What does nondimensionalization tell us
about the spreading of
Myxococcus xanthus?
Angela Gallegos
University of California at Davis,
Occidental College
Park City Mathematics Institute
5 July 2005
Acknowledgements
• Alex Mogilner, UC Davis
• Bori Mazzag, University of Utah/Humboldt
State University
• RTG-NSF-DBI-9602226, NSF VIGRE
grants, UCD Chancellors Fellowship, NSF
Award DMS-0073828.
OUTLINE
• What is Myxococcus xanthus?
• Problem Motivation:
• Experimental
• Theoretical
• Our Model
• How non-dimensionalization helps!
OUTLINE
• What is Myxococcus xanthus?
• Problem Motivation:
• Experimental
• Theoretical
• Our Model
• How non-dimensionalization helps!
Myxobacteria are:
• Rod-shaped bacteria
( 4 x 0.5 m )
Myxobacteria are:
• Rod-shaped bacteria
( 4 x 0.5 m )
• Bacterial omnivores:
sugar-eaters and
predators
Myxobacteria are:
• Rod-shaped bacteria
( 4 x 0.5 m )
• Bacterial omnivores:
sugar-eaters and
predators
• Found in animal dung
and organic-rich soils
Why Myxobacteria?
Why Myxobacteria?
• Motility Characteristics
• Adventurous Motility
– The ability to move individually
• Social Motility
– The ability to move in pairs and/or groups
Why Myxobacteria? Rate of Spread
Non-motile
4 Types of Motility
Adventurous Mutants
Social Mutants
Wild Type
OUTLINE
• What is Myxococcus xanthus?
• Problem Motivation:
• Experimental
• Theoretical
• Our Model
• How non-dimensionalization helps!
Experimental Motivation
• Experimental design
– Rate of spread
r0
r1
DIAMETER (MM)
100
80
WILD TYPE
A MUTANT
S MUTANT
60
40
20
Rate of Spread (MM/HR)
Experimental Motivation
0.5
0.4
0.3
0.2
0.1
0
0.07 0.1 0.14 0.22 0.32 0.45 0.71 1 1.41 2.24
0
TIME (HOURS)
Square Root of Nutrient (%)
*no dependence on initial cell density
*TIME SCALE: 50 – 250 HOURS (2-10 days)
Burchard, 1974
Experimental Motivation
* TIME SCALE: 50 – 250 MINUTES (1-4 hours)
Kaiser and Crosby, 1983
Experimental Motivation
Burchard
Linear rate of spread yes
Kaiser and
Crosby
yes
Cell motility level
yes
yes
Nutrient
concentration
Initial cell density
yes
no comment
no
yes
Time scale
days
hours
OUTLINE
• What is Myxococcus xanthus?
• Problem Motivation:
• Experimental
• Theoretical
• Our Model
• How non-dimensionalization helps!
Theoretical Motivation
• Non-motile cell
assumption
• Linear rate of increase in
colony growth
• Rate dependent upon
both nutrient
concentration and cell
motility, but not initial cell
density
Gray and Kirwan, 1974
r
Problem Motivation
Burchard
Kaiser and
Crosby
Gray and
Kirwan
Conditions
motile cells;
start only in
center of
dish
motile cells;
start only in
center of
dish
non-motile
cells
initially
everywhere
Linear rate of
spread
yes
yes
yes
Cell motility level
yes
yes
no
Nutrient
concentration
no
no comment
yes
Initial cell density
no
yes
no
Time scale
days
hours
long
Problem Motivation
Burchard
Kaiser and
Crosby
Gray and
Kirwan
Conditions
motile cells;
start only in
center of
dish
motile cells;
start only in
center of
dish
non-motile
cells
initially
everywhere
Linear rate of
spread
yes
yes
yes
Cell motility level
yes
yes
no
Nutrient
concentration
no
no comment
yes
Initial cell density
no
yes
no
Time scale
days
hours
long
Problem Motivation
• Can we explain the rate of spread data with more
relevant assumptions?
Burchard
Kaiser and
Crosby
Gray and
Kirwan
Gallegos,
Mazzag,
Mogilner
Conditions
motile cells;
start only in
center of
dish
motile cells;
start only in
center of
dish
non-motile
cells
initially
everywhere
motile cells;
start only in
center of
dish
Linear rate of
spread
yes
yes
yes
Cell motility level
yes
yes
no
Nutrient
concentration
no
no comment
yes
Initial cell density
no
yes
no
Time scale
days
hours
long
OUTLINE
• What is Myxococcus xanthus?
• Problem Motivation:
• Experimental
• Theoretical
• Our Model
• How non-dimensionalization helps!
Our Model
• Assumptions
• The Equations
Our Model
• Assumptions
• The Equations
Assumptions
• The cell colony behaves as a continuum
Assumptions
• The cell colony behaves as a continuum
• Nutrient consumption affects cell behavior
only through its effect on cell growth
Assumptions
• The cell colony behaves as a continuum
• Nutrient consumption affects cell behavior
only through its effect on cell growth
• Growth and nutrient consumption rates are
constant
Assumptions
• The cell colony behaves as a continuum
• Nutrient consumption affects cell behavior
only through its effect on cell growth
• Growth and nutrient consumption rates are
constant
• Spreading is radially symmetric
θ
r1
r2
r3
Assumptions
• The cell colony behaves as a continuum
• Nutrient consumption affects cell behavior
only through its effect on cell growth
• Growth and nutrient consumption rates are
constant

