The dynamics of complex adaptation

Transcription

1 The dynamics of complex adaptation Daniel Weissman Mar. 20, 2014

2 People Michael Desai, Marc Feldman, Daniel Fisher Joanna Masel, Meredith Trotter; Yoav Ram Other relevant work: Nick Barton, Shahin Rouhani; Lin Chao, Dan Weinreich; Freddy Christiansen, Sally Otto, Aviv Bergman; Rick Durrett, Deena Schmidt, Jason Schweinsberg; Lilach Hadany; Rutger Hermsen, Terry Hwa; Yoh Iwasa, Natalia Komarova, Franziska Michor, Martin Nowak; Michael Lynch; Yannis Michalakis, Monty Slatkin; Richard Neher, Boris Shraiman; Erik van Nimwegen, James Crutchfield; Maria Serra, Patsy Haccou; Arjan de Visser, Su-Chan Park, Kavita Jain, Joachim Krug;...

3 Complex adaptation 1+s Fitness 1 µ 0 µ 2 1-δ 1 1-δ 2 µ Number of Mutations 3 Need combination of K 2 mutations for benefit Fitness valley/plateau / Irreducible complexity

4 Why do we care? 1+s Fitness 1 µ 0 µ 2 1-δ 1 1-δ 2 µ Number of Mutations 3 Specific cases: signal-receptor, cancer,... Generally: When does evolution get stuck? Evolution by fittest mutations or fittest combinations? Space of genotypes grows exponentially with K

5 Problems Population has to: 1. Produce the combination 2. Fix it (incorporate it into everyone s genome)

6 Start with the second problem: When can a rare combination spread in a population?

7 Selection vs recombination Frequency x 1 of combination changes because of selection s, recombination r, etc ẋ = (s r)x + rf (mutant allele frequencies) + stochasticity +... { if r s : need f (allele freqs.) x to get ẋ > 0 if r < s : ẋ > 0 regardless of allele freqs. (Simplest (K = 2) case: f product of mutant allele frequencies)

8 Selection vs recombination: numbers Rare combination giving s = 1% more offspring/generation can spread faster than broken up by recombination if genes are within: Drosophila/human: 1Mb ( 100 genes in Drosophila, 10 genes in humans) Yeast: whole genome?? HIV within host: whole genome? E. coli: whole genome, all of the genes? Cancer: whole genome

9 Selection vs stochasticity Trajectories of mutant lineages n(t): Near-critical branching process deterministic increase once n 1/s If alive at t < 1/s, usually n t descendants P(alive at time t) 1/t for t < 1/s p fix (s) s: If s = 1%, need to produce combo 100

10 Now address first problem: How can a population find an adaptation that needs K > 2 mutations to function?

11 Moderate K: hard but possible? Have to do exhaustive search impossible for large K But what about moderate K? Practically important: heterodimers, cancer, drug resistance... Number of potential genotypes also growing exponentially Population sizes, mutation rates, recombination rates vary over many orders of magnitude need to know which parameter combinations are important

12 Simplest toy model Focus on K = 2 mutants needed for beneficial combination, asexual Population size N Find the mean time τ for population to acquire combination not the relevant statistic for cancer 1+s Fitness 1 µ µ 1-δ ab Ab, ab Genotype AB

13 Asexual dynamical regimes already complicated 2s Deterministic [ N = δ1 µ0µ1 2s De Population size, N πµ0µ1 1 µ0 s 1 µ1s Neutral semi-deterministic tunneling Neutral stochastic tunneling Neutral sequential fixation [ N = 1 δ1 N = 2δ2 1 πµ0µ1s Deleterious tunneling Deleterious sequential fixation N = 1 [ ] δ 2 log 1 δ1 µ1s Population size, N µ0µ1 1 µ0 s 1 µ1s Neutral sem tun Neutral stoc tunnelin Neutral seq fixatio 1/s [ 1/s 2 µ1s 0 2 µ1s s Selective disadvantage of single-mutants, δ 1 2 µ1s 0 Selectiv Focus on plateau case: small δ

14 First guess Let x 2 (t) = frequency of double-mutants at time t x 2 (0) = 0, ẋ 2 (t) = µ 2 t + sx 2 1/τ s/ log(s/µ) Cheated: what if Nx 2 (t) < 1? How can we select on nothing? Need Nµ 2 s Generally: Nµ K K!s K 1

