To obtain our SuperSigs signatures, we analyzed sequencing data from 30 types of cancers recorded in The Cancer Genome Atlas (TCGA) database (see Materials and methods). Four key features distinguish our approach for identifying signatures.

1. Our primary methodological step is to use supervised machine learning, that is we learn the signatures from the data, by using the available annotation on clinical variables such as age, smoking status, and body mass index. By using this information explicitly, we expect to identify stronger associations and make better predictions.

2. We do not specify a pre-determined base length, such as 3-base pairs (Alexandrov et al., 2013a), as the fundamental unit of the mutational signatures. This provides greater flexibility because there is no reason to assume that all signatures are optimally described by the same base length units. In fact, a single signature may be defined on units of variable base lengths, featuring, for example, significantly elevated proportions of both C>A (i.e. a single-base substitution from C to A) and A[C>T]G (i.e. a single-base substitution from C to T with flanking bases A and G) mutations. We will restrict our present analysis to only 1, 2, and 3 base pairs lengths to simplify the presentation, as this is already sufficiently different from current methods using only trinucleotides, and leave the further extension to future work.

3. We employ a probabilistic approach to signature discovery. An important characteristic of any mutational process is its randomness. The mutational distribution caused by the same etiological factor varies greatly among exposed patients: a mutation type very frequent in some patients may not be common in others. From a biological point of view, it seems natural that each patient – and in fact each cell - may have her/his individualized signature characterizing a specific etiological factor. Our signatures are therefore built only on a subset of selected features that are robust across the exposed population, that is features with relatively low variance, thereby increasing their predictive power.

4. We do not force the assumption that a given mutational process must have the same mutational signature across tissues, contrary to the approach developed by Alexandrov et al. where a given signature (e.g. signature 1) is the same across all tissues.

Our method for deriving mutational signatures is based on several steps. First, we construct a nested tree containing all potential features, with all mutations as the root, and all six single-base substitutions (C>A, C>G, C>T, T>A, T>C, and T>G) as the first level, followed by single-base substitutions with one flanking base as the second level, and by single-base substitutions with two flanking bases as the third level, and where the edges are placed between features which share mutations (Figure 1). In principle, our method can be applied to a tree with height greater than 3, by adding additional flanking bases, but here for simplicity and for comparing with current methods, we will only consider three levels.

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After ‘pruning’ the tree in order to keep only the features that have counts significantly different from their expected values (one-sided binomial test with a 0.05 significance level, subject to Bonferroni correction), these remaining features are ranked based on their ability to classify a given exposure, that is to discriminate exposed patients from unexposed ones, as measured by the area under the receiver operating characteristic (ROC) curve (AUC). The set of n top features that provide the highest prediction performance in terms of AUC form our signature for a given exposure and are used for prediction (Figure 1). A detailed explanation of our method is provided in the Materials and methods section.

The value of a mutational signature can be assessed by its prediction accuracy (AUC) in classifying patients as exposed or not to the associated etiological factor, or by its correlation with exposure to that factor. In the next sections, we provide statistical evaluations for both, relying on the availability of clinical annotation for the etiological factor associated to that signature.

In addition to simple performance, it is also important to evaluate the degree to which a given mutational signature improves upon prior knowledge about the mutational effects of an exposure to an etiological factor (Figure 2a). For example, consider the case when clinical annotation is available and the main ‘peak’ of a mutational signature, that is its most common mutation, is already known before the mutational signature is obtained. The peak may be a nucleotide, a dinucleotide, or a trinucleotide, depending on the specific mutational process. For example, prior validated knowledge indicated that aging induces [C>T]G mutations, and smoking induces C>A mutations. The added value of a mutational signature then depends on the additional ‘information’ that that signature provides beyond this already-known peak. If a mutational signature yields additional mutations that, under the effects of a given exposure, the genome is enriched for—but was previously unknown to be—then that signature adds valuable information to prior knowledge. Mathematically, a mutational signature is represented by the set of ‘weights’ that that signature attributes to all mutations included in the analysis, with the larger weights associated to mutations the signature is more enriched for. If these weights enable a mutational signature to have a higher prediction accuracy, or correlation, than random weights do, then we say that that mutational signature provides ‘information’.

Figure 2

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To statistically evaluate the added value of the information provided by the signatures of Alexandrov and colleagues, hereafter termed ‘unsupervised’, as well as of our SuperSigs, we compared both of their performances against random alternatives carrying no additional knowledge beyond the known peak, for both aging and smoking (Materials and methods). We termed these prior knowledge signatures ‘random’ because they were purposely created to just reflect random noise around the already known peak (Figure 2b). Such random signatures are of course only meaningful when there is a peak that is already known and cannot be meaningfully constructed without prior knowledge.

