1. Active Walker Model for Bacterial Colonies: Pattern Formation and Growth Competition Shane Stafford Yan Li

2. Introduction: Bacterial colonies exhibit complex growth patterns on starvation conditions Experimental facts: The growth pattern and fractal dimension depend on both the nutrient concentration and roughness of the agar substrate Bacteria perform a random walk like movement on the substrate, within a well-defined envelope (lubrication layer) Under extreme adverse living conditions, patterns become dense again by chemo tactic signaling (not dealt with in our simulations) E Ben-Jacob et al, Nature,368, (47)1994,

3. Simulation overview: Model : Active Walker Model (AWM) Variables: Nutrient concentration P Surface roughness N c Number of inoculation points Results: Patterns under different growth conditions (single colony) Growth radius R g,, R max Fractal dimension d Patterns of two bacterial colonies

4. Algorithm: active walker model Walkers Each walker is a bacteria cluster (10 3 –10 4 individual bacterium) and is characterized by its location (x i, y i ) and internal energy w i Perform off-lattice random walk of step size d  [0,d max ] at an angle  [0,2  ] generated by two random numbers loses energy at a fixed metabolism rate e consumes nutrient at a fixed rate c r or the maximum amount available divides at threshold w i = t r,, becomes stationary when w i =0 Threshold collision time N c  roughness of substrate N c is changed from 2 to 10 in our simulation

5. Landscape: The landscape is the nutrient (pepton) concentration c(r,t) on lattice At each time step, the landscape is updated by solving the diffuision equation locally: Boundary conditions are needed to realistically represent the system Initial nutrient concentration P is varied from high (supporting 10 walkers on a lattice site) to low (supporting 1 walker on a site only) Scaling: Parameters scaling is important for simulation to reproduce the phenomena in real life in both the correct time and space scales Diffusivity of nutrient, step size of walkers, lattice size, time step

6. Sample parameter input_____________________ //general parameters size = 200 initWalkers = 20 totalSteps = 2000 diffusionSteps = 1 seed = 5 peptoneConc = 20. lambda = 1.44//lambda is D * dt / dx**2 (unitless) //walker parameters reproThresh = 1.0 inactThresh = 0.0 maxUptake = 0.2 metabolism = 0.0667 maxJump = 0.4 initEnergy = 0.33 reproEnergy = 0.30 envelHits = 6 //N c

7. Algorithm: fractal dimension Dimension of fractal structures –Between regularity and total randomness: self-similarity Box counting method –Divide the pattern into  grid and count N , the minimal number of blocks to cover the pattern. Mass distribution method –Up limit of R is the gyration radius defined as –Problem: high concentration at the center bias the dimension towards high values

8. Results: patterns 200*200 lattice, run time=2000 steps, ten runs per set of parameters 1.One inoculation points at the center (100,100) (a) Fixed surface roughness N c =6 and vary the initial nutrient concentration P=1.0, 3.0, 5.0, 7.0 and 9.0

9. P=9.0P=7.0 P=5.0 P=1.0 N c =6

10. Results: 200*200 lattice, run time=2000 steps, ten runs per set of parameters 1.One inoculation point at the center (100,100) (a) Fixed surface roughness N c =6 and vary the initial nutrient concentration P=9.0, 7.0, 5.0, 3.0 and 1.0 (b) )Fixed initial nutrient concentration P=2.0 and vary surface roughness N c =2,4,6,8 and10

11. N c =2N c =4 N c =6 N c =8 N c =10 P=2.0

12. Results: growth radius Fixed surface roughness N c =6

13. Fixed initial concentration level P=2.0

14. Results: fractal dimension Fixed surface roughness N c =6

15. Fixed initial concentration level P=2.0

16. Conclusion: (1)The growth radius R g and R max decrease when the nutrient level is lowered or the surface becomes harder, consistent with the observation from experiments (2) The structure becomes more ramified as the nutrient level decrease, as expected. (3) The change of fractal dimension is less obvious in the case when the surface roughness is varied. –Possible reason: the range of surface hardness is not large enough –Need a faster algorithm to generate same size of patterns under extreme hard surface.

17. Competition between two colonies 1 N c =6, p=5.0; Inoculation points (40,100) and (160,100),  d=80

18. 2. N c =6, p=5.0; Inoculation points (75,100) and (125,100),  d=50

19. e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- Bacterial Capacitors I am happy with what I have Drive for food Invader, Run!!

20. Further investigation High level interaction? –Bacterial colonies interact not only locally, but also indirectly via marks left on the agar surface and chemical (chemo tactic) signaling. Patterns become dense at extreme low food level. –Two landscapes: nutrient concentration and chemical concentration –Inactive walkers generate a communicating field to attract active ones More realistic parameters –Variable metabolism rate and consumption rate –Need to obtain more insight into the physics in the growth process Speed up the code! –Optimization of template instantiation and random number mapping –Better diffusion solver