplets without information about transcription direction.

Important: Please be aware that signatures are initially not determined in isolation but as a set of signatures derived from a commonly large set of tumor genomes (Alexandrov et al. Cell Rep. 3:246–259, 2013; Alexandrov et al. Nature 500:415–421, 2013; Shiraishi et al. PLoS Genet. 11:e1005657, 2015). Therefore, the biological meaning of converting signatures is not well defined, and the approach should be taken only with caution! As an example for the possible outcome, please see our paper (Krüger and Piro, 2018).

## 3.2 Somatic mutations from individual tumors

Information on the somatic mutations found in a tumor can be given in the following formats.

### 3.2.1 Variant Call Format (VCF)

The somatic mutations of a tumor genome can be loaded from a VCF file. For detailed information on this format (including an example), see https://samtools.github.io/hts-specs/VCFv4.2.pdf.

Mutations from multiple tumor genomes can also be loaded from a multi-sample VCF file.

As an example, the decompTumor2Sig package provides the somatic mutations for the set of 21 breast cancer genomes originally published by Nik-Zainal et al (Cell 149:979–993, 2012). The dataset has been converted from MPF (see below) to VCF.

Example workflow:

# load the reference genome and the transcript annotation database
refGenome <- BSgenome.Hsapiens.UCSC.hg19::BSgenome.Hsapiens.UCSC.hg19
transcriptAnno <- TxDb.Hsapiens.UCSC.hg19.knownGene::TxDb.Hsapiens.UCSC.hg19.knownGene

# take the breast cancer genomes from Nik-Zainal et al (PMID: 22608084)
gfile <- system.file("extdata",
"Nik-Zainal_PMID_22608084-VCF-convertedfromMPF.vcf.gz",
package="decompTumor2Sig")

# load the cancer genomes in VCF format
trDir=TRUE, refGenome=refGenome, transcriptAnno=transcriptAnno, verbose=FALSE)

When loading somatic mutations of tumor genomes with loadGenomesFromVCF, they are preprocessed to determine mutation frequencies according to specific sequence characteristics which can be controlled by the following arguments:

• type: type of signatures that will be used with the genomes, “Shiraishi” or “Alexandrov”
• numBases: the number of nucleotides/bases of the mutated sequence (where the mutated base is at the center)
• trDir: whether the transcription direction should be taken into account (default: TRUE)
• refGenome: the reference genome from which to obtain the flanking bases of the mutated base
• transcriptAnno: the transcript annotation database from which to obtain the transcription direction (if needed)

### 3.2.2 Mutation Position Format (MPF)

Alternatively, somatic mutations can be loaded from an MPF file.

Example:

patient1   chr1   809687    G   C
patient1   chr1   819245    G   T
patient1   chr2   2818266   A   G
patient1   chr2   3433314   G   A
patient2   chr1   2927666   A   G
patient2   chr1   3359791   C   T

The five columns contain

1. the name of the sample (or tumor ID);
2. the chromosome name;
3. the position on the chromosome;
4. the reference base at that position (A, C, G, or T);
5. the alternate or variant base (A, C, G, or T).

Hence, with an MPF file, too, multiple tumors can be described.

As an example, the decompTumor2Sig package provides the somatic mutations for the set of 21 breast cancer genomes originally published by Nik-Zainal et al (Cell 149:979–993, 2012).

Example workflow:

# load the reference genome and the transcript annotation database
refGenome <- BSgenome.Hsapiens.UCSC.hg19::BSgenome.Hsapiens.UCSC.hg19
transcriptAnno <- TxDb.Hsapiens.UCSC.hg19.knownGene::TxDb.Hsapiens.UCSC.hg19.knownGene

# take the breast cancer genomes from Nik-Zainal et al (PMID: 22608084)
gfile <- system.file("extdata", "Nik-Zainal_PMID_22608084-MPF.txt.gz",
package="decompTumor2Sig")

# load the cancer genomes in MPF format
trDir=TRUE, refGenome=refGenome, transcriptAnno=transcriptAnno, verbose=FALSE)

Note: the preprocessing of the somatic mutations into mutation frequencies can be controlled with the same function arguments that have been described above for loadGenomesFromVCF.

### 3.2.3 Get somatic mutations from the package pmsignature

The third possibility to get the somatic mutations from one or more tumor genomes is to convert them directly from a tumor data object loaded using the pmsignature package (Shiraishi et al. PLoS Genet. 11:e1005657, 2015).

