RELATION BETWEEN ROBUSTNESS, EVOLVABILITY AND FITNESS · RELATION BETWEEN ROBUSTNESS, EVOLVABILITY AND FITNESS. RELATION BETWEEN ROBUSTNESS, EVOLVABILITY AND FITNESS SUPPORTED BY

Thomas Fink

Centre National de la Recherche Scientifique

London Institute for Mathematical Sciences

R E L A T I O N B E T W E E N R O B U S T N E S S ,

E V O L V A B I L I T Y A N D F I T N E S S

R E L A T I O N B E T W E E N R O B U S T N E S S ,

E V O L V A B I L I T Y A N D F I T N E S S

S U P P O R T E D B Y

I N C O L L A B O R AT I O N W I T H

Defense Advanced Research Projects Agency

(DARPA)

“Predictive biology: adaptability, robustness

and the fundamental laws of biology”

Jamie Blundell Alexis Gallagher

Postdoc Postdoc

R O B U S T N E S S , E V O LVA B I L I T Y, A N D F I T N E S S

1 P R O L O G U E

1.1 a paradox

2 W O R D G A M E S

2.1 carroll’s doublets

2.2 knuth’s word webs

2.3 universal word webs

3 G E N O T Y P E S & P H E N O T Y P E S

3.1 rna secondary structure

3.2 a paradox resolved

4 M U T A T I O N G R A P H S

4.1 size, perimeter and egress

4.2 a null model

5 N E U T R A L N E T W O R K S

5.1 size

5.2 theory vs simulation

6 R O B U S T N E S S

6.1 egress


7 E V O L V A B I L I T Y

7.1 colours in the perimeter

7.2 cereal box prize problem


8 F I T N E S S

8.1 effective fitness and pagerank

A system is robust if it is able to function as usual after the introduction of errors in its parts.

A system is evolvable if errors in its parts enable it to explore new and possibily beneficial functions.

1 P R O L O G U E 1.1 a paradox ••••••••••••••••••••••••



Meaning of “system” and “error” depends on level of organization we are studying:

System Error (or “mutation”) (Wagner 2008)

RNA secondary structure Change of nucleotide

Protein molecule Change of amino acid

Genetic circuit Change of regulatory logic

Genome-scale network Changes in enzymatic reactions

Whole organism Hox gene and others




Meaning of “system” and “error” depends on level of organization we are studying:

System Error (or “mutation”) (Wagner 2008)

RNA secondary structure Change of nucleotide

Protein molecule Change of amino acid

Genetic circuit Change of regulatory logic

Genome-scale network Changes in enzymatic reactions

Whole organism Hox gene and others

Robust: system is resilient to mutation. Evolvable: system is not resilient to mutation.

But there is evidence that robustness and evolvability are not antagonistic.

Bloom “Protein stabilitypromotes evolvability”, PNAS (2006).

Draghi, “Mutational robustness can facilitate adaptation”, Nature (2010).

Wagner, “Robustness and evolvability: a paradox resolved”, Proc. Roy. Soc. (2008).


Doublet: Two words of the same length where one can

be changed into the other by changing one letter at a time

such that each intermediate word is an English word.

1 W O R D G A M E S 1.1 lewis carroll’s doublets ••••••••••••••••••••••••

cat

dog




1 W O R D G A M E S 1.1 lewis carroll’s doublets

cat

dog

cat

cot

dot

dog

head

tail




1 W O R D G A M E S 1.1 lewis carroll’s doublets ••••••••••••••••••••••••

cat

dog

cat

cot

dot

dog

head

tail

head

heal

teal

tell

tall

tail

flour

bread





cat

dog

cat

cot

dot

dog

head

tail

head

heal

teal

tell

tall

tail

flour

bread

flour

floor

flood

blood

brood

broad

bread

winter

summer





cat

dog

cat

cot

dot

dog

head

tail

head

heal

teal

tell

tall

tail

flour

bread

flour

floor

flood

blood

brood

broad

bread

winter

summer

winter

winder

wander

warder

warner

warned

darned

damned

dammed

hammed

hummed

bummed

summed

summer


be changed into the other by changing one letter at a time.

