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Range Expansions with Competition or Cooperation
Large mammals expand over ~104 km
Bacteria (in a Petri dish) expand ~ 1 cm
In 500 generations….
Range Expansions with Competition or Cooperation
RED GREEN
Large mammals expand over ~104 km
Red and Green Strains….1. Could be neutral….2. Could have different doubling times3. One or both could secrete toxins
that impede the other…4. One or both could secrete amino
acids useful to the other (mutualism)Bacteria (in a Petri dish) expand ~ 1 cm
In 500 generations….
Competition and Cooperation at FrontiersWhy Frontiers?
--Range expansions are very common in biology… Number fluctuations very large at the edge of a population wave
-- Can we test theories of frontier evolution and cooperation with colored bacterial strains with variable “mutualism”?
Stepping Stone Models of Competition and Cooperation
-- Frequency-dependent selection (prisoner’s dilemma, snow drift, coordination games)
-- Phase transitions in 1+1 dimensions as the degree of cooperation is varied….
K. Korolev & drn
O. HallatschekJ. XavierK. FosterN. KarohanA. MurrayM. MuellerM. Lavrentovich
Genetic drift for a neutralmutation, (M. Kimura)
Finite populations go to fixation for long times(using, e.g., Fisher-Wright population sampling)
u(p,t)= probability allele A has frequency p at time t.
N = 20Populus 5.3
Genetic drift for a neutralmutation, (M. Kimura)
2
2
[ ( , )]( , )
( ) (1 ) /(4 )
G
G
D u p tu p tt p
D p p p N
Finite populations go to fixation for long times(using, e.g., Fisher-Wright population sampling)
u(p,t)= probability allele A has frequency p at time t.
N = 20Populus 5.3
Genetic drift for a neutralmutation, (M. Kimura)
2
2
[ ( , )]( , )
( ) (1 ) /(4 )
G
G
D u p tu p tt p
D p p p N
Finite populations go to fixation for long times(using, e.g., Fisher-Wright population sampling)
u(p,t)= probability allele A has frequency p at time t.
Finite populations go to fixation for long times
Probability of fixation of a single neutral mutation in a population of size N is just 1/N
But N is small in the vicinity of an expanding population front!
N = 20Populus 5.3
Mutation-Selection Balance
Linear inoculants (razor blade inculation) 50%-50% mixtures
100X, 6h after innoculation
HCB1553/pVS133Background DH5
HCB1550/pVS130Background DH5
PTrc99A: ecfpA206K
Amp(R)
Ori pBR
PtrcPTrc99A: yfpA206K
Amp(R)
Ptrc
Ori pBR
Genetic Demixing of Escherichia coli
P(
Phase transitions at frontiers…(mutation-selection balance)
0.5
0.5
1 -µ
G
R
p
Red wins
Green wins
Directed percolation
( | ) 0.5 selective advantage
( | ) 1( | ) 0
is a mutation rate;
these are if 0.5( | ) 0, ( | ) 1
(no back mutations, )
G R
G R
P G RG p ssP R RG pP R GG
deleterious pP G RR P R RR
R G
Domany-Kinzel model
0.5, 0G Rp 0.7, 0.1G Rp
Max LavrentovichOskar Hallatschek
R G
0.5p s
0.5
0.0
0 1.0
Compact DP Directed Percolation Phase Transition(mutation-selection balance)
Cluster of 100 cells (Max Lavrentovich)
Linear inoculants (razor blade inculation) 50%-50% mixtures
100X, 6h after innoculation
HCB1553/pVS133Background DH5
HCB1550/pVS130Background DH5
PTrc99A: ecfpA206K
Amp(R)
Ori pBR
PtrcPTrc99A: yfpA206K
Amp(R)
Ptrc
Ori pBR
Genetic Demixing of Escherichia coli
Razor blade inoculations are like massively parallel serial dilution experiments…
In effect, a moving population front is a serial dilution experiment in a well mixed test tube
N = 20: mutant with selective advantage s takes over after ~70 generations
For a “zero-dimensional” frontier, f(t), the fraction of red cells with selective advantage s at time t obeys
f(t)
t
Gen X Gen Y Gen Z ….
