caused by the covariance between 
and 
. In this section we
study the question: to what extent do the results from the 2-locus/2-allele
system carry over to systems with more than two loci? In particular we
consider models with four and six loci, in which half of the loci have
positive and half have negative pleiotropic effects, just as in
the two locus case considered above. The recombination terms of the
dynamical equations were derived by a Mathematica program
described in a separate paper (Baatz, submitted). It
is shown that the constraints on evolution, as prediced by the two
locus models, also apply to the multilocus situation. In addition
there are some cooperative effects which diminish the constraint due to
mutual compensation of opposing pleiotropic effects on character 2.
We partition the set of loci into non-overlapping sets of two
loci adjacent on the chromosome and call each set a `pair'. A 4-locus
system thus consists of 2 pairs and a 6-locus system is
described as a model of 3 pairs of loci. The loci with positive
effects on the second character and those with negative effects
alternate along the chromosome. Due to the shape of the fitness
function, only a simultaneous substitution at two loci with opposing
pleiotropic effects leads to progress along the ridge.
Table 3: Allelic effects of a certain pair k of loci
Figure 4:
Evolution of gamete frequencies of a simulation starting with symmetrical
initial conditions near fixed point 
. The frequencies
of gametes with cooperative effects (gametes 0110 and 1001) are shown in grey,
and that of gametes with anti-cooperative
effects (gametes 0101 and 1010) in black. Note that the frequency of the
cooperative effects is
higher than the frequency of non-cooperative effects. Parameters are:
, 
and
.
With the allelic effects on the different pairs of loci, the
covariance 
can be written in a good approximation
as
where 
gives the sum of
frequencies of all of the repulsion gametes of pair i with positive
or negative effect 
on character 2. The fact that only
repulsion gametes enter this equation is due to the symmetry of gene
effects. The negative terms in the first line of (15) account for the
gametes with opposite effects, which therefore reduce the variance of 
.
We call these gametes 'cooperative.' These negative terms demonstrate
that the total variance caused by segregation at 
loci can be less
than 
times the variance produced by a corresponding two locus system. This is an instance of the socalled "Bulmer-effect" (see chapter 9 in Bulmer, 1980).
Multilocus systems have the potential to 'hide' some of the pleiotropic
variance via the interaction between cooperative gametes. Below it is
demonstrated that exactly this is the case. In Fig. 4 the frequencies of
the cooperative gametes 0110, 1001 and of the anti-cooperative gametes
0101,1010 are plotted over time for a model with four loci.
Figure 5: Dynamics
of the relative covariances (see text),
starting with symmetrical initial conditions near fixed point 
,
or 
, respectively. Note that the maximum
covariance in the four locus system is less than twice the covariance of
the two locus system, and the maximum covariance in the six locus system is
less than three times the covariance of the two locus system. Paramters
are 
, 
and 
; from the right to the left:
1, 2 and 3 pairs of loci.
The frequency of cooperative gamete combinations is higher
than the frequency of the anti-cooperative ones, so that the negative terms
in (15) dominate over the positive ones. The number of cooperative
gamete combinations grows by 
with the number of pairs of
loci 
, which implies that the importance of cooperative effects may
increase more than proportionally to the number of loci.
The effect of the cooperation is shown in Fig. 5, which compares the
dynamics of the covariances for one, two and three pairs of loci under
the same selective conditions.
To estimate the magnitude of the cooperative effect the 
values have been divided by the maximal covariance of the two-locus two allele system times the number of gene pairs. This scaling operation makes the covariance values of the two locus two allele system less equal one. Fig. 5 shows that the maximal covariance of a four locus system is less than twice the maximal covariance of the two locus system. Analogously the maximal covariance of the six locus system is less than three times that of the two locus system. Furthermore, the more loci participate, the earlier
the transition occurs from a regime where the pleiotropic effects are
inhibiting evolution (positive covariance), to the regime where
pleiotropic effects are speeding up evolution (negative covariance).
Thus, cooperative effects due to multi-loci interactions
lead to higher rates of evolution. To a certain degree, they
compensate the inhibiting effect of hidden pleiotropy.