Problem statement

Let n be the number of single-nucleotide variants, or simply mutations, identified from initial bulk sequencing data of a matched normal and tumor biopsy sample. For each mutation i, we observe the variant allele frequency (VAF), i.e. the fraction of aligned reads that harbor the tumor allele at the locus of mutation i. Specialized methods exist that combine copy number information and VAFs to infer a cancer cell fraction f_i for each mutation i, which is the proportion of cells in the tumor biopsy that contain at least one copy of the mutation [3, 28–30]. Here, we refer to cancer cell fractions as frequencies. Typically, phylogenies inferred by current methods from frequencies f = [f_i] adhere to the infinite sites assumption. That is, each mutation i is introduced exactly once at vertex v_i and never subsequently lost.

When we sequence a single cell from the same tumor biopsy, assuming no errors, we identify a clone of the tumor. In other words, we observe a set of mutations that must form a connected path in the unknown true phylogeny T*. By repeatedly sequencing single cells until we observe all clones in the tumor, we will have observed all root-to-vertex paths of T*, thus identifying tree T* itself. We assume that (i) the true unknown phylogeny T* is among the trees in and that (ii) mutations among single cells that we sample from the tumor biopsy follow the same distribution as f. These assumptions are important for the mathematical derivation of PhyDOSE but it is typical for violations to occur in practice. Through simulations, we explore the impact of violating these assumptions and show that our approach is robust to many realistic scenarios.

This leads to the following question and problem statement with respect to these two assumptions. How many single cells do we need to identify T* with confidence level γ?

Problem 1 (SCS Power Calculation (SCS-PC)). Given a set of candidate phylogenies, frequencies f and confidence level γ, find the minimum number k* of single cells needed to determine the true phylogeny T* among with probability at least γ.

Clearly, we do not know which phylogeny in is the true underlying phylogeny T* of the tumor. Thus, we consider a slightly different problem: In the T-SCS-PC problem (defined formally at the end of the section), we are given an arbitrary phylogeny and want to perform a similar power calculation when conditioning on T being the true phylogeny. By solving the T-SCS-PC problem for all trees , we obtain the numbers of single cells needed for each tree. As T* is in , the maximum number among is an upper bound on the number of required SCS experiments to identify T* with probability at least γ. To solve the T-SCS-PC problem, we need to reason for which SCS experiments we can conclude that T is the true phylogeny.

Observe that each tree T in describes a unique set of clones, corresponding to the sets of mutations encountered in all root-to-vertex paths of T (Fig 1). Thus, if we observe all clones of a phylogeny T in our SCS experiments, we may conclude that T is the true phylogeny. What is the probability of doing so? To answer this question, we must compute the prevalence of each clone in the tumor biopsy.

For phylogenies that adhere to the infinite sites assumption, the prevalence u(T, f) = [u_i] of the clones in the tumor biopsy are uniquely determined by the phylogeny T and frequencies f as (1) where δ_T(i) is the set of children of the node where mutation i was introduced [9].

Tumor phylogeny inference methods guarantee that the inferred phylogenies from frequencies f have clonal prevalence u(T, f) = [u_i] that are nonnegative and that , where the remainder is the prevalence of the normal clone. Thus, conditioning on a phylogeny T and frequencies f, sequencing one cell from the tumor will lead us to observe one of the n + 1 clones of T with probabilities (u₀, …, u_n). In other words, the outcome of this SCS experiment with one cell is a draw from the categorical distribution Cat(u₀, …, u_n). The possible outcomes of an SCS experiment composed of k cells thus follow a multinomial distribution Mult(u₀, …, u_n). Thus, the probability of observing all tumor clones of T in such an SCS experiment with k cells corresponds to the tail probability of the multinomial where each of the n tumor clones is observed at least once.

The corresponding power calculation is to determine the smallest number for k where the tail probability is greater or equal to the confidence level γ. Note that this power calculation for observing all clones has been previously introduced [26].

Importantly, in many cases we need not observe all clones of T to distinguish T from the remaining phylogenies (Fig 2). This means that we may conclude that T is the true phylogeny with an SCS experiment with fewer cells. To formalize this notion, we start by defining a featurette.

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Fig 2. The SCS power calculation for phylogeny T (T-SCS-PC) problem.

