Review Turing-Gierer-Meinhardt models Local excitation, global inhibition 2 2 2 2 22 x i Diak t i x a Da i a kr t a iii aaaa ? ? +?= ? ? ? ? +?+= ? ? γ γ a: concentration activator i: concentration inhibitor t: time x: position variables r a : basal activator synthesis rate k a , k i : rate constant for synthesis γ a ,γ i : decay rates D a , D i : diffusion constants constants (parameters) 1 2 2 2 2 22 x i Diak t i x a Da i a kr t a iii aaaa ? ? +?= ? ? ? ? +?+= ? ? γ γ choose dimensionless variable normalize 4 variables () 2 2 2 2 22 1 s I PIAQ τ I s A A I A R τ A ? ? +?= ? ? ? ? +?+= ? ? only one fixed point, since both A and I >0 2 )1( 1 += += RI RA homogeneous solution 0// =??=?? ts 2 homogeneous solution 0// =??=?? ts A 3 A s I I s stability of homogeneous solution ? ? ? ? ? ? ? ? ?+ + ? + ? = ? ? ? ? ? ? ? ? ? ?? QQR R R R R QQA I AR I AR )1(2 )1(1 1 2 1 2 2 2 2 trace < 0 det > 0 0 1 1 > < + ? Q Q R R or in general real part of eigenvalues > 0 ),('),( ),('),( ττ ττ sIIsI sAAsA += += inhomogeneous solution: 4 inhomogeneous solution 5 A s s A I I’(s,τ) ),('),( ),('),( ττ ττ sIIsI sAAsA += += I 2 2 2 2 2 ' '')1(2 ' ' ' )1( ' 1 1' s I PQIARQ I s A I R R A R RA ? ? +?+= ? ? ? ? + + ? + ? = ? ? τ τ ),('),( ),('),( ττ ττ sIIsI sAAsA += += )cos()( ? ),(' )cos()( ? ),(' l l s IsI s AsA ττ ττ = = trial solution: 6 7 )cos()( ? ),(' )cos()( ? ),(' l l s IsI s AsA ττ ττ = = A I s s A I I’(s,τ) ),('),( ),('),( ττ ττ sIIsI sAAsA += += I P QARQ d Id I R R A R R d Ad ?? )1(2 ? ? )1( ? 1 1 1 ? 2 22 ? ? ? ? ? ? +?+= + ? ? ? ? ? ? ? ? + ? = l l τ τ )cos()( ? ),(' )cos()( ? ),(' l l s IsI s AsA ττ ττ = = 0 1 1 1 0 1 21 1 1 22 22 < ? ? ? ? ? ? ? + ? ?+ > + + ? ? ? ? ? ? + ? ? ? ? ? ? ? + ? ? ll ll R RP Q R QRP Q R R stability inhomogeneous solution 1 1 + ? > R R P Q 8 1 1 + ? > R R Q homogeneous stability: stability against spatial distrubance: 1 1 + ? > R R P Q s I I’(s,τ) I 9 if P < 1 (D i <D a ), systems is always stable, against any perturbation both spatial and temporal s I I homogeneously stable: I relaxes back to previous value after small uniform disturbance s I I I’ relaxes back to after small spatial disturbance I stable against spatial disturbance: 10 Introducing the molecules: - FtsZ function: Assembly of a polymeric ring of the tubulin-like GTPase FtsZ (Z ring). The Z-ring is localized to the center by the actions of the MinC, MinD, and MinE proteins. - MinC inhibits the initiation of the Z ring. MinC colocalizes with MinD. In wild-type (WT) cells, MinC/D forms a polar pattern that oscillates between the poles, keeping the center free for initiation of cell division. Thus, virtually all of MinC/D dynamically assembles on the membrane in the shape of a test tube covering the membrane from one pole up to approximately midcell. 12 Most of MinE accumulates at the rim of this tube, in the shape of a ring (the E ring). The rim of the MinC/D tube and associated E ring move from a central position to the cell pole until both the tube and ring vanish. Meanwhile, a new MinC/D tube and associated E ring form in the opposite cell half, and the process repeats, resulting in a pole-to-pole oscillation cycle of the division inhibitor. A full cycle takes about 50 s. Image removed due to copyright considerations. 