Example workflow:

# get breast cancer genomes from Nik-Zainal et al (PMID: 22608084) in the format produced by
# pmsignature (PMID: 26630308)
pmsigdata <- system.file("extdata",
"Nik-Zainal_PMID_22608084-pmsignature-G.Rdata",
package="decompTumor2Sig")

# extract the genomes from the pmsignature G object
genomes <- getGenomesFromMutationFeatureData(G, normalize=TRUE)

Please note that pmsignature is neither part of CRAN nor of Bioconductor and must be downloaded and installed manually (available at: https://github.com/friend1ws/pmsignature). To load somatic mutations without pmsignature being installed, see the previous sections.

Important: the argument normalize, that can be specified for getGenomesFromMutationFeatureData, controls whether the function should simply count the number of occurrences or whether it provides (normalized) fractions/percentages of mutations among the total set. Normalization is the default and is what should be used for determining the contribution of individual signatures to the mutational load of a tumor. normalize=FALSE should be used only in case you are interest how many somatic mutations of the single signature categories can be found in a tumor; it should not be used for further processing with decompTumor2Sig!

### 3.2.4 Get somatic mutations from a VRanges object

The Bioconductor package VariantAnnotation provides the class VRanges which can be used to store mutation information. decompTumor2Sig allows to extract single nucleotide variants (SNVs) from such an object and convert them into the tumor genomes’ mutation frequencies using the function convertGenomesFromVRanges, as in the following example workflow:

# load the reference genome and the transcript annotation database
refGenome <- BSgenome.Hsapiens.UCSC.hg19::BSgenome.Hsapiens.UCSC.hg19
transcriptAnno <- TxDb.Hsapiens.UCSC.hg19.knownGene::TxDb.Hsapiens.UCSC.hg19.knownGene

# take the breast cancer genomes from Nik-Zainal et al (PMID: 22608084)
gfile <- system.file("extdata",
"Nik-Zainal_PMID_22608084-VCF-convertedfromMPF.vcf.gz",
package="decompTumor2Sig")

# get the corresponding VRanges object (using the VariantAnnotation package)
library(VariantAnnotation)

# convert the VRanges object to the decompTumor2Sig format
genomes <- convertGenomesFromVRanges(vr, numBases=5, type="Shiraishi",
trDir=TRUE, refGenome=refGenome, transcriptAnno=transcriptAnno, verbose=FALSE)

# 4 Workflow

(Note: The following examples are for illustrative purpose only and need not be biologically meaningful.)

Once both the tumor genome(s) and the given mutational signatures have been loaded (see above), the contribution of the given signatures to the somatic mutations in individual tumors can be determined using the following workflow.

In the following, we assume to have the signatures in a list object named signatures and the mutation frequencies of the tumor genome(s) in another list object named genomes.

Taking, for example, Shiraishi-type signatures (example from Section 3.1.2) and loading the mutation data accordingly (e.g., example from Section 3.2.1), we get:

signatures:

$Nik-Zainal_PMID_22608084-pmsignature-sig1.tsv [,1] [,2] [,3] [,4] [,5] [,6] [1,] 0.05606328 7.038910e-02 0.3933106 0.1310378 0.2079748 0.1412245 [2,] 0.27758477 2.107542e-01 0.2354323 0.2762287 0.0000000 0.0000000 [3,] 0.33081021 2.534743e-01 0.2366254 0.1790902 0.0000000 0.0000000 [4,] 0.21472304 2.665627e-09 0.5523105 0.2329664 0.0000000 0.0000000 [5,] 0.22640542 2.002424e-01 0.3211338 0.2522184 0.0000000 0.0000000 [6,] 0.50140066 4.985993e-01 0.0000000 0.0000000 0.0000000 0.0000000$Nik-Zainal_PMID_22608084-pmsignature-sig2.tsv
[,1]       [,2]         [,3]         [,4]        [,5]        [,6]
[1,] 3.918463e-05 0.99844829 8.008569e-07 6.100149e-33 2.50988e-33 0.001511729
[2,] 1.887881e-01 0.45515661 9.147057e-02 2.645848e-01 0.00000e+00 0.000000000
[3,] 1.076492e-02 0.01850276 2.897822e-03 9.678345e-01 0.00000e+00 0.000000000
[4,] 3.904620e-01 0.08405818 1.403961e-02 5.114402e-01 0.00000e+00 0.000000000
[5,] 3.775239e-01 0.13305720 2.424492e-01 2.469697e-01 0.00000e+00 0.000000000
[6,] 5.136780e-01 0.48632204 0.000000e+00 0.000000e+00 0.00000e+00 0.000000000

