Epistasis and Pleiotropy in Evolving Populations

CitationJerison, Elizabeth. 2016. Epistasis and Pleiotropy in Evolving Populations. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences.

Permanent linkhttp://nrs.harvard.edu/urn-3:HUL.InstRepos:33840657

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1

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s > 10−2

10−7

1s

1s

11/s = s n n − 1

1s

s s ≪ 1

s

103

.5%

1%

pfix

104 107

s r

N = 104

⟨n(t+ τ)⟩ = Nn(t)e(r+s)τ

ne(r+s)τ + (N − n)erτ

≈ n(t)(1 + sτ − n(t)

Nsτ)

τ ≈ 10

sτ ≪ 1 s ≈ 10−2

s =1

tln

x1fx2f

x2ix1i

i f x1i

i t

τ

⟨(∆n(t+ τ))2⟩ ≈ n(t)(1− n(t)

N)

sτ ≪ 1

n t

n(t+ δt) = n(t) + n(t)s(1− n(t)

N)δt+

√n(t)(1− n(t)

N)√δtη(t)

η(t) ⟨η(t)⟩ = 0 ⟨η2(t)⟩ = 1 N s = sτ

∂tx = Nsx(1− x) +√

x(1− x)η(t)

x = nN

N

S = Ns

x ≫ 1Ns

Ns ≪ 1

pfix =1− e−2s

1− e−2Ns.

Ns ≫ 1

s s ≪ 1

Ns ≫ 1s ≈ 10−2 s ≈

10−1 N ≈ 104 1Ns ≈ 10−3

H(z, t) = ⟨ex(t)z(t)⟩

∂H(z, t)

∂t= (sz +

z2

2)∂H

∂x+ (sz +

z2

2)∂2H

∂x2

z = −2Ns ∂H(z,t)∂t = 0

x = 1 x = 0

H(2Ns, t = 0) = H(2Ns, t → ∞) → e−2Nsx0 = pfixe−2Ns + (1− pfix)

pfix =1− e−2Nsx0

1− e−2Ns

Ns ≪ 1 pfix ≈ 1N

1Ns

t = 1s

Ns

∂tx = Ns(t)x(1− x) +√

x(1− x)η(t).

s

1s

s(t) s

s

NUb

Ub s

1Ns T ≈ 1

s lnNs

1NUbs

≫ 1s lnNs

NUb lnNs ≫ 1

5 − 10%

s > 10−2 10−7

Ub ≈ 5× 10−5

NUb lnNs ≈ 5

2

104 303 3 2 1 1

2 3 1∆ 2

±

104

µl

104

104

3

s = 1τ ln(

nft

nfr

nirnit

) τ

nit nft

nfr nir

dij =∑8

k=1 |uik − ujk| uik i

k

dij

µl

µl µl

µl

µl

µl

h = −∑

i pilogpi pi i

h h > 3

pi = niτi∑i niτi

ni i

τi i

X

g m = (0, 1)

g

I(X, g) =∑

X

= (X1, ...Xn)∑

m=(0,1)

p(m|X) log2p(m|X)

p(m).

M(X, g) = I(X, g) − I(X, g) I(X, g)

pi =niτi∑i niτi

M(X, g)

M(X, g)

yjk

j k

sk =1∑

i,j ωij

m∑

j=1,j =k

nj∑

i=1

ωij(yijk− yjk)−1∑

i,j(1− ωij)

m∑

j=1

nj∑

i=1

(1−ωij)(yijk− yjk)

ij j sk

k ωij ij

sk

fsfgL fs

fg

L fs fg

L

χ2 =∑

i(Oi−Ei)2

EiOi i

Ei i

χ2

pi =niτi∑i niτi

ni i τi

i

dN/dS = 1

χ2

χ2

3 104

3 104

104

3 37

3

104

3

3

3

10 α 104 3 104

3

3

20 α 17 α 7 α

20 17 7

10ρ0 17ρ0 7ρ0 20ρ0 10 17 7

20

17 α 104 20 α 104

7 α 104 10 α 17 10 α 7 10 α

20 17 α 17 20 α 20 7 α 7 10

α 104 10ρ0 α 17 10ρ0 α 20 10ρ0

α 7 10 α 17ρ0 10 α 20ρ0 10 α

7ρ0

10 α 20ρ0

104

10 α 17 10ρ0 α 17 10

α 17 ρ0

µ

µ

µ

µ

104 104 104

1 : 105 µ

104

104

104 104

104

104

104 104 104

104 104

104

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3

∼ 50, 000 ∼ 25, 000

µ

1 : 210

µ

1 : 29

Ne ≈ 105

−80

s = 1τ ln(

ne,f

nr,f

nr,i

ne,i) τ

ne,i, ne,f nr,i, nr,f

≥

H2

f0 H2f0

f0 k

i

σ2g σ2

ϵ

H2f0 =

σ2g

σ2ϵ + σ2

g,

σ2ϵ =

1

ng

ng∑

i=1

σ2ϵ,i, σ2

g =1

ng

ng∑

i=1

[(Xi − X

)2 − σ2ϵ,i

].

ng σ2ϵ,i =

1nr,i(nr,i−1)

∑nr,i

k=1(Xki− Xi)2

i nr,i Xi

i X

ng H2

H2∆f

∆f

σ2p

H2∆f =

σ2g

σ2g + σ2

p + σ2ϵ.

j

k i Yijk

k Yij j

i

σ2ϵ =

1

n

ng∑

i=1

np,i∑

j=1

σ2ϵ,ij ,

σ2ϵ,ij = 1

nr,ij(nr,ij−1)

∑nr,ij

k=1 (Yijk − Yij)2 + σ2ϵ,i nr,ij

j i np,i

i n =∑ng

i=1 np,i

∆f

∆Xij ∆Xi

σ2p =

1

n

ng∑

i=1

np∑

i=1

(∆Xij −∆X i)2 − σ2

ϵ,j ,

σ2g = V ar(∆Xij)− σ2

p − σ2ϵ .

Xi ∆Xij j

i ζi

ζi = α+m∑

l=1

glial + ϵ,

m gli l i

1m − 1

m ϵ N(0,σ2ϵ ) al 0 σ2

a

V ar(ζi) mσ2a

⟨(ζi − ζp)2⟩ = −2mσ2

a

∑

l

gliglp + δ,

−∑

l gliglp i p

(ζi − ζp)2 = β

∑

l

gliglp + δ,

−∑

l gliglp −β2

ζi = Xi h2 = −β2

1V ar(Xi)

∆Xij h2 = −β2

1V ar(∆Xij)

h2

X i

∆Xij j i

(i) =1

n

log(1− r2i )

2 log 10

ri i

ζi

ζi = α+Glial + ϵi,

Gli i l

p < .05

ζi

W W1, ...Wn

W1 = W2 =

gl m 1

0

W gl

I(W, gl) =∑

W=(W1,...Wn)

p(W )∑

m=(0,1)

p(m|W ) log2p(m|W )

p(m)

p(m)

gl p(m|W = Wj)

Wj p(W = Wj)

Wj

Z

I(W, gl|Z) =∑

Z=(Z1,..Zq)

p(Z)∑

W=(W1,...Wn)

p(W |Z)∑

m=(0,1)

p(m|W,Z) log2p(m|W,Z)

p(m|Z)

M(W |Z) =∑

gl

I(W, gl|Z)

W = Kr W = E

W = F

M(Kr) M(E|Kr) M(F |Kr,E)

M(Kr) − Mp(Kr) Mp(Kr) M(E|Kr) −

Mp(E|Kr) M(F |Kr,E) − Mp(F |Kr,E)

0.5%

w0

wf

H2

h2

H2 − h2

H2 − h2

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EURDG�VHQVH KHULWDELOLW\�H2 =σ2tot−σ2

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σ2tot

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H2w0

H2w0

= 0.98

H2w0

= 0.99

h2w0h2w0

≈ 0.65

h2w0= 0.80 h2

H2w0

H2∆w ∆w =

wf − w0

H2∆w

∆w

H2∆w ≈ 0.75

h2∆w h2∆w = 0.36

h2∆w = 0.40

Yi i

Yi = µ+L∑

j=1

Gijaj + ϵi.

