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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)