1. Scott Williamson and Carlos Bustamante
Non-stationary population genetic models with selection: Theory and Inference
Scott Williamson
and Carlos Bustamante
Cornell University

2. Inferring natural selection from samples
Statistical tests of the neutral theory (lots)
Methods for detecting selective sweeps (lots)
Parametric inference: estimating selection parameters, etc.
Quantification of selective constraint, deleterious mutation

3. The demography problem
Many existing methods assume random mating, constant population size
These assumptions don’t apply in most natural populations
The effect of demography can mimic the effect of natural selection

4. Natural selection and population growth
Inferring selection from the frequency spectrum while correcting for demography
The McDonald-Kreitman test: does recent population growth cause you to misidentify negative selection as adaptive evolution?

5. The frequency spectrum: an example
Site A G C T
163
975
1972
2188
3529
4424
4961
5286
7019 1 2 3 4 5
Sequence
Count
Frequency class:
Frequency class
Ancestral
Derived

6. Natural selection and the frequency spectrum
Equilibrium neutral and positively selected
frequency spectra
Neutral
2Ns=2
Count
Frequency class

7. Natural selection and the frequency spectrum
Equilibrium neutral and negatively selected
frequency spectra
Neutral
2Ns=-2
Count
Frequency class

8. Natural selection vs. demography
Non-stationary neutral and equilibrium selected
frequency spectra
Population growth, neutral
Equilibrium, 2Ns=-2
Count
Frequency class

9. How do we distinguish selection from demography?
McDonald-Kreitman approach:
Use a priori information to classify changes as “neutral” (e.g. synonymous, non-coding) or “potentially selected” (e.g. non-synonymous)
Putatively neutral changes are treated as a standard for patterns of neutral evolution in a particular sample
Potentially selected sites are compared to the neutral standard
Can we develop a neutral standard for the frequency spectrum?

10. Comparing frequency spectra for different classes of mutation
Observed frequency spectra
This talk:
Likelihood ratio test of neutrality at potentially selected sites, using information from the neutral sites
Biologically meaningful measure of the difference between the two spectra
Putatively neutral
Potentially selected
Count
Frequency class

11. Comparing frequency spectra for different classes of mutation
Observed frequency spectra
A model-based approach:
Fit a neutral demographic model to estimate demographic parameters
Putatively neutral
Potentially selected
Count
Given those parameter estimates, fit a selective demographic model to estimate selection parameters, test hypotheses
Frequency class

12. Comparing frequency spectra for different classes of mutation
Observed frequency spectra
Requirements:
Demographic model
Frequency spectrum predictions from the model under neutrality
Frequency spectrum predictions from the model subject to natural selection
Putatively neutral
Potentially selected
Count
Frequency class

13. Theory: population growth model
2-epoch model  NC
Population size NA
=NA/NC
time
now
Model parameters: ,

14. Theory: predicting the frequency spectrum
Definitions: xi
Number of sites in frequency class i
f(q,t;)
Distribution of allele frequency, q, at time t n
Sample size
Predictions:

15. Theory: the distribution of allele frequency
Poisson Random Field approach (Sawyer and Hartl 1992):
Use single-locus diffusion theory to predict the distribution of allele-frequency
If sites are independent (i.e. in linkage equilibrium) and identically distributed, then the single-locus theory applies across sites
To get f, we need to solve the diffusion equation:

16. Theory: time-dependent solution, neutral case
The forward equation under neutrality:
Kimura’s (1964) solution, given some initial allele frequency, p:

17. Theory: time-dependent solution, neutral case
Applying Kimura’s solution to the 2-epoch model: ancestral mutations
Kimura’s (1964) solution, given some initial allele frequency, p:
Distribution of allele frequency:

18. Theory: time-dependent solution, neutral case
Expected frequency spectrum after
a change in population size (=0.01)
0.8
0.6
P(i,n;,0.01)
0.4
0.2 1 2 3 4 5 6 7 8 9
frequency class

19. Theory: time-dependent solution, neutral case
Multinomial likelihood:
 Maximum likelihood estimates of  and 
 Likelihood ratio test of population growth

20. Comparing frequency spectra for different classes of mutation
Observed frequency spectra
Requirements:
Demographic model
Frequency spectrum predictions from the model under neutrality
Frequency spectrum predictions from the model subject to natural selection
Putatively neutral
Potentially selected
Count
Frequency class

