Muller’s 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

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 generationUb: beneficial mutation rate“Mutation load”, “mutation number” k: the number of less-fit alleles in a genome as compared to the best possible genomeSelection coefficient s: small relative fitness gain/loss per mutation.

Special notation:

V = dkav/dt: average substitution rate of beneficial mutationsv = (1/U) dkav/dt: normalized rachet rate (substitution rate of deleterious mutations) = s/Uf(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= Ub/U: ratio of the beneficial mutation rate to the total mutation ratex = kkav: relative mutation numberk0, x0: the minimum values of k and x in a population (for the best-fit class)u = exp(‘(x0)]

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.

k1: mutation load of the reference strain of virusL: total number of sitesMgen: generations per passager: expansion ratio per generation

Fixation of deleterious mutations at very small population sizes

Mutation rate per genome is usually small for all organisms, U =10-3-10-1. 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 kav is the average number of deleterious alleles in genome.

2s e2Ns 1

dkav /dt 2NUs e2Ns 1 U,

Many-site effect: Muller’s ratchet at N >> eU/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 kav= U/s.

Zero-mutation class contains, on the average, n0=Nf(0) = Nexp(-U/s) genomes. Random fluctuations cause its eventual loss.

Distribution shifts by one notch in k: one click of “Muller’s 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)

Diffusion approach at N >> eU/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 kav. 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, feq=exp(-U/s)

where a(f) is the average change of f per generation, a(feq)=0.

f < feq -> 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 feq 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).

t

1

2N

2

df 2( f)

df

[a( f )]

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:

fk (t 1) fk (t) U(1 ) fk 1(t) fk (t) U s(k kav ) fk (t)

k 1,2,...,L 1

Ub

U bk

bk d (L k)

Early approach

Tsimring, Levine and Kessler, Phys. Rev. Lett., 1996: very similar model

Approximation: fk(t) is smooth in k

fk1(t) fk (t) f /k (1/2)2 f /k 2

Continuous set of soliton-like solutions fk(t)= FV(k-kav(t)) labeled by the “velocity”, V = dkav/dt, related to the soliton width, stdk.

Choice of the solution (physics: “lifting degeneracy”): Cutoff of distribution at the high-fitness edge at f(k) < 1/N.

Smooth logarithm of the distribution and the diffusion equation for the best-fit classDistribution fk(t) is not smooth in k in the tail (which is very important): fk/fk-1 ~ 1 or >> 1 (Rouzine et al 2003). Better:

as long as the scale in k is large, kavk0 >> 1, where k0 =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 = fk0 :

ln fk1(t) ln fk (t) ln fk (t) /dk

t

1

2N

2

df 2( f)

df

[ Sf M(t) ]

S U s(k0 kav ), M(t) Ub fk0 1(t)

which yields

df

dtM(t) Sf ,

dV f

dt

f

N 2SV f , V f f 2 f 2

“Stochastic threshold”: Best-fit class is lost or established:

V f f exp((x0))

Note: S is not zero, effect ofchange in f on S can be neglected

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 < 12 with different widths stdk:

Variance stdk2 = (12)/1/2 = (12)(U/s)1/2: equilibrium, v=0

Broader distribution: fixation of beneficial mutations dominates, v < 0Narrower distributionratchet dominates, 0 < v < 12

ln fk (t) (x), x k kav (t)

x (1 )e '(x ) e '(x ) v'(x) 1

s /U, v 1

U

dkav

dt

v 1 2 stdk2

General expression for the substitution rate

u 1

2 v v 2 4(1 ) 1

x0 1

1 2u v ln u v 0

Each solution exists in a finite interval x > x0,where [dx/d’ ]x=x0 = 0. At the boundary, ’(x0)=ln u, where

Thus, the deterministic distribution has a high-fitness edge at the relative mutation number x0.

We can integrate the balance equation in ’

(0) (x0) d 'ln u

0

'dx

d '

(0) (x0) 1 2 v2

ln2(eu) 1 2uln u

(0) 1 2 stdk , stdk 1

1, stdk 1

How (x0) depends on N, is determined by the stochastic best-fit class.

Muller’s ratchet at s << U

Muller’s ratchet with rate Uv: 0 < v < 1.Beneficial mutations are not important at low kav:

Ub /U 0

'(x0) ln u ln(1/v)

x0 1

1 v ln

e

v

1

The general result for v simplifies:

(0) (x0) 1v

2ln2 e

v1

(0) 1

2ln

2 (1 v)

For the continuous approach to work, we need |x0| >> 1, hence, = s/U << 1. The wave is also broad: stdk ~ 1/s1/2.

At s << U, the distribution of genomes in k is not formed. Single fixation events rule?

