Learning mutational graphs of individual tumor evolution from multi-sample sequencing data

Abstract

Phylogenetic techniques quantify intra-tumor heterogeneity by deconvolving either clonal or mutational trees from multi-sample sequencing data of individual tumors. Most of these methods rely on the well-known infinite sites assumption, and are limited to process either multi-region or single-cell sequencing data. Here, we improve over those methods with TRaIT, a unified statistical framework for the inference of the accumulation order of multiple types of genomic alterations driving tumor development. TRaIT supports both multi-region and single-cell sequencing data, and output mutational graphs accounting for violations of the infinite sites assumption due to convergent evolution, and other complex phenomena that cannot be detected with phylogenetic tools. Our method displays better accuracy, performance and robustness to noise and small sample size than state-of-the-art phylogenetic methods. We show with single-cell data from breast cancer and multi-region data from colorectal cancer that TRaIT can quantify the extent of intra-tumor heterogeneity and generate new testable experimental hypotheses.

Introduction

Intra-tumor heterogeneity (ITH) is the final product of the complex interplay arising from competition, selection and neutral evolution of cancer cell subpopulations, and currently represents a major hurdle in the development of effective diagnostic and therapeutic strategies for most cancer (sub)types [1–7]. For this reason, in the last years an impressive number of computational approaches to reconstruct the evolutionary history of tumors have been devised (see [8, 9] for recent reviews), taking advantage of the ever increasing resolution and availability of genomic data, as provided especially by multi-region and single-cell sequencing (SCS) experiments.

Such multi-sample datasets allow to overcome some limitations of single-sample bulk sequencing, which returns a noisy mixture of signals from the tumor subpopulations detected in the sequenced biopsy (e.g., the TCGA data [10]). In particular, the analysis of multiple spatially-separated regions of a tumor and its metastases is recently producing a better and clearer picture of ITH in various tumor types [11–16]. Accordingly, a number of algorithms to infer tumor phylogenies from allele frequencies, which were originally ideated for single-sample datasets, have been extended to multi-sample data [17–24].

Among multi-sample datasets, SCS data are likely to become predominant in the next few years, as they provide the highest possible resolution [25]. Yet, this technology still suffers from several technical challenges in cell isolation and genome amplification of single cells, which produce a broad range of data-specific errors, e.g., allelic dropouts, false alleles, missing data, non-uniform coverage and doublets [26–32]. This state of affairs prevents straightforward applications of perfect phylogeny algorithms to SCS data [33], and a growing number of cancer-specific probabilistic approaches to infer phylogenies from SCS data have being proposed [34–41]. Some of these approaches estimate the temporal ordering of accumulation of genomic alterations via mutational trees [36–38], whereas others deconvolve clones and their evolutionary relations [39–41].

Most of these methods rely on the Infinite Sites Assumption (ISA), according to which each mutation occurs at most once during the evolutionary history of a tumor, and is never lost. Unfortunately, violations of the ISA are more frequent than originally expected, due to chromosomal deletions and loss of heterozygosity, which could lead to back mutations, and to convergent evolution, in which the same mutation is observed in independent clones or lineages(i.e., parallel mutations) [34]. More in general, such methods usually depend on a large number of ad-hoc technical assumptions and parameters, e.g., noise model, search schemes, etc., which need to be opportunely tuned, often requiring computationally demanding automated procedures and/ or prior biological knowledge about the underlying phenomenon. Similarly, in many cases arbitrary heuristics are needed to disambiguate equivalently optimal solutions, e.g., when seeking for a maximum parsimony phylogenetic tree.

Here we introduce TRaIT (Temporal oRder of Individual Tumors), a computational framework to infer the order of accumulation of mutations in single tumors that unifies several distinct approaches.

• TRaIT is the first method that explicitly supports both multi-region and SCS data within a unique statistical framework, with remarkable performances with both data types without the need for a model of noise specific to them;

• TRaIT extends mutational trees by accounting for violations of the ISA due to convergent evolution, for certain values of the observed probabilities. In Figures 5-B the analysis of a multi-region colorectal cancer dataset via TRaIT is shown, in which parallel mutations of a driver gene hit independent lineages. As the general validity of the ISA is debated [34], this represents a major advantage of TRaIT with respect to tree-based phylogenetic techniques.

• TRaIT can detect other complex phenomena underlying ITH such as the presence of multiple independent trajectories in the same tumor, or confounding factors annotated in the data. In the former case, these could be due either to multiple cells of origin [42], or to tumour initiation triggered by epigenetic states not annotated in the data (e.g., methylations). The latter relates to the annotation, in the data, of events that are unrelated to the progression.

• TRaIT can process any kind of genomic lesion, e.g., somatic mutations, copy number alterations, fusions, etc., allowing for an unprecedented information integration among data types;

• TRaIT outperforms techniques specifically tailored for multi-region or SCS data, in terms of accuracy and robustness to data-specific errors and small sample size.