We obtained sequencing data for thirty tumor types, from the TCGA Genomics Commons (https://portal.gdc.cancer.gov). After splitting each dataset randomly into training and test partitions, we applied the method above to derive signatures of aging and smoking in the training data, evaluating performance in the test data. Our SuperSigs aging signatures, applied to classify patients in a binary fashion (i.e. young versus old) yielded a median AUC of 0.73, calculated over 30 tumor types, outperforming our random aging signature (single peak; median AUC = 0.63), which was built on the well-supported observation that over time, cytosines will consistently deaminate to thymine in the CpG context (Figure 3a, Figure 3—figure supplement 1, Supplementary file 1). When the signatures are used in a regression setting, to predict age as a continuous variable, the median correlation for SuperSig predictions was rho = 0.39 (Supplementary file 1). Our analysis on the same data yielded a median AUC = 0.57, and rho = 0.25, for the unsupervised aging Signature 1 (Figure 3a, Figure 3—figure supplement 1, Supplementary file 1). The combination of the ‘clock-wise’ unsupervised Signatures 1 and 5 (Alexandrov et al., 2015) performed slightly better (median AUC = 0.61), although it did not improve on the random signature (Supplementary file 1). Unsupervised signatures for aging were not present in four of the tissues, while all tissues had aging SuperSigs.

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We next evaluated the performance of these signatures with respect to smoking status across eight tissues known to be significantly affected by smoking and for which there was smoking status information in the TCGA database. Specifically, bladder (BLCA), cervical (CESC), esophageal (ESCAD and ESCSQ), head and neck (HNSCC), kidney (KIRP), lung (LUAD), and pancreatic (PAAD) cancers. The SuperSigs added value to prior knowledge while the unsupervised signatures did not, as the AUCs obtained by the SuperSigs were higher than the ones achieved by the random single peak but not so for the unsupervised ones (median AUCs for smoking: SuperSigs = 0.68, single peak = 0.55, unsupervised = 0.56) (Figure 3b, Figure 3—figure supplement 1, and Supplementary file 1). The correlation with smoking packs of the SuperSigs was 0.27 versus 0.23 when using the unsupervised smoking signatures. These results were confirmed with cross-validation, and even when using with the SuperSigs the same prediction method, non-negative least squares (NNLS), that was used by Alexandrov et al. (Figure 3—figure supplement 1, Supplementary file 1, and Materials and methods).

These data do not indicate that unsupervised signatures for aging and smoking are meaningless. However, the data indicate that the unsupervised signatures do not add any information to prior knowledge of a peak at [C>T]G for aging and at C>A for smoking. Optimally, an algorithm based on genome-wide cancer genomic sequencing data should add information that was not available from prior studies, and SuperSigs indeed added such information that goes beyond the previously known mutational peaks (Figure 2a).

Do supervised signatures perform better than unsupervised ones when no prior knowledge about an etiologic factor is available (second scenario in Figure 2a)? For factors (other than age) which could be evaluated by unsupervised methods, the median AUC of the unsupervised method was 0.77, while the median AUC for SuperSigs was 0.90 (Figure 3b, Supplementary file 1, Materials and methods).

Can we predict whether an individual patient was ‘exposed’ to a given etiologic factor simply from the SuperSigs in that patient's cancer genome sequencing data? In several cases, this was possible with high accuracy. For example, the cross-validated AUC was 0.90 when classifying patients with lung adenocarcinomas (LUAD) as smokers versus never-smokers. Similarly, the cross-validated AUC was 0.96 when classifying bladder urothelial carcinomas (BLCA) patients who were exposed or not to aristolochic acid (Supplementary file 1). At the same time, some exposures provided weaker performances: the cross-validated AUC was 0.62 when classifying patients with liver cancers (LICH) as drinking alcohol more than once per week vs. less than once per week.

When clinical annotation is not available for an etiologic factor (Figure 2a), the unsupervised method may appear to be the only viable approach. One limitation of any supervised approach is indeed that it cannot learn signatures of factors for which no annotation is currently available. And it may be desirable to have a method that is able to discover patterns of exposures, even when they are unknown. In this case, are we forced to use an exclusively unsupervised methodology? The answer is no. While it is true that our supervised methodology requires clinical annotation, there are many cases where we may have annotation available for at least some factors—for example patient’s age is typically available for each sample—and not for others. In that case, that is any time some annotated factors are present in a sample, it is better to take care of them first, by identifying them using a supervised approach and removing their effects, to then apply an unsupervised methodology on the mutational 'leftover', rather than only using the unsupervised methodology on the whole. That is, we can first take advantage of our supervised knowledge of all exposures with available annotations. After learning those SuperSigs, we can 'subtract' their effects from the mutational load of the patients exposed to those annotated factors, and then perform an unsupervised analysis, such as non-negative matrix factorization (NMF), on the leftover, to investigate the presence of further mutational patterns. We term this approach ‘partially supervised’ and report its results in Figure 3c, showing that this method indeed achieves higher AUCs than the exclusively unsupervised approach (see also Figure 3—figure supplement 36). We provide the technical details of this partially supervised extension of our method in the Materials and methods section.