rg
g gr
g
( ) (1 )(1 ) ( )
( ) ( ') ( ') (Ito calculus)
df t f fsf f tdt N
t t t t
One Dimensional Stepping Stone Model of Population Genetics
( , ) red fraction at position x, time 1 ( , ) green fraction at position x, time
m, spatial diffusion constant
f x t tf x t t
D
population size on an island
N
2
2
( , ) (1 ) (1 ) / 2 ( , )
( , ) ( ', ') 2 ( ') ( ')
f x t fD sf f f f N x tt xx t x t t t x x
Describes number fluctuations(i.e., genetic drift) on each island
Assume(1) the interface remains flat & (2)cells stop growing behind the frontier
Then invoke “dimensional reduction” and the….
x
Mutalistic yeast strains(a small step towards multicellular life)
Mutualism: survival requires exchanging amino acids, if leucine and tryptophan are not already present in the natural environment
Frequency-dependent selection2
2
( , ) ( , ) ( ) (1 ) (1 ) / 2 ( , )f x t f x tD s f f f f f N x tt x
Let w and w be the offspring produced during one generation at a given point on the frontier...
then, ( ) 2
R G
R G
R G
w ws fw w
Describe mutualism by...( , ) (1 ( , ))( , ) ( , )
assume ,
R
G
w x t g f x tw x t g f x t
g
fV(f) 1.0
-s0/32
f*=0.5
2
2
*0
( , ) ( , ) ( ) (1 ) / 2 ( , )
( ) ( ) (1 )
f x t f x t dV fD f f N x tt x df
dV f s f f f fdf
*0
0*
( ) ( )( ) /
/ ( )
s f s f fs g
f
s(f)
f
f* 1.00.0
-/g
/g
Connection with game theory (g = 1, zero dimensions)
2*
02
( , ) ( , ) ( ) (1 ) (1 ) / 2 ( , )f x t f x tD s f f f f f f N x tt x
0
*
( )
/ ( )
s
f
M. Nowak et al., Nature 428, 646 (2004)J. Gore et al. Nature 459, 253 (2009)E. Frey et al., Phys. Rev. Lett. 105, 178101 (2010)
*0 ( ) (1 )s f f f f
In a well-mixed culture, the evolutionarily stable strategy (ESS) for mutualists leads to (transient) mixing….
f*
Selective Dominance
Selective Dominance
(competitiveexclusion)
(mutualism)
Evolutionary Stable Strategy is reached independent of the initial condition!
Fitness Landscape in the (α,β)-plane
Mutualism prevails
Genetic demixing
Selective dominance: Yellow prevails
Selective dominance: Blue prevails
2*
02
( , ) ( , ) ( ) (1 ) (1 ) / 2 ( , )f x t f x tD s f f f f f f N x tt x
Mutualists on CSM-Leu-Trp
yMM26:yMM31
CSM-Leu-Trp→ Obligate mutualism→ No demixing
CSMMutualism unimportant→ Demixing
Intermediate [Leu], [Trp]→ Facultative mutualism→ Phase transition?
mutation
• Enforcing mutualism enhances mutation rate• Mutations appear as jumps in the fitness landscape…
[Trp]
[Leu
]
[Trp]
[Leu]
0.0x 0.2x 0.4x 0.6x 0.8x 1.0x
0.0x
0.1x
0.2x
0.3x
0.4x
0.5x
LeuFBR Trp-
Leu- TrpFBR
Phasespace for
experiments
Computer simulations: Can mutalism prevent genetic demixing?
==0.00s0 = 0
==1.20s0 = 0.20
==0.05s0 = 0.05
=, L=100, N=30, mN=2
unstable mutualism near transition stable mutualism
every 1000 generations
every 100 generations
Time
Mutualistic “Smoke”