Bioinformatics I, WS 14/15, D. Huson, December 15,

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1 Bioinformatics I, WS 4/5, D. Huson, December 5, Introduction to Population Genetics This chapter is closely based on a tutorial given by Stephan Schiffels (currently Sanger Institute) at the Australian Centre for Ancient DNA in November 204. This text is based very closely on his script, with his permission. 7. Questions The aim of today s lecture is to answer three questions:. What can we say about the time to the most recent common ancestor between you and the Queen of England? 2. How different or similar is the DNA sequence of you and the Queen of England? 3. How did our ancestral population size change through time? 7.2 Ancestors In a simple model of human populations, the number of ancestors that an individual has doubles when going back one generation: Number of ancestors as a function of number of generations g: A(g) = 2 g. Problem: After 32 generations this exceeds the number of living humans. 7.3 Coalescence events In a family tree in a finite population, it will occasionally happen that two (or more) different ancestors in the same generation share an ancestor in the previous generation. This is called a coalescence event:

2 08 Bioinformatics I, WS 4/5, D. Huson (this part by S. Schiffels) December 5, 204 Model of growth: Here is a more realistic model of growth of number of ancestors as a function of number of generations g: { 2 g if 2 A(g) = g N eff, N eff else where N eff is the so-called effective population size. 7.4 Effective population size The effective population size reflects the long-term population size of a population (in humans that is across several hundred thousands of years) reflects the effective number of people that you randomly choose your mates from; a so-called panmictic population: well-mixed randomly mating population. Some concrete numbers: North-Europeans: N eff West-Africans: N eff Native Americans: N eff Answer to question Question : Time to recent common ancestor: What can we say about the time to the most recent common ancestor between you and the Queen of England? The approximate probability of sharing an ancestor with someone else g generations ago is: This is approximately in only 7 generations: (2 g ) 2 N eff. (2 g ) 2 = (27 ) 2 N eff = = How strongly does this answer depend on the assumed effective population size? While the number of ancestors that lived g generations ago depends strongly on this, the number of generations that one must go back to find a last common ancestor does not:

3 Bioinformatics I, WS 4/5, D. Huson (this part by S. Schiffels) December 5, Ancestry of two or more genes We now model the genealogy of two or more genes (or, more precisely, alleles of a gene) backward until their most recent common ancestor is found: This looks complicated. A central idea is to ignore all genes that are not passed down to the currentday set of interest. This is also called looking backward in time. To study the genealogy of a set of genes, we start at the present, and move backward in time, generation by generation, modeling individual coalescence events:

4 0 Bioinformatics I, WS 4/5, D. Huson (this part by S. Schiffels) December 5, Coalescence theory with a pair of samples Definition 7.7. (Basic coalescense theory) Basic assumptions of coalescence theory of a pair of samples : Population has size N, with 2N gene copies. Recall 2 : The probability P (t) of two genes not having the same ancestor in t generations is given by P (t) = ( 2N )t. In the limit for very large populations, N, we have P (t) = e t 2N. So the waiting time to a coalescence event between two lineages is exponentially distributed with mean T 2 = t coal = 2N If the cumulative distribution function of an exponential { distribution is: e λx for x 0 F (x; λ) =, 0 else then the mean is. λ 7.8 Genetic diversity To model genetic diversity, we add mutations to our simple model. Mutations occur with probability µ per generation per site. The mean tmrca (time to the most recent common ancestor) between two genes is 2N generations ago. So, the number of mutations that we expect between two genes is 4Nµ. The site heterozygosity is given by Θ = 4Nµ. Estimator for population size: This gives us an estimator for population size: Fraction of heterozygote positions in the genome Θ = 4Nµ Effective population size This simple formula encapsulates a deep relationship between a purely genomic property (the heterozygosity) and a population level quantity (the effective population size). J.F.C. Kingman, On the Genealogy of Large Populations, J. of Applied Probability, 9:27-43 (982) 2

5 Bioinformatics I, WS 4/5, D. Huson (this part by S. Schiffels) December 5, Answer to question 2 Question 2: Sequence similarity: How similar is the DNA sequence of the Queen of England and of you? Consider a single chromosome and compare the Queen s copy with your copy. Using N = and µ = , we get: Hence: Θ = 4Nµ = = The Queen s and your chromosome differ at about in 333 sites. (Note that 333 = ) 7.0 Mutations on a coalescence tree Recall that the probability of two samples not coalescencing in time t is: ( P 2 (t) = ) t e t 2N. 2N The probability of i samples not coalescencing in time t is: ( ( ) ) i t ( i(i ) P i (t) = = 2 2N 2 2N ) t e i(i ) 4N t. Mean waiting time for coalescence events: So, the waiting times T i are exponentially distributed with mean 3 is: T i = 4N i(i ). Given n samples, and times T i, the total branch length is: T = n i T i = 4N i=2 n i=2 n i = 4N i. Hence, the expected number of mutations anywhere on the tree is: 3 Kingman, 982 S = µ T = µ4n n i= n i = Θ i= i= i.

6 2 Bioinformatics I, WS 4/5, D. Huson (this part by S. Schiffels) December 5, Two famous estimators of genetic diversity How to estimate the quantity Θ = 4Nµ (heterozygosity) from genome data? Consider n sequences of length L. Definition 7.. (Tajima s estimator) Tajima s estimator is the mean proportion of pairwise differences between any two sequences: Θ π = nr of pairwise differences. L Definition 7..2 (Watterson estimator) The Watterson estimator is the number of segregating sites: nr of segregating sites Θ W = L n i= /i. (figures by Stephan Schiffels) 7.2 Answer to question 3 Question 3: demographic history: How did our ancestral population size change through time? Three possible simple answers: The population has been (a) constant (b) declining (c) expanding Interestingly, we can destinguish between these three possible scenarios by only comparing existing genomes:

7 Bioinformatics I, WS 4/5, D. Huson (this part by S. Schiffels) December 5, Determining demographic history We compare Tajima s estimator and Watterson s estimator to get a useful measure: Definition 7.3. (Tajima s D) Define D = Θ π Θ W Var(Θπ Θ W ) If the population size is constant, then should have D 0. If the population size has been increasing, then more mutations will have occurred on leaf edges, thus effecting less pairs, causing D to be negative. If the population size has been decreasing, then more mutations will have occurred on inner edges, thus effecting more pairs, causing D to be positive. So, Taijima s D tells us something about the history of a population. 7.4 Summary The simple coalescence model allows us to: Estimate the time to the last common ancestor of individuals of a population. Estimate how similar the DNA of different individuals of a population is. Make statements about the shape of the recent history of a population.