connected doublets


cat

dog

cat

cot

dot

dog

head

tail

head

heal

teal

tell

tall

tail

flour

bread

flour

floor

flood

blood

brood

broad

bread

winter

summer

winter

winder

wander

warder

warner

warned

darned

damned

dammed

hammed

hummed

bummed

summed

summer

dead

deaf

leaf

loaf

loan

lend

lead

lean

dean

dead

loon

send

dead

mead

mean

moan

moon

moor

boor

boar

boat

beat

seat

sent

pent

pant

pane

bane

bone

boon

spat

dane

hair

fair

fail

foil

fool

tool

toon

slit

slot

spot

dame

toll

slid

damn

dead

deal

dell

tell

sled

cool

coop

comp

camp

damp

seal

sell

fell

seed

coil

dead

read

real

feel

feed

need

deed

toil

dead

tail

1 W O R D G A M E S 1.2 donald knuth’s word webs••••••••••••••••••••••••

There are 5,757 5-letter English

words (out of 11,881,376 poss.).

Connected by 14,135 lines

Word webs of different size s:

671 1-clusters (aloof words)

103 2-clusters (word pairs)

.

.

.

1 3000-cluster (approx.)








.

.

.


aloot

s=1

earta

s=1

aloaf

aloof

aroof

darth

earth

eerth

allof

earoh








.

.

.


aloot

s=1

earta

s=1

edium

epium

s=2

gonat

monat

s=2

aloaf

aloof

aroof

darth

earth

eerth

odiug

gonad

monad

spium

bonad

odium

opium

momad

allof

earoh

odius

opius

gohad

mohad








.

.

.


aloot

s=1

earta

s=1

edium

epium

s=2

gonat

monat

s=2

aloaf

aloof

aroof

darth

earth

eerth

odiug

gonad

monad

spium

bonad

odium

opium

momad

allof

earoh

odius

opius

gohad

mohad

salve

beast

boast

cones

s>41

rogue

vogue

vague

value

valve

halve

helve

heave

leave

lease

least

ropes

copes

cores

cords

words

raids

varve

heavy

popes

rails

hails

hairs

harry

hurry

curry

curvy

curve

carve

heady

heads

heals

hills

pills

piles

poles

pores

1 W O R D G A M E S 1.3 entangled word webs••••••••••••••••••••••••

English xxxx

French xxxx

German xxxx

Italian xxxx

Latin xxxx

Dutch xxxx

Nonsense xxxx

Let’s play the same game but with multiple languages:

1 W O R D G A M E S 1.3 entangled word webs

xxxx

wird

sind

bind

bird

xxxx

xxxx

xxxx

xxxx

bind

xxxx

mise

dico

xxxx

xxxx

xxxx

zoon

boon

boos

bios

bins

xxxx

mire

xxxx

dice

xxxx

loan

lean

leak

leuk

xxxx

xxxx

xxxx

xxxx

bids

xxxx

mite

xxxx

lice

xxxx

loon

xxxx

xxxx

arto

arts

ants

ands

aids

rids

ride

rite

rime

lime

limn

lion

lien

lieu

dieu

xxxx

ante

xxxx

xxxx

xxxx

tide

xxxx

rive

xxxx

xxxx

xxxx

mien

xxxx

xxxx

tine

xxxx

riva

miel

viel

vier

cena

cent

tent

tint

xxxx

ciel

xxxx

xxxx

xxxx

English xxxx

French xxxx

German xxxx

Italian xxxx

Latin xxxx

Dutch xxxx

Nonsense xxxx



xxxx

wird

sind

bind

bird

xxxx

xxxx

xxxx

xxxx

bind

xxxx

mise

dico

xxxx

xxxx

xxxx

zoon

boon

boos

bios

bins

xxxx

mire

xxxx

dice

xxxx

loan

lean

leak

leuk

xxxx

xxxx

xxxx

xxxx

bids

xxxx

mite

xxxx

lice

xxxx

loon

xxxx

xxxx

arto

arts

ants

ands

aids

rids

ride

rite

rime

lime

limn

lion

lien

lieu

dieu

xxxx

ante

xxxx

xxxx

xxxx

tide

xxxx

rive

xxxx

xxxx

xxxx

mien

xxxx

xxxx

tine

xxxx

riva

miel

viel

vier

cena

cent

tent

tint

xxxx

ciel

xxxx

xxxx

xxxx

pas

pao

fap

yan

pan

pat

xat

cip

cap

yap

pap

pat

sat

qat

nas

nab

nap

lap

map

mat

cat

dat

das

dab

dap

sap

rap

rat

bat

uat

dyb

dib

dip

sip

rip

rlt

bau

dij

diu

rrp

rrp

English xxxx

French xxxx

German xxxx

Italian xxxx

Latin xxxx

Dutch xxxx

Nonsense xxxx



xxxx

wird

sind

bind

bird

xxxx

xxxx

xxxx

xxxx

bind

xxxx

mise

dico

xxxx

xxxx

xxxx

zoon

boon

boos

bios

bins

xxxx

mire

xxxx

dice

xxxx

loan

lean

leak

leuk

xxxx

xxxx

xxxx

xxxx

bids

xxxx

mite

xxxx

lice

xxxx

loon

xxxx

xxxx

arto

arts

ants

ands

aids

rids

ride

rite

rime

lime

limn

lion

lien

lieu

dieu

xxxx

ante

xxxx

xxxx

xxxx

tide

xxxx

rive

xxxx

xxxx

xxxx

mien

xxxx

xxxx

tine

xxxx

riva

miel

viel

vier

cena

cent

tent

tint

xxxx

ciel

xxxx

xxxx

xxxx

pas

pao

fap

yan

pan

pat

xat

cip

cap

yap

pap

pat

sat

qat

nas

nab

nap

lap

map

mat

cat

dat

das

dab

dap

sap

rap

rat

bat

uat

dyb

dib

dip

sip

rip

rlt

bau

dij

diu

rrp

rrp

English xxxx

French xxxx

German xxxx

Italian xxxx

Latin xxxx

Dutch xxxx

Nonsense xxxx


Are many languages accessible from English? Or just a few?

2 G E N O T Y P E S & P H E N O T Y P E S 2.1 rna secondary structure

Now let’s look at RNA sequences of length l.

Sequences are neighbours if they differ by a single letter.