It has long been known that certain types of mutations, such as C>T transitions resulting from cytosine deamination, accumulate with age. We wondered whether other mutational signatures of aging were present in cancers and whether they varied among tissue types. To avoid confounding factors as much as possible, the analysis was confined to patients without known cancer-associated environmental exposures and without known germline predispositions to cancer.

We thereby obtained SuperSigs associated with aging for each cancer type analyzed, examples of which are shown in Figure 4a (Figure 4—figure supplements 1–30, and Supplementary file 2 for the full set). Not surprisingly, we found C>T transitions to be present in large fractions in many cancer types. However, others, such as C>A transversions in stomach and prostate adenocarcinomas, and adrenocortical carcinomas, T>C transitions in liver hepatocellular carcinomas, C>G transversions in colorectal adenocarcinomas, head and neck squamous cell carcinomas, prostate adenocarcinomas, renal clear cell carcinomas, testicular germ cell tumors, and uterine corpus carcinoma, and any mutations of the T pyrimidine in prostate and kidney cancers, and testicular tumors, had not been previously described as major age-associated mutations (Figure 4a, and Figure 4—figure supplements 1–30).

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We also sought to identify tissue-specific SuperSigs associated with specific environmental carcinogens. The analysis was performed after controlling for age and for other relevant covariates (Materials and methods). We obtained tissue-specific SuperSigs for smoking, alcohol, hepatitis B and C virus infection (HBV, HCV), aristolochic acid (AA), asbestos, and ultraviolet (UV) light (Figure 4b, Figure 4—figure supplements 31–67, Supplementary file 2, and Materials and methods). We also wanted to identify mutational signatures associated with defective DNA polymerization or repair, controlling for age, and other relevant covariates. We were thus able to obtain tissue-specific SuperSigs for mismatch repair deficiency, mutations in DNA polymerase delta or epsilon genes, mutations in the breast cancer susceptibility genes BRCA1 or BRCA2, methylation of the MGMT and IDH1 genes, and APOBEC (Figure 4b, Figure 4—figure supplements 31–67, Supplementary file 2, and Materials and methods).

In several cases, the SuperSigs associated with the same mutational factors varied across tissues, just as they did with aging. For example, the SuperSigs associated with smoking were very different in bladder, head and neck, and lung cancers (Figure 4c). And the SuperSigs associated with BRCA gene mutations were considerably different between breast and ovarian cancers (Figure 4—figure supplements 36–37). There were, however, SuperSigs that did not vary much among tissue types, for example those based on mismatch repair deficiency, and some of those associated with inherited factors (Figure 4—figure supplements 31–67).

Note that tissue specific differences with respect to etiologic factors are not possible to discover with the unsupervised approach described by Alexandrov et al. because the identity of a given signature across multiple tissues was a key theoretical assumption underpinning their approach.

We define the mutational landscape of an exposure in a tissue as the 96-long vector (96 trinucleotide mutations) where each entry is given by the average count of that mutation type, in the cohort of the samples with that exposure, divided by the average age in that cohort. The mutational landscape of aging is obtained in the same way using the cohort of samples without any known exposure (‘unexposed’). Consider now the distance between any two mutational landscapes as given by the Pearson’s correlation between the two mutational landscapes. The heatmap in Figure 5 shows the ‘closeness’ - as measured by their correlation - between the mutational landscapes of any two cohorts of patients across all cancer types, clustering the more similar ones with each other (Figure 3—figure supplements 2–18 and Materials and methods). The distances obtained by this alternative analysis indicate that the mutational landscapes produced by aging are spread all across the range, providing further evidence that the mutational processes associated with aging vary greatly with tissue type. This remained true even when subtracting the aging effect from the mutational landscape of the exposed cohort (Figure 3—figure supplements 19–35 and Materials and methods).

Figure 5

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Moreover, in several cases, the tissue-specific mutational landscape associated with an environmental factor was similar to the aging mutational landscape of the same tissue (Figure 5 and Figure 3—figure supplements 2–35). For example, the mutational landscape in smokers was more similar to the aging one in the corresponding tissue than to the ones of smokers in other tissues (Figure 3—figure supplements 2–35). This again remained true for bladder, cervical, esophageal, and kidney cancers even when subtracting the aging effect from the mutational landscape of the exposed cohort (Figure 3—figure supplements 19–35 and Materials and methods).

These analyses then suggest that a major effect of some environmental factors may simply be to increase the rate of cell division. This increase would induce a linearly proportional increase in mutation rate, but with a mutation pattern that remains similar to the one caused by normal aging, that is it would not be associated with new signatures such as those caused by direct interaction of carcinogens with DNA. Increases in the rate of cell division are known to occur when tissues are damaged or inflamed (Cheah et al., 2015; Walser et al., 2008). These observed similarities between the environmental and aging signatures then support the idea that, in certain tissues, the environmental factor’s main effect is to induce inflammation in that tissue, thus increasing its cells’ division rate.

