Exploring spatial pattern formation using a simple individual-based model

Exploring microbial patterns formation using a simple IBM

Exploring microbial patterns formation using asimple IBM

Nabil Mabrouk

www.cemagref.fr

15 decembre, 2009


Introduction

Introduction

Microscopic observation of microbial systems reveals adiversity of spatial patterns


Introduction

Introduction

Our aim: investigate how these large-scale patterns emerge

Our approach: individual-based modeling

Represent the individuals explicitlySimulate the pattern formation under different conditions


Introduction

Introduction



Represent the individuals explicitly

Simulate the pattern formation under different conditions


Introduction

Introduction



Represent the individuals explicitlySimulate the pattern formation under different conditions


A simple birth-death model

Model description

Simple is beautiful, and necessary (Deffuant et al., 2003)




Overview:

2D domain with individualsrepresented as point particles

Two processes:

death with a probability dbirth with a probability b

We are interested in the case:

wb << L : local birthb = d = constant

mean-field limit (for large N):




Overview:


Two processes:

death with a probability d

birth with a probability b







Overview:


Two processes:








Overview:


Two processes:



wb << L : local birth

b = d = constant





Overview:


Two processes:








Overview:


Two processes:




mean-field limit (for large N):dNdt = (b − d)N



Simulation with wb/L = 0.015

Figure: t = 0




Figure: t = 400




Overview:

Two processes:

death with a probability di ,i = 1..Nbirth with a probability b


wb << L : local birthbirth probability b isconstant

death probabilities dependon the neighborhood (thepattern)

di = d1 + d2∑

j Kd

(||xi−xj ||

wb

)wb << wd , b > d1 and d2 > 0




Overview:

Two processes:



wb << L : local birthbirth probability b isconstantdeath probabilities dependon the neighborhood (thepattern)

di = d1 + d2∑

j Kd

(||xi−xj ||

wb

)wb << wd , b > d1 and d2 > 0




Overview:

Two processes:




di = d1 + d2∑

j Kd

(||xi−xj ||

wb

)

wb << wd , b > d1 and d2 > 0




Overview:

Two processes:




di = d1 + d2∑

j Kd

(||xi−xj ||

wb

)wb << wd , b > d1 and d2 > 0



Simulation with wb/L = 0.015 and wd >> wb

Figure: t = 0



Simulation with wb/L = 0.015 and wd >> wb

Figure: t = 800


Birth-death model with motility

A birth-death model with motility

Overview:

Three processes:

death with a probability di ,i = 1..Nbirth with a probability bmotility with a probabilitymi , i = 1..N


motility probabilities dependon the neighborhood

mi = m1−m2∑

j Kv

(||xi−xj ||

wv

)



Parameters

9 parameters:

wb, wd , wm, wv

b, d1, d2, m1 and m2

Additional assumptions:

wb (birth) << wd (death)wm (mobility) >> wb (birth)wv (”viscosity’) > wd (death)b >> d1 m1 = 1.0 and d2, m2 > 0



Simulation results

Figure: t = 0



Simulation results

Figure: t = 800



Are these patterns realistic?

Figure: (Xavier et al., 2009) Fluorescent microscopy of yellow[U+FB02]uorescent protein-labeled biofilm shows cells in spatial patternswith holes, labyrinths, and wormlike shapes.



Are these patterns realistic?

Figure: (Xavier et al., 2009) Continuous variation of spatial patternsacross the surface of the coverslip is produced by the systematic variationof nutrient concentration. This image is a montage of four contiguousphase-contrast microscopy images.


Conclusion

”A change without pattern is beyond Science” (Zeide, 1991)

Experimental data contains: meaningful pattern andmisleading noise

IBM (modeling) can help in extracting patterns andunderstanding how they form and impact the population

Perspectives ...


Conclusion

The end!

Exploring spatial pattern formation using a simple individual-based model

Education

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