Enforcing Temporal Consistency in Migration History Inference

In addition to undergoing evolution, members of biological populations may also migrate between locations. Examples include the spread of tumor cells from the primary tumor to distant metastases or the spread of pathogens from one host to another. One may represent migration histories by assigning a location label to each vertex of a given phylogenetic tree such that an edge connecting vertices with distinct locations represents a migration. Some biological populations undergo comigration, a phenomenon where multiple taxa from distinct lineages simultaneously comigrate from one location to another. In this work, we show that a previous problem statement for inferring migration histories that are parsimonious in terms of migrations and comigrations may lead to temporally inconsistent solutions. To remedy this deficiency, we introduce precise definitions of temporal consistency of comigrations in a phylogenetic tree, leading to three successive problems. First, we formulate the temporally consistent comigration problem to check if a set of comigrations is temporally consistent and provide a linear time algorithm for solving this problem. Second, we formulate the parsimonious consistent comigrations (PCC) problem, which aims to find comigrations given a location labeling of a phylogenetic tree. We show that PCC is NP-hard. Third, we formulate the parsimonious consistent comigration history (PCCH) problem, which infers the migration history given a phylogenetic tree and locations of its extant vertices only. We show that PCCH is NP-hard as well. On the positive side, we propose integer linear programming models to solve the PCC and PCCH problems. We demonstrate our algorithms on simulated and real data.