Understanding Genetic Drift: Key Concepts and Models
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University of California, Los Angeles**We aren't endorsed by this school
Course
EE BIOL 135
Subject
Biology
Date
Dec 12, 2024
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37
Uploaded by MegaKoalaPerson1287
EEB 135/235: Population geneticsDr. Kirk Lohmueller Department of Ecology and Evolutionary Biology April 18, 2023Genetic drift
Learning objectives• Define genetic drift & Wright Fisher model • Understand Markov chains • Describe how the population size influences the magnitude of genetic drift
What happens to genetic variation with finite population size?• Thus far, we have assumed the population is infinite in size. Allele frequencies do not change over time. • In finite populations, alleles can change in frequency. This is due to genetic drift.
Genetic drift•Randomchanges in allele frequency as a consequence of finite population size. • Drift occurs because, by chance, some individuals leave more/less offspring than other individuals. • This differs from natural selection: –With selection, some genotypes systematically produce more/less offspring than other individuals.Gpassive/random
Wright-Fisher model of genetic variation• Discrete generations • Non-overlapping generations • Individuals in the next generation are sampled (with replacement) from individuals in the current generation. • This random sampling allows for changes in allele frequency.
Wright-Fisher model in actionickedoutofbag3thenputbackinagainPtostimulaterandom#ofoffspring
Genetic drift
Discussion: What do you notice?MostLendoutatOor100.
010002000300040000.00.20.40.60.81.0GenerationsFrequency10 replicates overlaid on each otherAfter 4000 generations: 7 loci = aa 2 loci = AA 1 locus polymorphic Drift alone leads to a loss of all variation.. Driftasonlyevolutionary
Genetic drift is a Markov Chain• Markov Chain: –A sequence of random variables where the value at one time step depends only upon the one immediately before. Eallelef
Genetic drift is a Markov Chain• Markov Chain: –A sequence of random variables where the value at one time step depends only upon the one immediately before. • Let Xtdenote the value at time t: –X1-->X2-->X3-->X4....Xt–In a Markov Chain, the value of X4depends only on the value X3(and not on X2or X1)
More on Markov Chains• Example: –“The probability that X4= 1 given that X3 = 6” –More generally, • The probability of going from one state to another is called the “transition probability”. • The “transition probability” is the key feature of the Markov Chain.P(Xt+1=j|Xt=i)P(X4=1|X3=6)
Markov model of genetic drift• Adults produce a large pool of gametes which will be joined randomly to form children • Let iout of 2Nadult chromosomes have allele A. The the proportion p=i/2Ngametes will be type A• Then 2Ngametes are sampled at random from the pool to make up offspring in the next generation • Note, the pool of gametes is large enough that gametes from some adults can contribute to multiple childrencountofA Ndiploidindividuals=finitebutconstantpopulationsize
Transition probabilities for genetic drift• Given the model on the previous slide, • where = all the ways that 2Ngametes can be chosen to have exactly jgametes of type A. P(Xt+1=j|Xt=i)=2Nj!"#$%&pj(1−p)2N−j2Nj!"#$%&⊥alleleI↓Icountweregoingto
Transition probabilities for genetic drift• Given the model on the previous slide, • where = all the ways that 2Ngametes can be chosen to have exactly jgametes of type A. • This is the binomial distribution. • For a given allele count (i) in the present generation, you can calculate the probability of transitioning to any other allele count j (jcan range from 0 to 2N).P(Xt+1=j|Xt=i)=2Nj!"#$%&pj(1−p)2N−j2Nj!"#$%&
=(ii)-nchoose Ithi!n=some ojellybearelike4otherIblueR=#ofredjellybeams=zwant#ofwaystoarrangethe4jellybeam=6orderings*#ofwaystoordernthingsgiventhatRareonetype&(n-K)aresomeother
pi(1-p)2n-;ifwepick;ofonetypeofalleleshavetopickzo-;oftheotheralleepicpickAjtimes(1-pepicka2N-;time-for1specificordering(2)togetallorderings
Example:2n=4,X+=2,P=.5Gent+1outcomes?Xt+1=30,43=jP(X++=0(x+=2)=(b)(.5)(:5)"=-0625-zN=c,p=.S,j=0P(X+=+(x+=2)=(2)(.5)(.5)=25p(Xt+=2(xt=2)=(2)(.S)2(.5)=-375P(X++==3(xt=2)=(j) (.S))(.S)"=25P(X++=4)y+=2)=(4) (.5) "(.5)"=0625
=
18162636465666768696Pr(X(t+1)=j | X(t) = 40), 2N=1000.000.020.040.060.08Binomial transition probabilities12345678910Pr(X(t+1)=j | X(t) = 4), 2N=100.000.050.100.150.200.25i=4, 2N=10, p=0.4i=40, 2N=100, p=0.4fromprobofnextgengoingcount4tocountother#
Quantifying genetic drift• What is the expected change in allele frequency (p) from one generation to the next? • What is the variance of the change in allele frequency from one generation to the next? –Variance is the expected “spread” of change in allele frequency. –It quantifies the amount of “randomness”