(A) We are given frequencies f and a tree T₁ that we want to distinguish from the other trees {T₂, T₃}. The pair (T₁, f) uniquely determine clonal prevalence u(T₁, f). (B) Featurettes of T₁ correspond to root-to-vertex paths, yielding distinguishing features Π₁ and Π₂, each with one featurette absent in T₂ and another absent in T₃. (C) With k = 2 cells, we must observe clones from either Π₁ or Π₂ for a successful outcome, resulting in probability Pr(Y₂∣u(T₁, f)) ≈ 0.12. (d) To increase this probability to γ = 0.95, we need k* = 32 cells.

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Definition 1. A featurette τ is a subset of mutations.

We say that a featurette τ is present in a phylogeny T if the nodes/mutations of τ form a connected path of T starting at the root node, otherwise we say that τ is absent in T. The same featurette, however, may be present in more that one phylogeny. Thus, multiple featurettes may be required to distinguish a phylogeny T from the remaining phylogenies .

Definition 2. A set Π of featurettes is a distinguishing feature of T if (i) for all featurettes τ ∈ Π it holds that τ is present in T, and (ii) for each remaining phylogeny there exists a featurette τ′ ∈ Π where τ′ is absent in T′.

Thus, an SCS experiment where we observe one cell from each clone of a distinguishing feature Π of T enables us to conclude that phylogeny T is the true phylogeny. As discussed, every phylogeny T has a trivial distinguishing feature, which is composed of all featurettes present in T. Moreover, T may have multiple distinguishing features. Therefore, we must consider the complete set of all distinguishing features, which we call the distinguishing feature family.

Definition 3. The set composed of all distinguishing features of T with respect to is a distinguishing feature family of T.

Let (c₀, …, c_n) be the outcome of an SCS experiment of k cells, where c_i ≥ 0 is the number of cells observed of clone i and . This experiment is successful if, among the k sequenced cells, we observe the clones of at least one distinguishing feature —i.e. c_i > 0 for all clones i in some distinguishing feature . As discussed, conditioning on frequencies f and T being the true phylogeny, outcomes (c₀, …, c_n) of SCS experiments of k cells follow a multinomial distribution Mult(k, u₀, …, u_n) where u(T, f) = [u_i] is defined as in (1). Let Y_k denote the event of a successful outcome. We are interested in computing the probability Pr(Y_k∣u(T, f)), which equals the sum of the probabilities of all successful outcomes. More specifically, we want to determine the smallest number k* of single cells to sequence such that Pr(Y_k*∣u(T, f)) is at least the prescribed confidence level γ (Fig 2).

Problem 2 (SCS Power Calculation for Phylogeny T (T-SCS-PC)). Given a set of candidate phylogenies and a phylogeny , frequencies f and confidence level γ, find the minimum number k* of single cells needed such that Pr(Y_k*∣u(T, f)) ≥ γ.

In Section A.1 in S1 Text, we prove that the above problem is NP-hard.

Theorem 1. T-SCS-PC is NP-hard.

Multinomial power calculation

To solve the T-SCS-PC problem, it suffices to have an algorithm that computes Pr(Y_k∣u(T, f)), which is the probability of concluding that T is the true phylogeny. Using this algorithm we identify k* by starting from k = 0 and simply incrementing k until the corresponding probability Pr(Y_k∣u(T, f)) exceeds the prescribed confidence level γ. In the following, we describe how to efficiently compute Pr(Y_k∣u(T, f)).

Recall that the outcome of an SCS experiment composed of k cells corresponds to a vector c = [c_i], where c_i ≥ 0 is the number of cells that we observe from clone i and . In a successful outcome c we observe at least one cell for each featurette in at least one distinguishing feature , where is the distinguishing feature family. For brevity, we will write Φ rather than .

Let c(Π, k) denote the set of all outcomes where we observe at least one cell for each featurette in a distinguishing feature Π—i.e. , and for all i ∈ {0, …, n} it holds that c_i > 0 if clone i is a featurette in Π and c_i ≥ 0 otherwise. The set c(Φ, k) of successful outcomes is defined as the union ⋃_Π∈Φ c(Π, k). The probability of any SCS outcome c = (c₀, …, c_n) is distributed according to Mult(k, u(T, f)). Since successful outcomes enable us to conclude that T is the true phylogeny, we have (2)

