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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? 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? • 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
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 • 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
OUTLINE • What is Myxococcus xanthus? • Problem Motivation: • Experimental • Theoretical • Our Model • How non-dimensionalization helps!
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 EquationsThings to notice: • Fewer parameters: p is gone, g is gone • remains, suggesting the ratio of cell diffusion to nutrient diffusion matters