Two continuum models for the spreading of myxobacteria swarms

Angela Gallegos1, Barbara Mazzag2, Alex Mogilner1¤1 Department of Mathematics,University of California, Davis, CA 95616, USA2 Department of Mathematics,University of Utah, Salt Lake City, Utah 84112, USA

• We analyze the phenomenon of spreading of a Myxococcus xanthus bacterial colony on plates coated with nutrient. The bacteria spread by gliding on the surface.

• On the time scale of tens of hours, effective diffusion of the bacteria combined with cell division and growth causes a constant linear increase of the colony’s radius.

• Mathematical analysis and numerical solution of reaction-diffusion equations describing the bacterial and nutrient dynamics demonstrate that, in this regime, the spreading rate is proportional to the square root of both the effective diffusion coeffcient and the nutrient concentration.

• Placing a droplet of a concentrated solution of cells on an agar filled with nutrient. The droplet dry and form the initial bacterial colony of radius r=0.1um. After that the thin zone spread in a radially symmetric way.

• By Measurement, we find that

• We call constant v the spreading rate.

0( )r t r vt

• In the first few hours, the M.bacteria first spread through a thin reticulum of cells.

• Kaiser and Crosby observed the radius of colony (the averaged distance from the center to the outermost tips) increases linearly with time.

1

( ) [1 (1 )exp( )]c

CD C D

C

density indepedent diffusion coefficient

maximal diffusion coefficient

• They found the effective diffusion coefficient is density dependent.

maximal diffusion coefficientD

• C is the average cell density• C1 is constant we call characteristics diffusion density

Model A: short time

• We model cell density dynamics using a 1D diffusion equation, r=0 corresponds to the edge of initial colony.

• A(r,t): fraction of the surface occupied by cells• A=1: the whole surface is covered with cells.• A=0: empty surface

By the conservation law for cell number (Edelstein-Keshet):

[ ( , ) ( , )] ( , ) ( , ), ( , ) ( , ) ( )

represent cell flux

c

A r t C r t J r t C r tJ r t A r t D C

t r rJ

0

( ln ) (ln )[ ] [ ] [ ] (*)

( , ): density of the peninsula tips, : merge rate of the tips

( , ) ( )

Assume all tip emerge together and the widt

c

t

C C A C AD Dc C

t r r r r rP r t

P Pv P P r t P e r vt

t r

0 /0

h of pensinsula doubled after merge

0 at ( , ) ( , ) { ( )

at

(ln ) ( ln ) 1so 0, and we simplify (*) as:

[ ] we can rewrite the second term of R.H

t

r l

c

r vt vA r t A P r d l

e r vt

A A

t r lC C Dc C

Dt r r l r

11

.S.as:

( )[ ( ) ], ( ) [1 (1 ) (1 )]

the bacteria undergo outward drift with rate V(C)

Cc CD C dC CDc C V C C D

V C el r r lC l C

• We look for the traveling wave solution of (*)

( , ) ( ) ( ), [ ] [ ( ) ]c

dC d dC dC r t C z C r vt v D V C C

dz dz dz dz

• We introduce some dimensionless quantities

0 1 0/ , * / , / .

[ ( ) ( ( )) ] 0,

*1 (1 )exp( ) 1 (1 ) (1 exp( ))

* *boundary conditions:

(x ,c 0, / 0), ( , 1, / 0)

c C C c C C x z l

d dcD c v V c c

dx dxc c c

D Vc c c

dc dx x c dc dx

• Integration and we first get:(1) ( )

( )

dc V V cc

dx D c

• Solve this numerically we get the Figure (at some time t>0)

• Second (by boundary condition) we get:

1/ 201

0 1

(1) (dimentionless variable)

( / ) (1) (dimentional variable)

= [1 (1 ) (1 exp( ))]

v V

v D l V

CCv D

C C

Model B: long time• We describe the dynamics of cell density C(r,t) and nutrient con

centration N(r,t) by reaction-diffusion equations as:

2

2

1[ ( ) ] ( )

1[ ]

c c

n

C C CD C D C pCN

t t t r t

N N ND gpCN

t r r r

• The first two terms on R.H.S are radial diffusion of cells and nutrient. Dn is constant diffusion coefficient.

• The upper third term means exponential growth of cell density with rate pN, p is the growth rate per nutrient concentration unit.

• The lower third term means corresponding nutrient depletion. And g is the nutrient uptake per new cell.

• Assumption (Burchard,1974) : Only cell growth depend on nutrient concentration. And gliding speed don’t depend on nutrient concentration.

• Again we introduce some dimensionless variables,

1/( ), means the constant initial nutrient concentration

, / , / , / , ' / , ' /

we get:

[ ( ) ] ( )

[

n

n n

T pN N

L D pN C N g c C C n N N t t T r r L

c D c D cD c D c cn

t D t t D r r

n

t

2

2

1]

n ncn

r r r

• T~ 5 hours: characteristics time scale• L~0.2 mm: characteristics distance of nutrient diffusion over T• C: the appropriate scale of cell density

• To investigate the swarming behavior, we solved the equations numerically. We use explicit Forward-Time Centered-Space method. We get the

• Billingham and Needham (1991) showed rigorously that there is a stable traveling wave evolved with dimensionless velocity

2n

Dv

D

n.02,30,60,90,120,150 time unit,D/D 0.2, 0.05, * 0.03t c

/ nD D

/ nD D *c

• To investigate the traveling wave speed dependence on parameters, we ran the simulations at different values of parameters

( , *) ( , *) ~ 1v c DpN c

Thank you