  The extent of parallel evolution at the genotypic level is quantitatively
linked to the distribution of beneficial fitness effects (DBFE) of mutations.
The standard view, based on light-tailed distributions (i.e. distributions with
finite moments), is that the probability of parallel evolution in duplicate
populations is inversely proportional to the number of available mutations, and
moreover that the DBFE is sufficient to determine the probability when the
number of available mutations is large. Here we show that when the DBFE is
heavy-tailed, as found in several recent experiments, these expectations are
defied. The probability of parallel evolution decays anomalously slowly in the
number of mutations or even becomes independent of it, implying higher
repeatability of evolution. At the same time, the probability of parallel
evolution is non-self-averaging, that is, it does not converge to its mean
value even when a large number of mutations are involved. This behavior arises
because the evolutionary process is dominated by only a few mutations of high
weight. Consequently, the probability varies widely across systems with the
same DBFE. Contrary to the standard view, the DBFE is no longer sufficient to
determine the extent of parallel evolution, making it much less predictable. We
illustrate these ideas theoretically and through analysis of empirical data on
antibiotic resistance evolution.
