  Linked beneficial and deleterious mutations are known to decrease the
fixation probability of a favorable mutation in large asexual populations.
While the hindering effect of strongly deleterious mutations on adaptive
evolution has been well studied, how weak deleterious mutations, either in
isolation or with superior beneficial mutations, influence the fixation of a
beneficial mutation has not been fully explored. Here, using a multitype
branching process, we obtain an accurate analytical expression for the fixation
probability when deleterious effects are weak, and exploit this result along
with the clonal interference theory to investigate the joint effect of linked
beneficial and deleterious mutations on the rate of adaptation. We find that
when the mutation rate is increased beyond the beneficial fitness effect, the
fixation probability of the beneficial mutant decreases from Haldane's
classical result towards zero. This has the consequence that above a critical
mutation rate that may depend on the population size, the adaptation rate
decreases exponentially with the mutation rate and is independent of the
population size. In addition, we find that for a range of mutation rates, both
beneficial and deleterious mutations interfere and impede the adaptation
process in large populations. We also study the evolution of mutation rates in
adapting asexual populations, and conclude that linked beneficial mutations
have a stronger influence on mutator fixation than the deleterious mutations.
