lecture notes for evolutionary ecology 548. lecture #2: …snuismer/nuismer_lab/54… · ·...

Lecture Notes for Evolutionary Ecology 548. Lecture #2: Fitness, Selection, and Adaptation

I. Defining Fitness

What is fitness?

Imagine a simple case where a population consists of individuals who have phenotype z1 and

individuals who have phenotype z2. The offspring of these individuals always have a phenotype

identical to their parent. What determines which of these individuals ultimately predominates

within the population?

Case 1: Annual organisms

Assume individuals survive to reproduce with probability l(z) and that surviving indidviduals

produce, on average, m(z) offspring. How do the relative frequencies of the types change over

time?

Start by determining the number of individuals of each type in the next generation

(1a)

(1b)

What do equations (1) tell us about the fate of the two phenotypes?

(2a)

(2b)

The quantity W entirely determines the relative abundances of the two phenotypes at any

point in the future and is thus a complete description of fitness

Conclusion: For annual organisms, the change in frequencies of types depends on only the

product of the survival and fertility of the types. Thus, NS may favor decreased fertility or

decreased survival if doing so increases their product (fitness) through the presence of

fundamental life history trade-offs


Case 2: Fitness in perennial organisms with age structure

Assume individuals survive to age x with probability lx and produce an expected number of

offspring equal to mx in age x. How do the relative frequencies of the types change over time?

Phenotype z1

x lx mx lxmx

1 1.00 0.00 0.00

2 0.75 0.00 0.00

3 0.50 1.00 0.50

4 0.25 4.00 1.00

How many offspring is an individual expected to produce over its lifetime?

k

x

xxml1

For annual organisms we found that this quantity, R0, which combines survival and fertility is

equal to fitness and completely determines how the frequency of genotypes/phenotypes changes

over time through selection.

However, for perennial organisms with age structure this simple quantity, R0, does not

completely describe fitness. If a stable age distribution has been reached, we can however,

describe fitness using another relatively simple quantity:

where τ is the generation time:

Calculating r for the example shown above reveals an important point: In age structured

populations, generation time is a fundamental component of fitness. Specifically, even though

individuals of the two phenotypes produce, on average, identical numbers of offspring, the

growth rate of z2 is greater. Also, note the remarkable fact that once z2 becomes fixed in the

population, individuals die at younger ages the population has evolved early senescence, a

fundamental shift in life history

Phenotype z2

x lx mx lxmx

1 1.00 1.00 1.00

2 0.75 0.67 0.50

3 0.00 0.00 0.00

4 0.00 0.00 0.00


Conclusion: In perennial populations with age structure, natural selection favors phenotypes

that increase r. Thus, NS favors phenotypes with effects on life histories that increase r,

potentially favoring reductions in survival and/or fertility.

II. How do we connect fitness to evolution?

We would like to predict how the phenotype distribution of the population will change over time.

A reasonable starting point would be to predict the mean phenotype of the population in the next

generation. How can we do this?

i. The number of individuals in the next generation that will be produced by individuals with

phenotype in the current generation is:

ii. The total number of individuals in the next generation is:

iii. The frequency of individuals in the next generation that will be produced by individuals with

phenotype in the current generation is:

iv. We can now use this information to calculate the mean phenotype in the next generation

v. However, we need to relate the phenotype of offspring (zo) to the phenotype of their parent.

We will assume the following linear relationship:

vi. With this assumption the mean in the next generation is:


vii. And the change in the mean over a single generation is:

(1)

LECTURE 2 ENDED HERE WHEN TIME RAN OUT

I. How else could we write equation (1)?

(2)

or

(3)

where:

G = additive genetic variance


II. What do these expressions tell us about evolution by natural selection?

Q1: What determines the rate of evolution?

Q2: What determines the direction of evolution?

Q3: What role does phenotypic variation play?

W

z


III. How does adaptation influence ecology?

We saw in section two that phenotypic variation for a trait influencing fitness causes the

population mean phenotype to evolve. Can we also predict how population size will change in a

phenotypically variable and evolving population?

