Joint Analysis of Single Cell and Bulk Tissue Sequencing Data to Infer Intra-Tumor Heterogeneity

Summary

Many computational methods have been developed to discern intratumor heterogeneity (ITH) using DNA sequence data from bulk tumor samples. These methods share an assumption that two mutations arise from the same subclone if they have similar mutant allele-frequencies (MAFs), and thus it is difficult or impossible to distinguish two subclones with similar MAFs. Single cell DNA sequencing (scDNA-seq) data can be very informative for ITH inference. However, due to the difficulty of DNA amplification, scDNA-seq data are often very noisy. A promising new study design is to collect both bulk and single cell DNA-seq data and jointly analyze them to mitigate the limitations of each data type. To address the analytic challenges of this new study design, we propose a computational method named BaSiC (Bulk tumor and Single Cell), to discern ITH by jointly analyzing DNA-seq data from bulk tumor and single cells. We demonstrate that BaSiC has comparable or better performance than the methods using either data type. We further evaluate BaSiC using bulk tumor and single cell DNA-seq data from a breast cancer patient and several leukemia patients.

1. Introduction

Somatic mutations, which are defined as the mutations occurred in any cells except the germ cells (i.e., sperm and egg), may occur in every mitosis. A recent estimate of somatic mutation rate in human is 2.66×10⁻⁹ mutations per bp per mitosis (Milholland et al., 2017), and thus on average one mitosis introduces 8 mutations in the 3 billion bp human genome. Therefore, human cells carry a mosaic of somatic mutations, and each cell may have some private somatic mutations. Some somatic mutations in tumor cells bring survival advantage to their host cells, and thus are selected and shared among a subset of tumor cells. More specifically, by the clonal expansion theory of tumor growth (Nowell, 1976), when a tumor cell acquires a set of somatic mutations that improve its survival, its descendant cells may outgrow other tumor cells and form a subclone. All the tumor cells within this subclone share all the somatic mutations of their common ancestor, including driver mutations that promote tumor growth and passenger mutations without functional impact. Each tumor cell of this sublcone may also have some additional somatic mutations accumulated along its evolutionary path from the common ancestor of this subclone.

Intra-tumor heterogeneity (ITH) refers to the fact that the tumor cells within a tumor sample are genetically heterogeneous, and they can often be divided into several subclones. Many computational methods have been developed to discern ITH (i.e., to estimate the number of subclones and the somatic mutations within each subclone) using DNA sequencing data from one or multiple bulk tumor samples (Roth et al., 2014; Shen and Seshan, 2016; Jiang et al., 2016; Lee et al., 2016; McGranahan and Swanton, 2017). As the techniques of single cell sequencing (SCS) are becoming mature, some computational methods have been proposed to study ITH using single cell sequencing data (Van Loo and Voet, 2014; Deshwar et al., 2015; Jahn et al., 2016; Ross and Markowetz, 2016; Roth et al., 2016; Salehi et al., 2017; Malikic et al., 2017). Next we briefly describe the data from bulk and single cell DNA sequencing, and then justify why it is necessary to jointly analyze these two types of data.

For a bulk tumor sample, the mutations that arise from one subclone can be identified by clustering mutations based on their mutant allele frequencies (MAFs) (Figure 1). The mean MAF of a mutation cluster can be used to infer its cellular frequency: the proportion of cells that harbor these somatic mutations. Based on a commonly used “infinite site assumption” (two mutations are unlikely occur on the same site due to the large size of human genome), we assume a somatic mutation occurs in one of the two alleles in a diploid genome. If we further assume there is no copy number alteration, the MAF of a somatic mutation is half of its cellular frequency (Figure 1(A)). A somatic mutation cluster may be shared by multiple subclones. For example, assuming an evolutionary tree for somatic mutations: C ← (A → B) → D, the mutation group A is shared by all 4 subclones (Figure 1(A)–(B)). Therefore, the cellular frequency of any mutation in A is the summation of the cellular frequencies across these four subclones. Some existing methods for ITH inference of bulk tumor samples focus on clustering somatic mutations (Roth et al., 2014), while other methods aim to infer the underlying phylogenetic tree (Deshwar et al., 2015).

Figure 1.

Mutant allele frequency and cellular frequency. (A) An toy example of tumor samples with five groups of cells: normal cells, and tumor cells from three subclones. The 1,000 somatic mutations can be divided into four groups, with 600, 200, 100, and 100 somatic mutations in groups A, B, C, and D respectively. If a group of somatic mutations only occur in one subclone, their mutation allele frequency (MAF) is one half of their cellular frequencies. If they occur in more than one subclone (e.g., mutations in group A), their MAF is one half of the summation of the cellular frequencies of relevant subclones. (B) The assumed clonal expansion history. (C) the distribution of MAF across all 1,000 mutations is a mixture of four components corresponding to the four sets of mutations. (D) Simulated distribution of mutant allele frequency given copy number alteration. This figure appears in color in the electronic version of this article, and any mention of color refers to that version.

The observed somatic mutation data from single cell DNA sequencing (scDNA-seq) can be written as a matrix with each row for a locus and each column for a cell. The (i, j)-th entry of this matrix is the somatic mutation call at the i-th locus of the j-th cell. Then subclones are identified by clustering either rows or columns of this matrix. Due to technical challenges to faithfully amplify DNA sequences from a single cell, scDNA-seq data are often very noisy, resulting missing or mistaken mutation calls. The most common type of sequencing errors is allele dropout (ADO), when one of the two alleles is missed (dropped out) from the sequencing data. If the mutant allele of a heterozygous locus is dropped, it leads to a false negative mutation call. In contrast, a false positive mutation call may arise due to false detection of mutant reads from a locus of homozygous reference allele.

The sequencing data from bulk tumor samples cannot distinguish two subclones with similar cellular frequencies. The scDNA-seq data can overcome this identifiability issue because the cells from different subclones have different sets of mutations. On the other hand, the low quality mutation calls from scDNA-seq data can be improved by borrowing information from bulk DNA-seq data. Motivated by such complementary nature of these two data types, we propose to jointly model the somatic mutation data from bulk tumor samples and single cells to study ITH. We call our method BaSiC: Bulk tumor and Single Cell.