µ Gij = ±1 i

j L aj

j ϵi

18%

∆w

67%

49%

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h2w0

H2w0

w0

w0

w0

w0

30%

w0

∆w

w0

60%

49%

67%

50%

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E M p < 10−4

G

M p = 10−3

0.1 E G

∆w

∼ 25, 000 ∼ 50, 000

90%

4

Gen 0Gen 250Gen 500

Inte

rqua

rtile

inte

rval

, %

012345

−10 −5 0 5 10 15 20 250

20

40

60

80

100

Relative fitness, %

Num

ber o

f pop

ulat

ions

17%

49%

34%31%

3%

21%

29%50%

46%

4%

Founder fitnessFounderEvolutionarystochasticityMeasurement error

Founder genotype

Gen

250

Gen

500

Initial relative fitness, %

Fina

l rel

ativ

e fit

ness

, %

R2 = 0.81R2 = 0.59

−10 −5 0 5 10−10

−5

0

5

10

15

Gen 250Gen 500

Initial relative fitness, %

Fina

l rel

ativ

e fit

ness

, %

Gen 250Gen 500

−10 −5 0 5 10−10

−5

0

5

10

15

R2 = 0.42R2 = 0.23

A B

C D

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D Number of populations with mutationACE2 (12)

IRA1 (8)

SUR2 (3)GAT2 (3)

CDC28 (3)DIG1 (3)PIN4 (3)

SUP35 (3)CSI1 (3)

HSL1 (3)MSH3 (3)

ECM21 (3)MIH1 (4)EGT2 (4)IRC8 (4)PTR2 (4)FUS3 (4)

KEL1* (4)GRR1* (5)

WHI2 (5)HMG1 (6)

IRA2 (6)SUN4 (6)SFL1 (6)

Mul

tihit

gene

s (n

umbe

r of h

its)

0 1 2 3

Foun

der m

utat

ions

L003 S121 S028 L096aL096b L041 L094 L048 L098 L102a L013S002 L102

MAG

2

KEL1

*RN

H70

TRZ1

MRH

4TM

A10

TUP1

UBP3

GRR

1*G

SP1

ENA1

ENP1

RPL3

5B P

AU3

TIM

8M

AL33

CIT

2HH

F2CD

C40

ZIP2

UBP1

4PH

M7

ESL2

FM

P43

YLL0

65W

RPO

31

YDR

089W

STO

1PH

O89

MG

A2 A

RO2

NUP8

5 DC

D1

RAD3

3 SP

C2O

SW1

RIF1

RAD

33 S

PC2

ASI1

OSW

1

putatively neutral

putatively functional

intron (11)

synonymous(133)

intergenic(187)

promoter(226)

non-synonymous

(489)

premature stop (54)

frameshift(49)

Founders (fitness, %)

0

5

10

15

20

25

Num

ber o

f put

ative

ly fu

nctio

nal m

utat

ions Clones

Means

L003

(–2.3)

S121

(–1.1)

S028

(0.7

)L0

96a

(1.4

)L0

96b

(2.4

)S0

02 (3

.0)

L041

(3.1

)L0

94 (3

.3)

L048

(5.9

)L0

98 (6

.4)

L102

(6.7

)L1

02a

(6.9

)L0

13 (8

.3)

L003

L003

S121

S121

S028

S028

L096

a

L096a

L096

b

L096b

S002

S002

L041

L041

L094

L094

L048

L048

L098

L098

L102

L102

L102

a

L102a

L013

L013

0 0.07 0.13 0.20 0.27 0.30

2.9 3.8 4.7 5.6 6.5 7.4

Mean number of shared multihit genes

Mean number of shared GO Slim terms

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1 2 2

∆ ∆ ∆

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Fitness of background strain, %−2 0 2 4 6 8

−2

0

2

4

6

8whi2∆sfl1∆

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Generations, t

�s + s

0

s + s

Fitness,s(t)

T0 T1 T2 T3 T4

s

A

Epoch length, T

0Proba

bility,

f(T

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�⌧

⌧B

0

1Ns

0.01

0.5

0.99

1 � 1Ns

1Frequ

ency,x(t)

C

one cycle

�x

0 2000 4000 6000 8000 10000

Generations, t

0

1Ns

0.01

0.5

0.99

1 � 1Ns

1

Frequ

ency,x(t)

one cycle

�x

D

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N

t x(t)

f(x, t)

∂f

∂t= − ∂

∂x[s(t)x(1− x)f ] +

1

2

∂2

∂x2

[x(1− x)

Nf

].

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s(t) s1

s2

s ≫ |s| Ns ≫ 1

N |s| ≫ 1

τ δτ2

µ

Nµ ≪ 1

Nµ ≫ 1

Selection

Genetic drift

Selection andseasonal drift

Dominant force

window of opportunity

Freq

uenc

y of

mut

ant a

llele deleterious

epochbeneficial

epoch

2⌧c

2⌧c

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x1/2

0

1

time

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xsel

xseas

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xseas

x1/2

0

1 11

1� xseas

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1� xseas

1� xsel

1� xsel

1� xsel

xsel

B C DA

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τ

x t = 1/s√

x2N t =

√x

2Ns

xsel =1

2Ns xsel

xsel

x < xsel x

xselx

xselx = 1

N

1N

1xsel

= 2s[x(t) = 1

sest/

(1 + 1

s (est − 1)

)]

2s log(Ns)

xsel O(1s

)

pfix ≈ 1

2· 2s = s.

sτ ≫ 2 log(Ns)

sτ ≪ 2 log(Ns)

sτ > 1

sτ ≪ 2 log(Ns)

∂f

∂k= − ∂

∂x[⟨δx⟩ f(x, k)] + 1

2

∂2

∂x2[⟨δx2

⟩f(x, k)

],

x

0 = ⟨δx⟩ ∂p(x)∂x

+1

2

⟨δx2

⟩ ∂2p(x)

∂x2,

p(x) x ⟨δx⟩⟨δx2

⟩

x

pfix p(x)

⟨δx⟩⟨δx2

⟩

pfix s s τ δτ N

s = 0, δτ = 0

τ

1/s

τ 1/s

s

1/s 1/s

−s

∼ 1/s

τc = 1/s

τc/τ ≪ 1

Ns ≫ 1 τc = 1/s

x ≪ e−sτ/2

x

∼ τc ∼ τc

x

τc/τ

e−sτ/2

e−sτ/2

x1/2 = e−sτ/2

p(x1/2) = 1/2

x1/2

x1/2

p(x1/2) = 1/2

x1/2

∼ τc/τ

1N

x1/2 ≈ 1/Nx1/2

pfix ≈ τcτ

· 1/Nx1/2

· 12≈ 2 esτ/2

πNsτ,

O(1) 4/π

sτ ≪ 1

τc = τ x1/2 = 1/2

pfix = 1/N sτ ≪ 1 1 ≪ sτ ≪ log(Ns)

1/N s

sτ ≫ log(Ns)

s = 0

e2sτ

2sτx |s|τ ≪ 1 12sτ

x√

2τcx2N

12sτ

≈√

τcx2Nsτ

xsel =1

2N |s|τcτ |s| xsel ≫ x1/2

x1/2

xsel ≪ x1/2 xsel ≈ τcτ · 1/Nxsel

≈ 2|s|

s

|s| = s∗

s∗ ≡ τc/τ

4Nx1/2≈

⎧⎪⎪⎨

⎪⎪⎩

12N sτ ≪ 1

esτ/2

πNsτ sτ ≫ 1

1/2

pfix ≈

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

2s s ≫ s∗

2s∗ |s| ≪ s∗,

0 −s ≫ s∗

−s∗ < s < s∗

sτ ≫ 1 s∗

s∗ ∼ 1N

s ≪ s∗

s = 0

es∆T

es∆T

sδτ ≪ 1 sδτx√

2τcx/2N

xseas =τc

N(sδτ)2

xseas ≫ x1/2

xseas ≪ x1/2

(sδτ)2 ≫ τcNx1/2

≈

⎧⎪⎪⎨

⎪⎪⎩

2τN sτ ≪ 1

esτ/2

Ns sτ ≫ 1

sδτ sτ

xseas

x1/2 xseas log (x/xseas) / log(x1/2/xseas

)

c · xseas

c τcτ

1N

1xseas

∼ (sδτ)2

τ

pfix ∼ p

(1

N→ c · xseas

)· p

(c · xseas → x1/2

)

∼ [sδτ ]2

τ· 1

log[N(sδτ)2x1/2/τc

] .

