Alternative views on gradient sensing:- Postma and van Haastert. ‘A diffusion-translocation modelfor gradient sensing by chemotactic cells.’Biophys. J. 81, 1314 (2001).- Levchenko and Iglesias. ‘Models of eukaryotic gradientsensing: applications to chemotaxis of amoeba and neutrophils’Biophys. J. 82, 50 (2002). Main point: - how to prevent cells to polarize ‘inreversibly’?1PmkxmDdtdmm+−∂∂=−122Dm~ 1 µm2s-1(membrane protein. lipid)Dm~ 100 µm2s-1(cytosolic small molecule)For a second messenger toestablish and maintain agradient the dispersionrange λ should be smallerthan cell sizeImages removed due to copyright considerations.See Postma, M., and P. J. Van Haastert."A diffusion-translocation model for gradient sensingby chemotactic cells." Biophys J. 81, no. 3 (Sep, 2001): 1314-23.mLskkDmµλ101111===−−−2⎟⎠⎞⎜⎝⎛∆−=+−∂∂=−rxRRkxPxPmkxmDdtdmRm**122)()(Second mesenger production in a gradientDm~ 1 µm2s-1(membrane protein. lipid)Dm~ 100 µm2s-1(cytosolic small molecule)Diffusion flattens internalgradientGain is < 1 (the largerDmthe smaller the gain)How to amplify ?3Images removed due to copyright considerations.See Postma, M., and P. J. Van Haastert."A diffusion-translocation model for gradient sensingby chemotactic cells." Biophys J. 81, no. 3 (Sep, 2001): 1314-23.Amplification by positive feedback4A. Before receptor stimulation only a small number of effectors (inactive) bound to membraneB. After receptor stimulation, membrane bound effectors will be stimulated to produce more phospholipid second mesengersC. Local phospholipid increase leads to increased translocation of effector moleculesD. receptor can signal to more effectors leading to even more phospholipid production and further depletion of cytosoliceffector molecules.Em: effector concentration in membraneEc: effector concentration in cytosol.)()()()(*122xExRkkxPxPmkxmDdtdmmEom+=+−∂∂=−Images removed due to copyright considerations.See Postma, M., and P. J. Van Haastert."A diffusion-translocation model for gradient sensingby chemotactic cells." Biophys J. 81, no. 3 (Sep, 2001): 1314-23.5Images removed due to copyright considerations.See Postma, M., and P. J. Van Haastert."A diffusion-translocation model for gradient sensingby 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)6Perfect adaptation module:R*A*kRk-RI*k-AkA’kI’k-IR7AIS)()(*'*'***'*'******IISkIkSIkIkdtdIAASkAkSAkAkdtdARAkRIkdtdRtotIIIItotAAAARR−+−=+−=−+−=+−=+−=−−−−−Main assumption: k-A& k-I>> k’A& k’I(Atot>>A*, Itot>>I*)SkIkdtdISkAkdtdARAkRIkdtdRIIAARR+−=+−=+−=−−−******totIItotAAIkkAkk==''8Steady state:RssssRssssRssIIssAAsskIAkIAkRSkkISkkA−−−+===*******//Image removed due to copyright considerations.for the rest of the calculationsignore ‘*’ for I and A !9Now introduce diffusion:- only I diffuses, other components are local22),(),(),(),(xtxIDtxSktxIkttxIII∂∂++−=∂∂−- assume signal S varies linearly with SxssxSo 1)(+=- no flux boundary conditions for I0),1(),0(=∂∂=∂∂xtIxtIin steady state,this system can be solvedanalytically !1022),(),(),(),(xtxIDtxSktxIkttxIII∂∂++−=∂∂−[]cxbxaIxxIxssDkxIDkxxIoII−−=∂∂+−=∂∂−)()()()(22122steady-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⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−+−+=−σσσσσσsinh1coshcoshsinh)(1xxxsskkxIoII12xI(x)kI/k-I=1s0=1 µMs1=0.1 µM σ=0.25 (µm)-1DkI/−≡σRemember: Perfect adaptation module:diffusesfixed in spaceR*A*kRk-RI*k-AkA’kI’k-IR13AISSteady state:RssssRssssRssIIssAAsskIAkIAkRSkkISkkA−−−+===*******//independent of S,perfect adaptationA does not diffuse, so A(x) directly reflects S(x)For finding R* only the ratio A/I is important14()110111sinhsinh1coshcosh1)()(sinh1coshcoshsinh)()(−−−−−⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−−++=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−+−+=+=σσσσσσσσσσσσxxxssskkkkxIxAxxxsskkxIxsskkxAIAIAoIIoAA154.0~/ DkI−≡σsmallwell mixed, A/I directly reflects