21. Theory: time-dependent solution, selected case
The forward equation with selection:
where =2NCs
Initial condition:

22. Theory: time-dependent solution, selected case
Numerically solve the forward equation using the Crank-Nicolson finite differencing scheme
Use this approximation of f to evaluate the likelihood function:
Fix  and  to their MLEs from the neutral data
Optimize the likelihood for . Likelihood ratio test of neutrality:

23. Theory: time-dependent solution, selected case
How can we be sure that the numerical solution actually works?
Von Neumann stability analysis: solution is unconditionally stable
Numerical solution converges to the stationary distribution after ~4NC generations
Comparison with time-dependent neutral predictions: Kimura, Crank, and Nicolson all agree with each other

24. Human Polymorphism Data
From Stephens et al. (2001)
80 individuals, geographically diverse ancestry
313 genes, 720 kb sequenced
~3000 SNPs (72% non-coding, 13% synonymous, 15% non-synonymous)

25. Results for non-coding changes, assuming neutrality
Model
MLEs
ln(L)
2-epoch
 = 0.016
 = 0.13
Equilibrium neutral
(P0, d.f. 2)
Goodness-of-fit
(P=0.54, d.f. 76)

26. Results for non-synonymous changes, categorized by Grantham’s distance
Category S
P-value
conservative
136
-2.24
0.52
moderate
137
-6.08
0.07
radical
107
-8.44
0.02
all nonsyn
380
-4.88
0.10

27. Ongoing work and future directions
Simulate, simulate, simulate
How robust is the method to different types of demographic forces?
How does linkage among some sites affect the analysis?
How does estimation error affect the LRTs?
Numerical solution for different demographic scenarios (e.g. bottleneck, population structure)
Variable selective effects among new mutations

28. The McDonald-Kreitman test
Sn Number of non-synonymous segregating sites
Dn Number of non-synonymous fixed differences
Ss Number of synonymous segregating sites
Ds Number of synonymous fixed differences
Adaptive evolution
Negative selection
Extensions: Sawyer and Hartl (1992), Rand and Kann (1996), Smith and Eyre-Walker (2002), Bustamante et al. (2002), others

29. Demography and the McDonald-Kreitman test
Robust to different demographic scenarios because it implicitly conditions on the underlying genealogy (see Nielsen 2001)
However, under some demographic scenarios it’s possible to misidentify the type of selection
Weak negative selection with population growth
When the population size is small, non-synonymous deleterious mutations might be fixed by drift
Once the population size becomes large, the level of non-synonymous polymorphism would be reduced (relative to the level of synonymous polymorphism)

30. Demography and the McDonald-Kreitman test
Over what range of parameter values might you misidentify negative selection as adaptive evolution?
How large is the effect?
Eyre-Walker (2002):
Addressed these questions, finding that recent population growth or bottlenecks can cause you to misidentify negative selection
Assumed that levels of polymorphism and fixation rates changed instantaneously with population size

31. Demography and the McDonald-Kreitman test
where tdiv is the divergence time, measured in 2NC generations

32. Demography and the McDonald-Kreitman test
=0.1, tdiv=10 10 10
=0.1, tdiv=4 1 1
0.01
0.1 1
0.01
0.1 1
Expected Neutrality Index (NI)
=1, tdiv=4
=1, tdiv=10 10 10 1 1
0.01
0.1 1
0.01
0.1 1
 (=NA/NC)

33. Demography and the McDonald-Kreitman test: Preliminary results
It is possible to misidentify negative selection for some parameter combinations
But…the parameter range over which this is true is probably smaller than previously thought, as is the magnitude of the effect

34. Summary
Model-based approach to correcting for demography while inferring selection
Evidence for very recent population growth in humans
Reasonable estimates of selection parameters for classes of non-synonymous changes
McDonald-Kreitman test: negative selection + population growth problem not as severe as previously thought
Numerical methods for solving the diffusion are fast, accurate, and fun!

35. Acknowledgements
Collaborator: Carlos Bustamante
Data: Genaissance Pharmaceuticals
Helpful discussions: Bret Payseur, Rasmus Nielsen, Matt Dimmic, Jim Crow, Hiroshi Akashi, Graham Coop