• TRaIT displays a significant improvement in terms of computational time and scalability, which represents another key advantage in anticipation of the increasing availability of large-scale studies/ datasets on single tumor evolution.

TRaIT’s models are assembled by combining simple “building blocks”: if a model contains edge x → y, then x⁺ mutants are ancestral to y⁺ ones, and y⁺ mutants are statistically associated to x⁺. Such conditions describe the underlying clock among x and y, and are estimated via simple inequalities from data; their confidence is assessed via several statistical approaches, such as testing, bootstrap and cross-validation.

The simplicity of the underlying theoretical framework has several computational advantages, which we exploited to implement a suite of efficient algorithms that can model complex temporal structures, up to direct acyclic graphs (DAGs) with disconnected components, hence capturing different key aspects of tumor evolution and allowing for ISA violations (Figures 1-D). On the overall, despite supporting a wider range of data and models compared to standard phylogenies, our methods have state-of-the-art accuracy and more stable performance with small sample size and different data types, as well as lower computational complexity and higher scalability, as extensive tests on synthetic data suggest.

Figure 1:

A. A phylogenetic model of tumor progression describes the order of accumulation of somatic mutations, CNAs, etc. The model describes a set of possible genotype signatures, which are observed with an unknown spatial and density distribution in a real tumor. B. Multi-region bulk sequencing processes a signal mixed from different tumor subpopulations, with potential contamination of non-tumor cells. Thus, a sample will be likely annotated with lesions from different tumor lineages (green, red), creating spurious correlations in the data. In this case, we expect the rate of false positives and negatives in the calling to be symmetric. C. If we sequence genomes of single cells we can, in principle, have a precise signal from each subpopulation. However, the inference with these data is made harder by high levels of asymmetric noise, and errors in the calling. D. We are interested in studying temporal models of cancer progression in 4 possible scenarios. (i) when all annotated mutations are related to the progression, (ii) when data harbours confounding factors, (iii) when a tumor might have multiple cells of origin and, accordingly, multiple independent progressions and (iv) when independent evolutionary trajectories converge toward a certain mutation (i.e., a confluence). Case (iv) is when the Infinite Sites Assumption (ISA) is violated, as red mutations appear in two distinct evolutionary trajectories.

TRaIT can be used to infer the order of accumulating mutations in individual tumors, but not to deconvolve tumor clones’ signatures. Thus, we show in this paper how to couple TRaIT to methods for clones detection, so to draw a new and all-encompassing pictures of tumor evolution and ITH, where one can infer which clones were present in the input data (signatures), and how mutations accumulated within each clone.

Materials and methods

TRaIT includes 4 optimal polynomial-time algorithms (Figures 2) that process a binary matrix D with n columns and m rows [37]. D stores n variables (mutations, CNAs, etc.) detected across m samples (single cells or multi-region samples). If an entry in D is 1, then the associated variable is detected in the sample. Missing data in SCS are handled by a standard EM procedure with multiple imputations [43]. A priori estimates of false positives/ negatives rates ϵ₊, ϵ₋ ≥ 0 in D can be provided to each algorithm. All the algorithms are implemented in R, in the TRONCO tool for Translational Oncology [44, 45].

Figure 2:

A. TRaIT’s input data is a binary matrix that stores the presence/absence of a variable in a sample (e.g., a mutation, a CNA, or a persistent epigenomic lesion). Some of these observations harbour false positives and negatives (noise). B. We estimate via bootstrap the prima facie ordering relation that satisfies Suppes’ conditions for probabilistic causation, here used as estimators of temporal orderings. This, in turn, induces a graph G over variables x_i. G can be weighted by information-theoretic measures for variables’ association. C. we weight G with pointwise mutual information (pmi), and we use an heuristic that makes G acyclic by removing less confident edges (see [46]); then we use Edmonds’ optimal solution for minimum directed spanning trees. D. if we weight G with mutual information (mi) and disregard edges’ orientation, we can use Prim’s optimal solution for minimum undirected spanning trees. To find a direction for each edge, we use mutations’ frequencies. With this algorithm, we can accomodate violations of the ISA due to confluent trajectories when Suppes’ conditions are still valid among the parent/ child nodes (see Methods). E. if we use Gabow’s optimal solution for path traversals of cyclic component, we can detect the best tree that makes G acyclic. Then, we can again use Edmonds’ algorithm for spanning trees. F. A Bayesian optimal mode-selection strategy can compute the Chow-Liu tree that induces the distribution with minimum divergence from the true one. In this case, we process G as in PRIM to detect edges’ direction. As for PRIM algorithm, these models can accomodate violations of the ISA due to confluent trajectories according to the parameter values (see Methods).