1/N

sτ ≪ 1 pfix N

s = 0

|s|

xsel ≪ xseas xsel ≈ 2|s|

xseas

x1/2 xsel ≫ xseas

2sτ (sδτ)2 s = 0

log2(x1/2/xseas)/(sδτ)2 xseas x1/2

s

x1/2

s ≪ s∗

s∗ ≡ [sδτ ]2

4τ

1

log[N(sδτ)2x1/2/τc

] ,

1/2

s ≫ s∗

2s 0

s∗

sτ ≪ 1 sδτ ≪ 1

sτ ≪ 1

⟨δx⟩ = x(1− x)[2sτ + (1− 2x)(sδτ)2

],

⟨δx2

⟩= x(1− x)

2τ

N+ 2x2(1− x)2(sδτ)2.

δτ = 0

s 2τ

δt > 0 δx

sτ ≫ 1

x ≪ 1 − esτ/Ns

⟨δx⟩ = x (2sτ) + x (sδτ)2 + x22esτ

Ns,

⟨δx2

⟩= 2x2(sδτ)2 +

2x

Ns

(1 + x2esτ

).

x ≪ x1/2

τc/τ = 1/(sτ) x ! x1/2

δx

x ! 1− esτ/Ns

s → −s

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p(x)

x = x1/2

pfix(s;N, s, τ, δτ) =2s

1− e−s/s∗,

s∗

s s∗

s N s τ δτ

s∗

|s| ≪ s∗

2s∗ s ≫ s∗ 2s

|s| ≫ s∗ 2|s|e−|s|/s∗

pfix(−s)/pfix(s)

s

pfix(−s)

pfix(s)= e−s/s∗ ,

|s|

s∗

|s| ! s∗ s∗

s∗ = 12N

Ne = 2/s∗

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2/2τ

s < 0

1/N

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p(−s, τ > 0)

p(s, τ = 0)=

1

ess∗ − 1

≈

⎧⎪⎪⎨

⎪⎪⎩

s∗

s s ≪ s∗

e−ss∗ s ≫ s∗

.

s∗

pfix ≈ s

dN/dS

s ≈ ±10%

τ ≈ 10 N 106

1 ! sτ ≪ 2 log(Ns)

N

sτ < 2 log(Ns) sτ ≪ 1

sδτ ≪√

τ/N

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sτ

s τ

sτ ≈ 1

N ≈ 105

δτ/τ ≫ 0.01

1/τ

sδτ ! 1

6

N U

s

ϵ ϵ

ωc

ωc

ϵ

ϵc

ϵ > ϵc

ϵc

ϵ < ϵc

ϵc

ωc ϵc

ϵc

Fitness (ω)

Bene

ficia

l fra

ctio

n of

mut

atio

ns (ε

) ω ≈ ω c ϵ (U, s, N)c

Adaptionω< ω c

ω> ω c

ϵ> ϵ c

ϵ< ϵ c

Muller’s ratchet

Attractor

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N

s s > 0 k

k

k s(k − k)

nk

U ϵ

Ub = Uϵ

Ud = U(1− ϵ)

k

dnk

dt= s(k − k)nk︸ ︷︷ ︸

I

−Unk︸︷︷︸II

+Udnk−1︸ ︷︷ ︸III

+Ubnk+1︸ ︷︷ ︸IV

.

k

s ≪ 1

ϵ k

k k ϵ

k ϵ

ϵc

nk

k ϵ

nkdnkdt = 0 k

ϵ = 0

nk = Ne−λ λk

k! λ = U/s k = 0

0

1.0

0

250

500

0.5

100 150 200 250 300 350 400 450

Nose fitnessNose occupancy

Nos

e occ

upan

cy

Nos

e fitn

essExtinction Establishment

Numb

er of

indivi

duals

Fitness class

n

n

0

1

Deleterious mutations U(1−ϵ)Beneficial mutations Uϵ

“Nose”

A

B

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ϵc = 0

ϵc > 0

ϵ > 0

ϵc

k

k = λ

nk = Ne−λ(1−2ϵ)+k log

√1−ϵϵ Jk(2α),

−350 −300 −250 −200 −150100

102

104

106

108

Fitness class

Num

ber of individuals

k = λ k⋆

SimulationTheory: nkGaussianPoisson

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Jk k α ≡ λ√

ϵ(1− ϵ)

k k

σ2k = λ(1 − 2ϵ)

ϵ > 0

k⋆

nk k > k⋆

k⋆ k < k⋆

k⋆ k⋆ − 1

III (k− k⋆−λ)nk⋆ +λϵnk⋆+1 = 0

k⋆

k⋆Jk⋆(2α) = αJk⋆+1(2α)

k⋆ ≈

⎧⎪⎪⎨

⎪⎪⎩

λ2ϵ ϵλ2 ≪ 1

2λ√

ϵ(1− ϵ) ϵλ2 ≫ 1.

k⋆ k⋆ = 0

ϵc = 0

e−λ Ne−λ

λ

k⋆ k⋆

ϵ

ϵ = ϵc N

nk⋆ U s ϵ nk⋆

ϵc(N, s, U)

nk⋆ nk⋆

nk⋆

nk⋆

nk⋆s > 1

ϵλ2 < 1 nk⋆ ≈ Ne−λ k⋆ ≈ 0

r−

r− ≈ e−γsnk⋆γs√

γsnk⋆/π γ

r+

r+ = nk⋆UbPest k⋆ ≈ 0

Pest ≈ 2s

ϵc =γ2

2λ

e−γsNe−λ

√γsNe−λπ

.

ϵc

λϵc(U,N, s)/γ2 ∼ f(γsNe−λ)

γ

γ = 1/√λ λ

γ ≈ 0.6 0.4 ! 1/√λ ! 0.8

0.1 ! 1/√λ ! 1

γ λ = U/s

ϵc Nse−λ λ

ϵ

nk⋆s ! 1

r−

r− ≈ 1/(2nk⋆)

r+ = ϵUnk⋆Pest Pest

12(k⋆+1)s

Pest = 2(k⋆ + 1)s r− = r+

nk⋆ ≈ 1√4ϵUsk⋆

.

ϵc(N, s, U)

0 2 4 6 810−6

10−4

10−2

100

102

Nse−λ/√

λ

λ2ϵ c

A Slow ratchet regime

1

2

4

8

16

32

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

λ−1 logNs

ϵ c

ϵc (simulation)

ϵ c(n

umer

ics)

B Fast ratchet regime

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

16

32

64

128

256

512

1024

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λ � DQG HYHQ LQ

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ϵc

λ nk⋆s ! 1

ϵc ≪ 12 λ λ

ϵcλ2 ≫ 1 ϵc ≪ 12

√ϵc log ϵc = λ−1 log

(Nse−λ(1−2ϵc)

)

ϵc ≈z2

log2 [z/ log (z−1)], 2z ≡ 1− logNs

λ.

