Review
Turing-Gierer-Meinhardt models
Local excitation, global inhibition
∂ 2a
∂a
a2
= ra + k a
− γ a a + Da 2
∂x
∂t
i
∂i
∂ 2i
2
= ki a − γ i i + Di 2
∂t
∂x
a:
i:
t:
x:
concentration activator
concentration inhibitor
time
position
ra:
basal activator synthesis rate
ka, ki: rate constant for synthesis
γa,γi : decay rates
Da, Di: diffusion constants
variables
constants
(parameters)
1
∂ 2a
∂a
a2
= ra + k a
− γ a a + Da 2
∂x
∂t
i
2
∂i
∂
i
2
= ki a − γ i i + Di 2
∂t
∂x
choose
dimensionless
variable
normalize
4 variables
A2
∂A
∂2 A
= 1+ R
− A+ 2
I
∂τ
∂s
2
I
∂I
∂
2
=Q A −I +P 2
∂τ
∂s
(
homogeneous
solution
∂ / ∂s = ∂ / ∂t = 0
)
A = R +1
I = ( R + 1) 2
only one fixed
point, since both
A and I >0
2
homogeneous
solution
A
A
∂ / ∂s = ∂ / ∂t = 0
s
I
I
s
3
stability of homogeneous solution
R ⎤
⎡ 2 RA
RA 2 ⎤ ⎡ R − 1
−
−1 − 2 ⎥ ⎢
⎢
=
R +1
( R + 1) 2 ⎥
I
I
⎢
⎥ ⎢
⎥
2
(
R
+
1
)
Q
−
Q
2
A
Q
−
Q
⎦
⎣
⎦ ⎣
R −1
<Q
R +1
Q>0
inhomogeneous
solution:
trace < 0
det > 0
or in general
real part of eigenvalues > 0
A( s,τ ) = A + A' ( s,τ )
I ( s,τ ) = I + I ' ( s,τ )
4
inhomogeneous
solution
A( s,τ ) = A + A' ( s,τ )
I ( s,τ ) = I + I ' ( s,τ )
A
A
s
I
I
I’(s,τ)
s
5
A( s,τ ) = A + A' ( s,τ )
∂A' R − 1
R
∂ 2 A'
=
A'−
I '+ 2
2
∂τ R + 1
(1 + R)
∂s
I ( s,τ ) = I + I ' ( s,τ )
∂I '
∂2I '
= 2Q (1 + R) A'−QI '+ P 2
∂τ
∂s
trial solution:
s
ˆ
A' ( s,τ ) = A(τ ) cos( )
l
s
ˆ
I ' ( s,τ ) = I (τ ) cos( )
l
6
A
A
s
ˆ
A' ( s,τ ) = A(τ ) cos( )
l
s
I ' ( s,τ ) = Iˆ(τ ) cos( )
l
s
I
I
I’(s,τ)
s
A( s,τ ) = A + A' ( s,τ )
I ( s,τ ) = I + I ' ( s,τ7 )
s
ˆ
A' ( s,τ ) = A(τ ) cos( )
l
s
ˆ
I ' ( s,τ ) = I (τ ) cos( )
l
stability
inhomogeneous
solution
dAˆ ⎛ R − 1 1 ⎞ ˆ
R
Iˆ
=⎜
− 2 ⎟A −
2
(1 + R )
dτ ⎝ R + 1 l ⎠
dIˆ
P ⎞ˆ
⎛
ˆ
= 2Q(1 + R) A − ⎜ Q + 2 ⎟ I
dτ
l ⎠
⎝
P ⎞ 2QR
⎛ R − 1 1 ⎞⎛
−⎜
− 2 ⎟⎜ Q + 2 ⎟ +
>0
l ⎠ 1+ R
⎝ R + 1 l ⎠⎝
P ⎛ R −1 1 ⎞
Q + 2 −⎜
− 2⎟<0
l ⎝ R +1 l ⎠
Q R −1
>
P R +1
8
homogeneous stability:
stability against spatial distrubance:
R −1
Q>
R +1
Q R −1
>
P R +1
I
I
I’(s,τ)
s
if P < 1 (Di<Da), systems is always stable, against any
perturbation both spatial and temporal
9
I
I
s
homogeneously stable:
I relaxes back to
previous value after
small uniform disturbance
I
I
s
stable against spatial
disturbance:
I’ relaxes back to
after small spatial
disturbance
I
10
Introducing the molecules:
- FtsZ function: Assembly of a polymeric ring of the
tubulin-like GTPase FtsZ (Z ring).
