Biol 54H
Luke Habegger
Meredith Johnston
Danny Willner
The Effects of Natural Selection During Apple Maggot Fly Development
Introduction:
The apple maggot fly, Rhagoletis pomonella, possesses a unique life cycle that offers a chance to
examine the effects of different natural selection regimes at various stages of its development.
Found in the northeastern and north central United States, these flies cause millions of dollars in
damage to apple crops each year. The life cycle we used in our examination consisted of four stages.
Apple maggot flies lay their eggs in unripe fruit that is still attached to the tree. These eggs hatch
after seven weeks and subsequently mature into the first larval stage within the fruit—we called this
our first life stage. After the fruit falls to the ground, the larvae leave the fruit, burrowing into the
soil where they pupate and remain dormant for up to several years. This dormant larva in the soil
was our second life stage. Our third life stage consisted of those larvae that have emerged from the
ground as adults but are not sexually mature. After 7-10 days, these adults become sexually mature
and compose the breeding stage, the fourth and final life stage examined in this model.
In our model we assumed that we were studying a population that was isolated from others,
that is, there is no immigration or emigration from this population as the larvae develop into
breeding adults. Additionally, we assumed that no mutation was occurring in the population.
Using these four stages, we were able to examine the effects of any number of selection
regimes as the apple maggot flies completed their life cycle. In particular, we were interested in the
effects of underdominance, overdominance, as well as the effects of selecting for different alleles at
different stages in the life cycle. Additionally, the effect of varying the starting population allele
frequencies was an area explored using this model. Ultimately, we sought to determine conditions at
which the population grew, stabilized and declined given a particular selection regime.
Additionally, we sough to examine the ability of a stage class matrix to accurately predict the
number of individuals in a population after a given number of generations when particular
genotypes were selected against, since stage class matrices utilize generalized survival rates for all
organisms at a particular stage in the life cycle. To do this, we created an additional model
incorporating what we knew of stage class matrices to compare the two methods.
The Model:
Our model attempts to track the allele frequencies at a single locus through any number of
generations. We assigned this allele the letter A and the genotypes as follows: dom (AA), het (Aa)
and rec (aa), though it ultimately depends on the selection regime when determining which genes are
dominant and recessive, if any. The model tracks the number of apple maggot flies with each
genotype as they progress through the four life stages, and uses the final allele frequencies in
determining the number of apple maggot flies of each genotype in the ensuing generation.
The first variable of the model is the number of individuals of each genotype that compose
the original population. The model is organized in a way that allows the user to input the number of
AA, Aa and aa individuals to determine the effects of the founding population’s genotype
distribution.
The second variable manipulated in the model was the selection against apple maggot flies at
each stage of their development. At each stage of the apple maggot fly’s life, the selection against
different forms was varied by using r(#), s(#) and t(#), where r corresponded to selection against
AA, s corresponded to selection against Aa and t corresponded to selection against aa. The #
indicates that these were present and variable at each stage, so r2 corresponded to selection against
AA at the first larval stage, r3 corresponded to selection against AA at the second larval stage, r4
corresponded to selection against AA at the third larval stage, and r5 corresponded to selection
against AA in the breeding stage.. The model began with r2 because r1 would have been the
number of offspring and this was a variable adjusted in the model by the value rep. There was no
selection at this stage. This ability to manipulate the selection against every genotype at every stage
in the apple maggot fly development allowed for the testing of countless selection regimes. In this
particular instance we examined those models that we thought to be most biologically realistic.
Our final variable in the model was the average number of offspring produced by a apple
maggot fly in the breeding stage. While most literature cited apple maggot flies as capable of
producing upwards of 200 young per female, we found that in our model a much lower number of
offspring was necessary to keep the population from exploding, perhaps because our selection
coefficients were much less than those found in the wild.