There are two existing methods for ITH inference using scDNA-seq data: SCITE (Single Cell Inference of Tumor Evolution) (Jahn et al., 2016) and OncoNEM (oncogenetic nested effects model) (Ross and Markowetz, 2016). Both methods model the observed mutation calls given the underlying true genotypes by Bernoulli distributions, parameterized by the sensitivity and specificity of somatic mutation calls. Both SCITE and OncoNEM use likelihood-based approaches to infer the phylogenetic tree of tumor cell evolution. SCITE searches across all the mutation trees (where each node corresponds to a mutation) by an MCMC approach. OncoNEM searches across all the cell lineage trees (where each node corresponds to a cell) by a heuristic approach followed by clustering cells into subclones.

Our method is motivated and built on these two methods. In addition to mutation call sensitivity and specificity, we also model the read count data. Read counts can provide additional information when mutation calls are missing or have non-ignorable uncertainty, as shown in an earlier study for germline variants (Hu et al., 2016). We found that low read-depth in scDNA-seq data is a major factor for the discrepancy between bulk and scDNA-seq data. If a mutation has relatively high MAF from bulk tumor, it is likely a clonal mutation and thus should appear in all the single cells. If such a mutation is not called in many cells, it is very likely due to low read-depth in those cells (Figure 2). Therefore, for joint analysis of bulk and scDNA-seq data, it is important to recover mutation calls even if read-depth is low, and our method provides a rigorous approach to do so. In addition, both SCITE and OncoNEM face the challenge of searching across huge space of trees, which makes it very difficult to study large datasets with many mutations or many cells. Our method employs a clustering approach that is computationally much more tractable and scales very well for large datasets. The trade-off is that we focus on clustering somatic mutations instead of inferring phylogenetic trees.

Figure 2.

Relation between read-depth and mutation call status. We examined the read-depth for 1,267 loci in 16 cells. Note that these loci were selected with at least two mutations in the 16 cells. We first divided these 1,267 loci into four groups based on the MAFs of these loci in bulk tumor sample. Then within each MAF group, we examined the read-depths for the cases where somatic mutations are called (Yes) versus the cases where somatic mutations are not called (No). The vertical line indicate read-depth cutoff 20. This figure appears in color in the electronic version of this article, and any mention of color refers to that version.

While we were working on the BaSiC method, we noted two other methods for jointly analyzing bulk and single cell DNAseq data for ITH study. ddClone (Salehi et al., 2017) uses the distances of somatic mutations derived from scDNA-seq data as prior information to refine the clustering of somatic mutations by their MAFs in a bulk sample. B-SCITE (Malikic et al., 2017) combines the likelihood of bulk tumor data and scDNA-seq data to infer mutation trees. BaSiC is different from ddClone as we use both data types for clustering instead of using one data type as prior. It is also different from B-SCITE since we aim to cluster a relatively large number of mutations and B-SCITE seeks to infer mutation trees for a relatively small number of mutations.

2. Method

Data Processing and Mutation Calling

We used a sequencing dataset from a triple-negative breast cancer patient (Wang et al., 2014) to illustrate some important characteristics of scDNA-seq data. This dataset includes exome-seq data from a bulk tumor sample, a bulk normal sample, 16 single tumor cells, and 16 single normal cells. First, calling somatic mutations from SCS data is a challenging task. We downloaded raw sequencing data (fastq files) from the NIH sequence read archive (SRA, with accession ID SRA053195), mapped the sequence reads to the human reference genome (hg38) using BWA (Li and Durbin, 2009), removed duplicated reads, and realigned reads around known indels (See Supporting Information Section A.1 for details).

There is no consensus on the best software package for somatic mutation calling, even for bulk tumor samples. We chose to use Strelka (Saunders et al., 2012) and Mutect (Cibulskis et al., 2013) because they are among the best performing packages in a third party evaluation (Xu et al., 2014). Calling somatic mutations requires paired tumor and normal samples. We called somatic mutations in the bulk tumor sample and each of the 32 single cells using the bulk normal sample as the paired-normal sample. Then we took the intersection of the mutation calls from Strelka and Mutect. The number of somatic mutation calls from the 16 single normal cells are used as indirect measurements of false positives. Surprisingly, a large number of somatic mutations were called in some single normal cells (e.g., > 10,000 somatic mutation calls). A filter based on read-depth (total depth ≥ 20 and number of mutant reads ≥ 5) and MAF (MAF ≥ 0.1) removed part of the somatic mutation calls in normal cells. The additional filter that removed the vast majority of mutation calls in normal cells was to require a mutation called in at least two cells (i.e., at least two of the 16 tumor cells or two of the 16 normal cells) (Supplementary Figure 2). 1,910 somatic mutations passed all the filters and were called in at least two tumor cells. We intersected them with 1,802 somatic mutations called by both Strelka and Mutect in the bulk tumor sample, and ended up with 1,267 somatic mutations for further analysis. Note that taking intersection of the somatic mutations called in bulk tumor and single cells is necessary for joint analysis of both types of data, though the mutations called in one but not both data types can be useful for further analysis.

Likelihood Models for DNA-seq Data from Bulk Tumor Samples

A bulk tumor sample is often composed of both tumor cells and non-tumor cells, such as infiltrating immune cells. The proportion of tumor cells within a bulk tumor sample is referred to as tumor purity. Assume there are h subclones with cellular frequencies η = (η₁,...,η_h)^T such that ${\sum }_{l=1}^h{\eta }_l=\rho$ where ρ is the tumor purity of this bulk tumor sample. Denote the total number of reads and the number of mutant reads in the bulk tumor sample across n mutant loci by Z = (Z₁,...,Z_n)^T and Y = (Y₁,...,Y_n)^T, respectively. We first consider the situation when there is no copy number alteration. The distribution of Y given Z can be modeled by a mixture of binomial distributions:

$$p(\mathbf{Y}|\mathbf{Z},\mathbf{\tau })=\prod _{i=1}^n\sum _{k=1}^h{\nu }_k{f}_{bin}\left(Y_i|Z_i,{\tau }_k\right),$$ (1)

where f_bin denotes the density of a binomial distribution with success probability τ_k, and τ = (τ₁,...,τ_h)^T. Since we aim to cluster somatic mutations instead of inferring phylogenic trees, we are interested in the inference of τ_k (mean mutant allele frequencies) instead of η (the cellular frequencies of subclones). Inference of η and phylogenic tree may be performed after obtaining mutation clusters, and treat each cluster as a pseudo-mutation.