At their core, TRaIT’s algorithms exploit a bootstrap procedure to compute 2 p-values per edge – one for temporal direction, one for association’s strength, according to Suppes’ theory of probabilistic causation [47] (see Methods). Two algorithms infer mutational trees (Edm, Gbw) and two DAGs (ChL, PRIM) (see Methods and Supplementary Information) – for this reason our framework supports mutational graphs. All algorithms can return a model with separate components, suggesting that data lacks statistically significant associations, or harbours multiple progressions which can be inputed to tumour triggers not annotated in the data (e.g., epigenetic lesions).

For these reasons, TRaIT can be used (Figures 1-D):

1. when all D’s variables are actually involved in the progression (i.e., all the events are drivers);

2. when some of D’s variables are confounding factors (i.e., miscalled mutations);

3. when D contains multiple independent progressions (i.e., multiple cells of tumor origin);

4. when distinct evolutionary trajectories converge to the same variable (i.e., certain drivers are shared by distinct (sub)clonal histories).

Edm and Gbw infer models for (i,…, iii), whereas only ChL and PRIM can explicitly account for (iv); the latter case happens when the ISA is violated. One can choose which TRaIT’s algorithm to use according to, e.g., research goals or prior knowledge on the evolutionary process. A rule of thumb might be the application of all TRaIT’s algorithms, followed by a comparative analysis of their output models – as we show in the case studies. The creation of a consensus model could be also effective to this end. In the Results section we present results from simulations of different experimental conditions and data; these tests allowed to assess the performance of the algorithms in the four scenarios, thus providing indications to an appropriate choice according to the specific case.

The detailed mathematical description of all TRaIT’s algorithm is included in the Supplementary Information, whereas in the following some details on the overall theoretical framework are provided.

Suppes’ probabilistic causation [47]

Let p(·) be multinomial estimates of the probabilities in D. For every pair of variables x and y in D, x is a plausible cause of y if

The former condition acts as an Infinite Sites Assumption (ISA), as we are assuming that lesions are persistent; which can be weakened in certain situations (see below). So, we estimate temporal precedence by marginal probabilities. The latter implies condition statistical dependence: p(x, y) ≠ p(x)p(y) [48]. For an edge to be part of our models, both conditions must be satisfied. When this is not the case, an edge is included and a non-significant p-value returned (see below). By iterating this approach, we can create models.

This tool is the core ingredient of successful causal approaches for cancer evolutionary inference [46]. It represents a necessary but not sufficient estimator of selective advantage, and combined with a statistical frameworks to disentangle true from spurious associations, can detect selection [49]. With data from a single patient, we limit its power to predict just temporal orderings (see the Supplementary Information).

Working scenarios (Figures 1-D)

Theres is a huge deal of variability in cancer data types, in cancer ITH, as well as in our ability to call mutations etc. Besides, several aspects of cancer evolution are yet undeciphered, so we considered four working scenarios representative of different biologically and technologically-motivated assumptions, and defined corresponding algorithms.

The simplest setting is (i) when all D’s variables are involved in the progression. Then, we generalize this to when (ii) some of D’s variables are annotated in D, but irrelevant to tumor progression (e.g., calling uncertainty or other confounding factors). Besides, we account for a (iii) a tumor with multiple cells of origin, and we aim at identifying multiple independent models from a unique dataset. The fourth case is that of a tumor that shows (iv) selective pressures that converge towards variable x. It seems reasonable to consider (i, ii) more common than (iii, iv).

Algorithms (Figures 2)

TRaIT’s algorithms use a non-parametric bootstrap strategy to assess Suppes’ conditions among variables pairs, and include them in a direct graph G. Then, four different strategies can compute, from G, a model. Output models can be interpreted as Suppes-Bayes Causal Networks [50–52], an extension of Bayesian Networks, with maximum likelihood estimates of the parameters θ [53] – bearing in mind that usually such models are used in truly causal approaches. The output is the Maximum A Posteriori probabilistic model that best explains D.

The algorithms are inspired by (i − iv), which require us to infer (i) a mutational tree T, (ii) T and some detached nodes for the variables identified as confounding, (iii) a set of trees, {T_i}, usually called a forest and (iv) a direct acyclic graph G (since confluent trajectories lead to a node with multiple incoming edges).

TRaIT implements two algorithms to infer trees: Edm (Edmonds) (Figures 2-C), Gbw (Gabow) (Figures 2-E), based on weighted directed minimum spanning tree reconstruction. These algorithms scan G to identify the T that maximizes the edges’ weights, which are computed via information-theoretic measures of the degree of association of variables – e.g., (pointwise) mutual information. Edm and Gbw differ from the way they order strongly connected components [54, 55] that appear in G because of finite sample bias. Computationally, Gbw is more expensive and general than Edm.

Two additional algorithms, ChL (Chow-Liu) (Figures 2-F), PRIM (Figures 2-D), are available to infer direct acyclic graphs. ChL is a Bayesian model-selection method to factorize a joint distribution over the input variables [56]. PRIM is the equivalent to Edm for undirected structures, is applied by rendering G undirected, and weighting it with mutual information (which is symmetric). By assigning a posteriori an ordering to the undirected spanning trees that is consistent with the variables’ frequency, we can retrieve confluent relations that capture violations of the ISA. In TRaIT these cases can be detected under certain conditions of the parameters: if – in the true progression – x and y converge to z, we will not detect two confluent trajectories if p(x) ≤ p(z) or p(y) ≤ p(z); see the Supplementary Information.

