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

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

Angela Gallegos

University of California at Davis,

Occidental College

Park City Mathematics Institute

5 July 2005

- 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.

- What is Myxococcus xanthus?
- Problem Motivation:
- Experimental
- Theoretical

- What is Myxococcus xanthus?
- Problem Motivation:
- Experimental
- Theoretical

Rod-shaped bacteria

- Rod-shaped bacteria
- Bacterial omnivores: sugar-eaters and predators

- Rod-shaped bacteria
- Bacterial omnivores: sugar-eaters and predators
- Found in animal dung and organic-rich soils

- Motility Characteristics
- Adventurous Motility
- The ability to move individually

- Social Motility
- The ability to move in pairs and/or groups

- Adventurous Motility

Non-motile

4 Types of Motility

Adventurous Mutants

Social Mutants

Wild Type

- What is Myxococcus xanthus?
- Problem Motivation:
- Experimental
- Theoretical

- Experimental design
- Rate of spread

r0

r1

*no dependence on initial cell density

*TIME SCALE: 50 – 250 HOURS (2-10 days)

Burchard, 1974

* TIME SCALE: 50 – 250 MINUTES (1-4 hours)

Kaiser and Crosby, 1983

- What is Myxococcus xanthus?
- Problem Motivation:
- Experimental
- Theoretical

- 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

r

Gray and Kirwan, 1974

- Can we explain the rate of spread data with more relevant assumptions?

- What is Myxococcus xanthus?
- Problem Motivation:
- Experimental
- Theoretical

- Assumptions
- The Equations

- Assumptions
- The Equations

- The cell colony behaves as a continuum

- The cell colony behaves as a continuum
- Nutrient consumption affects cell behavior only through its effect on cell growth

- 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

- 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

r1

r2

r3

- 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

r1

r2

r3

- Assumptions
- The Equations

- Reaction-diffusion equations
- continuous
- partial differential equations

- 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)

c

J := flux expressionc := cell density

J(x1,t)

- Now the time rate of change is due to the flux as well as a reaction term

J(x0,t)

c

J := flux expressionc := cell density

f := reaction terms

J(x1,t)

f(c,x,t)

- Flux form allows for density dependence:
- Cells grow at a rate proportional to nutrient concentration

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)

flux terms

reaction terms:

cell growth

- Flux is not density dependent:
- Nutrient is depleted at a rate proportional to the uptake per new cell

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)

flux terms

reaction terms:

nutrient depletion

OUTLINE

- What is Myxococcus xanthus?
- Problem Motivation:
- Experimental
- Theoretical

- Reduces the number of parameters
- Can indicate which combination of parameters is important
- Allows for more computational ease
- Explains experimental phenomena

where

are dimensionless, and

are the scalings (with dimension or units)

is the constant initial nutrient concentration with units of mass/volume.

is the cell density scale since g nutrient is consumed per new cell; the units are:

is the time scale with units of

is the spatial scale with units of

- Fewer parameters: p is gone, g is gone
- remains, suggesting the ratio of cell diffusion to nutrient
diffusion matters

- Velocity scale
- Depends on diffusion
- Depends on nutrient concentration

- 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

THE END!

Thank You!