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discovery of novel
mechanisms and
function
Stripe formation
In
Phenotype
an expanding
bacterial colony
with
density-suppressed
motility
Lei-Han Tang
Beijing Computational Science
Research Center
(structure and
spatiotemporal dynamics)
Traditional
biological
research
(painstaking)
Synthetic
biology
Molecular mechanisms
(players and their interactions)
GENETICS
BIOCHEMISTRY
and Hong Kong Baptist U
The 5th KIAS Conference on Statistical Physics: Nonequilibrium Statistical Physics of Complex Systems
3-6 July 2012, Seoul, Korea
The Team
HKU
Chenli Liu
(Biochem)
HKBU
Xuefei Li
Lei-Han Tang
UCSD:
Terry Hwa
Xiongfei Fu Dr Jiandong Huang
(physics)
(Biochem)
Marburg:
Peter Lenz
C. Liu et al, Science 334, 238 (2011); X. Fu et al., Phys Rev Lett 108, 198102 (2012)
Periodic stripe patterns in biology
fruit fly embryo
dicty
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
Components characterization
challenging in the native context
Growth
Sensing/Signaling
Movement
Differentiation
All play a role in the
observed pattern at
the population level
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
CheY-P
low
CheY-P
high
cheZ needed
for running
Extended run along
attractant gradient
=> chemotaxis
Couple cell density to cell motility

Low density
High density
cheZ
expression
normal
cheZ
expression
suppressed
Genetic Circuits
AHL
LuxR
AHL
cI
LuxI
PluxI
CI
luxR
luxI
Plac/ara-1
cheZ
Pλ(R-O12)
Motility control
module
CheZ
Quorum
sensing
module
Experiments done at HKU
WT control
Seeded at plate center at t = 0 min
300 min
400 min
500 min
600 min
engr strain
200 min
300 min
700 min
900 min
1100 min
• Colony size expands three times slower
• Nearly perfect rings at fixed positions once formed!
1400 min
Phase diagram
Increase basal cI expression => decrease cheZ
expression => reduction of overall bacterial motility
many rings => few rings => no ring
Simulation
Experiments at different aTc (cI inducer) concentrations
Qualitative and quantitative issues
• How patterns form?
• Anything new in this pattern formation process?
• Robustness?
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
Fisher/Keller-Segel:
Logistic growth + diffusion



  1    2  D  
t
 s 
No stripes!
Traveling wave solution
 ( x, t )  ˆ ( x  ct )

Exponential front
ρs

c
e ( x ct )
c  2  D  ,    / D 
1/ 2
x
1/ 2
Growth equations for
engineered bacteria
3-component model
Bacteria
(activator)

 n2 
2
  [  ( h)  ]  2
t
n  Kn
AHL-dependent motility
nutrient-limited growth
AHL
(repressor)
h
 Dh2 h     h
t
Nutrient
k n n 2 
n
2
 Dn n  2
t
n  Kn
Sequential
stripe formation
from numerical
solution of the
equations
front propagation
aggregation
behind the front
propagating front
unperturbed
Band formation
Analytic solution: 2-component model
random walk
immotile
μ(h)
Nonmotile
motile
low density/AHL
Bacteria
high density/AHL



  2x [  (h)  ]   1  
t
 s 
Growth rate
AHL
h
 Dh  2x h     h
t
Degradation rate
0
Kh-ε
 D

 D ( K  h)
 ( h)    h


0

Kh
h
for h  K h  
for K h    h  K h
for h  K h
Steady travelling wave solution (no stripes)
Moving frame, z = x - ct
Solution strategy
2

[

(
h
)

]

c
  (1   )  0
2
s
z
z
2h
h
Dh 2  c     h  0
z
z
i) Identify dimensionless parameters
ii) Exact solution in the linear case
iii) Perturbative treatment for growth with
saturation
Solution of the ho-eqn in two regions
Stability limit
Solution of the h-eqn using Green’s fn

hˆ( zˆ)  ˆ  dz1ˆ ( z )Gh ( z  z1 )

where Gh ( z ) 
1
4  4ˆ d
e
z/d
e

4  4 ˆ d
z
2d
Cell depletion zone
Motile front
“Phase Diagram” from the stability limit
Characteristic lengths
Cell density profile
L  D 
AHL diffusion
Lh  Dh 
Stability boundary:
Lh/Lρ 5
Key parameters governing the stability of the solution
Bacteria profile
L  D 
AHL profile
Lh  Dh 
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)
Life is complex!
Biology goes quantitative
Close
collaboration
key to success
New problems for
statistical physicists
This work
Cell:
reaction-diffusion
dynamics
5m
Biological game:
precise control of pattern
through molecular circuits
Population:
pattern
formation
5mm
Acknowledgements:
The RGC of the HKSAR Collaborative
Research Grant HKU1/CRF/10
HKBU Strategic Development Fund
Thank you for your attention!
Turing patterns
Ingredients: two diffusing species,
one activating, one repressing
u
 Du  2u  F (u , v)
t
v
 Dv  2 v  G (u, v)
t
control circuit
(reaction)
The Chemical Basis of Morphogenesis
A. M. Turing
Philosophical Transactions of the Royal
Society of London. Series B, Biological
Sciences 237, 37-72 (1952)
Pattern formation (concentration modulation) requires
i)
Slow diffusion of the active species (short-range
positive feedback)
ii) Fast diffusion of the repressive species (longrange negative feedback)
S Kondo and T Miura, Science 329, 1616 (2010)