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Stripe formation
In
an expanding bacterial colony
with density-suppressed
motility
The 5th 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 TangBeijing 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 LiLei-Han Tang
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
GrowthSensing/SignalingMovementDifferentiation
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)
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 densityLow density
cheZ expression normal
cheZ expression
suppressed
Genetic Circuits
CheZ
luxR luxI
Plac/ara-1
cIPluxI
CI
LuxR
LuxI
cheZPλ(R-O12)
AHL
AHL
Quorum sensing module
Motility control module
200 min 300 min 400 min 500 min 600 min
WT
con
trol
Experiments done at HKU
Seeded at plate center at t = 0 min
300 min 700 min 900 min 1400 min1100 min
engr
str
ain
• 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
Front propagation in bacteria growth
21s
Dt
Fisher/Keller-Segel:
Logistic growth + diffusion
x
ρs
c
Traveling wave solution
ˆ( , ) ( )x t x ct
( )x cte
Exponential front
1/ 2 1/ 22 , /c D D
No stripes!
22
2[ ( ) ]
n
nh
t n K
Growth equations for engineered bacteria
3-component model
Bacteria (activator)
22
2n
nn
k nnD n
t n K
h
tD
h2h hAHL
(repressor)
Nutrient
AHL-dependent motility nutrient-limited growth
Sequential stripe formation from numerical solution of the equations
front propagation
Band formation
propagating frontunperturbed
aggregationbehind the front
Analytic solution: 2-component model
Kh-ε
μ(h)
hKh0
motile Non-motile
for
( )( ) for
0 for
h
hh h
h
D h K
D K hh K h K
h K
Bacteria
AHL
2[ ( ) ] 1xs
ht
2h x
hD h h
t
random walk immotile
high density/AHLlow density/AHL
Growth rate
Degradation rate
Moving frame, z = x - ct
2
2
2
2
[ ( ) ] (1 ) 0
0
s
h
h cz z
h hD c h
z z
Steady travelling wave solution (no stripes)
Solution strategy
i) Identify dimensionless parametersii) Exact solution in the linear caseiii) Perturbative treatment for growth with
saturation
1 1ˆ ˆ ˆˆ( ) ( ) ( )hh z dz z G z z
ˆ4 4/ 21
where ( )ˆ4 4
dzz d d
hG z e ed
Solution of the rho-eqn in two regions
Solution of the h-eqn using Green’s fn
Stability limit
Motile frontCell depletion zone
“Phase Diagram” from the stability limit
Characteristic lengths
Cell density profile
AHL diffusion
L D
h hL D
Stability boundary:
Lh/Lρ ≈ 0.3-0.5
Key parameters governing the stability of the solution
h hL 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
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
5mm
This work
Acknowledgements:
The RGC of the HKSAR Collaborative Research Grant HKU1/CRF/10
HKBU Strategic Development Fund
Turing patterns The Chemical Basis of MorphogenesisA. M. TuringPhilosophical 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
vD
vv
Ft
uu u
ut
vD G
control circuit (reaction)
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