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An introduction to modeling evolutionary dynamics
In mathematics you don't understand things.
You just get used to them.
John von Neumann
Types of models in evolutionary biology
• Conceptual models  What determines what will happen?
• When does natural selection overwhelm genetic drift?
• When is recombination important?
• When will sex evolve?
• Predictive models  What will happen?
• Which strain of influenza will be dominant next year?
• What selection differential must be applied to increase milk yield by 10%?
• How quickly will insecticide resistance spread in the European Corn Borer?
• Statistical inferential models  What did happen?
• Has Influenza hemaglutinin evolved in response to natural selection or drift?
• Did speciation in Heliconius occur in sympatry or allopatry?
• Was differential pollinator visitation responsible for stabilizing selection?
Types of models in evolutionary biology
Statistical inferential
models
Conceptual
models
Few parameters
and variables
 Simple equations
Predictive
models
Many parameters
and variables
 Greater accuracy?
Conceptual models
• The goal is conceptual insight, not precise quantitative prediction
• This requires a simple model yielding analytical tractable equations
• This in turn, requires considering only a subset of variables and parameters
How to build a conceptual model
“With four parameters I can fit an elephant, and with five I can make him wiggle
his trunk”
John von Neumann
The challenge is to decide which four build the elephant!
How to choose the parameters and variables that matter
• Develop a simple and well defined question
For example:
• What types of selection maintain polymorphism at a single locus?
• Does directional selection favor increased recombination?
• Be willing to make risky or even obviously incorrect assumptions
For example:
• Infinite population size
• Free recombination
• Random mating
• No selection
Treat modeling as an ongoing process
Develop a specific question
Identify the minimal set of parameters
and variables needed to address your
question
Develop simulations based on a more
complete and realistic model
Develop a mathematical model based
on this set
Analyze the model and develop testable
predictions
Test predictions using simulations
Answer to question
qualitatively incorrect
add parameters or
variables
Answer to question
qualitatively correct
Test predictions using empirical data
(experiments, field studies, literature
surveys)
Answer to question
qualitatively incorrect
add parameters or
variables
Answer to question
qualitatively correct
Nobel Prize
Example 1: Modeling the evolutionary dynamics of sickle
cell mediated malarial resistance
Malaria in red blood cells
Genotype
Phenotype
AA
Normal red blood cells, malaria susceptible
Aa
Mostly normal red blood cells, malaria
resistant
aa
Mostly sickled cells, very sick
A ‘sickled’ red blood cell
Empirical background
An Example: Sickle cell and Malaria resistance.
(sAA = .11, sSS = .8)
1.1
Fitness
0.9
• Two alleles, A and S that differ at only a single amino acid position
0.7
0.5
0.3
0.1
AA
• AA Individuals are susceptible to Malaria
• AS Individuals are resistant to Malaria and have only mild anemia
• SS Individuals have severe anemia.
AS
Genotype
SS
Develop simple, well-defined questions
• Will genetic polymorphism be maintained?
• How much genetic polymorphism will exist at equilibrium?
• At equilibrium, what proportion of the population will
experience sickle cell anemia?
Make risky or even incorrect assumptions
• Infinite population size
 Need to follow expectations only (higher moments disappear)
• Random mating
 Can utilize Hardy-Weinberg Equilibrium (1 dynamical equation)
• No mutation
 Saves a parameter; yields simpler equations
• No gene flow
 Can consider only local dynamics (1 dynamical equation)
• Constant population size
 R0 can be used as an index of fitness
Develop consistent notation
WX
= The fitness of genotype X
pS
= The frequency of the sickle allele S
pi´
= The frequency of the sickle allele S in the next generation
W
= The mean fitness of the population
p̂S
= The equilibrium frequency of the sickle allele S
Write down dynamical equations using the notation
Solve for equilibria
What do these equilibria tell us biologically?
Next time we will use local stability analyses to answer the
remainder of our questions
What is fitness?
Fitness – The fitness of a genotype is the average per capita lifetime contribution of
individuals of that genotype to the population after one or more generations*
60
50
40
R0
30
20
10
0
AA
Aa
aa
Genotype
* Note that R0 is a good measure of an organisms fitness only in a population with a stable size.
Things are more complicated in growing populations!
Overdominant selection on single loci
Predicted evolutionary trajectories
(Stabilizing selection/Overdominance)
1.1
Fitness
0.9
0.7
0.5
0.3
0.1
AA
AS
SS
Genotype
Frequency of sickle cell
allele, p
s1 = .11, s2 = .8
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
150
Generations
The actual frequency of the
A allele is in the ballpark
of our estimate of .879
Frequency of the S allele in African populations
200
An introduction to conceptual models
“Truth is much too complicated to allow anything but approximations”
John von Neumann
“There's no sense in being precise when you don't even know what you're
talking about”
The goal is conceptual insight, not precise quantitative prediction
An introduction to modeling evolutionary dynamics
In mathematics you don't understand things.
You just get used to them.
John von Neumann