Who laughs last?
perturbation theory of games
Tibor Antal, Program for Evolutionary Dynamics, Harvard
- dynamics, question
- perturbation method: two key aspects
- simplest case: well mixed 2*2
- further examples: n strategies, structured populations
- general results
with: C Tarnita, H Otsuki, J Wakeley, P Taylor, A Traulsen, M Nowak
Pleiotropy as a mechanism
to stabilize cooperation
Cooperation
Kevin R. Foster1*, Gad Shaulsky2, Joan E. Strassmann1,
David C. Queller1 & Chris R. L. Thompson2,3*
1
Ecology and Evolution, Rice University, Houston, Texas 77005, USA
Department of Molecular and Human Genetics, Baylor College of Medicine,
One Baylor Plaza, Houston, Texas 77030, USA
3
School of Biological Sciences, University of Manchester, Manchester M13 9PT, UK
2
humans, bacteria, trees, slime molds, ...
* These authors contributed equally to this work.
.............................................................................................................................................................................
Most genes affect many traits1–4. This phenomenon, known as
pleiotropy, is a major constraint on evolution because adaptive
change in one trait may be prevented because it would compromise other traits affected by the same genes2,4. Here we show that
pleiotropy can have an unexpected effect and benefit one of the
most enigmatic of adaptations—cooperation. A spectacular act
of cooperation occurs in the social amoeba Dictyostelium discoideum, in which some cells die to form a stalk that holds the
other cells aloft as reproductive spores5,6. We have identified a
gene, dimA7, in D. discoideum that has two contrasting effects. It
is required to receive the signalling molecule DIF-1 that causes
differentiation into prestalk cells. Ignoring DIF-1 and not
becoming prestalk should allow cells to cheat by avoiding the
stalk. However, we find that in aggregations containing the wildtype cells, lack of the dimA gene results in exclusion from spores.
This pleiotropic linkage of stalk and spore formation limits the
potential for cheating in D. discoideum because defecting on
prestalk cell production results in an even greater reduction in
spores. We propose that the evolution of pleiotropic links
between cheating and personal costs can stabilize cooperative
adaptations.
Acts of cooperation such as stalk formation in D. discoideum are a
challenge for evolutionary biologists because of the potential for
disruptive cheaters8–12. When starving, the normally solitary amoebae of D. discoideum aggregate together to form a migratory slug
and then a fruiting body in which most cells become spores but
around one-fifth die to form a supporting stalk5,6. Genetically
different clones of D. discoideum will aggregate together and form
chimaeric fruiting bodies5 and chimaerism seems to be common in
nature because multiple clones co-occur in tiny soil samples13. This
suggests that a defector12 that produces fewer stalk cells can gain a
selfish advantage5 and raises the question of how stalk formation is
