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