ϵc Ns λ

ϵc Nse−λ

ϵ = 12

ϵc λ

ϵc ≈1

2−

(3

4λlogNs

)1/3

.

ϵmax ϵc < ϵmax

λ

logNs≤ 3

4

[1

0.5− ϵmax

]3,

ϵmax ! 0.2 ϵmax = 1/4 48 ϵmax = 1/3

Ns ! 1

U ≫ s

ϵc

ϵc

ϵc

ϵc → 0

ϵc → 1/2

ϵc

ϵc Ns

λ ϵc

Ns λ

Nse−λ = 1

ϵ ≪ ϵc ϵ ≫ ϵc ϵ →

ϵc ϵc

ϵc

U/s N

k⋆

Ns ∼ 103

U/s ∼ 8 2%

ϵ > ϵc ϵ < ϵc

1e−10

0.001

0.02

0.05

0.1

0.2

0.3

0.4

102 104 106 108 1010 1012 1014

Ns

λFast ratchet regime

sNe−� < 1

Slow ratchet regimesNe−� > 1

HIV0.5%

1.5%1.0%

VSV

E. coli (WT)

E. coli (MUT)

PhageYeast

2

8

32

128

512

2048

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2% 0.2% N 100 200

N ! 30

ϵc

ϵc ϵc ≪ 10−10

s

ϵ

ϵ

N ϵ

ϵ

ϵ

nk k

nk

es(k−k)−α nk

k (k − k) k α =∑

nk−NN

N

±s

Uϵ U(1 − ϵ)

P (i, j) i j

nk nk

k

nk

104

ϵc(U,N, s) ϵ v

v ϵc

ϵ k

k ϵc

A

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B

∆

α∆

∆

x y z x y

z

∆ WT ∆

+30

1 : 215

1 : 25

−80

−0.45 [−3.2%, 0.49%]

0.14

[−2.08%, 2.11%]

−4.9% 1.0% −0.41%

−3.1% 3.4%

0.14%

−2.3% 3.0% −0.67%

1 : 25

−80

1 : 25

1 : 1

1 : 25

F =1

t2 − t1log

(n(t2)

nr(t2)

/n(t1)

nr(t1)

),

n(t) nr(t) t

t1 = 10 t2 = 30

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∆ ∆

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Θ

Θ ∼ Beta(α,β)

α = 1ϵ−4 β = 0.271 α = 1ϵ−4

β = 0.434

> 5%

R A

ai

A i ki

A R mi

i

mi =

⎧⎪⎪⎨

⎪⎪⎩

1, i,

0, .

m = (m1,m2, . . . ,mN ) ∈ {0, 1}N N

m

Ppost(m|a,k) m

k = (k1, ..., kN ) A a =

(a1, ..., aN )

Ppost(m|a,k) = Pprior(m) L(a|m,k)∑m

Pprior(m) L(a|m,k),

L(a|m,k) A

m k Pprior(m)

m

Θ A Θ fΘ(θ)

Ppost(m|a,k) =Pprior(m)

∫ 10 L(a|m,k, θ) fΘ(θ) dθ

∑m

Pprior(m)∫ 10 L(a|m,k, θ) fΘ(θ) dθ

.

L(a|m,k, θ) a m

k θ

mi = 0 i

θ mi = 1

1− θ

L(a|m,k, θ) =N∏

i=1

P (ai|mi, θ, ki)

=N∏

i=1

(kiai

)θmi(ki−ai)+(1−mi)ai(1− θ)miai+(1−mi)(ki−ai)

= θ∑

i[mi(ki−ai)+(1−mi)ai](1− θ)∑

i[miai+(1−mi)(ki−ai)]N∏

i=1

(kiai

).

fΘ α β

fΘ(θ) =θα−1(1− θ)β−1

B(α, β),

B(α, β)

Ppost(m|a,k) =Pprior(m)

∫ 10 θα(m,k)−1(1− θ)β(m,k)−1dθ

∑m

Pprior(m)∫ 10 θα(m,k)−1(1− θ)β(m,k)−1dθ

=Pprior(m) B (α(m,k),β(m,k))∑

mPprior(m) B (α(m,k),β(m,k))

,

α(m,k) = α+N∑

i=1

[mi(ki − ai) + (1−mi)ai] ,

β(m,k) = β +N∑

i=1

[miai + (1−mi)(ki − ai)] .

m

10−6

Pprior(m) = 10−6 m

Pprior(m) = 10−12

m

m0 =

(0, 0, . . . , 0)

Pprior(m0)∑

m Pprior(m) = 1

m

m0

t = 0, 250 500

X(t)ijk k j

i t i = 1, 2, . . . nF = 66 j = 1, 2, . . . , npop(i) k = 1, 2, . . . , nrep(i, j)

npop(i) = 10 nrep(i, j) = 2

i t = 0

xi =1

∑nF(i)j=1 nrep(i, j)

nF(i)∑

j=1

nrep(i,j)∑

k=1

X(0)ijk

i xi

k j i t

Y (t)ijk = X(t)

ijk − xi.

t

fN (x;µ,σ2)

µ σ2 x

Yij· =1

nrep(i, j)

nrep(i,j)∑

k=1

Yijk

j i

VijY =1

nrep(i, j)

nrep(i,j)∑

k=1

(Yijk − Yij·

)2

j i

Yi·· =1

∑npop(i)j=1 nrep(i, j)

npop(i)∑

j=1

nrep(i,j)∑

k=1

Yijk

i

ViY =1

∑npop(i)j=1 nrep(i, j)

npop(i)∑

j=1

nrep(i,j)∑

k=1

(Yijk − Yi··

)2

i

Y

Yijk = α+ εijk,

α εijk

σ2e

α σ2e

L1A(Y ;α,σ2

e

)=

nFG∏

i=1

npop(i)∏

j=1

nrep(i,j)∏

k=1

fN (Yijk;α,σ2e).

Yijk = α+ βxi + εijk,

α + βxi i

α β σ2e

L1B(Y ;α,β,σ2

e

)=

nF∏

i=1

npop(i)∏

j=1

nrep(i,j)∏

k=1

fN (Yijk;α+ βxi,σ2e),

Yijk = α+Aij + εijk,

α + Aij j i

Aij σ2s

α σ2s σ2

e

L2A(Y ;α,σ2

s ,σ2e

)=

nF∏

i=1

npop(i)∏

j=1

Arep(i, j) fN(Yij·;α, σ

2ij

),

Arep(i, j) =(2πσ2

e

)−nrep(i,j)−12 (nrep(i, j))

− 12 exp

{− VijY

2σ2e/nrep(i, j)

},

σ2ij = σ2

e/nrep(i, j) + σ2s .

Yijk = α+ βxi +Aij + εijk.

