In a separate set of fate-mapping experiments, dorsal r2 from quail was grafted orthotopically into r2 or heterotopically into r1 of host chick embryos, which were reincubated for 2 to 5 days.
We consider X to be the set of potential fates; fates that are potentially relevant to our study.
Because for each given profile the global interpretation of the rank-order method generates a complete weak ordering over the whole of X, which will be used to determine the tissue fate over the set of actual fates, the corresponding lineage-tissue mapping satisfies the consistency condition but fails to satisfy IIA and regularity.
In order to ensure that our analysis is general (law-like), we will distinguish between X, the set of potential fates, and a set A ∈ K of actual fates; fates that are actually considered in a particular context.
However, for regularity it is the set of potential fates that is reduced in such a way that the original set of actually considered fates is still included in the set of potential fates after the reduction.
Given X, the set of alternative fates, which may potentially be relevant to our analysis, and the set A of actually considered fates for the cell lineages (at a particular point in time), the lineage-tissue mapping F maps elements from K × D into K such that for all A ∈ K and all (R1,..., R n) ∈ D: F A, R1,..., R n)) ⊆ A, where with D we denote the set of logically possible profiles.
The local interpretation of the rank-order method does satisfy regularity and IIA but fails to satisfy the consistency condition because to determine the ranking over a set of actual fates A, only the lineages' propensities restricted to A are taken into account (no information about their propensities over the potential-but-not-actually-considered fates is used).
The Hh, Wnt, TGFβ, FGF, and Notch pathways therefore constitute a core set of cell fate regulators, and small molecules that selectively modulate these signaling processes are valuable research and clinical tools.
To require a cross-level principle to be regular, or a lineage-tissue mapping to satisfy IIA, means that for any fixed context, that is, any given set of actually considered fates, the ranking of tissue fates should be independent of the very existence of alternatives outside the context, that is, independent of whether potential-but-actually-not-considered fates exist.