CAGU…

CAAU…

CACU…

CACG…

CUCG…

CUGG…

CUGA…

AUGA…

ACGA…

ACGU…

AGGU…

AGGC…

CAUU…

CAUU…

CAUU…

CAUG…

CUUG…

CUUG…

CUUA…

AUUA…

ACUA…

ACUU…

AGUU…

AGUC…

CACU…

CAGU…

CAUU…

CAUG…

CAUG…

CAUG…

CAUA…

AAUA…

AAUG…

AAUU…

AAUC…

CAUC…

CACC…

CAGC…

CAUC…

CAUA…

CAUG…

CAUU…

CAUC…

AAUC…

AAUG…

AAUC…

AAUC…

CAUC…

UACC…

UAGC…

GAUC…

UAUU…

CAUU…

UCUU…

UCUC…

UCUC…

UAUG…

GAUC…

CAUC…

UAUC…

UACC…

UAGC…

GAGC…

UAGU…

CAGU…

UCUU…

UCGA…

UCGC…

UAGG…

GAGC…

CAGC…

UACC…

GAGC…

UAGC…

GAGU…

GAGU…

GCUU…

GCGA…

GCGC…

GAGG…

UAGC…

GAGC…

CAGC…

GACC…




CAGU…

CAAU…

CACU…

CACG…

CUCG…

CUGG…

CUGA…

AUGA…

ACGA…

ACGU…

AGGU…

AGGC…

CAUU…

CAUU…

CAUU…

CAUG…

CUUG…

CUUG…

CUUA…

AUUA…

ACUA…

ACUU…

AGUU…

AGUC…

CACU…

CAGU…

CAUU…

CAUG…

CAUG…

CAUG…

CAUA…

AAUA…

AAUG…

AAUU…

AAUC…

CAUC…

CACC…

CAGC…

CAUC…

CAUA…

CAUG…

CAUU…

CAUC…

AAUC…

AAUG…

AAUC…

AAUC…

CAUC…

UACC…

UAGC…

GAUC…

UAUU…

CAUU…

UCUU…

UCUC…

UCUC…

UAUG…

GAUC…

CAUC…

UAUC…

UACC…

UAGC…

GAGC…

UAGU…

CAGU…

UCUU…

UCGA…

UCGC…

UAGG…

GAGC…

CAGC…

UACC…

GAGC…

UAGC…

GAGU…

GAGU…

GCUU…

GCGA…

GCGC…

GAGG…

UAGC…

GAGC…

CAGC…

GACC…

Stem-loop motif

2 G E N O T Y P E S & P H E N O T Y P E S 2.1 rna secondary structure ••••••••••••••••••••••••



Big dipper

fold

Pi fold

Cloverleaf fold, etc.

Stem-loop motif

CAGU…

CAAU…

CACU…

CACG…

CUCG…

CUGG…

CUGA…

AUGA…

ACGA…

ACGU…

AGGU…

AGGC…

CAUU…

CAUU…

CAUU…

CAUG…

CUUG…

CUUG…

CUUA…

AUUA…

ACUA…

ACUU…

AGUU…

AGUC…

CACU…

CAGU…

CAUU…

CAUG…

CAUG…

CAUG…

CAUA…

AAUA…

AAUG…

AAUU…

AAUC…

CAUC…

CACC…

CAGC…

CAUC…

CAUA…

CAUG…

CAUU…

CAUC…

AAUC…

AAUG…

AAUC…

AAUC…

CAUC…

UACC…

UAGC…

GAUC…

UAUU…

CAUU…

UCUU…

UCUC…

UCUC…

UAUG…

GAUC…

CAUC…

UAUC…

UACC…

UAGC…

GAGC…

UAGU…

CAGU…

UCUU…

UCGA…

UCGC…

UAGG…

GAGC…

CAGC…

UACC…

GAGC…

UAGC…

GAGU…

GAGU…

GCUU…

GCGA…

GCGC…

GAGG…

UAGC…

GAGC…

CAGC…

GACC…

No fold




CAGU…

CAAU…

CACU…

CACG…

CUCG…

CUGG…

CUGA…

AUGA…

ACGA…

ACGU…

AGGU…

AGGC…

CAUU…

CAUU…

CAUU…

CAUG…

CUUG…

CUUG…

CUUA…

AUUA…

ACUA…

ACUU…

AGUU…

AGUC…

CACU…

CAGU…

CAUU…

CAUG…

CAUG…

CAUG…

CAUA…

AAUA…

AAUG…

AAUU…

AAUC…

CAUC…

CACC…

CAGC…

CAUC…

CAUA…

CAUG…

CAUU…

CAUC…

AAUC…

AAUG…

AAUC…

AAUC…

CAUC…

UACC…

UAGC…

GAUC…

UAUU…

CAUU…

UCUU…

UCUC…

UCUC…

UAUG…

GAUC…

CAUC…

UAUC…

UACC…

UAGC…

GAGC…

UAGU…

CAGU…

UCUU…

UCGA…

UCGC…

UAGG…

GAGC…

CAGC…

UACC…

GAGC…

UAGC…

GAGU…

GAGU…

GCUU…

GCGA…

GCGC…

GAGG…

UAGC…

GAGC…

CAGC…

GACC…

Are many structures accessible from the big dipper fold?

Stem-loop motif

Big dipper

fold

Pi fold

Cloverleaf fold, etc.

