The deterministic analysis shows that evolution along the corridor is
possible if the evolvability criterion is fulfilled, 
.
This condition corresponds to the conditions in which each single locus
heterozygote with a plus allele for 
has a selective advantage.
Progress along the corridor proceeds in two phases. First a long phase
with little progress, and then a rapid transition due to the cooperative
effects among complementary pairs of alleles (see Fig. 1). The total rate
of evolution is largely determined by the time it takes to reach the
threshold to the second phase. In the stochastic model we ignore the
time it takes to fix complementary pairs of mutations once they have
reached the threshold of cooperativity.
As long as the gene
frequency is at undercritical levels, the dynamics of each mutant is
assumed to be independent of complementary genes. We use this
assumption to employ an approach pioneered by William Hill to predict the
rate of evolution due to new mutations and sustained directional selection
(Hill, 1982). It consists of predicting the average rate of evolution from
the expected change of the character per gene substitution and the
probabilities that a new mutation occurs and that it becomes fixed. In our case we
also have to assume that there are always complementary mutations
segregating, such that on the average the mean genotypic value of the
second character stays close to the ridge of the corridor. The probability
that a mutation with heterzygote selective advantage hs and initial
frequency 
gets fixed, 
is given by
Let
be the per locus mutation rate, f the average fertility per
parental pair, 
the size of the parental populaton, and 
the average change of 
per gene substitution,
then the expected average rate of evolution can be calculated as
To test the feasibility of this approach we specialize this general formula
to the parameters of our model simulations. For most of our simulations
`downhill mutations' have little chance of getting fixed. Therefore the
average change in 
due to a gene substitution is
Figure 6: Comparison between the
predicted rate of evolution in the corridor with hidden pleiotropic effects
(Eq. (27)) with the results of stochastic simulations. The simulation
results are the same as used for Table 5, except those with 
and
. For these values Eq. (27) predicts negative rates of
evolution. The remaining simulations are in very good agreement with the
prediction. The two values deviating from the prediction at the left hand
side of the graph are parameter combinations where Hill's (1982)
approximation for the probability of gene substitution 
is
inaccurate.
because the mutational
effects on the y values are all 
distributed. The coefficient
is the average of the first column of the B-matrix 
. Because we ignore
downhill mutations, the rate of mutation has to be halved if we want to
apply this formula to our simulations. If on the average for a mutation
the value 
, then the probability of substitution is
(Hill, 1982). If the mean
genotypic value of the second character is about zero, then the average
selective advantage of a heterozygote in our model is 
.
The complete formula for predicting the average rate of evolution in our
simulations then is
where U is the genomic mutation rate, 
.
Fig. 6 shows a comparison of our low mutation rate simulations with the
predicted rate of evolution. Note the excellent agreement between
simulation and prediction. The two data points that clearly deviate from
the predicition correspond to parameter values where the condition 
was not fulfilled. Excluded from this figure are the simulations
where the evolution criterion, 
was not fulfilled. In
this case the approximate stochastic model predicts negative rates of
evolution.
As shown in the last section, the rate of
evolution scales with population size. This property is clearly reflected
in the approximate analytical model. In addition, the rate of evolution
scales with the genomic mutation rate, but not directly with the mutational
variance, as in the case of characters without pleiotropic effects.
Another difference concerns the relationship between the strength of
directional selection and the rate of change. Without pleiotropic effects
the rate of change scales linearly with 
, but because of the
interaction with stabilizing selection, the relationship with pleiotropic
effects is exponential.
With weak directional selection the rate of change is much smaller than
without stabilizing selection on pleiotropic effects. With stronger
directional selection the rate of evolution grows progressively, because
stronger directional selection compensates for the effects of stabilizing
selection on pleiotropic effects. This prediction can be used to
empirically test for the presence of significant amounts of deleterious
pleiotropic effects (see Discussion).