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Mullers ratchet and fixation of beneficial alleles: the soliton approach to many-site problem Igor Rouzine Department of Molecular Biology and Microbiology Tufts University Boston, USA

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Basic terminology and notation N: haploid population size (number of genomes) Allele: variant of a site in a genome. Can be better-fit or less-fit (2-allele model). Fitness w: relative average number of progeny of a genome. Mutation event: DNA transcription error. Can be deleterious or beneficial. U: mutation rate per genome per generation U b : beneficial mutation rate Mutation load, mutation number k: the number of less-fit alleles in a genome as compared to the best possible genome Selection coefficient s: small relative fitness gain/loss per mutation. Special notation: V = dk av /dt: average substitution rate of beneficial mutations v = (1/U) dk av /dt: normalized rachet rate (substitution rate of deleterious mutations) = s/U f(k,t): frequency of a class of genomes with mutation number k (f,t): probability of having frequency f of a class at time t = ln f = U b /U: ratio of the beneficial mutation rate to the total mutation rate x = k k av : relative mutation number k 0, x 0 : the minimum values of k and x in a population (for the best-fit class) u = exp( (x 0 )]

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Experiment on steady state fitness of a virus (VSV) versus population size Experiment on measuring fitness of steady state fitness of vesicular stomatitis virus passaged at fixed number of infectious units N. One-site theory based on selection- mutation balance does not work. k 1 : mutation load of the reference strain of virus L: total number of sites M gen : generations per passage r: expansion ratio per generation

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Fixation of deleterious mutations at very small population sizes Mutation rate per genome is usually small for all organisms, U = At very small population sizes N, mutation events are rare and separated in time. Fixation of separate deleterious mutations is effectively opposed by selection. One-site, 2-allele model, diffusion equation (Lande 1994; 1998 based on Crow and Kimura 1970): A single genome containing a deleterious mutation with selection coefficient s will be fixed (spread to all population) with probability The average substitution rate is exponentially small at Ns << 1 as Where k av is the average number of deleterious alleles in genome.

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Many-site effect: Mullers ratchet at N >> e U/s N not too small: time overlap between fixation of mutations at different sites, accumulation of deleterious mutations is rapid even at N >> 1/s, provided U >> s. Case N >> exp(U/s) (Stephan 1993; Charlesworth and Charlesworth 1997; Gordo and Charlesworth 2000): Selection-mutation equilibrium: Poisson distribution of genomes f(k) with k av = U/s. Zero-mutation class contains, on the average, n 0 =Nf(0) = Nexp(-U/s) genomes. Random fluctuations cause its eventual loss. Distribution shifts by one notch in k: one click of Mullers ratchet (Muller 1964; Felsenstein 1974). Stopping ratchet: recombination (absent in Y chromosome or asexual organisms); beneficial mutation (not efficient at small k/L); epistasis (biointeracation between sites)

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Diffusion approach at N >> e U/s Clicks are infrequent due to large Nf(0). Calculating the average time between ratchet clicks. Assumptions: All classes but zero class are at deterministic equilibrium with current k av. In a transitional time interval between clicks, zero-class is out of equilibrium. Diffusion equation for f = f(0), the random frequency of zero-class, f eq =exp(-U/s) where a(f) is the average change of f per generation, a(f eq )=0. f decrease in the average fitness of population -> decrease in relative fitness of the zero class -> a(f) > 0 -> increase in f If f falls to 0, it never comes back. Estimate of a(f) for f far from f eq is far from trivial (Gordo and Charlesworth 2000). The average time between clicks is a complex function of N, s, U, not only of the zero-class size Nf(0).