When there are somatic copy number alterations (SCNAs), we assume that the copy number of the i-th mutation is e_i. If this copy number event occurs before the accumulation of any somatic point mutation, the multiplicity of any somatic mutation is 1/e_i, and the likelihood function becomes

$$p(\mathbf{Y}|\mathbf{Z},\mathbf{\tau })=\prod _{i=1}^n\sum _{k=1}^h{\nu }_k{f}_{bin}\left(Y_i|Z_i,2{\tau }_k/e_i\right).$$ (2)

If a copy number event happens after a somatic point mutation, the multiplicity of a somatic mutation may be e_iA/e_i or e_iB/e_i, where e_iA and e_iB are allele-specific copy numbers and e_i = e_iA + e_iB. Other multiplicities of somatic mutations are also possible, for example, due a series of three events, copy number alteration, somatic point mutation, followed by another copy number alteration. In such cases, the model become much more flexible and may lead to some identifiability issues. In our analysis, we impose a restrictive assumption that the multiplicity of a somatic mutation maybe 1/e_i, e_iA/e_i, or e_iB/e_i. We also assume the allele-specific copy numbers are pre-determined by another method.

Likelihood Models for scDNA-seq Data

To distinguish the notations for bulk DNA-seq data and scDNA-seq data, we denote the total number of reads and the number of mutant reads at the i-th locus of the j-th cell by $R_{ij}^T$ and $R_{ij}^A$, respectively. Let I_ij be an indicator whether a mutation is called, e.g., $I_{ij}=\mathbf{I}(R_{ij}^T\ge 20)$, where I is an indicator function. Note that 20 is an arbitrary cutoff and should be replaced by the threshold used in mutation calling/filtering. Denote the observed mutation call at the i-th locus and the j-th cell by D_ij, and denote the corresponding unobserved true mutation by E_ij. We model the scDNA-seq data conditioning on I_ij. If I_ij = 1, we only model D_ij and ignore read-depth information. This is because mutation call D_ij already incorporates the read-depth information, in addition to other information such as sequencing quality and alignment quality in nearby region. If I_ij = 0, we model read counts $R_{ij}^T$ and $R_{ij}^A$.

If I_ij = 1, we assume D_ij follows a Bernoulli distribution (f_ber) with success probability being α₁ or β₁ for E_ij = 0 or 1, respectively:

$$p\left(D_{ij}|E_{ij},I_{ij}=1\right)=\{\begin{array}{ll}{f}_{ber}\left(D_{ij}|{\alpha }_1\right),\hfill & \text{if}{E}_{ij}=0\hfill \\ f_{ber}\left(D_{ij}|{\beta }_1\right),\hfill & \text{if}{E}_{ij}=1\hfill \end{array},$$ (3)

1 − α₁ and β₁ is the specificity and sensitivity of mutation calls, respectively. While the specificity (1 − α₁) is often high because of stringent filtering, the sensitivity (β₁) can be low due to allele drop out (ADO) (Gawad et al., 2014).

If I_ij = 0, we model the mutant read count $R_{ij}^A$ given the total read count $R_{ij}^T$ by a mixture of beta-binomial distributions. There are a few different ways to parameterize a beta-binomial distribution. A generative model is to assume $R_{ij}^A$ given $R_{ij}^T$ follows a binomial distribution where the success probability follows a beta-distribution with parameter a > 0 and b > 0. Then the beta-binomial density is $f_{bb}(R_{ij}^A|R_{ij}^T,a,b)=\left(\begin{array}{c}{R}_{ij}^T\\ R_{ij}^A\end{array}\right)\frac{B\left(R_{ij}^A+a,R_{ij}^T-R_{ij}^A+b\right)}{B(a,b)}$, where B(a,b) is the beta function with parameters a and b. We choose the parameterization π = a/(a + b) and ρ = 1/(1 + a + b), where π is the expected proportion of mutant reads, and ρ is an over-dispersion parameter. ρ → 0 if there is no over-dispersion and the beta-binomial distribution is reduced to a binomial distribution.

We model $R_{ij}^A$ given $R_{ij}^T$ by a mixture of three beta-binomial distributions:

$$p(R_{ij}^A|R_{ij}^T)=\sum _{r=1}^3{\psi }_r{f}_{bb}(R_{ij}^A|R_{ij}^T,{\pi }_r,{\rho }_r),$$ (4)

with constraints 10⁻⁴ ≤ π₁ ≤ 0.15, 0.25 ≤ π₂ ≤ 0.75, and 0.85 ≤ π₃ ≤ 0.9999, respectively. These three mixture components correspond to three types of genotypes: homozygous reference allele, heterozygous, and homozygous mutant allele, and thus the 1st mixture component corresponds to D_ij = 0, and the next two components correspond to D_ij = 1. The expected values for π₁, π₂ and π₃ are 0, 0.5, and 1, respectively. We set the constraints to be around these expected values, but with big enough range to allow sequencing noise. The probability p(D_ij = 0) and p(D_ij = 1) can be estimated by ψ₁, and ψ₂ + ψ₃, respectively. The probability of read counts given D_ij can be written as

$$p(R_{ij}^A|R_{ij}^T,D_{ij})=\{\begin{array}{ll}{f}_{bb}(R_{ij}^A|R_{ij}^T,{\pi }_1,{\rho }_1)\hfill & \text{if}{D}_{ij}=0\hfill \\ {\psi }_2^{\prime }{f}_{bb}(R_{ij}^A|R_{ij}^T,{\pi }_2,{\rho }_2)+{\psi }_3^{\prime }{f}_{bb}(R_{ij}^A|R_{ij}^T,{\pi }_3,{\rho }_3)\hfill & \text{if}{D}_{ij}=1,\hfill \end{array}$$ (5)

where ${\psi }_2^{\prime }={\psi }_2/({\psi }_2+{\psi }_3)$, and ${\psi }_3^{\prime }={\psi }_3/({\psi }_2+{\psi }_3)$.

