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What does non-dimensionalization tell us
about the spreading of Myxococcus xanthus?
Angela GallegosUniversity of California at Davis,
Occidental College
Park City Mathematics Institute5 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)5.04( mx
Myxobacteria are:
• Rod-shaped bacteria
• Bacterial omnivores: sugar-eaters and predators
)5.04( mx
Myxobacteria are:
• Rod-shaped bacteria
• Bacterial omnivores: sugar-eaters and predators
• Found in animal dung and organic-rich soils
)5.04( mx
Why Myxobacteria?
Why Myxobacteria?
• Motility Characteristics
• Adventurous Motility– The ability to move individually
• Social Motility– The ability to move in pairs and/or groups
A-motility.mov
S-motility.mov
Why Myxobacteria? Rate of Spread
4 Types of Motility
Wild Type
Social MutantsAdventurous Mutants
Non-motile
OUTLINE
• What is Myxococcus xanthus?
• Problem Motivation:• Experimental• Theoretical
• Our Model
• How non-dimensionalization helps!
Experimental Motivation
• Experimental design– Rate of spread
r0 r1
Experimental Motivation
0
20
40
60
80
100
TIME (HOURS)
DIA
MET
ER (M
M)
WILD TYPEA MUTANTS MUTANT
0
0.1
0.2
0.3
0.4
0.5
0.07 0.1 0.14 0.22 0.32 0.45 0.71 1 1.41 2.24
Square Root of Nutrient (%)
Rate
of S
prea
d (M
M/H
R)*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 Kaiser and Crosby
Linear rate of spread yes yes
Cell motility level yes yes
Nutrient concentration
yes no comment
Initial cell density 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 MotivationBurchard Kaiser and
CrosbyGray 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 MotivationBurchard Kaiser and
CrosbyGray 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
CrosbyGray 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 symmetricr1
θ
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
• Spreading is radially symmetricr1
r2
r3
0
Our Model
• Assumptions
• The Equations
The Equations
• Reaction-diffusion equations– continuous– partial differential equations
The Equations: Diffusion
• the time rate of change of a substance in a volume is equal to the total flux of that substance into the volume
J(x0,t)J(x1,t)
x
J
t
c
J := flux expressionc := cell density
c
The Equations: Reaction-Diffusion
• Now the time rate of change is due to the flux as well as a reaction term
),,( txcfx
J
t
c
J(x0,t)J(x1,t)c
f(c,x,t)
J := flux expressionc := cell density f := reaction terms
The Equations: Cell concentration
• Flux form allows for density dependence:
• Cells grow at a rate proportional to nutrient concentration
ccDJ )(
The Equations: Cell Concentration
pcnr
ccD
rr
ccD
rt
c
)(1
)(
c := cell concentration (cells/volume)t := time coordinateD(c) := effective cell “diffusion” coefficientr := radial (space) coordinatep := growth rate per unit of nutrient
(pcn is the amount of new cells appearing)n := nutrient concentration (amount of nutrient/volume)
The Equations: Cell ConcentrationThings to notice
pcnr
ccD
rr
ccD
rt
c
)(1
)(
flux terms
reaction terms:cell growth
The Equations: Nutrient Concentration
• Flux is not density dependent:
• Nutrient is depleted at a rate proportional to the uptake per new cell
nDJ n
The Equations: Nutrient Concentration
gpcnr
n
rr
nD
t
nn
1
2
2
n:= nutrient concentration (nutrient amount/volume)t := time coordinateDn := effective nutrient diffusion coefficientr := radial (space) coordinateg := nutrient uptake per new cell made
(pcn is the number of new cells appearing)p := growth rate per unit of nutrientc := cell concentration (cells/volume)
The Equations: Nutrient Concentration Things to notice:
gpcnr
n
rr
nD
t
nn
1
2
2
flux terms
reaction terms:nutrient depletion
The Equations: Reaction-Diffusion System
gpcnr
n
rr
nD
t
n
pcnr
ccD
rr
ccD
rt
c
n
1
)(1
)(
2
2
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
r
r
t
t
c
cc
n
nn ,,~,~
where
are dimensionless, and
are the scalings (with dimension or units)
,,~,~ cn
rtcn ,,,
What are the scalings?
is the constant initial nutrient concentration with units of mass/volume.
n
What are the scalings?
is the cell density scale since g nutrient is consumed per new cell; the units are:
g
nc
volume
cell
cellmass
volumemass
What are the scalings?
is the time scale with units of
npt
1
time
timevolumemass
timevolumemass
11
11
What are the scalings?
is the spatial scale with units of
t
Dr n
time
disttime
timedist
..2
Non-dimensionalization:Dimensionless Equations
ncnnn
ncc
Dc
cDDc
~~~1~~
~~~1~
)(~~
2
2
Non-dimensionalization: Dimensionless Equations Things to notice:
• Fewer parameters: p is gone, g is gone
• remains, suggesting the ratio of cell diffusion to nutrient
diffusion matters
nD
DD
ncnnn
ncc
Dc
cDDc
~~~1~~
~~~1~
)(~~
2
2
Non-dimensionalization:What can the scalings tell us?
Non-dimensionalization:What can the scalings tell us?
• Velocity scale• Depends on diffusion• Depends on nutrient concentration
npDt
rn
Non-dimensionalization:What have we done?
• Non-dimensionalization offers an explanation for effect of nutrient concentration on rate of colony spread
• Non-dimensionalization indicates cell motility will play a role in rate of spread
• Simplified our equations
0
0.1
0.2
0.3
0.4
0.5
0.07 0.1 0.14 0.22 0.32 0.45 0.71 1 1.41 2.24
Square Root of Nutrient (%)
Rate
of S
prea
d (M
M/H
R)
0
20
40
60
80
100
TIME (HOURS)D
IAM
ETER
(MM
)
WILD TYPEA MUTANTS MUTANT
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!