Random variables (review)• Expected value of a random variable: –Is the average –Written as E[X] –Intuitively, it is each value that Xcan take on, weighted by the probability that Xtakes on that value
Random variables (review)• Expected value of a random variable: –Is the average –Written as E[X] –Intuitively, it is each value that Xcan take on, weighted by the probability that Xtakes on that value • Example: Throw a die. What is the expected outcome? –Values Xcan take on: 1,2,3,4,5,6 -> P(X=x)=1/6 E[X]=xP(X=x)x=16∑E[X]=1(1/ 6)+2(1/ 6)+3(1/ 6)+4(1/ 6)+5(1/ 6)+6(1/ 6)E[X]=3.5
Random variables (review)• Variance of a random variable: –Is the expected spread of the values –Written as V[X]=E[(X-E[X])2] –Also written as: V[X]=E[X2]-E[X]2
Random variables (review)• Variance of a random variable: –Is the expected spread of the values –Written as V[X]=E[(X-E[X])2] –Also written as: V[X]=E[X2]-E[X]2• Example: Flip a coin. Let X=1 if heads; X=0 if tails –Values Xcan take on: 0,1; P(X=1)=p; P(X=0)=1-pE[X]=xP(X=x)x=01∑=0(1−p)+1(p)=pE[X]2=p2E[X2]=x2P(X=x)x=01∑=02(1−p)+12(p)=pV[X]=p−p2=p(1−p)
• Let Xt+1be the count of Ain generation t+1. Let pt=Xt/2N. •Xt+1is binomially distributed with parameters ptand 2N.• Then, from what we just derived: • The expected allele frequency does not change over time. Change in allele frequency in WF model
• Let Xt+1be the count of Ain generation t+1. Let pt=Xt/2N. •Xt+1is binomially distributed with parameters ptand 2N.• Then, from what we just derived: • The expected allele frequency does not change over time. E[Xt+1]=np=2NXt2N!"#$%&=XtChange in allele frequency in WF model
• Let Xt+1be the count of Ain generation t+1. Let pt=Xt/2N. •Xt+1is binomially distributed with parameters ptand 2N.• Then, from what we just derived: • The expected allele frequency does not change over time. E[Xt+1]=np=2NXt2N!"#$%&=XtV[Xt+1]=np(1−p)=2NXt2N!"#$%&1−Xt2N!"#$%&=2Npt(1−pt)Change in allele frequency in WF model
• Let Xt+1be the count of Ain generation t+1. Let pt=Xt/2N. •Xt+1is binomially distributed with parameters ptand 2N.• Then, from what we just derived: • The expected allele frequency does not change over time. • But, alleles will fluctuate in frequency from generation to generation (genetic drift). • The degree of fluctuation is depended on 2N.• Eventually, pt+1=0 or pt+1=1. E[Xt+1]=np=2NXt2N!"#$%&=XtV[Xt+1]=np(1−p)=2NXt2N!"#$%&1−Xt2N!"#$%&=2Npt(1−pt)Change in allele frequency in WF model
• Quantities for the allele count Xt: • We can also find E[pt+1] and V[pt+1]: Change in allele frequency in WF modelE[pt+1]=EXt+12N!"#$%&=E[Xt+1]2N=Xt2N=ptXV[X]2Np(1−p)p(1−p)E[Xt+1]=np=2NXt2N!"#$%&=XtV[Xt+1]=np(1−p)=2NXt2N!"#$%&1−Xt2N!"#$%&=2Npt(1−pt)
• Quantities for the allele count Xt: • We can also find E[pt+1] and V[pt+1]: Change in allele frequency in WF modelE[pt+1]=EXt+12N!"#$%&=E[Xt+1]2N=Xt2N=ptV[pt+1]=VXt+12N!"#$%&=V[Xt+1]4N2=2Npt(1−pt)4N2=pt(1−pt)2NE[Xt+1]=np=2NXt2N!"#$%&=XtV[Xt+1]=np(1−p)=2NXt2N!"#$%&1−Xt2N!"#$%&=2Npt(1−pt)
So what?• Allele frequencies are not expectedto change with genetic drift. • However, they can and do! • Why? –With finite population size, there is a stochastic (i.e. random) component. –The variance measures how big that component is. • Variance in allele frequency change is determined by 2N. • As 2Ngets small, variance of pincreases. –Should see bigger changes in pover time in small populations!
Greater genetic drift with smaller population size• “Drunkard’s walk” analogy: –Randomly step to left and right –Eventually, may fall in the ditch (corresponds to loss of allele) –The size of the steps depends on the population size. Larger steps with smaller population size. –Thus, with smaller population size, fewer steps till loss.
Genetic drift leads to a loss of genetic diversity over time• Assume no new mutations • Once an allele hits count 0 or 2N, it stays there
What is the probability of fixation?•Fixation:All individuals carry the allele. The allele reaches a frequency of 1 in the population • In a neutral model, 1 copy of an allele will eventually take over and reach fixation • The probability that it is a particular allele is simply the allele frequency: –Why? –The higher the frequency, the greater probability of fixation. –P(fixation for a new mutation)=1/2N –Most new mutations will be lost! • Natural selection will shift this probability •
Take-home points• Genetic drift allows alleles to change in frequency due to random sampling. • Genetic drift as the only evolutionary force will lead to the loss of genetic diversity. • The degree of fluctuation of allele frequencies per generation depends on the population size. –More drift in smaller populations.
Take-home points• Genetic drift allows alleles to change in frequency due to random sampling. • Genetic drift as the only evolutionary force will lead to the loss of genetic diversity. • The degree of fluctuation of allele frequencies per generation depends on the population size. –More drift in smaller populations.