dt dm

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

P m

x k D m

dt dm

m

− +

∂

= ∂

2s^-1

(membrane protein. lipid) D_m ~ 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

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.

m L

s k

k D_m

µ λ

10 1 ¹

1

1

=

=

=

− −

−

2

⎟⎠

⎜ ⎞

⎝

⎛ − ∆

=

+

∂ −

= ∂ ₋

r R x R

k x

P

x P m

x k D m

dt dm

R m

*

* 2 1 2

) (

) ( Second mesenger production

in a gradient

D_m ~ 1 µm²s^-1 (membrane protein. lipid) D_m ~ 100 µm²s^-1

(cytosolic small molecule)

Diffusion flattens internal gradient

Gain is < 1 (the larger D_m the smaller the gain) How to amplify ?

3 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.

Amplification by positive feedback

4

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.

E_m: effector concentration in membrane

E_c: effector concentration in cytosol.

) ( )

( )

(

) (

* 2 1 2

x E

x R k k

x P

x P m

x k D m

dt dm

m E

o m

+

=

+

∂ −

= ∂ ₋

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 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.

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* k_R k_-R I*

k_-A

k_A’ R k_-I k_I’

7

A I

S

) (

) (

* '

* '

*

*

* '

* '

*

*

*

*

*

*

I I

S k I

k SI

k I

dt k dI

A A

S k A

k SA

k A

dt k dA

R A k R

I dt k

dR

tot I

I I

I

tot A

A A

A

R R

− +

−

= +

−

=

− +

−

= +

−

=

+

−

=

−

−

−

−

−

Main assumption: k_-A & k_-I >> k^’_A & k^’_I (A_tot>>A*, I_tot>>I*)

S k I

dt k dI

S k A dt k

dA

R A k R

I dt k

dR

I I

A A

R R

+

−

=

+

−

=

+

−

=

−

−

−

*

*

*

*

*

*

tot I I

tot A A

I k k

A k k

=

=

' '

8

Steady state:

R ss

ss R

ss ss

R ss

I I ss

A A ss

k I

A k

I A

R k

k S I k

k S A k

−

−

−

= +

=

=

*

*

*

*

*

*

*

/ /

Image removed due to copyright considerations.

for the rest of the calculations

ignore ‘*’ for I and A ! ⁹

Now introduce diffusion:

- only I diffuses, other components are local

2 2

( , ) )

, ( )

, ) (

, (

x t x D I

t x S k t

x I t k

t x I

I

I

∂

+ ∂ +

−

∂ =

∂

−

- assume signal S varies linearly with S

x s s

x

S ( ) =

+

- no flux boundary conditions for I

) 0 , 1 ( )

, 0

( =

∂

= ∂

∂

∂

x t I

x t I

in steady state,this system can be solved analytically !

10

2 2

( , ) )

, ( )

, ) (

, (

x t x D I

t x S k t

x I t k

t x I

I

I

∂

+ ∂ +

−

∂ =

∂

−

[ ]

cx b

x x aI

x I

x s D s

x k D I

k x

x I

o I

I

−

−

∂ =

∂

+

−

∂ =

∂

) ) (

(

) ) (

(

2 2

2 1 2

steady-state:

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

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛ − + −

+

=

−

σ

σ σ

σ σ

σ

sinh

1 cosh

cosh ) sinh

(

x x

x s

k s x k

I

I I

12

x I(x)

k_I/k_-I=1 s₀=1 µM s₁=0.1 µM

σ=0.25 (µm)^-1

D k

/

σ ≡

Remember: Perfect adaptation module:

diffuses fixed in space

R*

A* k_R k_-R I*

k_-A

k_A’ R k_-I k_I’

13

A I

S

Steady state:

R ss

ss R

ss ss

R ss

I I ss

A A ss

k I

A k

I A

R k

k S I k

k S A k

−

−

−

= +

=

=

*

*

*

*

*

*

*

/

/ 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

( )

1

1 0

1 1

1

sinh sinh

1 cosh

1 cosh )

( ) (

sinh

1 cosh

cosh ) sinh

( ) (

−

−

−

−

−

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛ − −

+ +

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛ − + −

+

=

+

=

σ σ σ

σ σ

σ

σ σ σ

σ σ

σ

x x

x s s

s k

k k k x

I x A

x x x

s k s

x k I

x s k s

x k A

I A

I A

o I

I

o A

A

15

4 . 0

~ / D k

σ ≡

small

well mixed, A/I directly reflects signal

A(x)/I(x)~R*

x

16

x

) ( / ) ( )

(

) ( )

(

) ( )

(

*

x A S I S

R

S A x

A

const S

I x

I

=

=

=

= I(x)