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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
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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 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)]
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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.
k1: mutation load of the reference strain of virusL: total number of sitesMgen: generations per passager: 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 =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,
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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)
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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 )]
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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:
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)
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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.
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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
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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 < 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
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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.
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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?
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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)
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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.
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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
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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
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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: 3Equilibrium 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: 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.
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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)
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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):
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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.
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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)
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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 = <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.
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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.
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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/Ub.
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