A sample of our program is provided below:
Dom=10;
Het=10;
Rec=10;
r2=.2;
s2=0;
t2=.2;
Wdom2=1.0-r2;
Whet2=1.0-s2;
Wrec2=1.0-t2;
r3=.2;
s3=0;
t3=.2;
Wdom3=1.0-r3;
Whet3=1.0-s3;
Wrec3=1.0-t3;
r4=.2;
s4=0;
t4=.2;
Wdom4=1.0-r4;
Whet4=1.0-s4;
Wrec4=1.0-t4;
r5=.2;
s5=0;
t5=.2;
Wdom5=1.0-r5;
Whet5=1.0-s5;
Wrec5=1.0-t5;
n=100
tabDom=Table[0, {i,n}];
tabHet=Table[0, {i,n}];
tabRec=Table[0, {i,n}];
Do[
tabDom[[i]]=Dom;
tabHet[[i]]=Het;
tabRec[[i]]=Rec;
Dom2=Wdom2*Dom;
Het2=Whet2*Het;
Rec2=Wrec2*Rec;
Dom3=Wdom3*Dom2;
Het3=Whet3*Het2;
Rec3=Wrec3*Rec2;
Dom4=Wdom4*Dom3;
Het4=Whet4*Het3;
Rec4=Wrec4*Rec3;
Dom5=Wdom5*Dom4;
Het5=Whet5*Het4;
Rec5=Wrec5*Rec4;
Rep=2;
FreDom=(Dom5/(Dom5+Het5+Rec5));
FreHet=(Het5/(Dom5+Het5+Rec5));
FreRec=(Rec5/(Dom5+Het5+Rec5));
A=(FreDom+(1/2)*FreHet);
a=((1/2)*FreHet+FreRec);
Domt1=A^2*Rep*(Dom5+Het5+Rec5);
Hett1=2*A*a*Rep*(Dom5+Het5+Rec5);
Rect1=a^2*Rep*(Dom5+Het5+Rec5);
Dom=Domt1;
Het=Hett1;
Rec=Rect1, {i, n}];
We began our model with a particular number of each of the three different genotypes and
determined a selection regime. At each life stage we derived the fitness using the method displayed
below:
r2=0;
s2=0.2;
t2=0;
Wdom2=1.0-r2;
Whet2=1.0-s2;
Wrec2=1.0-t2;
In this case W symbolizes the fitness of the genotype and the 2 symbolizes the second life stage.
Later in the model Wdom# is multiplied by the number of dom individuals at that stage to
determine the number of individuals that progress to the subsequent life stage. Following the
definition of the selection coefficients, we inserted the variable n, which denoted the number of
generations to cycle the model through, and gave the command to create tables for each of the
different genotypes that corresponded to that particular number of generations. For our project we
chose 100 as the standard number of generations, and that portion of the model is shown below:
n=10;
tabDom=Table[0, {i,n}];
tabHet=Table[0, {i,n}];
tabRec=Table[0, {i,n}];
In order to cycle through the appropriate number of generations, we introduced a “do” loop into
the model as a means of generating the results for the number of generations we selected. Within
the “do” loop we inserted a command to generate the number of individuals in a subsequent
generation given the number of individuals in the previous generation. An example is provided
below:
Dom5=Wdom5*Dom4;
Het5=Whet5*Het4;
Rec5=Wrec5*Rec4;
In this instance the number of individuals of genotypes in the previous generation is multiplied by
the fitness of the genotype in the current generation to create the number of individuals that survive
that life stage and progress to the subsequent generation. Following the completion of this step for
all four transitions, we defined Rep, the average number of offspring per apple maggot fly, and set to
work generating the allele frequencies from the number of flies of each genotype at the end of a
generation. We defined the percentage of each genotype as FreDom (AA), FreHet (Aa), and FreRec
(aa) using the equation Fre(insert genotype) = (genotype)5/(Dom5+Het5+Rec5). Next we defined
the allele frequencies of A and a using the following equations:
A=(FreDom+(1/2)*FreHet)
a=((1/2)*FreHet+FreRec)
Determining the number of individuals of each genotype to begin the next generation was
accomplished with the following formulas:
Domt1=A^2*Rep*(Dom5+Het5+Rec5)
Hett1=2*A*a*Rep*(Dom5+Het5+Rec5)
Rect1=a^2*Rep*(Dom5+Het5+Rec5)
These equations multiply the frequency of a given allele by the average number of offspring by the
total number of individuals after the last life stage to generate the number of individuals of each
genotype in the next generation (t1). After setting Domt1=Dom, Hett1=Het and Rect1=Rec, our
“do” loop was complete and functional.
Results:
All Genotypes Fitness of 1
In order to test our model we tested it with values of 10 for each of the three beginning
genotype classes, 0 for all the r, s and t values, and an average number of offspring equaling 1. We
expected the population to move to Hardy-Weinberg equilibrium where the heterozygous genotype
was twice as prevalent. In this case, with thirty individuals and the average offspring equaling 1, we
predicted a heterozygous population of 15 and homozygous populations of AA and aa at 7.5
individuals. As one can see from the graph at right, our hypothesis was correct and our model was
working correctly. Whenever the average number of offspring was moved above one, we observed
a steady growth in the population. In cases where the average number of offspring was less than
one, the population moved towards extinction. All these results were consistent with our predicted
results from this model. For all graphs displayed, the x-axis denotes the number of individuals, and
the y-axis corresponds to the number of generations.
