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. Acknowledgements. Alex Mogilner, UC Davis

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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


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!


  • Rod-shaped bacteria

    Myxobacteria are:


    Myxobacteria are:

    • Rod-shaped bacteria

    • Bacterial omnivores: sugar-eaters and predators


    Myxobacteria are:

    • Rod-shaped bacteria

    • 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


    Experimental Motivation

    *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


    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

    r

    Gray and Kirwan, 1974


    Problem Motivation


    Problem Motivation


    Problem Motivation

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


    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

    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

    • Spreading is radially symmetric

    r1

    r2

    r3


    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)

    c

    J := flux expressionc := cell density

    J(x1,t)


    The Equations: Reaction-Diffusion

    • 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)


    The Equations: Cell concentration

    • Flux form allows for density dependence:

    • Cells grow at a rate proportional to nutrient 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 ConcentrationThings to notice

    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


    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:

    flux terms

    reaction terms:

    nutrient depletion


    The Equations: Reaction-Diffusion System


    Our Model: What will it give us?


    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

    where

    are dimensionless, and

    are the scalings (with dimension or units)


    What are the scalings?

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


    What are the scalings?

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


    What are the scalings?

    is the time scale with units of


    What are the scalings?

    is the spatial scale with units of


    Non-dimensionalization:Dimensionless Equations


    Non-dimensionalization: Dimensionless EquationsThings 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?

    • Velocity scale

      • Depends on diffusion

      • Depends on nutrient concentration


    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?


    THE END!

    Thank You!


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