maintained9,11. Around half of prestalk cells in D. discoideum,
known as the prestalk O cells, are induced to differentiate by the
signalling molecule DIF-1 (ref. 14), that is released by the neighbouring prespore cells15 (Fig. 1a). A viable cheating strategy, therefore, would be to ignore DIF-1 and overproduce spore cells in
chimaeras. This is supported by the observation that cells with
lowered DIF sensitivity, resulting from growth with glucose, cheat
and are over represented in the spores when mixed with cells grown
without glucose16,17.
We used a knockout mutant to examine the effects of ignoring
DIF-1. dimA encodes a central component of the DIF response
pathway. A mutant with this gene disrupted (dimA 2) ignores DIF-1
Foster ‘04
prisoner's dilemma
7
C D
!
"
C
10 1
.
D 11 2
defectors win!
Figure 1 dimA 2 cells occupy the prespore zone and behave like a cheater in chimaeras.
a, A schematic diagram of a slug of the slime mould Dictyostelium discoideum showing
three of the major cell types: prestalk A (pstA), prestalk O (pstO) and prespore cells.
Prespore cells release DIF-1, which induces pstO cell differentiation. b, Schematic
Evolutionary Dynamics
Moran process, N players
Game
replication
payoff of
random
death
0
1
when playing against
A
A a11
B a21
B
a12
a22
µm λm
2
3
m
N-1 N
+
t1
−
tN −1
+
t1
=
−
tN −1
TA, Scheuring '06
What is the question?
Two strategies: A and B: Which one is better?
B
John Forbes Nash,
John Maynard Smith
fixation probabilities ...
A
Or: Which outnumbers the other in the long run?
with two way mutation u
!x" > 1/2
Kandori '93
u→0
x
fixation probabilities
ρA > ρB
x
general
time
u
time
Perturbation method: 2 key points
when playing against
A
Payoff = 1 + δ ×
payoff of
B
A a11
B a21
Wright-Fisher
a12
a22
x
δ
u
frequency of A
selection strength
mutation probability
t
selection
∆x
mutation
∆x
sel
mut
t+1
∆x
tot
= ∆x
sel
u
u
sel
− (x + ∆x ) + (1 − x − ∆xsel )
2
2
1 1−u
!x" = +
!∆xsel "
2
u
1
sel
!x" > ⇐⇒ !∆x " > 0
2
Perturbation method: 2 key points
when playing against
A
Payoff = 1 + δ ×
payoff of
A a11
B a21
B
a12
a22
x
δ
u
frequency of A
selection strength
mutation probability
1
sel
!x" > ⇐⇒ !∆x " > 0
2
Easy perturbation method for small δ
!
(1)
∆xi = 0 + δ∆xi
!∆x" =
∆xi πi
(0)
πi = πi
!∆x" = δ
!
(1)
∆xi
(0)
πi
+ O(δ )
2
(1)
+ δπi
neutral probabilities only !
Simplest example
payoffs:
!
1
T
S
0
"
well-mixed
fA = 1 + δ [X − 1 + S(N − X)]
fB = 1 + δ(XT + 0)
∆x
sel
! 3
"
2
2
= δN (x − x )(1 − S − T ) + (x − x )(S − 1/N )
average in the neutral stationary state
!∆xsel " > 0
!∆x" = δ
!
!
1
T <1−S+ S−
N
2
T <S+1−
N
Kandori '93,
TA, Traulsen, Nowak '09
Example:
(1)
(0)
∆xi πi
"
+ O(δ 2 )
!x" − !x2 "
!x2 " − !x3 "
=2
neutral correlations
from coalescence
1
T = 1, S =
2
A wins for N=5,
but B wins for N=3
!x" − !x2 "
= 2
2
3
!x " − !x "
Correlations from coalescent
1
!x" =
2
1
!x " = Pr(Sk = Sl )
2
2
1
!x " = Pr(Sk = Sl = Sq )
2
3
MRCA
time
time
Pr(Sk = Sl ) =
!
t
Pr(Sk = Sl |T = t)Pr(T = t)
n strategies: when is k better than average?