Detection of k independent progressions is a feature available for all algorithms, as it is enforced by the bootstrap when G has k disconnected components. Each group describes evolutions as triggered by multiple initiation cells. Thus, by G’s estimation and by picking the proper TRaIT’s algorithm, one can easily cover scenarios (i,…, iv).

Complexity

We observe that all TRaIT’s algorithms are optimal polynomial-time algorithmic solutions to each of their corresponding combinatorial problems. Thus, they scale well with sample size, a problem sometimes observed with Bayesian approaches that cannot compute a full posterior and need to sample. TRaIT’s algorithms, however, do not have a rich description of uncertainty since they return a single model, but can be however paired with a posteriori forecasts’ assessment strategies (e.g., cross-validation/ bootstrap) rather easily [49].

Data types

TRaIT’s algorithms work with both SCS and multi-region data. We expect D to contain noisy observations of the unknown true genotypes (Figures 1-A-C). The algorithms can be informed of the usual false positives and negatives rates ϵ₊ ≥ 0 and ϵ₋ ≥ 0, respectively. This adds no overhead to the computation, but prevents to learn noise rates from D, as it is instead possible with techniques as SCITE [37]. Since the algorithms show stable performance for slight variations in the input noise rates, avoidance of complex noise-estimation schemas seems a plus, especially when reasonable estimates of ϵ₊ and ϵ₋ are known a priori. This strategy is also used in OncoNEM [40].

For SCS with missing data we use a standard Expectation Maximization approach to input missing values; we repeat it n times, and then perform inference and select the MAP best model out of the n trials.

Code availability

All the algorithms included in TRaIT are implemented in R, and are available in the TRONCO tool for TRanslational ONCOlogy [44,45]. TRONCO is available under a GPL3 license at its webpage: https://sites.google.com/site/troncopackage or at Bioconductor: https://bioconductor.org.

Data availability

All data used in this paper are available from the supplementary material of [30] and [57]. To allow the reviewers to replicate our case studies we provide the source code as well as the input data at https://goo.gl/Ku13MM.

Additional file 1 — Supplementary Information

The detailed description of TRaIT’s algorithmic framework and the results of extensive tests both on simulated data and several real datasets are provided in the Supplementary Information.

Results

Simulations

We assessed the performance of TRaIT’s algorithms with simulated single cell and multi-region data.

In particular, we generated multiple batches of independent synthetic datasets from random phylogenies (generative models), with 5 ≤ n ≤ 20 nodes and different levels of topological complexity (Figures 1-D). SCS datasets with 10 ≤ m ≤ 100 cells and multi-region datasets with 5 ≤ m ≤ 50 regions (accounting for sampling bias) were created. To test the robustness against imperfect data, false positives, false negatives (highly asymmetric for SCS) and/or missing data were introduced in the true genotypes, consistently with previous studies [37]. Multiple configurations of parameters were scanned, and we measured the ability to infer true edges (sensitivity), and discriminate false ones (specificity); further details on data generation are available as Supplementary Information.

We compared our methods to SCITE, the state-of-the-art for phylogenetic inference of mutational trees from SCS data [37]. In the test, we also included previously developed approaches to causal inference from single-sample data (CAPRESE [48] and CAPRI [46]).

Full results are in Supplementary Figures S3 and S5–S15. Here we show four simulations in Figures 3; these settings are consistent with the results across all tests. Figures 3 displays the results for TRaIT and SCITE¹ in canonical settings of noise and sample size, for case (i) (SCS and multi-region data), for case (ii) (multi-region data), and for case (iii) (SCS data).

Figure 3:

We estimate from simulations the rate of detection of true positives (sensitivity) and negatives (specificity), visualized as box-plots from 100 independent points. We compare TRaIT’s algorithms and SCITE, the state-of-the-art for mutational trees inference. For each data type, here we show here a mild-noise setting with canonical sample size: in SCS data noise is ϵ₊ = 5 × 10⁻3; ϵ₋ = 5 × 10⁻2, in multi-region ϵ₋ = 5 × 10⁻2. Extensive results for different models, data type, noise and sample size are in Supplementary Figures 3–14. A. Here we use a generative model from [39] (Supplementary Figure5). (left) SCS datasets with m = 50 single cells, for a tumor with n = 11 mutations. (right) Multi-region datasets with m = 10 spatially separated regions, for a tumor with n = 11 mutations. B. We augment the setting in A-right with 2 random variables (with random marginal probabilty) to model confounding factors, and generated SCS data. C. We generated multi-region data from a tumor with n = 21 mutations, and a random number of 2 or 3 distinct cells of origin to model independent progressions. D. Spectrum of average sensitivity and specificity for Gbw algorithm estimated from 100 independent SCS datasets sampled from the generative model in Supplementary Figure 5 (m = 75, n = 11). The true noise rates are ϵ₊ = 5 × 10⁻3; ϵ₋ = 5 × 10⁻2; we scan input ϵ₊ and ϵ₋ in the ranges: ϵ₊ = (3, 4, 5, 6, 7) × 10⁻3 and ϵ₋ = (3, 4, 5, 6, 7) × 10⁻2.

