Genomic mutation rates that neutralize adaptive evolution and natural\n selection

When mutation rates are low, natural selection remains effective, and\nincreasing the mutation rate can give rise to an increase in adaptation rate.\nWhen mutation rates are high to begin with, however, increasing the mutation\nrate may have a detrimental effect because of the overwhelming presence of\ndeleterious mutations. Indeed, if mutation rates are high enough: 1) adaptation\nrate can become negative despite the continued availability of adaptive and/or\ncompensatory mutations, or 2) natural selection may be disabled because\nadaptive and/or compensatory mutations -- whether established or newly-arising\n-- are eroded by excessive mutation and decline in frequency. We apply these\ntwo criteria to a standard model of asexual adaptive evolution and derive\nmathematical expressions -- some new, some old in new guise -- delineating the\nmutation rates under which either adaptive evolution or natural selection is\nneutralized. The expressions are simple and require no \\emph{a priori}\nknowledge of organism- and/or environment-specific parameters. Our discussion\nconnects these results to each other and to previous theory, showing\nconvergence or equivalence of the different results in most cases.\n