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

Exploring microbial patterns formation using a simple IBM

Introduction

Introduction

Microscopic observation of microbial systems reveals adiversity of spatial patterns

Exploring microbial patterns formation using a simple IBM

Introduction

Introduction

Microscopic observation of microbial systems reveals adiversity of spatial patterns

Exploring microbial patterns formation using a simple IBM

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

Exploring microbial patterns formation using a simple IBM

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

Exploring microbial patterns formation using a simple IBM

Introduction

Introduction

Our aim: investigate how these large-scale patterns emerge

Our approach: individual-based modeling

Represent the individuals explicitly

Simulate the pattern formation under different conditions

Exploring microbial patterns formation using a simple IBM

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

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

Model description

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

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

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):

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

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):

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

Overview:

2D domain with individualsrepresented as point particles

Two processes:

death with a probability d

birth with a probability b

We are interested in the case:

wb << L : local birthb = d = constant

mean-field limit (for large N):

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

Overview:

2D domain with individualsrepresented as point particles

Two processes:

death with a probability d

birth with a probability b

We are interested in the case:

wb << L : local birthb = d = constant

mean-field limit (for large N):

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

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):

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

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):

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

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 birth

b = d = constant

mean-field limit (for large N):

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

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):

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

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):dNdt = (b − d)N

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

Simulation with wb/L = 0.015

Figure: t = 0

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

Simulation with wb/L = 0.015

Figure: t = 400

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

Simulation with wb/L = 0.1

Figure: t = 400

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

Overview:

Two processes:

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

We are interested in the case:

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

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

Overview:

Two processes:

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

We are interested in the case:

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

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

Overview:

Two processes:

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

We are interested in the case:

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

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

A simple birth-death model

Overview:

Two processes:

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

We are interested in the case:

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

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

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

Figure: t = 0

Exploring microbial patterns formation using a simple IBM

A simple birth-death model

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

Figure: t = 800

Exploring microbial patterns formation using a simple IBM

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

We are interested in the case:

motility probabilities dependon the neighborhood

mi = m1−m2∑

j Kv

(||xi−xj ||

wv

)

Exploring microbial patterns formation using a simple IBM

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

We are interested in the case:

motility probabilities dependon the neighborhood

mi = m1−m2∑

j Kv

(||xi−xj ||

wv

)

Exploring microbial patterns formation using a simple IBM

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

We are interested in the case:

motility probabilities dependon the neighborhood

mi = m1−m2∑

j Kv

(||xi−xj ||

wv

)

Exploring microbial patterns formation using a simple IBM

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

We are interested in the case:

motility probabilities dependon the neighborhood

mi = m1−m2∑

j Kv

(||xi−xj ||

wv

)

Exploring microbial patterns formation using a simple IBM

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

We are interested in the case:

motility probabilities dependon the neighborhood

mi = m1−m2∑

j Kv

(||xi−xj ||

wv

)

Exploring microbial patterns formation using a simple IBM

Birth-death model with motility

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

Exploring microbial patterns formation using a simple IBM

Birth-death model with motility

Simulation results

Figure: t = 0

Exploring microbial patterns formation using a simple IBM

Birth-death model with motility

Simulation results

Figure: t = 800

Exploring microbial patterns formation using a simple IBM

Birth-death model with motility

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.

Exploring microbial patterns formation using a simple IBM

Birth-death model with motility

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.

Exploring microbial patterns formation using a simple IBM

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 ...

Exploring microbial patterns formation using a simple IBM

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 ...

Exploring microbial patterns formation using a simple IBM

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 ...

Exploring microbial patterns formation using a simple IBM

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 ...

Exploring microbial patterns formation using a simple IBM

Conclusion

The end!