13 How does this work ? modeling efforts: ? Meinhardt and de Boer, PNAS 98, 14202 (2001); ? Howard et al., Phys. Rev. Let. 87, 278102 (2001); ?Kruse, Biophys. J. 82, 618 (2002); ? Huang, Meir, and Wingreen, PNAS 100, 12724 (2003). 14 Summary of main functions of proteins: polymerizes in a contractile Z-ring that initiates septum formation FtsZ inhibits formation of Z-ring MinC membrane associated protein that recruits minC and minE to membrane MinD ejects minC/minD from membrane into cytoplasm MinE 15 Howard et al. model (PRL) mind minD cytoplasm membrane e D ρσ ρσ ' 1 1 1+ ed ρρσ 2 mine minE D e ρσ ρσ ' 4 4 1+ ED ρρσ 3 16 in words: - first order reactions for own species - e inhibits membrane association of D (MM) - e enhances membrane dissociation of d (linear) - D enhances membrane association of E (recruitment, linear) - D inhibits membrane dissociation of E (MM) - d and e do not diffuse - D and E diffuse Howard et al. model (PRL) mind minD cytoplasm membrane e D ρσ ρσ ' 1 1 1+ ed ρρσ 2 mine minE D e ρσ ρσ ' 4 4 1+ ED ρρσ 3 association of cytoplasmic minD with membrane is inhibited by mine in membrane MM takes care of singularity as minE goes to zero. biological interpretation: mine in membrane spatially blocks membrane for minD similar to minC blocking FtZ association with membrane 17 Howard et al. model (PRL) mind minD cytoplasm membrane e D ρσ ρσ ' 1 1 1+ ed ρρσ 2 mine minE D e ρσ ρσ ' 4 4 1+ ED ρρσ 3 dissociation of membrane mind is stimulated by mine in membrane, after mind is ejected mine stays in membrane biological interpretation: binding of mine to mind lowers affinity of mind with membrane but membrane affinity of mine remains unchanged 18 Howard et al. model (PRL) mind minD cytoplasm membrane e D ρσ ρσ ' 1 1 1+ ed ρρσ 2 mine minE D e ρσ ρσ ' 4 4 1+ ED ρρσ 3 dissociation of membrane mine is inhibited by minD in cytoplasm MM takes care of singularity biological interpretation: ? 19 Howard et al. model (PRL) mind minD cytoplasm membrane e D ρσ ρσ ' 1 1 1+ ed ρρσ 2 mine minE D e ρσ ρσ ' 4 4 1+ ED ρρσ 3 association of cytoplasmic minE with membrane is stimulated by minD in cytoplasm after delivery of minE to the membrane, minD dives back in the cytoplasm biological interpretation: minD-minE complex has high affinity to membrane since the diffusion of this complex doesn’t appear in the model it should be very fast. 20 system of equations: D e ED e D e ED E E E de e Dd de e DD D D t x D t t x D t ρσ ρσ ρρσ ρ ρσ ρσ ρρσ ρρ ρρσ ρσ ρσρ ρρσ ρσ ρσρρ ' 4 4 3 ' 4 4 3 2 2 2 ' 1 1 2 ' 1 1 2 2 1 1 1 1 + ?= ? ? + +? ? ? = ? ? ? + = ? ? + + ? ? ? = ? ? 21 stability analysis 1. find fixed point (e.g. numerically: how_homog.m) different random initial conditions relax to same fixed point 0 0 = ? ? = ? ? x t result: one fixed point: d = 1383 e = 82 D = 117 E = 3 22 2. find stability matrix (Jacobian) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ?+ + + + ?? + ? ? + ?? + + ++ ? = D DE D e D DE D e d e D e e d e D e e A ' 4 4 33 2' 4 ' 44 ' 4 4 33 2' 4 ' 44 2 2' 1 ' 11 2 ' 1 1 2 2' 1 ' 11 2 ' 1 1 1 0 )1( 1 0 )1( )1( 0 1 )1( 0 1 σ σ σσ σ σσ σ σ σσ σ σσ σ σ σσ σ σ σ σ σ σσ σ σ σ 23 3. test stability of fluctuations around homogeneous solution )cos()( ? ),( )cos()( ? ),( )cos()( ? ),( )cos()( ? ),( qxtdtxd qxtDtxD qxtetxe qxtEtxE = = = = δ δ δ δ x δD(x,t) D 24 3. test stability of fluctuations around homogeneous solution ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ?