Introduction To Natural Selection

Introduction to NaturalSelectionRyan HernandezTim O’Connor1

Goals Learn about the population genetics of naturalselection How to write a simple simulation with naturalselection2

Basic BiologyFunctional non-coding ’genetrans-regulatory region(enhancers)Un-Translated Regions (UTRs)cis-regulatory region(promoters: transcription factor binding sites)noncoding RNA:snoRNAs, siRNAs,piRNAs, longncRNAs“Promoter”3 Overall, 2% of the humangenome is protein coding 5% of genome is obviouslyfunctional 80% of genome has “functionalactivity”

Life CycleSexualselectionParentsGametes(sperm & es4Compatibilityselection

Modern Human Genomics:A case for rare variants?1.1 10-8 6 109 66 [muts / person]66[muts/p] 130M [p/y] 3B [bp]2.86 muts/bp/yr5

Sequencing DataChromosomeSNP 1SNP 2SNP 3SNP 4SNP 5SNP 61ACAGCC2ATGACT3GTGATT4ACGACT# Pairwisedifferences343333! average pairwise diversity6

Sequencing DataChromosomeSNP 1SNP 2SNP 3SNP 4SNP 5SNP 61ACAGCC2ATGACT3GTGATT4ACGACT# Pairwisedifferences343333# Compared666666! average pairwise diversity7

Sequencing DataChromosomeSNP 1SNP 2SNP 3SNP 4SNP 5SNP 61ACAGCC2ATGACT3GTGATT4ACGACT# Pairwisedifferences343333# Compared6666660.50.50.5Avg.Pairwise Diff0.5 0.67 0.5Number of variants: 6 SNPsDiversity (π): 3.1667/L8

Sequencing DataSNP 1SNP 2SNP 3SNP 4SNP 5SNP .250.50.250.250.250.25MAF 51Numberof SNPsChromosome52.5091/42/5Minor Allele Frequency

Sequencing DataSNP 1SNP 2SNP 3SNP 4SNP 5SNP .250.50.250.250.250.25MAF 51Fractionof SNPsChromosome0.90.450101/42/5Minor Allele Frequency

Sequencing DataChromosomeSNP 1SNP 2SNP 3SNP 4SNP 5SNP 61ACAGCC2ATGACT3GTGATT4ACGACTChimpACAGCT11

Sequencing DataChromosomeSNP 1SNP 2SNP 3SNP 4SNP 5SNP 61ACAGCC2ATGACT3GTGATT4ACGACTChimpACAGCT12

Sequencing DataSNP 1SNP 2SNP 3SNP 4SNP 5SNP nt123311Proportionof SNPsChromosome130.50.3330.1670123Derived frequency in sampleSite-FrequencySpectrum (SFS)

Site-Frequency Spectrum1*CA*T *TC*G *AA G*C *G *A *TC*A *G *G *CT2C ATTT G A G A C G AT C A G G C T3C G TTT G A G A C G AT4C AT C G A G A C G AoutgroupTTA C C C A G G A G AT*A *T *AAT AT A G G C C AT AT C A G G C TAT AA C G C ATATTT non-coding synonymous* - Substitution betweenspecies nonsynonymous14

Site-Frequency Spectrum0.50.40.30.20.10.0proportion of SNPsThe proportion of SNPs at each frequency ina sample of chromosomes.1152345678910derived count 15in sample of 12 chrs.11

Site-Frequency Spectrum0.50.40.10.20.3Characteristicsignature of rapidadaptation116Rufi (rice)Indica (rice)Japonica (rice)0.0proportion of SNPsSNMAfAm (Human)Ch (RheMac)In (RheMac)2345678910derived count 16in sample of 12 chrs.11

Population Genetics Imagine a population of diploid individuals 17PQRPrinciples of random mating: Any two individuals are equally likely to mate andreproduce to populate the next generation. Either chromosome is equally likely to be passed on.

Hardy-WeinbergPrinciple Assumptions: Diploid organismSexual reproductionNon-overlapping generationsOnly two allelesGodfrey H. Hardy:1877-1947Wilhelm Weinberg:1862-1937 Identical frequencies inmales/females Infinite population sizeNo migrationNo mutationNo natural selectionRandom matingConclusion 1:2P pBoth allele AND genotype frequencies willQ 2p(1-p)remain constant at HWE generation after2R (1-p)18generation. forever!

