Genetic Variation in Finite Populations

Size: px
Start display at page:

Download "Genetic Variation in Finite Populations"

Transcription

1 Genetic Variation in Finite Populations The amount of genetic variation found in a population is influenced by two opposing forces: mutation and genetic drift. 1 Mutation tends to increase variation. 2 Genetic drift tends to reduce variation. In particular, if both the mutation rate and the effective population size are relatively stable, then the amount of genetic variation will tend towards an equilibrium known as mutation-drift balance at which the rate at which variation is lost through drift is equal to the rate at which new variation is created by mutation. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

2 Multilocus Surveys Reveal Limited Variation in Nucleotide Diversity ( π 0.1) Source: Leffler et al. (2012): Revisiting an Old Riddle: What Determines Genetic Diversity Levels within Species? Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

3 Mutation-Drift Balance and Identity by Descent Mutation and drift have opposing effects on the probabilities that individuals are identical by descent (Cotterman 1940, Malecot 1941). 1 We say that two haploid individuals are identical in state at a locus if they carry the same allele. 2 We say that two haploid individuals are identical by descent at a locus if they share the same allele and if they inherited that allele without mutation (or recombination) from their most recent common ancestor. Figure: Individuals can be identical by state even when they are not identical by descent (homoplasy). A1 A 1 A 2 A1 A2 A2 A2 Identity by state A1 A1 A1 Identity by descent Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

4 Suppose that we sample two chromosomes at random from generation t and let F t be the probability that they are identical by descent. We can derive a recursive equation relating F t+1 to F t by considering the parentage of the sampled individuals. For simplicity, we will make the following assumptions: 1 The population is diploid, with coalescent effective population size N e. 2 Mutation is governed by the infinite-allele model (IAM), which assumes that every mutation generates a unique allele (no back mutation). 3 The mutation rate is µ per chromosome per generation. In that case, F t+1 = 1 (1 µ) 2 2N } e {{} same parent ( N e ) F t (1 µ) 2. } {{ } different parents Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

5 As t increases, these probabilities tend to a limit F t F, which is the probability of identity by descent at equilibrium. This quantity satisfies the following equation: F = 1 ( (1 µ) ) 2N e 2N F (1 µ) 2. e Rearranging gives { ( F ) } (1 µ) 2 = 1 (1 µ) 2 2N e 2N e which can then be solved for F F = 1 1 2N e (1 µ) 2 ( ) N e (1 µ) 2 Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

6 If we assume that µ 1, then, at equilibrium, the probability of identity by descent in a diploid population is given by the following approximate expression Identity by descent at mutation-drift equilibrium in the IAM F N = 1 eµ 1 + Θ. F only depends on the parameter Θ = 4N eµ (population mutation rate). Increasing µ reduces F because individuals are more likely to have inherited alleles that are mutated from their ancestral state. Increasing N e reduces F because pairs of randomly sampled individuals are less likely to be closely related in a large population than in a small population. In other words, genetic drift reduces variation by increasing the relatedness of the members of a population. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

7 Since, in a randomly-mating population, the heterozygosity H is simply equal to 1 F, we also obtain the following classical result: Heterozygosity at mutation-drift equilibrium in the IAM H Θ 1 + Θ. Competing rates interpretation: As we trace two lineages backwards in time, there are two possible events The two lineages coalesce at rate 1/2N e; One of the lineages mutates, at total rate 2µ. The two chromosomes will carry different alleles if one of the lineages experiences a mutation before the two coalesce. This occurs with probability P(mutation first) = 2µ 2µ + 1/2N e = 4Neµ 1 + 4N eµ = Θ 1 + Θ. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

8 Example: Coyne (1976) detected 23 electrophoretically distinguishable alleles at the xanthine dehydrogenase locus in a sample of 60 D. persimilis chromosomes with the following frequencies: p 1 = p 2 = = p 18 = 1/60 (singletons) p 19 = p 20 = p 21 = 1/30 p 22 = 1/15 p 23 = 8/15 We can use this data to estimate both the probability of identity by descent at this locus and the population mutation rate Θ: ˆF = 23 i=1 p 2 i ˆΘ = 1 ˆF ˆF 2.37 However, without additional information, we cannot separately estimate µ and N e. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

9 Mutation-Drift Balance for Microsatellite Loci For certain kinds of loci, the infinite-alleles model is unsuitable and these predictions need to be modified. This will often be true for example of tandemly-repeated DNA sequences such as microsatellite loci. Microsatellite repeats are 2-7 bp in length. The number of repeats can vary greatly between individuals. These loci tend to mutate at very high rates and homoplasy may be common. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

10 Replication slippage leads to changes in copy number Replication slippage occurs when the parent and daughter strands partially separate during replication and then incorrectly re-anneal. Slippage usually leads to a gain or a loss of a single repeat, although larger changes sometimes occur. Mutation rates can be on the order of 1 event per 1000 generations. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

11 Copy-number change in microsatellite loci is often modeled using the stepwise mutation model (SMM), which assumes that the number of repeats can only increase or decrease by one per mutation event. For this model, Ohta & Kimura (1973) showed that Heterozygosity at mutation-drift equilibrium in the SMM H = 1 1 (1 + 2Θ) 1/2 The equilibrium heterozygosity under the SMM is less than that under the IAM. This prediction ignores the possibility of copy number changes involving more than one repeat, which may be common at some loci. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