+ + + + ??? + ? ? + ?? + + + ? + ? = D DE D e D qDDE D e d e D e e d e D eqD e A E D ' 4 4 33 2' 4 ' 44 ' 4 4 2 33 2' 4 ' 44 2 2' 1 ' 11 2 ' 1 1 2 2' 1 ' 11 2 2 ' 1 1 1 0 )1( 1 0 )1( )1( 0 1 )1( 0 1 ? σ σ σσ σ σσ σ σ σσ σ σσ σ σ σσ σ σ σ σ σ σσ σ σ σ 25 4. - determine eigenvalues of stability matrix, - find real part of eigenvalues, - plot the largest as a function of q. (e.g. how_eig.m) q = 1.5 (μm) -1 λ = 2π/q = 4.2 μm q = 2.3 (μm) -1 λ = 2π/q = 2.7 μm q Max(Real(Eigenvalues)) 1/s 26 Howard et al.: Results 27 Image removed due to copyright considerations. Huang, Meir, and Wingreen, PNAS 100, 12724 (2003). main differences: - ATP cycle - 1D versus 3D (projected on 2D) Image removed due to copyright considerations. 28 Image removed due to copyright considerations. ρ d : membrane bound minD:ATP complexes ρ de : membrane bound minD:minE:ATP complexes ρ D:ADP : concentration cytoplasmic minD bound to ADP ρ D:ATP :concentration cytoplasmic minD bound to ATP ρ E : concentration cytoplasmic minE 29 only minD-ATP can associate with membrane minE only binds minD-ATP oligomers in membrane only minD-minE-ATPcomplex can dissociate from membrane Reaction 1: minD-ATP binds both linearly and autocatalytically to minD-ATP in membrane minD forms polymers in membrane Image removed due to copyright considerations. dedeD:ADP ATPADP D ADPD D D:ADP ρσρσ dx ρd D dt dρ +?= → 2 : 2 ()[] ATPDdeddDDD:ADP ATPADP D ATPD D D:ATP ρρσ dx ρd D dt dρ : 2 : 2 ρρσσ ++?+= → EdEede E E E ρρσρσ dx ρd D dt dρ ?+= 2 2 ()[] ATPDdeddDDEdE d ρ dt dρ : ρρσσρρσ +++?= EdEdede de ρ dt dρ ρσρσ +?= 30 Reaction 2: minE binds minD-ATP in membrane ~ [minE]*[mind] Image removed due to copyright considerations. dedeD:ADP ATPADP D ADPD D D:ADP ρσρσ dx ρd D dt dρ +?= → 2 : 2 ()[] ATPDdeddDDD:ADP ATPADP D ATPD D D:ATP ρρσ dx ρd D dt dρ : 2 : 2 ρρσσ ++?+= → EdEede E E E ρρσρσ dx ρd D dt dρ ?+= 2 2 ()[] ATPDdeddDDEdE d ρ dt dρ : ρρσσρρσ +++?= EdEdede de ρ dt dρ ρσρσ +?= 31 Reaction 3: minD-minE-ATP complex disassociates from membrane hydrolyzing ATP ~ [mine] Image removed due to copyright considerations. dedeD:ADP ATPADP D ADPD D D:ADP ρσρσ dx ρd D dt dρ +?= → 2 : 2 ()[] ATPDdeddDDD:ADP ATPADP D ATPD D D:ATP ρρσ dx ρd D dt dρ : 2 : 2 ρρσσ ++?+= → EdEdede E E E ρρσρσ dx ρd D dt dρ ?+= 2 2 ()[] ATPDdeddDDEdE d ρ dt dρ : ρρσσρρσ +++?= EdEdede de ρ dt dρ ρσρσ +?= 32 Reaction 4: charging of minD in cytoplasm from ADP to ATP bound Image removed due to copyright considerations. dedeD:ADP ATPADP D ADPD D D:ADP ρσρσ dx ρd D dt dρ +?= → 2 : 2 ()[] ATPDdeddDDD:ADP ATPADP D ATPD D D:ATP ρρσ dx ρd D dt dρ : 2 : 2 ρρσσ ++?+= → EdEdede E E E ρρσρσ dx ρd D dt dρ ?+= 2 2 ()[] ATPDdeddDDEdE d ρ dt dρ : ρρσσρρσ +++?= EdEdede de ρ dt dρ ρσρσ +?= 33 ADPDPDA D D D dx d dt d : 2 2 ρσρσ ρρ +?=D () eeATPDddD d s dt d ρσρρσ ρ ?+= : () DAATPDddD ATPD D ATPD s dx d dt d ρσρρσ ρρ ++?= : 2 : 2 : D () eeEeddE e dt d ρσρρρσ ρ ??= () eeEeddE E E E dx d dt d ρσρρρσ ρρ +??= 2 2 D eeADPDP ADPD D ADPD dx d dt d ρσρσ ρρ +?= : 2 : 2 : D 34