15 Deterministic for very large population sizes Population size, N 1 µ 0 s 1 µ1s 1/s Deterministic 1/ s log(s/µ) 2s µ 0µ 1 p Neutral semi-deterministic tunneling Neutral stochastic tunneling Neutral sequential fixation N = µ 0µ 1s 2 µ 2 1s 0 µ 1s s Selective disadvantage of single-mutants, 1 N = 1 1 N = 1 2 log 1 1 µ1s N = 1 µ0µ1 Deleterious tunneling Deleterious sequential fixation

16 Second guess: need to treat double-mutants stochastically τ time to produce first successful double-mutant Single-mutant frequency x 1 (t) µt, so τ satisfies: Nµ 2 τ 2 1/s 1/τ µ Ns Ignored stochasticity in the single-mutants is this ok? Need x1 (τ) fluctuations Third guess: treat all mutants stochastically

17 Single-mutant lineage Total # of individuals (area) = # of mutational opportunities Prob(success) (# double-mutants produced) p fix (s) area µ p fix (s)

18 Distribution of total progeny Prob(success area) area µ p fix (s) Critical branching process: If alive at t N, usually n t descendants P(alive at time t) 1/t for t N P(area > a) P(alive at time a) 1/ a Long-tailed distribution of progeny large fluctuations Prob(success) 1/ µs Most likely path to success: rare lineage that persists for t 1/ µs; occurs with prob µs 1/τ Nµ µs

19 Most likely path to success: one big lineage 8000 Ab ab AB Number of mutants Number of mutants Prob ~ µs ~ 1/ µs Time in generations ~ 1/ µs Prob(success area) area µs; Prob(area > a) 1/ a So wait for one big lineage that persists for t 1/ µs; occurs with prob µs K > 2 : 1/τ (Nµ 2 )(s/µ) 1/2K 1 1/τ Nµ µs

20 Range of behaviors over different population sizes Population size, N 1 µ 0 s 1 µ1s 1/s Deterministic 1/ s log(s/µ) 2s µ 0µ 1 p Neutral semi-deterministic tunneling 1/ µ p Ns p Neutral stochastic tunneling 1/ Nµ p µs Neutral sequential fixation N = µ 0µ 1s 2 µ 2 1s 0 µ 1s s Selective disadvantage of single-mutants, 1 N = 1 1 N = 1 2 log 1 1 µ1s N = 1 µ0µ1 Deleterious tunneling Deleterious sequential fixation

21 When is complex adaptation likely? Population size, N 2s µ0µ1 1 µ0 s 1 µ1s 1/s Deterministic 1/ s log(s/µ) p Neutral semi-deterministic tunneling 1/ µ p Ns p Neutral stochastic tunneling 1/ Nµ p µs Neutral sequential fixation N = µ0µ1s 2 µ1s 0 2 µ1s s Selective disadvantage of single-mutants, 1 N = 1 1 N = 1 2 log 1 1 µ1s N = 1 µ0µ1 Deleterious tunneling Deleterious sequential fixation At least medium-sized population: N > 1/ µs Neutral single mutants: δ < µs condition on δ relaxed for larger N

22 Numbers To be able to see combo of two individually neutral point mutations with s = 0.01, need N > 10/ µ neutral : δ < µ/10 E. coli: µ N 10 6 ( in you) RNA virus: µ 10 4 N 10 3

23 What about sex?

24 Sex helps for r, δ < s/2

25 Putting it all together K = 2, δ = 10 3, µ = r=0.5 Time to cross valley r=0.025 r=0 s= x10 2 1x10 3 1x10 4 1x10 5 1x10 6 1x10 7 1x10 8 Population Size Complicated, but understandable

26 Conclusion Summary: Adaptation can spread without intermediate genotypes if advantage s > recombination rate r Moderately complex adaptation is easy if: Population is large (N > 1/ µs, N > 1/µ, etc) Intermediate genotypes not too deleterious (δ < µs, etc) Moderate recombination r s No reason why it shouldn t be happening in natural populations Questions: Effect of sex for K > 2? Interaction with simple adaptation? Real populations/fitness landscapes? Thanks for listening!