If there is only one distinguishing feature Π, i.e. Φ = {Π}, then the desired probability is a standard tail probability of the multinomial where we sum up the probabilities of outcomes c(Π, k) = [c_i] such that , c_i > 0 if clone i is a featurette of Π and c_i ≥ 0 otherwise. A fast calculation of this tail probability was developed using a connection to the conditional probability of independent Poisson random variables [26, 31]. If there are multiple distinguishing features but they are pairwise disjoint—i.e. no two distinct distinguishing features share the same featurette—then we simply have (3) and we can apply the fast computation [26] to obtain each independent tail probability. However, the equality in the above equation does not hold if the family Φ is composed of distinguishing features with overlapping featurettes. Incorrectly applying this equation will lead us to overestimate the value of k*. Since single-cell sequencing is expensive, overestimating the number of cells to sequence in an SCS experiment can be costly and unnecessary. One naive way would be to simply brute force all (n + 1)^k SCS outcomes, but this will not scale. Instead, to calculate Pr(Y_k∣u(T, f)) exactly, we propose to use the inclusion-exclusion principle as follows. (4) where I(Φ′) is the set of all featurettes in Φ′, i.e. I(Φ′) = ⋃_Π∈Φ′ Π (Fig 3A).

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Fig 3. PhyDOSE implementation details.

(A) To account for minimal distinguishing features that share featurettes, we use the inclusion-exclusion principle to compute Pr(Y_k∣u(T, f)). Here, Π₁ (red) and Π₂ (blue) share a featurette (with ‘triangle’ and ‘heart’ mutations). (B) To enumerate the set Φ* of minimal distinguishing features of T₁, we reduce the problem to Set Cover and repeatedly identify minimum covers. Here, the universe U is composed of trees {T₂, T₃} and there is a subset in for each featurette τ of T₁ composed of the trees where τ is absent.

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Thus, we need to compute 2^|Φ| − 1 tail probabilities, which each can be done using the fast calculation in SCOPIT [26].

In the worst case, Φ has O(2ⁿ) distinguishing features resulting in O(2ⁿ) tail probabilities. We now describe one final optimization that will significantly reduce the number of required computations. This is based on the following observation.

Observation 1. If Π is a distinguishing feature of T then for all featurettes τ present in T it holds that Π ∪ {τ} is a distinguishing feature of T.

This means that distinguishing features in Φ form a partially ordered set under the set inclusion relation. We call a distinguishing feature Π minimal if there does not exist another distinguishing feature Π′ ∈ Φ that is a proper subset of Π, i.e. Π′ ⊊ Π.

A direct consequence of Observation 1 is that the outcome of an SCS experiment is successful when we observe all featurettes of a distinguishing feature Π, and remains so even if we observe additional featurettes τ′ ∉ Π.

As such, successful outcomes w.r.t. Φ equal those w.r.t. the set Φ* of all minimal distinguishing features of T.

Observation 2. It holds that c(Φ*, k) = c(Φ, k).

Therefore, it suffices to restrict our attention to only Φ* rather than the complete family Φ when computing Pr(Y_k∣u(T, f)) using (4). Section B.1 in S1 Text describes how to find Φ* by reducing the problem to that of finding all minimal set covers, which we solve in an iterative fashion using integer linear programming.

Power calculation for multiple biopsies

We now discuss the T-Mul-SCS-PC, which is the generalization of the T-SCS-PC problem to b ≥ 1 biopsies. The key probability is Pr(Y_k∣U(T, F)), i.e. the probability of concluding that T is the true tree when sequencing k = [k₁, …, k_b]^T cells from each biopsy. In the following, we discuss an exact (but computationally intensive) approach to compute this probability as well as a fast heuristic.

Given the numbers k = [k₁, …, k_b]^T of cells to sequence from each biopsy, an outcome of the corresponding SCS experiment across b biopsies is defined as a matrix C = [c_pi] such that is the number of cells that we observe from clone i in biopsy p, and ∑_{i = 0} c_pi = k_p for all biopsies p. A successful outcome for a distinguishing feature Π is an outcome where we observe at least one cell for each featurette in Π. Let C(Π, k) denote the set of all successful outcomes for distinguishing feature Π. The set C(Φ*, k) of successful outcomes for the minimal distinguishing feature family Φ* is defined as the union ⋃_Π∈Φ* C(Π, k). Since sequencing of each biopsies proceeds independently, the probability of a observing an outcome C = [c₁, …, c_b]^T across b biopsies equals (5)

Hence, the desired tail probability of successful outcomes for the minimal distinguishing feature family Φ* equals (6)

For a single biopsy (b = 1), the probability Pr(Y_k∣U(T, F)) corresponds to sums of multinomial tail probabilities, enabling fast calculation using [26] as discussed in the previous section. This is no longer the case for b > 1 biopsies where the tail probability is over b independent multinomial distributions (see product in Eq (6)). A naive way to compute Pr(Y_k∣U(T, F)) would be to exhaustively enumerate all SCS outcomes with k cells, which scales exponentially in k. As such, we develop a heuristic approach, which selects a subset of required featurettes/clones in each biopsy that together form a distinguishing feature in Φ* and achieve the smallest total number of cells with confidence level γ. Section B.4 in S1 Text provides more details and Figure B in S1 Text provides an example where the heuristic returns a suboptimal solution.