[...]

genomes:

$PD3851a [C>A] [C>G] [C>T] [T>A] [T>C] [T>G] mut 0.1913161 0.1031208 0.3487110 0.1166893 0.156038 0.08412483 -2 0.2605156 0.2306649 0.1831750 0.3256445 0.000000 0.00000000 -1 0.2862958 0.2388060 0.1791045 0.2957938 0.000000 0.00000000 1 0.2795115 0.1886024 0.2686567 0.2632293 0.000000 0.00000000 2 0.2727273 0.2184532 0.1981004 0.3107191 0.000000 0.00000000 tr 0.4735414 0.5264586 0.0000000 0.0000000 0.000000 0.00000000$PD3890a
[C>A]     [C>G]     [C>T]     [T>A]     [T>C]     [T>G]
mut 0.1543062 0.2500000 0.2312600 0.1164274 0.1311802 0.1168262
-2  0.3026316 0.2109250 0.1925837 0.2938596 0.0000000 0.0000000
-1  0.2360447 0.2296651 0.1646730 0.3696172 0.0000000 0.0000000
1   0.2990431 0.2085327 0.1527113 0.3397129 0.0000000 0.0000000
2   0.2910686 0.1981659 0.1885965 0.3221691 0.0000000 0.0000000
tr  0.4860447 0.5139553 0.0000000 0.0000000 0.0000000 0.0000000

[...]

Important note: it is imperative that the mutation frequencies represented by both signatures and genomes have been computed with the same characteristics. That is, if the signatures refer to sequences of 5 nucleotides/bases (with the mutated base at the center), then also the genomes must have been loaded for 5 bases. If the signatures have been produced taking the transcription direction into account, then also the genomes must have been loaded taking the transcription direction into account, and so on.

## 4.1 Visualizing genome characteristics and mutational signatures

(Note: this functionality is not essential for decompTumor2Sig, we therefore only wrap the visualization capabilities of the package pmsignature which is neither part of CRAN nor of Bioconductor and must be downloaded and installed manually from https://github.com/friend1ws/pmsignature.)

Given that the signatures and the genomes (if loaded appropriately) have the same format and contain the same type of information (fractions/percentages of somatic mutations that have specific features, e.g., specific neighboring bases), both can essentially be visualized in the same way.

The function plotMutationDistribution takes as an input either a single signature or the mutation frequencies of an individual tumor genome (either of Alexandrov- or of Shiraishi-type) and converts them such that they can be plotted using the functionalities provided by the package pmsignature.

To plot, for example, Alexandrov signature 1 (obtained as described in Section 3.1.1):

signatures <- loadAlexandrovSignatures()

plotMutationDistribution(signatures[[1]])

To plot the first of the Shiraishi signatures provided with this package (see Section 3.1.2):

sigfiles <- system.file("extdata",
paste0("Nik-Zainal_PMID_22608084-pmsignature-sig",1:4,".tsv"),
package="decompTumor2Sig")

plotMutationDistribution(signatures[[1]])

For comparison, the following example loads the tumor genomes provided with this package using the Alexandrov model (see Section 3.2.1) and plots the mutation frequencies of the first genome:

refGenome <- BSgenome.Hsapiens.UCSC.hg19::BSgenome.Hsapiens.UCSC.hg19
gfile <- system.file("extdata",
"Nik-Zainal_PMID_22608084-VCF-convertedfromMPF.vcf.gz",
package="decompTumor2Sig")
trDir=FALSE, refGenome=refGenome, verbose=FALSE)

plotMutationDistribution(genomes[[1]])

## 4.2 Explained variance as a function of the number of signatures

In many cases already a small subset of the given signatures is sufficient to explain the major fraction of the variance of the mutation frequencies observed in a single tumor genome.