We model D_ij given the underlying true genotype E_ij by two Bernoulli distributions, with parameters α₂ and β₂ that are different from the parameters while I_ij = 1 (α₁ and β₁):

$$p\left(D_{ij}|E_{ij},I_{ij}=0\right)=\{\begin{array}{ll}{f}_{ber}\left(D_{ij}|{\alpha }_2\right),\hfill & \text{if}{E}_{ij}=0\hfill \\ f_{ber}\left(D_{ij}|{\beta }_2\right),\hfill & \text{if}{E}_{ij}=1\hfill \end{array}.$$ (6)

Since D_ij is unobserved, we sum over two possible values of D_ij and get

$$p(R_{ij}^A|R_{ij}^T,E_{ij},I_{ij}=0)=\sum _{D_{ij}=0,1}p(R_{ij}^A|R_{ij}^T,D_{ij})p(D_{ij}|E_{ij},I_{ij}=0).$$ (7)

Next we model the variation of E_ij within each cluster of mutations. Within the k-th mutation cluster, we assume E_ij follows a Bernoulli distribution with success probability γ_jk. Let γ_k = (γ_1k,...,γ_mk)^T be the Bernoulli parameters across the m cells. Write E_i = (E_i1,...,E_im)^T. Given the γ_jk’s, the E_ij’s are independent across cells (j’s), and

$$p\left(E_i\right)=\sum _{k=1}^h{\nu }_{k}p\left(E_i|{\gamma }_k\right)=\sum _{k=1}^h{\nu }_k\prod _{j=1}^m{f}_{ber}\left(E_{ij}|{\gamma }_{jk}\right),$$ (8)

where ν_k is the mixture proportion of the k-th cluster, the same as the mixture proportion parameter for bulk tumor data (equation (2)).

Finally, the likelihood for somatic mutations in single cells is given by

$$\sum _{E_{ij}}\left\{\prod _{i=1}^n\left[\prod _{j=1}^{m}p{(R_{ij}^A|R_{ij}^T,E_{ij},I_{ij}=0)}^{\mathbf{I}\left(I_{ij}=0\right)}p{\left(D_{ij}|E_{ij},I_{ij}=1\right)}^{\mathbf{I}\left(I_{ij}=1\right)}\right]p\left(E_i\right)\right\},$$ (9)

where I() is an indicator function. Here the summation is over all possible values of E_ij and thus the observed data likelihood is not feasible to compute when the number of mutations is very large. We use an EM algorithm to estimate the parameters.

We chose not to consider copy number alterations in single cell data because the single cell sequencing protocol to generate accurate somatic point mutation calls has uneven coverage, and thus it is not appropriate to study copy number alterations. For example, Wang et al. (2014) generated two sets of scDNA-seq data using different protocols: one for calling somatic point mutations and one for quantifying copy number alterations. It is possible that a somatic mutation is lost due to copy number loss. However, such event cannot be distinguished with the ADO in scDNA-seq data. Therefore we do not need to separately model the effect of copy number loss on mutation calls.

Clustering mutations based on bulk and scDNA-seq data

In the previous subsections, we have described the likelihood models for bulk and scDNA-seq data. Now we can combine these two types of data to cluster mutations. This is essentially a problem of clustering the latent variables E_ij. In the following algorithm, we iteratively estimate the E_ij’s and classify somatic mutations into h clusters. We will discuss how to select the number of clusters in the next subsection.

1. Initialize E by $\widehat{E}$.

   • If mutation is called (i.e., I_ij = 1), ${\widehat{E}}_{ij}=D_{ij}$
   • If mutation is not called and the read-depth is not too low (i.e., I_ij = 0 and $R_{ij}^T\ge 3$), ${\widehat{E}}_{ij}=p(D_{ij}=1|R_{ij}^A,R_{ij}^T)=\frac{{\widehat{\psi }}_2{f}_{bb}\left(R_{ij}^A|R_{ij}^T,{\widehat{\pi }}_2,{\widehat{\rho }}_2\right)+{\widehat{\psi }}_3{f}_{bb}\left(R_{ij}^A|R_{ij}^T,{\widehat{\pi }}_3,{\widehat{\rho }}_3\right)}{{\sum }_{t=1}^3{\widehat{\psi }}_t{f}_{bb}\left(R_{ij}^A|R_{ij}^T,{\widehat{\pi }}_t,{\widehat{\rho }}_t\right)}$. Recall that ${\widehat{\psi }}_t$, ${\widehat{\pi }}_t$, and ${\widehat{\rho }}_t$ can be estimated by fitting a mixture distribution as specified in equation (4). In practice, we first fit the mixture distribution using $R_{ij}^A$ and $R_{ij}^T$ from all mutations and all cells, and then re-estimate the mixture proportions using the data with missing mutation calls. When bulk tumor data are available, by default, we stratify somatic mutations based on MAFs of bulk tumor samples, and fit these mixture distributions within each stratum defined by bulk tumor MAF.
   • If the read depth is very low (i.e., $R_{ij}^T<3$), ${\widehat{E}}_{ij}$ is set to be the mutation frequency across all the cells, i.e., the the average of the ${\widehat{E}}_{ik}$’s for all k’s such that $R_{ik}^T\ge 3$.

2. Calculate the distance between two loci i and r by $d(i,r)={\sum }_{j=1}^m{({\widehat{E}}_{ij}-{\widehat{E}}_{rj})}^2+\log \left({\ell }_{ir}\right)$, where ${\ell }_{ir}$ is the likelihood ratio statistic for testing the hypothesis that the mutant allele frequencies of the two loci i and r from the bulk tumor sample are the same. See Supporting Information Section D.1 for details.