12 Mutation-Drift Balance in the Infinite Sites Model The infinite alleles model was useful in the pre-sequencing era when allelic variation could only be discriminated using biochemical means. However, to handle DNA sequence data, we need a more refined model. The infinite sites model (ISM) was introduced by Kimura (1969). It assumes that there are infinitely many sites, each of which is equally likely to mutate and that no site mutates more than once. This simplification is reasonable if the mutation rate per site is low and the sequences being analyzed are not too distantly related, i.e., for intraspecific polymorphism, but not for interspecific divergence. With the ISM, we can ask questions about the number of segregating sites and their frequencies at mutation-drift balance. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

13 Suppose that n chromosomes are sampled from a population with coalescent effective population size N e and let S n be the number of segregating sites. Then Expected number of segregating sites at equilibrium under the ISM n 1 1 E[S n] = Θ i. i=1 Here Θ = 4N eµ, where µ is the locus-wide mutation rate of the region sequenced. When n is large, E[S n] Θ log(n). This grows very slowly with n, meaning that very large sample sizes will often be needed to discover new segregating sites, e.g., E[S 10] 2.83Θ, E[S 100] 5.18Θ, E[S 1000] 7.48Θ, E[S 10000] 9.79Θ Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

14 We can turn this last result into an estimator of Θ using the method of moments. Watterson s estimator / n 1 1 Θ W = S n i. i=1 Θ W is unbiased and asymptotically normal as n. However, the variance of the estimator is fairly large and does not go to 0 as n. Nonetheless, Θ W is sometimes useful when estimates are needed on the fly, e.g., migrate-n uses Θ W to estimate initial effective population sizes that are then refined through more computationally intensive procedures. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

15 The nucleotide diversity of a locus is defined to be the probability that two randomly chosen individuals differ at a randomly chosen site within that locus. This can be estimated from a sample of chromosomes, in which case the sample nucleotide diversity is usually denoted π. Equilibrium nucleotide diversity under the ISM E[π] = Θ 1 + Θ. Here Θ = 4N eµ, where µ is the mutation rate per site per generation. This result can be derived using the competing rates calculation that we saw previously. The expected value does not depend on the sample size, but its variance does. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

16 Example: Kreitman (1983) sequenced a 768 bp region of the ADH locus in 11 chromosomes sampled from D. melanogaster and found a total of 6 alleles containing 14 segregating sites, shown below. Name Ref T C C C C C T C C A C T A G Wa-S. T T. A A C Fl-1S. T T. A A C Af-S A Fr-S A Fl-2S G Ja-S G T. T. C A Fl-F G G T C T C C. Fr-F G G T C T C C. Wa-F G G T C T C C. Af-F G G T C T C C. Ja-F G.. A... G T C T C C. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

17 For this data set, the population mutation rate Θ = 4N eµ can be estimated in three ways, using S 11, F or π. Here we will estimate the per-site population mutation rates, so we will have to divide the first two estimates (which are locus-wide) by the number of sites: S 11 = 14 ˆΘ W = 1 ( ) F = ˆΘ F = 1 ( ) 1 F F π = ˆΘ π = π 1 π Remarks: With a mutation rate of µ = 10 8 mutations per site per generation, these calculations give estimates of N e 450, , 000. The variation between estimates has several possible sources: estimation error (noise), use of different information from the data, and model misspecification. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

18 Mutation-Drift Balance in Bi-allelic Models We can incorporate bi-allelic mutation into the Wright-Fisher model by making the following modifications: 1 Mutations occur only during reproduction and are independently transmitted to each offspring. 2 Each descendant of an A parent inherits a mutant a allele with probability v. Similarly, each descendant of an a parent inherits a mutant A allele with probability u. All other assumptions remain unchanged, i.e., non-overlapping generations, constant population size, binomial sampling and neutrality. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

19 Mutation changes the behavior of the Wright-Fisher model in several ways. It is no longer the case that the average frequency of allele A is constant. Instead, ] E [ p t = u (1 p t) v p t, which shows that A will tend to increase in frequency when rare and decrease in frequency when common. Although alleles may be transiently lost from the population, they will eventually be reintroduced by mutation. 1 N=10 3, µ= N=10 3, µ= N=10 4, µ=10 4 p 0.5 p 0.5 p Generation Generation Generation Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

20 Stationary Distribution of Allele Frequencies under Mutation-Drift Balance If the mutation rates are positive, then the allele frequencies will never settle into fixed values. On the other hand, it can be shown that the distribution of p t will converge to a limiting distribution which we call the stationary distribution. The limiting distribution does not depend on the initial frequency of A. It takes 4N e generations for the population to forget the initial frequency p = 0.01 t = t = t =100 Stationary behavior of the Wright-Fisher process: (N = 100, u = 0.02) density p = 0.5 p = 0.9 density density p p p Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

21 p Two interpretations of the stationary distribution: If we run a large number of independent simulations or experiments, then after a sufficient number of generations, the distribution of allele frequencies across trials will be given by the stationary distribution. Alternatively, if we run a single simulation or experiment for a very long time, then the proportion of time when the allele frequency is equal to p will be proportional to the stationary density of p. 1 Neutral Wright Fisher model 0.9 Ergodic behavior of the Wright-Fisher process: (N = 100, u = 0.02) Generation x 10 Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

22 Provided that N e is sufficiently large (N e 100), the stationary distribution at a neutral bi-allelic locus in a population with coalescent effective population size N e is given by a Beta distribution. Stationary distribution of allele frequencies The stationary distribution can be approximated by a Beta distribution with parameters 4N eu and 4N ev, which has the following density: π(p) = 1 C p4ne u 1 (1 p) 4Ne v 1, 0 p 1. In particular, if we sample the population at some sufficiently large time t, then the probability that the allele frequency p(t) at that time is between a and b will be approximately: P(a < p(t) < b) b a π(p)dp. Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

23 The stationary distribution reflects the competing effects of genetic drift, which eliminates variation, and mutation, which generates variation Nu = Nu = p p When 4N eu, 4N ev < 1, drift dominates mutation and the stationary distribution is bimodal, with peaks at the boundaries (one allele is common and one rare). When 4N eu, 4N ev > 1, mutation dominates drift and the stationary distribution is peaked about its mean (both alleles are common). Jay Taylor (ASU) Mutation-Drift Balance 25 Jan / 23

The Wright-Fisher Model and Genetic Drift

The Wright-Fisher Model and Genetic Drift The Wright-Fisher Model and Genetic Drift January 22, 2015 1 1 Hardy-Weinberg Equilibrium Our goal is to understand the dynamics of allele and genotype frequencies in an infinite, randomlymating population

More information

How robust are the predictions of the W-F Model?