All the techniques achieve high sensitivity and specificity scores from SCS generated by phylogenies with drivers only – Edm and Gbw highlighting the best results (medians approx. 0.8 and 1). When we sampled multi-region data from the same topology, performances worsened for all methods likely due to the smaller sample size and the mixed bulk signal. The introduction of confounding factors (2 out of n = 13 variables), does not to impact the performance significantly, and all algorithms mostly discriminate the true generative model. Finally, the inference of tumors with multiple independent progressions proves to be a harder task, as sensitivity decreases and the performance of all methods are similar. Notice that SCITE, in all tests, achieves the lowest specificity; this might point at a mild-tendency to overfitting, probably due to the combination of its search scheme and noise-learning model (see also Conclusion).

General conclusions can be drawn from the whole set of tests that we carried out. As expected, performances improve with lower noise and larger datasets. In particular, with SCS data Gbw, Edm and SCITE seem the best algorithms; they generally achieve very similar sensitivity, even though the latter presents (on average) lower specificity. For SCS data, all the tested algorithms seem very efficient up to 20/30% of missing data, with SCITE showing a slightly greater robustness (Supplementary Figure S11).

Results on multi-region data display similar trends, with Gbw and Edm showing the overall best performance. In this case, however, SCITE is less effective in retrieving both the true and the false relations, especially with small datasets and/or low noise levels.

Interestingly, by systematically analyzing the impact of a variation of the input ϵ₊ and ϵ₋ with respect to the true noise values, we discovered that the performance is rather stable for all algorithms (in Figures 3-D we show Gbw algorithm); this supports our choice – in line with other tools [40]. – of not implementing sophisticate noise-learning strategies in TRaIT.

Finally, a computation time assessment allowed to record a 3× speedup of all the algorithms included in TRaIT with respect to SCITE, on standard CPUs (Supplementary Table 10).

SCS data: Triple-Negative Breast Cancer

We applied TRaIT to a SCS dataset of Triple-Negative Breast Cancer (patient TNBC in [30]). The input data consists of single-nucleus exome sequencing of 32 cells: 8 aneuploid (A) cells, 8 hypodiploid (H) cells and 16 normal cells (N) (Figures 4-A).

Figure 4:

A. Input data from single-nucleus sequencing of a triple-negative breast cancer [30] (32 cells). The rate of missing values for this dataset is very low (around 1%), allelic dropout has rate 9.73 × 10⁻², and false discovery 1.24×10⁻⁶. B Manually curated phylogenetic tree estimated in [30]. Mutations are annotated to the trunk if they are ubiquitous across cells, and detected also in a bulk control sample [30]. Subclonal mutations are those appearing only in more than one cell. C. Temporal mutational tree obtained with Edm algorithm; p-values are obtained by 3 tests for Suppes’ conditions and overlap (hypergeo-metric test), and edges annotated with a posteriori non-parametric bootstrap scores (100 estimates). For these data, all TRaIT’s algorithms return trees (Supplementary Figure 16), consistently with the manually curated phylogeny (A). Most edges are highly confident (p <0.05>), expect for groups of variables with the same frequency which have unknown ordering (red edges). The ordering of mutations in subclones A₁, A₂ and tumor initiation has high bootstrap estimates (> 75%). D. We perform a concerted analysis to estimate both clones’ signatures and their formation, at least for mutations with a clear statistical signal in these data. We do this by computing a clonal tree with OncoNEM, which predicts 10 clones. Mutations are assigned to clones via maximum a posteriori estimates. The mutational ordering of the early clonal expansion of the tumor, which involves mutations in pten, tbx3, notch2, is consistent among both models. The same happens for most of the late subclonal events, e.g., mutations in map2k7, map3k4, ppp2r1a, syne2 and aurka in subclone A1. However, the temporal ordering of intermediate events has weaker support as they have the same marginal probability, and any permutation of their ordering would be equivalent for our method.

In [30], with a bulk sequencing control, mutations detected both in the bulk and in the majority of the cancer cells were annotated as clonal, whereas those undetected in the bulk as subclonal. The authors then manually curated a qualitative phylogenetic tree (Figures 4-B). We here run TRaIT with the mutational profiles of each single cell describing the presence/ absence of nonsynonymous point mutations of the 22 genes selected in [30]. The rate of missing values in this dataset is very low (around 1%); as suggested in [30], we used 9.73 × 10⁻² for allelic dropout rate and 1.24 × 10⁻6 for false positive rate.