α β σ2s σ2

e

L2B(Y ;α,β,σ2

s ,σ2e

)=

nF∏

i=1

npop(i)∏

j=1

Arep(i, j) fN(Yij·;α+ βxi, σ

2ij

),

Arep(i, j) σ2ij

Yijk = α+Bi +Aij + εijk,

α + Bi i Bi

σ2g

α σ2g σ2

s σ2e

L3A(Y ;α,σ2

g ,σ2s ,σ

2e

)=

nF∏

i=1

⎛

⎝npop(i)∏

j=1

Arep(i, j)

⎞

⎠Apop(i) fN(Yi··;α,σ

2g + σ2

i

),

Apop(i) = (2π)−npop(i)−1

2

⎛

⎝ σ2i∏npop(i)

j=1 σ2ij

⎞

⎠

12

exp

{−ViY

2σ2i

},

σ2i =

⎛

⎝npop(i)∑

j=1

1

σ2ij

⎞

⎠−1

,

Yi·· = σ2i

npop(i)∑

j=1

Yij·σ2ij

,

ViY = σ2i

npop(i)∑

j=1

(Yij·

)2

σ2ij

−(Yi··

)2,

Arep(i, j) σ2ij

Yijk = α+ βxi +Bi +Aij + εijk,

α + βxi + Bi i

α β σ2g σ2

s σ2e

L3B(Y ;α,β,σ2

g ,σ2s ,σ

2e

) nF∏

i=1

⎛

⎝npop(i)∏

j=1

Arep(i, j)

⎞

⎠Apop(i) fN(Yi··;α+ βxi,σ

2g + σ2

i

),

Arep(i, j) Apop(i) σ2i Yi··

σ2e σ2

s σ2g

σ2f = VY − σ2

g − σ2s − σ2

e ,

VY =1

∑nFi=1

∑nF(i)j=1 nrep(i, j)

nF∑

i=1

nF(i)∑

j=1

nrep(i,j)∑

k=1

(Yijk − Y

)2

Y =1

∑nFi=1

∑nF(i)j=1 nrep(i, j)

nF∑

i=1

nF(i)∑

j=1

nrep(i,j)∑

k=1

Yijk

P ≪ 10−3

P = 0.018

σ2g σ2

f σ2s

c nsyn(c) nnon(c)

nstop(c)

νc

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fi =19

∑c νcni(c) i = syn, non, stop

fstop

χ2

χ2

P < 0.001 < 0.001

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pgene = C(Lgene + Lprom),

Lgene Lprom = 500

C

818

M0 =

815 nk

k k k = 0, 1, 2, . . .

k k > 1

k

i Mi

M0 k

Bk(Mi) = nk − nk(Mi), nk(Mi) k Bk(Mi) > 0

k > 1 kBk(Mi) M0

M0−∑

k>1 kBk(Mi) M0−∑

k>1 kBk(Mi) > Mi

Mi nk(Mi)

k Mi+1 = ⌈M0 −∑

k>1 kBk(Mi)⌉

|Mi+1 −Mi| < 1∑

k>1 kBk(Mi)

i = 4 129

Mij j i

Mij = Bij +Nij ,

Bij Nij

Bij ∼ Poisson(βi) Nij ∼ Poisson(ν)

k k

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RI PXOWL�KLW JHQHV KDYHP < 0.01�

M = N−1∑ij Mij , EM ≡ λ = β + ν,

β = N−1∑iNiβi Ni

i N =∑

iNi = 104

λ = 7.9

β ν λ

β = fλ ν = (1 − f)ν f = N−1∑iNifi

fi = βi/λ

i

fi

fi = f − α

λ

xi − x

xmax − xmin,

xi i x = N−1∑iNixi = 3.2%

α > 0

xmin = −2.3%

xmax = 8.3% α

f α

0 ≤ fi ≤ 1

α ≤ αmax(f) ≡ min

{λf

xmax − xmin

xmax − x,λ(1− f)

xmax − xmin

x− xmin

}.

f

fi

fi f = (xmax − x)/(xmax − xmin ) = 0.48

1000 f = 0.1, 0.25, 0.5, 0.75, 0.9 α

0 αmax(f)

Mij

P

α f

Mij

Var Mij = βi + ν = λf − αxi − x

xmax − xmin+ λ− λf = λ− α

xi − x

xmax − xmin

f

0.60 P 0.05 α ≥ 2.1

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c(i, j)

i j CI

PI

CI =2

N(N − 1)

∑

i

∑

j>i

c(i, j),

P I =2∑

k Nk(Nk − 1)

∑

i

∑

j>i:φ(i)=φ(j)

c(i, j).

φ(i) i N = 104 Nk

k

CI

CI

P < 10−4 P = 0.013

PI

PI CI

PI P = 0.055

P = 0.75

Yij

j i

mij

Yij = α+ βmij + εij ,

εij σ2e

α β σ2e

L1A(Y ;α,β,σ2

e

)=

nF∏

i=1

npop(i)∏

j=1

fN (Yij ;α+ βmij ,σ2e).

β = 0

L0(Y ;α,σ2

e

)=

nF∏

i=1

npop(i)∏

j=1

fN (Yij ;α,σ2e),

xi i

Yij = α+ βmij + γxi + εij .

α β γ σ2e

L2A(Y ;α,β, γ,σ2

e

)=

nF∏

i=1

npop(i)∏

j=1

fN (Yij ;α+ βmij + γxi,σ2e).

β = 0

L1B(Y ;α, γ,σ2

e

)=

nF∏

i=1

npop(i)∏

j=1

fN (Yij ;α+ γxi,σ2e),

Yij = α+Ai + βmij + εij ,

Ai σ2g

α σ2g β σ2

e

L2B(Y ;α,β,σ2

g ,σ2e

)=

nFG∏

i=1

A(i) fN(Yi·;α+ βmi·, σ

2i

),

A(i) =(2πσ2

n

)−npop(i)−12 (npop(i))

− 12 exp

{−ViY − 2β Ci [Y,m] + β2 Vim

2σ2e/npop(i)

},

σ2i = σ2

e/npop(i) + σ2g ,

Ci [Y,m] =1

npop(i)

npop(i)∑

j=1

(Yij − Yi·

)(mij − mi·)

i

Yij = α+Ai + εij ,

L1C(Y ;α,σ2

g ,σ2e

)=

nFG∏

i=1

A(i) fN(Yi·;α, σ

2i

),

∼ 6

∼ 5 × 10−10

! 106

fitnesslow high

DivAnc

Diversification240 gen

Adaptation500 gen

Founder 1

Founder 2

Founder 64

...

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Initial relative fitness, %

Fina

l rel

ativ

e fit

ness

, %

−2 0 2 4 6

2

4

6

8

10

12 L041

L094

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SRSXODWLRQV ZHUH LQGLVWLQJXLVKDEOH LQLWLDOO\ �1HVWHG $129$�F1,37 = 0.03, P = 0.86�� EXW VLJQLĆFDQWO\ GLIIHUHQWDIWHU ��� JHQHUDWLRQV RI HYROXWLRQ �F1,37 = 15.95, P < 10−3�� 6HH DOVR 7DEOH %��� (UURU EDUV VKRZ±1 VHP�

All

0

20

40

60

80

Num

ber o

f mut

atio

ns

PutativelyNeutral

0

5

10

15

20

25

PutativelyFunctional

0

10

20

30

40

50

60

Likely mutatorsDiploids

Founder

L125

L034

L003

S121

S028

L096

a

L096

b

S002

L041

L094

L048

L098

L102

L102

a

L013

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Number of mutations

Fitness incre

ment, %

Fitness incre

ment, %

Mutations in genes

hit 3 or more times

Conservative set of

beneficial mutationsA

NC

OV

AM

ultip

le r

egre

ssio

n

0 1 2 3 4 5 6 7

−2

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7

−2

0

2

4

6

8

10

12

14

0 1 2 3 4 5

−2

0

2

4

6

8

10

12

14

0 1 2 3 4 5

−2

0

2

4

6

8

10

12

14L003S121S028L096aL096bS002L041L094L048L098L102L102aL013

L003S121S028L096aL096bS002L041L094L048L098L102L102aL013

L003S121S028L096aL096bS002L041L094L048L098L102L102aL013

L003S121S028L096aL096bS002L041L094L048L098L102L102aL013

A

C

B

D

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Mutations in geneshit 3 or more times

