6 Introduction to Population Genetics

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1 70 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, Introduction to Population Genetics This chapter is based on: J. Hein, M.H. Schierup and C. Wuif, Gene genealogies, variation and evolution, Oxford University Press, Population genetics: Given a current-day population of individuals, represented by a collection of genes, the goal of population genetics is to infer details of the evolutionary processes that produced the population. In this chapter we will take a look at the very basic Wright-Fischer model of populations, which was introduced by Wright (1931) and Fischer (1930). While this field has very old roots, it has again become a hot topic because of an increased focus on sequencing many individuals of a population, made possible by recent advances in DNA sequencing technologies. For example, the aim of the 1000 Genomes Project is to produce genome sequences of 1000 different humans, whereas the 1001 Genomes Project will produce 1001 genome sequences of different eco-types of the model plant Arabidopsis thaliana. Many studies focus on SNPs (single nucleotide polymorphisms) to compare different genomes. (Source: The history of a population is shaped by many factors, such as: mutational processes, mixing of variations through recombination, population size, migration and geographic or social structure. 6.1 Human population genetics Migration is believed to have played a major part in population genetics of humans:

2 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, (Source: Simple models of population assume that the population size is constant, which is clearly not true for humans: (Source: Different models of population growth There are three different models of population growth: (a) Constant (b) Explosion (c) Exponential The width between the thin lines represents the size of the population and the trees represent the

3 72 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, 2011 history of a set of genes. The direction of time is from top to bottom. Population growth influences the types of genes observed in a sample. When a gene is passed from parent to offspring, there is a small chance of it acquiring a mutation. In growth scenarios (b) and (c) this will lead to many singleton genes that each carry one mutation, whereas in (a) genes will usually carry multiple mutations. 6.3 Haploid vs. diploid populations Diploid populations have two copies of most genes. Recombinations reshuffle parts of genes and thus introduce mathematical difficulties in the analysis: While the effect of recombination is an important topic, will not consider it in this course. In the 1001 Genomes Project, the plants considered are inbred lines, that is, they have been bred to make then isogenetic (having only one allele per gene). To allow comparison of haploid and diploid populations, we will assume a population size of 2N genes, representing N diploid or 2N haploid individuals. This is the haploid reproduction model:!" #" $" %" #!" &'(')*+,("#"!" #" $" %" #!" &'(')*+,("#-!" In the haploid model, each gene i in generation t + 1 is found by randomly choosing a gene g in generation t. Any gene not chosen dies out. This is a diploid reproduction model:,%-(.%/# -(.%/#!" #"! "#!" #"! %# $%&%'()*&"$# $%&%'()*&"$+!"!" #"! "#!" #"! %# In the diploid model, let us assume that the species has two sexes, females and males. Let N f and N m denote the size of the female and male subpopulations, respectively, with N = N f + N m.

4 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, Each individual in generation t+1 randomly chooses a male (father) and female (mother) in generation t. Then a gene is chosen for each parent, with equal odds. In this model, each gene has one parent gene and each individual has two parents. For large values of N, N f and N m, one can approximate the diploid model by the haploid model. Thus, we will only consider the haploid model. 6.4 Wright-Fischer model Wright-Fischer model: The Wright-Fischer population model makes the following simplifying assumptions: 1. Discrete and non-overlapping generations. 2. Haploid individuals. 3. Constant population size. 4. All individuals are equally fit. 5. The population has no geographic or social structure. 6. No recombinations of genes (or sequences). 6.5 The coalescent process Calculations in population genetics are often based on the following concept: Coalescent process: The basic idea of the coalescent process is to make inferences by moving backward in time. One starts with the present-day generation and constructs previous generations by randomly choosing parents in the previous generation. Assume we are given a set G of n present-day genes. If we trace back their lineage of ancestors to their most recent common ancestor, then we obtain a tree called the genealogy of G Binomial distribution Consider a population of size 2N under the Wright-Fischer model. Let v i be the expected number v i of descendants that a gene i will have. The probability that this equals k is P (v i ) = ( ) ( ) 2N 1 k ( 1 1 ) 2N k, k 2N 2N the Binomial distribution Bi(m, p), with parameters m = 2N and p = 1 2N. For the mean value and variance we have: and respectively. E(v i ) = m p = 2N 1 2N = 1 V ar(v i ) = mp(1 p) = 2N 1 ( 1 1 ) = 1 1 2N 2N 2N,