Best-fit class

. Stochastic threshold condition (Rouzine et al 2003):

V fk 0(t) f k 0

2(t) e2 (x0 )

Solving the equations for the variance and the average without beneficial mutations:

e (x0 ) 1

NS

S Uv lne

v

(well-known “stochastic threshold” from 1-site theory)

(Note: effective selection coefficient S=0 at v=0)

(x0) ln NUv ln(e /v)

Best-fit class: another methodFinding the average time to the loss of the class

A best-fit class with k01 mutationsis lost at t = 0.

Ratchet click time:

Answer:

fk 0(t) fk 0(0)e St

1

Uv

1

Sln

fk 0(0)

1/(NS)

(x0) 1

2ln fk0(0) ln

1

NS

ln NUv 3 / 2 lne

v

Cf. previous method: extra factor v1/2 in the logarithm.

Edge effect on continuity

Continuity approximation

requires

At x ~ x0, but not |x-x0| << x0, the condition is equivalent to 1/ >> 1 and is met.At x = x0, 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:

ln fk1 ln fk ln f /dk

ln f /dk 2 ln f /dk 2 'dx

d'1

dfk0

dt Sfk0

, S Uv lne

vdfk

dt Sfk Ufk 1, k k0 1

Periodic initial condition:

Numeric solution:

fk 1(0) fk (1/Uv)

ln fk (t) ln fk 0(t) ln(1/v)(k k0) ln 1.2(k k0) 1.0

at kk0 = x0, edge correction to (x0)

Continuous result

Final result

ln(NU 3 / 2) 1 v

2ln2 e

v1

lnv 3

2 (1 v)

ln(e /v)

1.2 1 v ln(e /v)

= s/U, v=V/U

Simulation vs analytic theory

Simulation vs analytic theory: 2

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

Simulation vs analytic theory: 4

Simulation vs analytic theory: 5

Conclusions: Muller’s ratchet1. 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 (Muller’s 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.

Fixation of beneficial alleles: v < 0Deleterious 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, Ub,and V=-vU:

The high-fitness tail length, the edge derivative, and the distribution maximum:

v 0, v 1

s (0) (x0) V

2ln2 V

eUb

1

,

x0 V

sln

V

Ub

, '(x0) ln u lnV

Ub

(0) (1/2) ln(s /2V ), V s

(0) (x0) 1v

2ln2 e

v1

, 0 v 1

(0) (x0) v

2ln2 v

e1

,

and 1 otherwise.

V U

V s /ln(V /Ub)

Beneficial mutations: the high-fitness edge

df

dtM(t) Sf ,

dV f

dt

f

N 2SV f

S V lnV

Ub

, M(t) Ub fk0 1(t)

Again, from diffusion equation for fk0 = f :

Unlike in the ratchet case, S > 0, and M is not zero:

V f (t) f (t) e (x0 )

(x0) ln NUb ln2 V

Ub

“Stochastic threshold” approach (Rouzine et al 2003):

High-fitness edge: method 2

Sfk

Mln

V

Ub

1 fk 0(t) eSt

Note: for an established class, beneficial mutations are not critically important:

We have 1/S = (1/V)/ln(V/Ub) (from the continuous part) >> 1/V. Beneficial mutation creates a new class k0 within time interval ~ 1/S:

(Rouzine & Coffin 2005, recombination model;Desai & Fisher 2006, this model, preprint online)

S(Nfmax )(1/S)Ub ~ 1

fmax ~1

NUb

(x0) 1

2ln fmax ln

1

NS

ln N UbV ln(V /Ub)

N to N (V /Ub) /ln3(V /Ub)New:

Example: V/Ub=104, N=103: change in lnN is 0.18.

Final answer in the limit when U is not important

(0) 1/( 2 stdk ) :

V 2sln N Ub ln(V /Ub)

ln2 V

eUb

1, V max(U,Ub,s)

max U,s

ln(V /Ub

V s : stdk 1 (0) 1

Intermediate V (Desai & Fisher 2006, preprint online) :

The same as for large V, except N is replaced with N(V/s)1/2 ~ N ln1/2(NUb)]/ln(s/Ub),

i.e., relatively small difference in V.

Large V:

Transition to 1-site theory starts at |x0| ~ kav, and ends at stdk ~ kav1/2:

V ~ skav/ln(skav/Ub) to V ~ skav (1-site result)

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 = <s> 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 <s> 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.

Comparison with simulation

Difference between green and blueline, mostly, due to neglecting U.

Dependence in simulation on kav at U=0.05 and small kav is due to |x0| = 53 at N = 1013. We assumed |x0| << kav.

No transition to 1-site theory yet (expected at V/s = kav= 50).

Possible reasons for the difference with simulation at kav=500:

1) S = s|x0| = 0.5, we assumed S<<1. 2) Edge effects on the continuity of lnfk.

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/Ub.

5. Good agreement with Monte-Carlo simulation is obtained for some parameters

relevant to viral populations.

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

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