Genome evolution

1. Genome evolution Lecture 4: population genetics III: selection

2. Population genetics Drift: The process by which allele frequencies are changing through generations Mutation: The process by which new alleles are being introduced Recombination: the process by which multi-allelic genomes are mixed Selection: the effect of fitness on the dynamics of allele drift Epistasis: the drift effects of fitness dependencies among different alleles “Organismal” effects: Ecology, Geography, Behavior

3. Wright-Fischer model for genetic drift N individuals ∞ gametes N individuals ∞ gametes We follow the frequency of an allele in the population, until fixation (f=2N) or loss (f=0) We can model the frequency as a Markov process on a variable X (the number of A alleles) with transition probabilities: Sampling j alleles from a population 2N population with i alleles. In larger population the frequency would change more slowly (the variance of the binomial variable is pq/2N – so sampling wouldn’t change that much) 0 1 Loss 2N 2N-1 Fixation

4. The Moran model Instead of working with discrete generation, we replace at most one individual at each time step A A A Replace by sampling from the current population a a X A A A a a a A A A A A A We assume time steps are small, what kind of mathematical models is describing the process?

5. The Moran model A A A Replace by sampling from the current population a a X A A A a a a A A A A A A Assume the rate of replacement for each individual is 1, We derive a model similar to Wright-Fischer, but in continuous time. A process on a random variable counting the number of allele A: i i-1 i+1 0 1 Loss 2N 2N-1 Fixation “Birth” Rates: “Death”

6. Fixation probability i i-1 i+1 0 1 Loss 2N 2N-1 Fixation “Birth” Rates: “Death” In fact, in the limit, the Moran model converge to the Wright-Fischer model, for example: Theorem: When going backward in time, the Moran model generate the same distribution of genealogy as Wright-Fischer, only that the time is twice as fast Theorem: In the Moran model, the probability that A becomes fixed when there are initially I copies is i/2N Proof: like the proof for the Wright-Fischer model. The expected X value is unchanged since the probability of births and deaths is the same

7. Fixation time Expected fixation time assuming fixation Theorem: In the Moran model, let p = i / 2N, then: Proof: not here..

8. Selection Fitness: the relative reproductive success of an individual (or genome) Fitness is only defined with respect to the current population. Fitness is unlikely to remain constant in all conditions and environments Sampling probability is multiplied by a selection factor 1+s Mutations can change fitness A deleterious mutation decrease fitness. It would therefore be selected against. This process is called negative or purifying selection. A advantageous or beneficial mutation increase fitness. It would therefore be subject to positive selection. A neutral mutation is one that do not change the fitness.

9. Adaptive evolution in a tumor model Selection Human fibroblasts + telomerase Passaged in the lab for many months Spontaneously increasing growth rate V. Rotter

10. Selection in haploids: infinite populations, discrete generations • This is a common situation: • Bacteria gaining antibiotic residence • Yeast evolving to adapt to a new environment • Tumors cells taking over a tissue Allele Frequency Relative fitness Fitness represent the relative growth rate of the strain with the allele A It is common to use s as w=1+s, defining the selection coefficient Gamete after selection Generation t: Ratio as a function of time:

11. Selection in haploid populations: dynamics Growth = 1.5 We can model it in continuous time: Growth = 1.2 In infinite population, we can just consider the ratios:

12. Computing w Example (Hartl Dykhuizen 81): E.Coli with two gnd alleles. One allele is beneficial for growth on Gluconate. A population of E.coli was tracked for 35 generations, evolving on two mediums, the observed frequencies were: Gluconate: 0.4555  0.898 Ribose: 0.594  0.587 For Gluconate: log(0.898/0.102) - log(0.455/0.545) = 35logw log(w) = 0.292, w=1.0696 Compare to w=0.999 in Ribose.

13. Fixation probability: selection in the Moran model When population is finite, we should consider the effect of selection more carefully i i-1 i+1 0 1 Loss 2N 2N-1 Fixation The models assume the fitness is the probability of the offspring to be viable. If it is not, then there will not be any replacement “Birth” Rates: “Death” Theorem: In the Moran model, with selection s>0

14. Fixation probability: selection in the Moran model Theorem: In the Moran model, with selection s>0 Note: Note: Variant (Kimura 62): The probability of fixation in the Wright-Fischer model with selection is: Reminder: we should be using the effective population size Ne

15. Fixation probability: selection in the Moran model Theorem: In the Moran model, with selection s>0 Proof: First define: Hitting time Fixation given initial i “A”s The rates of births is bi and of deaths is di, so the probability a birth occur before a death is bi/(bi+di). Therefore:

16. Fixation probabilities and population size

17. Selection and fixation Recall that the fixation time for a mutation (assuming fixation occurred) is equal the coalescent time: Theorem: In the Moran model: Theorem (Kimura): (As said: twice slower) Fixation process: 1.Allele is rare – Number of A’s are a superciritcal branching process” Selection 2. Alelle 0<<p<<1 – Logistic differential equation – generally deterministic 3. Alelle close to fixation – Number of a’s are a subcritical branching process Drift

18. Selection in diploids Assume: Genotype Fitness Frequency (Hardy Weinberg!) There are different alternative for interaction between alleles: a is completely dominant: one a is enough – f(Aa) = f(aa) a is Complete recessive: f(Aa) = f(AA) codominance: f(AA)=1, f(Aa)=1+s, f(aa)=1+2s overdominance: f(Aa) > f(AA),f(aa) The simple (linear) cases are not qualitatively different from the haploid scenario