Figure 2 shows that, under asexuality, even if a ben-
eficial mutation is doomed to extinction by the pro-
cesses just described, it may rise to a high copy number
before it is eliminated from the population. This occurs
because asexuality produces a high level of correlation
in the average of fitness of beneficial-mutation-bearing
individuals from one generation to the next. Thus, if a
beneficial mutation is initially associated with a relatively
fit genotype, it is likely that adults carrying the beneficial
mutation will continue to be relatively fit for a substantial
number of generations, even if their fitness eventually
erodes. It should be recognized, however, that the pat-
tern shown in Figure 2 might not appear in a relatively
small population. As the number of copies of a benefi-
cial mutation increases into the thousands, it can cause
an increase in the average fitness of a population that
contains, say, only ten or twenty thousand members.
This will cause a decrease in the relative fitness of in-
dividuals that carry the beneficial mutation. As a result,
in relatively small populations, beneficial mutations that
will eventually become extinct might not achieve the
sorts of copy numbers that are sometimes observed in
the extremely large (effectively infinite) populations
studies here.
One problem with the approach taken in this study is
that I have assumed an infinite population (N = 03). In
an infinite population a beneficial mutation that initially
arises in a single copy can never (in finite time) attain
a substantial frequency because its initial frequency is
infinitesimal. If the mutation becomes established, then
its numbers will continually grow, but the frequency of
the mutation will always be vanishingly small, except in
the limit as the time since the initial appearance of the
mutation becomes infinite. Another problem is that, un-
der a biologically reasonable mechanism of mutation,
one might expect the constant appearance of all pos-
sible beneficial mutations during each successive gen-
eration in an infinite population. The probability that a
given type of beneficial mutation will arise de nom may
be tiny for a given newborn individual, but so long as this
probability is finite, we can expect an infinite number of
new occurrences in an infinite population. This is con-
trary to my assumption that beneficial mutations are
rare.
Fortunately, these diffkulties with the N = a assump-
tion are more apparent than real. They tell us what we
already know, that infinite populations are biologically
unrealistic, and that they should only be used to approxi-
mate the case of very large populations. In a large (but
not infinite) population, established mutations will certainly achieve a substantial frequency in finite time. Fur-
thermore, we have data that strongly suggests that all
possible beneficial mutations will not necessarily arise at
one time even in very large microbial populations (e.g.,
PAQUIN and ADAMS 1983; LENSKI et al. 1991). The calcu-
lations made in this paper will be approximately correct
so long as the distribution of deleterious mutations is
similar to a Poisson distribution, and this requirement
can be fulfilled by a finite population, so long as it is
sufflciently large (HAIGH 1978). Thus, there is no real
problem with the N = ~0 assumption, as long as one
keeps in mind that the calculations are meant to be a
useful approximation for the case of very large (but fi-
nite) populations.
It may be worth pointing out that, while the popula-
tion is assumed to have an infinite number of members,
the fate of beneficial mutations (ie., establishment or
loss) is a random process. This may seem surprising to
some readers, as infinite populations are usually as-
sociated with deterministic processes. The reason for
the non-deterministic character of evolution in the
case of the models studied here is that the number of
copies of a beneficial mutation is finite at any given
(finite) time after the initial appearance of the
mutation.