The Z-ring is localized to the center by the actions of
the MinC, MinD, and MinE proteins.
- MinC inhibits the initiation of the Z ring.
MinC colocalizes with MinD.
In wild-type (WT) cells, MinC/D forms a polar pattern
that oscillates between the poles, keeping the center
free for initiation of cell division.
Thus, virtually all of MinC/D dynamically assembles on the
membrane in the shape of a test tube covering the membrane
from one pole up to approximately midcell.
12
Most of MinE accumulates at the rim of this tube, in the shape
of a ring (the E ring). The rim of the MinC/D tube and
associated E ring move from a central position to the cell
pole until both the tube and ring vanish. Meanwhile, a new
MinC/D tube and associated E ring form in the opposite cell
half, and the process repeats, resulting in a pole-to-pole
oscillation cycle of the division inhibitor.
A full cycle takes about 50 s.
Image removed due to copyright considerations.
13
How does this work ?
modeling efforts:
• Meinhardt and de Boer, PNAS 98, 14202 (2001);
• Howard et al., Phys. Rev. Let. 87, 278102 (2001);
• Kruse, Biophys. J. 82, 618 (2002);
• Huang, Meir, and Wingreen, PNAS 100, 12724 (2003).
14
Summary of main functions of proteins:
FtsZ
MinC
MinD
MinE
polymerizes in a contractile Z-ring
that initiates septum formation
inhibits formation of Z-ring
membrane associated protein that
recruits minC and minE to membrane
ejects minC/minD from membrane into
cytoplasm
15
Howard et al. model (PRL)
mind
σ 1ρ D
1+ σ 1' ρ e
σ 2 ρ d ρe
minD
minE
σ 4 ρe
1+ σ 4' ρ D
σ 3ρD ρE
mine
membrane
in words:
cytoplasm
- first order reactions
for own species
- e inhibits membrane
association of D (MM)
- e enhances membrane
dissociation of d
(linear)
- D enhances membrane
association of E
(recruitment, linear)
- D inhibits membrane
dissociation of E (MM)
- d and e do not diffuse
16
- D and E diffuse
Howard et al. model (PRL)
membrane
mind
σ 1ρ D
1+ σ 1' ρ e
σ 2 ρ d ρe
minD
cytoplasm
association of cytoplasmic
minD with membrane is
inhibited by mine in membrane
MM takes care of singularity
as minE goes to zero.
biological interpretation:
minE
σ 4 ρe
1+ σ 4' ρ D
σ 3ρD ρE
mine in membrane spatially
blocks membrane for minD
similar to minC blocking FtZ
association with membrane
mine
17
Howard et al. model (PRL)
membrane
mind
σ 1ρ D
1+ σ 1' ρ e
σ 2 ρ d ρe
cytoplasm
dissociation of membrane
mind is stimulated by mine
in membrane, after mind is
ejected mine stays in membrane
minD
biological interpretation:
minE
σ 4 ρe
1+ σ 4' ρ D
σ 3ρD ρE
binding of mine to mind lowers
affinity of mind with membrane
but membrane affinity of mine
remains unchanged
mine
18
Howard et al. model (PRL)
membrane
mind
σ 1ρ D
1+ σ 1' ρ e
σ 2 ρ d ρe
cytoplasm
dissociation of membrane
mine is inhibited by minD
in cytoplasm
MM takes care of singularity
minD
biological interpretation:
minE
σ 4 ρe
1+ σ 4' ρ D
?
σ 3ρD ρE
mine
19
Howard et al. model (PRL)
membrane
mind
σ 1ρ D
1+ σ 1' ρ e
σ 2 ρ d ρe
minD
minE
σ 4 ρe
1+ σ 4' ρ D
σ 3ρD ρE
mine
cytoplasm
association of cytoplasmic
minE with membrane is
stimulated by minD in cytoplasm
after delivery of minE to the
membrane, minD dives back
in the cytoplasm
biological interpretation:
minD-minE complex has high
affinity to membrane
since the diffusion of this complex
doesn’t appear in the model it
should be very fast.
