Range Expansions &
Population Genetics on Land
Oskar Hallatschek & drn, (E. coli)
In 500 generations….
Large mammals expand over ~104 km
Bacteria (in a Petri dish) expand ~ 1 cm
K. Korolev et al., Reviews of
Modern Physics 82, 1691 (2010)
K. Foster, J. Xavier, K. Korolev, drn
(genetic demixing in P. Aeruginosa)
What would happen if we could “replay the tape of life”?
can infer the radius R 0
of the homeland from
data at the boundary:
N sec   R0 v / 2 DW
Chiral range expansions! KC Huang et al.
But… life probably evolved
first in a liquid environment
•~2-3 billion years ago, water covered most of
the earth
•Fossilized, oxygen-producing cyanobacteria
have been dated at ~2 billion years ago.
Cyanobacterium Synechococcus
www.dr-ralf-wagner.de/Blaualgen-englisch.htm
•Oxygenic cyanobacteria transformed the
atmosphere via photosynthesis
•Spatial growth and evolutionary competition
took place in liquid environments at both high
low Reynolds numbers
•These photosynthetic organisms can control
their buoyancy to resist down welling currents
and stay close to the ocean surface.
Bloom of cyanobacteria
in Lake Atitlán, Guatemala
NASA Earth observatory
Life at High Reynolds Number
Competition and cooperation on land
and at sea…
--Range expansions and genetic competitions
on solid surfaces
--Cyanobacteria (& plankton) in the ocean
Effect of compressible turbulent
advection on the Fisher Equation–
--Quasilocalization on velocity sinks
--Greatly reduced carrying capacity
Effect on turbulence on population
genetics –
--Population bottlenecks and
altered fixation times
R. Benzi
M. Jensen
P. Perlecar
S. Pigolotti
F. Toschi
P. Perlekar, R. Benzi, F. Toschi and drn,
Population Dynamics at High
Reynolds Numbers, Phys. Rev. Lett.
105, 144501 (2010)
S. Pigolotti, R. Benzi, M. Jensen and
drn. Population Genetics in
Compressible Flows Phys. Rev. Lett.
108, 128102 (2012).
S. Pigolotti et al. Growth, competition
and cooperation in spatial population
genetics, Theoretical Population
Biology 84, 72 (2013).
Fisher Waves and Population Dynamics on Land
c(t )  population of species at time t in region 
dc(t )
 births - deaths + saturation + migration
dt
1798 T.R. Matthus
dc(t )
 ac(t ), a  0
dt
1836 P.F. Verhulst
dc(t )
 ac(t )  bc 2 (t )
dt
stable
population

size: c  a / b
biomass of yeast
c*
you are here
J. Maynard Smith,
Evolutionary Genetics
B
time (hours)
1937 R.A. Fisher

c  c(r , t )
BB
B
BB
BB
B

c(r , t )



 D 2c(r , t )  ac(r , t )  bc 2 (r , t )
t
Population & Genetic Waves In One Dimension
(R. A. Fisher, Kolmogorov-Petrovsky-Piscounov, 1937)

2
c( x, t )  D 2 c( x, t )  ac( x, t )  bc 2 ( x, t ); let c( x, t )  f ( x  vt )
t
x
c(x,t)
c(x,t)
a/b
a/b
c(x,t)
a/b
e at
v
2Dt
Schematic time development of a wavefront solution of Fisher’s equation on the infinite line. (J.D. Murray,
Mathematical Biology)
Interface velocity = 2 Da
Interface width =
D/a
Genetic fluctuations in a
neutral population, (M. Kimura)
Populus 5.3
N = 20
a
A
u(p,t)= probability allele A has
frequency p at time t.
Finite populations go to fixation for long times
(using, e.g., Fisher-Wright population sampling)
u ( p, t )  2 [ DG u ( p, t )]

t
p 2
DG ( p )  p (1  p ) /(4 N )
df (t ) / dt 
fi (1  fi ) / 2 N (t )
 i (t ) j (t ')  2 (t  t ')
(use Ito calculus...)
Survival of the Luckiest
(M. Kimura)
Populus 5.3
N = 20
u(p,t)= probability allele A has frequency
p at time t.
Finite populations go to fixation for long
times(using, e.g., Fisher-Wright
population sampling)
u ( p, t )  2 [ DG u ( p, t )]