How robust are the predictions of the W-F Model? How robust are the predictions of the W-F Model? As simplistic as the Wright-Fisher model may be, it accurately describes the behavior of many other models incorporating additional complexity. Many population

More information

6 Introduction to Population Genetics

6 Introduction to Population Genetics Grundlagen der Bioinformatik, SoSe 14, D. Huson, May 18, 2014 67 6 Introduction to Population Genetics This chapter is based on: J. Hein, M.H. Schierup and C. Wuif, Gene genealogies, variation and evolution,

More information

Computational Systems Biology: Biology X

Computational Systems Biology: Biology X Bud Mishra Room 1002, 715 Broadway, Courant Institute, NYU, New York, USA Human Population Genomics Outline 1 2 Damn the Human Genomes. Small initial populations; genes too distant; pestered with transposons;

More information

6 Introduction to Population Genetics

6 Introduction to Population Genetics 70 Grundlagen der Bioinformatik, SoSe 11, D. Huson, May 19, 2011 6 Introduction to Population Genetics This chapter is based on: J. Hein, M.H. Schierup and C. Wuif, Gene genealogies, variation and evolution,

More information

Selection and Population Genetics

Selection and Population Genetics Selection and Population Genetics Evolution by natural selection can occur when three conditions are satisfied: Variation within populations - individuals have different traits (phenotypes). height and

More information

Solutions to Even-Numbered Exercises to accompany An Introduction to Population Genetics: Theory and Applications Rasmus Nielsen Montgomery Slatkin

Solutions to Even-Numbered Exercises to accompany An Introduction to Population Genetics: Theory and Applications Rasmus Nielsen Montgomery Slatkin Solutions to Even-Numbered Exercises to accompany An Introduction to Population Genetics: Theory and Applications Rasmus Nielsen Montgomery Slatkin CHAPTER 1 1.2 The expected homozygosity, given allele

More information

NEUTRAL EVOLUTION IN ONE- AND TWO-LOCUS SYSTEMS

NEUTRAL EVOLUTION IN ONE- AND TWO-LOCUS SYSTEMS æ 2 NEUTRAL EVOLUTION IN ONE- AND TWO-LOCUS SYSTEMS 19 May 2014 Variations neither useful nor injurious would not be affected by natural selection, and would be left either a fluctuating element, as perhaps

More information

Population Structure

Population Structure Ch 4: Population Subdivision Population Structure v most natural populations exist across a landscape (or seascape) that is more or less divided into areas of suitable habitat v to the extent that populations

More information

Population Genetics I. Bio

Population Genetics I. Bio Population Genetics I. Bio5488-2018 Don Conrad dconrad@genetics.wustl.edu Why study population genetics? Functional Inference Demographic inference: History of mankind is written in our DNA. We can learn

More information

Gene Genealogies Coalescence Theory. Annabelle Haudry Glasgow, July 2009

Gene Genealogies Coalescence Theory. Annabelle Haudry Glasgow, July 2009 Gene Genealogies Coalescence Theory Annabelle Haudry Glasgow, July 2009 What could tell a gene genealogy? How much diversity in the population? Has the demographic size of the population changed? How?

More information

Effective population size and patterns of molecular evolution and variation

Effective population size and patterns of molecular evolution and variation FunDamental concepts in genetics Effective population size and patterns of molecular evolution and variation Brian Charlesworth Abstract The effective size of a population,, determines the rate of change

More information

Frequency Spectra and Inference in Population Genetics

Frequency Spectra and Inference in Population Genetics Frequency Spectra and Inference in Population Genetics Although coalescent models have come to play a central role in population genetics, there are some situations where genealogies may not lead to efficient

More information

URN MODELS: the Ewens Sampling Lemma

URN MODELS: the Ewens Sampling Lemma Department of Computer Science Brown University, Providence sorin@cs.brown.edu October 3, 2014 1 2 3 4 Mutation Mutation: typical values for parameters Equilibrium Probability of fixation 5 6 Ewens Sampling

More information

Processes of Evolution

Processes of Evolution 15 Processes of Evolution Forces of Evolution Concept 15.4 Selection Can Be Stabilizing, Directional, or Disruptive Natural selection can act on quantitative traits in three ways: Stabilizing selection

More information

Demography April 10, 2015

Demography April 10, 2015 Demography April 0, 205 Effective Population Size The Wright-Fisher model makes a number of strong assumptions which are clearly violated in many populations. For example, it is unlikely that any population

More information

Outline of lectures 3-6

Outline of lectures 3-6 GENOME 453 J. Felsenstein Evolutionary Genetics Autumn, 009 Population genetics Outline of lectures 3-6 1. We want to know what theory says about the reproduction of genotypes in a population. This results

