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 Gallegos
University of California at Davis,
Occidental College
Park City Mathematics Institute
5 July 2005
Acknowledgements : 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 : OUTLINE What is Myxococcus xanthus?
Problem Motivation:
Experimental
Theoretical
Our Model
How non-dimensionalization helps!
OUTLINE : OUTLINE What is Myxococcus xanthus?
Problem Motivation:
Experimental
Theoretical
Our Model
How non-dimensionalization helps!
Myxobacteria are: : Myxobacteria are: Rod-shaped bacteria
Myxobacteria are: : Myxobacteria are: Rod-shaped bacteria
Bacterial omnivores: sugar-eaters and predators
Myxobacteria are: : Myxobacteria are: Rod-shaped bacteria
Bacterial omnivores: sugar-eaters and predators
Found in animal dung and organic-rich soils
Why Myxobacteria? : Why Myxobacteria?
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 : Why Myxobacteria? Rate of Spread 4 Types of Motility Wild Type Social Mutants Adventurous Mutants Non-motile
OUTLINE : OUTLINE What is Myxococcus xanthus?
Problem Motivation:
Experimental
Theoretical
Our Model
How non-dimensionalization helps!
Experimental Motivation : Experimental Motivation Experimental design
Rate of spread r0 r1
Experimental Motivation : Experimental Motivation *no dependence on initial cell density
*TIME SCALE: 50 – 250 HOURS (2-10 days) Burchard, 1974
Experimental Motivation : Experimental Motivation * TIME SCALE: 50 – 250 MINUTES (1-4 hours) Kaiser and Crosby, 1983
Experimental Motivation : Experimental Motivation
OUTLINE : OUTLINE What is Myxococcus xanthus?
Problem Motivation:
Experimental
Theoretical
Our Model
How non-dimensionalization helps!
Theoretical Motivation : 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 : Problem Motivation
Problem Motivation : Problem Motivation
Problem Motivation : Problem Motivation Can we explain the rate of spread data with more relevant assumptions?
OUTLINE : OUTLINE What is Myxococcus xanthus?
Problem Motivation:
Experimental
Theoretical
Our Model
How non-dimensionalization helps!
Our Model : Our Model
Assumptions
The Equations
Our Model : Our Model
Assumptions
The Equations
Assumptions : Assumptions The cell colony behaves as a continuum
Assumptions : Assumptions The cell colony behaves as a continuum
Nutrient consumption affects cell behavior only through its effect on cell growth
Assumptions : 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 : 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 r1 r2 r3
Assumptions : 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
r1 r2 r3
Our Model : Our Model
Assumptions
The Equations
The Equations : The Equations
Reaction-diffusion equations
continuous
partial differential equations
The Equations: Diffusion : 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) J := flux expression c := cell density c
The Equations: Reaction-Diffusion : The Equations: Reaction-Diffusion
Now the time rate of change is due to the flux as well as a reaction term J(x0,t) J(x1,t) c f(c,x,t) J := flux expression c := cell density
f := reaction terms
The Equations: Cell concentration : The Equations: Cell concentration Flux form allows for density dependence:
Cells grow at a rate proportional to nutrient concentration
The Equations: Cell Concentration : The Equations: Cell 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)
The Equations: Cell Concentration Things to notice : The Equations: Cell Concentration Things to notice flux terms reaction terms:
cell growth
The Equations: Nutrient Concentration : The Equations: Nutrient Concentration Flux is not density dependent:
Nutrient is depleted at a rate proportional to the uptake per new cell
The Equations: Nutrient Concentration : The Equations: Nutrient Concentration 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: : The Equations: Nutrient Concentration Things to notice: flux terms reaction terms:
nutrient depletion
The Equations: Reaction-Diffusion System : The Equations: Reaction-Diffusion System
Our Model: What will it give us? : Our Model: What will it give us?
Slide41 : OUTLINE What is Myxococcus xanthus?
Problem Motivation:
Experimental
Theoretical
Our Model
How non-dimensionalization helps!
Non-dimensionalization: Why? : Non-dimensionalization: Why?
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 : Non-dimensionalization: Rewrite the variables where
are dimensionless, and
are the scalings (with dimension or units)
What are the scalings? : What are the scalings?
is the constant initial nutrient concentration with units of mass/volume.
What are the scalings? : What are the scalings?
is the cell density scale since g nutrient is consumed per new cell; the units are:
What are the scalings? : What are the scalings?
is the time scale with units of
What are the scalings? : What are the scalings?
is the spatial scale with units of
Non-dimensionalization: Dimensionless Equations : Non-dimensionalization: Dimensionless Equations
Non-dimensionalization: Dimensionless Equations Things to notice: : 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
Non-dimensionalization: What can the scalings tell us? : Non-dimensionalization: What can the scalings tell us?
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
Non-dimensionalization: What have we done? : 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
Non-dimensionalization: What have we done? : Non-dimensionalization: What have we done?
THE END! : THE END! Thank You!