Stripe formation
In
an expanding
bacterial colony
with
density

suppressed
motility
The 5
th
KIAS Conference on Statistical Physics:
Nonequilibrium
Statistical Physics of Complex Systems
3

6 July 2012, Seoul, Korea
Synthetic
biology
Phenotype
(structure and
spatiotemporal dynamics)
Molecular mechanisms
(players and their interactions)
Traditional
biological
research
(painstaking)
GENETICS
BIOCHEMISTRY
discovery of novel
mechanisms and
function
Lei

Han Tang
Beijing Computational Science
Research Center
and Hong Kong Baptist U
Chenli Liu
(Biochem)
Xiongfei Fu
(physics)
Dr
Jiandong
Huang
(
Biochem
)
The Team
HKU
UCSD
:
Terry
Hwa
Marburg
:
Peter
Lenz
C. Liu et al, Science
334
, 238 (2011); X. Fu et al.,
Phys Rev
Lett
108
, 198102 (2012)
HKBU
Xuefei
Li
Lei

Han Tang
Periodic stripe patterns in biology
dicty
fruit fly embryo
snake
Morphogenesis in biology:
two competing scenarios
•
Morphogen
gradient
(
Wolpert
1969)
–
Positional information laid
out
externally
–
Cells respond passively
(gene expression and
movement)
•
Reaction

diffusion
(Turing 1952)
–
Pattern formation
autonomous
–
Typically involve mutual
signaling and movement
Reaction

Diffusion Model as a Framework for Understanding
Biological Pattern Formation
,
S Kondo
and
T Miura
,
Science
329
, 1616 (2010)
Cells have complex physiology and behavior
Growth
Sensing/Signaling
Movement
Differentiation
All play a role in the
observed pattern at
the population level
Components characterization
challenging in the native context
Synthetic molecular circuit inserted into
well

characterized cells (E. coli)
Experiment
Swimming bacteria (Howard Berg)
Bacterial motility 1.0:
Run

and

tumble motion
~10 body length
in 1
sec
cheZ
needed
for running
Extended run along
attractant gradient
=>
chemotaxis
CheY

P
low
CheY

P
high
Couple cell density to cell motility
High density
Low density
cheZ
expression
normal
cheZ
expression
suppressed
Genetic Circuits
CheZ
luxR
luxI
Plac/ara

1
cI
PluxI
CI
LuxR
LuxI
cheZ
P
λ
(R

O12)
AHL
AHL
Quorum
sensing
module
Motility control
module
200 min
300 min
400 min
500 min
600 min
WT control
Experiments done at HKU
Seeded at plate center at t = 0 min
300 min
700 min
900 min
1400 min
1100 min
engr strain
•
Colony size expands three times slower
•
Nearly perfect rings at fixed positions once formed!
Phase diagram
Simulation
Experiments at different
aTc
(
cI
inducer) concentrations
Increase basal
cI
expression
=>
decrease
cheZ
expression
=>
reduction of overall bacterial motility
many rings => few rings => no ring
•
How patterns form?
•
Anything new in this pattern formation process?
•
Robustness?
Qualitative and quantitative issues
How patterns form
Initial low cell density,
motile population
Growth =>
high density region
=> Immotile zone
Expansion of immotile region via
growth and aggregation
Appearance of a
depletion zone
Same story
repeats
itself?
Sequential
stripe
formation
Modeling and analysis
Front propagation in bacteria growth
2
1
s
D
t
Fisher/Keller