a11
 ..
 .
an1
a12
...
...
...
Low mutation

a1n
.. 
. 
ann
n
!
1
Lk =
(akk + aki − aik − aii ) > 0
n i=1
High mutation
n = 2 : a11 + a12 > a21 + a22
n !
n
!
1
Hk = 2
(akj − aij ) > 0
n i=1 j=1
Arbitrary mutation
Lk + N uHk > 0
Example 1
λ = 4.6
0.340
x1
S2
0
λ
7
S3

13
8 
9
Abundances
S1

S1
1
S2  0
S3
0
N = 30
0.336
δ = 0.003
x2
0.332
µ = Nu
x3
0.328
0.1
1
Mutation Rate µ




S1
S1
1
S2  0
S3
0

S2
0
λ
7
S3

13
8 
9






















10
Example 2
Defectors beat Cooperators
since a11 + a12 < a21 + a22
C D
!
"
C
10 1
.
D 11 2
C

C
10
D  11
L
0
But lets add Loners
D
1
2
0
L

0
0
0
8
LC = ,
3
4
LD = ,
3
LL = −4
C win
HC = 1,
5
HD = ,
3
8
HL = −
3
D win
General, C win for µ < µ ≡ 1
∗
Example 3
m round Prisoner's dilemma
AllC

AllC (b − c)m
AllD  bm
TFT (b − c)m
AllD
−cm
0
−c

TFT

(b − c)m

b
(b − c)m



b
3m
=
c
m−1
b 5m + 1
=
c
m−1
b 3m + 1
=
c
m−1
4m − 1
b
=
c
m−1
b
2m + 1
=
c
m−1
AllC better
than average

5m + 1
b
=
c
2(m − 1)
b 2m + 1
=
c
m−1
b
m+2
=
c
m−1
b
m+1
=
c
m−1

b benefit, c cost


Islands: simplest structured population
C
C
DD
DC
D
u
β
strategy mutation
position mutation
D "
! C
C b − c −c
b
0
D
y = Pr(Sk = Sq )
z = Pr(Xk = Xq )
Sk
g = Pr(Sk = Sq , Xk = Xq )
Xk
h = Pr(Sl = Sk , Xk = Xq )
! "∗
b
z−h
=
c
g−h
strategy
position
! "∗
#
$
b
M +r+2+µ
M (1 + µ)
3+µ
1
=
+
−
c
M −1
(M − 1)(2 + µ)
r
r+2+µ
M=2 islands
20
µ = 2N u
r = 2N β
10
5
0
2
4
6
r
8
10
island size dependence (µ = 0)
10
M=2
3
5
10
oo
8
*
0
µ=2
µ=1
µ=0
6
(b/c)
(b/c)*
15
4
2
0
0
2
4
6
r
8
10
Evolution in phenotype space
Moran 75
birth
β 1 − 2β β
N =7
random
death
phenotype
Evolution in phenotype space
birth
β 1 − 2β β
N =7
random
death
phenotype
Evolution in phenotype space
N =7
phenotype
disperse or condense ?
Group of size
√
r diffuses as D = r/2
r = Nβ
u
Colors
Mutation
y = Pr(Sk = Sq )
z = Pr(Xk = Xq )
Sk
g = Pr(Sk = Sq , Xk = Xq )
Xk
strategy
position
h = Pr(Sl = Sk , Xk = Xq )
D "
! C
C b − c −c
b
0
D
onsite play
phenotype
! "∗
b
z−h
=
c
g−h
TA '09
β = 1/2, µ = 1/2
10
Simulation
Theory
8
8
6
6
(b/c)*
(b/c)*
10
4
4
√
1 + 12 2
7
2
0
0
10
r=1
r=2
r=4
r=∞
2
20
30
40
50
0
N
0
1
2
3
µ
µ = 2N u
r = 2N β
4
5
Coop
Def
Coop
!
1
T̂
Def
"
Ŝ
0
T̂ < Ŝ + 1 +
√
3
Fig 1
Higher dimensions
phenotype II
Sets
?
phenotype I
Graphs
Tarnita '09
Tarnita '09
One parameter to rule them all
A wins iff
when playing against
σa + b > c + σd
single parameter for all structures
payoff of
classical well mixed σ = 1
a+b>c+d
(risk dominance)
√
!A
A a
B c
B"
b
d
or σ = 1 − 2/N
phenotype game σ = 1 + 3
more strategies on structure?
# strategies
2
≥3
# parameters
1
2
Wage, Tarnita '10
Relations to relatedness
A wins iff
b
1
>
c
R
(Hamilton's rule)
same size islands
Pr(Sk = Sq |Xk = Xq ) − Pr(Sk = Sq )
R=
1 − Pr(Sk = Sq )
fluctuating size islands,
phenotype walk
! "∗
b
z−h
=
c
g−h
Pr(Sk = Sq | Xk = Xq ) − Pr(Sl = Sk | Xk = Xq )
R=
1 − Pr(Sl = Sk | Xk = Xq )
TA '09, Taylor '10
sets, more general
structures
no relatedness interpretation
of our general formulas
Final slide
general method to study weak selection
TA, Ohtsuki, Wakeley, Taylor, Nowak, PNAS '09
papers can be found on my website
thanks