More information

Notes on Population Genetics

Notes on Population Genetics Notes on Population Genetics Graham Coop 1 1 Department of Evolution and Ecology & Center for Population Biology, University of California, Davis. To whom correspondence should be addressed: gmcoop@ucdavis.edu

More information

Outline of lectures 3-6

Outline of lectures 3-6 GENOME 453 J. Felsenstein Evolutionary Genetics Autumn, 007 Population genetics Outline of lectures 3-6 1. We want to know what theory says about the reproduction of genotypes in a population. This results

More information

An introduction to mathematical modeling of the genealogical process of genes

An introduction to mathematical modeling of the genealogical process of genes An introduction to mathematical modeling of the genealogical process of genes Rikard Hellman Kandidatuppsats i matematisk statistik Bachelor Thesis in Mathematical Statistics Kandidatuppsats 2009:3 Matematisk

More information

Mutation, Selection, Gene Flow, Genetic Drift, and Nonrandom Mating Results in Evolution

Mutation, Selection, Gene Flow, Genetic Drift, and Nonrandom Mating Results in Evolution Mutation, Selection, Gene Flow, Genetic Drift, and Nonrandom Mating Results in Evolution 15.2 Intro In biology, evolution refers specifically to changes in the genetic makeup of populations over time.

More information

DISTRIBUTION OF NUCLEOTIDE DIFFERENCES BETWEEN TWO RANDOMLY CHOSEN CISTRONS 1N A F'INITE POPULATION'

DISTRIBUTION OF NUCLEOTIDE DIFFERENCES BETWEEN TWO RANDOMLY CHOSEN CISTRONS 1N A F'INITE POPULATION' DISTRIBUTION OF NUCLEOTIDE DIFFERENCES BETWEEN TWO RANDOMLY CHOSEN CISTRONS 1N A F'INITE POPULATION' WEN-HSIUNG LI Center for Demographic and Population Genetics, University of Texas Health Science Center,

More information

Problems for 3505 (2011)

Problems for 3505 (2011) Problems for 505 (2011) 1. In the simplex of genotype distributions x + y + z = 1, for two alleles, the Hardy- Weinberg distributions x = p 2, y = 2pq, z = q 2 (p + q = 1) are characterized by y 2 = 4xz.

More information

Outline of lectures 3-6

Outline of lectures 3-6 GENOME 453 J. Felsenstein Evolutionary Genetics Autumn, 013 Population genetics Outline of lectures 3-6 1. We ant to kno hat theory says about the reproduction of genotypes in a population. This results

More information

1. The diagram below shows two processes (A and B) involved in sexual reproduction in plants and animals.

1. The diagram below shows two processes (A and B) involved in sexual reproduction in plants and animals. 1. The diagram below shows two processes (A and B) involved in sexual reproduction in plants and animals. Which statement best explains how these processes often produce offspring that have traits not

More information

Estimating effective population size from samples of sequences: inefficiency of pairwise and segregating sites as compared to phylogenetic estimates

Estimating effective population size from samples of sequences: inefficiency of pairwise and segregating sites as compared to phylogenetic estimates Estimating effective population size from samples of sequences: inefficiency of pairwise and segregating sites as compared to phylogenetic estimates JOSEPH FELSENSTEIN Department of Genetics SK-50, University

More information

122 9 NEUTRALITY TESTS

122 9 NEUTRALITY TESTS 122 9 NEUTRALITY TESTS 9 Neutrality Tests Up to now, we calculated different things from various models and compared our findings with data. But to be able to state, with some quantifiable certainty, that

More information

Mathematical Population Genetics II

Mathematical Population Genetics II Mathematical Population Genetics II Lecture Notes Joachim Hermisson March 20, 2015 University of Vienna Mathematics Department Oskar-Morgenstern-Platz 1 1090 Vienna, Austria Copyright (c) 2013/14/15 Joachim

More information

Major questions of evolutionary genetics. Experimental tools of evolutionary genetics. Theoretical population genetics.

Major questions of evolutionary genetics. Experimental tools of evolutionary genetics. Theoretical population genetics. Evolutionary Genetics (for Encyclopedia of Biodiversity) Sergey Gavrilets Departments of Ecology and Evolutionary Biology and Mathematics, University of Tennessee, Knoxville, TN 37996-6 USA Evolutionary

More information

AEC 550 Conservation Genetics Lecture #2 Probability, Random mating, HW Expectations, & Genetic Diversity,

AEC 550 Conservation Genetics Lecture #2 Probability, Random mating, HW Expectations, & Genetic Diversity, AEC 550 Conservation Genetics Lecture #2 Probability, Random mating, HW Expectations, & Genetic Diversity, Today: Review Probability in Populatin Genetics Review basic statistics Population Definition

More information

Life Cycles, Meiosis and Genetic Variability24/02/2015 2:26 PM

Life Cycles, Meiosis and Genetic Variability24/02/2015 2:26 PM Life Cycles, Meiosis and Genetic Variability iclicker: 1. A chromosome just before mitosis contains two double stranded DNA molecules. 2. This replicated chromosome contains DNA from only one of your parents

More information

Mathematical Population Genetics II

Mathematical Population Genetics II Mathematical Population Genetics II Lecture Notes Joachim Hermisson June 9, 2018 University of Vienna Mathematics Department Oskar-Morgenstern-Platz 1 1090 Vienna, Austria Copyright (c) 2013/14/15/18 Joachim

More information

Introduction to population genetics & evolution

Introduction to population genetics & evolution Introduction to population genetics & evolution Course Organization Exam dates: Feb 19 March 1st Has everybody registered? Did you get the email with the exam schedule Summer seminar: Hot topics in Bioinformatics