Num

ber o

f mut

atio

ns

0

2

4

6

8

10 Conservative set ofbeneficial mutations

Expanded set ofbeneficial mutations

L003

S121

S028

L096

aL0

96b

S002

L041

L094

L048

L098

L102

L102

aL0

13

L003

S121

S028

L096

aL0

96b

S002

L041

L094

L048

L098

L102

L102

aL0

13

L003

S121

S028

L096

aL0

96b

S002

L041

L094

L048

L098

L102

L102

aL0

13

ClonesMeans

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ℓ P1 P2

α β γ σ2g σ2

e

4.72 11.14 544.8

3.89 0.42 11.02 542.7 0.148

7.08 −0.73 5.49 470.2

4.48 5.44 4.45 487.3

5.75 0.74 −0.77 4.86 456.5 ≪ 10−6 2ϵ−4

3.27 0.62 5.81 3.92 475.5 ≪ 10−6 2ϵ−4

4.72 11.14 544.8

4.08 0.61 10.94 541.9 0.090

7.08 −0.73 5.49 470.2

4.48 5.44 4.45 487.3

6.11 1.08 −0.78 4.60 450.8 ≪ 10−6 1ϵ−5

3.40 1.02 6.13 3.60 471.5 ≪ 10−6 7ϵ−5

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Mutations in geneshit 3 or more times

0

1

2

3

4

5 Conservative set ofbeneficial mutations

0

1

2

3

4

5

6

Expanded set ofbeneficial mutations

Num

ber o

f mut

atio

ns

Founder fitness relative to DivAnc, %−3 −2 −1 0 1 2 3 4 5 6 7 8 90

2

4

6

8

10 All putativelyfunctional mutations

−3 −2 −1 0 1 2 3 4 5 6 7 8 90

5

10

15

20

25

ClonesMeans

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0 0.02 0.04 0.06 0.08 0.1 0.120

0.02

0.04

0.06

0.08

0.10

4 4.2 4.4 4.6 4.8 5 5.20

0.002

0.004

0.006

0.008

0.010

0.012

0 0.01 0.02 0.03 0.04 0.05 0.060

0.01

0.02

0.03

0.04

0.05

3 3.5 4 4.5 5 5.50

0.01

0.02

0.03

0.04

Genes GO Slim

Genes GO Slim

Parallelism

Convergence

P < 10–4 P = 0.013

P = 0.055 P = 0.75

PI

CI CI

PI

Pro

bability

Pro

bability

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JHQFH�

R2 = 0.79

Fitness relative to DivAncCit, %

Fitn

ess

rela

tive

to R

mR

ef, %

−2 0 2 4 6 8−4

−2

0

2

4

6

8

10

–1.1%

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Distance between mutations, bp

Num

ber o

f mut

atio

ns in

a c

lone

100

101

102

103

104

105

106

0

400

800

1200

1600

2000

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2 param

3 param

4 param

5 param

Model 3Aα σg2 σs2 σe2

Model 1Aα σe2

α

Model 2Aσs2 σe2

withoutcovariate

Model 3Bα β σg2 σs2 σe2

Model 2Bα β σs2 σe2

Model 1Bα β σe2

withcovariate

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0 20 40 60 80 1000

100

200

300

400

500

600

700

800

Num

ber o

f gen

es w

ith m

utat

ion

Number of sampled clones

DataRandomization

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0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Strength of trend�  α

Pow

er

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C

sτ ≪ 2 log(Ns) ,

sτ ≪ 1 ,

sδτ ≪ 1 ,

Ns ≫ 1

sτ ≪ 1 ≪ 2 log(Ns)

τ

1/s

dx

dt= x(1− x)s(t) +

√x(1− x)

Nη(t),

η(t) ⟨η(t)⟩ = 0 ⟨η(t)η(t′)⟩ = δ(t− t′)

dx = x(1− x)s(t)dt+

√x(1− x) dt

Nη(t).

δx =∫epoch dx ≪ x x

δx =x(1− x)

(e(s+s)T1+(−s+s)T2 − 1

)

1 + x(e(s+s)T1+(−s+s)T2 − 1

) +

√x(1− x)(T1 + T2)

Nη

≈ 2sT2 x(1− x) + s(T2 − T1)x(1− x) +1

2s2(T2 − T1)

2x(1− x)(1− 2x)

+

√x(1− x)(T1 + T2)

Nη.

T1 T2 δx

⟨δx⟩ = x(1− x)2sτ + x(1− x)(1− 2x)(sδτ)2

⟨δx2

⟩= x(1− x)

2τ

N+ 2x2(1− x)2(sδτ)2,

⟨δx⟩ = x(1− x)τ

N

[1

xsel+

1

xseas(1− 2x)

]

⟨δx2

⟩= x(1− x)

2τ

N

[1 +

1

xseasx(1− x)

].

x

[1

xsel+

1

xseas(1− 2x)

]∂p

∂x+

[1 +

1

xseasx(1− x)

]∂2p

∂x2= 0,

∂x log

[∂xp(x)

∣∣∣∣1 +1

xseasx(1− x)

∣∣∣∣

]= −

1xsel

1 + 1xseas

x(1− x).

x± 1+x(1−x)/xseas x± = (1±√1 + 4xseas)/2

∂xp(x) = C1

|(x− x+)(x− x−)|

∣∣∣∣x+ − x

x− x−

∣∣∣∣λ

= C|x+ − x|λ−1

|x− x−|λ+1,

λ = 1xsel(x+−x−) p(0) = 0

p(1) = 1

p(x) =

1−∣∣∣∣1− x/x+x/x− − 1

∣∣∣∣λ

1− |x−/x+|2λ.

sτ ≪ 1 1N ≪ x±

x

p(x) ≈λ x

x+ − x−x−x+

1− |x−/x+|2λ=

2sNx

1− exp

(2λ log

∣∣∣∣x−x+

∣∣∣∣

) .

pfix = ⟨p(x)⟩ = 2s

1− e−s/s∗,

⟨x⟩ = 1N +O(sτ, sτ, sδτ) s∗

s∗ =

(sδτ)2

4τ

√1 + 4τ

N(sδτ)2

log

⎡

⎣

√1 + 4τ

N(sδτ)2 + 1√

1 + 4τN(sδτ)2 − 1

⎤

⎦

.

s∗ ≈

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

12N

N(sδτ)2

4τ ≪ 1

(sδτ)2

4τ · 1

log[N(sδτ)2

τ

] N(sδτ)2

4τ ≫ 1

s∗

1 ≪ sτ ≪ 2 log(Ns)

sτ ≫ 1

sτ ≪ 1 sδτ ≪ 1

sτ ≫ 1 s ≪ s δτ ≪ τ

s δτ

χ = x/(1 − x)

∂f

∂t= − ∂

∂χ[s(t)χf ] +

1

2

∂2

∂χ2

[χ(1 + χ)2

Nf

].

χ ≪

(Ns)−1 χ ≫ Ns χ ≪ Ns

∂f

∂t= − ∂

∂χ[s(t)χf ] +

1

2

∂2

∂χ2

[ χN

f].

χ′ = 1/(1− x)

∂f

∂t= − ∂

∂χ′[−s(t)χ′f

]+

1

2

∂2

∂χ′2

[−χ′

Nf

].

χ ≤ 1

χ′ ≤ 1 χ = χ′ = 1,

χ = χ0 ≪ 1 T1

δχ χ

T2 χ(t) Hχ(z, t) =

⟨exp (−zχ(t))⟩ χ(0) = χ0 s(t)

Hχ(z|χ0) = exp

⎡

⎣ −z χ0 e∫ t0 s(t

′)dt′

1 + z2N e

∫ t0 s(t

′)dt′∫ t0e

∫ t′0 −s(t′′)dt′′dt′

⎤

⎦.

t 1s ≪ t ≪ T1

χ

Hχ (z, t |χ0) = exp

[− zχ0e(s+s)t

1 + z2Nse

(s+s)t

].

ν1 χ(t) = ν1e(s+s)t ν1

χ ν1

Hν1(z, t|χ0) = ⟨exp (−zν1) |χ0⟩ = Hχ

(ze−(s+s)t, t |χ0

)≈ exp

[− zχ0

1 + z2Ns

],

⟨ν1⟩ = χ0, var (ν1) =χ0

Ns.