3. Given d(i, r), we cluster the somatic mutations using hierarchical clustering, and estimate cluster means for mutations in single cells (γ_jk’s) and MAFs of the bulk tumor sample (τ_k’s). Specifically, γ_jk (for the j-th cell in the k-th cluster) is estimated by ${\sum }_{i\in R_k}{\widehat{E}}_{ij}/\left|R_k\right|$, where R_k represents all the mutations of the k-th cluster and ∣R_k∣ is the size of R_k. τ_k is estimated by maximizing the likelihood, see Supporting Information Section D.2 for details.

4. Initialize ν_k by ${\widehat{\nu }}_k=1/h$ for the first iteration or use ${\widehat{\nu }}_k$ from the previous iteration. Denote the estimate of all the parameters by $\widehat{\mathrm{\Theta }}$, which include α’s, β’s, γ_jk’s, and τ_k’s. Given $\widehat{E}$ and $\widehat{\mathrm{\Theta }}$, estimate the probability that locus i belongs to the k-th cluster,

   $$w_{ik}=p({\delta }_i=k|\widehat{\mathrm{\Theta }})=\frac{{\widehat{\nu }}_k{f}_{bin}\left(Y_i|Z_i,2{\tau }_k/e_i\right){\prod }_{j=1}^{m}p\left({\widehat{E}}_{ij}|{\widehat{\gamma }}_{jk}\right)}{{\sum }_{g=1}^h{\widehat{\nu }}_g{f}_{bin}\left(Y_i|Z_i,2{\tau }_g/e_i\right){\prod }_{j=1}^{m}p\left({\widehat{E}}_{ij}|{\widehat{\gamma }}_{jg}\right)},$$ (10)

   where $p({\widehat{E}}_{ij}|{\widehat{\gamma }}_{jk})=(1-{\widehat{E}}_{ij})\left(1-{\widehat{\gamma }}_{jk}\right)+{\widehat{E}}_{ij}{\widehat{\gamma }}_{jk}$.

5. Given the w_ik’s, re-estimate the ν_k’s: ${\widehat{\nu }}_k={\sum }_i{w}_{ik}/n$.

6. Given $\widehat{\mathrm{\Theta }}$ and the w_ik’s, estimate ${\widehat{E}}_{ij}=p(E_{ij}=1|D_{ij},R_{ij}^A,R_{ij}^T,I_{ij},\widehat{\mathrm{\Theta }},\left\{w_{ik}\right\})$:

   $${\widehat{E}}_{ij}=\frac{p\left(D_{ij},R_{ij}^A|R_{ij}^T,I_{ij},E_{ij}=1,\widehat{\mathrm{\Theta }}\right)p\left(E_{ij}=1|\widehat{\mathrm{\Theta }},\left\{w_{ik}\right\}\right)}{{\sum }_{t=0,1}p\left(D_{ij},R_{ij}^A|R_{ij}^T,I_{ij},E_{ij}=t,\widehat{\mathrm{\Theta }}\right)p\left(E_{ij}=t|\widehat{\mathrm{\Theta }},\left\{w_{ik}\right\}\right)},$$

   where

   $$\begin{gathered} p\left(D_{ij},R_{ij}^A|R_{ij}^T,I_{ij},E_{ij},\widehat{\mathrm{\Theta }}\right)=p{\left(R_{ij}^A|R_{ij}^T,E_{ij},I_{ij}=0\right)}^{\mathbf{I}\left(I_{ij}=0\right)}p{\left(D_{ij}|E_{ij},I_{ij}=1\right)}^{\mathbf{I}\left(I_{ij}=1\right)}, \\ p\left(E_{ij}=1|\widehat{\mathrm{\Theta }},\left\{w_{ik}\right\}\right)=\sum _k{w}_{ik}{\widehat{\gamma }}_{jk},\text{and}p\left(E_{ij}=0|\widehat{\mathrm{\Theta }},\left\{w_{ik}\right\}\right)=\sum _k{w}_{ik}\left(1-{\widehat{\gamma }}_{jk}\right). \end{gathered}$$
7. Given $\widehat{E}$, estimate Θ. Let Ω_t = {(i, j) ∶ I_ij = t} for t = 0 or 1, and ${\widehat{D}}_{ij}=p\left(D_{ij}=1|R_{ij}^A,R_{ij}^T\right)$, which can be estimated based on equation (5).

   1. ${\widehat{\alpha }}_2={\sum }_{i,j\in {\mathrm{\Omega }}_0}{\widehat{D}}_{ij}\left(1-{\widehat{E}}_{ij}\right)/{\sum }_{i,j\in {\mathrm{\Omega }}_0}\left(1-{\widehat{E}}_{ij}\right)$, ${\widehat{\beta }}_2={\sum }_{i,j\in {\mathrm{\Omega }}_0}{\widehat{D}}_{ij}{\widehat{E}}_{ij}/{\sum }_{i,j\in {\mathrm{\Omega }}_0}{\widehat{E}}_{ij}$.
   2. ${\widehat{\alpha }}_1={\sum }_{i,j\in {\mathrm{\Omega }}_1}{D}_{ij}\left(1-{\widehat{E}}_{ij}\right)/{\sum }_{i,j\in {\mathrm{\Omega }}_1}\left(1-{\widehat{E}}_{ij}\right)$, ${\widehat{\beta }}_1={\sum }_{i,j\in {\mathrm{\Omega }}_1}{D}_{ij}{\widehat{E}}_{ij}/{\sum }_{i,j\in {\mathrm{\Omega }}_1}{\widehat{E}}_{ij}$.
   3. Estimate the γ_k’s for single cells and the τ_k’s for the bulk tumor sample.