Hardy-Weinberg PrincipleImagine a population of diploid individualsA AA aa aQR 0.10.00.88110.60.81.0P0.20.4AAAaaa0.0frequency 246Generation19810

Hardy-Weinberg PrincipleImagine a population of diploid individuals2a aA aA Ap 0.3025QPR 2p(1p) 0.495 0.10.40.5(1 p)2 0.2025p P Q/2 0.551.0 0.60.40.00.2frequency0.8AAAaaa2 20Conclusion 2:46Generation810A single round of random mating will return thepopulation to HWE frequencies!

Hardy-WeinbergPrinciple Assumptions: Diploid organismSexual reproductionNon-overlapping generationsOnly two allelesRandom mating21Godfrey H. Hardy:1877-1947Wilhelm Weinberg:1862-1937 Identical frequencies inmales/females Infinite population sizeNo migrationNo mutationNo natural selection

Hardy-Weinberg Equilibrium22

Hardy-Weinberg Equilibrium23Graham Coop

Genetic Drift In finite populations, allele frequencies can and do changeover time. In fact, EVERY genetic variant will either be lost from thepopulation (p 0) or fixed in the population (p 1) some timein the future. The most common model for finite populations is theWright-Fisher model. This model makes explicit use of the binomial distribution.24

Bernoulli DistributionJacob Bernoulli1655-1705 One of the simplest probability distributions A discrete probability distribution Classic example: tossing a coin If a coin toss comes up heads with probability p, itresults in tails with probability 1-p. If X is a Bernoulli Random(Variable, x is an observation we write:f (x p) p1pif x 1if x 0The Expected Value is E[X] p, and the Variance is V[X] p(1-p).25

Binomial Distribution We introduced the Bernoulli Distribution, where weimagine a coin flip resulting in heads with probability p. But if we flipped the coin N times, how many headswould we expect? What is the probability that we get heads all N times? The number of “successes” in a fixed number of trials isdescribed by the Binomial Distribution. Written out, if the probability of each success is p, thenthe probability we observe j successes in N trials is: N jP (j N, p) p (1jp)26N NN!j; jj!(N j)!

Binomial Mean and Variance The mean of a Binomial Random Variable is: E[J] Np With variance: V[J] p(1-p)/N27

Wright-Fisher ModelSewall Wright:1889-1988 28Sir Ronald Fisher1890-1962Suppose a population of N individuals.Let X(t) be the #chromosomes carrying an allele A in generation t: N jN jp (1 p)P (X(t 1) j X(t) i) j j N jNiN i Bin(j N, i/N ) jNN

Wright-Fisher Model A simple R function to simulation genetic drift:WF function(N, p, G){t array(,dim G);t[1] p;for(i in 2:G){t[i] rbinom(1,N,t[i-1])/N;}return(t);} Run it in R using:f WF(100, 0.5, 200)plot(f)29

000.8500.4Allele frequency0.4Allele frequency0.400.81502000.4100150Allele frequency501000.80500.4Allele frequency0.4Allele frequency00.00.00.0Allele frequency0.00.00.00.80.80.40.8Allele frequency0.4Allele frequency0.4Allele frequencyWright-Fisher ion100Generation100Generation150200150200150200

Demographic Effects What do you think will happen if apopulation grows? Or shrinks?31

Wright-Fisher ModelSewall Wright:1889-1988 32Sir Ronald Fisher1890-1962Suppose a population of N individuals.Let X(t) be the #chromosomes carrying an allele A in generation t: N jN jp (1 p)P (X(t 1) j X(t) i) j j N jNiN i Bin(j N, i/N ) jNN

Wright-Fisher Model A simple R function to simulation genetic drift:WFdemog function(N, p, G, Gd, v){t array(,dim G);t[1] p;for(i in 2:G){if(i Gd){N N*v;}t[i] rbinom(1,N,t[i-1])/N;}return(t);}33

ation0505050100100100150150150Run it using: WFdemog(100, 0.5, 200, 50, 100)Generation2000.0 0.2 0.4 0.6 0.8 1.0200Allele frequency1500.0 0.2 0.4 0.6 0.8 1.050100Allele frequency0500.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequency00.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequencyWright-Fisher Model with 0

ation0505050100100100150150150Run it using: WFdemog(100, 0.5, 200, 50, 0.1)Generation2000.0 0.2 0.4 0.6 0.8 1.0200Allele frequency1500.0 0.2 0.4 0.6 0.8 1.050100Allele frequency0500.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequency00.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequency0.0 0.2 0.4 0.6 0.8 1.0Allele frequencyWright-Fisher Model with 200

Hardy-Weinberg PrincipleAssumptions: Diploid organismSexual reproductionNon-overlapping generationsOnly two allelesRandom mating Identical frequencies inmales/females Infinite population sizeNo migrationNo mutationNo natural selectionWhat happens when we allow natural selection to occur?Alleles change frequency!36