No fold

2 G E N O T Y P E S & P H E N O T Y P E S 2.2 a paradox resolved,

Genotype: a particular set of instructions for producing some function.

Phenotype: a particular function.

Many genotypes produce the same phenotype.


Genotype: a particular set of instructions for producing some function.

Phenotype: a particular function.


Genotype (instructions) Phenotype (function)

Sequence of DNA nucleotides Sequence of amino acids

Sequence of RNA nucleotides RNA secondary structure

Sequence of amino acids Protein native conformation

Set of regulatory logics Period of genetic oscillator

Set of enzymatic reactions Resistance to specific toxin


Genotype: a particular set of instructions for producing some function or structure.

Phenotype: a particular function or sturcture.


Genotype (instructions) Phenotype (function)

Sequence of DNA nucleotides Sequence of amino acids

Sequence of RNA nucleotides RNA secondary structure

Sequence of amino acids Protein native conformation

Set of regulatory logics Period of genetic oscillator

Set of enzymatic reactions Resistance to specific toxin

Genotype robustness: number RG (or fraction rG) of neutral mutants of genotype G.

Genotype evolvability: number EG (or fraction eG) of different phenotypes found in the mutants of G.

Phenotype robustness: number RP (or fraction rP) of neutral mutants averaged over all genotypes

G with a given phenotype.

Phenotype evolvability: number EP (or fraction eP) of different phenotypes found in the mutants

of all genotypes G with a given phenotype. (Wagner 2008)


