Download Differences between populations

Differences between populations
7 December 2016.
Joe Felsenstein
Biology 550D
Differences between populations – p.1/19
Evolutionary forces acting on populations
Natural selection toward optima (which can differ between
populations)
Differences between populations – p.2/19
Evolutionary forces acting on populations
Natural selection toward optima (which can differ between
populations)
Genetic drift, making populations differ
Differences between populations – p.2/19
Evolutionary forces acting on populations
Natural selection toward optima (which can differ between
populations)
Genetic drift, making populations differ
Migration, making adjacent populations more similar
Differences between populations – p.2/19
Evolutionary forces acting on populations
Natural selection toward optima (which can differ between
populations)
Genetic drift, making populations differ
Migration, making adjacent populations more similar
Differences between populations – p.2/19
An simulated example with no migration
character mean
20
15
10
N = 100
m=0
f = 0.002
5
0
0
20
40
60
population
80
100
Differences between populations – p.3/19
An simulated example with migration
20
15
10
N = 100
m = 0.005
f = 0.002
5
0
20
40
60
80
100
population
Differences between populations – p.4/19
A geographic region with local migration
Differences between populations – p.5/19
Phenotypes resulting from genetic drift alone
An example from simulation
Scale:
−6.14
2
10
18
26
34.46
Differences between populations – p.6/19
A more general migration matrix
2
1
5
3
6
4
7
a separate m between each pair of connected populations
ij
Differences between populations – p.7/19
13 populations and an environmental variable
100
100
90
90
80
80
70
70
60
50
40
30
20
Simulated with N = 100, m = 0.2, a = 10, b = 0.1, σ = 0.0001
Differences between populations – p.8/19
phenotype plotted against the environmental variable
18.4
phenotype
18.2
18.0
17.8
17.6
17.4
20
40
60
80
environmental variable
100
Differences between populations – p.9/19
contrasts of phenotype versus constrast of environment
contrast of phenotype
2
1
0
−1
−800
−600 −400
−200
0
200
contrast of environmental variable
400
Differences between populations – p.10/19
A numerical example with a 5-population matrix
0.3
1
2
0.25 0.25
5
0.25
3
0.4
0.25
0.3
4
Differences between populations – p.11/19
The contrasts from that 5-population matrix
key:
++ + 0
− −−
Differences between populations – p.12/19
A simulated example
3.06
mean phenotype
3.04
3.02
3.00
2.98
2.96
2.94
0
5
10
15
20
population
(No selection, with m = 0.01 and β = 0.02)
Differences between populations – p.13/19
Its contrasts
standardized contrast score
3
2
1
0
−1
−2
−3
0
5
10
15
20
contrast number
Differences between populations – p.14/19
Fourier transform of characters
When we have a linear “stepping-stone" model of population structure, the
appropriate transform is simply the discrete Fourier transform. Migration
damps the sine waves of different wavelengths differently. Here is a linear
continuum with the optimum phenotype curve the sum of three sine
waves, and the corresponding curve of the phenotypic means that results
when migration damps them:
2.0
1.5
phenotypic mean
1.0
0.5
0.0
−0.5
−1.0
−1.5
−2.0
−4
−2
0
2
4
geographic coordinate
Differences between populations – p.15/19
A worry: direct effects of the environment
If the responses to the environment are not genetic but are the direct
effect on the phenotype, we might get wrong inferences about selection.
We could escape this by “common garden” experiments but this is
hard.
There are statistical ways of detecting this – basically that migration
among neighbors will not make them more similar than
corresponding pairs of populations that have the same two
environments but are farther away from each other. Power of
inference?
Differences between populations – p.16/19
Previous work
Previous papers on this:
Malécot (1949) and Kimura and Weiss (1964) used Fourier
transforms to solve for identity by descent in one- and
two-dimensional infinite stepping stone models of gene frequencies.
Much subsequent work by Takeo Maruyama (1971ff.) on this for
finite one- and two-dimensional stepping stone models.
Hansen, Armbruster and Antonsen (American Naturalist, 2000) used
Fourier transforms to correct for migration in analyzing phenotypic
data in one-dimensional stepping stone models. This work is in
effect a generalization of theirs to the general migration matrix
model of Bodmer and Cavalli-Sforza (1965).
Differences between populations – p.17/19
Previous work, continued
Much of the present algebra was already published by me:
Felsenstein, J. 2002. Contrasts for a within-species
comparative method. pp. 118-129 in M. Slatkin and M.
Veuille, Modern Developments in Theoretical Population
Genetics. Oxford University Press, Oxford.
See also relevant agonizing about what we can infer about migration
matrices in my paper in Journal of Theoretical Biology, 1981.
Inference of migration matrices by sampling (IS and MCMC)
methods: papers by Beerli and Felsenstein (1999, 2001) and Bahlo
and Griffiths (2000).
Differences between populations – p.18/19
Furthermore ...
Is there some way of thinking about how we can infer, from distributions of
gene frequencies, the whole set of migration matrices and historical
branching events that are not rejected? This raises some very interesting
algebraic issues.
It is an example (like pedigree estimation and invariants of phylogenies) of
a highly structured set of hypotheses, all with the same number of degrees
of freedom, where we need to know which ones the data supports.
In this sense the problem is not so statistically simpleminded.
Differences between populations – p.19/19