χ = 1 − log(ν1)s+s T1 + log(ν1)

s+s

χ′ ν1 ν2

χ′(t′) = ν2e(s−s)t′ t′

χ′ = 1 (T1 + log(ν1)/(s + s))

ν2 t′ 1s ≪ t′ ≪ T2

Hν2(z, t|ν1, T1) = exp

[−z 1ν1e−(s+s)T1

1 + zNs

],

ν2

⟨ν2|ν1, T1⟩ =1

ν1e−(s+s)T1 , var (ν2|ν1, T1) =

2

ν1Nse−(s+s)T1 .

χ

1 T2+log(ν2)/(s−s)

Hχ (z|T1, T2, ν1, ν2) = exp

[−z 1ν2e(−s+s)T2

1 + z2Ns

].

δχ Hδχ(z, t) = ezχ0Hχ(z, t)

δχ

⟨δχ|T1, T2, ν1, ν2⟩ =1

ν2e−(s+s)T2 − χ0,

⟨δχ2|T1, T2, ν1, ν2

⟩=

x0Ns

+2

2Ns

(1

ν2e−(s+s)T2 − χ0

)+

(1

ν2e−(s+s)T2 − χ0

)2

.

ν1, ν2, T1, T2

T1 T2 ν1 T2 ν2

sτ ≪ 1 sτ ≪ 1

⟨δχ⟩ = χ0

[2sτ + (sδτ)2 +

2esτ

Nsχ0

]

⟨δχ2

⟩= 2χ0

[χ0(sδτ)

2 +1

Ns

(1 + χ2

0esτ)]

.

x ≪ 1 χ ≈ x δx

⟨δx⟩ = x

[2sτ + (sδτ)2 + x

2esτ

Ns

]

⟨δx2

⟩= 2x

[x(sδτ)2 +

1

Ns

(1 + x2esτ

)],

⟨δx⟩ = xτ

N

τcτ

[1

xsel+

1

xseas+ 2

x

x21/2

]

⟨δx2

⟩= 2x

τ

N

τcτ

[1 +

x

xseas+

(x

x1/2

)2].

x−21/2

p(x) x±

1 + xxseas

+(

xx1/2

)2λ =

x21/2

xsel(x+−x−)

p(x) = C

(1− x

x−

1− xx+

)λ

+D.

p(0) = 0 x1/2

p(x) =

(1− x

x−1− x

x+

)λ

− 1

(1−

x1/2x−

1−x1/2x+

)2λ

− 1

,

p(x) =x

xsel(1−

x1/2x−

1−x1/2x+

)2λ

− 1

x ≪ 1

δττ δ

p(T ) ≈ δ(T − τ),

⟨χ⟩ = 1

2×

⟨∫ T1

0

dt

τ

1

NesT1−sT2

⟩

︸ ︷︷ ︸+

1

2×

⟨∫ T2

0

dt

τ

1

Ne−st

⟩

︸ ︷︷ ︸=

1

Nsτ+O (sτ, sδτ) .

pfix = p(⟨x⟩) = 2s

1− e−s/s∗,

s∗ =

x+−x−2Nsτx2

1/2

log

[1− x+

x1/2

1− x−x1/2

] ≈

⎧⎪⎪⎨

⎪⎪⎩

esτ/2

πNsτ (sδτ)2 ≪ esτ/2

Ns

(sδτ)2

4τ1

log(Nse−sτ/2(sδτ)2)(sδτ)2 ≫ esτ/2.

Ns

τ1 = τ2

s s τ δτ s1 s2 τ1 τ2 δτ1 δτ2

τ ≡ τ1 + τ22

,

s ≡ s1τ1 + s2τ22τ

,

s ≡ s1τ1 − s2τ22τ

,

δτ ≡√

(s1δτ1)2 + (s2δτ2)2

2s2,

Nµ ≫ 1

µ

ν

µ = ν

Ns ≫ 1 µ ≪

s

Nµ → 0

Nµ ≫ 1

N ττcx

x µ ·N ττcx ·

τcτ = Nµ · x τc/τ

Nµ ≫ 1

s · N ττcx · x

(sδτ)2

τ · N ττcx · x xsel = µ 1

|s|τcτ

xseas = µ 1(sδτ)2 τc

xsel

xsel

xsel

xsel xsel < x1/2

12

1−xsel

12 xsel < x1/2 xsel ! x1/2

12 s∗ = µ τc

τ1

x1/2

s∗ = 1N

τcτ

1x1/2

|s| ! s∗

12

12

xseas 1 − xseas

xseas x1/2 1− xseas 1− x1/2

|s| ! s∗ ∼ (sδτ)2

τ1

log(x1/2/xseas)

Nµ ≫ 1

Nµ ≪ 1 N 1/µ

s∗

xseas 1 − xseas

0 =∂

∂x[−2sτ x(1− x)f(x)] +

∂2

∂x2[2(sδτ)2 x2(1− x)2f(x)

].

ξ = log(

x1−x

)

f(ξ) ξ1/2 = log(

x1/2

1−x1/2

)−ξ1/2

f(ξ) ∝

⎧⎪⎪⎨

⎪⎪⎩

exp[

s(sδτ)2 ξ

], ξseas < ξ < ξ1/2

exp[2 s(sδτ)2 ξ1/2

]exp

[s

(sδτ)2 ξ], −ξ1/2 < ξ < −ξseas,

f(x)

f(x) ∝ 1

x(1− x)exp

[s

(sδτ)2log

(x

1− x

)],

xseas 1 − xseas f(x)

xseas 1− xseas

f(xseas)

f(1− xseas)= exp

[2s

(sδτ)2log

(xseas

1− xseas

1− x1/2x1/2

)]≈ exp

(− s

2s∗

).

log(

x1−x

)

12

2T

⟨log

(x

1− x

)⟩

cycle

=1

2T

∫ 2T

0dt log

(x(t)

1− x(t)

)= log

(x(0)

1− x(0)

)+

sT

2.

⟨log

(x

1− x

)⟩=

⟨log

(x(0)

1− x(0)

)⟩+

sτ

2.

⟨log

(x

1−x

)⟩

⟨log

(x

1− x

)⟩=

∫ ξ1/2ξseas

(ξ + sτ2 )f(ξ)dξ +

∫ ξseas−ξ1/2

(ξ − sτ2 )f(ξ)dξ

∫ ξ1/2ξseas

f(ξ)dξ +∫ ξseas−ξ1/2

f(ξ)dξ,

⟨log

(x

1−x

)⟩

ξ1/2 − ξseas= coth

( s

2s∗

)− 2s∗

s=

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

1, s ≫ s∗

0, |s| ≪ s∗

−1, s ≪ −s∗

.

12

s∗

|s| > s∗

sτ ≪ 2 log(Ns) ,

sτ ≪ 1 ,

sδτ ≪ 1 .

1 ≪ 2 log(Ns) ≪ sτ

sτ ≫ 2 log(Ns) ≫ 1

δττ δ

f(T ) ≈ δ(T − τ) .

δ

log(Ns)−1

f(Tfix) ≈ δ

(T − 2

slog[Ns]

).

pfix ≈ 1

2︸︷︷︸×

︷ ︸︸ ︷P [T0 > Tfix] × 2s︸︷︷︸

s

,

T0

P [T0 > Tfix]

pfix ≈ s

[∫ τ− 2s log(Ns)

0

dT0

τ

]≈ s

[1− 2 log(Ns)

sτ

].

sτ !

2 log(Ns)

sτ

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sτ ! 1

sτ

s ∼ s s1 s

pfix s

s/s∗ = 1/ (s∗τ) ≫ 1

s

x ≪ 1

∂x

∂t≈ s(t)x+

√x

Nη(t) .

pfix = 1−⟨exp

[− 2Nx∫∞0 e−

∫ t0 s(t′)dt′dt

]⟩

{T},x

,

B

B ≡∫ ∞

0e−

∫ t0 s(t′)dt′dt .

