What does non-dimensionalization tell us about the spreading of Myxococcus xanthus ?

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!