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Population Genetics Modeling Using Populus
A. Lyman-Buttler
Background
This activity is intended to be completed after AP Biology Investigation 2, “Mathematical Modeling: Hardy-Weinberg”.
Please see the AP Biology Lab Manual for background information.
Installing Populus—Populus is a population modeling program developed at the University of Minnesota. If you are
doing this lab activity in class, it should already be installed on your computer; if you're doing it at home, a basic level of
“computer savvy” is assumed. You can get the program at http://www.cbs.umn.edu/populus/. If you need help, ask Mr.
Lyman-Buttler. Populus requires the Java Runtime Environment to be installed in order to run; you can check to see
whether this is the case at http://www.java.com/en/download/testjava.jsp.
The Modern Synthesis
Natural selection then became a process that altered the frequency of genes in a population and this defined evolution.
This point of view held sway for many decades but more recently the classic Neo-Darwinian view has been replaced by a
new concept which includes several other mechanisms in addition to natural selection. Current ideas on evolution are
usually referred to as the Modern Synthesis which is described by Futuyma:
"The major tenets of the evolutionary synthesis, then, were that populations contain genetic variation that arises by
random (ie. not adaptively directed) mutation and recombination; that populations evolve by changes in gene frequency
brought about by random genetic drift, gene flow, and especially natural selection; that most adaptive genetic variants
have individually slight phenotypic effects so that phenotypic changes are gradual (although some alleles with discrete
effects may be advantageous, as in certain color polymorphisms); that diversification comes about by speciation, which
normally entails the gradual evolution of reproductive isolation among populations; and that these processes, continued for
sufficiently long, give rise to changes of such great magnitude as to warrant the designation of higher taxonomic levels
(genera, families, and so forth)."
- Futuyma, D.J. in Evolutionary Biology, Sinauer Associates, 1986; p.12
This view may be best described as “neo-Darwinian” since Darwin was unaware of genes and principles of Mendelian
inheritance. Biochemist Larry Moran summarizes:
“The modern theory of the mechanism of evolution differs from Darwinism in three important respects:
1.
2.
3.
It recognizes several mechanisms of evolution in addition to natural selection. One of these, random genetic drift, may
be as important as natural selection.
It recognizes that characteristics are inherited as discrete entities called genes. Variation within a population is due to
the presence of multiple alleles of a gene.
It postulates that speciation is (usually) due to the gradual accumulation of small genetic changes. This is equivalent to
saying that macroevolution is simply a lot of microevolution.
In other words, the Modern Synthesis is a theory about how evolution works at the level of genes, phenotypes, and populations
whereas Darwinism was concerned mainly with organisms, speciation and individuals.” (Moran 1993)
Genetic Drift: Monte Carlo Model
This simulation uses a random number generator to sample genes from a parental population and pass them on to
offspring. Population size is assumed to remain constant from generation to generation. (Which Hardy-Weinberg
assumption does this imply?)
Suppose that a population consists of one male and one female, and that both are heterozygous (Aa) at a locus. There
are four alleles in the gene pool, 2 A and 2 a, so p = q = 0.5. The female will produce A and a eggs in equal proportions,
and the male will produce half A and half a sperm. If gametes are chosen at random and used to form a filial population
of 2 individuals, the probability that the first individual will be AA is (0.5*0.5) = 0.25, and the probability that both
progeny will be AA is (0.5*0.5)*(0.5*0.5) = 0.0625. Thus if population size = 2, there is 1 chance in 16 that allelic
frequency will change from p = 0.5 to p = 1.0 in a single generation simply through the random sampling of gametes.
(Note that all subsequent generations can inherit only the A allele, and will always have genotype AA, because a no
longer exists in the population. The A allele is said to be fixed.) (Alstad 2003)
In addition to the drift from p = 0.5 to p = 1.0, there are other possible outcomes; p could change to 0, 0.25, or 0.75, and
the likelihood of these events is calculated similarly (you should be able to do it). A computer model which uses random
numbers to mimic this stochastic sampling process is called a "Monte Carlo Simulation" (Alstad 2003).