15
0.00014
14
0.00012
13
0.0001
12
0.00008
11
0.00006
10
0.00004
9
0.00002
2
4
6
8
20
10
40
7´1015
6´1015
15
4´1015
3´1015
2´1015
1´10
80
Rep = 0.5; Population Declines
Rep = 1; Equilibrium Reached
Green = Aa
Blue = AA, aa
5´10
60
15
20
40
60
80
Rep = 1.5; Population Grows
100
100
Underdominance
In our next model we tested the effects of underdominance on the population of apple
maggot flies, where selection against heterozygosity was set at 0.2 at each stage of development. In
this case, we again began the population with 10 of each different genotype and sought a value of
average number of offspring that would place the population in equilibrium. In this case we found
the equilibrium to be at 1.419, and after one generation, the population moved into a HardyWeinberg equilibrium. Stabilization of genotypes in this instance was around 26 for heterozygous
individuals and 13 for each of the homozygous genotypes. However, when we started with 30
heterozygous individuals in the population, we observed a significant drop in these values. While
the average number of offspring equilibrium point remained the same, the number of individuals at
each genotype was much lower; heterozygous individuals stabilized at around 8 individuals and the
homozygous stabilized around 4.
15
25
12.5
22.5
10
20
7.5
17.5
15
5
12.5
2.5
2
4
6
8
Rep = 1.419; Equilibrium Reached
10 individuals of each genotype
Blue = AA, aa
Green = Aa
10
2
4
6
8
10
Rep = 1.419; Equilibrium Reached
30 heterozygous individuals
Overdominance
For a model applying overdominance to the population, we again used a 0.2 selection
coefficient, but this time applied it to the homozygous populations at each stage, so each r and t
value was set to 0.2. We wanted to again determine the average number of offspring at which
equilibrium would be reached, and again found this value to be 1.419. However, the number of
individuals at each of the genotype classes was lower at this equilibrium than in the underdominance
model. Two interesting questions were raised by this result.
First, we wanted to discover why the equilibrium would be the same. Eventually we realized
that in a Hardy Weinberg equilibrium, there are equal numbers of homozygous and heterozygous
individuals (i.e. p^2 + 2pq + q^2) since p and q are equal and heterozygous individuals are twice as
prevalent as either p or q. In this way, selection against either heterozygous or homozygous
individuals with the same selection coefficient would be selecting against the two alleles in the same
way. If twelve individuals possess Aa in a population and six possess AA and aa, then selection
against the homozygous would essentially result in 12 selections against A and twelve selections
against a. Since homozygous individuals possess two of the same allele, selections against them
would be doubled; selecting against the six AA individuals would constitute 12 selections against A,
the same as selection against the heterozygous. For this reason, the graphs supported only two
colors for underdominance and overdominance; blue represents AA and aa while green represents
Aa.
However, there was also the issue of why the numbers of individuals were lower at the
equilibrium point in the case of overdominance. This is resolved when one examines the first
generation’s development through the selection regime. Since the population does not start at
Hardy-Weinberg equilibrium, but with equal numbers of all three genotypes, overdominance selects
against more individuals in the first generation and the subsequent equilibrium values for the three
genotypes are lower once the population does enter Hardy-Weinberg equilibrium.
10
9
Rep = 1.419; Equilibrium Reached
10 individuals of each genotype in each generation
Blue = AA, aa
Green = Aa
8
7
6
5
4
2
4
6
8
10
The first model used was one with 10 of each genotype, and we found an equilibrium when
the average offspring was 1.419. At this point, AA = aa = 2.9 and Aa = 5.8. We completed a
separate model where we started with 30 AA , 10 Aa, and 30 aa. In this case, the numbers were
almost quadrupled. Equilibrium was reached at the same point, but there were 12.5 AA and aa and
25 Aa. This is due not only to the larger initial population, but Aa was selected for so it increased the
numbers of both alleles which were very prominent to begin with. Our next model sought to answer
what would happen if we kept the same 10, 10, 10 population but varied the selection coefficient
against AA and aa to be .8, giving Aa and even higher fitness compared to AA and aa? At the
previous equilibrium point of 1.419, the population crashes. This is a reasonable result since AA and
aa have such a low fitness. We determined that the reproduction rate would need to be higher to
reach an equilibrium, and determined that rep value to be 1.99. Essentially, this population would
be able to survive even with selection coefficients against AA and aa of 0.8 if each fly yielded an
average of 2 offspring.