The explained variance can be determined by comparing the true mutation frequencies $$g_i$$ of the tumor genome—where $$i$$ is one of the mutation patterns of the signature model (e.g., triplet mutations for Alexandrov signatures, or base changes or flanking bases for Shiraishi signatures)—to the mutation frequencies $$\hat{g}_i$$ obtained when recomposing/reconstructing the mutation frequencies of the tumor genome from the mutational signatures and their computed exposures/contributions. (See Section 4.3 for computing the exposures/contributions, and Section 4.4 for reconstructing the mutation frequencies of a tumor.)

For the Alexandrov model, the explained variance can be estimated as:

$\mathrm{evar} = 1 - \frac{\sum_i{(g_i - \hat{g}_i)^2}}{\sum_i({g_i} - \bar{g})^2}$

where the numerator is the residual sum of squares (RSS) between the predicted and true mutation frequencies of the tumor genome, and the denominator can be interpreted as the deviation from a tumor genome with a uniform mutation frequency of 1/96 (which is precisely the average mutation frequency $$\bar{g}$$).

For the Shiraishi model, $$\bar{g}$$ does not describe a tumor genome with uniform mutation frequencies, and hence the explained variance is estimated as:

$\mathrm{evar} = 1 - \frac{\sum_i{(g_i - \hat{g}_i)^2}}{\sum_i({g_i} - {g}^{*}_i)^2}$

where $$g^{*}$$ is a uniform tumor model that uses mutation frequencies of 1/6 for the six possible base changes, 1/4 for each of the possible flanking bases, and 1/2 for each of the two possible transcription-strand directions. For more details, please see Krüger and Piro (2018).

The function plotExplainedVariance allows to visually analyze how many signatures are necessary to explain certain fractions of the variance of a tumor genome’s mutational load.

For an increasing number K of signatures, the highest variance explained by any possible subset of K signatures will be plotted. This can help to evaluate what minimum threshold for the explained variance could be used to decompose tumor genomes with the function decomposeTumorGenomes (see below).

### 4.2.1 Example: input data

As a simple example, load the set of 15 Shiraishi-type signatures (object signatures) provided with this package, that we obtained with the package pmsignature from 435 tumor genomes with at least 100 somatic mutations from ten different tumor entities (data obtained from Alexandrov et al. Nature 500:415-421, 2013; for the analysis, see our papers mentioned in Section 1.1):

# take the 15 Shiraishi signatures obtained from 435 tumor genomes from Alexandrov et al.
sfile <- system.file("extdata",
"Alexandrov_PMID_23945592_435_tumors-pmsignature-15sig.Rdata",
package="decompTumor2Sig")
load(sfile)

This provides an object signatures with 15 Shiraishi signatures for mutated subsequences of length 5 (five nucleotides with the mutated base at the center) and taking transcription direction into account.

Now load the tumor genomes (object genomes) provided with this package, as described in Section 3.2.1.

### 4.2.2 Example: plot the explained variance

The explained variance can be plotted only for a single tumor genome. Plotting the explained variance of the first tumor genome when using increasing subsets of the 15 mutational signatures (see above), for example, can be done with the following command:

plotExplainedVariance(genomes[[1]], signatures, minExplainedVariance=0.9,
minNumSignatures=2, maxNumSignatures=NULL)

The function plotExplainedVariance takes the following arguments:

• genome: the mutation frequencies of a single tumor genome (according to the Alexandrov or Shiraishi model)
• signatures: the set of given signatures (must be of the same model as the genome)
• minExplainedVariance (default is NULL): if specified, the smallest subset of signatures necessary to explain at least this fraction of the variance is highlighted in red (including the list of the signatures in the subset)
• minNumSignatures (default is 2): take at least this minimum number of signatures
• maxNumSignatures (default is NULL, i.e., all): if specified, take at most this maximum number of signatures

Note: the function plotExplainedVariance will evaluate for each number of signatures K all possible subsets of K signatures to compute the highest explained variance for K. This is can take very long if the total number of given signatures is too high.

## 4.3 Decomposing tumor genomes to determine the contributions of the mutational signatures

This is the heart of the functionality of decompTumor2Sig, its main purpose.

Given a tumor genome and a set of mutational signatures (that represent mutational processes like UV light, smoking, etc; see Alexandrov and Stratton, Curr Opin Genet Dev 24:52-60, 2014), we would like to estimate how strongly the different signatures (processes) contributed to the overall mutational load observed in the tumor.