CACU…

CACG…

CUCG…

CUGG…

CUGA…

AUGA…

ACGA…

ACGU…

AGGU…

AGGC…

CAUU…

CAUG…

CUUG…

CUUG…

CUUA…

AUUA…

ACUA…

ACUU…

AGUU…

AGUC…

CAUU…

CAUG…

CAUG…

CAUG…

CAUACAUA……

AAUA…

AAUG…

AAUU…

AAUC…

CAUC…

CAUC…

CAUA…

CAUG…

CAUUCAUU……

CAUC…

AAUCAAUC……

AAUG…

AAUC…

AAUC…

CAUC…

GAUC…

UAUU…

CAUU…

UCUU…

UCUC…

UCUC…

UAUG…

GAUC…

CAUC…

UAUC…

CACU…

CACG…

CUCG…

CUGG…

CUGA…

AUGA…

ACGA…

ACGU…

AGGU…

AGGC…

CAUU…

CAUG…

CUUG…

CUUG…

CUUA…

AUUA…

ACUA…

ACUU…

AGUU…

AGUC…

CAUU…

CAUG…

CAUG…

CAUG…

CAUA…

AAUA…

AAUG…

AAUU…

AAUC…

CAUC…

CAUC…

CAUA…

CAUG…

CAUU…

CAUC…

AAUC…

AAUG…

AAUC…

AAUC…

CAUC…

GAUC…

UAUU…

CAUU…

UCUU…

UCUC…UCUC…

UCUC…

UAUG…

GAUC…

CAUC…

UAUC…

CACU…

CACG…

CUCG…

CUGG…

CUGA…

AUGA…

ACGA…

ACGU…

AGGU…

AGGC…

CAUU…

CAUG…

CUUG…

CUUG…

CUUA…

AUUA…AUUA…

ACUA…

ACUU…

AGUU…

AGUC…

CAUU…

CAUG…

CAUG…

CAUG…

CAUA…

AAUA…

AAUG…

AAUU…

AAUC…

CAUC…

CAUC…

CAUA…

CAUG…

CAUU…

CAUC…

AAUC…

AAUG…

AAUC…

AAUC…

CAUC…

GAUC…

UAUU…

CAUU…

UCUU…

UCUC…UCUC…

UCUC…UCUC…

UAUG…

GAUC…

CAUC…

UAUC…

CACU…

CACG…

CUCG…

CUGG…

CUGA…

AUGA…

ACGA…

ACGU…

AGGU…

AGGC…

CAUU…

CAUG…

CUUG…CUUG…

CUUG…

CUUA…

AUUA…

ACUA…

ACUU…

AGUU…

AGUC…

CAUU…

CAUG…CAUG…

CAUG…CAUG…

CAUG…CAUG…

CAUA…CAUA…

AAUA…AAUA…

AAUG…AAUG…

AAUU…AAUU…

AAUC…AAUC…

CAUC…

CAUC…

CAUA…

CAUG…

CAUU…CAUU…

CAUC…CAUC…

AAUC…AAUC…

AAUG…

AAUC…

AAUC…

CAUC…

GAUC…

UAUU…

CAUU…

UCUU…

UCUC…

UCUC…

UAUG…

GAUC…

CAUC…

UAUC…

Genotype robustness: number RG (or fraction rG)

of neutral mutants of genotype G. 3Genotype evolvability: number EG (or fraction eG) of dif-

ferent phenotypes found in mutants of genotype G. 1

Phenotype robustness: number RP (or fraction rP)

of neutral mutants averaged over all genotypes G

with a given phenotype. 11/6

Phenotype evolvability: number EP (or fraction eP)

of different structures found in mutants of all geno-

types G with a given phenotype. 3

4 M U T A T I O N G R A P H S 4.1 size, perimeter and egress

0000

0001

0010

0011

0000

0001

0010

0011

0100

0101

0110

0111

1100

1101

1110

1111

G4,2

l = 4, a = 2. So 16 genotypes:

4 M U T A T I O N G R A P H S 4.1 size, perimeter and egress••••••••••••••••••••••••

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

G4,2

l = 4, a = 2. So 16 genotypes: m = 4. So paint 4 colours:


0000

0001

0010

0011

0000

0001

0010

0011

0100

0101

0110

0111

1100

1101

1110

1111

G4,2



0000

0001

0010

0011

0000

0001

0010

0011

0100

0101

0110

0111

1100

1101

1110

1111

G4,2

cluster size s = 5



0000

0001

0010

0011

0000

0001

0010

0011

0100

0101

0110

0111

1100

1101

1110

1111

G4,2

cluster size s = 5 perimeter t = 8



0000

0001

0010

0011

0000

0001

0010

0011

0100

0101

0110

0111

1100

1101

1110

1111

G4,2

cluster size s = 5 perimeter t = 8 egress u = 12



0000

0001

0010

0011

0000

0001

0010

0011

0100

0101

0110

0111

1100

1101

1110

1111

G4,2

cluster size s = 5 perimeter t = 8 egress u = 12

evolvability E = 2 robustness r = (s l - u)/sl = 0.4


4 M U T A T I O N G R A P H S 4.2 a null model••••••••••••••••••••••••

How do evolvability and robustness depend on the number of colours m? ...

a=2

a=3

a=4

4 M U T A T I O N G R A P H S 4.2 a null model••••••••••••••••••••••••

l=2 l=3 l=4 l=5

How do evolvability and robustness depend on the number of colours m? ...

How do they depend

on the genome length

l and alphabet size a?

5 N E U T R A L N E T W O R K S 5.1 size••••••••••••••••••••••••

How big is a typical neutral network?

5 N E U T R A L N E T W O R K S 5.1 size••••••••••••••••••••••••

How big is a typical neutral network?

In the large l limit, the probability that an occupied

(say, red) site belongs to a cluster of size s is

The mean neutral network size is

5 N E U T R A L N E T W O R K S 5.2 theory vs simulation ••••••••••••••••••••••••

• x/(x–l), left plot

• 1/(1–x), right plot• l = 5• l = 10• l = 20• l = 40

ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ

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ææææ

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æ

æ

æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ

1 2 5 10 20 50 100 200

1

5

10

50

100

500

1000

m

S

æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ

æææææææææææææææææææææææææææææææææææææææææææ

ææææææææææææææææææææææææææææææææææææææ

æ

ææææææææ

æææææææææ

æææææææææææ


ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ

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0.1

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2.0

5.0

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m�d

S-1

a = 2 for all curves

S(m) = m/(m–l)

Critical point at mc = l

Finite dimensional scaling: mc-eff → mc as l → ∞?