The Mendelian genetics problems you’ve been solving involve a lot of genetic diversity (blood types, fur color, etc.), so
you may be surprised to learn that all human individuals are nearly identical with respect to coding sequences. In our
models, any genetic variation from the “wild type” must theoretically have its origin in a single germ-line mutation in a
single individual; before this event, all members of the population were identical with respect to the locus in question.
For example, the “wild type” human eye color is brown; all people with blue eyes inherited from a single ancestor a
mutation in the OCA2 gene, which directs the synthesis of melanin in the iris. The mutation arose between 6000 and
10000 years ago, and would have affected just one chromosome—which means that before there were any blue-eyed
people, there must have been a brown-eyed heterozygote who, by chance, happened to pass this mutant allele on to
more than one offspring. Only later—probably generations later—would some of these heterozygote descendants have
had children with each other, and some of these children (25% on average) would have blue eyes.
Setting Up Populus
1) Select “Model” from the main menu. Go to the Mendelian Genetics menu and select “Genetic Drift.” This will
bring up a box that lets you set initial conditions (population size, starting allele frequency, etc.)
2) Set the population size to 10 and the number of loci to 10. Leave the other parameters as they are.
3) Click the “View” button, and observe the graph. Notice how the allele frequencies change randomly. Sometimes,
an allele frequency will reach 1.0 or 0.0. When this happens, an allele has become “fixed” in the population: only
one allele for that trait remains, and the other one has been eliminated, never to return.
4) To run another iteration, click “View” again (or “Iterate”—it does the same thing)
5) For your investigation, you’ll need to keep track of the output of the model; use a spreadsheet to do this. One
way to compare outputs is to determine the average number of alleles fixed per generation. Note that the
simulation is set to run for 3 times as many generations as the population size; so, for example, for a population
of 50, 150 generations are simulated. This will be the same for every iteration. Therefore, in this case, you can
just take the number of alleles fixed during the course of the simulation, and divide 150.
Practice Problem
Determine the probability of fixation for an allele in populations of size 10, 100, 250, and 500. We will calculate the
probability of fixation (PF) on a per-generation basis using the formula
𝑃𝐹 =
𝐹
𝐿𝑡
where F is the number of alleles fixed, L is the number of loci, and t is the number of generations. This formula allows us
to make apples-to-apples comparisons between models regarding the rate at which genetic diversity is eliminated from
a population. For the practice problem, fixation will be defined as an allele frequency reaching 1.0 or 0.0 before the end
of the run. Later, this definition may change depending on the investigation.
Using a spreadsheet, plot probability of fixation vs. population size, and add a trendline with equation to your graph.
(You should remember how to do this from 9th grade bio; ask Mr. LB for help if you need it.) What is the relationship
between these two variables? Now, e-mail a copy of your spreadsheet to [email protected].
Drift and Selection
Now we are going to examine two different causes of evolution at the same time. This gets our model a little closer to a
real-world population, and it will also help us understand the relative importance of different causes of evolution a little
better. Unlike the previous model, this one is programmed to track only a single allele. Because selection is involved,
there are three new variables: WAA, WAa, and Waa, which stand for the relative fitness of the homozygous dominant,
heterozygous, and homozygous recessive genotypes, respectively.
What does relative fitness mean? ______________________________________________________________________
Many new alleles in a gene pool originate as recessive mutations. These mutations are usually deleterious, that is to say,
they reduce the relative fitness of the organism. Suppose we model the emergence of a new deleterious allele in a
population, and track its progress through future generations. Set a population size of 500, an initial allele frequency of
0.999 (this corresponds to 499 normal individuals and 1 heterozygous individual with 1 normal + 1 mutant allele), and
relative fitness values of 1.0. 1.0, and 0.5 for AA, Aa, and aa, respectively. Click “View” to run the simulation. Click “View”
again to run it again.
