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

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!


  • Outline1

    OUTLINE

    • What is Myxococcus xanthus?

    • Problem Motivation:

      • Experimental

      • Theoretical

  • Our Model

  • How non-dimensionalization helps!


  • Myxobacteria are

    Rod-shaped bacteria

    Myxobacteria are:


    Myxobacteria are1

    Myxobacteria are:

    • Rod-shaped bacteria

    • Bacterial omnivores: sugar-eaters and predators


    Myxobacteria are2

    Myxobacteria are:

    • Rod-shaped bacteria

    • Bacterial omnivores: sugar-eaters and predators

    • Found in animal dung and organic-rich soils


    Why myxobacteria

    Why Myxobacteria?


    Why myxobacteria1

    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

    Non-motile

    4 Types of Motility

    Adventurous Mutants

    Social Mutants

    Wild Type


    Outline2

    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 motivation1

    Experimental Motivation

    *no dependence on initial cell density

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

    Burchard, 1974


    Experimental motivation2

    Experimental Motivation

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

    Kaiser and Crosby, 1983


    Experimental motivation3

    Experimental Motivation


    Outline3

    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

    r

    Gray and Kirwan, 1974


    Problem motivation

    Problem Motivation


    Problem motivation1

    Problem Motivation


    Problem motivation2

    Problem Motivation

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


    Outline4

    OUTLINE

    • What is Myxococcus xanthus?

    • Problem Motivation:

      • Experimental

      • Theoretical

  • Our Model

  • How non-dimensionalization helps!


  • Our model

    Our Model

    • Assumptions

    • The Equations


    Our model1

    Our Model

    • Assumptions

    • The Equations


    Assumptions

    Assumptions

    • The cell colony behaves as a continuum


    Assumptions1

    Assumptions

    • The cell colony behaves as a continuum

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


    Assumptions2

    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


    Assumptions3

    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


    Assumptions4

    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 model2

    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)

    c

    J := flux expressionc := cell density

    J(x1,t)


    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)

    c

    J := flux expressionc := cell density

    f := reaction terms

    J(x1,t)

    f(c,x,t)


    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 concentration1

    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 ConcentrationThings 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 concentration1

    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?


    What does non dimensionalization tell us about the spreading of myxococcus xanthus

    OUTLINE

    • What is Myxococcus xanthus?

    • Problem Motivation:

      • Experimental

      • Theoretical

  • Our Model

  • How non-dimensionalization helps!


  • Non dimensionalization why

    Non-dimensionalization: Why?


    Non dimensionalization why1

    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 scalings1

    What are the scalings?

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


    What are the scalings2

    What are the scalings?

    is the time scale with units of


    What are the scalings3

    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 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?


    Non dimensionalization what can the scalings tell us1

    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 done1

    Non-dimensionalization:What have we done?


    The end

    THE END!

    Thank You!


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