Obesity (as measured by a body mass index, BMI, greater than 30) has emerged as the major lifestyle factor contributing to cancer in general (Giovannucci et al., 1995; Hruby et al., 2016; Song and Giovannucci, 2016). How obesity contributes to cancer risk, however, is unknown. For example, obesity could lead to cancer by inducing mutations or by stimulating the growth of neoplastic cells that have already acquired mutations (Song et al., 2018). If the former explanation were valid, there might be a mutational signature associated with obesity, but no such signature has been previously identified. Four cancer types associated with obesity in which adequate number of samples and body mass index data for a supervised machine learning approach were available: colon, esophageal, kidney, and uterine cancer. We were able to identify SuperSigs for obesity in all these cancer types (Figure 4—figure supplements 49–52 and Supplementary file 2). Two of them, however, had an unreliable performance (AUC < 0.60) in cross-validation. For the other two (Figure 6), our ability to predict which patients were obese simply by the SuperSigs in their cancers – as measured by the apparent AUC – was 0.80 in kidney cancer (kidney renal papillary cell carcinoma - KIRP), and 0.66 in uterine cancer (UCEC) (Supplementary file 1). The obesity SuperSigs varied among the four cancer types, again suggesting the tissue specificity of mutational signatures associated with the same risk factor. The finding of a negative difference in the rate of T[C>G]T mutations in obese patients with uterine cancer (Figure 6) suggests an explanation for the observation that often the total number of somatic mutations found in cancers of obese patients is not significantly different from that of non-obese patients, when controlling for age. Only the mutational spectrum is different. Obesity could then induce interaction effects among mutational processes that go beyond the usual additive effects.

Figure 6

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We applied the supervised approach to estimate the proportion of the overall mutational load that can be attributable to normal aging rather than to other mutational processes. This proportion is directly provided by the contribution of the age SuperSigs in each patient (see Materials and methods). When considering all 30 tissues, we estimate that on average 69% of the mutations can be attributable to the normal endogenous mutational processes associated with aging, that is normal DNA replication (Supplementary file 3). This estimate is consistent with what previously reported in Tomasetti et al., 2017b. The proportion varied widely across tissues, for example it is 2% on average in endometrial cancers (UCEC) of patients with POLe mutations to 87% in pancreatic cancer (PAAD) patients who smoke. This estimated proportion may be an overestimate given the lack of full annotation for all environmental and inherited factors. At the same time, only non-annotated effects that are both common in the population of patients analyzed and that increase with age may erroneously end up in the age SuperSigs. Overall, the fact that the estimates for the contribution of the age signature are consistently large across the many tumor types analyzed points to a major role of the normal endogenous mutational processes.

To validate our results, we tested the performance of the SuperSigs via four different analyses: cross-validation, random label shuffling, partial label switching, and validation on external data. First, we performed five iterations of 5-fold cross-validation of our SuperSigs (Materials and methods and Supplementary file 1). The cross-validation confirmed the positive performance of the SuperSigs as we have already reported in the previous sections. Overall, there was a drop in the median AUC of only 3 and 4 percentage points for the SuperSigs of age and of all other exposures, respectively, indicating a lack of major overfitting. As expected, there were a few signatures whose apparent AUCs were >0.60 but cross-validated AUC were <0.60 (6 out of 30 for age and 10 out of 37 for other exposures, see Supplementary file 1), and those should not be considered reliable. We then performed another test against overfitting, by randomly shuffling all the labels and then re-running the entire framework from feature selection to test prediction. Rather than obtaining similar AUCs between the shuffled and unshuffled datasets, the AUCs dropped to 0.50 in cross-validation on the shuffled data, providing evidence that our SuperSigs do not suffer from major overfitting (Supplementary file 4). Next, we tested the robustness of our methodology by considering scenarios where 5, 10, 20, and 25% of the clinical annotations for all etiological factors used in the training set were mislabeled. Even so, our performance in terms of AUC remained higher than the unsupervised one even with the highest percentage of mislabeling (25%; see the ‘Robustness analysis with respect to mislabeling’ section in the Materials and methods and Supplementary file 5).

Finally, we externally validated our results on an independent whole-genome sequencing (WGS) dataset from the International Cancer Genome Consortium (ICGC) database (downloaded from https://dcc.icgc.org/releases/PCAWG/). Only ICGC datasets that were not present in the TCGA database were used. All signatures that had previously achieved a cross-validated AUC of at least 0.6 on TCGA data were tested, as the others should not be considered reliable. Using, for each exposure, the associated logistic regression model trained on TCGA data only, and containing its SuperSig, we predicted the unexposed versus exposed status for all available factors and tissues in the ICGC data. The AUC for every of those signatures is reported in Table 1. We found that the predictive power of the SuperSigs observed in the TCGA dataset was retained when predicting on ICGC data, with relatively modest drops in AUC for a few signatures. The only exception was the age signature in melanoma (SKCM). This signature, however, had a borderline performance already in the original analysis (AUC = 0.61), probably due to the inability of the methodology to properly remove sun exposure as a confounding. At the same time, there were several SuperSigs that achieved better accuracies in the external validation ICGC dataset than in the original cross-validated TCGA, with gains up to 16–18 percentage points for the age signatures in ovarian (OV) and prostate (PRAD) cancers.