The result will be a vector of weights/contributions (or “exposures”) which indicate the fractions/percentages of somatic mutations which can likely be attributed to the given signatures.

Lets take, for example, the signature and tumor data used for the example in Section 4.2 (see there).

To compute the contributions of the 15 given signatures to the first tumor genome, we can run the following command …

exposure <- decomposeTumorGenomes(genomes[1], signatures)

… and get the following exposures (contributions) for the 15 signatures:

> exposure
$PD3851a sign_1 sign_2 sign_3 sign_4 sign_5 sign_6 4.874215e-02 1.237708e-01 5.984978e-02 5.764690e-02 1.426703e-20 1.109195e-01 sign_7 sign_8 sign_9 sign_10 sign_11 sign_12 8.766150e-02 5.115825e-02 2.758243e-02 9.180615e-02 3.070582e-02 3.256016e-02 sign_13 sign_14 sign_15 3.608152e-02 1.953982e-01 4.611685e-02  The exposures/contributions for a single tumor genome can also be plotted: plotDecomposedContribution(exposure) In some cases, multiple tumor genomes need to be decomposed (each one, however, individually). In this case, a set of tumor genomes can be passed to decomposeTumorGenomes: exposures <- decomposeTumorGenomes(genomes, signatures) exposures: > exposures$PD3851a
sign_1       sign_2       sign_3       sign_4       sign_5       sign_6
4.874215e-02 1.237708e-01 5.984978e-02 5.764690e-02 1.426703e-20 1.109195e-01
sign_7       sign_8       sign_9      sign_10      sign_11      sign_12
8.766150e-02 5.115825e-02 2.758243e-02 9.180615e-02 3.070582e-02 3.256016e-02
sign_13      sign_14      sign_15
3.608152e-02 1.953982e-01 4.611685e-02

$PD3890a sign_1 sign_2 sign_3 sign_4 sign_5 sign_6 sign_7 0.01525397 0.02791685 0.07609293 0.07239998 0.00815879 0.17830407 0.04543191 sign_8 sign_9 sign_10 sign_11 sign_12 sign_13 sign_14 0.09344739 0.14291157 0.13844086 0.05390118 0.00000000 0.03827808 0.08551768 sign_15 0.02394473 [...] ### 4.3.1 Finding exposures for a subset of signatures with a minimum explained variance Instead of decomposing a tumor genome precisely into the given set of signatures (15 in the example above), the function decomposeTumorGenomes can alternatively be used to search for subsets of signatures for which the decomposition satisfies a minimum threshold of explained variance: exposures <- decomposeTumorGenomes(genomes, signatures, minExplainedVariance=0.9, minNumSignatures=2, maxNumSignatures=NULL, greedySearch=FALSE, verbose=TRUE) For each tumor genome, the minimum subset of signatures that explain at least minExplainedVariance percent of the variance of the mutation frequencies will be identified (the exposures of all other signatures will be set to NA): Decomposing genome 1 (PD3851a) with 2 signatures ... with 3 signatures ... with 4 signatures ... with 5 signatures ... with 6 signatures ... with 7 signatures ... explained variance: 0.903349612580512 with 3 signatures ... with 4 signatures ... with 5 signatures ... with 6 signatures ... with 7 signatures ... explained variance: 0.908539373424795 [...] > exposures$PD3851a
sign_1     sign_2     sign_3     sign_4     sign_5     sign_6     sign_7
NA 0.14986576 0.08839117 0.11203026         NA 0.18900885 0.16543502
sign_8     sign_9    sign_10    sign_11    sign_12    sign_13    sign_14
NA         NA 0.09461580         NA         NA         NA 0.20065314
sign_15
NA

