The Evolution of Mutation Rates - Department of Zoology, …johnson/seminar.pdf · Evolution of new...

The Evolution of Mutation Rates

Toby Johnson

[email protected]

University of British Columbia, Vancouver, Canada

1

Why are mutation rates interesting ?

(1) Mutations are the ultimate source of heritable variation

2



(2) Deleterious mutations may be an important evolutionary

force:

◦ Evolution of sex and recombination (e.g. Kondrashov 1988, Otto &

Barton 1997)

◦ Evolution of inbreeding avoidance (e.g. Charlesworth & Charlesworth 1998)

◦ Evolution of ploidy level (e.g. Kirkpatrick 1994, Orr 1995, Mable & Otto 1998)

◦ Degeneration of Y chromosomes (e.g. Rice 1987, Charlesworth 1996)

◦ Extinction of small populations (e.g. Lande 1994, Lynch et al. 1995)

3



(2) Deleterious mutations may be an important evolutionary force

(3) Mutation may help explain observed level of:

◦ Variation in quantitative traits (e.g. Lande 1975, Turelli 1984)

◦ Variation in life history traits (e.g. Charlesworth & Hughes 1996)

4



(2) Deleterious mutations may be an important evolutionary force

(3) Mutation may help explain observed level of variation

(4) Cells with atypical mutation rates have been implicated in

◦ Cancer (e.g. Loeb and Loeb 2000)

◦ Evolution of new pathogenic strains of bacteria (e.g. LeClerc et al.

1996, Picard et al. 2001)

◦ Survival of bacteria in sterilised foods (Anderson et al. 1956, White et al.

1999)

5

Mutation Rates and Drake’s Rule

103 104 105 106 107 108 109 1010

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Genome size (bp)

perbase-pairperreplication

pergenomeperreplication

d

t

ddddd

t

dd

t

d

t

d

t

M13 d

t

λ

d

t

T2d

t

T4 d

t

Sac

ddd

t

Eco

dd

t

Sce

dd

t

Ncrddddd

t

Cel

d

t

Dme

d

t

Mmu

d

t

Hsa

Key: RNA viruses (Rhinovirus, Poliovirus, Vesicular stomatitis virus and Measles virus). DNAviruses: M13, λ, T2, T4. Archaea: Sac Sulfolobus acidocaldarius. Bacteria: Eco E. coli. Eukaryoticmicrobes: Sce S. cerevisiae, Ncr Neurospora crassa. Metazoa: Cel C. elegans, Dme D.melanogaster, Mmu Mouse, Hsa Human.

(Data from Drake et. al. 1998 and Drake & Holland 1999)

6

Genetic basis of mutation rates

Many loci involved in DNA replication, proofreading and mismatch

repair; they control the mutation rate over the whole genome

Example: In vitro analysis of DNA polymerase purified from phage T4

0.00 0.05 0.100.0

0.5

1.0

Error rate

NTPswastedperbasepolymerised

t

t

antimutatorT4

twildtypeT4

t t

mutatorT4

(Data from Bessman et al. 1974)

7

The Evolution of Mutation Rates

• “Why is the mutation rate not reduced to zero ?” (Sturtevante 1937)

• There is a large class of deleterious and lethal mutations, and

hence decreased mutation rates are favoured

• Forces which could oppose reduction in the mutation rate include

◦ Beneficial mutations

◦ Costs of high accuracy replication

• Question: Which of these forces are important . . .

◦ . . . in sexual populations ?

◦ . . . in asexual populations ?

8

How do Mutation Rates Evolve ?Group Selection ?

• There is a mutation rate which maximises population mean

fitness (Kimura 1960)

◦ Mean fitness is reduced by the load due to deleterious

mutations, but

◦ Mean fitness is increased by the potential to adapt to a

changing environment

• Competition between groups could lead to evolution of the

optimum mutation rate, BUT

◦ In sexual populations, group selection is likely to be

overwhelmed by individual selection (Leigh 1973)

◦ In asexual populations, group selection is equivalent to

inter-clone selection, so optimality arguments were thought to

apply (Maynard Smith 1978)

9

How do Mutation Rates Evolve ?Answers from Modifier Models

din��

��

ModifierlocusM/m

SelectedlocusA/a

recombinationat rate r

• Suppose M causes a higher mutation rate than m

• Frequency of M allele may change by indirect selection, if there is:

◦ Selection at the A/a locus, and

◦ Linkage disequilibrium between the M/m and A/a loci

10

Example I. Effect of deleterious allele A

Population polymorphic for modifier M/m:

��

��M A

��

��M A

��

��M A

��

��M A�

��M a�

��M a �

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11


Mutation occurs (more with M than with m):

��

��M A

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12


Selection eliminates some A alleles:

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��m A

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��m a

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��m a

13

The effect of deleterious mutation is:

to indirectly select against a modifier which increases themutation rate

This effect is reduced by recombination

14

Example II. Effect of beneficial allele A in asexuals

Allele A arises in (say) M individual:

��

��

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��M A

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��M a

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15


Later:

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16


Much later:

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17

The effect of beneficial mutation in asexuals is:

the modifier “responsible” hitchhikes to fixation;a modifier increasing the rate is more likely to be fixed.