More information

BIRS workshop Sept 6 11, 2009

BIRS workshop Sept 6 11, 2009 Diploid biparental Moran model with large offspring numbers and recombination Bjarki Eldon New mathematical challenges from molecular biology and genetics BIRS workshop Sept 6, 2009 Mendel s Laws The First

More information

Neutral Theory of Molecular Evolution

Neutral Theory of Molecular Evolution Neutral Theory of Molecular Evolution Kimura Nature (968) 7:64-66 King and Jukes Science (969) 64:788-798 (Non-Darwinian Evolution) Neutral Theory of Molecular Evolution Describes the source of variation

More information

Endowed with an Extra Sense : Mathematics and Evolution

Endowed with an Extra Sense : Mathematics and Evolution Endowed with an Extra Sense : Mathematics and Evolution Todd Parsons Laboratoire de Probabilités et Modèles Aléatoires - Université Pierre et Marie Curie Center for Interdisciplinary Research in Biology

More information

Q1) Explain how background selection and genetic hitchhiking could explain the positive correlation between genetic diversity and recombination rate.

Q1) Explain how background selection and genetic hitchhiking could explain the positive correlation between genetic diversity and recombination rate. OEB 242 Exam Practice Problems Answer Key Q1) Explain how background selection and genetic hitchhiking could explain the positive correlation between genetic diversity and recombination rate. First, recall

More information

Evolution in a spatial continuum

Evolution in a spatial continuum Evolution in a spatial continuum Drift, draft and structure Alison Etheridge University of Oxford Joint work with Nick Barton (Edinburgh) and Tom Kurtz (Wisconsin) New York, Sept. 2007 p.1 Kingman s Coalescent

More information

Population Genetics: a tutorial

Population Genetics: a tutorial : a tutorial Institute for Science and Technology Austria ThRaSh 2014 provides the basic mathematical foundation of evolutionary theory allows a better understanding of experiments allows the development

More information

The coalescent process

The coalescent process The coalescent process Introduction Random drift can be seen in several ways Forwards in time: variation in allele frequency Backwards in time: a process of inbreeding//coalescence Allele frequencies Random

More information

Mathematical models in population genetics II

Mathematical models in population genetics II Mathematical models in population genetics II Anand Bhaskar Evolutionary Biology and Theory of Computing Bootcamp January 1, 014 Quick recap Large discrete-time randomly mating Wright-Fisher population

More information

Darwinian Selection. Chapter 7 Selection I 12/5/14. v evolution vs. natural selection? v evolution. v natural selection

Darwinian Selection. Chapter 7 Selection I 12/5/14. v evolution vs. natural selection? v evolution. v natural selection Chapter 7 Selection I Selection in Haploids Selection in Diploids Mutation-Selection Balance Darwinian Selection v evolution vs. natural selection? v evolution ² descent with modification ² change in allele

More information

STAT 536: Genetic Statistics

STAT 536: Genetic Statistics STAT 536: Genetic Statistics Frequency Estimation Karin S. Dorman Department of Statistics Iowa State University August 28, 2006 Fundamental rules of genetics Law of Segregation a diploid parent is equally

More information

The Combinatorial Interpretation of Formulas in Coalescent Theory

The Combinatorial Interpretation of Formulas in Coalescent Theory The Combinatorial Interpretation of Formulas in Coalescent Theory John L. Spouge National Center for Biotechnology Information NLM, NIH, DHHS spouge@ncbi.nlm.nih.gov Bldg. A, Rm. N 0 NCBI, NLM, NIH Bethesda

More information

The neutral theory of molecular evolution

The neutral theory of molecular evolution The neutral theory of molecular evolution Introduction I didn t make a big deal of it in what we just went over, but in deriving the Jukes-Cantor equation I used the phrase substitution rate instead of

More information

Lecture 18 : Ewens sampling formula

Lecture 18 : Ewens sampling formula Lecture 8 : Ewens sampling formula MATH85K - Spring 00 Lecturer: Sebastien Roch References: [Dur08, Chapter.3]. Previous class In the previous lecture, we introduced Kingman s coalescent as a limit of

More information

CHAPTER 23 THE EVOLUTIONS OF POPULATIONS. Section C: Genetic Variation, the Substrate for Natural Selection

CHAPTER 23 THE EVOLUTIONS OF POPULATIONS. Section C: Genetic Variation, the Substrate for Natural Selection CHAPTER 23 THE EVOLUTIONS OF POPULATIONS Section C: Genetic Variation, the Substrate for Natural Selection 1. Genetic variation occurs within and between populations 2. Mutation and sexual recombination

More information

Population Genetics of Selection

Population Genetics of Selection Population Genetics of Selection Jay Taylor School of Mathematical and Statistical Sciences Arizona State University Jay Taylor (Arizona State University) Population Genetics of Selection 2009 1 / 50 Historical

More information

Darwinian Selection. Chapter 6 Natural Selection Basics 3/25/13. v evolution vs. natural selection? v evolution. v natural selection

Darwinian Selection. Chapter 6 Natural Selection Basics 3/25/13. v evolution vs. natural selection? v evolution. v natural selection Chapter 6 Natural Selection Basics Natural Selection Haploid Diploid, Sexual Results for a Diallelic Locus Fisher s Fundamental Theorem Darwinian Selection v evolution vs. natural selection? v evolution

More information

Microsatellite evolution in Adélie penguins

Microsatellite evolution in Adélie penguins Microsatellite evolution in Adélie penguins Bennet McComish School of Mathematics and Physics Microsatellites Tandem repeats of motifs up to 6bp, e.g. (AC) 6 = ACACACACACAC Length is highly polymorphic.

