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 Centerand 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)

HKBUXuefei LiLei-Han Tang

Periodic stripe patterns in biology

dictyfruit 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 movementReaction-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)

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 densityLow density

cheZ expression normal

cheZ expression

suppressed

Genetic Circuits

CheZ

luxR luxIPlac/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

cont

rol

Experiments done at HKU

Seeded at plate center at t = 0 min

300 min 700 min 900 min 1400 min1100 min

engr

stra

in

• 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

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

nht n K

Growth equations for engineered bacteria3-component model

Bacteria (activator)

22

2n

nn

k nn D nt n K

ht

Dh2h 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

h D h ht

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 hz 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/ 21where ( )

ˆ4 4

dzz d d

hG z e ed

Solution of the ho-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 ≈ 0.30.5

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

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 stripesanalysis of the immotile band formation in the motile zone

Robustness of the pattern formation schemeResidual chemotaxisInhomogeneous cell population

Cell-based modelingSharpness of the zonesMultiscale 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

5m

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 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) requiresi) 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

v v

Ftu u u

ut

vD G

control circuit (reaction)