5 74 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, 2011!"!"#"$%&'#(#" $!"!"#"$%&'#(#)*( Note: If we change the model such that E(v i ) < 1 or > 1, then the population size with decrease or increase. The covariance for two different genes i and j is: and the correlation coefficient is: Cov(v i, v j ) = E(v i v j ) E(v i )E(v j ) = 1 2N Cor(v i, v j ) = Cov(v i, v j ) V ar(vi )V ar(v j ) = 1 2N 1. Note: The numbers of offspring for i and j are negatively correlated. The larger the population size, the lower the correlation. Binomial distribution Recall that the binomial distribution describes the probability of having k successes in a probabilistic experiment that is repeated m times. Binomial distribution for m = 20 and p = 0.1 (blue), p = 0.5 (green) and p = 0.8 (red): (Source: Distribution.PNG) If 2N >> 0, then v i is approximately Poisson distributed: P (v i = k) 1 k! e 1, with mean = 1 and variance = 1. (For large numbers, the Binomial and Poisson distributions are very similar.) The probability that a given gene i does not have a descendant in the next generation is P (v i = 0) = e The proportion of genes that have a descendant is 1 e

6 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, So, the present-day population descends from a small fraction of genes a few generations t ago, namely 0.63 t. For example, if 2N = and t = 15, then the number of ancestral genes = All other genes have died out. Poisson distribution Recall that the Poisson distribution describes the probability P (k) = λk k! e λ of seeing k occurrences in a fixed time interval, for a given mean value λ. It looks like this: (Source: pmf.svg/1000px-poisson pmf.svg.png) Geometric distribution We want to answer the following question: Most recent common ancestor: How many generations back is the most recent common ancestor of two present-day genes? Assume that time is discrete, measured in generations. Above we saw that the binomial distribution can be used to compute the probability of having k successes in a repeated probabilistic experiment. Note that we can use the geometric distribution to compute the probability that we have to repeat the experiment k times to get a first success. (The exponential distribution is used for continuous time.) Let X i, i = 1, 2... be a series of i.i.d. probablistic experiments with a probability p of success (1) and probability 1 p of failure (0). (In population genetics, each experiment is going back one generation and success would be finding a common ancestor of two present-day genes.) Let T be the waiting time until a first success, that is: The geometric distribution implies: T = min{i X i = 1, i = 1, 2,... }. P (T = j) = (1 p) j 1 p,

7 76 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, 2011 based on j 1 failures followed by 1 success. We will use T Geo(p) to indicate that T is geometrically distributed with parameter p. Some simple properties: E(T ) = 1 p and V ar(t ) = 1 p p 2. For t 2 > t 1 we have: P (T > t 2 T > t 1 ) = P (T > t 2 t 1 ). 6.6 Coalescence of two genes With the help of the geometric distribution we will answer the question posed earlier: Coalescence time for two genes: Consider a haploid model with 2N genes. What is the waiting time T 2 for the most recent common ancestor (MRCA) of two present-day genes i and j? In other words, when do i and j coalesce? +,-.%/0%!%"'1%2%!"#$%&'%()*% Consider two present-day genes i and j.!" #" The probability that T 2 = 1 equals 1 2N. To see this, note that one gene can choose its parent freely and then the other must choose the same parent, with probability 1 2N. The probability that i and j have different parents is 1 1 2N. Under the discrete-time coalescent model of haploid reproduction of size 2N, we have the following result: Lemma (Coalescence time for two genes) The probability that two lineages find a common ancestor t generations back is P (T 2 = t) = ( 1 1 ) t 1 1 2N 2N. Proof: The choice of parents is independent between generations. In the first t 1 generations back we always choose different parents with probability (1 1 2N ) and then in the last generation we choose the same parent with probability 1 2N. 6.7 Coalescence of a sample of k genes Consider the discrete-time coalescent model of haploid reproduction of size 2N.

8 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, Coalescence time for k genes: What is the coalescence time of k genes, that is, how many generations ago did the most recent common ancestor of all k genes exist? The probability that k genes have k different parents is given by if k 2N. 1 2N 1 2N 2N 2 k 1 2N (k 1) = (1 i 2N 2N 2N ) k 1 = 1 i=1 i 2N + O( 1 N 2 ) = 1 1 ( ) k 1 2 2N, i=1 ( ) k 1 2 2N + O( 1 N 2 ) Hence, the probability of a coalescence event in one generation is ( ( ) ) ( ) k 1 k = 2 2N 2 2N. In consequence, the probability that two genes out of the k genes find a common ancestor T k = j, j = 1, 2,..., generations ago is ( ( ) ) k 1 j 1 ( ) ( ( ) k k 1 P (T k = j) 1 2) 2 2N 2 2N Geo. 2N Different time epochs in a coalescent tree: The coalescence time for k genes is given by T k + T k T 2, where all terms are independent and geometrically distributed. 6.8 Continuous time coalescent model So far, we have considered time to be a discrete quantity measured in generations. We will now consider an extension of the Wright-Fischer model in which time is continuous and is scaled so that one unit of time corresponds to the average time for two genes to find a common ancestor, which is 2N. A continuous time genealogy with time in units of generations (left) and in units of continuous time (right):