B

B =

∫ T1

0e−s1t dt+

∫ T1+T2

T1

e−s1T1−s2t dt+

∫ ∞

0e−s1T1−s2T2−

∫ t0 s(t′+T1+T2) dt′ dt ,

=1− e−s1T1

s1+ e−s1T1

1− e−s2T2

s2+ e−s1T1−s2T2B .

=1− e−(s+s)τ

s+ s+ e−2sτ 1− e−(s−s)τ

s− s+ e−2sτB

sδτ (sτ)−1

B

B

B =1−e−(s+s)τ

s+s + e−2sτ 1−e−(s−s)τ

s−s

1− e−2sτ,

pfix = 1−⟨exp

[−x · 2N(1− e−2sτ )

1−e−(s+s)τ

s+s + e−2sτ 1−e−(s−s)τ

s−s

]⟩

x

,

= 1−Hx

(z =

2N(1− e−2sτ )1−e−(s+s)τ

s+s + e−2sτ 1−e−(s−s)τ

s−s

).

Hx(z)

sδτ

Hx(z) =

∫ τ

0

dt

2τexp

[−

zN es1(τ−t)−|s2|τ

1 + z2N |s2|

(1− e−|s2|τ

)+ z

2Ns1e−|s2|τ

(es1(τ−t) − 1

)]

+

∫ τ

0

dt

2τexp

[−

zN e−|s2|(τ−t)

1 + z2N |s2|

(1− e−|s2|(τ−t)

)]

≈ 1 +1

τ

{log

[1 +

z

2N |s2|

(1− e−|s2|τ

)+

z

2Ns1e−|s2|τ

(es1(τ−t) − 1

)]τ

0

+ log

[1 +

z

2N |s2|

(1− e−|s2|(τ−t)

)]τ

0

= 1− 1

τlog

[1 +

z

2N |s2|

(1− e−|s2|τ

)+

z

2Ns1e−|s2|τ (es1τ − 1)

],

z < ∞

pfix =1

τlog

[1 +

e2sτ − e−(s−s)τ + s+ss−s

(1− e−(s−s)τ

)

1− e−(s+s)τ + s+ss−s · e−2sτ

(1− e−(s−s)τ

)(1− e−2sτ

)]

=1

τlog

[e2sτ

]= 2s.

sδτ ≫ 1

sδτ ≫ 1

δτ ∼ τ

sδτ ≫

1

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sδτ =

sτ ≫ 1

sδτ ≫ 1

δτ ! τ

δτ

δτ ≫ τ

f(T ) ∝ T (τδτ )

2−1e−T τ/δτ2

τ δτ2/τ ≫ τ

f(T |mutation) ∝ (µT )f(T ) ≈ τ

δτ2e−T τ/δτ2

τ δτ2

τ ≫ 2s log(Ns) ≫ τ

δτ/τ → ∞

pfix ≈ s

x = 0 x = 1

O(s2)

s ≪ 1

s

N →

∞ s → 0 Ns

1 ≪ τ ≪ 1/s

s(t)

1/N

D

0 = (k − k)nk − λnk + λ(1− ϵ)nk−1 + λϵnk+1

n(q) =∑

k

e−iqknk

n(q)

d

dqn(q) = −iλ[(1− ϵ)e−iq + ϵeiq]n(q).

ln n(q) = λ[(1− ϵ)e−iq − ϵeiq

]− λ(1− 2ϵ)

∑k nk = 1

nk

nk = e−λ(1−2ϵ)∫ π

−π

dq

2πexp

[ikq + λ(1− ϵ)e−iq − λϵeiq

].

Q∗ = ln√

(1− ϵ)/ϵ q = q′ − iQ∗

nk = e−λ(1−2ϵ)+k log

√1−ϵϵ

∫ π

−π

dq′

2πe

{ikq′+α[e−iq′−eiq

′]}

= e−λ(1−2ϵ)+k log

√1−ϵϵ Jk(2α)

α ≡ λ√

ϵ(1− ϵ)

ϵ → 0 ϵλ2 << 1

nk = e−λ(1−2ϵ)+k log[λ(1−ϵ)]∑∞j=0

(−1)jα2j

j!(k+j)!

= [λ(1−ϵ)]ke−λ(1−2ϵ)

k!

{1− λ2ϵ(1−ϵ)

k+1 + ...}

ϵ → 0

∂tp(x, t) = −∂x [D1(x)p(x, t)] + ∂2x [D2(x)p(x, t)]

p(x) = p(nk⋆/N = x) xk⋆ =

e−U/s D1(x) = sx(1 − x/xk⋆) D2(x) = x(1 −

x)/2N x = 1 x = 0

ϕ(t;x)

∂tϕ(t;x) = D1(x)∂xϕ(t;x) +D2(x)∂2xϕ(t;x)

t x

x = y t = 0 t(y) =∫∞0 tϕ(t; y)dt

− 1 = D1(y)∂y t(y) +D2(y)∂2y t(y).

ϕ(x) = e∫ x0 dz

D1(z)D2(z)

t(xk⋆) =

∫ xk⋆

0dy

1

ϕ(y)

∫ 1

ydζ

ϕ(ζ)

D2(ζ),

ϕ(x) = (1− ζ)2Ns(1−xk⋆ )

xk⋆ e2Nsζxk⋆ ,

Ns ≫ 1

2N

∫ 1

ydζ

(1− ζ)2Ns(1−xk⋆ )

xk⋆ e2Nsζxk⋆

ζ(1− ζ)≈ 2N

∫ 1

ydζ

e2Nsxk⋆

(2ζxk⋆

− ζ2

x2k⋆

)

ζ

x ϕ(x)

ϕ(x) ≈ e2Nsxk⋆

(2ζxk⋆

− ζ2

x2k⋆

)

α = Nsxk⋆ , nk⋆ = Nxk⋆

η = y/xk⋆ − 1, z = ζ/xk⋆ − 1

t(xk⋆) ≈ 2nk⋆

∫ 0

−1dηeαη

2∫ 1/k⋆−1

ηdz

e−αz2

1 + z

α ≫ 1

t(xk⋆) ≈ 2nk⋆

√π

4α

∫ 0

−1dηeαη

2 [erf(

√αβ)− erf(

√αη)

]

β = 1k⋆

− 1 η2 = 1 − θ2/α α ≫ 1

t(xk⋆) ≈ nk⋆

√πα−3/2eα.

r− = 1/t(xk⋆)

ϵc ϵc

1

σ√ϵk⋆

= Ne−λ(1−2λ)+ k⋆2 log 1−ϵ

ϵ Jk⋆(2α).

α ≫ 1 k⋆ + 1 ≈ α

k⋆ ∼ k2/3⋆

λ(1− 2ϵ)− λ√

ϵ(1− ϵ) log1− ϵ

ϵ= logNs

16 O(1) ϵ ≪ 1

λ2ϵ ≫ 1

1− 2ϵ+√ϵ log ϵ = λ−1 logNs

√ϵ log ϵ ≈ λ−1 log(Ns)− 1 = −2z

z ϵ

ϵc =z2

W (−z)2≈ z2

(log(z)− log(− log(z)))2

W (x) −1 W (x)eW (x) = x

ϵi+1 =(z + ϵi)2

W (−z + ϵi)2.

C ϵ C W (x)

ϵ2

ϵ → 1/2

δ = 12 − ϵ

1− 2ϵ−√

ϵ(1− ϵ) log1− ϵ

ϵ=

4

3δ3 +

44

15δ5 +O(δ7) = λ−1 logNs.

ϵ ≈ 1

2−

(3

4λlog(Ns)

)1/3

.

δ5

0.0 0.1 0.2 0.3 0.4 0.5 0.6sU log(Ns)

0.0

0.1

0.2

0.3

0.4

0.5

✏ c

12 �

�3s4U logNs

�1/3z2

W (z)2 , z = s2U logNs� 1

2

Numerical solution

1

10

102

103

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sU logNs

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φ

+

φ