Active Walker Model for Bacterial Colonies:Pattern Formation and Growth Competition

Shane Stafford

Yan Li

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,

Simulation overview:•Model: Active Walker Model (AWM)

•Variables:

•Nutrient concentration P

•Surface roughness Nc

•Number of inoculation points

•Results:

•Patterns under different growth conditions (single colony)

• Growth radius Rg, ,Rmax

• Fractal dimension d

•Patterns of two bacterial colonies

Algorithm: active walker model•Walkers

• Each walker is a bacteria cluster (103 –104 individual bacterium) and is characterized by its location (xi, yi) and internal energy wi

•Perform off-lattice random walk of step size d[0,dmax] at an angle

[0,2] generated by two random numbers

•loses energy at a fixed metabolism rate e

•consumes nutrient at a fixed rate cr or the maximum amount available

•divides at threshold wi= tr,, becomes stationary when wi=0

•Threshold collision time Ncroughness of substrate

Nc is changed from 2 to 10 in our simulation

ri

irii

tnw

enrccnwnw

)(0

)),(,min()()1(

•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

)),(,min()(),(2 trccrrtrcDt

cri

actives

• 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 //Nc

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

))log(/)log( Nd 0,~ dN

dRM ~ )log(/)log( RMd R

2ig rR

Results: patterns200*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 Nc=6 and vary the initial nutrient

concentration P=1.0, 3.0, 5.0, 7.0 and 9.0

P=9.0 P=7.0

P=5.0 P=5.0

P=1.0

Nc=6

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 Nc=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 Nc=2,4,6,8 and10

Nc=2 Nc=4

Nc=6 Nc=8

Nc=10

P=2.0

Results: growth radiusFixed surface roughness Nc=6

Fixed initial concentration level P=2.0

Results: fractal dimensionFixed surface roughness Nc=6

Fixed initial concentration level P=2.0

Conclusion:(1) The growth radius Rg and Rmax 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.

Competition between two colonies1    Nc=6, p=5.0;

Inoculation points (40,100) and (160,100), d=80

2.    Nc=6, p=5.0;

Inoculation points (75,100) and (125,100), d=50

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 CapacitorsI am happy with

what I have

Drive for food

Drive for food

Invader, Run!!

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