8. Repeat steps 4–7 until convergence.

Choice of the number of clusters

To guide the choice of h, the number of clusters, we calculate an AIC-like quantity. For scDNA-seq only analysis,

$$\begin{gathered} \text{AIC}=-2\sum _{i,j}\left\{\log \left[p{\left(R_{ij}^A|R_{ij}^T,{\widehat{E}}_{ij},I_{ij}=0\right)}^{\text{I}\left(I_{ij}=0\right)}\right]+\log \left[p{\left(D_{ij}|{\widehat{E}}_{ij},I_{ij}=1\right)}^{\text{I}\left(I_{ij}=1\right)}\right]\right\} \\ -2\sum _{i=1}^n\log \left[p\left({\widehat{E}}_i\right)\right]+2(mh+h-1), \end{gathered}$$ (11)

Where $p\left(R_{ij}^A|R_{ij}^T,{\widehat{E}}_{ij},I_{ij}=0\right)={\omega }_0\left(1-{\widehat{E}}_{ij}\right)+{\omega }_1{\widehat{E}}_{ij}$, ${\omega }_t=p\left(R_{ij}^A|R_{ij}^T,E_{ij}=t,I_{ij}=0\right)$ for t = 0 or 1, and $p\left(R_{ij}^A|R_{ij}^T,E_{ij},I_{ij}=0\right)$ is specified by equation (7). The first two terms quantify the goodness of fit, though they are not exactly the log likelihood function because there is no summation across all possible values of E_ij for all i’s and j’s. The third term 2(mh + h − 1) is the penalty for model complexity, where mh + h − 1 is the degrees of freedom for cluster means of single cell data (mh) and mixture proportions (h − 1).

For the joint analysis of bulk tumor and scDNA-seq data, the AIC-like quantity is

$$\begin{gathered} \text{AIC}=-2\sum _{i,j}\left\{\log \left[p{\left(R_{ij}^A|R_{ij}^T,{\widehat{E}}_{ij},I_{ij}=0\right)}^{\mathbf{I}\left(I_{ij}=0\right)}\right]+\log \left[p{\left(D_{ij}|{\widehat{E}}_{ij},I_{ij}=1\right)}^{\mathbf{I}\left(I_{ij}=1\right)}\right]\right\} \\ -2\sum _{i=1}^n\log \left[p\left({\widehat{E}}_i\right)\right]-2\sum _{i=1}^n\log \left[\sum _{k=1}^h{\nu }_k{f}_{bin}\left(Y_i|Z_i,2{\tau }_k/e_i\right)\right]+2(mh+2h-2), \end{gathered}$$ (12)

where mh+2h−2 is the degrees of freedom for cluster means of single cell data (mh), mixture proportions (h − 1), and the mean values for binomial mixtures for bulk tumor data (h − 1).

3. Application

3.1. Simulation Setup

We set up our simulations to mimic the features observed in the real data, such as read depth, MAF etc. We assumed a phylogenetic tree S₃ ← (S₁ → S₂) → S₄ (Figure 1(B)), and set the cellular frequencies of subclones S₁, S₂, S₃, and S₄ to be 0.28, 0.06, 0.20, and 0.18 (Figure 1(A)). Thus, the tumor purity was 0.72. Without SCNA and with mutations occurred on one of the two alleles (by the infinite site assumption), the MAFs of these four subclones were 0.36, 0.22, 0.10, and 0.09, respectively (Figure 1(A)). Therefore, it is very challenging to separate the mutations of subclones S₃ and S₄ based on their MAFs (Figure 1(C)).

We simulated 1,000 point mutations in total with 600, 200, 100, and 100 mutations first occuring in subclones S₁, S₂, S₃, and S₄, respectively. For each locus, we simulated the read-depth (Z_i in equation (2)) by a negative binomial distribution, guided by the observations from real data (Supplementary Figure 3). For each subclone, we chose 30% of the loci with deletion (CN=1) and 15% of the loci with amplification (CN=3). The read-depth for the mutant allele was simulated from a binomial distribution with mean values 2τ_k/e_i, where τ_k was MAF without copy number changes and e_i was the simulated copy number (equation (2)). After incorporating copy number changes, the distribution of MAFs showed an extra mode due to deletions (Figure 1(D)).

Next we simulated data for 20 single cells, with 2, 2, 8, and 8 cells for subclones S₁ to S₄, respectively. The mutation profile of one locus is a binary vector of length 20. For example, if a mutation is observed in all the 20 cells, its mutation profile is a vector of 20 one’s. Since the cells within one subclone may have slight different mutation profiles, we simulated the true mutation E of each cell by a binomial distribution with success probability 0.05 or 0.95, for the cases with subclonal-level mutation being 0 or 1, respectively. Next we simulated observed mutation data D by a binomial distribution with success probability α_k(1 − E_ij) + β_kE_ij for k = 1,2, where β_k and 1 − α_k is sensitivity and specificity of mutation calls, respectively. To decide a reasonable sensitivity of mutation calls, we note that previous study has given an estimation of median ADO rate of 24% in scDNA-seq (Gawad et al., 2014). If half of ADO events lead to drop of mutant allele, ADO rate of 24% will set the maximum sensitivity to be around 0.88, Other sequencing or calling errors may further reduce sensitivity and specificity. To be consistent with our real data analysis, we consider the somatic mutations that have passed some filters, such as being called in both single cells and bulk tumor, and such mutation calls can have sensitivity around 0.8–0.9 (without considering ADO), and specificity above 0.95 (Xu et al., 2014). Therefore, we set specificity (1 − α₁) to be 0.95 and sensitivity β₁ to be 0.7 for mutations with read-depth ≥ 20, and specificity 0.9 and sensitivity 0.6 for mutations with read-depth < 20. Finally we simulated read depth $R_{ij}^T$ and the number of mutant reads $R_{ij}^A$ by a negative binomial distribution and a mixture of beta-binomial distributions, respectively. The parameters of the mixture distribution were estimated from real data (Supporting Information Section B.3).

3.2. Simulation Results

It is apparent that we cannot separate subclones S₃ and S₄ using the bulk tumor data only (Figure 1(D)) because these two subclones have very similar cellular frequencies. We confirmed this by running PhyloWGS on this dataset (Supplementary Figure S7). So we compare the performance of three alternative approaches that involve single cell sequencing (SCS) data: (1) to only use the SCS data, (2) to combine the bulk tumor and the SCS data but ignoring copy number changes, and (3) to combine the bulk tumor and the SCS data and acknowledge copy number changes. We evaluate the results by two criteria: how well the grouping of these 1,000 somatic mutations matches with the true subclone structure, and how well the mutation profiles across those 20 single cells can be estimated.