We downloaded somatic exomic mutational data from the TCGA Bioportal (https://portal.gdc.cancer.gov) and filtered out the mutations which have less than 5% Variant Allele Frequency (VAF). Out of the total thirty-three datasets available, large B-cell lymphoma (DLBC) was not included in the analysis because of the small number of samples available, while lung squamous cell carcinoma (LUSC) and mesothelioma (MESO) were excluded because of the extremely small number of patients unexposed to smoking and asbestos, respectively. For ovarian cancer (OV) and acute myeloid leukemia (LAML) whole genome sequencing data were used. The human genome reference build hg38 was used to determine the context (flanking bases) for each mutation. The clinical information was downloaded from the website Cbioportal (http://www.cbioportal.org). For calculating the background frequency of each trinucleotide on both the exome and the genome, we used the R package, deconstructSigs. For the Unsupervised Signature method (Alexandrov et al., 2013a), we downloaded the signatures from the Cosmic Signature website (http://cancer.sanger.ac.uk/cosmic/signatures) and used the table http://cancer.sanger.ac.uk/signatures/matrix.png in order to determine which signatures were present in which tissue.

All analyses were performed using R version 3.5.2. Logistic regression was performed using glm from the STATS package. LDA was performed using the function lda from the package MASS. Non-negative matrix factorization (NMF) was performed using the function nmf with method ‘Lee’ from the package NMF.

To reduce the effect of confounding factors, we applied several filtering criteria. In each tissue type, we divided samples into two categories: (1) ‘unexposed’, meaning that no exposure to a known environmental factor was recorded, according to the available clinical annotation, and (2) ‘exposed’. To mitigate the effects of other unknown factors in the unexposed group, we removed any sample with a mutational load more than 3 times higher than the median number of mutations found among the unexposed samples. We also excluded samples if the total number of mutations was equal to zero on the exome, a probable indication of low neoplastic cell content. We removed samples with microsatellite instability (MSI) or with a mutation in POLE/POLE2/POLE3/POLE4 or POLD1/POLD2/POLD3/POLD4 genes - except for when the signature for the specific effects of those mutations was the objective of the analysis - because of the known large increase in the number of mutations they induce. A tissue type was divided into subtypes whenever possible. Acute Myeloid Leukemia (AML) patients younger than 40 years old were not considered because we were interested in adult rather than childhood AML. Among the ‘exposed’ samples we excluded samples with known multi-factor exposures to minimize confounding factors and only evaluated samples with a single known exposure. Samples with unknown exposure were treated as unexposed.

Mutation counts are used to characterize mutational burden when considering predictors of aging. For all other exposures, mutation rates (i.e. counts/age) are used. In a patient exposed only to time, that is unexposed to any known environmental or inherited factor, the rate of a mutation type is expected to remain constant irrespective of age – as dictated by the aging signature – while the absolute count is expected to increase with age. In contrast, in a patient exposed to an environmental or inherited factor, the rate of a mutation type as well as the count may change with respect to the age signature.

We provide here the details for the method we developed to obtain the supervised mutational signatures (see flowchart in Figure 1 of the main text).

At its simplest, a mutational signature of exposure is nothing more than a set of substitutions that characteristically occur at different rates in exposed tissue than in unexposed tissue. In practice, though, a few considerations suggested by prior biological knowledge quickly turn a simple calculation into a complex engineering problem. Specifically, a key principle of the SuperSig approach is that signatures may not be optimally described by the same base length units. Accordingly, we consider all single-base substitutions, with or without the flanking context bases, as potential, signature features. In addition to six single base substitutions: C>A, C>G, C>T, T>A, T>C, and T>G, named according to the pyrimidine of the mutated Watson–Crick base pair, there are 48 dinucleotides, in which the substitution is paired with a specific base as a prefix or as a suffix but not both (e.g. A[C>T] or [C>T]G), as well as 96 trinucleotides (e.g. A[C>T]G), which include both flanking bases as context. Hence, we have a list of 151 potential features (6 + 48 + 96 + 1).

The resulting flexibility carries a price, however, as features are no longer independent. The simple substitution C>T spawns dinucleotide children, such as A[C>T], and trinucleotide grandchildren like A[C>T]G. Frequent, exposure-driven A[C>T] substitutions would increase the observed rates of both the C>T parent and the trinucleotide children, making it difficult to assign ownership to the correct generation. The next section, ContextMatters, describes our approach to solving this problem, while the sequel, FeaturesSelection, describes how candidate signature features are combined to create a final signature.

