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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, USA
2 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
r (t )  r0  vt
• We call constant v the spreading rate.
• 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.
• They found the effective diffusion coefficient is
density dependent.
C
Dc (C )  D[1  (1   ) exp( )]
C1
• C is the average cell density
• C1 is constant we call characteristics diffusion density
D  maximal diffusion coefficient

density indepedent diffusion coefficient
maximal diffusion coefficient
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):
[ A(r , t )C (r , t )]
J (r , t )
C (r , t )

, J (r , t )   A(r , t ) Dc (C )
t
r
r
J represent cell flux
 ( ln A) C  (ln A)
C
C 
]C (*)
[
]
]  [ Dc
 [ Dc
r
r
r
r
t r
P(r , t ): density of the peninsula tips,  : merge rate of the tips
P
P
  P  P (r , t )  P0e  t (r  vt )
 v
r
t
Assume all tip emerge together and the width of pensinsula doubled after merge
t
0 at r  vt
v
(l  )
A(r , t )  A0  P(r , ) d  {  r / l
0
at r  vt
e

 ( ln A) 1
 (ln A)
 and we simplify (*) as:
 0,
so
l
r
t
C Dc C
C 
we can rewrite the second term of R.H.S.as:
]
 [ Dc
l r
r
t r
C

dC
)
C
(
D
C
D
Dc C [V (C )C ]
c

 [1  (1   ) 1 (1  e C1 )]
,V (C ) 

C
l
lC
r
l r
the bacteria undergo outward drift with rate V(C)
• We look for the traveling wave solution of (*)
C (r , t )  C ( z )  C (r  vt ),  v
dC d
dC
d
 [ Dc
]  [V (C )C ]
dz dz
dz
dz
• We introduce some dimensionless quantities
c  C / C0 , c*  C1 / C0 , x  z / l.
d
dc
[ D(c)  (v  V (c))c]  0,
dx
dx
c
c*
c
D  1  (1   ) exp( ) V  1  (1   ) (1  exp( ))
c*
c
c*
boundary conditions:
(x  ,c  0, dc / dx  0), ( x  , c  1, dc / dx  0)
• Integration and we first get:
dc
V (1)  V (c)

c
dx
D (c )
• Solve this numerically we get the Figure (at some time t>0)
• Second (by boundary condition) we get:
v  V (1) (dimentionless variable)
v  ( D / l )V (1) (dimentional variable)
C0 1/ 2
C1
 v = D [1  (1   ) (1  exp( ))]
C0
C1
Model B: long time
• We describe the dynamics of cell density C(r,t) and nutrient
concentration N(r,t) by reaction-diffusion equations as:
C 
C 1
C
 [ Dc (C ) ]  Dc (C )
 pCN
t t
t
r
t
N
 2 N 1 N
 Dn [ 2 
]  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,
T  1/( pN ), N means the constant initial nutrient concentration
L  Dn pN , C  N / g , c  C / C , n  N / N , t '  t / T , r '  r / L
we get:
c D 
c
D
c

[ D (c ) ] 
D(c)  cn
t Dn t
t
Dn r
r
n  2 n 1 n
[ 2 
]  cn
t
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
• Billingham and Needham (1991) showed rigorously that
there is a stable traveling wave evolved with
dimensionless velocity
D
v2
Dn
• To investigate the swarming behavior, we solved the
equations numerically. We use explicit Forward-Time
Centered-Space method. We get the
t  .02,30,60,90,120,150 time unit,D/Dn  0.2,   0.05, c*  0.03
• To investigate the traveling wave speed dependence on
parameters, we ran the simulations at different values of
parameters
D / Dn
D / Dn

c*
v   ( , c*) DpN  ( , c*) ~ 1
Thank you