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) GENETICSBIOCHEMISTRY 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, (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 Pλ(R-O12) AHL Quorum sensing module Motility control module

200 min300 min400 min 500 min 600 min WT control Experiments done at HKU Seeded at plate center at t = 0 min 300 min700 min900 min1400 min1100 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 Fisher/Keller-Segel: Logistic growth + diffusion  x ρsρs c Traveling wave solution Exponential front No stripes!

Growth equations for engineered bacteria 3-component model Bacteria (activator) 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 Kh-εKh-ε μ(h)μ(h) hKhKh 0 motile Non- motile Bacteria AHL random walkimmotile high density/AHLlow density/AHL Growth rate Degradation rate

Moving frame, z = x - ct 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 Solution of the  -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 Stability boundary: L h /L ρ 

Key parameters governing the stability of the solution 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m5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, (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) control circuit (reaction)