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An introduction to modeling evolutionary dynamics

John von Neumann

In mathematics you don't understand things.You just get used to them.

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

Conceptualmodels

Predictivemodels

Statistical inferentialmodels

Many parametersand variables

Greater accuracy?

Few parametersand variables

Simple equations

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

Identify the minimal set of parameters and variables needed to address your

question

Develop a mathematical model based on this set

Analyze the model and develop testable predictions

Develop simulations based on a more complete and realistic model

Test predictions using simulations

Test predictions using empirical data (experiments, field studies, literature

surveys)

Develop a specific question

Answer to question qualitatively correct

Answer to question qualitatively incorrect

add parameters or variables

Answer to question qualitatively correct

Nobel Prize

Answer to question qualitatively incorrect

add parameters or variables

Example 1: Modeling the evolutionary dynamics of sickle cell mediated malarial resistance

Malaria in red blood cells

A ‘sickled’ red blood cell

Genotype Phenotype

AA Normal red blood cells, malaria susceptible

Aa Mostly normal red blood cells, malaria resistant

aa Mostly sickled cells, very sick

Empirical background

An Example: Sickle cell and Malaria resistance.

0.1

0.3

0.5

0.7

0.9

1.1

AA AS SS

Genotype

(sAA = .11, sSS = .8)

Fit

ness

• Two alleles, A and S that differ at only a single amino acid position

• AA Individuals are susceptible to Malaria

• AS Individuals are resistant to Malaria and have only mild anemia • SS Individuals have severe anemia.

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

= The mean fitness of the population

= The equilibrium frequency of the sickle allele S

W

Sp̂

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*

0

10

20

30

40

50

60

AA Aa aa

Genotype

R0

* 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

0.1

0.3

0.5

0.7

0.9

1.1

AA AS SS

Genotype

s1 = .11, s2 = .8(Stabilizing selection/Overdominance)

Fit

ness

00.10.20.30.40.50.60.70.80.9

1

0 50 100 150 200

GenerationsF

req

uen

cy o

f si

ckle

cel

l al

lele

, p

Predicted evolutionary trajectories

The actual frequency of the A allele is in the ballparkof our estimate of .879

Frequency of the S allele in African populations

An introduction to conceptual models

The goal is conceptual insight, not precise quantitative prediction

“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”

An introduction to modeling evolutionary dynamics

John von Neumann

In mathematics you don't understand things.You just get used to them.