  The ratio of non-synonymous to synonymous substitutions
$\omega(=d_{N}/d_{S})$ has been widely used as a measure of adaptive evolution
in protein coding genes. Omega can be defined in terms of population genetics
parameters as the fixation ratio of selected vs. neutral mutants. Here it is
argued that approaches based on the infinite sites model are not appropriate to
define $\omega$ for single codon locations. Simple models of amino acid
substitution with reversible mutation and selection are analysed, and used to
define $\omega$ under several evolutionary scenarios. In most practical cases
$\omega<1$ when selection is constant throughout time. However, it is shown
that when the pattern of selection on amino acids changes, for example after an
environment shift, a temporary burst of adaptive evolution ($\omega\gg1$) can
be observed. The fixation probability of a novel mutant under frequency
dependent selection is calculated, and it is used to show why $\omega>1$ can be
sometimes expected for single locations at equilibrium. An example with
influenza data is discussed.