More information

that does not happen during mitosis?

that does not happen during mitosis? Review! What is a somatic cell?! What is a sex cell?! What is a haploid cell?! What is a diploid cell?! Why is cell division important?! What are the different types of cell division?! What are these useful

More information

A. Correct! Genetically a female is XX, and has 22 pairs of autosomes.

A. Correct! Genetically a female is XX, and has 22 pairs of autosomes. MCAT Biology - Problem Drill 08: Meiosis and Genetic Variability Question No. 1 of 10 1. A human female has pairs of autosomes and her sex chromosomes are. Question #01 (A) 22, XX. (B) 23, X. (C) 23, XX.

More information

Linking levels of selection with genetic modifiers

Linking levels of selection with genetic modifiers Linking levels of selection with genetic modifiers Sally Otto Department of Zoology & Biodiversity Research Centre University of British Columbia @sarperotto @sse_evolution @sse.evolution Sally Otto Department

More information

Theory a well supported testable explanation of phenomenon occurring in the natural world.

Theory a well supported testable explanation of phenomenon occurring in the natural world. Evolution Theory of Evolution Theory a well supported testable explanation of phenomenon occurring in the natural world. Evolution the process by which modern organisms changed over time from ancient common

More information

This course is about VARIATION: its causes, effects, and history.

This course is about VARIATION: its causes, effects, and history. This course is about VARIATION: its causes, effects, and history. For thousands of years, western thought had accepted the Platonic view that an object s ultimate reality was its essence or ideal type.

More information

Neutral behavior of shared polymorphism

Neutral behavior of shared polymorphism Proc. Natl. Acad. Sci. USA Vol. 94, pp. 7730 7734, July 1997 Colloquium Paper This paper was presented at a colloquium entitled Genetics and the Origin of Species, organized by Francisco J. Ayala (Co-chair)

More information

Observation: we continue to observe large amounts of genetic variation in natural populations

Observation: we continue to observe large amounts of genetic variation in natural populations MUTATION AND GENETIC VARIATION Observation: we continue to observe large amounts of genetic variation in natural populations Problem: How does this variation arise and how is it maintained. Here, we look

More information

Homework Assignment, Evolutionary Systems Biology, Spring Homework Part I: Phylogenetics:

Homework Assignment, Evolutionary Systems Biology, Spring Homework Part I: Phylogenetics: Homework Assignment, Evolutionary Systems Biology, Spring 2009. Homework Part I: Phylogenetics: Introduction. The objective of this assignment is to understand the basics of phylogenetic relationships

More information

2. Map genetic distance between markers

2. Map genetic distance between markers Chapter 5. Linkage Analysis Linkage is an important tool for the mapping of genetic loci and a method for mapping disease loci. With the availability of numerous DNA markers throughout the human genome,

More information

Notes for MCTP Week 2, 2014

Notes for MCTP Week 2, 2014 Notes for MCTP Week 2, 2014 Lecture 1: Biological background Evolutionary biology and population genetics are highly interdisciplinary areas of research, with many contributions being made from mathematics,

More information

Chapter 13 Meiosis and Sexual Reproduction

Chapter 13 Meiosis and Sexual Reproduction Biology 110 Sec. 11 J. Greg Doheny Chapter 13 Meiosis and Sexual Reproduction Quiz Questions: 1. What word do you use to describe a chromosome or gene allele that we inherit from our Mother? From our Father?

More information

Wright-Fisher Models, Approximations, and Minimum Increments of Evolution

Wright-Fisher Models, Approximations, and Minimum Increments of Evolution Wright-Fisher Models, Approximations, and Minimum Increments of Evolution William H. Press The University of Texas at Austin January 10, 2011 1 Introduction Wright-Fisher models [1] are idealized models

More information

Evolutionary change. Evolution and Diversity. Two British naturalists, one revolutionary idea. Darwin observed organisms in many environments

Evolutionary change. Evolution and Diversity. Two British naturalists, one revolutionary idea. Darwin observed organisms in many environments Evolutionary change Evolution and Diversity Ch 13 How populations evolve Organisms change over time In baby steps Species (including humans) are descended from other species Two British naturalists, one

More information

Australian bird data set comparison between Arlequin and other programs

Australian bird data set comparison between Arlequin and other programs Australian bird data set comparison between Arlequin and other programs Peter Beerli, Kevin Rowe March 7, 2006 1 Data set We used a data set of Australian birds in 5 populations. Kevin ran the program

More information

Statistical Genetics I: STAT/BIOST 550 Spring Quarter, 2014

Statistical Genetics I: STAT/BIOST 550 Spring Quarter, 2014 Overview - 1 Statistical Genetics I: STAT/BIOST 550 Spring Quarter, 2014 Elizabeth Thompson University of Washington Seattle, WA, USA MWF 8:30-9:20; THO 211 Web page: www.stat.washington.edu/ thompson/stat550/

More information

Evolutionary Theory. Sinauer Associates, Inc. Publishers Sunderland, Massachusetts U.S.A.

Evolutionary Theory. Sinauer Associates, Inc. Publishers Sunderland, Massachusetts U.S.A. Evolutionary Theory Mathematical and Conceptual Foundations Sean H. Rice Sinauer Associates, Inc. Publishers Sunderland, Massachusetts U.S.A. Contents Preface ix Introduction 1 CHAPTER 1 Selection on One

More information

Population Genetics. with implications for Linkage Disequilibrium. Chiara Sabatti, Human Genetics 6357a Gonda

Population Genetics. with implications for Linkage Disequilibrium. Chiara Sabatti, Human Genetics 6357a Gonda 1 Population Genetics with implications for Linkage Disequilibrium Chiara Sabatti, Human Genetics 6357a Gonda csabatti@mednet.ucla.edu 2 Hardy-Weinberg Hypotheses: infinite populations; no inbreeding;

More information

Parents can produce many types of offspring. Families will have resemblances, but no two are exactly alike. Why is that?