This effect is very sensitive to recombination . . .

18

Example III. Effect of beneficial allele A in sexuals

Allele A arises in (say) M individual:

��

��

��

��M A

��

��M a

��

��M a

��

��M a

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��M a

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��m a

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��m a

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��m a

19


Later (recombination has occurred):

��

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20


Much later (more recombination has occurred):

��

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21

The effect of beneficial mutation in sexuals is:

the modifier “responsible” hitchhikes to higher frequency;a modifier increasing the rate is indirectly selected for.

This effect is negligible when r > s.

22

Finite vs. Infinite Population Models

Why are they different ?

At beginning of a selective sweep:

(A is more fit than a)

Population size Genotypes Linkage disequilibrium(which drives indirect selection)

“Infinite” (N � 1µbp

) Am and AM weak and +ve

Finite (N < 1µbp

) Am or AM strong but +ve or −ve

23

Finite vs. Infinite Population Models

0.2

0.4

0.6

0.8

1.0

2000 4000 6000 8000 10000

0.2

0.4

0.6

0.8

1.00.0

0.00

Allele Frequency

Allele Frequency

Time in generations

Infinite N(Leigh 1973)

Finite N(Johnson 1999)

Solid lines: selected alleles A/a, s = 0.01, r/s = 0.1

Dashed lines: modifier allele M which doubles the mutation rate

24

Question

When finite population effects are taken into account, can rare

beneficial mutations favor the evolution of higher mutation rates in

sexual populations ?

25

Stochastic model for the evolution of mutation rates insexual populations

(Johnson 1999 Genetics 151:1621–31)

• Model allows:

◦ Many loci scattered over n chromosomes each M Morgans

long

◦ Beneficial mutations and deleterious mutations and the cost

of accuracy

◦ Stochastic effects when beneficial alleles are rare or unique

• Find evolutionarily stable (ESS) mutation rate by invasion

analysis, assuming:

◦ Modifier M has weak effect ∆U

◦ Fate of modifier determined while rare

26

Stochastic model for the evolution of mutation rates insexual populations

• Analysis showed that ESS mutation rate depends on a function

describing the cost of accuracy, but there are no data about this

function

• Proceeded by ignoring the cost of accuracy altogether !!!

• When there is more than one crossover per genome per

generation, the “no cost” ESS mutation rate is

Unc ' K sbsd

1

ln [4Nesb](nM+ln

[1

2sd

]) � K

• Where:K is rate of selective sweepssb is selection coefficient for beneficial mutationsU is genomic deleterious mutation ratesd is selection coefficient against deleterious mutationsnM is map size of whole genome, assume nM > 1Ne is effective population size

27

Theory compared to data

• Theory shows that with no cost of accuracy, Unc � K

• The available data show that

◦ For human–chimpanzee-old world monkey comparisons

U ' 3.0 (Keightley & Eyre-Walker 2000)

K ' 0.05 (Fay et al. 2001)

◦ For mouse–rat comparison


K < 0.1 ? (Divergence in coding regions only)

◦ For Drosophila melanogaster–simulans comparison


K < 0.01 ? (Divergence in coding regions only)

• This inconsistency implies that the cost of accuracy must

maintain the observed U for these species

28

Question

When finite population effects are taken into account, can rare

beneficial mutations favor the evolution of higher mutation rates in

sexual populations ?

Answer

The effect is negligible relative to the physiological costs of accuracy

29

How do Mutation Rates Evolve ?Group Selection ?

• There is a mutation rate which maximises population mean

fitness (Kimura 1960)

◦ Mean fitness is reduced by the load due to deleterious

mutations, but

◦ Mean fitness is increased by the potential to adapt to a

changing environment

• Competition between groups could lead to evolution of the

optimum mutation rate

◦ In sexual populations, group selection is likely to be

overwhelmed by individual selection (Leigh 1973)

◦ In asexual populations, group selection is equivalent to

inter-clone selection, so optimality arguments were

thought to apply (Maynard Smith 1978)

30

Previous Modeling Approaches for Asexuals

'

&

$

%

=

'

&

$

%

U = U1

w = f (U1)+

'

&

$

%

+ . . .

U = U2

w = f (U2)

v

vvvv vvv

vv

vv

vvv

vv

vv vv v

vv

v

v

vv

vvv v

vv

v

vv

vv

vvv

vv vvv

vvvv v vv

vv

vv

vv

vvv

vvv

v

• An asexual population polymorphic at a locus controlling the mutation rate isan ensemble of independent subpopulations

• The subpopulation with highest mean fitness has the “optimum mutation rate”(Uopt) and is assumed to eventually drive other subpopulations to extinction

• Results are robust to many of the model parameters, e.g.