For every factor other than age, we applied the above feature-engineering (ContextMatters) step separately to samples from patients that were respectively unexposed or exposed to the factor under consideration. We then combined these two lists of second-phase candidate features, which are both partitions, by considering all intersections and relative complements of the elements in the two original partitions, to form the minimal refinement of the two, and define this final list as the list of candidate features.

For aging signatures, we applied the feature engineering steps described above only to samples from patients who were unexposed to any known environmental or inherited factor. Therefore, we skip this step of combining partitions, because we only have one partition, that is its second-phase candidate features, which automatically provided its ‘candidate features’ list.

We performed the supervised feature engineering step described in the previous section in an inner loop of 5 iterations of 3-fold cross-validation. Each feature in each fold was ranked according to its ability to discriminate exposed samples from unexposed, based on the rates for that feature (or counts, as appropriate for the exposure). Discriminatory performance was measured by the area under the receiver operating characteristic (ROC) curve (AUC).

The fifteen lists of ‘candidate features’ obtained across the 3 folds of each of the five iterations was then combined by considering all intersections and relative complements of the features in the original lists. The minimal refinement thus obtained constitutes the final list of candidate features. These candidate features were then ranked by their median AUC across the 15 folds.

Also, for each fold, the number n corresponding to the top n ranked features that provided the highest AUC - using a multivariate, logistic regression classifier (LR) - was selected. The median of those 15 n values, defined as n*, was then used to select the first n* candidate features according to their median AUC ranking. Out of these n* features, the ones for which the median AUC ≤ 0.6 were discarded, and the remaining features are what we defined as the predictive features for a given exposure.

The set of predictive features selected above form the supervised signature (SuperSig). Two values are associated to each one of these predictive features: (1) the difference in mean counts (age) or rates (all other exposures) between the exposed and unexposed cohorts, and (2) the beta (β) coefficient for that feature as estimated by logistic regression. Both vectors yield critical information.

The difference in means for each feature, which is the only constraint used by logistic regression in maximizing entropy over the dataset, provide a natural measure of the difference in counts or rates for that feature induced by a given exposure. We report these values in the figures of the main text and in Figure 4—figure supplements 1–67.

The beta coefficients of the features in a logistic regression have also an intuitive interpretation, since the logarithm of the odds of being in the exposed class C versus the unexposed one, given the mutational data (counts or rates), is given by

Therefore, e^β of a feature is the factor by which the odds of being in the exposed class increase for every extra unit increase in that feature, when all other features are kept constant. The β coefficients of the mutational signatures for each factor (aging or exposure) can be found in Supplementary file 2 and are depicted in Figure 4—figure supplements 1–67.

We used Logistic Regression (LR) to test the predictive accuracy of each set of features representing a mutational signature as measured by AUC. We also report the performance of Linear Discriminant Analysis (LDA), and Random Forest (RF), when applied to both feature selection and prediction (Supplementary file 1). In both LR and LDA models, the mean vectors equal the empirical mean vector. In addition, LDA also accounts for the dependencies among the features. All methods yielded relatively comparable results in cross-validation.

In order to compare the prediction accuracy (AUC) of all three sets of signatures (Alexandrov et al., single peak, and SuperSigs), we first apply the same prediction methodology that was previously used in Alexandrov et al. to determine the contribution of each signature in each patient: non-negative least squares (NNLS).

More specifically, to determine in a given patient the respective proportional contributions (used as a score) X_i of each mutational signature i = 1,…, k, where a total of k signatures are present in that tissue, NNLS is applied to

that is Y = AX in matrix form, where Y_i is the total number of mutations of type i, and A_ij is the relative frequency (for Alexandrov et al. and single peak signatures) or the difference in mean count (SuperSigs for age) or rate (SuperSigs for all other etiological factors) of mutation type i in the mutational signature j, across each one of the k signatures present in that tissue.

The performance of the various methodologies is presented in Figure 3, Figure 3—figure supplement 1, and Supplementary file 1.

For Alexandrov et al. their Signature one was used for predicting age in one comparison (Figure 3b), and the combination of the ‘clock-wise’ unsupervised Signatures 1 and 5 as determined in Alexandrov et al., 2015 was used in the other comparison. The specific combination of signatures used for Alexandrov et al. in predicting smoking status was instead determined by the specific combinations provided for each tissue in Alexandrov et al., 2016. The accuracies (AUC) of all predictions can be found in Supplementary file 1.

