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In an asexual population, the fate of a beneficial mutation depends on how its lineage competes against other mutant lineages in the population. With high beneficial mutation rates or large population sizes, competition between contending mutations is strong, and successful lineages can accumulate multiple mutations before any single one achieves fixation. Most current theory about asexual population dynamics either neglects this multiple-mutations regime or introduces simplifying assumptions that may not apply. Here, we develop a theoretical framework that describes the dynamics of adaptation and substitution over all mutation-rate regimes by conceptualizing the population as a collection of continuously adapting lineages. This model of “lineage interference” shows that each new mutant’s advantage over the rest of the population must be above a critical threshold in order to likely achieve fixation, and we derive a simple expression for that threshold. We apply this framework to examine the role of beneficial mutations with different effect sizes across the transition to the multiple-mutations regime.

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