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 , NSF VIGRE grants, UCD Chancellors Fellowship, NSF Award DMS

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!

Myxobacteria are: Rod-shaped bacteria

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 4 Types of Motility Wild Type Social Mutants Adventurous Mutants Non-motile

OUTLINE What is Myxococcus xanthus? Problem Motivation: Experimental Theoretical Our Model How non-dimensionalization helps!

Experimental Motivation Experimental design –Rate of spread r0r0 r1r1

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 BurchardKaiser and Crosby Linear rate of spreadyes Cell motility levelyes Nutrient concentration yesno comment Initial cell densitynoyes Time scaledayshours

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 Gray and Kirwan, 1974 r

Problem Motivation BurchardKaiser and Crosby Gray and Kirwan Conditionsmotile cells; start only in center of dish motile cells; start only in center of dish non-motile cells initially everywhere Linear rate of spread yes Cell motility levelyes no Nutrient concentration nono commentyes Initial cell densitynoyesno Time scaledayshourslong

Problem Motivation BurchardKaiser and Crosby Gray and Kirwan Conditionsmotile cells; start only in center of dish motile cells; start only in center of dish non-motile cells initially everywhere Linear rate of spread yes Cell motility levelyes no Nutrient concentration nono commentyes Initial cell densitynoyesno Time scaledayshourslong

Problem Motivation Can we explain the rate of spread data with more relevant assumptions? BurchardKaiser and Crosby Gray and Kirwan Gallegos, Mazzag, Mogilner Conditionsmotile cells; start only in center of dish motile cells; start only in center of dish non-motile cells initially everywhere motile cells; start only in center of dish Linear rate of spread yes Cell motility levelyes no Nutrient concentration nono commentyes Initial cell densitynoyesno Time scaledayshourslong

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 r1r1 r2r2 r3r3

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 r1r1 r2r2 r3r3

Our Model Assumptions 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(x 0,t) J(x 1,t) J := flux expression c := cell density c

The Equations: Reaction-Diffusion Now the time rate of change is due to the flux as well as a reaction term J(x 0,t) J(x 1,t) c f(c,x,t) J := flux expression c := cell density f := reaction terms

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 Concentration Things 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 D n := 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? BurchardKaiser and Crosby Gray and Kirwan Gallegos, Mazzag, Mogilner Conditionsmotile cells; start only in center of dish motile cells; start only in center of dish non-motile cells initially everywhere motile cells; start only in center of dish Linear rate of spread yes ? Cell motility levelyes no? Nutrient concentration nono commentyes? Initial cell densitynoyesno? Time scaledayshourslong?

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 Equations

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?

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? BurchardKaiser and Crosby Gray and Kirwan Gallegos, Mazzag, Mogilner Conditionsmotile cells; start only in center of dish motile cells; start only in center of dish non-motile cells initially everywhere motile cells; start only in center of dish Linear rate of spread yes ? Cell motility levelyes noyes Nutrient concentration nono commentyes Initial cell densitynoyesno? Time scaledayshourslong

THE END! Thank You!