t
p 2
DG ( p )  p (1  p ) /(4 N )
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!
But… life probably evolved
first in a liquid environment
•~2-3 billion years ago, water covered most of
the earth
•Fossilized, oxygen-producing cyanobacteria
have been dated at ~2.8-3.5 billion years ago.
Cyanobacterium Synechococcus
www.dr-ralf-wagner.de/Blaualgen-englisch.htm
•Oxygenic cyanobacteria transformed the
atmosphere via photosynthesis
•Spatial growth and evolutionary competition
took place at high Reynolds numbers
•These photosynthetic organisms can control
their bouyancy to resist down welling currents
and stay close to the ocean surface.
Swimming Synechococcus from Berg Lab
Why do they swim at all???
Navier-Stokes Equations and Reynolds Numbers
 
u (r , t ) = velocity field (incompressible)

   

 t u  (u )u  (1/  0 )p    2 u
 

  u  0, p (r , t ) = pressure
0  density;   viscosity
Navier &
Stokes ~1841
Kolmogorov
Energy
Cascade
~1941
Reynolds Number
Re  L0u0 / 
http://en.wikipedia.org/wiki/
Reynolds_number
Re ~ 102-103
Re ~ 108-109
BP Gulf oil spill
Life at High Reynolds Number (Roberto Benzi & drn)
Phytoplankton blooms at high Reynolds number in the Norwegian Sea and near Iceland
110 km
http://visibleearth.nasa.gov/cgi-bin/viewrecord?5278
.see also, Tel. et al. Phys. Rep. 413, 91 (2005).
A. P. Martin, Prog. Oceanography 57, 125 (2003)
mixing layer ≈ 25-100 m.
Phytoplankton
(see also zooplankton
& bacterioplankton)
http://earthobservatory.nasa.gov/Ex
periments/ICE/Channel_Islands/
Re  LU /   108  109
Large eddy turnover time  50 days
Small eddy turnover time  5 minutes
Plankton doubling time  12 hours
(in the middle of this range!)
PNAS 107,
18366 (2010)
Chlorophyll map
Velocity field from altimetry
diatoms (green)
Prochlorococcus (red)
Synechococcus (dark blue)
nanoeukaryotes (yellow)
Phaeocystis (magenta)
coccolithophorids (cyan).
Dominant species types
Compressible advection of microorganism density c(x,t)
 
 




c( x , t )    [u ( x , t )c( x , t )]  D 2 c( x , t )   c( x , t )[1  c( x , t )]
t
  
  u ( x, t )  0

u ( x , t ) is an effective 2d compressible turbulent velocity field....
 is the growth rate...
•Advection by an effectively compressible two dimensional
velocity field results for organisms that actively control their
buoyancy to stay close to the ocean surface.
accumulation points
Uop.whoi.edu/projects/projects.htm
Steady
Effect of compressibility in two dimensions
incompressible u ( x, y )   F sin(2 y / L)
u x ( x, y )   F sin(2 x / L)
x
flow
u ( x, y )   F sin(2 y / L)
u ( x, y )  F sin(2 x / L)
y
y
 
u  0
 
u  0
 
u  0
 
u  0
time
cell division &
competitive
death
Steady
compressible
flow
Fisher Equation with Compressible Turbulence in One Dimension


 2 c ( x, t )
c( x, t )  [u ( x, t )c( x, t )]  D
  c( x, t )[1  c( x, t )]
2
t
x
x
•Turbulent advection described by a synthetic velocity field u(x,t)
described by a shell model in Fourier space
u ( x, t )  F  [un (t )eikn x  un* (t )e  ikn x ],
n
d
2
*
*

k
u
(
t
)

i
(
k
u
u

k
u


n  n
n 1 n 1 n  2
n n 1un 1

 dt

(1   )kn 1un 1un  2 )  f n
u ( x, t ), one dimension
F controls advection strength
 controls intermittancy
f n    n ,0 injects energy
at long wavelengths
Typically, microorganism doubling time
~-1 exceeds small scale Kolmogorov
eddy turnover time  ….
  ( /  )1/2   1  L2/3 /  1/3
L
The effect of compressible
turbulence on the microorganism
dynamics is dramatic…
“Quasilocalization” in one dimension
c ( x, t )
See 1d movie
u ( x, t )
•Long wavelength features of source/sink
landscape remain fixed as bacteria grow
and saturate
•Effectively frozen velocity field u(x,t)
acts like a random force field that
localizes organisms at sinks
• At various intervals, intermittent bursts
of velocity reorganize pattern of sources
and sinks and pattern shifts….
•Landscape  Seascape!
•Populations constantly overshoot
carrying capacity at sinks, leading to
localized boom and bust cycles
•Dramatic reductions in carrying
capacity
x
Intermittent microorganism fluctuations in two dimensions
(Prasad Perlecar, F. Toschi, R. Benzi, )
 
