Alternative views on gradient sensing:
- Postma and van Haastert. ‘A diffusion-translocation model
for gradient sensing by chemotactic cells.’
Biophys. J. 81, 1314 (2001).
- Levchenko and Iglesias. ‘Models of eukaryotic gradient
sensing: applications to chemotaxis of amoeba and neutrophils’
Biophys. J. 82, 50 (2002).
Main point:
- how to prevent cells to polarize ‘inreversibly’?
1
dm
∂ m
= Dm 2 − k −1m + P
dt
∂x
2
Images removed due to copyright considerations.
See Postma, M., and P. J. Van Haastert.
"A diffusion-translocation model for gradient sensing
by chemotactic cells." Biophys J. 81, no. 3 (Sep, 2001): 1314-23.
Dm ~ 1 µm2s-1
(membrane protein. lipid)
Dm ~ 100 µm2s-1
(cytosolic small molecule)
For a second messenger to
establish and maintain a
gradient the dispersion
range λ should be smaller
than cell size
λ=
Dm
k −1
k −1 = 1s −1
L = 10 µm
2
Second mesenger production
in a gradient
Dm ~ 1 µm2s-1 (membrane protein. lipid)
Dm ~ 100 µm2s-1
(cytosolic small molecule)
dm
∂ 2m
= Dm 2 − k −1m + P ( x)
dt
∂x
⎛ *
* x⎞
P ( x ) = k R ⎜ R − ∆R ⎟
r⎠
⎝
Images removed due to copyright considerations.
See Postma, M., and P. J. Van Haastert.
"A diffusion-translocation model for gradient sensing
by chemotactic cells." Biophys J. 81, no. 3 (Sep, 2001): 1314-23.
Diffusion flattens internal
gradient
Gain is < 1 (the larger
Dm the smaller the gain)
How to amplify ?
3
Amplification by positive feedback
Images removed due to copyright considerations.
See Postma, M., and P. J. Van Haastert.
"A diffusion-translocation model for gradient sensing
by chemotactic cells." Biophys J.
81, no. 3 (Sep, 2001): 1314-23.
∂ 2m
dm
= Dm 2 − k −1m + P( x)
∂x
dt
P ( x ) = k o + k E R * ( x ) Em ( x )
A. Before receptor stimulation
only a small number of effectors
(inactive) bound to membrane
B. After receptor stimulation,
membrane bound effectors will be
stimulated to produce more
phospholipid second mesengers
C. Local phospholipid increase
leads to increased translocation of
effector molecules
D. receptor can signal to more
effectors leading to even more
phospholipid production and
further depletion of cytosolic
effector molecules.
Em: effector concentration in
membrane
Ec: effector concentration in
cytosol.
4
Images removed due to copyright considerations. See Postma, M., and P. J. Van Haastert.
"A diffusion-translocation model for gradient sensing by chemotactic cells." Biophys J.
81, no. 3 (Sep, 2001): 1314-23.
5
Molecules ??
Image removed due to copyright considerations. See Levchenko, A., and P. A. Iglesias.
"Models of eukaryotic gradient sensing: application to chemotaxis of amoebae and neutrophils."
Biophys J. 82 (1 Pt 1)(Jan 2002): 50-63.
receptor binding →
G-protein activation →
activation of PI3K (activator) →
activation of PTEN (inhibitor) →
P3 ~ R* (binding PH domains)
6
Perfect adaptation module:
R*
A*
kA’ k-A
kR
k-R
R
A
I*
k-I
kI’
I
S
7
dR *
= − k − R I * R * + k R A* R
dt
dA*
= − k − A A* + k A' SA = − k − A A* + k A' S ( Atot − A* )
dt
dI *
= − k − I I * + k I' SI = − k − I I * + k I' S ( I tot − I * )
dt
Main assumption: k-A & k-I >> k’A & k’I (Atot>>A*, Itot>>I*)
dR *
= −k − R I * R * + k R A* R
dt
dA*
= −k − A A + k A S
dt
dI *
= −k − I I + k I S
dt
k A' = k A Atot
k I' = k I I tot
8
Steady state:
kA
A =
S
k− A
*
ss
Image removed due to copyright considerations.
kI
I =
S
k− I
*
ss
*
*
k
A
/
I
R ss
ss
Rss* =
k R Ass* / I ss* + k − R
for the rest of the calculations
ignore ‘*’ for I and A !
9
Now introduce diffusion:
- only I diffuses, other components are local
∂I ( x, t )
∂ 2 I ( x, t )
= − k − I I ( x, t ) + k I S ( x, t ) + D
∂t
∂x 2
- assume signal S varies linearly with S
S ( x) = so + s1 x
- no flux boundary conditions for I
∂I (0, t ) ∂I (1, t )
=
=0
∂x
∂x
in steady state,this system can be solved
analytically !
10
∂I ( x, t )
∂ 2 I ( x, t )
= − k − I I ( x, t ) + k I S ( x, t ) + D
∂t
∂x 2
steady-state:
∂ 2 I ( x) k − I
kI
I ( x) − [so + s1 x ]
=
2
D
∂x
D
∂ 2 I ( x)
= aI ( x) − b − cx
2
∂x
MATLAB can solve this for you:
>> dsolve('D2x=a*x-b-c*t','Dx(0)=0,Dx(1)=0')
ans =
(b+c*t)/a+c*(-1+cosh(a^(1/2)))/a^(3/2)/sinh(a^(1/2))*cosh(a^(1/2)*t)
-c/a^(3/2)*sinh(a^(1/2)*t)
11
kI
I ( x) =
k− I
⎛
sinh σx cosh σx cosh σ − 1 ⎞ ⎞
⎛
+
⎜⎜ so + s1 ⎜ x −
⎟ ⎟⎟
σ
σ
sinh σ ⎠ ⎠
⎝
⎝
kI/k-I=1
s0=1 µM
s1=0.1 µM
σ=0.25 (µm)-1
I(x)
σ ≡ k−I / D
x
12
Remember: Perfect adaptation module:
diffuses
fixed in space
A*
kA’ k-A
R*
kR
k-R
R
A
I*
k-I
kI’
I
S
13
Steady state:
kA
A =
S
k− A
*
ss
kI
I =
S
k− I
*
ss
*
*
k
A
/
I
R ss
ss
Rss* =
k R Ass* / I ss* + k − R
independent of S,
perfect adaptation
A does not diffuse, so
A(x) directly reflects S(x)
For finding R* only the ratio A/I is important
14
kA
(so + s1 x )
A( x) =
k− A
kI
I ( x) =
k− I
⎛
sinh σx cosh σx cosh σ − 1 ⎞ ⎞
⎛
⎜⎜ so + s1 ⎜ x −
+
⎟ ⎟⎟
sinh σ ⎠ ⎠
σ
σ
⎝
⎝
A( x) k A k − I
=
I ( x) k − A k I
⎛
s1 ⎛ cosh σx cosh σ − 1 sinh σx ⎞ ⎞
⎜⎜1 +
−
⎜
⎟ ⎟⎟
sinh σ
σ ⎠⎠
⎝ s0 + s1 x ⎝ σ
15
−1
small
σ ≡ k − I / D ~ 0 .4
well mixed, A/I directly reflects signal
A(x)/I(x)~R*
I(x)
x
I ( x) = I ( S ) = const
x
A( x) = A( S )
R * ( x) = A( S ) / I ( S )
16