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Far from equilibrium: ratchet at N < exp(U/s) and fixation of beneficial mutations All classes are way out of equilibrium (e.g., ratchet clicks overlap in time). Soliton approach (Tsimring et al, Phys. Rev. Lett. 1996; Rouzine et al 2003) Basic model including beneficial mutations: Deterministic detailed balance equation for the class frequencies:

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Early approach Tsimring, Levine and Kessler, Phys. Rev. Lett., 1996: very similar model Approximation: f k (t) is smooth in k Continuous set of soliton-like solutions f k (t)= F V (k-k av (t)) labeled by the velocity, V = dk av /dt, related to the soliton width, std k. Choice of the solution (physics: lifting degeneracy): Cutoff of distribution at the high-fitness edge at f(k) < 1/N.

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Smooth logarithm of the distribution and the diffusion equation for the best-fit class Distribution f k (t) is not smooth in k in the tail (which is very important): f k /f k-1 ~ 1 or >> 1 (Rouzine et al 2003). Better: as long as the scale in k is large, k av k 0 >> 1, where k 0 =min(k) is the mutation number for the best-fit class. All groups are deterministic but the best-fit class. Diffusion equation for the best-fit class frequency f = f k0 : which yields Stochastic threshold: Best-fit class is lost or established: Note: S is not zero, effect of change in f on S can be neglected

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Solitary wave solution Seeking solution in the soliton form the balance equation becomes I cannot solve it for (x), but can find all I need without solving it. A continuous set of solutions at any v < 1 2 with different widths std k : Variance std k 2 = (1 2 )/ 1/2 = (1 2 )(U/s) 1/2 : equilibrium, v=0 Broader distribution: fixation of beneficial mutations dominates, v < 0 Narrower distribution ratchet dominates, 0 < v < 1 2

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General expression for the substitution rate Each solution exists in a finite interval x > x 0, where [dx/d ] x=x0 = 0. At the boundary, (x 0 )=ln u, where Thus, the deterministic distribution has a high- fitness edge at the relative mutation number x 0. We can integrate the balance equation in How (x 0 ) depends on N, is determined by the stochastic best-fit class.

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Mullers ratchet at s << U Mullers ratchet with rate Uv: 0 < v < 1. Beneficial mutations are not important at low k av : The general result for v simplifies: For the continuous approach to work, we need |x 0 | >> 1, hence, = s/U << 1. The wave is also broad: std k ~ 1/s 1/2. At s << U, the distribution of genomes in k is not formed. Single fixation events rule?

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Best-fit class. Stochastic threshold condition (Rouzine et al 2003): Solving the equations for the variance and the average without beneficial mutations: (well-known stochastic threshold from 1-site theory) (Note: effective selection coefficient S=0 at v=0)

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Best-fit class: another method Finding the average time to the loss of the class A best-fit class with k 0 1 mutations is lost at t = 0. Ratchet click time: Answer: Cf. previous method: extra factor v 1/2 in the logarithm.

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Edge effect on continuity Continuity approximation requires At x ~ x 0, but not |x-x 0 | > 1 and is met. At x = x 0, dx/d =0, so the condition is violated close to the edge. Edge creates perturbation that spreads inward. The effect is deterministic. Balance equations near the edge: Periodic initial condition: Numeric solution: at k k 0 = x 0, edge correction to (x 0 ) Continuous result

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Final result = s/U, v=V/U

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Simulation vs analytic theory

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Simulation vs analytic theory: 2

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Simulation vs analytic theory: 3 Equilibrium best-fit class size = 1

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Simulation vs analytic theory: 4

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Simulation vs analytic theory: 5

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Conclusions: Mullers ratchet 1.At U >> s and high average fitness, an approach based on continuous deterministic treatment of the logarithm of the mutation number distribution combined with stochastic treatment of the best-fit class has been developed. 2.In a broad interval of population sizes from s << N << exp(U/s), we predict enhanced, versus one-site theory, accumulation of deleterious mutations (Mullers ratchet) in the form of travelling wave for the mutation number distribution, 3.At moderately small s/U, the edge correction to the continuity approximation is important for the numeric accuracy. 4.Two methods of edge treatment based on the diffusion equation yield different factors multiplying N, with a small numeric difference for relevant parameter values. 5.In the entire range of N, a very good agreement with Monte-Carlo simulation results is obtained. 6.At larger N, the distribution is close to equilibrium, and the earlier separate-click approach applies. At small N ~ s, the results match that of single-mutation model.