Consideration of SCS error rates

One current challenge with SCS is that the false negative rate per site is quite high with typical rates up to 0.4 for the commonly used multiple displacement amplification (MDA) method [32]. On the other hand, current false positive rates are low and are typically less than 0.0005 for MDA-based whole-genome amplification [32]. A false negative is defined as not observing a mutation that is present in the cell. A false positive occurs when we observe the presence of a mutation that did not occur in that cell.

With PhyDOSE, we propose one possible method for incorporating the false negative rate β when it is known. Specifically, sampled cells follow a categorical distribution u = [u₀, …, u_n] when conditioned on tree T. Hence, the probability of sampling a cell from clone i equals u_i. True positives, i.e. correctly observing a mutation in a clone, follow a Bernoulli distribution with parameter 1 − β. To observe a featurette/clone i that has n_i mutations and a prevalence of u_i, we thus need to have n_i true positives. In other words, assuming independence among mutations, we require n_i successful draws from a Bernoulli distribution parameterized by 1 − β. As such, we derive new clonal prevalence from u(T, f) = [u_i]. Additionally assuming independence between the events of a cell being sampled from clone i and the absence of false negatives, we set where n_i is the number of mutations in featurette/clone i. We set to be equal to . This adjustment results in a reduction of the clonal prevalence and ultimately increases the value of k*. The issue of false positives is less serious as error rates are low enough to be negligible.

Prioritizing candidate trees post SCS experiment

The final step is to prioritize candidate trees after performing an SCS experiment with the number k* of cells computed by PhyDOSE. To this end, we compute the support of each tree . Intuitively, support(T) is the number of cells that support the conclusion that T is the actual phylogeny. Formally, we say that a distinguishing feature Π of a tree T is observed if each featurette of Π is observed in at least one cell. Using this, we define support(T) as the number of cells that correspond to featurettes of an observed distinguishing feature Π of T. Per Observation 1, it suffices to restrict our attention to the set Π* of minimal distinguishing features.

There are two outcomes of an SCS experiment with k* cells. Either there is no tree with non-zero support or there are one or more trees with non-zero support. In the former case, the SCS experiment has failed, which is expected to occur with probability 1 − γ. In the latter case, which may occur in the presence of false negatives and false positives, we return the set of trees with maximum support.

Alternatively, we may use existing methods that infer tumor phylogeny from SCS data [14, 16, 33] or a combination of SCS and bulk data [19, 20].

k* confidence interval

A common challenge in bulk sequencing is uncertainty in the cancer cell fractions f due to sampling of reads as well as sequencing and mapping errors. Following standard practice [9, 10, 34], we account for this uncertainty by taking confidence intervals [f⁻, f⁺] as input. Typically, such confidence intervals are obtained by viewing variant read counts as draws from a binomial or beta-binomial distribution. Importantly, uncertainty in the cancer cell fraction leads to uncertainty in clonal prevalences. Therefore, for each tree , we utilize [f⁻, f⁺] to construct an interval of the number of single cells reflecting the extreme values the clonal prevalences may assume. To find these values, we must consider the frequencies [f⁻, f⁺] using constraints from the tree T following the sum condition (1) and the featurettes in a distinguishing feature of T. We do this using a heuristic that we describe in Section B.2 in S1 Text. We set the overall interval conservatively as the confidence interval from the tree T with the maximum among all trees .

phydoser R package

We developed PhyDOSE and the associated optimizations into a freely available R package named phydoser. The functions in the phydoser R package are grouped into four areas: (i) I/O support (ii) pre and post processing (iii) PhyDOSE implementation iv) visualization. For I/O support, phydoser offers a suite of functions to read and convert external data into the data structures required by the R implementation of PhyDOSE. The pre and post processing capabilities include generation of the distinguishing feature families of each tree, the frequency inputs for the k* confidence interval and the computation of the support metric for a completed SCS experiment. The implementation of PhyDOSE includes both a single biopsy and a multiple biopsy mode. Lastly, the visualization functions facilitate creation of high resolution tree graphics while annotating the distinguishing feature or a specific featurette of a tree. phydoser is available at https://github.com/elkebir-group/phydoser.