◦ Uopt = min{U} in an unchanging environment, e.g. Liberman & Feldman 1986

◦ Uopt = K = rate of adaptation required “to keep up”, Kimura 1967

◦ Uopt = Eh(sd) = harmonic mean selection against deleterious alleles, Orr 2000

◦ Uopt is set by the physiological cost of accuracy, Kondrashov 1995, Dawson 1998

• Any of these results could explain the observed constant rate of mutation

31

Previous Modeling Approaches for Asexuals

'

&

$

%

=

'

&

$

%

U = U1

w = f (U1)+

'

&

$

%

+ . . .

U = U2

w = f (U2)

v

vvvv vvv

vv

vv

vvv

vv

vv vv v

vv

v

v

vv

vvv v

vv

v

vv

vv

vvv

vv vvv

vvvv v vv

vv

vv

vv

vvv

vvv

v

Problems:

• Previous approaches assumed no gene flow, BUT

◦ Mutation at modifier locus moves cells between subpopulations, so thesubpopulations do not evolve independently (Johnson 1999 Proc. R. Soc. B

266:2389–97)

• Previous approaches assumed infinite subpopulation sizes, BUT

◦ Subpopulations may be small, and the fate of a new modifier may bedetermined while it is rare

• Previous approaches only considered long term mean fitness, BUT

◦ Some subpopulations may go extinct in the short term

32

Questions

• Can beneficial mutations drive the evolution of higher wild type

mutation rates ?

(Experiments show that beneficial mutations can cause the evolution of

greatly elevated rates (mutator strains), but evolution of small increases has

never been observed )

• Will a finite asexual population evolve towards its optimum

mutation rate ?

• Can optimality based arguments explain Drake’s observation ?

(The organisms involved are, to an approximation, asexual)

33

Stochastic model for the evolution of mutation rates inasexual populations

(work in progress)

Assumptions:

• Stochastic, drift with population size N

• A modifier locus controls the mutation rate

◦ A set of selectively neutral modifier alleles

effect xi = 0.1 1. 10.

. . . . . .~��~��~��~�� ~��~��~��~��~��~��~�� ~��~��~��~��

◦ Each individual carries one allele with effect xi

◦ Modifier controls its own rate of mutation

◦ Results shown here assume mutation between “adjacent” modifiers only

• Modifier locus controls mutation rate at loci under selection

◦ Deleterious mutation, rate xi U and effect sd

◦ Beneficial mutation, rate xi k U and effect sb

34

‘Typical’ simulation results

0 106

10-3

10-2

10-1

Population mean mutation rate

time (generations)

← Uopt

k = 10−8 N = 1010

sd = 10−2

sb = 10−2

mi i−1 = 10−9U

mi i+1 = 10−6U

10 replicates are shown for eachchoice of initial conditions

35

Analytical results (taking a SSWM type limit)

0 106

10-3

10-2

10-1


time (generations)

← Uopt

probability

0.03 0.10 0.32 1.00

k = 10−8 N = 1010

sd = 10−2

sb = 10−2

mi i−1 = 10−9U

mi i+1 = 10−6U

36

Analytical and simulation results

0 106°

10-3

10-2

10-1


time (generations)

← Uopt

probability

0.03 0.10 0.32 1.00

∞

6← Uequil.

80%

20%

k = 10−8 N = 1010

sd = 10−2

sb = 10−2

mi i−1 = 10−9U

mi i+1 = 10−6U

37

Predicted equilibrium distributions of mutation rates

106 108 1010 101210-3

10-2

10-1

10-4 10-3 10-210-3

10-2

10-1

Population meanmutation rate

Population meanmutation rate

population size N selection for beneficial mutations sb

← Uopt

probability

0.03 0.10 0.32 1.00

k = 10−8 N = 1010

sd = 10−2

sb = 10−2

mi i−1 = 10−9U

mi i+1 = 10−6U

38

Questions and Answers

• Can beneficial mutations drive the evolution of higher wild type

mutation rates ?

◦ Sort of – they can prevent the evolution of lower rates

• Will a finite asexual population evolve towards its optimum

mutation rate ?

◦ Probably not

• Can optimality based arguments explain Drake’s observation ?

◦ Probably not

39

Summary

• The mutation rate is under genetic control and can therefore

evolve. To understand its evolution I have studied theoretical

models of individual selection in finite sized populations.

• In sexual populations the mutation rate is set largely by a balance

between indirect selection due to deleterious mutations and direct

selection due to the cost of fidelity. The effect of beneficial

mutations is negligible.

• In asexual populations beneficial mutations may have a role in

setting the mutation rate, but the result is unlikely to be optimal.

• There is no plausible explanation for Drake’s observation !

40

Further work:

What is the explanation for Drake’s observation ?

• More empirical studies are needed

◦ Estimate mutation rates for a range of microbes with diverse

life histories

• Theoretical work

◦ Many microbes exchange genes at low rates, previous models

have ignored this

◦ Real populations are spatially structured, previous models have

ignored this

41

Acknowledgements

Philip AwadallaNick Barton

Brian CharlesworthKevin DawsonPeter KeightleyAlex Kondrashov

Gil McVeanSally Otto

Rosie RedfieldPaul SniegowskiOlivier Tenaillon

BBSRC (UK)NSERC (Canada)

Wellcome Trust (UK)

42

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