30
10
9
25
8
20
7
6
15
5
4
2
4
6
8
Rep = 1.419; Equilibrium Reached
Blue = AA, aa
Green = Aa
10 individuals of each genotype
Selection coefficient 0.2 for r#, t#
10
2
4
6
8
Rep = 1.419; Equilibrium Reached
30 individuals AA, aa
10 individuals Aa
Selection Coefficient 0.2 for r#, t#
10
10
Rep = 1.419
Selection Coefficient 0.8 for r#, t#
10 individuals of each genotype
8
6
4
2
2
4
6
8
10
Special Cases
The first special case we tested was a selection regime where a different genotype was
selected against in each of the first three generations, and in the fourth generation the genotypes
possessed equal fitness. We again chose 0.2 as our selection coefficient and selected against AA in
the first generation, Aa in the second generation, and aa in the third. We found that 1.25 average
offspring gave us an equilibrium point where 15 Aa’s and 7.5 AA and aa’s were present, equaling the
original 30 individuals we began with, as one can see in the graph at right. As a slight alteration of
this case, we tested the effects of reversing this selection regime to test whether it would have any
effect on the outcome of the equilibrium point. Since there is no breeding in between the four
stages, we did not expect there to be any difference, and the results were the exact same in both
cases. This is similar to a case where all the fitnesses are equal to one since there is equal overall
selection against each of the genotypes, but since there was some selection, the rep equilibrium was
higher.
15
14
Rep = 1.25; Equilbrium Reached
Green = Aa
Blue = AA, aa
13
12
11
10
9
2
4
6
8
10
Next we tested the effects of a population where overdominance was present in the first two
larval stages, and overdominance was present in the final two. Again, we decided to test the reverse
to see whether the results would be the same. In both cases, equilibrium was reached when the
average number of offspring was 1.563, and at this point there are 15 Aa’s and 7.5 AA and aa’s in
each generation at equilibrium. As an additional test of our model, we looked at alternating between
overdominance and underdominance so that overdominance was present in the first and third life
stages and underdominance was present in the second and fourth. Again, the equilibrium was found
to be at 1.563, which was expected when no breeding was occurring between the developmental
stages.
Our final special case was one where different genotypes were selected for at different
stages, but with different selection coefficients. In our first model, we selected against AA in the
first life stage by setting its fitness at 0.8, then selected against Aa in the second life stage by setting
its fitness at 0.7. The third life stage was used to select against aa with its fitness set at 0.6, and the
fourth life stage saw all the genotypes with equal fitness. The result of this model was that each of
the genotypes had a specific equilibrium value for average number of offspring. Since all the
genotypes were selected against with different fitnesses, they required a different number of
offspring produced by the flies to remain in the gene pool. Since AA was selected against the least,
we expected its equilibrium value to be the lowest, and it was—1.25. The equilibrium value for Aa
was a bit higher at 1.428, and the equilibrium value for aa was the highest at 1.633. The three graphs
displaying these values are shown below, where red corresponds to AA, green corresponds to Aa,
and blue corresponds to aa.
20
14
17.5
12
15
10
12.5
8
10
6
7.5
4
5
2
2.5
5
10
15
20
25
30
5
AA (red) equilibrium – Aa, aa decreasing
10
15
20
25
30
Aa (green) equilibrium – AA increasing, aa
decreasing
20
17.5
15
12.5
10
7.5
5
2.5
5
10
15
20
25
30
aa (blue) equilibrium – AA, Aa increasing
Discussion:
Our model largely confirmed many of the expectations we had for it. Since there was no
breeding in between the life stages, the order of any particular selection regime could be altered but
still provide the same results—repeated tests confirmed this. However, varying the numbers of
individuals that we began the model with generated some interesting results. When the population
began with more of those individuals that were selected against, it did not affect the average number
of offspring equilibrium point, but it lowered the number of individuals at each genotype when the
population did actually reach equilibrium. Additionally, we were able to make a model that
possessed three different equilibrium points for the three different genotypes which, given the
fitness of each genotype at each stage, could predict whether a certain genotype would go towards
extinction or growth based on the number of offspring each fly would produce.
While few animals have the sort of life stages of the apple maggot fly, the idea of selection
for different genotypes at different life stages remains an intriguing one, but one that could perhaps
be modeled to develop insight into a population of such a species. Expanding our model to allow
for multiple breeding stages, disease, mutation and migration from other nearby populations might
make this a more biologically realistic model, but there are still valuable lessons to be learned from
the model presented in this paper.
Reference:
“Rhagoletis pomonella (Walsh) - Apple Maggot.” Plant Pest Surveillance Unit. 2005-10-24. Accessed:
2006-04-15. <http://www.inspection.gc.ca/english/sci/surv/data/rhapome.shtml>