$PD3890a sign_1 sign_2 sign_3 sign_4 sign_5 sign_6 sign_7 sign_8 NA NA NA NA NA 0.1910117 0.1074078 0.1395364 sign_9 sign_10 sign_11 sign_12 sign_13 sign_14 sign_15 0.1604152 0.1696167 NA NA 0.1231509 0.1088613 NA [...] plotDecomposedContribution(exposures[[1]]) [If plotDecomposedContribution is run with removeNA=FALSE, also signatures with an NA value as exposure will be included in the x-axis of the plot. Additionally, the signatures can be passed to the function using the parameter signatures; this allows to use the correct signature names for the plot.] Important note: although a subset of signatures may explain the somatic mutations observed in the tumor genomes reasonably well, they need not necessarily be the signatures with the highest contribution when taking the entire set of signatures. Important note: if for a tumor genome no (sub)set of signatures is sufficient to explain at least minExplainedVariance percent of the variance, no result (NULL) is returned for that tumor. The search behavior of decomposeTumorGenomes when finding a subset of signatures to explain the somatic mutations of a tumor can be influenced by the following arguments: • minExplainedVariance (default is NULL): if not specified, all signatures are taken for the decomposition; if specified, the smallest subset of signatures which explains at least minExplainedVariance of the variance is taken for the decomposition • minNumSignatures (default is 2): if a search for a subset of signatures is performed, at least minNumSignatures will be taken • maxNumSignatures (default is NULL, i.e., all): if a search for a subset of signatures is performed, at most maxNumSignatures will be taken; if NULL, all given signatures will be taken as the maximum; if maxNumSignatures is specified but minExplainedVariance=NULL (no search), then exactly the best maxNumSignatures will be taken • greedySearch (default is FALSE): if FALSE, a full search will be performed: for increasing numbers K of signatures, all possible subsets of K signatures will be tested and the subset with the highest explained variance is chosen, increasing K until the threshold of minExplainedVariance is satisfied; if TRUE, a much faster, greedy search is performed: first, all possible subsets with K=minNumSignatures are evaluated, taking the subset with the highest explained variance, then stepwise one additional signature (highest increase of explained variance) is added to the already identified set until the threshold of minExplainedVariance is satisfied Performing, for example, a greedy search for the example above is much faster: exposures <- decomposeTumorGenomes(genomes, signatures, minExplainedVariance=0.95, minNumSignatures=2, maxNumSignatures=NULL, greedySearch=TRUE, verbose=TRUE) Decomposing genome 1 (PD3851a) with 2 signatures ... adding signature 3 ... adding signature 4 ... adding signature 5 ... adding signature 6 ... adding signature 7 ... adding signature 8 ... adding signature 9 ... adding signature 10 ... explained variance: 0.959544941532486 Decomposing genome 2 (PD3890a) with 2 signatures ... adding signature 3 ... adding signature 4 ... adding signature 5 ... adding signature 6 ... adding signature 7 ... adding signature 8 ... adding signature 9 ... adding signature 10 ... adding signature 11 ... explained variance: 0.962228749687402 [...] > exposures$PD3851a
sign_1     sign_2     sign_3     sign_4     sign_5     sign_6     sign_7
0.04423271 0.17855158 0.05436733 0.07616399         NA 0.12658971 0.17677283
sign_8     sign_9    sign_10    sign_11    sign_12    sign_13    sign_14
0.03854308 0.03481150 0.07507863         NA         NA         NA 0.19488865
sign_15
NA

