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”