For the Drift and Selection model, PF should be calculated differently than before. Simply divide the number of trials in
which the allele becomes fixed by the number of total trials.
How to model the introduction of a recessive deleterious allele:
1. Set the initial population size to whatever you want.
2. Multiply the initial population size by 2 (because there are 2 loci per individual). Divide 1 by this number. This
gives you the frequency of the new recessive allele (q). If you did the math right, it should be close to 0.
3. Because the graph plots p, and p + q = 1, solve for p and enter it into the Initial Allelic Frequency (p) box.
4. Set wAA and wAa to 1.0; set waa to something less, depending on how large a disadvantage you want.
How to model the introduction of a dominant beneficial allele:
1. Set the initial population size to whatever you want.
2. Multiply the initial population size by 2 (because there are 2 loci per individual). Divide 1 by this number. Now,
subtract this number from 1.000. This gives you the frequency of the new dominant allele (p). If you did the
math right, it should be close to 1.0. Enter it into the Initial Allelic Frequency (p) box.
3. Set wAA and wAa to 1.0; set waa to something less, depending on how large a disadvantage you want.
Designing Your Investigation
Now that you’ve test-driven the Genetic Drift and Drift and Selection models, you can
Some possible considerations:
 How many iterations do I need to run for each case to get statistically valid results?
 What assumptions does my scenario require?
 How will I collect data from my model?
 How will I graph my data?
 Does the answer to my question have real-world implications? (e.g. does it tell us how a trait may have evolved,
or how we should target our species-conservation efforts?)
Possible Investigation Questions
Here are some ideas to get you started. Feel free to choose your own question to investigate.

Genetic drift:
◦
What was the typical effective population size of a human population 10000 years ago? Based on this, how
many times did the OCA2 mutation that causes blue eyes probably originate before it became established?
◦

All other things being equal, how does the probability of fixation of a deleterious allele depend on
population size? On relative fitness?
Selective sweep:
◦
What is the relationship between an allele’s relative fitness and its tendency to become fixed in a
population?
◦
Selective sweeps are thought to be rare and play only a minor role in human evolution, but interesting
examples do exist: the evolution of lactase persistence (lactose tolerance) in Northern Europeans, or the
evolution of high-altitude-adapted hemoglobin in Tibetans.
◦

All other things being equal, how does the probability of fixation of a beneficial allele depend on population
size? On relative fitness?
Heterozygote advantage:
◦
Given what we know about sickle-cell allele frequencies in West Africa, what is the relative fitness of each
phenotype there? How does this compare to other populations (e.g. African Americans)? Is there evidence
of selection? (Hint: A problem like this needs to be worked backwards. First, research the real-world allele
frequency; then, tweak the relative fitness values until your model consistently stabilizes around that
frequency on the graph.)
Analysis questions
1)
2)
3)
4)
Generally speaking, what is the effect of drift on genetic diversity in a population?
Were the populations you simulated at Hardy-Weinberg equilibrium? Why or why not?
How do the effects of drift on genetic diversity very with population size?
Did it appear that any of the alleles conferred a selective advantage, or were they all equally likely to be lost?
Explain.
5) Which Hardy-Weinberg assumption(s) is/are violated in your simulation?
6) Describe 2 possible events in nature that could cause a population to experience a significant amount of drift.
Present your work using the mini-poster format.
References
Alstad, D. (2003). Populus and Populus help system. Retrieved from http://www.cbs.umn.edu/populus.
Kimura, M. On the probability of fixation of mutant genes in a population. (1962). Genetics, 47, 713-719.
Moran, L. (1993). The modern synthesis. Retrieved from http://www.talkorigins.org/faqs/modern-synthesis.html.
Otto, S., & Whitlock, M. (1997). The probability of fixation in populations of changing size. Genetics, 146, 723-733.