Simulations

Design.

We used simulations to assess (i) the benefit of PhyDOSE’s distinguishing feature analysis, (ii) robustness to uncertainty due to sequencing errors and (iii) robustness to violations of PhyDOSE’s model assumptions. We generated simulated data where the ground truth tree T* is known. Given a fixed number c of clones and n mutations, we first generated a ground truth tree T* with c vertices uniformly at random using Prüfer sequences [35] and randomly distributed the n mutations to the c clones while ensuring that every clone had at least one mutation. Next, we generated clonal prevalences u = [u_i] by drawing from a symmetric (n + 1)-dimensional Dirichlet distribution with concentration parameter 0.2. We used rejection sampling to ensure that each clonal prevalence u_i was at least 0.05. Let σ(i) be the set of clones that contain mutation i. We generated frequencies f = [f_i] by setting f_i = ∑_j∈σ(i) u_j for each mutation i ∈ {1, …, n}. We used the SPRUCE algorithm to enumerate the set of trees given frequencies f [9].

To account for common single-cell sequencing errors, we varied false negative rates β ∈ {0, 0.2} and doublet rates δ ∈ {0, 0.1}. We generated, for each simulation instance, 10, 000 single cells sampled under the specified false negative rates β and doublet rates δ according to the bulk clonal prevalence u. To account for uncertainty in bulk sequencing, we additionally obtained confidence intervals on the cancer cell fractions of simulation instances sim3a, using a binomial distribution (with confidence α = 0.05) and a mean coverage of 1000x (drawn from a Poisson distribution). Modeling additional uncertainty in bulk sequencing, sim4a instances consist of n = 100 mutations each and use PyClone [36] to cluster the 100 mutations before enumerating .

Recall that PhyDOSE has three model assumptions: (i) ground truth tree T* is among the candidates trees , (ii) correspondence between clonal prevalences in bulk and subsequent single-cell sequencing samples, and (iii) infinite sites assumption for mutations. To assess (i), for simulation conditions ‘b’ (sim1b, sim2b and sim3b), we randomly sampled 10% of the trees outputted by SPRUCE [9]. To assess (ii), we varied single-cell clonal prevalences from the bulk clonal prevalences by resampling . We tuned the parameter λ so that the clonal prevalence varied by an absolute average of 5% and 20% from the clones of the ground truth tree T*, which resulted in λ = 2000 and λ = 50, respectively (Figure C in S1 Text). To assess (iii), we introduced violations of the infinite sites assumption (ISA) in the form of mutation losses. Specifically, we introduced one mutation loss in each of the instances of sim1a as follows. First, we randomly picked two distinct mutations (i, j) where i is introduced prior to j in the ground truth tree T*. Then, we designated the descendant mutation j as a loss of mutation i in each candidate tree .

In total, we generated 100 simulation instances under eight varying conditions as specified in Table 1.

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Table 1. Simulation conditions.

We generate simulated data under eight conditions with 100 instances each. These conditions have varying subsets of candidate trees, number of clones, number of mutations per clone, clonal prevalence distortions, false negative and doublet rates. To analyze violations of the infinite sites assumption analysis, we introduced mutation losses in the sim1a instances. To analyze uncertainty in cancer cell fractions, we used the sim3a instances with a coverage of 1000x.

https://doi.org/10.1371/journal.pcbi.1008240.t001

The benefit of PhyDOSE’s distinguishing feature analysis.

We compared PhyDOSE against SCOPIT [26], an existing method to design SCS experiments which takes as input a confidence level and the prevalence rate of each clone. Since SCOPIT does not connect the clones to be observed with a phylogenetic tree, we ran SCOPIT under two regimes. In the first regime (called ‘SCOPIT’), we input the clonal prevalence rates for each and take the maximum SCOPIT output of all trees as an upper bound. In the second regime (called ‘SCOPIT (true clones)’), we supplied SCOPIT with the prevalence rates of the simulated ground truth clones. The comparison to PhyDOSE was conducted at confidence level γ = 0.95.

