Constructing a model

Step 1: Formulate the question

What do you want to know?
Describe the model in the form of a question.
Boil the question down!
Start with the simplest, biologically reasonable description of the problem.

Step 2: Determine the basic ingredients

Define the variables in the model.
Describe any constraints on the variables.
Describe any interactions between variables.
Decide whether you will treat time as discrete or continuous.
Choose a time scale (i.e., decide what a time step equals in discrete time and specify whether rates will be measured per second, minute, day, year, generation, etc.).
Define the parameters in the model.
Describe any constraints on the parameters.

Step 3: Qualitatively describe the biological system

Draw a life-cycle diagram (see Figure 2.2) for discrete-time models involving multiple events per time unit.
Draw a flow diagram to describe changes to the variables over time.
For models with many possible events, construct a table listing the outcome of every event.

Step 4: Quantitatively describe the biological system

Using the diagrams and tables as a guide, write down the equations.
Perform checks. Are the constraints on the variables still met as time passes? Make sure that the units of the right-hand side equal those on the left-hand side.
Think about whether results from the model can address the question.

Step 5: Analyze the equations

Start by using the equations to simulate and graph the changes to the system over time.
Choose and perform appropriate analyses.
Make sure that the analyses can address the problem.

Step 6: Checks and balances

Check the results against data or any known special cases.
Determine how general the results are.
Consider alternatives to the simplest model.
Extend or simplify the model, as appropriate, and repeat steps 2-5.

Step 7: Relate the results back to the question

Do the results answer the biological question?
Are the results counter-intuitive? Why?
Interpret the results verbally, and describe conceptually any new insights into the biological process.
Describe potential experiments.

Evolution and the Hardy-Weinberg Equilibrium

Darwin, in 1859, published the Origin of Species, arguing that organisms evolve over time by natural selection.

"As many more individuals of each species are born than can possibly survive, and as, consequently, there is a frequently recurring struggle for existence, it follows that any being, if it vary slightly in any manner profitable to itself, under the complex and sometimes varying conditions of life, will have a better chance of surviving, and thus be naturally selected."
-- Charles Darwin (1859)

The process of evolution rests upon three premises:

(2) Not all individuals survive and certain traits improve fitness.

(3) Traits may be inherited.

Parents with characteristics that improve fitness are more likely to have offspring. These characteristics therefore increase in frequency over time leading the population to evolve.

"Variations neither useful nor injurious would not be affected by natural selection, and would be left a fluctuating element, as perhaps we see in the species called polymorphic."
-- Charles Darwin (1859)

Evolution in the absence of selection

Step 1: Formulate the question

QUESTION: How do gene frequencies change over time in the absence of natural selection?

DESCRIPTION OF THE PROBLEM: In a diploid population with two variant alleles at a gene (A and a), how will the frequency, p, of the A allele change over time?

We assume that each diploid individual (AA, Aa, and aa) has equal fitness and that individuals reproduce and then die (non-overlapping generations).

We also assume that individuals produce haploid gametes that form a gamete pool. Gametes within the gamete pool unite at random to produce the next generation of diploid individuals.

Step 2: Determine the basic ingredients

Variables in this model:

x = frequency of AA individuals

y = frequency of Aa individuals

z = frequency of aa individuals

Constraints on these variables:

x, y, z are positive and less than one

x+y+z = 1. Why?

We will follow the genotype frequencies from one generation to the next, using a discrete-time model with a time scale set to a generation.

The frequency of allele A among these individuals? p =

The frequency of allele a among these individuals? q =

p+q=1.

Because all individuals are equally fit, the gamete allele frequencies are equal to parental allele frequencies.

Gamete Mating Table

Step 3: Qualitatively describe the biological system

Gametes unite at random in the gamete pool to produce diploid offspring (life-cycle diagram).

To calculate offspring frequencies, we use a mating table.

These are known as the Hardy-Weinberg frequencies.

Evolution in the absence of selection

Step 4: Quantitatively describe the biological system

x' = p² = (x+y/2)²
y' = 2 p q = 2 (x+y/2) (y/2+z)
z' = q² = (y/2+z)²

Checks: Does x'+y'+z' = 1?

Step 5: Analyze the equations

Point 1: Populations not at Hardy-Weinberg reach Hardy-Weinberg equilibrium after only one generation of random mating (as in the above example).

The frequency of allele A in the next generation? p' =

The frequency of allele a in the next generation? q' =

Point 2: In the absence of selection and mutation, allele frequencies stay constant. Meiosis and random mating do not, by themselves, change allele frequencies.

Step 6: Checks and balances

We have made a large number of assumptions. Changing these assumptions can alter the above results:

	Hardy-Weinberg attained in one generation	p stays constant
Individuals mate randomly rather than gametes	Yes	Yes
Adding mutation	Yes	No
Adding selection	No*	No
Overlapping generations	No	Yes
With different allele frequencies in the two sexes	No	Yes
In finite populations	No	No

*Hardy-Weinberg holds after random mating (among offspring) but not necessarily after selection (among adults).

Step 7: Relate the results back to the question

How do gene frequencies change over time in the absence of natural selection? They don't.

Data Example:

Are these at or near Hardy-Weinberg equilibrium?

x = 0.292
y = 0.496
z = 0.212

p = x+y/2 = 0.540
q = y/2+z = 0.460

p² = 0.2916
2pq = 0.4968
q² = 0.2116

The genotype frequencies appear to be in very good agreement with Hardy-Weinberg!

Some Definitions

Gene: Segment of the DNA, generally a region that codes for a single protein.

Locus: A site on a chromosome (usually synonymous with gene).

Allele: A variant of a gene (a particular sequence).

Haploid: Individuals that carry one copy of each gene.

Diploid: Individuals that carry two copies of each gene.

Gamete: The reproductive cell of a diploid sexual organism (eg sperm or egg).

Genotype: The alleles carried by an individual at a gene.

Homozygote: Individual that carries two identical alleles.

Heterozygote: Individual that carries two different alleles.

Fitness: The average contribution of one allele or genotype to the next generation.

Back to biology 301 home page.