In plants with a tristylous mating system, frequency-dependent natural selection is expected to cause all three morphs to be equally abundant at evolutionary equilibrium. This handout provides a justification for this assertion. The key to this justification is to recognize that a tristylous mating system automatically generates a rare-morph advantage in mating, a type of frequency-dependent selection that acts to maintain all three morphs in the population. To see that this is the case, we first consider a situation in which there are initially only two morphs, say short and medium, and show that whichever morph is rarer will transmit disproportionately more genes through pollen. We then extend this argument to three morphs by illustrating that a rare third morph will enjoy a similar transmission advantage when two other morphs are common.
I. Expected equilibrium morph frequencies when there are two morphs
We wish to determine whether each of the morphs will increase in frequency when rare. As an illustration, suppose that there are 50 plants in a population, 49 of which are S-morph plants and 1 of which is an M-morph plant. Let us assume, further, that each plant produces only one seed. (This assumption is not necessary, but a convenience to keep calculations simple.)
1. The total number of seeds produced by S plants is thus 49, while
the total number produced by M plants is 1
(Table 1, second row).
2. Because only inter-morph matings are possible in a tristylous
system with incompatibility barriers, all of the S plants
have only 1 M ovule available to fertilize. The total number
of successful pollen grains originating from all S plants
is thus 1. By contrast, the lone M plant has available 49 ovules
that its pollen can fertilize. (See Table 1, third row).
3. The total number of succesful gametes emanating from a particular
morph is just the sum of seeds (ovules) produced
plus the number of successful pollen grains, or 50 for each morph (Table
1, fourth row).
4. The total per-capita number of successful gametes is simply the
number of successful gametes for a morph
divided by the number of individuals of that morph (Table 1, row 5). For
the S morph, this is about 1 gamete per
individual, while for the M morph, this is 50 gametes per individual.
This much greater reproductive success of M
individuals is due to the fact that there are more mates available for
each individual than for each S individual and
results in very strong selection to increase the frequency of the genes
that are responsible for the M morph.
5. Our assumption that the morph was initially rare was completely
arbitrary. If we assume instead that the S morph is
initially rare, a completely equivalent argument shows that the S morph
will have a mating advantage and will increase
in frequency. Since the situation is completely symmetrical, either
morph will have a mating advantage if it is less
frequent than the other morph, and will thus tend to increase in frequency.
Consequently, the equilibrium occurs
when both morphs are equally abundant, i.e. when neither morph has a mating
advantage. In other words, the
equilibrium frequency of each morph is 1/2.
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Table 1.
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S morph M morph
Number of individuals 49 1
Total seeds produced 49 1
Total successful pollen
1
49
grains
Total successful gametes 50 50
Per-capita # successful
50/49 ~= 1 50/1 = 50
gametes
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Note: ~= means "is approximately equal to"
II. Expected equilibrium morph frequencies when there are three morphs
We now examine what will happen if a third morph is introduced at low frequency into an 2-morph equilibrium population. The calculations are a bit more complex, but conceptually similar to what we saw previously.
Let us suppose that there are 25 S-morph and 25 M-morph individuals and 1 L-morph individual.
1. The number of seeds produced per morph are then 25, 25 and 1 respectively,
assuming again one seed per plant
(Table 2, row 2).
2. The calculation of male success is a bit more complicated.
Consider first the S morph. There are a total of 25 ovules
of M-morph plants that S plants can fertilize, but the S plants are competing
with the lone L plant for these ovules.
Consequently a total of 26 plants are competing to fertilize 25 M ovules.
Assuming individual S plants and individual
L plants are equally good competitors, this means that on average the probability
that any individual M ovule will be
fertilized by pollen from an S plant is 25/26. Consequently, the
total number of M ovules fertilized by pollen from S
plants equals:
(prob. individual ovule will be fertilized by ) x (Number of ovules) = 25/26 x 25.
Still considering S plant success, we next ask how many L ovules are fertilized
by pollen from S plants. Competing
for each L ovule are 25 S plants and 25 M plants. Again assuming
equal competitive ability for the two morphs, the
probability that an L ovule will be fertilized by pollen from an S plant
is simply 25/50 = 1/2. The number of L ovules
available is 1, so the total number of L ovules fertilized by pollen from
S plants equals:
(prob. individual ovule will be fertilized by ) x (Number of ovules) = 25/50 x 1 .
Putting these two calculations together, the total number of ovules fertilized by S plants is
25/26 x 25 + 25/50 x 1
~= 25.5 (Table 2, row 3)
Calculations for the M morph male success are similar and yield the same
result. Now consider the male success of
the L morph. The lone L plant constitutes 1 out of 26 plants (the
other 25 are M plants) competing for S ovules.
The probability that an S ovule is fertilized by pollen from the L plant
is thus 1/26. Since there are 25 such ovules,
the total number of S ovules fertilized by L pollen is 1/26 x 25, which,
by similar reasoning, is also the number of M
ovules fertilized by L pollen. The total number of successful L pollen
grains is thus just
1/26 x 25 + 1/26 x 25 ~= 2 (Table 2, row 3).
3. Once more, for each morph we add up the number of successful gametes
(ovules and pollen grains) produced
(Table 2, row 4), then divide by the number of individuals of that morph
to obtain the per-capita successful gamete
production (Table 2, row 5). The result is that while each individual
S and M morph plant produces about 2 succesful
gametes, each L plant produces 3. This 50% reproductive advantage
constitutes a strong selective force that will
cause the L morph to increase in frequency.
4. Once again, the choice of which morph was initially rare was completely
arbitrary. Consequently the argument will
work for the S and M morphs when they are rare, i.e. whenever a
morph is rare, it enjoys a mating advantage and
will thus increase in frequency. This means that an equilibrium will
occur when all three morphs have equal
frequency and none has a mating advantage, i.e. when the frequency of each
morph is 1/3.
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Table 2.
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S morph M morph L morph
Number of individuals 25 25 1
Total seeds produced 25 25 1
Total successful pollen (25/26)x25+(25/50)x1
same(25.5) 25/26 + 25/26
grains
~= 25.5
~= 2
Total successful gametes ~= 50.5 ~= 50.5 ~= 3
Per-capita # successful
50.5/25 ~= 2
50.5/25 ~= 2
3/1 ~= 3
gametes
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