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

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 of

  • 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 of

  • What is Myxococcus xanthus?

  • Problem Motivation:

    • Experimental

    • Theoretical

  • Our Model

  • How non-dimensionalization helps!


  • Outline1
    OUTLINE of

    • What is Myxococcus xanthus?

    • Problem Motivation:

      • Experimental

      • Theoretical

  • Our Model

  • How non-dimensionalization helps!


  • Myxobacteria are

    Rod-shaped bacteria of

    Myxobacteria are:


    Myxobacteria are1
    Myxobacteria are: of

    • Rod-shaped bacteria

    • Bacterial omnivores: sugar-eaters and predators


    Myxobacteria are2
    Myxobacteria are: of

    • Rod-shaped bacteria

    • Bacterial omnivores: sugar-eaters and predators

    • Found in animal dung and organic-rich soils



    Why myxobacteria1
    Why Myxobacteria? of

    • 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 of

    Non-motile

    4 Types of Motility

    Adventurous Mutants

    Social Mutants

    Wild Type


    Outline2
    OUTLINE of

    • What is Myxococcus xanthus?

    • Problem Motivation:

      • Experimental

      • Theoretical

  • Our Model

  • How non-dimensionalization helps!


  • Experimental motivation
    Experimental Motivation of

    • Experimental design

      • Rate of spread

    r0

    r1


    Experimental motivation1
    Experimental Motivation of

    *no dependence on initial cell density

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

    Burchard, 1974


    Experimental motivation2
    Experimental Motivation of

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

    Kaiser and Crosby, 1983



    Outline3
    OUTLINE of

    • What is Myxococcus xanthus?

    • Problem Motivation:

      • Experimental

      • Theoretical

  • Our Model

  • How non-dimensionalization helps!


  • Theoretical motivation
    Theoretical Motivation of

    • 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 motivation2
    Problem Motivation of

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


    Outline4
    OUTLINE of

    • What is Myxococcus xanthus?

    • Problem Motivation:

      • Experimental

      • Theoretical

  • Our Model

  • How non-dimensionalization helps!


  • Our model
    Our Model of

    • Assumptions

    • The Equations


    Our model1
    Our Model of

    • Assumptions

    • The Equations


    Assumptions
    Assumptions of

    • The cell colony behaves as a continuum


    Assumptions1
    Assumptions of

    • The cell colony behaves as a continuum

    • Nutrient consumption affects cell behavior only through its effect on cell growth


    Assumptions2
    Assumptions of

    • 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


    Assumptions3
    Assumptions of

    • 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


    Assumptions4
    Assumptions of

    • 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 model2
    Our Model of

    • Assumptions

    • The Equations


    The equations
    The Equations of

    • Reaction-diffusion equations

      • continuous

      • partial differential equations


    The equations diffusion
    The Equations: Diffusion of

    • 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
    The Equations: Reaction-Diffusion of

    • 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
    The Equations: Cell concentration of

    • Flux form allows for density dependence:

    • Cells grow at a rate proportional to nutrient concentration


    The equations cell concentration1
    The Equations: Cell Concentration of

    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 of Things to notice

    flux terms

    reaction terms:

    cell growth


    The equations nutrient concentration
    The Equations: Nutrient Concentration of

    • Flux is not density dependent:

    • Nutrient is depleted at a rate proportional to the uptake per new cell


    The equations nutrient concentration1
    The Equations: Nutrient Concentration of

    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 of Things to notice:

    flux terms

    reaction terms:

    nutrient depletion


    The equations reaction diffusion system
    The Equations: of Reaction-Diffusion System



    OUTLINE of

    • What is Myxococcus xanthus?

    • Problem Motivation:

      • Experimental

      • Theoretical

  • Our Model

  • How non-dimensionalization helps!



  • Non dimensionalization why1
    Non-dimensionalization: Why? of

    • 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: of Rewrite the variables

    where

    are dimensionless, and

    are the scalings (with dimension or units)


    What are the scalings
    What are the scalings? of

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


    What are the scalings1
    What are the scalings? of

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


    What are the scalings2
    What are the scalings? of

    is the time scale with units of


    What are the scalings3
    What are the scalings? of

    is the spatial scale with units of


    Non dimensionalization dimensionless equations
    Non-dimensionalization: of Dimensionless Equations


    Non dimensionalization dimensionless equations things to notice
    Non-dimensionalization: Dimensionless Equations of 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: of What can the scalings tell us?


    Non dimensionalization what can the scalings tell us1
    Non-dimensionalization: of What can the scalings tell us?

    • Velocity scale

      • Depends on diffusion

      • Depends on nutrient concentration


    Non dimensionalization what have we done
    Non-dimensionalization: of 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 done1
    Non-dimensionalization: of What have we done?


    The end

    THE END! of

    Thank You!


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