Segel
:
Logistic
growth + diffusion
x
ρ
s
c
Traveling wave solution
ˆ
(,) ( )
x t x ct
( )
x ct
e
Exponential front
1/2 1/2
2,/
c D D
No stripes!
2
2
2
[
(
)
]
n
n
h
t
n
K
Growth equations for
engineered bacteria
3

component model
Bacteria
(activator)
2
2
2
n
n
n
k
n
n
D
n
t
n
K
h
t
D
h
2
h
h
AHL
(repressor)
Nutrient
AHL

dependent motility
nutrient

limited growth
Sequential
stripe formation
from numerical
solution of the
equations
front propagation
Band formation
propagating front
unperturbed
aggregation
behind the front
Analytic solution: 2

component model
K
h

ε
μ
(
h
)
h
K
h
0
motile
Non

motile
for
( )
( ) for
0 for
h
h
h h
h
D h K
D K h
h K h K
h K
Bacteria
AHL
2
[ ( ) ] 1
x
s
h
t
2
h x
h
D h h
t
random walk
immotile
high density/AHL
low density/AHL
Growth rate
Degradation rate
Moving frame
,
z
=
x

ct
2
2
2
2
[ ( ) ] (1 ) 0
0
s
h
h c
z z
h h
D c h
z z
Steady travelling wave
solution (no stripes)
Solution
strategy
i)
Identify dimensionless parameters
ii)
Exact solution in the linear case
iii)
Perturbative
treatment for growth with
saturation
1 1
ˆ
ˆ
ˆ
ˆ
( ) ( ) ( )
h
h z dz z G z z
ˆ
4 4
/
2
1
where ( )
ˆ
4 4
d
z
z d
d
h
G z e e
d
Solution of the
ho

eqn
in two regions
Solution of the
h

eqn
using Green’s fn
Stability limit
Motile front
Cell depletion zone
“Phase
Diagram” from the stability limit
Characteristic lengths
Cell
density profile
AHL diffusion
L D
h h
L D
Stability boundary:
L
h
/
L
ρ
5
Key
parameters governing the stability of the solution
h h
L D
L D
Bacteria profile
AHL profile
i)
AHL profile follows
the cell density profile
most of the time.
ii)
In the
depletion zone
,
AHL profile is
smoothened
compared to the cell
density profile. The
degree of
smoothening
determines if AHL
density can exceed
threshold value in the
motile zone.
iii)
If the latter occurs,
nucleation of high
density/immotile band
takes place
periodically
=>
formation of stripes
Discussion
The mathematics of biological pattern formation
Debate:
cells are much more complex than small molecules
=>
Deciphering necessary ingredients in the native
context challenging
Resort to synthetic biology (E. coli)
–
Minimal ingredients: cell growth, movement, signaling,
all well
characterized
–
Defined interaction: motility inhibited by cell density (aggregation)
Formation of sequential periodic stripes on semi

solid agar
Genetically tunable
Stripe formation in open geometry (new physics)
Theoretical analysis deepens understanding of the experimental
system in various parameter regimes
Open issues
Period of stripes
analysis of the immotile band formation in the motile zone
Robustness of the pattern formation scheme
Residual chemotaxis
Inhomogeneous cell population
Cell

based modeling
Sharpness of the zones
Multiscale treatment (cell => population)
Biology goes quantitative
New problems for
statistical physicists
Close
collaboration
key to success
Life is complex!
Biological game
:
precise control of pattern
through molecular circuits
Population
:
pattern
formation
5mm
Cell
:
reaction

diffusion
dynamics
5
m
This work
Acknowledgements:
The RGC of the HKSAR Collaborative
Research Grant HKU1/CRF/10
HKBU Strategic Development Fund
Thank you for your attention!
Turing patterns
The Chemical Basis of Morphogenesis
A. M. Turing
Philosophical Transactions of the Royal
Society of London. Series B, Biological
Sciences
237
, 37

72 (1952)
Ingredients
:
two
diffusing species,
one
activating
, one
repressing
S Kondo and T Miura, Science
329
, 1616 (2010)
Pattern formation (concentration modulation) requires
i)
Slow diffusion of the active species (short

range
positive feedback
)
ii)
Fast diffusion of the repressive species (long

range
negative feedback
)
2
2
(,)
(,)
u
v
v
D
v
v
F
t
u
u u
u
t
v
D G
control circuit
(reaction)
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