relation between robustness, evolvability and fitness · relation between robustness, evolvability...
TRANSCRIPT
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
6.2 theory vs simulation
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
7.3 theory vs simulation
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 ••••••••••••••••••••••••
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.
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
1 P R O L O G U E 1.1 a paradox ••••••••••••••••••••••••
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.
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).
1 P R O L O G U E 1.1 a paradox ••••••••••••••••••••••••
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
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
cat
cot
dot
dog
head
tail
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
cat
cot
dot
dog
head
tail
head
heal
teal
tell
tall
tail
flour
bread
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
cat
cot
dot
dog
head
tail
head
heal
teal
tell
tall
tail
flour
bread
flour
floor
flood
blood
brood
broad
bread
winter
summer
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
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
Doublet: Two words of the same length where one can
be changed into the other by changing one letter at a time.
connected doublets
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
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.)
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
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
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
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
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
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
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
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
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
Let’s play the same game but with multiple languages:
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…
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…
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 ••••••••••••••••••••••••
Now let’s look at RNA sequences of length l.
Sequences are neighbours if they differ by a single letter.
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
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…
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.
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 (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
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 or structure.
Phenotype: a particular function or sturcture.
Many genotypes produce the same phenotype.
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)
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,
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:
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: m = 4. So paint 4 colours:
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
cluster size s = 5
l = 4, a = 2. So 16 genotypes: m = 4. So paint 4 colours:
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
cluster size s = 5 perimeter t = 8
l = 4, a = 2. So 16 genotypes: m = 4. So paint 4 colours:
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
cluster size s = 5 perimeter t = 8 egress u = 12
l = 4, a = 2. So 16 genotypes: m = 4. So paint 4 colours:
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
cluster size s = 5 perimeter t = 8 egress u = 12
evolvability E = 2 robustness r = (s l - u)/sl = 0.4
l = 4, a = 2. So 16 genotypes: m = 4. So paint 4 colours:
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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æ
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æ
æ
ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
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æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
1 2 5 10 20 50 100 200
1
5
10
50
100
500
1000
m
S
æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
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ææææææææ
æææææææææ
æææææææææææ
ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
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1.0 10.05.02.0 20.03.01.5 15.07.0
0.1
0.2
0.5
1.0
2.0
5.0
10.0
20.0
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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ææ æ æ æ
ææææææææ
ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
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æ
ææææææææ
æææææææææ
1 2 5 10 20 50 100 200
0.01
0.02
0.05
0.10
0.20
0.50
1.00
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
ææææ
æ
æææææææ
æ
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ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
æææææææ
ææ
æææææææ
æææææ
ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
0 10 20 30
0.00
0.05
0.10
0.15
0.20
0.25
0.30
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.