Parents can produce many types of offspring. Families will have resemblances, but no two are exactly alike. Why is that? Parents can produce many types of offspring Families will have resemblances, but no two are exactly alike. Why is that? Meiosis and Genetic Linkage Objectives Recognize the significance of meiosis to sexual

More information

Wald Lecture 2 My Work in Genetics with Jason Schweinsbreg

Wald Lecture 2 My Work in Genetics with Jason Schweinsbreg Wald Lecture 2 My Work in Genetics with Jason Schweinsbreg Rick Durrett Rick Durrett (Cornell) Genetics with Jason 1 / 42 The Problem Given a population of size N, how long does it take until τ k the first

More information

Statistical population genetics

Statistical population genetics Statistical population genetics Lecture 7: Infinite alleles model Xavier Didelot Dept of Statistics, Univ of Oxford didelot@stats.ox.ac.uk Slide 111 of 161 Infinite alleles model We now discuss the effect

More information

Inventory Model (Karlin and Taylor, Sec. 2.3)

Inventory Model (Karlin and Taylor, Sec. 2.3) stochnotes091108 Page 1 Markov Chain Models and Basic Computations Thursday, September 11, 2008 11:50 AM Homework 1 is posted, due Monday, September 22. Two more examples. Inventory Model (Karlin and Taylor,

More information

Lesson Overview Meiosis

Lesson Overview Meiosis 11.4 As geneticists in the early 1900s applied Mendel s laws, they wondered where genes might be located. They expected genes to be carried on structures inside the cell, but which structures? What cellular

More information

Conservation Genetics. Outline

Conservation Genetics. Outline Conservation Genetics The basis for an evolutionary conservation Outline Introduction to conservation genetics Genetic diversity and measurement Genetic consequences of small population size and extinction.

More information

Sequence evolution within populations under multiple types of mutation

Sequence evolution within populations under multiple types of mutation Proc. Natl. Acad. Sci. USA Vol. 83, pp. 427-431, January 1986 Genetics Sequence evolution within populations under multiple types of mutation (transposable elements/deleterious selection/phylogenies) G.

More information

8. Genetic Diversity

8. Genetic Diversity 8. Genetic Diversity Many ways to measure the diversity of a population: For any measure of diversity, we expect an estimate to be: when only one kind of object is present; low when >1 kind of objects

More information

Breeding Values and Inbreeding. Breeding Values and Inbreeding

Breeding Values and Inbreeding. Breeding Values and Inbreeding Breeding Values and Inbreeding Genotypic Values For the bi-allelic single locus case, we previously defined the mean genotypic (or equivalently the mean phenotypic values) to be a if genotype is A 2 A

More information

Introduction to Natural Selection. Ryan Hernandez Tim O Connor

Introduction to Natural Selection. Ryan Hernandez Tim O Connor Introduction to Natural Selection Ryan Hernandez Tim O Connor 1 Goals Learn about the population genetics of natural selection How to write a simple simulation with natural selection 2 Basic Biology genome

More information

Heterozygosity is variance. How Drift Affects Heterozygosity. Decay of heterozygosity in Buri s two experiments

Heterozygosity is variance. How Drift Affects Heterozygosity. Decay of heterozygosity in Buri s two experiments eterozygosity is variance ow Drift Affects eterozygosity Alan R Rogers September 17, 2014 Assumptions Random mating Allele A has frequency p N diploid individuals Let X 0,1, or 2) be the number of copies

More information

Lesson Overview Meiosis

Lesson Overview Meiosis 11.4 THINK ABOUT IT As geneticists in the early 1900s applied Mendel s laws, they wondered where genes might be located. They expected genes to be carried on structures inside the cell, but which structures?

More information

EVOLUTIONARY DYNAMICS AND THE EVOLUTION OF MULTIPLAYER COOPERATION IN A SUBDIVIDED POPULATION

EVOLUTIONARY DYNAMICS AND THE EVOLUTION OF MULTIPLAYER COOPERATION IN A SUBDIVIDED POPULATION Friday, July 27th, 11:00 EVOLUTIONARY DYNAMICS AND THE EVOLUTION OF MULTIPLAYER COOPERATION IN A SUBDIVIDED POPULATION Karan Pattni karanp@liverpool.ac.uk University of Liverpool Joint work with Prof.

More information

Build a STRUCTURAL concept map of has part starting with cell cycle and using all of the following: Metaphase Prophase Interphase Cell division phase

Build a STRUCTURAL concept map of has part starting with cell cycle and using all of the following: Metaphase Prophase Interphase Cell division phase Build a STRUCTURAL concept map of has part starting with cell cycle and using all of the following: Metaphase Prophase Interphase Cell division phase Telophase S phase G1 phase G2 phase Anaphase Cytokinesis

More information

ROBUST METHODS FOR ESTIMATING ALLELE FREQUENCIES SHU-PANG HUANG

ROBUST METHODS FOR ESTIMATING ALLELE FREQUENCIES SHU-PANG HUANG ROBUST METHODS FOR ESTIMATING ALLELE FREQUENCIES SHU-PANG HUANG May 30, 2001 ABSTRACT HUANG, SHU-PANG. ROBUST METHODS FOR ESTIMATING ALLELE FREQUENCIES (Advisor: Bruce S. Weir) The distribution of allele

More information

Tutorial on Theoretical Population Genetics

Tutorial on Theoretical Population Genetics Tutorial on Theoretical Population Genetics Joe Felsenstein Department of Genome Sciences and Department of Biology University of Washington, Seattle Tutorial on Theoretical Population Genetics p.1/40

More information

So in terms of conditional probability densities, we have by differentiating this relationship:

So in terms of conditional probability densities, we have by differentiating this relationship: Modeling with Finite State Markov Chains Tuesday, September 27, 2011 1:54 PM Homework 1 due Friday, September 30 at 2 PM. Office hours on 09/28: Only 11 AM-12 PM (not at 3 PM) Important side remark about

More information

(Write your name on every page. One point will be deducted for every page without your name!)