For these data, all TRaIT’s algorithms return trees, suggesting consistency with the ISA (Supplementary Figures S16-S17); here, we discuss Edm’s one since that algorithm achieved the best performance in the simulations with drivers and confounding factors (Figures 4-C). To improve the analysis, from the same dataset we also deconvolve the signatures of putative clones with OncoNEM, and compare the predictions (Figures 4-D).

TRaIT allows to characterize the qualitative phylogeny provided in [30] by identifying the gradual accumulation of point mutations, expectedly due to defects in DNA repair or replication machineries, both in the clonal and subclonal histories of the tumor.

On the one hand, Edm’s model displays high-confidence branched evolution consistent with subclone A₁ (ppp2r1a, syne2 and aurka mutations), A₂ (ecm2, chrm5 and tgfb2 mutations), and H (nrrk1, aff4, ecm1, cbx4 mutations) [30]. On the other hand, TRaIT provides a notably higher resolution in the description of the mutations annotated as clonal in [30], e.g., pten, tbx3 and notch2, are suggested to trigger tumor initiation. These results are also consistent with the presence of different molecular clocks operating at different stages of tumour growth [30]. TRaIT allows to formulate new hypotheses about undetected subclones, possibly characterized by private mutations in akap9, or in jak1, setbp1 and cdh6, which however would require further experimental confirmations.

Clones’ analysis via OncoNEM detects 10 clones, their lineages and temporal relations, thus refining the qualitative analysis of [30]. Remarkably, such results are mostly consistent with ours, as the mutational ordering predicted by OncoNEM (obtained by estimating the assignment of mutations to clones, as suggested in [40]) largely overlaps with that inferred via TRaIT. This is particularly evident for early events, and for most of the late subclonal ones; exception made for subclone H, which is not detected by OncoNEM. As mutations in araf, akap9, notch3 and jak1 have the same marginal probability, their temporal ordering can not univocally determined from these data. TRaIT, in fact, provides a p-value p > 0.05 for the direction of those edges, suggesting that any permutation of their ordering would be possible. For this reason, unless more sequenced cells were available, we can not univocally match the clonal signatures obtained with OncoNEM and the temporal orderings identified by TRaIT for these temporally-intermediate events.

This result proves that the concerted use of techniques for the inference of mutational ordering, together with clonal deconvolution approaches, can provide a picture of tumor evolution and ITH at an unprecedented resolution and accuracy.

Multiple-biopsy data: MSI-high Colorectal Cancer

We applied TRaIT to a moderately-differentiated MSI-High colon cancer characterized by a primary tumor and a right hepatic lobe metastasis, with no prior treatments (patient “P3” in [57]). For this patient, targeted DNA resequencing of three regions of the primary tumor (P3-1, P3-2, and P3-3) and of two metastatic regions (L-1 and L-2), allowed to identify 47 nonsynonymous point mutations and 11 indels [57] (Figures 5-A).

Figure 5:

A. Multi-region sequencing data for a MSI-high colorectal cancer [57], with three regions of the primary cancer: p3-1, p3-2 and p3-3, and two of one metastasis: L-1 and L-2. To use this data with TRaIT we merge mutations that have the same signature across all regions, obtaining a clonal group (light blue) including 34 mutations and sublclonal group (light yellow) including: non-synonymous SNVs of smad4 and setd2, and non-framshift insertion of AR. B. Models obtained by Edm and PRIM algorithms, with their confidence annotated and the overlap in the predicted ordering obtained by SCITE. PRIM predicts convergent evolution – which violates the ISA – towards a non-synonymous mutation in cdkn2a, which is also predicted by ChL (Supplementary Figure 18). All edges, in all models, are statistically significant for Suppes’ conditions (temporal precedence and selection strengths). C. Four of the predicted ordering relations are consistently found across all TRaIT’s algorithm, which gives a high-confidence explanation for the formation of the L2 metastasis. This finding is also in agreement with predictions by SCITE (Supplementary Figure 19).

To process this dataset with TRaIT, we first grouped the mutations with the same signature across the five regions, hence obtaining: (a) a clonal group, including the 34 mutations detected in all the samples, (b) a subclonal group, including the 3 mutations detected only in the L regions, and (c) 8 mutations with different mutational profiles. Our methods will not resolve their ordering since their signals are statistically undistinguishable. However, we will be able to order the groups against the mutations.

With these data both PRIM and ChL predict confluent evolutionary trajectories (Supplementary Figures S19), suggesting a violation of the ISA. In Figures 5-B we show PRIM’s direct acyclic graph and Edm’s tree. Both models predict branched tumor evolution and high ITH among the subclonal populations, consistently with the phylogenetic analysis carried out in [57].