Based on the AIC defined by equations (11) and (12), we identified five clusters using SCS only, and four clusters based on the joint analysis bulk tumor and SCS, with or without SCNA information (Table 1). In terms of the proportion of somatic mutations that are correctly allocated, these three approaches give 60% (SCS only), 59% (Bulk tumor + SCS, without SCNA), and 93% (Bulk tumor + SCS, with SCNA). Therefore our proposed method performs much better than the approaches using SCS only or ignoring SCNA.

Table 1.

Clustering of somatic mutations in simulation study.

True subclone	SCS only			Bulk + SCS, w/o SCNA				Bulk + SCS, w/ SCNA
	2	4	5	1	2	3	4	1	2	3	4
1	570	18	12	282	308	4	6	560	40	0	0
2	190	6	4	49	123	2	26	18	178	0	4
3	4	1	95	0	4	93	3	0	0	99	1
4	3	96	1	0	1	4	95	1	0	1	98

Using SCS only, the recovered cluster 1–3 correspond to subclone 1, cluster 4 corresponds to subclone 4, and cluster 5 corresponds to subclone 3 (Table 1, Figure 3(B)). When combining bulk tumor and SCS, but ignoring SCNA, subclone 1 and 2 cannot be well separated (Figure 3(C)). Finally, using both the bulk tumor and the SCS data, as well as SCNA information, we can accurately cluster the somatic mutations (Table 1) and recover the true mutation profiles of the four groups (Figure 3(D)).

Figure 3.

True and estimated mutation profiles of mutation clusters across 20 single cells in simulation studies. Mutation profiles are summarized by a matrix, where each row corresponds to a mutation cluster (i.e., γ_k = (γ_1k,...,γ_mk)^T ) and each column corresponds to a cell. This figure illustrate the true mutation profiles (panel (A)) as well as three estimates: using single cell sequencing (SCS) data only (B), and jointly analyzing bulk tumor data and SCS data with or without copy number alteration information (C-D). As shown in the legend in upper-right corner, the color intensity indicate the probability the mutations within this cluster is mutated in a single cell. This figure appears in color in the electronic version of this article, and any mention of color refers to that version.

We also used this simulated dataset to compare the performance of BaSiC and B-SCITE. B-SCITE is designed for detailed analysis of a smaller number of mutations. Running B-SCITE on 1,000 mutations took more than 3 days and the results were unstable across runs. Therefore we evaluated BaSiC and B-SCITE on 100 randomly sampled mutations. When the number of mutations is smaller, clustering is more challenging. BaSiC suggest 5 clusters instead of 4 by splitting one true cluster into two. Overall the results of BaSiC is still good. In contrast, B-SCITE reported a tree with many branches. Even if we ignored the branches with one or two mutations, there are still 9 branches and they do not match well with know cluster membership, the GitHub URL for the analysis pipeline and results can be found in the Supporting Information section.

We have also conducted additional simulations to evaluate the sensitivity of our method to two factors. One is the read-depth cutoff to call somatic mutations. We have used read-depth of 20 as cutoff. In the sensitivity analysis, we showed the results are robust to cutoffs 16, 18, 22, and 24. See the URLs in Supporting Information. The other factors is the errors of copy number calls. We randomly selected 10% or 20% of copy number calls and randomly assign the copy numbers from 1 to 3. The results are similar to the situation when there is no errors in copy number calls. See the URLs in Supporting Information.

3.3. Analyzing Exome-seq Data from a Breast Cancer Patient

We first demonstrated our method using the sequencing dataset mentioned earlier (Wang et al., 2014). The experimental protocol to generate exome-seq data of the 16 tumor cells provides higher accuracy for calling somatic point mutations, but is not suitable for studying copy number changes. The authors have also used another protocol for whole genome sequencing of the 50 tumor cells to detect large-scale SCNAs. We included these 50 tumor cells in our analysis in order to infer subclonal copy number changes. Combining the information from the bulk tumor sample and these 50 tumor cells, we could confidently separate clonal and subclonal copy number alteration, but these data were not sufficient to accurately estimate a phylogenetic tree of subclonal SCNAs and somatic point mutations. Therefore, in our analysis, we only used the genomic regions without subclonal copy number changes, which cover 789 somatic point mutations. The details of the SCNA data processing steps were given in Supporting Information Section C.

The somatic mutation status of these 789 point mutations in the 16 cells can be saved in a matrix of size 789 × 16. Recall that these mutations were called in at least two of the 16 cells. If a mutation is not called in a specific cell, it could imply wild-type genotype or that there is no enough data to decide mutation status, e.g., the read-depth is lower than 20. In this mutation status matrix, there are altogether 789 × 16 = 12,624 entries, among them 4,562 (36.1%) have low read-depth, and 6,723/1,339 (53.3%/10.6%) have positive/negative mutation calls. Therefore, it is a challenging task that requires imputation of more than 1/3 of the mutation calls. We applied three approaches to analyze this dataset, including the single cell sequencing (SCS) only, and a joint analysis of bulk tumor and SCS (BaSiC) with or without SCNA information. Based on the AIC, SCS only and BaSiC with SCNA chose 2 clusters, and BaSiC without SCNA chose 3 clusters (Figure 4). The results of all three methods are similar in terms of allocation of somatic mutations into these 3 clusters (Table 2). The analysis of BaSiC without SCNA generates an extra cluster, though the size of that cluster is small.

Figure 4.

Estimated mutation profiles of two clusters across 16 single cells. This figure illustrates the mutation profiles of each cluster estimated using single cell sequencing (SCS) data only (A), and jointly analyzing bulk tumor data and SCS data with or without copy number alteration information (B-C), or naive clustering of the observed mutation calls from single cells without imputation (D). Panel (E) illustrates the MAFs of the two clusters reported in (B). As expected, the cluster that are mutated in 4 cells has much smaller MAFs than the cluster that are mutated in 16 cells. This figure appears in color in the electronic version of this article, and any mention of color refers to that version.

Table 2.

Comparison of the clustering of somatic mutations in real data analysis using the proposed method: Bulk + SCS, with SCNA versus two alternative approaches.