0
• Spreading is radially symmetric

r1
r2
r3
Our Model
• Assumptions
• The Equations
The Equations
• Reaction-diffusion equations
– continuous
– partial differential equations
The Equations: Diffusion
c
J

t
x
J := flux expression
c := cell density
J(x0,t)
c
J(x1,t)
• the time rate of change of a substance in a
volume is equal to the total flux of that substance
into the volume
The Equations: Reaction-Diffusion
c
J

 f (c, x, t )
t
x
J := flux expression
c := cell density
f := reaction terms
J(x0,t)
c
J(x1,t)
f(c,x,t)
• Now the time rate of change is due to the flux as
well as a reaction term
The Equations: Cell concentration
• Flux form allows for density dependence:
J  D(c )c
• Cells grow at a rate proportional to nutrient
concentration
The Equations: Cell Concentration
c
 
c  1
c

D
(
c
)

D
(
c
)

pcn


t r 
r  r
r
c := cell concentration (cells/volume)
t := time coordinate
D(c) := effective cell “diffusion” coefficient
r := radial (space) coordinate
p := growth rate per unit of nutrient
(pcn is the amount of new cells appearing)
n := nutrient concentration (amount of nutrient/volume)
The Equations: Cell Concentration
Things to notice
c
 
c  1
c

D
(
c
)

D
(
c
)

pcn


t r 
r  r
r
flux terms
reaction terms:
cell growth
The Equations: Nutrient Concentration
• Flux is not density dependent:
J  Dnn
• Nutrient is depleted at a rate proportional
to the uptake per new cell
The Equations: Nutrient Concentration
  n 1 n 
n
 Dn  2 
  gpcn
t
r r 
 r
2
n:= nutrient concentration (nutrient amount/volume)
t := time coordinate
Dn := effective nutrient diffusion coefficient
r := radial (space) coordinate
g := nutrient uptake per new cell made
(pcn is the number of new cells appearing)
p := growth rate per unit of nutrient
c := cell concentration (cells/volume)
The Equations: Nutrient Concentration
Things to notice:
  n 1 n 
n
 Dn  2 
  gpcn
t
r r 
 r
2
flux terms
reaction terms:
nutrient depletion
The Equations:
Reaction-Diffusion System
c  
c  1
c

D
(
c
)

D
(
c
)

pcn


t r 
r  r
r
2
  n 1 n 
n
 Dn  2 
  gpcn
t
r r 
 r
Our Model: What will it give us?
Burchard
Kaiser and
Crosby
Gray and
Kirwan
Gallegos,
Mazzag,
Mogilner
Conditions
motile cells;
start only in
center of
dish
motile cells;
start only in
center of
dish
non-motile
cells
initially
everywhere
motile cells;
start only in
center of
dish
Linear rate of
spread
yes
yes
yes
?
Cell motility level
yes
yes
no
?
Nutrient
concentration
no
no comment
yes
?
Initial cell density
no
yes
no
?
Time scale
days
hours
long
?
OUTLINE
• What is Myxococcus xanthus?
• Problem Motivation:
• Experimental
• Theoretical
• Our Model
• How non-dimensionalization helps!
Non-dimensionalization: Why?
Non-dimensionalization: Why?
• Reduces the number of parameters
• Can indicate which combination of
parameters is important
• Allows for more computational ease
• Explains experimental phenomena
Non-dimensionalization:
Rewrite the variables
n ~
c
t
r
~
n 
,c 
,  ,  
n
c
t
r
where
~
~
n, c , , 
are dimensionless, and
n, c , t , r
are the scalings (with dimension or units)
What are the scalings?
n
is the constant initial nutrient concentration
with units of mass/volume.
What are the scalings?
n
c
g
is the cell density scale since g nutrient is
consumed per new cell; the units are:
cell
mass mass

volume cell
volume
What are the scalings?
1
t 
pn
is the time scale with units of
1
1

 time
1
1




mass

 time
mass
volume



time

 volume 




What are the scalings?
r 
Dn
t
is the spatial scale with units of
2
dist.
time
dist.

time time
Non-dimensionalization:
Dimensionless Equations
~
~
~
c
c 
  ~
 1 c
~
~
D
D(c )   D
 cn


 
 
 
2~
~
~
n  n 1 n ~~
 2
 cn
 
 
Non-dimensionalization: Dimensionless Equations
~
c

~
n

Things to notice:
~
~



c
1

c
~

~n
~
 D
D
(
c
)

D

c
 
 
 
~
~
 2n
1 n
~n
~



c
 2
 
• Fewer parameters: p is gone, g is gone
•
D
D 
Dn

remains, suggesting the ratio of cell diffusion to nutrient
diffusion matters
Non-dimensionalization:
What can the scalings tell us?
Non-dimensionalization:
What can the scalings tell us?
r
 Dn pn
t
• Velocity scale
• Depends on diffusion
• Depends on nutrient concentration
Non-dimensionalization:
What have we done?
100
0.5
0.4
DIAMETER (MM)
Rate of Spread (MM/HR)
• Non-dimensionalization offers an explanation for effect of
nutrient concentration on rate of colony spread
0.3
0.2
0.1
0
0.07 0.1 0.14 0.22 0.32 0.45 0.71 1 1.41 2.24
Square Root of Nutrient (%)
80
WILD TYPE
A MUTANT
S MUTANT
60
40
20
0
TIME (HOURS)
• Non-dimensionalization indicates cell motility will play a
role in rate of spread
• Simplified our equations
Non-dimensionalization:
What have we done?
Burchard
Kaiser and
Crosby
Gray and
Kirwan
Gallegos,
Mazzag,
Mogilner
Conditions
motile cells;
start only in
center of
dish
motile cells;
start only in
center of
dish
non-motile
cells
initially
everywhere
motile cells;
start only in
center of
dish
Linear rate of
spread
yes
yes
yes
?
Cell motility level
yes
yes
no
yes
Nutrient
concentration
no
no comment
yes
yes
Initial cell density
no
yes
no
?
Time scale
days
hours
long
long
THE END!
Thank You!