Many-to-One Formula

Single-lineage models can misrepresent the whole population when rates depend on telomere lengths. The many-to-one formula represents the full-population first moment as a weighted expectation along a single trajectory. The paper derives a many-to-one formula connecting the first moment of the full branching population to a weighted single-particle process. The weighting is necessary because some lineages contribute disproportionately to the population. The single pure-jump process follows one typical lineage with modified jump, switching, and killing rates.