Crossing Rule
Definition
The crossing rule is a phylogenetic constraint that applies in multi-region tumor sequencing data: if clone A has higher cancer cell fraction (CCF) than clone B in one sampled region but lower CCF than clone B in another region, then A and B cannot be ancestor and descendant — they must be sibling clones in a branching phylogeny (Tarabichi et al., 2021).
Rationale
The rule follows from the definition of ancestry. If clone A is the ancestor of clone B, then every cell in clone B is descended from a cell in clone A. Consequently, clone B can never be present in a region where clone A is absent — the ancestor must be at least as abundant as the descendant in every sample. Formally, for any two clones A and B where A → B (A is ancestral to B):
This is the nesting constraint. When it is violated in multi-region data — CCF_A > CCF_B in region 1 but CCF_A < CCF_B in region 2 — the ancestor-descendant hypothesis is falsified. The two clones must be on separate branches of the phylogenetic-tree (sibling subclones descended from a common ancestor whose CCF is at least CCF_A + CCF_B in both regions).
Example
Consider a tumor sampled at two regions:
| Clone | CCF in Region 1 | CCF in Region 2 |
|---|---|---|
| Clone X | 0.80 | 0.30 |
| Clone Y | 0.40 | 0.70 |
In Region 1, X appears ancestral (CCF_X > CCF_Y). In Region 2, Y appears ancestral (CCF_Y > CCF_X). Both cannot be true. The crossing pattern — X dominates Region 1, Y dominates Region 2 — reveals that X and Y are sibling subclones with spatially variable abundance, not an ancestor-descendant pair. Their common ancestor clone (not directly observed) must have had CCF ≥ 0.80 in Region 1 and CCF ≥ 0.70 in Region 2.
Significance
The crossing rule is the strongest single constraint on phylogenetic topology available from bulk sequencing data. It transforms multi-region sampling from a convenience (more data) into a necessity (topological identifiability):
- Without multi-region data, the ancestor-descendant vs. sibling distinction is underdetermined. A single sample showing CCF_X > CCF_Y is consistent with X being ancestral to Y, X and Y being siblings with unequal abundance, or even Y being ancestral to X (if X carries a CNA that inflates its CCF).
- With multi-region data, crossing patterns directly identify branching topologies. This is the basis for the TRACERx studies’ ability to distinguish branching-evolution from clonal-sweep patterns and to demonstrate that branching architecture is the dominant pattern in solid tumors (Turajlic et al., 2019; intratumor-heterogeneity §4).
The rule is also why single-biopsy studies systematically underestimate branching architecture: without a second region to reveal crossing, clone pairs with crossing CCFs in space are misclassified as linear (ancestor-descendant) based on the single observed CCF ordering.
Relationship to the Pigeonhole Principle
The crossing rule is complemented by the pigeonhole principle for CCFs: the sum of CCFs of mutually exclusive (sibling) subclones cannot exceed 1.0. If CCF_A + CCF_B > 1.0 in any sample, A and B cannot be mutually exclusive sibling subclones — some cells must belong to both, meaning one must be ancestral to the other (or they share a nested relationship). The pigeonhole principle constrains sibling assignments; the crossing rule constrains ancestor-descendant assignments. Together, they are the two formal constraints that make phylogenetic inference from CCF data possible (Tarabichi et al., 2021).
Limitations
Requires multi-region data. The crossing rule is inapplicable to single-biopsy studies, which constitute the majority of clinical genomic cohorts. In single-sample data, phylogenetic topology is underdetermined — multiple topologies are consistent with the observed CCF distribution.
CCF measurement error can produce false crossings. If CCF estimates have wide confidence intervals, a statistically insignificant difference in CCF between clones can appear as a crossing when the true CCFs satisfy the nesting constraint. Robust application of the crossing rule requires confidence intervals on CCF estimates, not point estimates alone.
The common ancestor may be extinct or undetectable. The crossing rule requires that the common ancestor of the sibling clones be present at sufficient CCF to contain both. If the ancestor clone is extinct (eliminated by a sweep) or below the detection floor, crossing patterns may appear to violate both the crossing rule and the pigeonhole principle — a scenario that is biologically plausible (intermediate clones, intermediate-clones) but methodologically challenging.