PhyDOSE yielded a significant reduction in the number of cells to sequence compared to SCOPIT (Fig 4A). This is even the case when we provided the clonal prevalences of the ground truth tree to SCOPIT but not to PhyDOSE. In particular, for sim1a, SCOPIT required a median of 18.7 times as many cells than PhyDOSE, whereas SCOPIT (true clones) required 1.5 times as many cells (Fig 4A). In absolute numbers, PhyDOSE computed a median number of k* = 35 cells compared to 544 cells computed by SCOPIT and 59 cells computed by SCOPIT (true clones) (Figure D in S1 Text).

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Fig 4. Simulations demonstrate that PhyDOSE’s calculated number of single cells resolves tree ambiguity in bulk sequencing data.

We used confidence level γ = 0.95 to determine the number k* of single cells to sequence. (A) SCOPIT to PhyDOSE cell ratio on a log scale when considering SCOPIT in a worst case regime where the true phylogeny is unknown and a best case regime where SCOPIT utilizes the clonal prevalance of the clones in the simulated ground truth tree. (B) Recall metrics of the tree inferred by SPhyR [16] by randomly sampling k*/2, k* and 2k* simulated single cells. (C) Number |Π| of featurettes among minimal distinguishing features Φ* when compared between the enumerated candidate set (conditions a) and the downsampled candidate set (conditions b). (D) Number of trees in the candidate set when enumerated by SPRUCE [9] (condition a) and when downsampling the enumerated candidate set (condition b). (E) Number k* of cells identified by PhyDOSE.

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To assess the accuracy of PhyDOSE’s k* value, we generated follow-up in silico SCS experiments. Specifically, we ran our approach for prioritizing candidate trees and SPhyR [16] on sampled single cells. For the former, we performed 100 experiments for each simulation instance, reporting the number of experiments that successfully recovered the ground truth tree T*. We counted an experiment as successful if we correctly and uniquely selected the ground truth tree as T*. For the latter, we also considered the performance of SPhyR when sampling half and double the number k* of cells determined by PhyDOSE.

Figure F in S1 Text shows that the prioritization approach worked particularly well for sim1a with a median success rate of 96%. Similarly, SPhyR [16] was able to identify the true tree in the majority of cases after sampling PhyDOSE computed number k* of cells. To quantify the similarity between the tree T estimated by SPhyR and the true tree T*, we used two commonly-used tree distance metrics, ancestral and incomparable pair recall. Ancestral pair recall is defined as where () is the set of ordered pairs of mutations that occur on distinct edges of the same branch of T (T*). Incomparable pair recall is defined as where () is the set of unordered pairs of mutation that occur on edges in distinct branches in T (T*). For sim1a, the median of both metrics is 1 when sampling k* cells, reflecting that SPhyR identified the true tree in the majority of cases (Fig 4B). Moreover, we found greater gains in performance between sampling k* cells versus k*/2 cells than sampling 2k* cells versus k* cells.

Fig 4C shows the number of clones in each minimal distinguishing feature identified by PhyDOSE, ranging from 1 to 5 with a median of 3 for simulation conditions ‘a’. Importantly, this number is smaller than the total number of 7 clones, demonstrating that a distinguishing feature yields an efficient representation of a tree. This led to a smaller number k* of cells inferred by PhyDOSE compared to SCOPIT without sacrificing performance in the tree reconstruction from the follow-up SCS experiment.

Retrospective analysis of an acute lymphoblastic leukemia patient

We considered a cohort of six childhood acute lymphoblastic leukemia (ALL) patients whose blood was sequenced using bulk and targeted single-cell DNA sequencing [23]. The number of sequenced single cells per patient varied between 96 and 150. To validate our approach, we used PhyDOSE to calculate the number k(T*) of cells needed to identify the true phylogeny T* that is consistent with both data types, thereby retrospectively determining whether fewer single cells suffice to determine T*, decreasing the cost of replicate experiments. In addition, we assessed whether the calculated number k(T*) yielded T* using in silico SCS experiments.