20
system of equations:
∂ρ D
∂2ρD
σ 1ρ D
= DD
−
+ σ 2 ρe ρd
'
2
∂t
∂x
1 + σ 1ρe
∂ρ d
σ 1ρ D
=
− σ 2 ρe ρd
'
∂t 1 + σ 1 ρ e
σ 4 ρe
∂ρ E
∂2ρE
= DE
− σ 3ρD ρE +
2
∂t
∂x
1 + σ 4' ρ D
∂ρ e
σ 4 ρe
= σ 3ρD ρE −
∂t
1 + σ 4' ρ D
21
stability analysis
1. find fixed point
(e.g. numerically:
how_homog.m)
∂
=0
∂t
∂
=0
∂x
different random initial conditions relax to
same fixed point
result: one fixed point:
d = 1383
D = 117
e = 82
E= 3
22
2. find stability matrix (Jacobian)
⎤
⎡
σ 1 Dσ 1'
− σ1
σ 2e
0
+ σ 2d ⎥
⎢
'
2
'
1 + σ 1e
(1 + σ 1e)
⎥
⎢
'
σ1
σ 1 Dσ 1
⎥
⎢
σ
0
σ
−
−
−
e
d
2 ⎥
2
'
2
'
⎢
1 + σ 1e
(1 + σ 1e)
A=⎢
⎥
'
σ4
⎥
⎢ − σ 4 eσ 4 − σ E
0
σ
−
D
3
3
⎥
⎢ (1 + σ 4' D) 2
1 + σ 4' D
⎥
⎢
'
σ
σ
σ
e
4
4
4
⎥
⎢+
σ
0
σ
+
−
E
D
3
3
'
'
2
⎥⎦
⎢⎣ (1 + σ 4 D)
1+ σ 4D
23
3. test stability of fluctuations around homogeneous solution
δE ( x, t ) = Eˆ (t ) cos(qx)
δe( x, t ) = eˆ(t ) cos(qx)
δD( x, t ) = Dˆ (t ) cos(qx)
δd ( x, t ) = dˆ (t ) cos(qx)
D
δD(x,t)
x
24
3. test stability of fluctuations around homogeneous solution
⎡ − σ1
2
σ 2e
0
D
q
−
D
⎢
'
⎢ 1 + σ 1e
σ1
⎢
0
− σ 2e
'
⎢
1 + σ 1e
ˆA = ⎢
'
σ
σ
e
2
4
4
⎢−
σ
σ
0
−
E
−
D
−
D
q
E
3
3
⎢ (1 + σ 4' D) 2
⎢
'
σ
σ
e
4
4
⎢+
σ 3D
0
+ σ 3E
'
2
⎢⎣ (1 + σ 4 D)
⎤
σ 1 Dσ 1'
+ σ 2d ⎥
'
2
(1 + σ 1e)
⎥
'
σ 1 Dσ 1
⎥
σ
d
−
−
2 ⎥
(1 + σ 1' e) 2
⎥
σ4
⎥
⎥
1 + σ 4' D
⎥
σ4
⎥
−
'
⎥⎦
1+ σ 4D
25
Max(Real(Eigenvalues)) 1/s
4. - determine eigenvalues of stability matrix,
- find real part of eigenvalues,
- plot the largest as a function of q.
(e.g. how_eig.m)
q = 1.5 (µm)-1
λ = 2π/q = 4.2 µm
q = 2.3 (µm)-1
λ = 2π/q = 2.7 µm
q
26
Howard et al.: Results
Image removed due to copyright considerations.
27
Huang, Meir, and Wingreen, PNAS 100, 12724 (2003).
main differences:
- ATP cycle
- 1D versus 3D (projected on 2D)
Image removed due to copyright considerations.
28
Image removed due to copyright considerations.