Natural Selection Usually parameterized in terms of a dominancecoefficient (h), and a selection coefficient (s), withwildtype fitness set to 1:GenotypeFrequencyFitnessAAp21Aa2pq1 hsaaq21 s h 1 is completely dominant h 0 is completely recessive h 0.5 is “genic” selection, or “codominance”, or“additive” fitness37

Natural SelectionGenotypeFrequencyFitnessAAp21Aa2pq1 hsaaq21 s How do we model the change in allele frequencies? What is fitness?! Refers to the average number of offspringindividuals with a particular genotype will have. Wild-type individuals have on average 1 offspring,while homozygous derived individuals have onaverage 1 s offspring.38

Natural SelectionGenotypeFrequencyFitnessAAp21Aa2pq1 hsaaq21 s In this case, s is the absolute fitness. If the population size is fixed, then we need toconsider relative fitness. That is, how fit is an individual genotype relative tothe population. For this, we need to know average population fitness!2392w̄ p (1) 2pq(1 hs) q (1 s) 1 sq(2hp q)

Natural SelectionGenotypeFrequencyFitnessAAp21Aa2pq1 hsaaq21 s The expected frequency in the next generation (q’) isthen the density of offspring produced by carriers ofthe derived allele divided by the population fitness:2q (1 s) pq(1 hs)q 1 sq(2hp q)040

Natural Selection Trajectory of selected allele with various selection0.20.40.60.80.50.30.20.10.050.0Frequency of A allele1.0coefficients under genic selection (h 0.5) in an“infinite” population041204060Generation80100

Hardy-Weinberg PrincipleAssumptions: Diploid organismSexual reproductionNon-overlapping generationsOnly two allelesRandom mating Identical frequencies inmales/females Infinite population sizeNo migrationNo mutationNo natural selectionWhat happens with natural selection in a finite population? Directional selection AND drift!42

Simulating Natural Selection First write an R function for the change in allelefrequencies:fitfreq function(q, h, s){p 1-q;return((q 2*(1 s) p*q*(1 h*s))/( 1 s*q*(2*h*p q)));} Now use this in an updated WF simulator:WF.sel function(N, q, h, s, G){t array(,dim G);t[1] N*q;for(i in 2:G){t[i] rbinom(1,N,fitfreq(t[i-1]/N, h, s));}return(t);}43

Natural SelectionN 100; s 0.1; h 0.50.040608010004060GenerationGeneration50 simulations100 y200.8200.800440.4Frequency0.40.0Frequency0.810 simulations0.81 simulations204060Generation801000204060Generation

0.4 Simulated(1 exp( s))(1 exp( Ns)0.3 0.2 0.1Fixation Probability0.5Natural Selection 0.0 0.1 0.00.10.20.3selection coefficient (s)0.40.5 Estimating the probability of fixation of a new mutation(p0 1/N) 5000 simulations: N 100; h 0.5 Pr(Fixation s 0, p ) p !!4500

Natural SelectionTime-course data from artificial selection/ancient DNA Let’s estimate some selection coefficients! Given 2 alleles at a locus with frequencies pand q0, andfitnesses w1 and w2 (with w the population-wide fitness).0 Expected freq. in next generation is: p p’ p w /w. We can then pwrite:p w /w p w 11q1 01q0 w2 /w 01q0w201for any generation t: Using induction, you could prove 46ptp0 w1 /w qtq0 w2 /wp0q0w1w2t

Natural Selectionlog of this equation: Taking the natural logptqt logw1w2t logp0q0 Which is now a linear function of t, the number ofgenerations. 47Therefore, the ratio of the fitnesses w1/w2 eslope

Natural Selection Experiment: Set up a population of bacteria in achemostat, and let them reproduce. Sample roughly every 5 generations. A slope of 0.139 implies:15% fitness advantageover allele q!4 3 2 0 (simulated with 20% advantage)48 12 0 Assume w 1. Thus, allele p has a5 1.15ln(p/q)0.139w1 eslope 0.1395101520Generation253035

Summary Hardy-Weinberg Equilibrium requires manyassumptions, all of which are routinely violated innatural populations. Nevertheless, the vast majority of variants are in HWE. Deviations almost always due to technical artifacts! Simulating Wright-Fisher models is easy! Natural selection changes the expected allele frequencyin the next generation. But drift still acts in finite populations!49

Life Cycle 4 Parents Gametes (sperm & eggs) Zygotes Adults Gametic selection Compatibility selection Viability selection Sexual selection