The total population size in the next generation is:

Doing some simple algebra:

(4)

The change in population size depends on the mean ***absolute*** fitness of the population.

How does mean fitness change over time?

Equation (1) is valid for any trait. What if the trait of interest were fitness itself?

The frequency of individuals with absolute fitness W in the next generation is:

The population mean ***absolute*** fitness of the population in the next generation is


The change in the population mean ***absolute*** fitness over a generation is:

(5)

This is essentially Fisher's fundamental theorem of natural selection: the change in population

mean fitness is proportional to the additive genetic variance for fitness.

Note that equation (5) is simply equation (1) where the trait (z) of interest is fitness (W) itself.

What can we learn about the interface between ecology and evolution by combining

equations (4) and (5)?


IV. How can we use the theory to predict/understand adaptation in the real world?

What quantities would we want to measure?

How could we measure them?

What assumptions must we make?

How could we standardize the data so that selection could be compared across studies?

Total fitness vs. Fitness components and the realities of studying NS in natural pops

A. Measuring heritability

The heritability of a population, h2, can be estimated by calculating the slope of the linear

regression of mid-parent phenotype on offspring phenotype. Although sufficient for the purposes

of this class, this is a crude approach and much more sophisticated methods exist.

B. Estimating the strength and form of selection

Directional Selection

i. Binary fitness data (e.g., survived vs. died) can be easily analyzed by simply estimating the

difference between the mean of the surviving individuals and the mean of the entire population

prior to selection. The result is an estimate of the selection differential S appearing in equation

(2) of section II. Multiplying the selection differential by the heritability yields a prediction for

the change in the trait mean over a single generation.

ii. Continuous fitness data (e.g., fertility, seed set, growth rate, etc) can be analyzed by

calculating the slope of the regression of relative fitness (or a component of fitness) on trait

value. This slope is equal to the selection gradient β appearing in equation (3) of section II.

Multiplying this selection gradient by the additive genetic variance G, yields a prediction for the

change in the trait mean over a single generation.

Warnings and conventions

i. Warning: Fitness data rarely conforms to the assumptions of linear regression. Thus, it is often

tempting to transform the fitness data. DO NOT do this. Transforming the fitness data yields

incorrect prediction.

ii. Convention: Trait values are generally transformed into units of phenotypic standard

deviations. Although this does not influence the accuracy of prediction, it facilitates comparison

across studies where traits may have very different units of measurement.

LECTURE #3 ENDED HERE WHEN TIME RAN OUT


IV. Where do genetic constraints fit into the framework we have developed?

The only way genetic constraints can be manifested in our current framework is as limited or

absent additive genetic variance.

How else could genetic constraints be manifested in the real world?

What if selection acts on multiple traits?

(6)

where is now a vector of population means for a suite of traits, G is now the additive genetic

variance-covariance matrix, and β is now a vector of selection gradients.

Expanding (6) for insight yields:

Remembering linear algebra ("rows into columns") this yields:

What do these equations suggest about the nature of genetic constraints?

Which components of the G matrix comprise the genetic constraints?


V. A concrete example with two traits and a single bivariate phenotypic optimum


VI. Estimating the G-matrix

This can be accomplished in many ways, but all involve complicated breeding designs and

statistical techniques. If you are interested in finding more information on how G-matrices are

estimated, check out the following references:

VII. Estimating selection gradients


Appendix 1. Some Important Parameters and Variables for Evolutionary Ecology

Symbol Meaning

z The phenotype of an individual

The population mean phenotype

The number of individuals in the population

The number of individuals in the population with phenotype z

The frequency of individuals in the population with phenotype z

The number of offspring produced by an individual with phenotype z

The average number of offspring produced by individuals within the population

The number of offspring produced by an individual with phenotype z relative to

the average number produced by individuals within the population

h2 Heritability. The slope of the parent offspring regression. The proportion of

phenotypic variation attributable to the additive action of genes

lecture notes for evolutionary ecology 548. lecture #2: …snuismer/nuismer_lab/54… · ·...

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