(Write your name on every page. One point will be deducted for every page without your name!) POPULATION GENETICS AND MICROEVOLUTIONARY THEORY FINAL EXAMINATION (Write your name on every page. One point will be deducted for every page without your name!) 1. Briefly define (5 points each): a) Average

More information

Stochastic Demography, Coalescents, and Effective Population Size

Stochastic Demography, Coalescents, and Effective Population Size Demography Stochastic Demography, Coalescents, and Effective Population Size Steve Krone University of Idaho Department of Mathematics & IBEST Demographic effects bottlenecks, expansion, fluctuating population

More information

Segregation versus mitotic recombination APPENDIX

Segregation versus mitotic recombination APPENDIX APPENDIX Waiting time until the first successful mutation The first time lag, T 1, is the waiting time until the first successful mutant appears, creating an Aa individual within a population composed

More information

Heredity Variation Genetics Meiosis

Heredity Variation Genetics Meiosis Genetics Warm Up Exercise: -Using your previous knowledge of genetics, determine what maternal genotype would most likely yield offspring with such characteristics. -Use the genotype that you came up with

More information

The Origin of Species

The Origin of Species The Origin of Species Introduction A species can be defined as a group of organisms whose members can breed and produce fertile offspring, but who do not produce fertile offspring with members of other

More information

MEIOSIS C H A P T E R 1 3

MEIOSIS C H A P T E R 1 3 MEIOSIS CHAPTER 13 CENTRAL DOGMA OF BIOLOGY DNA RNA Protein OFFSPRING ACQUIRE GENES FROM PARENTS Genes are segments of DNA that program specific traits. Genetic info is transmitted as specific sequences

More information

Lecture Notes: BIOL2007 Molecular Evolution

Lecture Notes: BIOL2007 Molecular Evolution Lecture Notes: BIOL2007 Molecular Evolution Kanchon Dasmahapatra (k.dasmahapatra@ucl.ac.uk) Introduction By now we all are familiar and understand, or think we understand, how evolution works on traits

More information

Mathematical modelling of Population Genetics: Daniel Bichener

Mathematical modelling of Population Genetics: Daniel Bichener Mathematical modelling of Population Genetics: Daniel Bichener Contents 1 Introduction 3 2 Haploid Genetics 4 2.1 Allele Frequencies......................... 4 2.2 Natural Selection in Discrete Time...............

More information

Genetic hitch-hiking in a subdivided population

Genetic hitch-hiking in a subdivided population Genet. Res., Camb. (1998), 71, pp. 155 160. With 3 figures. Printed in the United Kingdom 1998 Cambridge University Press 155 Genetic hitch-hiking in a subdivided population MONTGOMERY SLATKIN* AND THOMAS

More information

Match probabilities in a finite, subdivided population

Match probabilities in a finite, subdivided population Match probabilities in a finite, subdivided population Anna-Sapfo Malaspinas a, Montgomery Slatkin a, Yun S. Song b, a Department of Integrative Biology, University of California, Berkeley, CA 94720, USA

More information

Challenges when applying stochastic models to reconstruct the demographic history of populations.

Challenges when applying stochastic models to reconstruct the demographic history of populations. Challenges when applying stochastic models to reconstruct the demographic history of populations. Willy Rodríguez Institut de Mathématiques de Toulouse October 11, 2017 Outline 1 Introduction 2 Inverse

More information

I of a gene sampled from a randomly mating popdation,

I of a gene sampled from a randomly mating popdation, Copyright 0 1987 by the Genetics Society of America Average Number of Nucleotide Differences in a From a Single Subpopulation: A Test for Population Subdivision Curtis Strobeck Department of Zoology, University

More information

Full file at CHAPTER 2 Genetics

Full file at   CHAPTER 2 Genetics CHAPTER 2 Genetics MULTIPLE CHOICE 1. Chromosomes are a. small linear bodies. b. contained in cells. c. replicated during cell division. 2. A cross between true-breeding plants bearing yellow seeds produces

More information

genome a specific characteristic that varies from one individual to another gene the passing of traits from one generation to the next

genome a specific characteristic that varies from one individual to another gene the passing of traits from one generation to the next genetics the study of heredity heredity sequence of DNA that codes for a protein and thus determines a trait genome a specific characteristic that varies from one individual to another gene trait the passing

More information

Bustamante et al., Supplementary Nature Manuscript # 1 out of 9 Information #

Bustamante et al., Supplementary Nature Manuscript # 1 out of 9 Information # Bustamante et al., Supplementary Nature Manuscript # 1 out of 9 Details of PRF Methodology In the Poisson Random Field PRF) model, it is assumed that non-synonymous mutations at a given gene are either

More information

Supporting Information

Supporting Information Supporting Information Hammer et al. 10.1073/pnas.1109300108 SI Materials and Methods Two-Population Model. Estimating demographic parameters. For each pair of sub-saharan African populations we consider

More information