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Fixation of beneficial alleles: v < 0 Deleterious mutations are not important in the general formula for v, if The result simplifies to (Rouzine et al 2003): Compare to the ratchet result: Because U is no longer important, we return to notation s, U b,and V=-vU: The high-fitness tail length, the edge derivative, and the distribution maximum: and 1 otherwise.

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Beneficial mutations: the high-fitness edge Again, from diffusion equation for f k0 = f : Unlike in the ratchet case, S > 0, and M is not zero: Stochastic threshold approach (Rouzine et al 2003):

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High-fitness edge: method 2 Note: for an established class, beneficial mutations are not critically important: We have 1/S = (1/V)/ln(V/U b ) (from the continuous part) >> 1/V. Beneficial mutation creates a new class k 0 within time interval ~ 1/S: (Rouzine & Coffin 2005, recombination model; Desai & Fisher 2006, this model, preprint online) New: Example: V/U b =10 4, N=10 3 : change in lnN is 0.18.

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Final answer in the limit when U is not important Intermediate V (Desai & Fisher 2006, preprint online) : The same as for large V, except N is replaced with N(V/s) 1/2 ~ N ln 1/2 (NU b )]/ln(s/U b ), i.e., relatively small difference in V. Large V: Transition to 1-site theory starts at |x 0 | ~ k av, and ends at std k ~ k av 1/2 : V ~ sk av /ln(sk av /U b ) to V ~ sk av (1-site result)

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Contrasting to two-clone approach If only two competing beneficial clones within a pre-existing virus variant are considered at a time, saturation of the fixation speed is predicted: V = at large N. (Maynard Smith, What use is sex? 1971; Gherish and Lenski 1998; Orr 2000) Variation in s is essential: A clone with larger s pushes out the previous one. Mutations with larger and larger s win, as N increases. Hence, the effective increase in for fixed mutations, and increase in the adaptation rate, sV. Solitary wave approach: additional mutations at other sites resolve clonal interference. Variation in s is not vitally important.

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Comparison with simulation Difference between green and blue line, mostly, due to neglecting U. Dependence in simulation on k av at U=0.05 and small k av is due to |x 0 | = 53 at N = We assumed |x 0 | << k av. No transition to 1-site theory yet (expected at V/s = k av = 50). Possible reasons for the difference with simulation at k av =500: 1) S = s|x 0 | = 0.5, we assumed S<<1. 2) Edge effects on the continuity of lnf k.

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Conclusions: accumulation of beneficial alelles 1.Using the same approach as in the ratchet case, at large population sizes or low fitness, we predict accumulation of beneficial mutations under Fisher-Muller- Hill-Robertson effect, in the form of traveling wave iof mutation number distribution. 2.In a very broad range of N, the substitution rate V is proportional to the logarithm of the population size (in contrast to the two-clone interference model result). 3.In the limit of large N, transition to the one-site deterministic theory is predicted (in contrast to the two-clone interference model result). 4.A more accurate treatment of the best-fit class based on the diffusion equation affects the factor multiplying N, which difference may be numerically detectable at moderately large N and large s/U b. 5.Good agreement with Monte-Carlo simulation is obtained for some parameters relevant to viral populations.

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Current work and future directions 1) Asexual populations: - variation of s among sites -linkage disequilibrium 2) Partly sexual haploid populations and sexual diploid populations: - accumulation rate of pre-existing beneficial alleles - correlations between genomes in fitness and site-site correlations - coalescent time - linkage disequilibrium - the fitness distribution of a far ancestor of a site - synergy between beneficial mutations and recombination

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Acknowledgements John Coffin, Tufts University, Boston, USA Alex Kondrashov, National Institutes of Health, MD, USA John Wakeley, Harvard University, Boston, USA Isabel Novella, Medical College of Ohio, Toledo, OH, USA

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