DISPOSICIONES GENERALES Consideración de interés

In addition to clustering mutations, many of the existing methods infer a clonal tree of tumor evolution. In some of the methods, clustering is the first step and is followed by inferring clonal trees (e.g., [97]), whereas in the others the inference of a clonal tree and clustering of mutations is performed simultaneously (e.g., [33]).

In addition to the weak parsimony assumption discussed above, most of the available methods also use the Infinite Sites Assumption. In order to constrain the search space, additional constraints derived from CP values are sometimes imposed. Below we discuss some of the most important of these assumptions/constraints:

1. Infinite sites assumption (ISA): The use of ISA is justified on the following grounds:

Due to (i) length of human genome (∼3 billion basepairs per haploid genome) and (ii) number of SNVs usually detected in tumors driven by SNVs (hundreds to thou- sands) and (iii) assuming that mutations occur uniformly across the genome, then it is very unlikely that a given genomic position will be hit by (any) mutation. The probability of recurrent mutation at the given genomic position is much lower and it is extremely unlikely that two independent SNVs occur at the same genomic position in two different cells or that mutation gained in some cell gets lost in some of its descendants.

ISA dates back to 1969 [77] when it was proposed for Mendelian populations. However, in cancer genomes, loss of heterozygosity (LOH) and deletions of genomic segments are commonly observed events in many cancer types [35, 112, 176, 113, 102]. Loss of a genomic segment harboring a mutation (i.e., variant allele) due to LOH or deletion results in the violation of ISA. In other words, the use of the ISA is very sensitive to non-detected deletions and losses of heterozygosity. Recent analysis of several single- cell datasets reported evidence for ISA violations in almost all of the analyzed cases [82]. Most of the reported violations are losses of early clonal SNVs due to subclonal deletion or LOH events occurring at later stages of tumor progression. While accurate copy number profiling can help in mitigating the effects of ISA violations, recurrent mutations remain more challenging. However, they are expected to be very rare based on the arguments provided above, although more research supporting this assumption will be required in the future.

Provided hundreds or thousands of SNVs used in the methods applicable to WES or WGS data, the effect of a small fraction of mutations for which ISA is violated can be negligible [97]. However, detection of all subclonal deletions is still very challenging and, in the studies involving small numbers of mutated loci and targeted sequencing, the effect of ISA violations is expected to be more pronounced. A recent method [8], where losses of mutations are allowed, suggests that in these cases clonal reconstruction methods can benefit from relaxation of ISA. The same conclusion can be derived from

our results for a colorectal cancer patient presented in Chapter 5 where we allow a subset of mutations to violate ISA.

Given that ISA holds, for two mutations, a and b, we define that a is an ancestor of b if and only if the subclone of the first occurrence of a is either the same as or is an ancestor of the subclone of the first occurrence of b. In this case, we also say that a and b are in

ancestor-descendantrelation or, equivalently, belong to the same lineage. Otherwise, a and

b are at different branches of tree of tumor evolution or, equivalently, belong to different

lineages.

If ISA holds true, then the lineage precedence rule [120] and the sum rule [70] should hold as well. Although the sum rule implies the lineage precedence rule, since independent use of the lineage precedence rule can be found in the literature [120], we describe both rules below. We also describe the strong parsimony assumption, the assumption of the unique tree of tumor evolution shared by multiple bulk samples and the crossing rule (which can be used only in the cases where multiple bulk samples are available).

2. Lineage precedence rule: If mutation a is an ancestor of mutation b then all cells

harboring b also harbor a. Consequently, CP(a) ≥ CP(b) must hold. If multiple bulk samples are available, lineage precedence rule must be fulfilled in each of the samples.

3. Sum rule: Given a mutation a occurring at the non-leaf subclone S. Let {S1, S2, . . . , Sn}

denote the set of all children of subclone S. Consider a set of mutations {a1, a2, . . . , an}

such that ai occurs for the first time at Si. Since mutation a is present in all

S, S1, S2, . . . , Sn and each pair of distinct mutations from {a1, a2, . . . , an} occurs at

different branches of the clonal tree, the following must hold: CP(a) ≥ CP(a1) +

CP(a2) + · · · + CP(an). If multiple bulk samples are available, lineage precedence rule

must be fulfilled in each of the samples. The sum rule is also known as pigeonhole

principle[119] and linear divergence rule [120].

4. Strong parsimony assumption: The strong parsimony assumption favors reconstruc-

tions with maximum number of non-populated subclones (i.e., subclones of zero preva- lence). Some further evidence for the validity of this assumption is required. Strong parsimony assumption is currently used in TrAp [159] and rec-BTP [60], two of the first methods for reconstructing clonal trees from single sample bulk sequencing data.

5. Weak parsimony assumption: This assumption was already presented above. The

weak parsimony assumption favors reconstructions with smaller numbers of subclones and is used by most of the available methods.

6. Unique tree of tumor evolution shared by all bulk samples: According to this

samples. The only values that differ between individual samples are subclonal fre- quencies. The assumption about the existence of a unique tree of tumor evolution is directly supported by the clonal theory of cancer evolution. In the case where cancer originates from a single cell, all cells in all tumor samples are descendants of a single cancer founding cell and are related through a shared tree. However, even in cases of multicentric tumors the same reasoning still applies.

7. Crossing rule: This rule is applicable only in cases where multiple samples are avail-

able. It states that, if a and b are two mutations such that there exist samples p and

q and CPp(a) > CPp(b) and CPq(a) < CPq(b) then a and b are not in an ancestor-

descendant relation (i.e., they belong to different branches of tree of tumor evolution).

Here, CPr(c) denotes CP value of mutation c in sample r. The crossing rule follows

directly from the lineage precedence rule and the assumption about a common tree of tumor evolution shared by all bulk samples.