Robustness: the number RP (or fraction rP) of neutral neighbours averaged over all genotypes G

with a given phenotype.

6 R O B U S T N E S S 6.1 exgress ••••••••••••••••••••••••

6 R O B U S T N E S S 6.2 theory vs simulation ••••••••••••••••••••••••

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1 2 5 10 20 50 100 200

0.01

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0.05

0.10

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m

rP

All trees have maximum egress u.

For s less than l,

6 E V O LVA B I L I T Y 6.1 number of colours in the perimeter ••••••••••••••••••••••••

Phenotype evolvability: number EP (or fraction eP) of different structures found in the mutants of

all genotypes G with a given phenotype.

How many different phenotypes EP are accessible in a perimeter of length t painted m colours?

In every box of cereal, there is one of m prizes. How many boxes of cereal t do we need to

buy to collect them all?

6 E V O LVA B I L I T Y 5.2 cereal box prize problem ••••••••••••••••••••••••

... ?

The probability of finding all m prizes in t > m boxes is

The expected number of prizes found in t boxes is

6 E V O LVA B I L I T Y 6.1 number of colours in the perimeter ••••••••••••••••••••••••

Phenotype evolvability: number EP (or fraction eP) of different structures found in the mutants of

all genotypes G with a given phenotype.

How many different phenotypes EP are accessible in a perimeter of length t painted m colours?

EP has different behaviour below and above the critical point m = d:

6 E V O LVA B I L I T Y 5.2 theory vs simulation

• + • EP(m/l), l = ∞• + • eP(m/l), l = ∞

• l = 5• l = 10• l = 20• l = 40

a = 2 for all curves

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æææææææ

æææææ

ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ

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0 10 20 30

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m�d

EP�d-1

Fitness

f(•) = 1/3f(•) = 1f(•) = 3

D E F

μ=ε μ=3ε μ=9ε

A B C

μ=0

A 8 genotypes are assigned to 3 phenotypes, with an

identical size population at each genotype to begin.

B Phenotype fitnesses, which dictate the relative

number of offspring.

C With mutation rate μ= 0, the phenotype with high-

est fitness (blue) eventually dominates.

D For small mutation rate, the less fit genotypes have

non-zero population.

E As mutation rate increases, the populations shift.

F The second-most-fit phenotype (red) dominates the

population for higher mutation rate because of the

advantageous cluster geometry.

8 F I T N E S S 8.1 effective fitness and pagerank

Each phenotype has fitness (reproductive rate) fi.Each genotype has sub-population nj.

At each time step,

(i) the sub-population nj → fi nj.

(ii) At each generation, (1–μ) of the population remains with the

same genotype and μ/3 is transmitted to each mutation.

8 F I T N E S S 8.1 effective fitness and pagerank

Each phenotype has fitness (reproductive rate) fi.Each genotype has sub-population nj.

At each time step,

(i) the sub-population nj → fi nj.

(ii) A fraction d of each sub-population is sent to all mutants.

H I G H L I G H T S A N D O P E N P R O B L E M S

m high: neutral networks small, system has low evolvability;

m moderate: neutral networks small, system has high evolvability;

m low neutral networks large, system has high evolvability.

Maximum evolvability occurs a fixed distance from the critical point (Emax).

Increasing robustness (via m, l or a) promotes increased robustness, apart from inside Emax.

What is the effect of a large fraction (1-ε) of mutants being deleterious (“grey words”)?

Effective fitness (“true” fitness?) depends strongly on cluster structure geometry.

Different cluster probabilities - real genotype-phenotype maps are highly varied in degeneracies.

RELATION BETWEEN ROBUSTNESS, EVOLVABILITY AND FITNESS · RELATION BETWEEN ROBUSTNESS, EVOLVABILITY AND FITNESS. RELATION BETWEEN ROBUSTNESS, EVOLVABILITY AND FITNESS SUPPORTED BY

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