$PD3890a sign_1 sign_2 sign_3 sign_4 sign_5 sign_6 sign_7 NA 0.01956464 0.05925686 0.08682446 NA 0.18705075 0.07223231 sign_8 sign_9 sign_10 sign_11 sign_12 sign_13 sign_14 0.10630698 0.14095155 0.14069602 0.05832093 NA 0.04276313 0.08603236 sign_15 NA [...] plotDecomposedContribution(exposures[[1]]) Important note: of course, a greedy search which starts from the best combination of minNumSignatures need not yield the same result as a full search because the latter finds the overall best subset while the greedy search can get stuck in a local optimum depending on the starting point of the search. The precise behavior depends on different factors, including for example the similarity between mutational signatures and the minimum required explained variance (lower thresholds can easily be satisfied by very different combinations of signatures). We recommend to test different settings (minimum numbers of signatures, thresholds for explained variance, etc) and learn about the behavior to ensure a meaningful biological interpretation of the results. ### 4.3.2 Computing the explained variance Given the mutation frequencies of one or more tumor genomes (genomes), a set of mutational signatures (signatures) and their computed exposures/contributions to the given tumor (exposures), the following command allows to compute—for each individual tumor—the variance of the mutation frequency data that the exposures explain. The following is an example taking the full set of tumors and signatures from Section 4.2.1: exposures <- decomposeTumorGenomes(genomes, signatures) computeExplainedVariance(exposures, signatures, genomes) Example output:  PD3851a PD3890a PD3904a PD3905a PD3945a PD4005a PD4006a PD4085a 0.9796806 0.9693211 0.9913913 0.9695197 0.9861931 0.9956940 0.8988394 0.9430757 PD4086a PD4088a PD4103a PD4107a PD4109a PD4115a PD4116a PD4120a 0.9912875 0.9693047 0.9843990 0.9910398 0.9931142 0.9607950 0.9937769 0.9995375 PD4192a PD4194a PD4198a PD4199a PD4248a 0.9764347 0.9678732 0.9866895 0.9987643 0.9841221  Note: for computing the variance explained for a single tumor by a set of signatures, the corresponding number of exposure values must be the same as the number of signatures. Undefined exposure values (NA), which can be present if a search for a subset of signatures has been performed as described above, will be set to zero such that the corresponding signature does not contribute. For genomes for which the minExplainedVariance could not be reached, and whose exposure vectors are NULL, the explained variance will be set to NA. ## 4.4 Recomposing/reconstructing tumor genomes from exposures and signatures Estimating the explained variance of the decomposition of a tumor genome (see Sections 4.2 and 4.3.2) and assessing its quality requires the mutation frequencies $$\hat{g}_i$$ of the tumor genome to be reconstructed, or predicted, from the mutational signatures $$S_j$$ and their exposures/contributions (or weights) $$w_j$$: $\hat{g}_i = \sum_j{w_j * (S_j)_i}$ This can be easily achieved using the function composeGenomesFromExposures. The following is an example taking the tumor genomes from Section 3.2, the signatures from Section 4.2.1, and the exposures as computed in Section 4.3: genomes_predicted <- composeGenomesFromExposures(exposures, signatures) Example output: > genomes_predicted [[1]] [,1] [,2] [,3] [,4] [,5] [,6] [1,] 0.1871191 0.1003964 0.3534400 0.1183363 0.1637962 0.07691213 [2,] 0.2789142 0.2232669 0.1935118 0.3043071 0.0000000 0.00000000 [3,] 0.2759081 0.2481861 0.1757908 0.3001150 0.0000000 0.00000000 [4,] 0.2757096 0.1900909 0.2655066 0.2686929 0.0000000 0.00000000 [5,] 0.2825315 0.2121257 0.2028239 0.3025190 0.0000000 0.00000000 [6,] 0.4704991 0.5295009 0.0000000 0.0000000 0.0000000 0.00000000 [...] Once genomes have been reconstructed, the function evaluateDecompositionQuality allows to compare the reconstructed genome mutation features to the originally observed features in order to assess the quality of the decomposition. The function is applied to an individual tumor genome, and can either return numerical quality measurements, or provide a quality plot which includes said measurements. Numerical measurements to compare the reconstructed/predicted and the original tumor mutation patterns are: • explainedVariance: the explained variance (see Section 4.2) • pearsonCorr: the Pearson correlation coefficient (PCC) between the predicted/reconstructed and the original mutation frequencies of the tumor genome. Although the PCC does usually not consider the amplitudes of two input data vectors—only their linear relationship—here, a high PCC also entails small absolute differences in the mutation frequencies because the frequencies in both the predicted and the original data sum up to 1, such that a high correlation automatically means very similar values. Example for obtaining numerical measurements: evaluateDecompositionQuality(exposures[[1]], signatures, genomes[[1]], plot=FALSE) Output: $explainedVariance
[1] 0.9796806

\$pearsonCorr
[1] 0.9989003

Example for obtaining a quality plot which compares the reconstructed and the original data:

evaluateDecompositionQuality(exposures[[1]], signatures, genomes[[1]], plot=TRUE)

## 4.5 Mapping and comparing sets of signatures

In some cases it may be of interest to compare or find a mapping between two sets of signatures, e.g., if they have been inferred from different datasets. For this purpose, decompTumor2Sig provides a set of additional functions.