 




c( x , t )  [u ( x , t )c( x , t )]  D 2 c( x , t )   c( x , t )[1  c( x , t )]
t
•Generate 2d compressible velocity field by assuming
microorganisms populate a fixed slice of a simulation of 3d
incompressible turbulence with Re  105
degree of compressibility
  (u ) 2  /  ( i u j ) 2  0.17
See 2d
movie
  
u (r , t )
Quasi localization in two dimensions
Bacterial localization length  :
1/4
D 
 




2
•One point closure
predicts a localization
length ξ ~ D1/2
D=0.05
•Average carrying capacity
<Z>  1 as μ  ∞
•<Z> suffers an 80%
reduction as μ  0.
Synechococcus: ~1/3
can swim at velocity
~25um/sec. Increases
Deff by factor of 1000…..
D=0.01
D=0.005
Competition, Cooperation and Fluid Mechanics
Particle-based
range expansions
for neutral variants
and with a selective
advantage of red
over green (no flow)
Melanie Mueller & Andrew Murray
Mutualists on various substrates (M. Mueller)
CSM (abundant Leu & Trp)
→ Mutualism unimportant
→ Strong genetic demixing
CSM-Leu-Trp (Leu, Trp missing)
→ Obligate mutualism
→ Slow growth, but genetic demixing
surpressed
Stochastic Fluid Mechanics of Two Interacting Species
See S. Pigolotti et al. Growth, competition and cooperation in spatial
population genetics,, Theoretical Population Biology 84, 72 (2013).
time
 
 [u ( x , t )c A ]
 
 [u ( x , t )cB ]
 A  B  
AA  BB  
AB   (1   A )
BA   (1   B )
 A and  B control the
strength of mutualism
Gillespie
algorithm
Effect of compressible turbulence on population
genetics with Lagrangian inertial particles (1d)
(R. Benzi, M. Jensen, S. Pigolotti, drn)
Long-lived “persistor states” can
have enormous fixation times…..
Start with well mixed initial
conditions and a single sink
in a converging flow
R|R vs. G|G vs. R|G vs. G|R
“Selective
Advantage
Meter”
S = 0.0
See also: A. Brandenburg and T. Multamaki, How long can left and right
handed life forms coexist?, Int. J. of Astrobiology 3, 209 (2004)
S = 0.3
A. Steady slightly compressible flow
u x ( x, y )  F sin(2 x / L)  F 'sin(2 y / L)
u y ( x, y )  F sin(2 y / L)  F 'sin(2 x / L)
  2
  (u )  /  ( i u j ) 2 
 F / ( F  F ' )  0.0027
2
2
2
Compressible flows with two
interacting species (P. Perlekar)
B. Compressible turbulent flow (Re ~105)
  2
  (u )  /  ( i u j ) 2 
 0.17
Life at High Reynolds Number
Competition and cooperation on land
and at sea…
--Range expansions and genetic competitions
on solid surfaces
--Cyanobacteria (& plankton) in the ocean
Effect of compressible turbulent
advection on the Fisher Equation–
--Quasilocalization on velocity sinks
--Greatly reduced carrying capacity
Effect on turbulence on population
genetics –
--Population bottlenecks and
altered fixation times
R. Benzi
M. Jensen
P. Perlecar
S. Pigolotti
F. Toschi
P. Perlekar, R. Benzi, F. Toschi and drn,
Population Dynamics at High
Reynolds Numbers, Phys. Rev. Lett.
105, 144501 (2010)
S. Pigolotti, R. Benzi, M. Jensen and
drn. Population Genetics in
Compressible Flows Phys. Rev. Lett.
108, 128102 (2012).
S. Pigolotti et al. Growth, competition
and cooperation in spatial population
genetics, Theoretical Population
Biology 84, 72 (2013).