Due to the absence of published copy-number aberration information for this dataset, we focused our attention on patient 2 whose single-cell phylogeny adhered to the infinite sites assumption and the variant allele frequencies suggested the absence of copy-number aberrations (as detailed in Section C in S1 Text). For this patient, 16 autosomal mutations in 115 cells were sequenced [23]. We note that the authors had no knowledge of the number of cells that would suffice to infer the tumor phylogeny of the patient. Using the infinite sites assumption and assuming the absence of copy-number aberrations, we define the cancer cell fraction, or frequency f_i of each mutation i in the bulk data as 2 ⋅ VAF(i). We define the SCS mutation frequency as the fraction of single cells that harbor the mutation. Strikingly, there is a clear correlation between the bulk and SCS mutation frequencies, supporting PhyDOSE’s first assumption (Fig 5A). We excluded mutation CMTM8 because of a notable discrepancy in frequencies (0.4 in bulk vs. 0.2 in SCS). Using SPRUCE [9], we enumerated the set of trees from the bulk data, yielding over 2.5 million trees. This number is mainly driven by 3 mutations (ATRNL1, LINC00052 and TRRAP) with a VAF less than 0.05. Excluding these 3 mutations resulted in a more tractable number of 2, 576 trees. We note that in practice we may similarly exclude mutations because of very low VAFs or less importance in downstream analyses. Fig 5B shows the single tree that was consistent with the cleaned single-cell data, supporting PhyDOSE’s second assumption.

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Fig 5. Retrospective analysis of ALL patient 2 [23] and AML cohort [27] demonstrates that fewer cells suffice for replication.

Panels (A)-(D) consider ALL patient 2 [23] and panel (E) considers the AML cohort [27]. (A) There is a strong correlation between bulk and single-cell mutation frequencies. Colors indicate mutation clusters from SCS data and excluded mutations are indicated by ‘x’. (B) Phylogeny T* that is consistent with the SCS and bulk data. (C) Percent of successful outcomes in 100 in silico SCS experiments, obtained by sampling from the 115 sequenced cells without replacement following PhyDOSE’s calculated number k(T*) of cells (103 for γ = 0.95 and 50 for γ = 0.75). Exclusive outcomes (yellow) uniquely identified T* whereas tied outcomes (purple) yielded a small set of candidate phylogenies that include T*. (D) Number of candidate phylogenies in the case of ties. (E) The distribution of PhyDOSE’s k* for γ ∈ {0.75, 0.95} of all patients in the AML cohort with as well as the number of cells that were originally sequenced.

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We ran PhyDOSE using varying confidence levels γ ∈ {0.75, 0.95} and an estimated false negative rate of β = 0.2 reported by the authors [23]. PhyDOSE calculated that k(T*) = 103 cells suffice to identify T* with confidence level γ = 0.95. Indeed, performing 100 in silico SCS experiments, by sampling k(T*) cells among the 115 sequenced cells without replacement, yielded a success rate of 99% (Fig 5C).

To reduce costs, we explored what would have happened retrospectively with a lower confidence level γ of 0.75. PhyDOSE calculated that k(T*) = 50 cells are needed for γ = 0.75, which is a significant cost savings over γ = 0.95. Performing 100 in silico SCS experiments yielded a success rate of uniquely identifying T* of 66%, which was lower than the expected rate of 75%. Furthermore, we noted that in an additional 26% of experiments the correct phylogeny T* was among the trees with the highest overall support (Fig 5C). The number of trees in the tied set of successes varied from 2 to 6 (Figure L in S1 Text), showing that although PhyDOSE did not uniquely identify the tree, it was able to significantly reduce the original set of 2576 trees (Figure L in S1 Text and Figure M in S1 Text).

In summary, this retrospective analysis shows that the true tree for patient 2 could have been identified confidently with fewer cells than the 115 cells initially sequenced [23]. With a lower confidence level γ, PhyDOSE computes that far fewer cells are required, significantly reducing costs but at the expense of a lower success rate of uniquely identifying the true phylogeny. Nevertheless, the resulting SCS experiment will eliminate a large fraction of the original set of candidate phylogenies due to the incorporation of distinguishing features in the PhyDOSE power calculation.

Retrospective analysis of an acute myeloid leukemia cohort

Morita et al. [27] performed high-throughput targeted microfluidic single-sequencing using the Tapestri platform [38] on a cohort of 77 patients with acute myeloid leukemia (AML). The authors additionally performed bulk sequencing in order to confirm the presence of a mutation in the single-cell data. We note that the authors restricted their analysis to somatic mutations (SNVs and indels) that did not occur in regions affected by additional copy number aberrations.