		SCS only		Bulk + SCS, w/o SCNA
		1	2	1	2	3
Bulk + SCS, with SCNA	1	767	4	766	5	0
	2	3	15	0	3	15

The clonal cluster identified by our analysis is similar to the one reported by Wang et al. (2014), but the subclonal ones are different. One reason is that we started from raw data and re-did the mutation calling and filtering, and thus had a slightly different set of somatic mutations. In addition, we further filtered out the somatic mutations only called in single cells or those located in copy number alteration regions. We reported the list of genes that have at least one non-silent mutation and belong to the cancer gene census (Sondka et al., 2018) in Supplementary Table 1. Cluster 1 includes many cancer driver genes such as TP53 or PTEN. Cluster 2, which is missed by Wang et al. (2014), has only one such gene FUBP1, which regulates the expression of c-myc, a well known oncogene.

We want to emphasize that a main contribution of our method is to provide a rigorous approach to impute the missing somatic mutation calls from single cells by borrowing information from bulk tumor data. Our method improves the accuracy to cluster somatic mutations. As a comparison to our method, we also performed a naive clustering of mutation calls from single cells without any imputation. As expected, the resulting clustering does not capture the pattern that the majority of the mutations/cells are clonal and a small subset of them are subclonal (Figure 4(D)).

3.4. Analyzing Target Sequencing Data from 5 Childhood Leukemia Patients

To evaluate BaSiC for the situations with larger number of single cells, we have applied BaSiC to analyze target sequencing data from 5 childhood leukemia patients (Gawad et al., 2014). BaSiC is designed for the DNA-seq data from whole exome or whole genome sequencing studies, where the number of mutations is large and read-depth can be low in some loci. In contrast, the data from Gawad et al. (2014) were generated by target sequencing for a small number of mutations (10 to 105 mutations per patient) with high depth, and thus there is no missing value due to low read-depth. Nevertheless, it is helpful to evaluate our method in this dataset. The original study includes 6 patients, we excluded patient 6 from our study because this patient has only 10 mutations and it is apparent that all the 10 mutations are clonal. The clusters identified by BaSiC fit observed data very well. Because the scDNA-seq data of this dataset have high quality, incorporating bulk tumor data does not lead to substantial changes (Supplementary Section E.2).

4. Discussion

We have developed a new computational method to cluster somatic mutations using mutant allele frequencies (MAFs) from bulk tumor and somatic mutation calls from single cells. The major challenge for analyzing single cell DNA sequencing data is the high error rate and low coverage in many genomic regions. We overcame this challenge by borrowing information from read counts as well as bulk tumor data. We expect the technique of single cell DNA sequencing will become more mature and more popular. However, it may still be infeasible to sequence most tumor cells within a tumor sample. Therefore, the joint analysis of bulk tumor samples and single cell data remains an attractive option. In our real data analyses, it appears that the single cells capture all the subclones. In practice, it is possible that the single cells cover some, but not all the subclones. In such cases, it is more important to incorporate bulk tumor data in the analysis.

Although our method does not directly estimate the number of subclones or mutation profile per subclone, our mutation clustering results can be considered as a cleaned and condensed version of input data for subclone inference. It is a cleaned version since missing mutation calls due to low read-depth have been imputed. The part of condensing refers to the fact that we obtain one mutation profile per cluster. We will briefly discus the inference of subclone information though its implementation is beyond the scope of our method. First, assume the tumor cells from all the subclones have been sampled by the single cell DNA-seq data or the cellular frequencies of any two subclones are sufficiently different to be detected by our model, then the number of subclones is the number of clusters.

Next we discuss the inference of subclone cellular frequencies, denoted by η = (η₁,...,η_h)^T and the mutation profiles of subclones. Their estimates depend on the underlying phylogenetic tree. For example, assume that there are three subclones and two possible phylogenetic trees: a branching tree of s₂ ← s₁ → s₃ and a linear tree of s₁ → s₂ → s₃. Let Q be a binary matrix that describe the assignment of each mutation cluster to a subclone. Each row of Q corresponds to a mutation cluster and each column corresponds to a subclone. The Q matrices corresponding to the branching and linear tree are

$$Q_1=Q\left(s_2\leftarrow s_1\to s_3\right)=\left(\begin{array}{ccc}1& 1& 1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\text{and}{Q}_1=Q\left(s_2\leftarrow s_1\to s_3\right)=\left(\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right),$$

respectively. Given Q, τ = Qη, and thus η = Q⁻¹τ. Then it is clear that the inference of subclonal frequencies depend on the phylogenetic trees. Given the mutation profiles γ_k from single cells, we may be able to distinguish the underlying phylogenetic tree. In the above example, if the cells harboring the mutations in cluster 2 and 3 do not have overlap, we can conclude the phylogenetic tree is the branching tree of s₂ ← s₁ → s₃. In general, it is not straightforward to evaluate whether the mutation profiles from single cells are consistent with a phylogenetic tree or not. One potential approach is to follow our recent work of SMASH (Little et al., 2019) to construct likelihood function based on a phylogenetic tree, and then either select the appropriate tree using BIC or consider multiple configurations in the next step of analysis. However, specification of the likelihood model is not trivial since the input data in this likelihood function are sample averages instead of empirically observed data. Therefore, we leave this as a potential future work.

In principal, we can estimate the variance of the parameters of our model by evaluating the Fisher’s information matrix of the observed data likelihood or using Louis formula (Louis, 1982). However, since the missing information of mutation status is high-dimensional, it is computationally demanding to evaluate the integration over all the missing variables. Furthermore, our focus is on clustering rather than parameter estimation. Therefore we do not provide parameter variance estimates by default. If needed, these variances can be evaluated using bootstrap method.

One limitation of our implementation is to assume allele-specific copy numbers are pre-estimated and they are clonal events. Modeling both subclonal copy number and somatic mutations is a very challenging task that warrants further studies. Another future development is to consider multiple bulk tumor samples from the same patient. This can be achieved by expanding the likelihood function of bulk tumor samples, though cautions should be taken to model the dependency and heterogeneity across bulk tumor samples.