### 4.5.1 Comparison of signatures of the same format

Let’s first load two distinct sets of Shiraishi signatures of the same format (5 bases, with transcript-strand direction):

# get 4 Shiraishi signatures from 21 breast cancers from Nik-Zainal et al (PMID: 22608084)
sigfiles <- system.file("extdata",
paste0("Nik-Zainal_PMID_22608084-pmsignature-sig",1:4,".tsv"),
package="decompTumor2Sig")

# get 15 Shiraishi signatures obtained from 435 tumor genomes from Alexandrov et al.
sfile <- system.file("extdata",
"Alexandrov_PMID_23945592_435_tumors-pmsignature-15sig.Rdata",
package="decompTumor2Sig")
sign_s15 <- signatures

Since these signatures have the same format, we can directly compare them. Using the function determineSignatureDistances, we can compute the distances of one target signature to all signatures from a set:

determineSignatureDistances(sign_s4[[1]], sign_s15, method="frobenius")

Output:

   sign_1    sign_2    sign_3    sign_4    sign_5    sign_6    sign_7    sign_8
0.7585028 1.1070702 1.1324959 0.7741531 1.2625974 0.9338330 0.6743049 1.0077730
sign_9   sign_10   sign_11   sign_12   sign_13   sign_14   sign_15
1.5575345 1.2758753 1.1879876 1.2602450 1.0191394 0.6792248 1.0914233 

Apart from the Frobenius distance, which is suitable to compare matrices and hence Shiraishi signatures, other distance metrics can be used: the “rss” (residual sum of squares = squared error) or any distance measure available for the function dist of the package stats.

If not the distances of one signature to an entire signature set is needed, but instead a mapping from one signature set to another, mapSignatureSets can be used.

mapSignatureSets(fromSignatures=sign_s4, toSignatures=sign_s15, method="frobenius", unique=FALSE)

Output:

Nik-Zainal_PMID_22608084-pmsignature-sig1.tsv
"sign_7"
Nik-Zainal_PMID_22608084-pmsignature-sig2.tsv
"sign_9"
Nik-Zainal_PMID_22608084-pmsignature-sig3.tsv
"sign_6"
Nik-Zainal_PMID_22608084-pmsignature-sig4.tsv
"sign_12" 

Like for determineSignatureDistances, with the function mapSignatureSets a mapping can be built based on different distance metrics. Additionally, the user can specify whether the mapping should be unique (one-to-one mapping), or not.

If unique=FALSE then for each signature of fromSignatures the best match (minimum distance) from toSignatures is selected. The selected signatures need not be unique, i.e., one signature of toSignatures may be the best match for multiple signatures of fromSignatures.

If unique=TRUE, i.e., if a unique (one-to-one) mapping is required, an iterative procedure is performed: in each step, the best matching pair from fromSignatures and toSignatures is mapped and then removed from the list of signatures that remain to be mapped, such that they cannot be selected again. In this case, of course, fromSignatures must not contain more signatures than toSignatures.

### 4.5.2 Comparison of signatures of different types or formats

Sometimes it may also be useful to compare different types or formats of signatures. For example, since Alexandrov signatures are comparably well studied, it might be interesting to determine which Alexandrov signature is closest to a Shiraishi signature of interest.

Since only signatures of the same format can be directly compared or mapped (see above), decompTumor2Sig provides two functions that transform signatures, such that two signatures, or two sets of signatures, can be converted to the same format.

One of these functions, convertAlexandrov2Shiraishi, has already been presented in Section 3.1.4 (see there for details). We can convert Alexandrov signatures to Shiraishi-type signatures with 3 bases (without transcription-strand direction):

sign_a <- loadAlexandrovSignatures()

sign_a2s <- convertAlexandrov2Shiraishi(sign_a)

Note: Of course, there is some information loss here (better: a loss of specificity), as we discuss in our paper (Krüger and Piro, 2018).

Additionally, the function downgradeShiraishiSignatures can be used to reduce the number of flanking bases and/or discard the information on transcription-strand direction from one or more Shiraishi signatures:

sign_sdown <- downgradeShiraishiSignatures(sign_s15, numBases=3, removeTrDir=TRUE)

Having obtained two signature sets of the same format (sequence triplets, but no transcription-strand direction), we can now map one set to the other:

mapSignatureSets(fromSignatures=sign_sdown, toSignatures=sign_a2s, method="frobenius", unique=TRUE)

Output:

        sign_1         sign_2         sign_3         sign_4         sign_5
"Signature.22"  "Signature.4" "Signature.10"  "Signature.5" "Signature.15"
sign_6         sign_7         sign_8         sign_9        sign_10
"Signature.8" "Signature.26" "Signature.28" "Signature.13" "Signature.30"
sign_11        sign_12        sign_13        sign_14        sign_15
"Signature.24"  "Signature.2" "Signature.18"  "Signature.1" "Signature.12"