First, the clonal lesions – the clonal root – trigger the first expansions of this tumor, with mutations in the key colorectal drivers apc, kras, pik3ca and tp53 [49]. These biomarkers are ubiquitous, and could not be used to disentangle the mutational spectrum of the primary tumor from the metastatic lesions, in accordance with [58].

Second, the models identify distinct branches outgoing from the trunk, which discriminate the different subclonal evolutions. In both models one subclonal trajectory is initiated by a stopgain SNV in the DNA damage repair gene atm. Edm, in particular, characterizes region P3-1 by a subsequent accumulation of inhba and cdkn2a nonsynonimous mutations, whereas P3-3 by smad4 (stop-gain SNV) and kmt2c (frameshift). Conversely, PRIM infers a more complex model, in which two confluent trajectories anticipate common late mutations in different regions: mutations of either inhba or of tgfbr2 may precede mutations of cdkn2a in region P3-1, whereas in region P3-3 alterations of either inhba or of smad4 might precede alterations of kmt2c. Interestingly, the alterations of cdnk2a might point to a cell cycle arrest hallmark for this tumor. Notice that the model inferred via PRIM exactly fits in scenario (iv), which could not be identified with canonical phylogenetic approaches.

Third, in both models the subclonal metastatic expansion is originated by a stopgain SNV in gnaq, anticipating mutations in smad4, setd2, ar (i.e., the subclonal group) and ppp2r1a. The models suggest canonical convergent evolution (i.e., parallel) towards smad4 (a stopgain SNV in the primary tumor, and a nonsynonimous mutation in the metastasis). The transducer of transforming growth factor-β superfamily signaling smad4 regulates cell proliferation, differentiation and apoptosis [59], and its loss is usually correlated with colorectal metastases [60]. ar is a transcription factor that regulates cell migration and inhibites hepatocellular carcinoma metastases [61]; its splice variants are known to promote metastasis in several tumor types [62]. Similarly, gnaq is supposed be relevant in metastases development in certain tumor types [63]. ppp2r1a is a negative regulator of signal transduction, gene expression and cell cycle [64], and its mutation influences tumor-endothelium interaction in melanoma metastases [65]. Notice that many other genes that are supposed to characterize MSI-high progression are wildtype in this tumor, e.g., fbxw7, braf, arid1a, fam123b, etc., [49], as a further evidence of the high level of ITH even within the same tumor subtype.

We finally compared the ordering estimated by TRaIT to the predictions obtained by SCITE (Supplementary Figure S20). Both approaches predict the same formation of the metastatic lesion, yet some significant differences are present. First, SCITE predicts that the mutation of atm triggers tumor initiation, prior to the mutations included in TRaIT’s clonal group, which are ordered in a 34 events-long linear chain. Yet, this specific order has score equivalent to several other models (Supplementary Figure S20) and, thus, might be unreliable.

Besides, in SCITE’s model tgfbr2 is associated to region P3-1 (in accordance with PRIM, but not with Edm), gnaq’s stopgain is upstream to both P3-3 and L branches, and some relations appear in inverted temporal ordering, e.g., between smad4 and kmt2c. Finally, by construction, SCITE can not infer any confluent evolutionary trajectory, as its statistical model relies on the ISA.

Conclusion

The increasing availability of high-resolution multi-sample sequencing data allows one to study ITH at an unprecedented resolution, to understand origination and development of tumors both at the genotype and phenotype level, and to better stratify and treat cancer patients.

Multi-region and SCS data harbours signals that can be informative of different aspects of tumor evolution. In fact, several techniques have been developed that either deconvolve clonal signatures, determine the ordering of growing clones or accumulating mutations, or estimate clonal fractions and cellular prevalence. The concerted application of these techniques allows to draw complex pictures of cancer evolution. As we show for a triple negative breast cancer, one can use the same dataset to detect both the signatures of the prevalent clones, and to infer the temporal precedence (i.e., ordering) of mutations that generated them. With the right tools, one can hence understand which clones are annotated in the data, and how they were shaped by evolutionary pressures.

The majority of techniques that perform such analyses ground their roots in standard phylogenetic theory, or in some of its cancer-specific derivations. These techniques are very effective, but sometimes they also implement a noteworthy deal of technical assumptions regarding sequence substitution models, alleles fixation, noise or search scheme etc. As a consequence, it could be hard to quantify how much the final predictions are shaped by the model and its assumptions, or actually suggested by the data. For instance, complex noise-learning models to leverage the imbalances of SCS data might resolve the ordering of clonal mutations in arbitrary ways. This manifests as long trunks whose actual order can not be estimated from current data, and the inclusion of subclonal mutations in dubious positions in the trunk (Supplementary Figure S19).

Similarly, when one seeks for a maximum parsimony phylogenetic tree of tumor evolution, several equivalent-scoring solutions could be returned. When that happens, one has to implement disambiguation heuristics to select one output model [11, 12]. This could be one of the computed trees, or a new tree that is a combination of those (e.g., a bootstrap consensus [66]). Despite these routines are often adopted, they are somewhat arbitrary and some deal of care should be warned.