It was not possible to cross-validate directly the unsupervised method of Alexandrov et al., 2013a, due to the fact that there are some manual or semi-automated steps in their signature creation, validation and association to the risk factors. For example, in their section ‘Samples and curation of freely available cancer data’ it is stated that: "All mutational signatures were clustered using unsupervised agglomerative hierarchical clustering and a threshold was selected to identify the set of consensus mutational signatures. Mis-clustering was avoided by manual examination (and whenever necessary re-assignment) of all signatures in all clusters. 27 consensus mutational signatures were identified across the 30 cancer types." Thus, we chose to use the core methodology used in Alexandrov et al., which is non-negative matrix factorization (NMF) (Lee and Seung, 1999), and approximate their method in two alternative ways in order to perform cross-validation: (1) ‘BestNMF’ and (2) ‘MatchedNMF’, as described below.

For both approaches, we applied NMF (Lee-Sung NMF_0.21.0) to the profile of the count mutations of the training samples, that is a matrix whose 96 rows represent mutation types and columns represent training samples. We set the rank parameter, r, of the NMF algorithm equal to what shown in Cosmic signature v2 (https://cancer.sanger.ac.uk/cosmic/signatures_v2) for the tissue of interest. We hardwired this parameter to help the unsupervised method to limit model misspecification (see also the model misspecification section below).

After obtaining the r signatures from NMF, we used two alternative methods to select among them the signature of a specific age or environmental factor: (1) for BestNMF, we chose the signature whose contributions had the highest AUC in classifying exposure to the environmental factor on the training set; (2) for MatchedNMF, we paired each of the identified signatures from the training set to exactly one of those listed in Cosmic v2 for this specific tissue. This pairing process was obtained by maximizing the sum of the cosine similarity for each pair.

Then, on the test set, we used an NNLS algorithm to estimate the contribution of each signature on the test set.

The performance of the various methodologies is presented in Figure 3, Figure 3—figure supplement 1, and Supplementary file 1.

We would like to note that we are being conservative against our own supervised results when using the alternative method called ‘BestNMF’ to cross-validate the unsupervised approach. In fact, BestNMF is more of a weakly supervised method rather than a purely unsupervised method, and therefore having an expected better performance. This can also be verified by comparing the apparent performances of Alexandrov and BestNMF in Supplementary file 1. Also, BestNMF provides us with signatures for cases that Alexandrov et al did not study like BMI, alcohol, etc. and it is not clear how we can generate those signatures in Alexandrov as the Cosmic signatures are pre-fixed for tissues. The closest we can get to the Alexandrov implementation is with what we termed ‘MatchedNMF’. It is still in favor of the Alexandrov implementation because the process for matching the signature uses the Cosmic signatures and those signatures indirectly used the exposure labels of test labels to identify the associated risk factor as well.

We explain here how the supervised learning of a mutational signature (specifically the aging signature in this example) can be used to improve the performance of an unsupervised approach by discounting the effects of that supervised signature on the test data. We refer hereafter to this methodology as ‘partially supervised’.

To simplify matters, we do not engineer features; rather, we use the 96 fundamental mutations as in Alexandrov et al., 2013b. We only use the datasets that show a higher average rate of mutation per year in the exposed samples than in the unexposed samples. This increase in the rate is required to conform to the premise of non-negativity and linearity in the NMF model. We use one half of the unexposed samples as the training set to learn the rate of each feature of the age signature (thus a supervised signature) so that we can discount the effect of age (i.e. controlling for age) on the test set. Next, we formed the test set by bootstrapping over the left-out half of the unexposed samples and all exposed ones.

We apply NMF (Lee-Sung ref) with rank equal to three to decompose the test set, Y, thus obtaining two matrices, A and X: one containing the unsupervised signatures (A) and a second one with the corresponding contributions of each of those signatures in each patient (X). These contributions have not been discounted for age yet. This is the standard unsupervised approach. However, in order to estimate the discounted contributions of a signature in each test sample, we now discount the effect of age of a patient on each unsupervised signature, by multiplying the learned supervised age signature by the age of the patient, times the estimated mutation rate, and then projecting this vector onto the directions identified by NMF using NNLS, and then subtracting these projected contributions of age from the contributions of the three unsupervised signatures obtained by NMF. To conform to the premises of NMF, we set the negative discounted contributions to zero.

We then choose the direction whose contribution, divided by the total number of mutations, is the most associated (in terms of the highest AUC) to the exposure status using the known ground-truth, for both the unsupervised and the partially supervised methods, by using the not discounted and discounted contributions, respectively. To obtain the ‘partially supervised signatures’ we now use again non-negative linear regression but this time where the contributions (X) are known and the signatures (A) are unknown. In other words, we still have the decomposition Y=AX, but now, Y and X are known and we want to estimate A.

We used the AUC to evaluate the association of the signature with the exposure status, for both the unsupervised and partially supervised approach, where the contribution of each signature has been divided by the number of total mutations. We repeated 50 times this whole process (from the random selection of half of the unexposed patients used to learn the age signature and so on) and take the average AUC over them to account for the effect of randomness. This is what is depicted in Figure 3—figure supplement 36, where the increase in performance of the partially supervised method with respect to the unsupervised is evident.

