Abstract
The Yampolsky-Stoltzfus (YS) model provided the first formal demonstration that a bias in the introduction of new mutations can shape the direction of adaptive evolution, even when mutation rates are small relative to selection coefficients. Despite its foundational role, the model has been characterized almost entirely by simulation and closed-form solutions for its central quantity, the probability PA that a mutationally favored but less-fit genotype fixes ahead of a rarer, fitter alternative, have been available only in the limiting regimes of vanishing mutation supply and infinite population size. Here we derive closed-form approximations for PA across the intermediate regime of finite population size and appreciable mutation supply that lies between these two limits. Under strong selection between the competing mutants, we combine an establishment-probability argument with clonal interference theory to obtain an explicit expression for PA that accounts for both the ongoing input of the mutationally favored mutation and the frequency-dependent establishment probability of its competitor. Under weak selection between competing mutants, we derive a complementary expression based on the fixation of a mixed population under Kimura's classical formula. Both expressions agree closely with direct simulation of the underlying Wright-Fisher process, recover the known limiting cases, and remain consistent with one another over a substantial intermediate range of parameters. These results are expected to provide a basis for assessing how the strength of mutation bias affects adaptation in empirical data.