On top of these, recent evidences on the violation of the ISA suggest that this assumption might not be always appropriate; the ISA is preponderant in the derivation of most inferential techniques, and future methods should find a way to consistently account for its violations [35].

In this paper, we deviate from phylogenetic methods and present the TRaIT computational framework, whose methods give statistically robust estimates of mutational orderings in a variety of settings. Our models are simple, and can be interpreted straightforwardly: if an edge connects two mutations (i) it statistically resolves their temporal ordering and (ii) the mutations are statistically dependent. Both conditions are estimated from data without using complex inferential models, and assessed via statistical testing, which leads to p-values. Further assessment of models’ confidence can be obtained by usual bootstrap or cross-validation approaches [49]. One should be warned that, as mutational trees or other phylogenies, TRaIT’s models will display in output all mutations annotated in the input data. So, x and y could be passengers mutations observed by hijacking in this patient; nonetheless, their temporal relation can be disentangled, but for a more thorough carachterization of divers against passenger mutations, one would arguably need more complex tools and data combined with causal approaches [46, 48, 49].

The simplicity of our framework has major advantages, both from an evolutionary and a computational point of view. First of all, TRaIT’s models can account for any variable that can be annotated in a tumor sample. Thus, with TRaIT one can introduce high-level information on pathways, hallmarks, phenotypic-triggering lesions or epigenetic states (e.g., methylations), as long as they are persistent during tumor evolution. Inclusion of these information in traditional sequence-based phylogenetic methods that work with sequences could be harder. Second, TRaIT implements four optimal (i.e., polynomial-time) algorithms that look for different types of signals in the sequencing data and can model more complex topologies than trees, such as direct acyclic graphs with disconnected components. Therefore, TRaIT can be used to investigate whether data suggest the presence of confounding factors, or if the tumor’s data harbours several progressions, or if late mutations associate to multiple evolutionary trajectories. This latter case is a first attempt at performing inference when the ISA is violated by convergent evolution, a possibility that is missing in classical phylogenetic methods that are limited to estimating single trees from data ². Thus, with TRaIT’s algorithms one can test a broad set of hypotheses on tumor evolution as we show in a colorectal cancer case study where we find convergent evolution towards cdkn2a, which might point to a cell cycle arrest hallmark for this tumor type.

The computational burden of our techniques is limited, compared to standard Bayesian approaches (which, however, include an estimation of uncertainty within the model). We do not compute a full posterior over our estimates, but rather a Maximum A Posteriori model constrained by Suppes’ conditions. These conditions impose minimum levels of significance to the ordering predicted by our models, and are enforced as empirical Bayes priors. In light of the increasingly available data – especially from SCS – TRaIT’s scalability properties represent an important algorithmic advancement over Bayesian computations that might become impractical with larger datasets. Our methods accommodate low-effort parallel implementations [67], which we provide in the TRONCO tool for TRanslational ONCOlogy [44, 45].

TRaIT could be improved in several ways. For instance, we could pair bulk sequencing samples to either SCS or multi-region inputs; in fact, the combination of these has been recently shown to improve the estimation of the mutational ordering [38]. Furthermore, we could extend our framework to infer, besides mutational orderings, clonal signatures and architectures, in the attempt of defining a unified framework for cancer evolutionary inference. A general extension to models where the ISA is violated could also be investigated. From a broader perspective, our methods build on our earlier contributions on tumor evolution from single-sample bulk sequencing data [46, 48, 49] (see the Supplementary Information for further discussion). These models allowed us to define the first automatic pipeline to quantify inter-tumor heterogeneity across multiple patients [49].

To conclude, we advocate the use of our methods as complementary to phylogenetic tools for clone deconvolution, in a joint effort to better quantify the extent of ITH. To this end TRaIT represents an innovative and powerful tool to raise precision and effectiveness of large-scale analyses of single tumor evolution.

Author contribution

DR, AG and GC designed the algorithmic framework. DR, LDS and GC implemented the tool. LDS carried out the simulations on synthetic data. Data gathering was performed by DR, AG, LDS and GC. All the authors analyzed the results and interpreted the models. DR, AG, MA and GC wrote the original draft of the paper, which all authors reviewed and revised in the final form.

Acknowledgements

We thank Bud Mishra for his valuable comments on the manuscript and his insights on the effects of parallel evolution to our framework. Furthermore, we thank Guido Sanguinetti and Yuanhua Huang for useful discussions on the preliminary version of this manuscript. This work was partially supported by the SysBioNet project, a Ministero dell’Istruzione, dell’Universitá e della Ricerca initiative for the Italian Roadmap of European Strategy Forum on Research Infrastructures.

Footnotes

• 1 SCITE can return a posterior with many equivalent-scoring mutational trees; in those cases, to compute its error, we selected the first of those models.

• 2 We refer to Supplementary Information for a discussion of the statistical complications arising from such a generalization, in our framework.