In this partially supervised extension, we have used NMF to easily compare with the unsupervised approach by Alexandrov et al. However, other methodologies (e.g. a classifier based on EM) may provide even better performance.

Pretend we had no annotation for the presence of defects in the gene POL-ε among patients with endometrial cancer in the UCEC-TCGA dataset and that we did not know the POL-ε signature. The normalized results for an NMF decomposition are depicted in Figure 3—figure supplement 37a. This example illustrates a critical weakness of unsupervised approaches in general. The POL-ε signature in Figure 3—figure supplement 37a was obtained by ‘telling’ NMF to search for one (i.e. rank = 1) pattern. If instead we had asked for two, three, or four signatures, respectively, NMF would have returned the patterns depicted in Figure 3—figure supplement 37b–d. Figure 3—figure supplement 37b–d show that the POL-ε signature has been parsed into multiple patterns: the more patterns the more the optimum signature is spread across different claimed signatures. Therefore, the quality of the results of NMF strongly depend on the number of signatures NMF is required to extract. Unfortunately, there is no fully satisfactory rule to determine a priori how many patterns should be found by NMF. This is a problem that all unsupervised approaches have because the researcher is blind to the actual number of different exposures that are present among the patients in the dataset during the discovery phase. As Hastie et al. put it in their classic treatise: ‘it is difficult to ascertain the validity of inferences drawn from the output of most unsupervised learning algorithms’, and their ‘effectiveness is a matter of opinion and cannot be verified directly’ (p. 487 of (21)). The problem may be so severe that at times it may be best to just consider the distribution of mutation types without using NMF at all.

Each predictive feature of our SuperSigs can be represented by its rate. For age, the ‘rate’ of feature i, r_i^a, is defined as the mean of the ratio:

in unexposed patients. This rate estimates the number of mutations of that particular feature accumulating per year and attributable to age. To estimate the proportion of mutations due to aging in each specific sample, we multiplied r_i^a of each feature i present in our SuperSig age signature by the patient’s age of that specific sample. The number obtained by summing the above counts for each feature in the age SuperSig is then divided by the total number of mutations observed in that sample. This resulting ratio, being forced to be not greater than 1, is our estimate for the proportion of somatic mutations attributable to age in that sample (see Supplementary file 3).

We define the mutational landscape of an exposure in a tissue as the 96-long vector (96 trinucleotide mutations) where each entry is given by the average count of that mutation type in the cohort of the samples with that exposure divided by the average age in that cohort. The mutational landscape of aging is obtained in the same way using the cohort of samples without any known exposure (‘unexposed’). Then, the distance between any two mutational landscapes is given by the Pearson’s correlation between the two mutational landscapes (see Figure 5 in the main text and Figure 3—figure supplements 2–18). For the results in Figure 3—figure supplements 19–35 the effect of age has been removed from the mutational landscape of all exposures but age, by subtracting the mutational landscape of age from the relevant exposed tissue. Replacing the distance based on correlation with one based on cosine similarity yields equivalent results.

To assess the robustness of the methodology with respect to the quality of the clinical annotation, we switch the labels from unexposed to exposed (or vice versa) for 5, 10, 20, and 25% of the samples in the training set. For example, non-smokers would be mislabeled as smokers and vice versa. Then we rerun our supervised method, including feature engineering and selection, on the training set to obtain new signatures. These new signatures are then used for prediction in the test set, where we used the original labels as the ground truth. The performance is reported in Supplementary file 5. We compare AUCs at the different mislabeling percentages and found that our method still outperforms the unsupervised method all the way up to a mislabeling proportion of 25%.

Heatmap of the distance, in terms of correlation, between any two etiological factors’ mutational landscapes in the corresponding tissues. Distance not discounted for age. The distance between any two mutational landscapes is given by 1- the Pearson’s correlation between the two mutational landscapes. See Materials and method section for details.

Heatmap of the distance, in terms of correlation, between any two etiological factors’ mutational landscapes in the corresponding tissues. Distance not discounted for age. The distance between any two mutational landscapes is given by 1- the Pearson’s correlation between the two mutational landscapes. See Materials and method section for details.

Correspondence and requests for materials should be addressed to C.T.: ctomasetti@jhu.edu. A repository for the R package containing the SuperSigs algorithm can be found at https://github.com/TomasettiLab/superSigs (Kuo and Zhang, 2020; copy archived at swh:1:rev:7d6aac85a1b3930cb93e0810039db4d65a242cca).

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Dr. Henry B Laufer for critical review and useful suggestions and comments, and Mahnoosh Mehrabani for useful comments. This research was supported by The John Templeton Foundation, the Lustgarten Foundation for Pancreatic Cancer Research, The Virginia and D K Ludwig Fund for Cancer Research, and NCI grants P30CA006973.

© 2021, Afsari et al.

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