9 78 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, 2011 Let t = j 2N be continuous time, where j is time measured in generations. We can convert from continuous time back to generations using j = 2N t. The (continuous) waiting time Tk c for k genes to have k 1 ancestors is exponentially distributed with parameter ( k 2), that is, (( )) k P (Tk c t) = 1 2)t e (k Exp. 2 Exponential distribution Recall that the exponential distribution describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate. The cumulative distribution function is given by P (T t) = 1 e λt, where λ is called the rate parameter. The function looks like this: (Source: cdf.svg/1000px-exponential cdf.svg.png) 6.9 Algorithm for sampling genealogies The following algorithm stochastically samples possible genealogies for genes: Algorithm (Sample genealogy for n genes)

10 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, Start with k = n genes while k > 1 do Choose waiting time Tk c for next event, according to Exp(( k 2) ) Choose random pair of genes i and j with 1 i < j k, uniformly from ( k 2) possible pairs Merge i and j into one gene Set k = k Calculating simple quantities on a coalescent tree Consider this genealogy (with n = 5): The height H n of the tree on n genes is the sum of the time epochs T j while there are j = n, n 1,..., 2 ancestors. The mean height is easier to compute: E(H n ) = n E(T j ) = 2 j=2 n j=2 ( 1 j(j 1) = ). n Hence, the expected height of the tree grows toward 2 for increasing n (scaled in 2N generations). This leads to a slightly counterintuitive fact: While the expected time for 2 genes to find their most recent common ancestor (MRCA) is 1, the time for n genes to find their MRCA is at most Genealogies with sequences We now want to extend the Wright-Fischer model so as to take sequences into account. To do so, consider each gene to be a DNA sequence. When a gene is copied from one generation to the next, then a mutation may occur according to a given mutational model. Copies of a gene that have different mutations are called different alleles of the gene. Example: Here, genes 4 and 5 have the same allele, while all others differ.

11 80 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, 2011 We will assume that mutations are selectively neutral, that is, they do not influence the probability with which genes are chosen to be copied from one generation to the next. (Hence, we keep the assumption that all genes are equally fit.) There are two different models for sequences: Finite sites model: In a simple finite sites model, all genes have the same finite length and are aligned without gaps. Each site is able to mutate zero, one or more times. Infinite sites model: In this model, a mutation in an existing allele will always create a new allele that is different to all existing ones. Any given mutation may only occur once, i.e. sequences must have infinitely many different positions to allow this. While the finite sites model is more realistic, the infinite sites model is in many cases much easier to calculate with. The infinite sites model is applicable to populations in which there are a low mutation rate and long sequences, where most of the sites in the sequences are constant and the others usually experience only one mutation. Example of infinite sites data: gene \ site a b d c 6.12 Wright-Fischer with mutations Assume that we are given a mutation probability u. This is the probability with which a gene will mutate once while being copied from parent to offspring. Consider the Wright-Fischer model for 2N haploid genes and the coalescent process. We modify this so as to incorporate mutations, as follows: In the extended model, we first decide which parent to choose for a given gene i and then decide whether to copy the gene without error, with probability 1 u, or to introduce a mutation, with probability u. Under the finite sites model, the mutation is applied to one of the given sites. Under the infinite sites model, the mutation is applied to a new site that has not yet been mutated.

12 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, The effective population size The Wright-Fischer model is only a very simple approximation of reality. In particular, reproductive structure due to geographic proximity or e.g. social constraints will result in unrealistically long coalescence times, because such restrictions result in a decreased effective population size of the model. For example, many human genes have a most recent common ancestor (MRCA) that is less than years ago. If we count one generation as 20 years, then: Generations = = = time to MRCA. For the expected time for present-day genes to coalesce we have: Hence 4N = and so N E(H n ) = 2 2N = 4N generations. In other words, if we take N = to be the population size for humans then the coalescence times for many human genes are predicted by the Wright-Fischer model correctly. The population size that one has to use to get good results is called the effective population size Summary There is renewed interest in population genetics due to ongoing sequencing of the whole genomes of large populations. The Wright-Fischer model provides a dynamic description of the evolution of an idealized population and the transmission of genes from one generation to the next. The coalescent process allows one to compute statistical features of interest, such as mean time to the most recent common ancestor of two genes, by going backward in time. The Wright-Fischer model does not take geographic or social structure into account and is thus unrealistically simple. However, this can be compensated, in part, by employing an adjusted effective population size, which can be much smaller than the true population size.

6 Introduction to Population Genetics

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