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 Pmk x m D dt dm m +? ? ? = ?1 2 2 D m ~ 1 μm 2 s -1 (membrane protein. lipid) D m ~ 100 μm 2 s -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. mL sk k D m μ λ 10 1 1 1 1 = = = ? ? ? 2 ? ? ? ? ? ? ??= +? ? ? = ? r x RRkxP xPmk x m D dt dm R m ** 1 2 2 )( )( Second mesenger production in a gradient D m ~ 1 μm 2 s -1 (membrane protein. lipid) D m ~ 100 μm 2 s -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. )()()( )( * 1 2 2 xExRkkxP xPmk x m D dt dm mEo 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’ k I’ k -I R 7 A I S )( )( *'*'* * *'*'* * *** * IISkIkSIkIk dt dI AASkAkSAkAk dt dA RAkRIk dt dR totIIII totAAAA RR ?+?=+?= ?+?=+?= +?= ?? ?? ? Main assumption: k -A & k -I >> k ’ A & k ’ I (A tot >>A*, I tot >>I*) SkIk dt dI SkAk dt dA RAkRIk dt dR II AA RR +?= +?= +?= ? ? ? * * *** * totII totAA Ikk Akk = = ' ' 8 Steady state: RssssR ssssR ss I I ss A A ss kIAk IAk R S k k I S k k A ? ? ? + = = = ** ** * * * / / 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 txI DtxSktxIk t txI II ? ? ++?= ? ? ? - assume signal S varies linearly with S xssxS o 1 )( += - no flux boundary conditions for I 0 ),1(),0( = ? ? = ? ? x tI x tI in steady state,this system can be solved analytically ! 10 2 2 ),( ),(),( ),( x txI DtxSktxIk t txI II ? ? ++?= ? ? ? [] cxbxaI x xI xss D k xI D k x xI o II ??= ? ? +?= ? ? ? )( )( )( )( 2 2 1 2 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 1coshcoshsinh )( 1 xx xss k k xI o I I 12 x I(x) k I /k -I =1 s 0 =1 μM s 1 =0.1 μM σ=0.25 (μm) -1 Dk I / ? ≡σ Remember: Perfect adaptation module: diffuses fixed in space R* A* k R k -R I* k -A k A’ k I’ k -I R 13 A I S Steady state: RssssR ssssR ss I I ss A A ss kIAk IAk R S k k I S k k A ? ? ? + = = = ** ** * * * / / 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 10 1 1 1 sinh sinh 1coshcosh 1 )( )( sinh 1coshcoshsinh )( )( ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + += ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? +?+= += σ σ σ σ σ σ σ σ σ σ σ σ xx xss s kk kk xI xA xx xss k k xI xss k k xA IA IA o I I o A A 15 4.0~/ Dk I? ≡σ small well mixed, A/I directly reflects signal A(x)/I(x)~R* x 16 x )(/)()( )()( )()( * SISAxR SAxA constSIxI = = == I(x)