ρd:
membrane bound minD:ATP complexes
ρde: membrane bound minD:minE:ATP complexes
ρD:ADP: concentration cytoplasmic minD bound to ADP
ρD:ATP :concentration cytoplasmic minD bound to ATP
ρE :
concentration cytoplasmic minE
only minD-ATP can associate with membrane
minE only binds minD-ATP oligomers in membrane
29
only minD-minE-ATPcomplex can dissociate from membrane
Reaction 1:
minD-ATP binds both linearly
and autocatalytically to minD-ATP
in membrane
Image removed due to copyright considerations.
minD forms polymers in membrane
dρD:ADP
d 2 ρD: ADP
ADP → ATP
= DD
−
σ
ρD:ADP + σ de ρde
D
2
dt
dx
dρD:ATP
d 2 ρD: ATP
ADP → ATP
= DD
+
σ
ρD:ADP − [σ D + σ dD (ρ d + ρ de )]ρD: ATP
D
2
dt
dx
dρE
d 2 ρE
= DE
+ σ de ρe − σ E ρd ρE
2
dt
dx
dρd
= −σ E ρ d ρ E + [σ D + σ dD (ρ d + ρ de )]ρD: ATP
dt
dρde
= −σ de ρ de + σ E ρ d ρE
dt
30
Reaction 2:
minE binds minD-ATP in membrane
~ [minE]*[mind]
Image removed due to copyright considerations.
dρD:ADP
d 2 ρD: ADP
ADP → ATP
= DD
−
σ
ρD:ADP + σ de ρde
D
2
dt
dx
dρD:ATP
d 2 ρD: ATP
ADP → ATP
= DD
+
σ
ρD:ADP − [σ D + σ dD (ρ d + ρ de )]ρD: ATP
D
2
dt
dx
dρE
d 2 ρE
= DE
+ σ de ρe − σ E ρd ρE
2
dt
dx
dρd
= −σ E ρ d ρ E + [σ D + σ dD (ρ d + ρ de )]ρD: ATP
dt
dρde
= −σ de ρ de + σ E ρ d ρE
dt
31
Reaction 3:
minD-minE-ATP complex disassociates
from membrane hydrolyzing ATP
~ [mine]
Image removed due to copyright considerations.
dρD:ADP
d 2 ρD: ADP
ADP → ATP
= DD
−
σ
ρD:ADP + σ de ρde
D
2
dt
dx
dρD:ATP
d 2 ρD: ATP
ADP → ATP
= DD
+
σ
ρD:ADP − [σ D + σ dD (ρ d + ρ de )]ρD: ATP
D
2
dt
dx
dρE
d 2 ρE
= DE
+ σ de ρde − σ E ρd ρE
2
dt
dx
dρd
= −σ E ρ d ρ E + [σ D + σ dD (ρ d + ρ de )]ρD: ATP
dt
dρde
= −σ de ρ de + σ E ρ d ρE
dt
32
Reaction 4:
charging of minD in cytoplasm
from ADP to ATP bound
Image removed due to copyright considerations.
dρD:ADP
d 2 ρD: ADP
ADP → ATP
= DD
−
σ
ρD:ADP + σ de ρde
D
2
dt
dx
dρD:ATP
d 2 ρD: ATP
ADP → ATP
= DD
+
σ
ρD:ADP − [σ D + σ dD (ρ d + ρ de )]ρD: ATP
D
2
dt
dx
dρE
d 2 ρE
= DE
+ σ de ρde − σ E ρd ρE
2
dt
dx
dρd
= −σ E ρ d ρ E + [σ D + σ dD (ρ d + ρ de )]ρD: ATP
dt
dρde
= −σ de ρ de + σ E ρ d ρE
dt
33
dρ D
d 2ρD
= DD
− σ A ρ D + σ P ρ D: ADP
2
dt
dx
dρ d
= (σ D + sd ρ d )ρ D: ATP − σ e ρ e
dt
dρ D: ATP
d 2 ρ D: ATP
= DD
− (σ D + sd ρ d )ρ D: ATP + σ A ρ D
2
dt
dx
dρ e
= σ dE (ρ d − ρ e )ρ E − σ e ρ e
dt
dρ E
d 2ρE
= DE
− σ dE (ρ d − ρ e )ρ E + σ e ρ e
2
dt
dx
dρ D: ADP
d 2 ρ D: ADP
= DD
− σ P ρ D: ADP + σ e ρ e
2
dt
dx
34