A graph homomorphism approach for unraveling histories of metastatic cancers and viral outbreaks under evolutionary constraints.

Viral infections and cancers are driven by evolution of populations of highly mutable genomic variants. A key evolutionary process in these populations is their migration or spread via transmission or metastasis. Understanding this process is crucial for research, clinical practice, and public health, yet tracing spread pathways is challenging. Phylogenetics offers the main methodological framework for this problem, with challenges including determining the conditions when a phylogenetic tree reflects the underlying migration tree structure, and balancing computational efficiency, flexibility, and biological realism. We tackle these challenges using the powerful machinery of graph homomorphisms, a mathematical concept describing how one graph can be mapped onto another while preserving its structure. We focus on metastatic migrations and viral host-to-host transmissions in outbreak settings. We investigate how structural constraints on migration patterns influence the relationship between phylogenetic and migration trees and propose algorithms to evaluate trees consistency under varying conditions. Leveraging our findings, we introduce a framework for inferring transmission/migration trees by sampling potential solutions from a prior random tree distribution and identifying a subsample consistent with a given phylogeny. By varying the prior distribution, this approach generalizes several existing models, offering a versatile strategy applicable in diverse settings.