Here, we utilized the published bulk sequencing VAFs of the SNVs in each patient, eliminating any mutations not detected via bulk sequencing, to enumerate a set of candidate trees using SPRUCE [9]. We restrict our analysis to the 24 patients where bulk sequencing data was available and SPRUCE identified more than one candidate tree. The median number of mutations for these patients was 4 (IQR: 3-5). We retrospectively used PhyDOSE at confidence levels γ ∈ {0.75, 0.95} to estimate the cells needed to perform an equivalent single-cell experiment. We used false negative rate β = 0.049, which is the mean of the per patient published false negative rate. In the original study, a median of 7, 584 cells per patient (IQR: 6, 194–8, 361) were sequenced. Fig 5E shows the distribution of PhyDOSE k* for the 24 patients (median is 2, IQR: 2–6, max is 316) at γ ∈ {0.75, 0.95} versus the total number of cells sequenced in [27]. For γ = 0.95, the median value of k* was 274 cells (IQR: 230–497). This is a significant reduction from the number of cells sequenced per patient in [27] with a median percent reduction at confidence level γ = 0.95 of 95.4% (IQR: 92.2%–98.0%) (Table A in S1 Text).

For these 24 patients, Morita et al. [27] sequenced 153, 558 cells while the PhyDOSE design at confidence level γ = 0.95 requires 8,144 cells (Table A in S1 Text). Using that a Tapestri run of 10,000 cells costs $795 and including additional sequencing costs of $200 per run with NovaSeq or $1000 per run with MiSeq [39, 40], we estimate the costs of the original study as $15,920 and the estimated costs of the PhyDOSE design as $1,995. This assumes the original study utilized 16 Tapestri runs with NovaSeq while PhyDOSE requires 1 Tapestri run with the more expensive MiSeq to avoid multiplexing. Thus, designing the experiment with PhyDOSE would have yielded a 93.75% cost savings. Further, we note that this study design requires targeted sequencing and that the potential cost savings when the experimental design requires whole genome sequencing would further increase.

Prospective analysis of a non-small cell lung cancer cohort

Using PhyDOSE, we prospectively determined the number of cells needed to uniquely identify the true phylogeny for the 25 out of 100 patients in the TRACERx non-small-cell lung cancer cohort that have multiple candidate trees [3]. The authors previously identified the set of candidate trees for each patient using CITUP [11] after clustering mutations with PyClone [36]. The authors also reported the cancer cell fraction of each mutation cluster in each bulk sample. The number of trees in the candidate set for each patient ranged from 2 to 17, with each containing mutation clusters with between 5 and 882 mutations (Table B in S1 Text).

Assuming high confidence on the co-occurrence of mutations in a cluster, mutation clusters alleviate the issue of false negatives, i.e. it suffices to only observe a small number of mutations to impute the presence of the other mutations in the same cluster. Here, with a typical SCS false negative rate of 0.2, the probability of all mutations in the smallest cluster (with size 5) dropping out thus equals 0.2⁵ = 0.00032, a probability that can be neglected. As such, we set β = 0. Unlike in the simulations and the previous real datasets, multiple bulk samples corresponding to distinct spatial locations were available for analysis per patient. In addition to the naive method where we select a single biopsy that minimizes k*, we used the multiple biopsy heuristic to infer numbers k* of cells for each biopsy. For both methods, we used confidence level γ = 0.95.

Following the naive approach, PhyDOSE returned a finite value of k* for 24 out of the 25 patients. The naive approach yielded k* = ∞ for patient CRUK0037 because for each of the 5 biopsies there is a tree where every distinguishing feature is not observable. That is, the clonal prevalence of one of the comprising featurettes is 0. By contrast, the heuristic calculated a total of 243 cells (R1: 36; R2: 49; R3: 60; R4: 38 and R5: 51 cells) for this patient. For patients CRUK0013 and CRUK0076, the naive approach required the sequencing of more cells from a single biopsy than the multiple biopsy heuristic (CRUK0013: 1, 051 vs. 215 cells; CRUK0076 47, 479 vs. 48 cells). For the remaining 22 patients, treating the samples independently yields the same number of cells from the selected biopsy as the heuristic. Table B in S1 Text and Fig 6 provide detailed numbers.

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Fig 6. PhyDOSE multiple biopsy heuristic calculated numbers k* of cells per biopsy for the lung cancer cohort [3] at confidence level γ = 0.95.

https://doi.org/10.1371/journal.pcbi.1008240.g006

These strikingly low values of total number of cells for the 25 patients with multiple candidate trees and multiple biopsies demonstrate the benefit of using PhyDOSE to strategically optimize the design of follow-up single cell experiments.