Formation, persistence, and breakdown of carrier-set topology in linkage disequilibrium: empirical structure in 1000 Genomes and a two-locus Wright–Fisher model

Linkage disequilibrium between two biallelic loci is usually summarized by scalar association measures such as r ² and D ^′ . These measures quantify how visible an allelic association is to a symmetric LD scan, but they do not directly represent the topology of carrier sets: whether the carriers of one variant are contained within, partially overlap with, or are disjoint from the carriers of the other. This distinction is structural. On the haplotype-frequency simplex, carrier-set inclusion corresponds to a boundary face where one haplotype class is absent. In the rare–common regime, a nested rare variant is further constrained by the ceiling r ² ≤ p _A / p _B , so that complete carrier-set inclusion can remain nearly invisible to r ² .

Here, as a companion to the Fisher-geometry preprint ¹ , we examine the empirical and dynamic behavior of this carrier-set topology. In 1000 Genomes Phase 3, across 156,604,320 SNP pairs from the MHC and NEGR1 regions, pairs on the ∣ D ^′ ∣= 1 boundary span a wide range of r ² and ∣ C ∣. Within fixed r ² strata, r ² poorly distinguishes nested from non-nested carrier-set configurations, with AUROC values of approximately 0.54–0.62, whereas the boundary-sensitive normalization ∣ D ^′ ∣ separates them much more effectively, with AUROC values of approximately 0.90–0.92. The empirical data also obey the predicted r ² ≤ p _A / p _B ceiling.

We then introduce a temporal axis using a two-locus Wright–Fisher model on the same simplex. Carrier-set topology evolves through three motions relative to the ∣ D ^′ ∣= 1 boundary: formation or persistence, in which recombination suppression establishes and maintains inclusion without requiring selection; visibility change, in which selection or drift moves r ² along the boundary while preserving the inclusion relation; and breaking, in which a recombination pulse introduces the previously absent haplotype and dissolves inclusion. A fourth mode, specificity erosion, expands the partner carrier set while preserving inclusion, thereby lowering P ( A ∣ B ) while keeping P ( B ∣ A ) and ∣ D ^′ ∣ equal to one. This mode shows that asymmetric conditional probabilities are best understood as diagnostic coordinates for carrier-set topology, not as the primary object itself.

Together, these results show that topology and visibility are separable axes of LD structure. Conventional r ² -based scans and carrier-set topology scans therefore answer complementary, not interchangeable, questions.